AN INTRODUCTION TO CUT-OFF GRADE ESTIMATION SECOND EDITION BY JEAN-MICHEL RENDU BLENDING CONSTRAINTS WASTE BLOCK CAVING STOCKPILES CLOSURE GRADE-TONNAGE PRICES MINE CUT-OFF LEACHING UNDERGROUND NPV SOCIOECONOMIC METAL EQUIVALENT OPEN PIT OPPORTUNITY POLYMETALLIC PROCESS PROFIT RECLAMATION SELECTIVITY STOPES UTILITY NSR VARIABLE RECOVERY NPV MILL CUT-OFF LEACHING BREAKEVEN STOCK STOCKPILES CLOSURE POLYMETALLIC UNDERGROUND NPV SOCIO-ECONOMIC METAL EQUIVALENT OPEN PIT OPPORTUNITY POLYMETALLIC PROCESS PROFIT BLENDING CONSTRAINTS WASTE BLOCK CAVING STOCKPILES CLOSURE GRADE-TONNAGE PRICES MINE CUT-OFF LEACHING UNDERGROUND NPV SOCIO- Published by the Society for Mining, Metallurgy & Exploration GRADE ESTIMATION SECOND EDITION AN INTRODUCTION TO CUT-OFF BY JEAN-MICHEL RENDU !SME_COGrade2013_2ndED_TitlePg_L3.indd 1 11/21/13 2:15 PM Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Society for Mining, Metallurgy & Exploration (SME) 12999 East Adam Aircraft Circle Englewood, Colorado 80112 (303) 948-4200 / (800) 763-3132 www.smenet.org SME advances the worldwide mining and minerals community through information exchange and professional development. SME is the world’s largest association of mining and minerals professionals. Copyright © 2014 Society for Mining, Metallurgy & Exploration Inc. Electronic edition published 2014. All Rights Reserved. Printed in the United States of America. Information contained in this work has been obtained by SME from sources believed to be reliable. However, neither SME nor the authors guarantee the accuracy or completeness of any information published herein, and neither SME nor the authors shall be responsible for any errors, omissions, or damages arising out of use of this information. 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An introduction to cut-off grade estimation / by Jean-Michel Rendu. -- Second edition. pages cm Includes bibliographical references and index. ISBN 978-0-87335-393-9 1. Ores--Grading. 2. Ores--Sampling and estimation. I. Title. TN560.R45 2014 622’.7--dc23 2013043932 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. iii Contents PREFACE TO 2014 EDITION. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . vii PREFACE TO 2008 EDITION. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . ix Chapter 1 INTRODUCTION. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ . . . . . 1 Chapter 2 GENERAL CONCEPTS. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . 5 Mathematical Formulation: Utility Function. . . . . . . . . . . . . . . 5 Cut-off Grade and Grade–Tonnage Relationship. . . . . . . . . . . 6 Direct Profit and Loss: Udir(x). . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . 8 Net Present Value: NPVi . . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . 10 Opportunity Costs and Benefits: Uopp(x). . . . . . . . . . . . . . . . . .å°“ 12 Other Costs and Benefits: Uoth(x). . . . . . . . . . . . . . . . . .å°“ . . . . . . 15 Cut-off Grade and Blending Strategy. . . . . . . . . . . . . . . . . .å°“ . . . 16 Chapter 3 BREAKEVEN CUT-OFF GRADE. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . 17 Cut-off Grade Between Ore and Waste. . . . . . . . . . . . . . . . . .å°“ . 17 Cut-off Grades in Open Pit Mines. . . . . . . . . . . . . . . . . .å°“ . . . . . . 20 Cut-off Grades in Underground Mines. . . . . . . . . . . . . . . . . .å°“ . 23 Cut-off Grade Between Two Processes. . . . . . . . . . . . . . . . . .å°“ . . 26 Cut-off Grade with Variable Recovery. . . . . . . . . . . . . . . . . .å°“ . . 28 Constant Tail Model of Variable Recovery. . . . . . . . . . . . . . . . 30 Breakeven Cut-off Grade and Polymetallic Deposits. . . . . . . 32 Opportunity Cost of Not Using the Optimum Cut-off Grade. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ . . . . . . . . 38 Cut-off Grade and Low-Grade Stockpiles. . . . . . . . . . . . . . . . . 40 Cut-off Grade and Optimization of Processing Plant Operating Conditions. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . 44 Chapter 4 CAPACITY CONSTRAINTS AND OPPORTUNITY COSTS. . . 53 When Marginal Analysis No Longer Applies: A Gold Leaching Operation. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . 53 Capacity Constraints and Increase in Cut-off Grade. . . . . . . 56 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. iv Contents Decreasing Cut-off Grade over Time: A Strategic Objective. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ . . . . . 61 Cut-off Grade Below Marginal Cut-off Grade. . . . . . . . . . . . . 63 Cut-off Grade Optimization with Opportunity Costs: Practical Considerations. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . 66 Chapter 5 GEOLOGIC CONSTRAINTS AND OPPORTUNITY COSTS. . 69 Cut-off Grade When Processing Capacity Is Fixed. . . . . . . . . 69 Cut-off Grade When Mining Capacity Is Fixed. . . . . . . . . . . . 76 Cut-off Grade When Volume of Sales Is Fixed. . . . . . . . . . . . . 78 Releasing Capacity Constraints: A Base Metal Example. . . . 81 Chapter 6 CUT-OFF GRADE AND MINE PLANNING. . . . . . . . . . . . . . . . . .å°“ 87 Open Pit Mine: Economic Valuation of a Pushback. . . . . . . . 87 Underground Mine: Economic Valuation of a Stope. . . . . . . 89 Similarities Between Open Pit and Underground Mine Planning. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ . . . . . . 90 Block and Panel Caving. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . 91 Relationship Between Mine Selectivity, Deposit Modeling, Ore Control, and Cut-off Grade. . . . . . . . . .98 Chapter 7 WHICH COSTS SHOULD BE INCLUDED IN CUT-OFF GRADE CALCULATIONS?. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . 101 General Considerations. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . 101 How Specific Costs Influence Cut-off Grade Calculations. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ . 103 Other Costs and Benefits. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . 105 Chapter 8 BLENDING STRATEGY. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . 109 Blending Two Material Types. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . 110 Blending Three Material Types. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . 114 Blending to Maximize Tonnage. . . . . . . . . . . . . . . . . .å°“ . . . . . . . 118 Remarks Concerning an Increasingly Complex Blending Problem. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ . . . . . 120 Chapter 9 CLOSING REMARKS. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . 123 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Contents v Appendix A Example 1. Net Present Value of Constant Cash Flow: Proof of Formula. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . 127 Example 2. Net Present Value of Perpetuity Cash Flow: Proof of Formula. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . 128 Example 3. Opportunity Cost of Mining a Peripheral Deposit. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ . . . . . 128 Example 4. Simplified Equation to Estimate Opportunity Cost: Verification of First-Order Approximation. . . . 130 Example 5. Opportunity Cost of Not Using the Optimal Cut-off Grade in a Copper Mine. . . . . . . . . . . . . . . . . .å°“ 131 Example 6. Stockpiling and Cut-off Determination in a Copper Mine. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ 132 Example 7. Properties of Constant-Metric-Tons Curves in Blending Diagrams. . . . . . . . . . . . . . . . . . . . . . . . . . . . .135 LIST OF SYMBOLS. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ 139 BIBLIOGRAPHY. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ . . . 145 INDEX. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . 149 ABOUT THE AUTHOR. . . . . . . . . . . . . . . . . .å°“ . . . . . . . . . . . . . . . . 157 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. vi DEDICATION I am dedicating this book to my wife Karla and my two sons, Yannick and Mikael. Life with a husband and father who spent too much time traveling to remote mines all over the world, and then returned home to work long hours in front of his computer, was not without challenges and disappointments. I am grateful for their patience, understanding, and unquestioning love. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. vii Preface to 2014 Edition The first edition of An Introduction to Cut-off Grade Estimation was published in 2008. Since then, many changes have occurred in the world economy. Prices and costs have changed significantly. The commodity megatrend that prevailed during the last decade changed management’s emphasis from cost control to growth in production. But this trend reached its inevitable end in 2012, resulting in a return to a high priority being given to efficiency and cost controls. The increasingly demanding political and socioeconomic environ￾ment has altered the way decisions are made. Such changes should not alter the validity of a reference manual. How￾ever, the need to update An Introduction to Cut-off Grade Estimation became apparent as a result of feedback received from those who read the first edition and who attended cut-off grade courses I have given over the years. It became apparent that the information was not always presented in a rational fashion. Some fundamental concepts, such as those related to opportunity costs and how those costs should be accounted for in decision making, needed clarifi￾cation. The focus of the 2008 edition was on metal mining, and, over time, many questions have been raised concerning application to industrial miner￾als, iron ore, and coal. This new edition has attempted to take this feedback into account. Chapters have been reorganized and their content clarified. The relationship between optimization of net present value, capacity constraints, and oppor￾tunity cost is explained in greater detail. A new section has been added that discusses blending strategies applicable to coal and iron ore mines, as well as in an increasing number of metal mines. The objective was to make this new edi￾tion easier to read and of greater practical interest to practitioners operating in a wider range of situations. In addition, the bibliography has been greatly expanded. You, the readers, will decide whether the effort that went into this new edition was justified. Many individuals contributed to this edition. I am particularly grateful to Professor Roussos Dimitrakopoulos of McGill University, Canada, who not only reviewed and commented on early drafts of this book, but also gave me the opportunity to give short courses on cut-off grade estimation at McGill University and in Perth, Australia. The challenging questions raised by the short-course attendees resulted in many of the changes made in this new edition. I also thank Professor Joao Felipe Costa of Universidade Federal do Rio Grande do Sul, Brazil, who also reviewed early drafts and helped correct Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. viii Preface to 2014 Edition some of the errors I made. I also recognize Steven Hoerger and Fred Seymour whom I have had the pleasure to work with for many years, and who contrib￾uted to the early development of ideas included in both the first and second editions. I also thank the Society for Mining, Metallurgy & Exploration, who accepted the manuscript, as well as Jane Olivier, manager of book publishing, and Diane Serafin, senior editor, who made publication possible. Bringing this book to successful publication required the effort of many more individuals not listed here, but they know who they are and I am grate￾ful to all of them. The responsibility for any errors, lapses, or absences of clarity that may remain from the first edition or that I may have introduced in the second edition is entirely mine. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. ix Preface to 2008 Edition This book started with a desire to understand how to answer an apparently simple but actually complex question faced by all those responsible for the development and operation of mines: How do we determine which cut-off grade should be used to separate material that should be processed from that which should be sent to the waste dump? The answer appears straightforward: If it is profitable to process one metric ton of material, this ton should be processed. But what is profitable? The cut-off grade has a direct bearing on the tonnage of material mined, the tonnage and average grade of material pro￾cessed, the size of the mining operation, and consequently capital costs, oper￾ating costs, environmental and socioeconomic impacts. Should we maximize cash flow, net present value, the life of the mining operation, the return to shareholders? How do we take into account economic, environmental, social, political, ethical and moral values, objectives, and regulations? Somewhat surprisingly, only one other book has been written exclusively on the subject of cut-off grade estimation: The Economic Definition of Ore: Cut-Off Grades in Theory and Practice by Ken Lane, published in 1988. Lane’s book was and will remain the standard for mathematical formulation of solutions to cut-off grade estimation when the objective is to maximize net present value. Concepts first formulated by Lane were used as the foundation of this book. Considerable progress has been made in the last twenty years to improve mine planning and optimize cut-off grades. Increasingly complex algorithms have been developed, and better, easier to use computer programs have been written to assist engineers and economists in analyzing mine plans, testing the options, and improving production schedules. Computer programs have become easier to use, but the assumptions made by those who write the pro￾grams are often lost to the end user. With this book I am hoping to bridge the gap between theory and practice, the ivory tower and engineers in the field, by describing the fundamental principles of cut-off grade estimation and pro￾viding concrete examples. This book started as notes written during the last thirty years. Eventu￾ally these notes turned into an introductory short course. Each time I gave the course, more and more questions were asked, concerning increasingly complex situations, demanding more practical examples and challenging the assumptions made. Each question resulted in corrections, additions, and more chapters. I am extremely thankful to all those who helped me in this respect. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. x Preface to 2008 Edition They include too many individuals over too many years to be listed here. They know who they are and I would not have continued this work without their probing and their interest in the subject. I am particularly grateful to Ernie Bohnet who kept on motivating me when I doubted that I had a story to tell or that there would be sufficient interest in continuing this effort to make it worthwhile. It is because of Ernie that I completed this book. I also want to thank the Society for Mining, Metallurgy & Exploration, and Jane Olivier, who accepted the manuscript and brought it to publication in record time. None of these people, of course, can be blamed for any errors or lapses that I may have made and for which I am fully responsible. My first book, An Introduction to Geostatistical Methods of Mineral Eval￾uation, was published in 1978 with the objective to clarify the already arcane science of geostatistics. It is only fitting that An Introduction to Cut-Off Grade Estimation be published, with similar objectives, in 2008, exactly thirty years later. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 1 A cut-off grade is generally defined as the minimum amount of valuable product or metal that one metric ton (i.e., 1,000 kg) of material must contain before this material is sent to the processing plant. This definition is used to distinguish material that should not be mined or should be wasted from that which should be processed. In complex geological environments, impurities may have to be considered to define the cut-off grade. Cut-off grades are also used to decide the routing of mined material when two or more processes are available, such as heap leaching and milling. Cut-off grades are used to decide whether material should be stockpiled for future processing or processed immediately. The need to separate material being mined according to its physical and chemical properties before it is processed was well understood by historical miners. In the year 1556, Georgius Agricola reminded the readers of De Re Metallica that Experienced miners, when they dig ore, sort the metalliferous material from earth, stones, and solidified juices before it is taken from the shafts and tunnels, and they put the valuable metal in trays and the waste into buckets.… To smelt waste together with an ore involves a loss, for some expenditure is thrown away, seeing that out of earth and stones only empty and useless slags are melted out, and further, the solidified juices also impede the smelting of the metals and cause loss. Agricola thus pointed out that, for economic reasons, both chemical and physical properties of material being mined must be taken into account, including impurities and deleterious elements (solidified juices) before this material is processed. Cut-off grades are calculated by comparing costs and benefits. In sim￾ple geological and metallurgical environments, a single number, such as a minimum metal content, is sufficient to define the cut-off grade. In more complex situations, the quality of the material being processed, including the Introduction Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 2 CHAPTER ONE amount of deleterious elements, must be taken into account. Sales contracts may impose quality requirements that can only be satisfied by stockpiling and blending material according to quality. In most situations, costs and recoveries, and therefore cut-off grades, vary with the geological characteristics of the material being mined. Grade is usually the most important factor but may not be the only one. If material is sent to a waste dump, the acid-generating potential of this material may have a direct impact on costs related to environmental controls. Sulfide content may be a critical—even overriding—factor for material sent to a roasting or flotation plant. Clay content may have a deleterious effect on the recovery and through￾put of a leaching plant. Phosphorous and silica grades must be considered in selecting iron ore. Ash and sulfur content influence decisions in coal mines. The cut-off grade defines the profitability of a mining operation as well as the mine life. A high cut-off grade can be used to increase short-term profit￾ability and the net present value (NPV) of a project, thus possibly enhancing the benefit to shareholders and other financial stakeholders, including the government and local communities. However, increasing the cut-off grade is also likely to decrease the life of the mine. A shorter mine life can reduce time-dependent opportunities such as those offered by price cycles. A shorter mine life can also result in higher socioeconomic impact with reduced long￾term employment and decreased benefits to employees and local communities. Increased cut-off grades may be considered to reduce political risk by ensuring a higher financial return over a shorter time period. The cut-off grade may be increased when metal prices go up if this is needed to strengthen the financial position of the company and reduce the risk of failure when metal prices fall. Conversely, cut-off grades may be decreased during periods of high prices to increase mine life and keep high-grade material available to main￾tain profitability when prices fall. Cut-off grades may also be constrained by economic or technical performance criteria imposed by bank loans and other financial institutions. In some instances, a conscious decision might be made to increase the mining capacity while keeping the processing capacity constant. This allows an increase in cut-off grade. Some of the lower-grade material may be stock￾piled for processing at a later date. Stockpiling may have a number of conse￾quences—some positive (such as increased useful life of processing facilities) and others negative (such as increased environmental risk and decreased metallurgical recovery of stockpiled material). Public reporting of reserves, which are dependent on the cut-off grade, is subject to the rules and regulations of the various stock exchanges and other Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Introduction 3 regulatory agencies. A link exists between published reserves and generally accepted accounting practices. Reserves enter into the calculation of capital depreciation, company book value, unit cost of production, and taxes. A link also exists between published reserves and the value that the financial market gives to a mining company. For some commodities there is a fairly widely held but arguably incorrect belief that this link is primarily a function of the mag￾nitude of the reserves while quality is of lesser significance. Low cut-off grades may be considered desirable by those calculating or publicly reporting reserves if personal bonuses are a function of the magnitude of the published reserves. As a result of these various links—some desirable, some not—there can be a desire to maximize the published reserves by using the lowest technically, financially, and legally defendable cut-off grade. However, one must always keep in mind that reserves are published for the purpose of informing inves￾tors and other stakeholders, and that processes and controls should be put in place to eliminate the influence of factors that could result in publication of misleading estimates. Both outsider and insider stakeholders have an interest in the cut-off grade and the reserves deriving from it. Outsiders include shareholders, financial institutions, local communities, environmentalists, regulators, gov￾ernmental and nongovernmental agencies, suppliers, and contractors and buyers of the product being sold. Insiders include company management and employees. The board of directors represents the interests of the shareholders and is often composed of both insiders and independent outsiders. Cut-off grades are and should be primarily calculated taking into account only tech￾nical and economic constraints. However, the often-conflicting interests and objectives of the many stakeholders must be understood and prioritized if the best decision is to be made concerning cut-off grade determination. The technical literature includes many publications on estimation and optimization of cut-off grades. The most comprehensive reference is Ken￾neth F. Lane’s The Economic Definition of Ore: Cut-Off Grades in Theory and Practice (Lane 1988). The objective most commonly accepted in cut-off grade optimization studies is optimization of the NPV of future cash flows. To reach this objective one must take into account space-related variables (such as the geographic location of the deposit and its geological characteristics), as well as time-related variables (including the order in which the material will be mined and processed and the resulting cash flow). The time–space nature of the problem is quite complex; consequently, so are the proposed mathe￾matical solutions to cut-off grade optimization. The bibliography provides detailed references to some of these solutions. This book attempts to explain Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 4 CHAPTER ONE basic concepts in a simple fashion, making them accessible to mine managers, analysts, geologists, mining engineers, accountants, and other practitioners. The book is divided into several chapters. General concepts are intro￾duced in Chapter 2, including those of utility function, breakeven cut-off grade, and opportunity cost. Each one of these concepts is then analyzed for its application in increasingly complex situations, starting with Chapter 3 in which estimation of breakeven cut-off grades is discussed. Examples are given for how to separate ore from waste, choose between different processes, ana￾lyze polymetallic deposits, and develop stockpiling strategies. The less intuitive but fundamental concept of opportunity cost is dis￾cussed in Chapters 4 and 5. Opportunity costs result from constraints that are present in all mining operations. As shown in Chapter 4, ignoring capacity constraints can result in underestimation of cut-off grades and the processing of uneconomic material. Formulas are developed and examples given to illus￾trate the relationship between opportunity costs and cut-off grades. Opportu￾nity costs decrease over time, and cut-off grades should decrease accordingly. Constraints imposed by the geology of the deposit are discussed in Chapter 5. Capacity constraints and geologic constraints must be taken into account jointly to determine the applicable cut-off grade. This complex relationship is illustrated by a number of examples. Chapter 6 shows that while the same fundamental concepts can be used to estimate cut-off grades in any situation, there are significant differences in the application of these concepts to different mining methods. The costs that should be included in cut-off grade calculations are further discussed in Chap￾ter 7. Situations are considered in Chapter 8 where there is a need to stockpile and blend material to satisfy constraints imposed by technical or marketing requirements. New methods are introduced to optimize blending depending on the objective to be reached. Closing remarks can be found in Chapter 9, which is followed by Appen￾dix A, a list of symbols, and a bibliography. Appendix A contains additional examples and solutions to problems referenced in the main text of this book. Measurements are usually given using the International System of Units (SI), such as metric tons (t), grams (g), or meters (m). However, examples are included that also make use of other units, such as ounces (oz) for gold or pounds (lb) for copper, which are commonly used in specific industries. One metric ton is equal to 1,000 kg (kilograms), or 2,205 lb. The word “ounce” refers to a troy ounce of gold and is equal to 31.1035 g. The word “short ton,” commonly used in the United States, is equal to 2,000 lb. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 5 2 Choosing a cut-off grade is equivalent to choosing the value of a geologically defined parameter or set of parameters that will be used to decide whether one metric ton (1 t) of material should be sent to one process or another. MATHEMATICAL FORMULATION: UTILITY FUNCTION Let x be the value of the parameter(s) that must be taken into account to determine the destination to which the material should be sent. In simple cases, a single parameter may be sufficient to define the destination such as copper grade or gold grade. In other cases, a set of parameters may have to be considered such as copper and gold grades, sulfide content, clay content, and percentage of deleterious elements. The value, or utility*, of sending one metric ton of material with parame￾ter value (grade) x to destination 1 (process 1) is U1 (x). The utility of sending the same material to destination 2 (process 2) is U2 (x). The destination that should be chosen is that which results in the highest utility Umax(x). Figure 2-1 shows the relationship between Umax(x), U1 (x), and U2 (x). The cut-off grade xc is the value of x for which U x U x 1 c 2 c () () = If U2 (x) exceeds U1 (x) for x greater than xc , then all material for which x is greater than xc should be sent to process 2. Figure 2-1 shows how the cut-off * The term utility is used in decision theory to represent the satisfaction gained from following a given course of action (Raiffa and Schlaifer 2000). This satisfaction is a function of preferences and values specific to the decision maker. The utility of a given cut-off grade strategy is a measure of the extent to which this strategy reaches the mining company’s objectives. General Concepts Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 6 CHAPTER TWO grade can be graphically determined by finding the intersection between U1 (x) and U2 (x). As indicated in the introduction, the choice of a cut-off grade is governed primarily by financial objectives. However, the consequences of choosing a given cut-off grade are complex and not all of a financial nature. When esti￾mating cut-off grades, all controlling variables must be taken into account. To facilitate this process, the utility U(x) of sending material of grade x to a given process is expressed as the sum of three parts: U x U x U x U x dir opp oth () () () () = + + In this equation, Udir(x) represents the direct profit or loss that will be incurred from handling one metric ton of material of grade x. Uopp(x) represents the opportunity cost or benefit of changing the processing schedule by adding one metric ton of grade x to the material flow. This opportunity cost only occurs when there are constraints that limit how many tons can be processed at a given time. Other factors, which must be taken into account in the calculation of cut-off grades but may not be quantifiable, are represented by Uoth(x). C U T- O F F G R A D E A N D G R A D E – TO N N A G E RELATIONSHI P The cut-off grade determines the tonnage and average grade of material delivered to a given process and therefore the amount of product sold. In first –40 –20 0 20 Grade x 40 60 0 2 4 6 8 10 Umax(x) Cut-off Grade xc U2 (x) U1 (x) Figure 2-1 Utility maximization and graphical determination of cut-off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Utility U(x) General Concepts 7 approximation, if T+c represents the tonnage and x+c symbolizes the average grade of material above the cut-off grade xc , the revenue from sales is equal to T+c ∙ x+c ∙ r ∙ V where r is the proportion of valuable product recovered during processing and V is the market value of the product sold. The cut-off grade also determines the tonnage of material mined that will not be processed. Fig￾ure 2-2 shows the relationship among cut-off grade and tonnage and average grade above cut-off grade. The curves on this graph are known as the grade– tonnage curves. Grade–tonnage curves are used extensively throughout this document to illustrate the impact that different cut-off grade strategies have on the economics of a mining operation. Another relationship that is useful in optimizing cut-off grades is shown in Figure  2-3. This figure  shows the quantity Q+c of valuable product contained in material above cut-off grade as a function of the cut-off grade: Q T + + c c = ⋅x +c As an example, when considering a copper mine, the grade x, the cut-off grade xc , and the average grade above cut-off x+c are expressed in % Cu; the tonnage T+c is expressed in metric tons; and the quantity Q+c is expressed in metric tons or pounds of copper (1 t equals 2,205 lb). In a gold mine, the grades are expressed in grams per metric ton (g/t) and the quantity of metal in metric tons or ounces of gold (1 troy ounce equals 31.1035 g). 0 2 8 6 4 Cut-off Grade 10 12 0.0 1.0 4.0 3.0 2.0 5.0 6.0 0 0.5 1 2.5 2 1.5 3 3.5 Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade x+c T+c xc Figure 2-2 Example of grade–tonnage curve: tonnage and grade above cut-off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade 8 CHAPTER TWO DIRECT PROFIT AND LOSS: Udir( x ) Direct profits or losses associated with one metric ton of material, Udir(x), are estimated by taking into account only costs and revenues that can be directly assigned to mining this material, processing it, and selling the final product. Breakeven cut-off grades are calculated taking into account only costs and rev￾enues directly resulting from mining and processing (or wasting) one metric ton of material at grade x. The impact, if any, on total tons mined, production schedule, and cost per metric ton is ignored. The grade–tonnage relationship is also ignored. Mathematical Formulation The direct profit or loss Udir(x) expected from processing one metric ton of material of grade x is Uore(x) expressed as follows: U x x r V R M P O ore o o o ( ) ( )= ⋅ ⋅ − ( )− + + where x = average grade r = recovery, or proportion of valuable product recovered from the mined material V = value of one unit of valuable product R = refining, transportation, cost of sales, and other costs incurred per unit of valuable product 0 2 8 6 4 Cut-off Grade 10 12 0 0.5 1 2.5 2 1.5 3 3.5 Metal Content Above Cut-off Grade xc Q+c Figure 2-3 Quantity of valuable product contained above cut-off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Metal Content Above Cut-off Grade General Concepts 9 Mo = mining cost per metric ton processed Po = processing cost per metric ton processed Oo = overhead cost per metric ton processed If the valuable product is a concentrate, V is the value of one unit of metal contained in the concentrate. For example, V can be the copper price expressed in dollars per pound of copper, or the gold price expressed in dollars per troy ounce of gold. r is the percentage of metal in concentrate that will be paid for. R includes transportation and refining costs, and other deductions and penalties to be deducted from V. When concentrate is sold to a smelter, the applicable values of V, r, and R may be negotiated between seller and buyer and specified in a smelter contract. If the material is to be wasted, the value of Udir(x) is Uwaste(x), expressed as follows: U x M P O waste w w w ( ) = −( ) + + Mw and Ow are mining and overhead costs, respectively, per metric ton of waste. Pw is the cost of processing one metric ton of waste as may be needed to avoid potential water contamination and acid generation, and to satisfy other applicable regulatory and environmental requirements. The cut-off grade between ore and waste is xc such that Uore(xc ) = Uwaste(xc ). Solving this equation gives the following result: x M P O M P O r V R c o o o w w w [ ] ( ) ( ) ( ) = + + − + +     ⋅ − The cut-off grade between ore and waste is equal to the difference between the cost of mining and processing ore and waste, (Mo + Po + Oo ) – (Mw + Pw + Ow), divided by the value of one unit of metal contained in one metric ton of material processed r ∙ (V – R). Precious Metal Example To illustrate how these formulas are used to calculate the cut-off grade, con￾sider an open pit gold mining operation characterized as follows: • For ore being processed: r = 90%, V = $1,400 per ounce of gold, R = $25.00 per ounce, Mo = $4.50 per metric ton mined and processed, Po = $75.00 per metric ton processed, and Oo = 20% of operating costs. • For wasted material: Mw + Pw = $5.00, and Ow = 20% of operating costs. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 10 CHAPTER TWO The cut-off grade between ore and waste is calculated as follows: [ ] [ ] ( ) ( ) = ⋅ + − ⋅ ⋅ − = = x 1.20 4.50 75.00 1.20 5.00 0.90 1,400 25 0.072 oz/t 2.25 g/t c NET PRESENT VALUE: NPVi The optimal cut-off grade strategy is that which maximizes the enterprise￾specific objective function. Such a function nearly always includes, but is not limited to, net present value optimization. The net present value (NPVi ) of a project is today’s cash equivalent of the cash flow that is expected to be gen￾erated by this project, assuming that money can be invested or borrowed at a specified discount rate i (e.g., i = 10%). Mathematical Formulation If n is the life of a project measured in years, and the (positive or negative) cash flow generated in year k is Ck (k = 0, 1, 2, …, n–1), the net present value of the project using the discount rate i is ∑ () () () ( ) = + + + + + + + = + − − = −  NPV C C 1 i C 1 i C 1 i NPV C 1 i i 0 1 2 2 n 1 n 1 i k k k 0,n 1 Note that in this formula, and throughout this book, the first year (year 0) is not discounted. Example of NPV Calculation Consider a project that is expected to generate the following cash flow over seven years. The discount rate to be used is i = 15%. Yearly cash flows are in millions of dollars. C 200, 20, 100, 120, 100, 50, 20 k , k 0,6 = −{ } − = The net present value is calculated in Table 2-1. The discounted cash flow for a given year is equal to the undiscounted cash flow multiplied by the dis￾count factor for this year. The total undiscounted cash flow is $170 million, whereas the discounted cash flow—the project net present value—is $45.30 million. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. General Concepts 11 NPV of Constant and Perpetuity Cash Flows In some situations a constant cash flow must be generated over a finite num￾ber of years to justify an investment such as sustaining capital. Some situations also exist where constant yearly costs are expected to perpetuity, such as envi￾ronmental remediation costs. Examples will be encountered throughout this book. In such situations, simplifying formulas can be used to calculate net present values. If a project is expected to generate a constant cash flow C over n years, the following equation can be used to calculate the net present value:  ()() () ( ) = + + + + + + +     = − NPV C 111 i 1 1 i 1 1 i C g i,n i 2 n 1 where g(i,n) = [1 – (1 + i)–n] (1 + i)/i If the cash flow is expected to be constant to perpetuity, the net present value is NPV C 1 i i i = + ( ) Proof of these formulas is given in Examples 1 and 2 in Appendix A for constant cash flows and cash flows to perpetuity, respectively. Table 2-2 shows how the net present value of a $1.00 constant cash flow varies depending of the life of the project and the discount rate. For example, when calculated at a 7% discount rate, the net present value of a yearly $1.00 cash flow over five years has a net present value of $4.39. The same cash flow over an infinite period has a net present value of $15.29. Table 2-1 Calculation of net present value using a 15% discount rate Year, k Undiscounted Cash Flow, millions, Ck Discount Factor, 1/(1 + i)k Discounted Cash Flow, millions, Ck /(1 + i)k 0 ($200.00) 1.0000 ($200.00) 1 $20.00 0.8696 $17.39 2 $100.00 0.7561 $75.61 3 $120.00 0.6575 $78.90 4 $100.00 0.5718 $57.18 5 $50.00 0.4972 $24.86 6 ($20.00) 0.4323 ($8.65) Total $170.00 $45.30 undiscounted discounted Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 12 CHAPTER TWO OPPORTUNITY COSTS AND BENEFITS: U opp( x ) Ken Lane (1964, 1988) was first to demonstrate the importance of opportunity costs in cut-off grade estimation. Opportunity cost is defined as the change in a project net present value that results from changing the production schedule. Capacity Constraints and Opportunity Cost Consider a project for which the net present value of future cash flows NPVi was calculated using the discount rate i. Assume that material was added to the production schedule that was not included in the original schedule. Also assume that there are production constraints, the implication being that by adding new material, the originally planned production schedule is postponed or slowed down. The net present value of the delayed production schedule is NPV'i , less than the original net present value NPVi . By defini￾tion, the opportunity cost of changing the production schedule is equal to the decrease in net present value, NPV'i – NPVi . Adding new material to the production schedule is economically justified only if the net present value of the cash flow generated by this new material exceeds the opportunity cost of changing the original production schedule. Consider a mining operation in which, under current operating condi￾tions, the processing plant capacity is fully utilized. If new material is added to the original production schedule and given priority in the processing plant, the material originally scheduled for processing will be delayed. The duration t of this delay is the time needed to process the new material. The net present value of the originally scheduled production will be reduced from NPVi to NPV'i : NPV ' NPV 1 i i i t ( ) = + Table 2-2 Calculation of net present value of a constant $1.00 cash flow over n years Discount Rate, i Project Life (n years) 5 10 15 20 Infinity NPVi = g(i,n) NPVi = (1 + i)/i 5% $4.55 $8.11 $10.90 $13.09 $21.00 7% $4.39 $7.52 $9.75 $11.34 $15.29 9% $4.24 $7.00 $8.79 $9.95 $12.11 11% $4.10 $6.54 $7.98 $8.84 $10.09 13% $3.97 $6.13 $7.30 $7.94 $8.69 15% $3.85 $5.77 $6.72 $7.20 $7.67 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. General Concepts 13 The opportunity cost of postponing mining of the already scheduled material is equal to the loss in NPV: NPV ' NPV NPV 1 i 1 i i i t ( ) − = + −    −  The new material should be mined or processed only if its net present value exceeds the opportunity cost. An example is given in Example 3 in Appendix A, which shows how to calculate the opportunity cost of mining a peripheral deposit when this deposit is given priority over previously sched￾uled production. Application to Cut-off Grade Estimation The cut-off grade between two options is obtained by comparing the utility as well as the advantages and disadvantages of choosing one option rather than the other. The conclusions are valid only if all costs and benefits are taken into account. Opportunity costs or benefits, Uopp(x), may result from mining and pro￾cessing one metric ton of material not previously scheduled for processing. No opportunity cost is incurred if the mine, mill, refining facilities, and volume of sales are not capacity constrained and if adding one more metric ton to the process has no impact on previously expected cash flows. If there is a capac￾ity constraint, the opportunity cost includes the cost of displacing material already scheduled for processing and postponing treatment of this material. Consider a project whose net present value of future cash flows NPVi was calculated on the basis of currently planned production. According to the current plan, there is no spare capacity in the processing plant. If one metric ton of material is added to the capacity-constrained processing plant, treat￾ment of the originally scheduled material is postponed by the time needed to process the additional ton. Processing one metric ton of material takes t units of time, where t is the inverse of the yearly plant capacity. Adding one new metric ton of material not currently planned to be processed will decrease the net present value of future cash flows from NPVi to NPV'i = NPVi / (1 + i)t . The difference, NPV'i – NPVi , is the opportunity cost of processing one more metric ton of material: U x NPV 1 i 1 opp i t ( ) ( ) = + −    −  In this equation, the time t is very small, as it is equal to the inverse of the yearly capacity of the constrained material flow. Calculus can be used to demonstrate the following simplified formula for the opportunity cost: Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 14 CHAPTER TWO ( ) = − ⋅ ⋅ U x i t NPV opp i A comparison of the exact formula NPVi [1 – (1 + i)–t] and its first-order approximation i∙t∙NPVi is given in Example 4 in Appendix A. This difference is less than 10% for a discount rate i = 15%, and decreases for smaller discount rates. The discount rate i is expressed as a percentage per year; the time t is expressed in years per unit of capacity-constrained material; the net present value is expressed in dollars. The opportunity cost is therefore expressed in dollars per unit of capacity-constrained material. The opportunity cost must be added to the direct cost of the process that is capacity constrained. If the mine is capacity constrained and a decision is made to mine an additional metric ton of material that was not scheduled to be mined, t is the time needed to mine this ton and the opportunity cost must be added to the mining cost Mo or Mw. If one new ton of ore is sent to a capacity-constrained mill, t is the time needed to mill this metric ton, and the opportunity cost must be added to the processing cost Po . If the refining process is capacity limited, t is the time needed to refine the concentrate produced from one metric ton of material at grade x, and the opportunity cost must be added to the refining cost R. Application to an Underground Gold Mine Consider an underground gold mine in which the net present value of future cash flows is NPVi = $800,000,000. This value was calculated using the discount rate i = 15%. The mineshaft is capacity constrained, with a maxi￾mum haulage capacity of 1,500,000 t/yr. Consideration is given to mining low-grade material on the periphery of high-grade stopes. This material was not included in calculation of the above-mentioned net present value. The time needed to mine and deliver one metric ton of material to the surface is t = 1/1,500,000 year. The opportunity cost of adding one metric ton to the production schedule is U x 15% $800,000,000 1,500,000 $80.00 per metric ton of ore mined opp ( )= − ⋅ = − The following parameters apply to ore being processed: r = 90% V = $1,400 per ounce of gold recovered Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. General Concepts 15 R = $20.00 per ounce of gold recovered Mo = $250.00 per metric ton mined Po = $85.00 per metric ton processed Oo = 20% of operating costs If only direct costs and revenues are taken into account, the cut-off grade between ore and waste is [ ] [ ] ( ) ( ) = ⋅ + ⋅ − = = x 1.20 250.00 85.00 0.90 1,400.00 20.00 0.324 oz/t 10.07 g/t c Adding the $80.00 opportunity cost to the mining cost results in a 2 g/t increase to the cut-off grade: [ ] [ ] ( ) ( ) = ⋅ + + ⋅ − = = x 1.20 250.00 85.00 80.00 0.90 1,400.00 20.00 0.388 oz/t 12.07 g/t c At this stage of the mine life, low-grade peripheral material should be mined only if it averages at least 12.07 g/t. When the mine approaches the end of its life, the net present value of future cash flows will decrease toward zero and so will the opportunity cost Uopp(x). The cut-off grade will then decrease to reach the marginal cut-off grade, 10.07 g/t. Multiple examples are given later that show how opportunity costs must be taken into consideration depending on prevailing situations. O T HER COSTS AND BENEFITS: U oth( x ) Cut-off grades play a critical role in defining tonnages mined and processed, average grade of mill feed, and project economic feasibility. In addition to the quantifiable financial impact that cut-off grade changes may have, other costs and benefits must be taken into account, although their impacts are often not easily quantifiable. Consideration must be given not only to changes in net present value and cash flow—as measured by Udir(x) and Uopp(x)—but also to all other impacts, Uoth(x), including those of an environmental, socioeco￾nomic, ethical, or political nature. Costs and benefits to all stakeholders must be evaluated, including share￾holders, financial institutions, analysts, employees, customers, suppliers, local communities, government, nongovernmental agencies, and future genera￾tions. Senior management must decide how to balance the needs, interests, Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 16 CHAPTER TWO and requirements of these stakeholders. Practical guidelines must be devel￾oped, including guidelines for cut-off grade determination, to ensure that the projects are designed to reach the company’s often conflicting objectives. Maximizing shareholder value is often quoted as a company’s primary objec￾tive. However, other objectives must be taken into account, which include recognition of responsibilities toward all stakeholders. CUT-OFF GRADE AND BLENDING STRATEGY When determining cut-off grades, the situation most commonly encountered is one where the decision whether a material type—defined by its physical and chemical properties—should be mined and processed is made solely on the basis of these properties. This does not take into account more complex situations where blending of different material types is needed to obtain a product that can be economically processed and sold. A material type may have no economic value if considered on its own, but it is needed for blending with other material to obtain an economically valuable product. For example, in gold mines where refractory ore is treated by roasting, material with a high sulfide content and low gold content may have no value if considered on its own. Yet the calorific value of this material may be needed to economically extract the gold contained in high-grade, low-sulfide ore. Circumstances also occur where one material type does not satisfy con￾straints imposed by sales contracts, but blending with other material will result in a product that exceeds contractual requirements. This can be the case in a coal mining operation where one seam has a low calorific value and high ash content and another seam has a high calorific value and low ash content. By appropriately mixing the two products, it may be possible to increase the tonnage of salable coal. Similarly, the product from one iron mine may not be salable on its own, but a salable product may be obtained by blending with the product of another mine that exceeds contractual requirements. A typical situation happens where stockpiles are available with known tonnages and geochemical properties. In which proportion should the stock￾piles be blended to satisfy specified quality requirements? A similar situation occurs when two or more mines are being designed whose products will have to be blended before being processed or sold. What should be the mining rate for each mine, and which cut-off grades should be applied to obtain a material blend that maximizes profits? Different blending strategies are discussed in this book. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 17 3 Minimum or breakeven cut-off grades are those that apply to situations where only direct operating costs are taken into account. Capacity constraints are ignored. Cash flows are not discounted. Opportunity costs are not taken into consideration and neither are other consequences, financial or otherwise, that changing the cut-off grade may have on mining and processing schedules and cash flows. CUT-OFF GRADE BETWEEN ORE AND WASTE Consider material for which the decision has already been made that it will be mined, so the remaining question is whether it should be sent to the pro￾cessing plant or wasted. Mathematical Formulation Using notations introduced previously, the utility of mining and processing one metric ton of ore-grade material, Uore(x), can be written as follows: ( ) ( )= ⋅ ⋅ − ( )− + + U x x r V R M P O ore o o o where x = average grade r = proportion of valuable product recovered from the mined material V = value of one unit of valuable product R = refining costs, defined as costs that are related to the unit of valuable material produced Mo = mining cost per metric ton of ore Po = processing cost per metric ton of ore Oo = overhead cost per metric ton of ore Breakeven Cut-off Grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 18 CHAPTER THREE The utility of mining and wasting one metric ton of waste material, Uwaste(x), can be written as follows: ( ) = −( ) + + U x M P O waste w w w where Mw = mining cost per metric ton of waste Pw = processing cost per metric ton of waste, as may be needed to avoid potential water contamination and acid generation Ow = overhead cost per metric ton of waste The minimum cut-off grade is the value xc of x for which [ ] () () ( ) ( ) ( ) = = + + − + +     ⋅ − U x U x x M P O M P O r V R ore c waste c c o o o w w w (EQ 3-1) In this formula, the numerator represents the difference between mining, processing, and overhead costs incurred when treating the material as ore and those incurred when treating the same material as waste. In the denominator, the metal recovery r must be that which applies to material of grade xc , which is not necessarily equal to the average recovery for all material sent to the processing plant. The relationship between utility functions and cut-off grade is graphically represented in Figure 3-1. Internal or Mill Cut-off Grade If material must be mined, it should be processed if its grade is high enough to pay for processing costs even if it does not pay for mining costs. If the costs of mining and shipping material to the waste dump or to the primary crusher are the same (Mo = Mw) and there are no significant addi￾tional costs in processing waste (Pw = 0 and Ow = 0), this cut-off grade is only a function of mill costs and recoveries and is independent of mining costs. Equation 3-1 can be written as follows: [ ] [ ] ( ) = + ⋅ − x P O r V R c o o (EQ 3-2) This cut-off grade applies to material located within the limits of an open pit mine or an underground stope. This material is internal to the current mine plan and must be mined. The only question is whether it should be Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Breakeven Cut-off Grade 19 processed. For these reasons it is often called internal or mill cut-off grade. In this formula the overhead cost Oo is that which applies to processing only. External or Mine Cut-off Grade Without Waste Stripping If the option is to leave the material in the ground at no cost, it should be mined only if its grade is high enough to pay for both mining and processing costs. ( ) ( ) ( ) ( ) = = ⋅ ⋅ − − + + U x 0 U x x r V R M P O waste ore o o o The cut-off grade between ore and waste is [ ] ( ) ( ) = + + ⋅ − x M P O r V R c o o o (EQ 3-3) This cut-off grade includes the mining cost and is often referred to as the external or mine cut-off grade*. The overhead cost Oo is that which applies to both mining and processing. External or Mine Cut-off Grade with Waste Stripping In the previous formula (Equation 3-3), the external or mine cut-off grade applies to one metric ton of material of grade x that is exposed to the surface * Other terms used in open pit mines include in-pit cut-off grade for material within the pit limits (the internal or mill cut-off grade) and ex-pit cut-off grade for mate￾rial not included in the pit limits (the external or mine cut-off grade). U(x) 0 Uwaste(x) = –(Mw + Pw + Ow) Uore(0) = –(Mo + Po + Oo) Uore(x) = x·r·(V – R) – (Mo + Po + Oo) Uore(xc ) = Uwaste(xc ) xc Figure 3-1 Relationship between utility functions and cut-off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 20 CHAPTER THREE but had not been included in the mine plan; this material can be mined with￾out waste stripping. This next example considers the case where some waste stripping is needed. The strip ratio is defined as follows: =s metric tons of waste per metric ton of ore If this material is not mined, the utility is zero: ( ) = U x 0 waste If the decision is made to mine the mineralized material, the utility is ( ) ( ) ( )= ⋅ ⋅ − ( )− + + − ⋅ + + U x x r V R M P O s M P O ore o o o w w w The cut-off grade is [ ] ( ) ( ) ( ) = + + + ⋅ + +     ⋅ − x M P O s M P O r V R c o o o w w w (EQ 3-4) CUT-OFF GRADES IN OPEN PIT MINES The following example shows how to calculate mine and mill cut-off grades, with and without waste stripping, in an open pit copper mine. The mine oper￾ating parameters are characterized as follows: r = 86% (includes mill and smelter recovery) V = $3.20 per pound of payable copper R = $0.60 per pound of payable copper (includes freight, smelting, and refining) Mo = $3.00 per metric ton of ore mined Po = $10.00 per metric ton of ore processed Oo = $1.70 per metric ton of ore processed Mw = $3.30 per metric ton of waste mined Pw = $0.50 per metric ton of waste mined (includes environmental control and remediation) Ow = $0.40 per metric ton of waste mined The opportunity cost of classifying one metric ton of material as ore or waste is as follows (1 t = 2,205 lb): Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Breakeven Cut-off Grade 21 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ⋅ ⋅ − − + + = ⋅ ⋅ − ⋅ − + + = − = − + + = − + + = − U x x r V R M P O x 0.86 3.20 0.60 2,205 3.00 10.00 1.70 4,929x $14.70 U x M P O 3.30 0.50 0.40 $4.20 ore o o o waste o o o For material internal to the pit limits, the mill cut-off grade is = − ( ) = x 14.70 4.20 4,929 0.21% Cu c (EQ 3-5) For material external to the pit limits, the mine cut-off grade is x = = 14.70 4,929 0.30% Cu c (EQ 3-6) The cut-off grades calculated in Equations 3-5 and 3-6 could have been obtained directly by substituting the appropriate values in Equations 3-1 and 3-3. The difference in cut-off grade, 0.09% Cu, represents the amount of additional copper needed to pay for the cost of mining material that was not scheduled to be mined. Figures 3-2 and 3-3 give graphical representations of the utility functions used to calculate the mill and mine cut-off grades, respectively. In the previous calculations leading to Equation 3-6, the external or mine cut-off grade is that which applies to one metric ton of material of grade x that is exposed to the surface but had not been included in the mine plan. This material can be mined without waste stripping. This next formula considers the case where some waste stripping is needed to mine this material. The ore to waste strip ratio is s = 1.5 t of waste per metric ton of mineralized material The utility of mining and processing the mineralized material is ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ⋅ ⋅ − − + + − ⋅ + + = ⋅ ⋅ − ⋅ − + + − ⋅ + + = − U x x r V R M P O s M P O x 0.86 3.20 0.60 2,205 3.00 10.00 1.70 1.5 3.30 0.50 0.40 4,929x $21.00 ore o o o w w w Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 22 CHAPTER THREE The corresponding cut-off grade is x = = 21.00 4,929 0.43%Cu c (EQ 3-7) This cut-off grade could have been calculated directly by substituting the appropriate values in Equation 3-4. Figure 3-4 gives a graphical representation of the utility functions used to calculate the mine cut-off grade with a 1.5 strip ratio. –$20 –$10 –$15 $5 0 –$5 Cut-off Grade, % Cu $10 $15 0.10% 0.20% 0.40% 0.30% 0.50% Material Mined and Wasted Material Mined and Processed Cut-off Grade 0.21% Cu Figure 3-2 Graphical estimation of internal (mill) cut-off grade –$20 –$10 –$15 $5 0 –$5 Cut-off Grade, % Cu $10 $15 0.10% 0.20% 0.40% 0.30% 0.50% Material Not Mined Material Mined and Processed Cut-off Grade 0.30% Cu Uore(x) Figure 3-3 Graphical estimation of external (mine) cut-off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Utility, Profit per Metric Ton Utility, Profit per Metric Ton Breakeven Cut-off Grade 23 CUT-OFF GRADES IN UNDERGROUND MINES General Considerations Capacity constraints are common in underground mines. These may include constraints imposed by ore body geometry, geotechnical conditions, shaft and haulage capacities, ventilation requirements, mining method, size and type of mining equipment, health and safety regulations, and other restrictions that limit production from a stope, a mine section, or the mine as a whole. Further￾more, many decisions made in an underground mine are capital intensive and take a significant amount of time to implement and result in positive cash flow. In such an environment, capacity constraints have a significant effect on the cut-off grades. Opportunity costs must be taken into account to quantify how the project net present value is modified over time by changing cut-off grade and mine plan. In this section, only undiscounted cash flows are taken into account, without discounting. Opportunity costs will be discussed in Chapter 4. The minimum conditions that must be satisfied to justify development of a new stope and how to determine the locations of stope boundaries, are two aspects of underground mining that are discussed in the next sections (see Figure 3-5). Minimum Tonnage and Grade A minimum grade is occasionally quoted when referring to the average grade that a stope must exceed before it is considered for mining. Strictly speaking, –$25 –$10 –$15 –$20 $5 0 –$5 Copper Grade, % Cu $10 $15 0.10% 0.20% 0.40% 0.30% 0.50% Material Not Mined Material Mined and Processed Cut-off Grade 0.43% Cu Uore(x) Figure 3-4 Graphical estimation of internal (mill) cut-off grade with strip ratio Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Utility, Profit per Metric Ton 24 CHAPTER THREE this is not a cut-off grade but an average grade, which must be linked to a tonnage. The minimum stope average grade depends on the size of the stope, its location with respect to existing facilities, ease of access, geotechnical conditions, and other stope-specific characteristics. This average grade is that for which the cost of developing and mining the stope is expected to be less than the profit made by processing the ore and selling the final product. This calculation must be made on a discounted basis, taking all physical constraints into account. Optimization of Stope Size When designing a stope, one must take into account the constraints imposed by mining method and geotechnical conditions. One must also determine whether lower-grade material located along the boundary of the stope should be included in the stope. Such material should be mined only if the expected value of the recoverable product it contains exceeds all incremental costs, including mining, haulage, processing, backfilling, and other costs. The min￾imum cut-off grade that defines boundary material which should be mined is the mine cut-off grade, and is estimated using a formula similar to that for material at the bottom of an open pit mine (Equation 3-3): [ ] ( ) ( ) = + + ⋅ − x M P O r V R c o o o Mineralized Zone Should we extend the top drift and make the stope wider? Proposed Accesses and Stope Boundaries Existing Accesses Figure 3-5 Estimation of average grade and cut-off grade in an underground mine Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Breakeven Cut-off Grade 25 As an example, consider an underground gold mine where the incremen￾tal mining cost is $120.00 per metric ton, the mill processing cost is $80.00 per metric ton, and the mining and processing overhead is $30.00 per metric ton. The mill recovery is 90%. Given a gold price of $1,400 per ounce and a sales cost of $80.00 per ounce, the minimum cut-off grade to be considered to locate a stope boundary is (1 troy ounce = 31.1035 g) [ ] [ ] ( ) = ⋅ + + ⋅ − = x 31.1035 120.00 80.00 30.00 0.90 1,400.00 80.00 6.02 g/t c This cut-off grade is the lowest average grade that a stope increment must have before it is considered for mining. Both planned and unplanned dilution must be taken into account. The mining cost should include the incremental development cost needed to make the stope wider. Opportunity costs such as those imposed by haulage capacity constraint should be taken into account, which will increase the cut-off grade. Opportunity costs will be discussed in Chapter 4. If low-grade material must be mined because it is located within a stope or within other planned openings such as shafts, raises, drifts, crosscuts, and so forth, a lower cut-off grade can be used to determine whether this material should be wasted or processed. For such material, blasting and haulage costs will be incurred whether the material is treated as ore or waste. Only incre￾mental costs need be considered. The minimum cut-off grade is estimated using the following general formula: [ ] ( ) ( ) ( ) = + + − + +     ⋅ − x M P O M P O r V R c o o o w w w If ore and waste mining costs, including overhead, are the same (Mo = Mw) and waste processing costs are negligible (Pw = 0), this formula can be written as follows: [ ] ( ) ( ) = + ⋅ − x P O r V R c o o The mill cut-off grade is recognized here (Equation 3-2). In this formula, the overhead cost Oo is that which applies to processing only. Applicable oppor￾tunity costs, which in this case are likely to be only those imposed by mill capacity constraints, should also be taken into account, which will increase the cut-off grade. As an example, consider exploration drifts in an underground gold mine. While mostly in waste, these drifts periodically encounter mineralized zones, and the decision must be made whether this material should be wasted or Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 26 CHAPTER THREE sent to the processing plant. Mining costs are Mo = Mw = $350.00 per metric ton, and mining overhead costs are Ow = $70.00 per metric ton, regardless of whether the material is hauled to the waste dump or the mill. The milling cost is Po = $79.00 per metric ton, with a processing overhead cost of $12.00 per metric ton. The total overhead cost per metric ton of ore, including mining and processing, is Oo = $70.00 + $12.00 = $82.00. The gold price is $1,450 per ounce, and the cost of sales is $90.00 per troy ounce. The mill recovery is 89%. The minimum cut-off grade that should be used to send material to the mill is [ ] [ ] ( ) ( ) ( ) ( ) ( ) = ⋅ + + − +       ⋅ − = ⋅ + ⋅ − = x 31.1035 350.00 79.00 82.00 350.00 70.00 0.89 1,450.00 90.00 31.1035 79.00 12.00 0.89 1,450.00 90.00 2.34 g/t c CUT-OFF GRADE BETWEEN TWO PROCESSES Mathematical Formulation If two processes are available to treat the same material, cut-off grades must be calculated to separate waste from ore being processed and to decide to which one of the two processes the ore should be sent. How to decide whether material should be processed or wasted was discussed previously. To decide between two processes, the utility of sending material of grade x to process 1 must be compared with that of sending the same material to process 2. Min￾ing costs, including haulage cost to the processing plant, may vary depending on the process. Processing costs will be different and so will metallurgical recoveries and overhead costs. If the product sold is a function of the process being used, even the revenue per metric ton produced may differ. The cut-off grade between two processes is calculated using the following formulas, where subscripts refer to the process number: ( ) ( ) () () ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ⋅ ⋅ − − + + = ⋅ ⋅ − − + + = = − + − + −     ⋅ − − ⋅ −     U x x r V R M P O U x x r V R M P O U x U x x M M P P O O r V R r V R 1 1 1 o1 o1 o1 2 2 2 o2 o2 o2 1 c 2 c c o1 o2 o1 o2 o1 o2 1 1 2 2 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Breakeven Cut-off Grade 27 In this formula, the numerator is the difference between costs incurred if one metric ton of material is sent to process 1 and costs incurred if the same metric ton is sent to process 2. The denominator is the difference between revenues resulting from sending one metric ton of material containing one unit of valuable product (e.g., one ounce of contained gold or one pound of contained copper) to process 1 and revenues resulting from sending the same metric ton to process 2. Precious Metal Example Consider a gold mining operation from which material can be sent to either of two processing facilities, a leach pad (process 1) or a mill (process 2). The costs associated with this operation are as follows: Mw = $3.00 mining cost per metric ton of waste Pw = $0.80 processing cost per metric ton of waste Ow = $0.40 overhead cost per metric ton of waste Mo1 = $5.00 mining cost per metric ton of ore leached Mo2 = $5.50 mining cost per metric ton of ore milled Po1 = $9.00 processing cost per metric ton leached Po2 = $35.00 processing cost per metric ton milled Oo1 = $1.20 overhead cost per metric ton leached Oo2 = $4.50 overhead cost per metric ton milled r1 = 60% average leach recovery r2 = 97% average mill recovery V = $1,600 per ounce of gold sold R1 = $50.00 per ounce of gold sold R2 = $50.00 per ounce of gold sold The utilities of sending one metric ton of material to the waste dump, the leach pad, or the mill are calculated below. Dollars per ounce was converted to dollars per gram (1 troy ounce = 31.1035 g) to obtain cut-off grades in grams per metric ton. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = − + + = − = ⋅ ⋅ − − + + = ⋅ − = ⋅ ⋅ − − + + = ⋅ − U x M P O $4.20 U x x r V R 31.1035 M P O $29.90 x $15.20 U x x r V R 31.1035 M P O $48.34 x $45.00 waste w w w 1 1 1 o1 o1 o1 2 2 2 o2 o2 o2 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 28 CHAPTER THREE These utility functions are plotted in Figure 3-6. The cut-off grades are ( ) ( ) ( ) ( ) = − = = − − = Waste-leach cut-off 15.20 4.20 29.90 0.37 g/t Leach-millcut-off 45.00 15.20 48.34 29.90 1.62 g/t CUT-OFF GRADE WITH VARIABLE RECOVERY Mathematical Formulation In the previous examples, it was assumed that the recovery achieved in the processing plant was a constant. For many processes and deposits, the recov￾ery r is a function r(x) of the head grade x. The value of Uore(x) must then be written as follows: ( ) () () = ⋅ ⋅ − ( )− + + U x x r x V R M P O ore o o o The value of Uwaste(x) remains independent of x: ( ) = −( ) + + U x M P O waste w w w ($60) $0 ($20) ($40) $20 Gold Grade, g/t $40 $60 $80 0.5 1.5 1 2 Waste-Leach Cut-off Leach-Mill Cut-off Leach Utility Waste Utility Mill Utility Figure 3-6 Graphical estimation of cut-off grade between wasted, leached, and milled material Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Utility, Profit per Metric Ton Breakeven Cut-off Grade 29 Calculating the cut-off grade requires finding the value of x such that Uore(x) = Uwaste(x). Nonlinear Recovery: A Precious Metal Example Consider a gold mining operation from which material can be sent to either of two processing facilities, a leach pad or a mill. Mining, processing, and overhead costs are $4.20 for waste, $15.20 for leach, and $45.00 for mill. The gold price is $1,600 per ounce from which must be deducted a $50.00-per￾ounce sales cost. The maximum gold recoveries are 60% for leaching and 90% for milling. Significantly lower recoveries are achieved for low-grade material. Figure 3-7 shows the relationship between recoveries and grade, as deter￾mined from metallurgical testing and historical production statistics. Figure 3-8 shows the profit per metric ton that will be made depending on whether material of grade x is wasted (Uwaste(x) = –$4.20), sent to the leach pad (U1 (x) = x ∙ r1 ∙ $49.83 – $15.20), or processed in the mill (U2 (x) = x ∙ r2 ∙ $49.83 – $45.00). ($49.83 is the maximum value of one gram of gold contained, assuming 100% recovery: (V – R)/32.1035). Figure 3-8 also shows how the cut-off grade can be determined by graphical method. The relation￾ship between the utility of leaching or milling material and the average grade of this material is no longer linear. The optimal process for material of grade x is that for which the utility is highest. The cut-off grades are the grades at which the curves intersect. If a constant 60% recovery for leached material and 90% recovery for milled material had been assumed, the waste-leach 0 10 20 30 40 50 60 70 80 90 100 Gold Grade, g/t 0 1 2 3 4 5 6 7 8 Mill Recovery Leach Recovery Figure 3-7 Relationship between recoveries and average grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Gold Recovery, % 30 CHAPTER THREE cut-off would have been estimated at 0.38 g/t and the leach-mill cut-off at 1.99 g/t. When variable recoveries are taken into account, the cut-offs are substantially higher, 1.16 g/t and 3.95 g/t respectively. CONSTANT TAIL MODEL OF VARIABLE RECOVERY Mathematical Formulation A model often used to represent the relationship between plant recovery and average grade of plant feed is the constant tail model. This model assumes that a fixed amount of metal cannot be recovered, whatever the grade of the material sent to the plant. If x is the average grade of one metric ton of material and c is the fixed amount that cannot be recovered, the recoverable amount is ( ) ( ) ⋅ = ⋅ − x r x r x c c where x = average grade of material sent to process r(x) = plant recovery if head grade is x rc = constant recovery after subtracting constant tail c = constant tail ($100) $0 ($50) $50 Gold Grade, g/t $100 $150 $200 $250 1 2 3 4 5 6 7 Waste-Leach Cut-off Leach-Mill Cut-off Leach Utility Mill Utility Figure 3-8 Graphical estimation of cut-off grade between wasted, leached, and milled material with variable recoveries Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Utility, Profit per Metric Ton Breakeven Cut-off Grade 31 The recovery function is as follows: ( ) ( ) ( ) < = > = ⋅ − If x c: r x 0 If x c: r x r 1 c x c The cut-off grade is determined as follows: [ ] ( ) ( ) ( ) ( ) ( ) ( ) () () ( ) ( ) ( ) ( ) = − + + = ⋅ ⋅ − − + + = ⋅ − ⋅ − − + + = + + − + +     ⋅ − + U x M P O U x x r x V R M P O r x c V R M P O x M P O M P O r V R c waste w w w ore o o o c o o o c o o o w w w c If x'c is the cut-off grade with constant recovery rc , the cut-off grade with constant tail is x x c c = +' c Constant Tail Model: A Precious Metal Example Consider a gold mining operation where material is being milled. Mining, processing, and overhead costs are $4.20 for waste and $45.00 for mill. The gold price is $1,600 per ounce from which must be deducted a $50.00-per￾ounce charge. As a result of metallurgical tests, the mill recovery was determined to be best represented using the constant tail model with parameters rc = 90% and c = 0.50 g/oz: ( ) ( ) ( ) ( ) < = > = ⋅ − = ⋅ − If x c: r x 0 If x c: r x r 1 c x 0.90 1 0.50 x c The relationship between recovery r(x) and average grade x is shown in Figure 3-9. The relationships between Uore(x) and Uwaste(x) and grade x are calculated as follows: ( ) ( ) ( ) () () ( ) ( ) ( ) = − + + = − = ⋅ ⋅ − − + + = ⋅ − ⋅ − − = − U x M P O 4.20 U x x r x V R M P O 0.90 x 0.50 1,600 50.00 31.1035 45.00 44.85x 67.43 waste w w w ore o o o Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 32 CHAPTER THREE These relationships are depicted in Figure 3-10. The cut-off grade between ore and waste is xc such that Uore(x) = Uwaste(x): = − ( ) = x 67.43 4.20 44.85 1.41 g/t c B R E A K E V E N C U T- O F F G R A D E A N D P O LY M E TA L L I C DEPOSITS Polymetallic deposits are defined as deposits that contain more than one metal of economic value. The formulas that must be used to calculate the utility of sending one metric ton of material to a given destination or process must consider the contribution of each metal. The decision whether one metric ton of material should be wasted or sent to the processing plant can no longer be made on the basis of grade alone. Dollar values must be calculated for each possible process, and the cut-off between ore and waste must be expressed in dollar terms. General Considerations Consider a metric ton of material that contains two valuable metals, copper and gold. Let x1 and x2 be the copper and gold grades, respectively. The pro￾cessing plant consists of crushing, grinding, and flotation circuits. A copper concentrate is produced, which is sold to a smelter. The flotation plant recov￾ery is r1 for copper and r2 for gold. Mining, processing, and overhead costs 0 10 20 30 40 50 60 70 80 90 100 Gold Grade, g/t 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 Asymptotic Recovery rc Constant Tail c Mill Recovery Figure 3-9 Relationship between recovery and average grade with constant tail Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Gold Recovery, % Breakeven Cut-off Grade 33 associated with one metric ton of material sent to the flotation plant are Mo , Po , and Oo , respectively. The corresponding costs per metric ton of waste are Mw, Pw, and Ow. According to the smelter contract, the value received for sale of the concentrate is p1 = 96.5% of the value of the copper contained in the concentrate after deducting d1 , and p2 = 99% of the gold contained. Smelter cost deductions are Cs per metric ton of concentrate. The concentration ratio K is the number of metric tons of material that must be processed to produce one metric ton of concentrate. The cost of shipping one metric ton of concen￾trate to the smelter is Ct . Metal prices are those quoted on the London Metal Exchange, V1 and V2 for copper and gold, respectively. Hence the value of one metric ton of material sent to the flotation plant: ( ) ( ) ( ) ( ) = − + −−− + + U x , x x r d p V x r p V C K C K M P O ore 1 2 1 1 1 1 1 2 2 2 2 s t o o o If the same metric ton is sent to the waste dump, the corresponding costs are = −( ) + + U M P O waste w w w The material should be sent to the processing plant if ( ) > U x , x U ore 1 2 waste ($60) $0 ($20) ($40) $20 Gold Grade, g/t $40 $60 0.5 1.5 1.0 2.0 2.5 3.0 3.5 Constant Tail c Waste-Mill Cut-off Waste Utility Mill Utility Figure 3-10 Graphical estimation of cut-off grade between wasted and milled material with constant tail Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Utility, Profit per Metric Ton 34 CHAPTER THREE These formulas show that many factors enter into the calculation of the cut￾off between ore and waste. Processing costs and recoveries are likely to be depen￾dent not only on metal content but also on other geological characteristics such as mineralogy, hardness, clay content, and degree of oxidation, which change depending on the area of the deposit being mined. Smelter contracts heavily penalize concentrates that are found to contain excessive amounts of specified deleterious elements. All these factors must be taken into account when esti￾mating the cut-off value applicable to one metric ton of mineralized material. Because the value of one metric ton of material is a function of more than one grade, it is no longer meaningful to talk about a “cut-off grade.” His￾torically, this multidimensional problem was reduced to a one-dimensional problem by defining a “metal equivalent.” With the advance of computers and the ease of use with which complex mathematical calculations can be made, one now talks of cut-off values, which are expressed in dollar terms and require calculation of a net smelter return. Net smelter return and metal equivalent are discussed in the following sections. Calculation of Cut-off Grades Using Net Smelter Return For polymetallic deposits, the utility of sending one metric ton of material to the smelter is best expressed in terms of net smelter return, or NSR. The net smelter return is defined as the return from sales of concentrates, expressed in dollars per metric ton of ore, excluding mining and processing costs. Mathematical Formulation In the previous copper-gold example, the net smelter return of one metric ton of ore with copper grade x1 and gold grade x2 is ( ) ( ) ( ) = − + − − NSR x , x x r d p V x r p V C K C K 1 2 1 1 1 1 1 2 2 2 2 s t The utility of sending this metric ton of ore to the processing plant is ( ) () () = − + + U x , x NSR x , x M P O ore 1 2 1 2 o o o Using NSR values simplifies the calculation of cut-off grades. The NSR cut-off between processing and wasting one metric ton of material is NSRc , obtained by setting Uore(x1 , x2 ) equal to Uwaste: ( ) ( ) = + + − + + NSR M P O M P O c o o o w w w In polymetallic deposits, cut-offs should not be expressed in terms of minimum metal grade; they should be expressed in terms of minimum NSR. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Breakeven Cut-off Grade 35 Calculation of NSR Cut-off: A Copper-Molybdenum Example Consider a copper molybdenum mining operation. In this section, the sub￾script 1 refers to copper and subscript 2 refers to molybdenum. Hence x1 is the copper grade and x2 is the molybdenum grade. The following parameters characterize the operation: r 1 = 89% copper flotation plant recovery p1 = 96.5% copper smelting recovery r2 = 61% molybdenum flotation plant recovery p2 = 99% molybdenum roasting recovery V1 = $1.20 value of one pound of copper sold V2 = $6.50 value of one pound of molybdenum sold R1 = $0.065 refining cost per pound of copper K = 72 metric tons of ore must be processed to produce one metric ton of concentrate Cs + Ct = $145 smelting and freight costs per metric ton of concentrate R2 = $0.95 conversion, roasting, and freight costs per pound of molybdenum Mo = $1.00 mining cost per metric ton of ore milled Po1 = $3.00 mill processing cost per metric ton milled Po2 = $0.15 incremental molybdenum processing cost per metric ton milled Oo = $0.50 overhead cost per metric ton milled Mw = $1.00 mining cost per metric ton wasted Pw = $0.05 processing cost per metric ton wasted Ow = $0.05 overhead cost per metric ton wasted The NSR of one metric ton of material with average grade x1 , x2 is calcu￾lated as follows: ( ) ( ) ( ) ( ) ( ) ( ) = − + − − + = ⋅ ⋅ − ⋅ ⋅ + ⋅ ⋅ − ⋅ ⋅ − = + − NSR x , x x r p V R x r p V R C C K 0.89 0.965 1.20 0.065 2,205 x 0.61 0.99 6.50 0.95 2,205 x 145.00 72 2,149x 7,390x 2.014 1 2 1 1 1 1 1 2 2 2 2 2 s t 1 2 1 2 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 36 CHAPTER THREE Therefore, the NSR of one metric ton of ore averaging x1 = 0.45% Cu and x2 = 0.035% Mo is $10.24. For material that must be mined but can be either wasted or processed, the cut-off NSR (mill or internal cut-off NSR) is NSRc calculated as follows: ( ) ( ) ( ) ( ) ( ) ( ) = + − + − + − = + −+−+ − = NSR P P P O O M M 3.00 0.15 0.05 0.50 0.05 1.00 1.00 $3.55 per metric ton c o1 o2 w o w o w For material that need not be mined (mine or external cut-off NSR), NSRc is calculated as follows: = + + + = + + + = NSR P P O M 3.00 0.15 0.50 1.00 $4.65 per metric ton c o1 o2 o o The relationship between NSRc , x1 , and x2 is shown in Figure 3-11. Calculation and Reporting of Metal Equivalent Before computers became widely used, it was common practice to talk about polymetallic deposits in terms of metal equivalent. If one metric ton of mate￾rial contains two metals, copper and molybdenum, with average grade x1 and 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10 %Cu 0.0 0.05 0.10 0.15 0.20 0.25 0.30 0.35 Waste Ore NSR = $4.65 (Mine Cut-off) NSR = $3.55 (Mill Cut-off) Figure 3-11 Relationship between cut-off NSR and metal grades Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. %Mo Breakeven Cut-off Grade 37 x2 respectively, the corresponding copper equivalent is defined as the copper grade x1e that one metric ton must contain to produce the same revenue, assuming no molybdenum. The revenue generated by mining and processing one metric ton of material with copper grade x1 and molybdenum grade x2 is NSR(x1 , x2 ). The revenue generated by mining and processing one metric ton of material with copper grade x1e and no molybdenum is NSR(x1e, 0.0). The copper equivalent is obtained by solving the following equation: ( ) = ( ) NSR x ,0.0 NSR x , x 1e 1 2 A molybdenum equivalent can be calculated instead of a copper equiva￾lent. The molybdenum equivalent is the molybdenum grade x2e that satisfies the following equation: ( ) = ( ) NSR 0.0, x NSR x , x 2e 1 2 In the previous copper-molybdenum example, the net smelter return was expressed as follows: ( ) ( ) ( ) ( ) = − + − − + NSR x , x x r p V R x r p V R C C K 1 2 1 1 1 1 1 2 2 2 2 2 s t Hence, ( ) ( ) ( ) = −−+ NSR x ,0.0 x r p V R C C K 1e 1e 1 1 1 1 s t The copper equivalent is ( )() = + −     −     x x x r p V R r p V R 1e 1 2 2 2 2 2 1 1 1 1 Similarly, the molybdenum equivalent is ( )() = + −     −     x x x r p V R r p V R 2e 2 1 1 1 1 1 2 2 2 2 Using the information listed previously concerning prices, cost, and recov￾eries, the copper and molybdenum equivalents can be calculated as follows: ( ) ( ) = + = + = + = + x x x 7,390 2,149 x 3.439x x x x 2,149 7,390 x 0.291x 1e 1 2 1 2 2e 2 1 2 1 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 38 CHAPTER THREE The copper equivalent of material averaging x1 = 0.45% Cu and x2 = 0.035% Mo is 0.57% Cu-equivalent. The molybdenum equivalent of the same mate￾rial is 0.166% Mo-equivalent. In practice, because of the complexity of the formulas to be used to correctly estimate the value of one metric ton of material, and because equivalence changes with metal price, recoveries, and refining costs, grade equivalence is rarely a useful tool in calculation of cut-off grades. Quoting the amount of metal equivalent contained in a deposit is of little use to investors. Publication of reserves in terms of metal equivalence is generally not accepted by regulatory agencies unless additional disclosures are made, including publi￾cation of the average grade of each metal and explanation of the formula used to calculate metal equivalence. OPPORTUNITY COST OF NOT USING THE O P T I M U M CUT-OFF GRADE If the optimum cut-off grade is not used, material is sent to a destination where the profit is less than could be made otherwise or the loss incurred is greater than necessary. Figure 3-12 shows the opportunity cost incurred per metric ton when a leach-mill cut-off grade of 3 g/t is used although the opti￾mal cut-off grade is 4 g/t. The loss is represented by the difference between the utility of the chosen process and that of the optimal process for the same average grade. Figure 3-13 shows the opportunity cost incurred per metric ton when a leach-mill cut-off grade of 5 g/t is used. Mathematical Formulation Let U1 (x) be the utility of leaching one metric ton of material of grade x and U2 (x) be the utility of milling the same metric ton. These utilities can be writ￾ten as follows (in these equations the cost of sales R is included in V, and the overhead costs Oo are included in Mo , Po1, and Po2): ( ) ( ) ( ) ( ) = ⋅ ⋅ − + = ⋅ ⋅ − + U x x r V M P U x x r V M P 1 1 o o1 2 2 o o2 The optimal cut-off grade is ( ) ( ) = − −     x P P r r V c o1 o2 1 2 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Breakeven Cut-off Grade 39 –20 0 –10 10 20 30 40 Grade, g/t 1 2 3 4 5 6 Area of Opportunity Loss If Cut-off = 3 g/t Cut-off Actually Used xs = 3 g/t Optimal Leach-Mill Cut-off xc = 4 g/t Figure 3-12 Opportunity cost of using a cut-off grade lower than the optimal cut￾off grade –20 0 –10 10 20 30 40 Grade, g/t 1 2 3 4 5 6 Area of Opportunity Loss If Cut-off = 5 g/t Optimal Leach-Mill Cut-off xc = 4 g/t Cut-off Actually Used xs = 5 g/t Figure 3-13 Opportunity cost of using a cut-off grade higher than the optimal cut￾off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Utility, $/t Utility, $/t 40 CHAPTER THREE Let xs be the selected cut-off grade, which is lower than the optimal cut￾off grade xc (Figure 3-12). Material with grade x between xs and xc is being milled but, ideally, should be leached. For each metric ton of grade x between xs and xc , the opportunity cost is ( ) ( ) () () − = ⋅ − ⋅ − − U x U x x r r V P P 2 1 2 1 o2 o1 Integrating this formula from x = xs to x = xc we obtain the total opportunity cost: ( ) ( ) ( ) ( ) ( ) ( ) = −    ⋅ − ⋅ − −    ⋅ − total opportunity cost Q x Q x r r V Tx Tx P P s c 2 1 s c o2 o1 In this formula, T(xs ) – T(xc ) is the tonnage of material with average grade between xs and xc , and Q(xs ) – Q(xc ) is the quantity of metal contained in this material. One could avoid this opportunity cost by increasing the mill capac￾ity by a tonnage amount equal to T(xs ) – T(xc ). Such an increase in capacity is justified if the cost of such an increase is expected to be less than the total opportunity cost. Similar equations are applicable if xs is higher than xc and material that should be milled is leached (Figure 3-13): ( ) ( ) ( ) ( ) ( ) ( ) = −    ⋅ − ⋅ − −    ⋅ − total opportunity cost Q x Q x r r V Tx Tx P P c s 1 2 c s o1 o2 An example of how to estimate the opportunity cost is given in Exam￾ple 5 in Appendix A. CUT-OFF GRADE AND LOW-GRADE STOCKPILES Consideration may be given to stockpiling low-grade material instead of wast￾ing it if such material is not currently economic to process but metal prices are expected to be higher at a later date. Stockpiling low-grade material may also be considered when capacity constraints prevent current processing of material that otherwise could be processed economically. To decide whether material of grade x should be wasted or stockpiled, one must compare the utility of wasting Uwaste(x) with that of stockpiling Ustp(x). The cut-off grade between stockpile and waste is the value xc of x for which U stp(x) = Uwaste(x). Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Breakeven Cut-off Grade 41 The utility of wasting material of grade x is ( ) = −( ) + + U x M P O waste w w w To calculate the utility of stockpiling, one must take into consideration stockpiling costs, and the cost of retrieving material from the stockpile and processing it at a later date. In addition, metallurgical recoveries of stockpiled material may differ from those of freshly mined material, and the price of the product sold may be different from that prevailing when the decision to stockpile is made: ( )= −( ) + + − − + U x M P O NPV(futurecosts of stockpile maintenance) NPV(futurere-handling and processing costs) NPV(futurerevenues from sales) stp stp stp stp where M stp = current mining costs per metric ton delivered to the low-grade stockpile P stp = current costs of stockpiling material that will be processed later, including the cost per metric ton of extending the stockpile area if required O stp = current overhead costs associated with mining and stockpiling NPV(future costs of stockpile maintenance) = net present value of yearly costs that will be incurred to maintain stockpiled material in an environmentally safe fashion until it is processed NPV(future re-handling and processing costs) = net present value of the one-time costs that will be incurred when the mate￾rial is retrieved from the stockpile and processed NPV(future revenues from sales) = net present value of revenues expected from sales when processed material is sold Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 42 CHAPTER THREE At the time the product recovered from the stockpiles is sold, the reve￾nues are x ∙ r stp ∙ (Vstp – R stp), where rstp is the expected recovery at the time of processing, Vstp is the dollar value of the product sold at the time it is sold, and R stp is the corresponding cost of sales per unit of product sold. The recovery r stp may be less or higher than that which would apply to the same material if processed when mined. Sulfide material is likely to oxidize during stockpiling. If a sulfide flotation process is to be used, oxidation will result in lower recov￾ery. Conversely, if an oxide leach process is to be applied to material that was not fully oxidized when mined, stockpiling may result in enhanced recovery. There are obvious difficulties in using these formulas, the main one being that future costs and revenues are difficult or impossible to estimate with accuracy. Furthermore, because processing of stockpiled material is likely to occur late in the mine life, the net present value of future revenues is likely to be small compared with costs incurred at the time of mining and to maintain the stockpile over time. For this reason, stockpiling low-grade material is often a strategic decision that takes into account expectations of future increases in metal prices, benefits associated with lengthening the mine life, and good management of mineral resources. An example of how cut-off grades can be estimated can be found in Example 6 in Appendix A. Special Case: Low-Cost Stockpiles The previous formulas are of a general nature and apply to stockpiles intended to be processed in the foreseeable future that are located in specially prepared areas. Some situations allow low-grade material to be stockpiled on top of waste dumps with no additional cost. Such stockpiles may or may not be pro￾cessed at a future date. The utility of stockpiling can then be written: ( )= −( ) + + − + U x M P O NPV(futurere-handling and processing costs) NPV(futurerevenues from sales) stp w w w The cut-off grade between waste and stockpile is the value of x for which NPV(futurerevenues from sales) = NPV(futurere-handling and processing costs) Because sales and stockpile re-handling happen at the same time, dis￾counting can be ignored and the cut-off grade is obtained by setting the expected revenues from sales equal to the expected re-handling and processing costs. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Breakeven Cut-off Grade 43 The expected revenues from sales are x ∙ rstp ∙ (Vstp – R stp). The cut-off grade between waste and stockpile is x = (expected re-handl ⋅ − ( ) ing and processing costs) r V R c stp stp stp Net present values do not enter in this equation. If the assumptions concern￾ing prices, costs, and recoveries that will prevail at the time of processing are independent of time, the cut-off grade will also be independent of when the stockpile is processed. Special Case: High-Cost Waste Dumps Other situations occur where the cost of maintaining mineralized waste dumps is as high as that of stockpiling low-grade material for future process￾ing. However, waste dumps must be maintained for life, whereas low-grade stockpiles will be processed. If the expected life of the stockpile is n years and C is the yearly cost of maintaining either the stockpile or the waste dump, the net present value of maintenance costs is* ( ) ( ) ( ) = + + = P C g i,n C 1 i i P C g i,n waste stp C g(i,n) is the net present value of a constant cash flow C over n years. C (1 + i)/i is the net present value of a constant cash flow C to perpetuity. The cut-off grade between waste and stockpile can be written as follows: ( ) ( ) = − + x (expected re-handling a ⋅ − nd processing costs) C 1 i i r V R c stp stp stp If the assumptions concerning prices, costs, and recoveries that will prevail at the time of processing are independent of time, the cut-off grade will also be independent of when the stockpile is processed. * It was shown earlier that the net present value of a constant cash-flow C over n years is NPVi = C g(i,n), where g(i,n) = [1 – (1 + i)–n] (1 + i)/i while the net present value to perpetuity is C (1 + i)/i. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 44 CHAPTER THREE C U T- O F F G R A D E A N D O P T I M I Z AT I O N O F PROCESSING PLANT OPERATING CONDITIONS In this section a method is developed to optimize a copper mining operation where mining capacity is fixed, but the capacity of the processing plant can be changed by changing grind size. Depending on the metallurgical properties of the ore, using a coarser grind will increase plant throughput while reducing cost per metric ton processed and decreasing recovery. Conversely, a finer grind can decrease plant capacity, increase processing cost and recovery. Mathematical Formulation The following notations are used in this section: r = processing plant recovery V = value of copper contained in concentrate, after deduction for smelter loss, freight, smelting, and refining costs Po = cost per metric ton of ore processed, including overhead xc = cut-off grade T+c = tonnage above cut-off grade to be processed in one year Q+c = quantity of copper to be processed in one year x+c = average grade above cut-off grade Since mining capacity and costs are fixed, the utility function that must be optimized to estimate the economically optimal grind size is only a func￾tion of mill operations and can be written as follows: () () () = ⋅ ⋅ − ⋅ + + + + + U T Q r T V T P T c c c c o c where U(T+c) = utility of running the plant at T+c capacity for one year r(T+c) = processing plant recovery, if plant capacity is T+c Po (T+c) = cost per metric ton of ore processed, if plant capacity is T+c Q+c is also a function of T+c. Both Q+c and T+c are functions of the cut-off grade xc . The optimal plant capacity is that for which U(T+c) reaches a maxi￾mum, and is calculated by setting the first derivative of U(T+c) equal to zero: dU T dT 0 dQ dT r T V P T Q dr T dT V T dP T dT c c c c c o c c c c c o c c ( ) () () ( ) ( ) = = ⋅ ⋅ − + ⋅ ⋅ − ⋅ + + + + + + + + + + + + Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Breakeven Cut-off Grade 45 If the tonnage processed is changed by a small amount dT+c because of a small change in cut-off grade xc , the amount of copper contained is increased from Q+c = T+c ∙ x+c to Q+c + dQ+c = T+c ∙ x+c + dT+c ∙ xc . Hence, dQ+c = dT+c ∙ xc and the optimal plant capacity is T+c such that () () () ( ) ⋅ ⋅ − + ⋅ ⋅ − ⋅ = + + + + + + + + x r T V P T Q dr T dT V T dP T dT 0 c c o c c c c c o c c If the recovery r and the processing cost Po were independent of T+c, this equation would be easily solved for xc : ()() [ ] = ⋅     = ⋅ + + x P T r T V P r V c o c c o In such a situation, the applicable cut-off grade is the mill cut-off grade. The term Q+c ∙ dr(T+c)/dT+c ∙ V represents the change in the value of the product sold in one year, which results from the change in recovery. The term T+c ∙ dPo (T+c)/dT+c represents the change in operating cost per year, which results from the change in processing cost per metric ton. In this formulation of the problem, it was assumed that the value V of the product sold is independent of the tonnage processed. This may not be the case if the quality of the concentrate varies with tonnage processed and head grade. It was also assumed that recovery is only a function of tonnage processed and is independent of head grade. More complex equations would apply if these assumptions could not be made. Example: Optimization of Grinding Circuit in a Copper Mine This example illustrates how plant capacity can be optimized when mine plans are fixed; no major change can be made to the processing plant, but the plant capacity can be increased by changing grind size. Mine production is fixed for at least one year, and the tonnage, grade, and metal content of copper-bearing material expected to be mined during this one-year period is as shown in Table 3-1 and illustrated in Figures 3-14 and 3-15. The ore is to be processed in a flotation plant. The mill was designed to operate at the rate of 39.5 million metric tons per year with an average copper recovery of 95%. Under these conditions, the mill operating costs are $5.24 per metric ton. When mine plans were finalized for the coming year, the expected value of product sold was $1.00 per pound of copper in concentrate, and the following mill cut-off grade was used for planning: ( ) = ⋅ ⋅ = x 5.24 0.95 1.00 2,205 0.25% Cu c Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 46 CHAPTER THREE Table 3-1 Grade–tonnage relationship for the coming year of mining Cut-off, % Cu Tonnage, million metric tons Average grade, % Cu Copper Content thousand metric tons Cu million lb Cu 0.15% 53.7 0.335% 180 397 0.16% 52.6 0.340% 179 395 0.17% 51.4 0.344% 177 390 0.18% 50.1 0.348% 174 384 0.19% 48.8 0.352% 172 378 0.20% 47.5 0.355% 168 372 0.21% 46.0 0.360% 165 365 0.22% 44.4 0.365% 162 357 0.23% 42.8 0.370% 159 349 0.24% 41.2 0.375% 155 341 0.25% 39.5 0.381% 150 332 0.26% 37.7 0.387% 146 322 0.27% 35.9 0.393% 141 311 0.28% 34.1 0.399% 136 300 0.29% 32.1 0.406% 131 288 0.30% 30.2 0.413% 125 275 0.31% 28.2 0.421% 119 262 25 35 30 40 45 55 50 60 0.32 0.36 0.34 0.38 0.40 0.42 0.44 Cut-off Grade, % Cu 0.12 0.16 0.20 0.24 0.28 0.32 Metric Tons of Material Average Grade Figure 3-14 Next year expected relationship between tonnage and grade above cut-off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Metric Tons of Material Above Cut-off, millions Average Grade Above Cut-off, % Cu Breakeven Cut-off Grade 47 As shown in Table 3-1, this cut-off grade implies that the mill feed will be 39.5 million metric tons averaging 0.381% Cu and containing 332 million pounds of copper. The value of the material sent to the mill, based on $1.00 per pound of recoverable copper and excluding mining costs, was expected to be () () () = ⋅ ⋅ − ⋅ = ⋅ ⋅ − ⋅ = + + + + + U T Q r T V T P T 332 0.95 1.00 39.5 5.24 $108 million c c c c o c Because of an unexpected increase in copper price, the mining company is investigating whether short-term changes could be made to the mill feed and throughput, which would result in increased utility. The copper price is now expected to be $1.50 per pound of copper in concentrate instead of the $1.00 that was used for planning. The mine plan cannot be changed for at least one year, and only changes in operating conditions can be made to the processing plant. One option is to operate the mine and mill as planned while selling the concentrate at the higher price. The value of the material sent to the mill, excluding mining costs, would increase from $108 million to ( ) = ⋅ ⋅ − ⋅ = + U T 332 0.95 1.50 39.5 5.24 $266 million c 100 120 140 160 180 200 0.32 0.36 0.34 0.38 0.40 0.42 0.44 Cut-off Grade, % Cu 0.12 0.16 0.20 0.24 0.28 0.32 Metric Tons of Contained Copper Average Grade Figure 3-15 Next year expected relationship between metal content and grade above cut-off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Metric Tons of Copper Above Cut-off, thousands Average Grade Above Cut-off, % Cu 48 CHAPTER THREE Alternatively, one could consider a decrease in cut-off grade. At $1.50 per pound of copper in concentrate, the minimum cut-off grade is ( ) = ⋅ ⋅ = x 5.24 0.95 1.50 2,205 0.17% Cu c Table 3-1 shows that 51.4 million metric tons of ore would be mined above this cut-off grade, averaging 0.344% Cu. Under current operating con￾ditions, the mill can only process 39.5 million metric tons. The higher-grade tons could be sent to the mill and the lower-grade tons could be stockpiled. But such an approach is likely to increase short-term costs without increasing revenues from concentrate sales. No advantage is taken of the higher copper price. Another option would consist of increasing mill throughput by increas￾ing grind size. The result would be a decrease in operating cost per metric ton. However, this is expected to result in a decrease in mill recovery. It has been determined that the mill operating costs are 55% fixed costs and 45% inversely proportional to the tonnage processed: ( ) = + + + P T 2.88 93.1 T o c c This relationship between operating cost per metric ton and tonnage processed per year is shown in Figure 3-16. The relationship between copper recovery and mill throughput is shown in Figure 3-17. This relationship is represented by the following equation: () () = − + + + + + r T 0.000232 T 0.01362T 0.7729 c c 2 c The function to be optimized is () () () = ⋅ ⋅ − ⋅ + + + + + U T Q r T V T P T c c c c o c In this equation, Q+c ∙ r(T+c) ∙ V represents the value of copper in concentrate, and T+c ∙ Po (T+c) represents the processing costs. The relationship between U(T+c) and the cut-off grades (which defines T+c) is easily calculated using Table 3-1 and the three preceding equations. The results are summarized in Table 3-2 and shown in Figure 3-18. The highest return is $272 million, $6 million higher than the $266 million calculated previously when plant capacity was kept at 39.5 million metric tons per year. This highest return is reached by increasing the plant capacity to approximately 44.4 million metric tons per year. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Breakeven Cut-off Grade 49 4.00 5.00 4.50 5.50 6.00 6.50 Metric Tons Processed per Year, millions 25 30 35 40 45 50 55 60 Po (T+c) = 2.88 + 93.1/T+c Figure 3-16 Relationship between mill operating cost per metric ton and tonnage processed per year 80 85 90 95 100 Metric Tons Processed per Year, millions 25 30 35 40 45 50 55 60 r(T+c) = –0.000232(T+c)2 + 0.013624(T+c) + 0.7729 Figure 3-17 Relationship between copper recovery and tonnage processed per year Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Operating Cost per Metric Ton Processed, $ Copper Recovery, % 50 CHAPTER THREE An alternative method of calculating the optimum processing rate con￾sists of solving the following equation: ( ) = dU T d + + T 0.0 c c which can be written as () () () ( ) ⋅ ⋅ − + ⋅ ⋅ − ⋅ = + + + + + + + + x r T V P T Q dr T dT V T dP T dT 0.0 c c o c c c c c o c c Table 3-2 Calculation of U(T+c) for various cut-off grades and corresponding tonnages of mill feed T+c V $/lb $1.50 $1.50 $1.50 $1.50 $1.50 $1.50 xc % Cu 0.20% 0.21% 0.22% 0.23% 0.24% 0.25% T+c million metric tons 47.5 46.0 44.4 42.8 41.2 39.5 x+c % Cu 0.355% 0.360% 0.365% 0.370% 0.375% 0.381% Q+c million pounds Cu 372 365 357 349 341 332 r(T+c) % 89.65% 90.86% 92.04% 93.09% 94.03% 94.90% Po (T+c) $/t $4.84 $4.90 $4.98 $5.06 $5.14 $5.24 Q+c · r(T+c) · V million $/yr $500 $497 $493 $487 $481 $473 –T+c · Po (T+c) million $/yr ($230) ($226) ($221) ($216) ($212) ($207) U(T+c) million $/yr $270 $271 $272 $271 $269 $266 200 220 240 260 280 300 Metric Tons Processed T+c, millions 25 30 35 40 45 50 55 60 U(T+c) Figure 3-18 Relationship between utility U(T+c) and tonnage of mill feed T+c Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Utility of Processing T Metric Tons, $ million +c Breakeven Cut-off Grade 51 In this equation, xc ∙ r(T+c) ∙ V – Po (T+c) represents the change in value assuming that recovery and costs remain constant; Q+c ∙ dr(T+c)/dT+c is attributable to the change in recovery; –T+c ∙ dPo (T+c)/dT+c is attributable to the change in costs. The derivatives of Po (T+c) and r(T+c) are easily calculated: () () ( ) = − = − ⋅ + + + + + + + dP T dT 93.1 T dr T dT 0.000464 T 0.01362 o c c c 2 c c c The relationship between dU(T+c)/dT+c and the cut-off grades is easily calculated using Table 3-2 and the three preceding equations. The results are summarized in Table 3-3 and plotted in Figure 3-19. The optimal return is obtained if the tonnage of mill feed is set slightly less than 45 million metric tons per year, the point where dU(T+c)/dT+c = 0.0 (Figure 3-19). Setting the cut-off grade at 0.22% Cu will reach this objective, producing 44.4 million Table 3-3 Calculation of dU(T+c)/dT+c for various cut-off grades and corresponding tonnages of mill feed T+c xc % Cu 0.20% 0.21% 0.22% 0.23% 0.24% 0.25% xc · r(T+c) · V – Po(T+c) $/t $1.09 $1.41 $1.72 $2.03 $2.32 $2.61 Q+c · dr(T+c)/dT+c · V $/t ($4.70) ($4.23) ($3.74) ($3.27) ($2.81) ($2.34) –T+c · dPo (T+c)/dT+c $/t $1.96 $2.02 $2.10 $2.18 $2.26 $2.36 dU(T+c)/dT+c $/t ($1.65) ($0.80) $0.08 $0.94 $1.77 $2.62 ($2) ($4) ($6) ($8) $4 $2 $- $6 $8 Metric Tons Processed T+c, millions 30 35 40 45 50 55 dU(T+c)/dT+c Figure 3-19 Relationship between incremental utility dU(T+c)/dT+c and tonnage of mill feed T+c Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Change in Utility per Additional Metric Tons Processed 52 CHAPTER THREE metric tons of mill feed (Table 3-2). The average mill head grade will be 0.365% Cu. Increasing the tonnage from 39.5 million metric tons to 44.4 mil￾lion metric tons will be achieved by decreasing recovery from 95% to 92%. This loss in recovery will be more than compensated by a decrease in operating costs from $5.24 to $4.98 per metric ton. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 53 4 Capacity constraints have a direct influence on the cut-off grade. Opportu￾nity costs are associated with each constraint, which implies that the cut-off grade must be higher than the marginal cut-off grade, if the primary objective is to maximize net present value. Decisions that are made without fully under￾standing the effects, direct and indirect, of changing cut-off grades can have dire consequences. This was the case when very low marginal cut-off grades were first introduced in low-grade gold leaching operations. WHE N M A R G I N A L A N A LY S I S N O L O N G E R A P P L I E S : A GOLD LEACHING OPERATION In the 1990s, many gold mining companies significantly increased the tonnage of material placed on their leach pads by lowering the cut-off grade. Marginal analysis of leaching costs indicated that already low cut-off grades, often less than 0.5 g/t, could be further reduced, sometimes down to 0.2 g/t. The expec￾tation was that, with more ounces being placed on the leach pad, the amount of gold recovered would increase on a monthly basis, as well as cumulatively over time, while the cost per metric ton placed would decrease. The results initially obtained were often disappointing. The tonnage of material added by lowering the cut-off grade was large, resulting in a short-term decrease in recovery, which in the worse cases meant a decrease in revenue instead of the expected increase. In addition, the long-term impact of adding large tonnages of very low-grade material to a leach pad was not fully understood. In some cases, the result seemed to be a decrease in overall pad recovery, not only post￾poning short-term revenues but showing no increase in cumulative revenues over the life of the project. On a discounted basis, the benefit of lowering the cut-off grade was significantly less than expected, if not negative. To illustrate this point, consider a gold mining operation where the total tonnage of ore and waste material scheduled to be mined in the com￾ing year was 10 million metric tons. This material was characterized by the Capacity Constraints and Opportunity Costs Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 54 CHAPTER FOUR grade–tonnage curve shown in Figure 4-1. Initially, the cut-off grade was set at 0.50 g/t, which corresponded to 6.59 million metric tons of leach-grade material averaging 1.36 g/t and containing 288,000 oz of gold. The leach recovery was expected to be 65%, resulting in the production of 187,000 oz in the coming year. A review of the previous year’s operating costs showed that the cut-off grade could be lowered to 0.40 g/t if the recovery could be maintained at 65%. Laboratory tests confirmed that recovery was independent of grade, and the decision was made to lower the cut-off grade and add the lower-grade material to the pad. After two months of operation, managers realized that the gold pro￾duction target for the year was not going to be met. If nothing changed, the amount of gold sold was going to be less than expected before the cut-off grade was decreased. Management immediately requested a review of the situation. The results of this review were as follows: • Metallurgical tests confirmed no decrease in recovery for lower-grade material. • Metallurgical tests and review of past operational conditions showed that the amount of gold recovered was an increasing function of the solution ratio, defined as the metric tons of cyanide solution used per metric ton of material placed on the pad. This relationship was as illustrated in Figure 4-2. 3 4 5 6 7 10 9 8 1.0 1.4 1.2 1.6 1.8 2.0 Cut-off Grade, g/t 0.2 0.3 0.4 0.5 0.6 0.7 0.8 Tonnage 7.51 Tonnage 6.59 Cut-off 0.4 Cut-off 0.5 Average Grade 1.25 Average Grade 1.36 Figure 4-1 Estimation of tonnage and grade above cut-off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Metric Tons Above Cut-off Grade, millions Average Grade Above Cut-off Grade Capacity Constraints and Opportunity Costs 55 • Provided that a four-month leaching cycle was adhered to, the origi￾nal solution ratio was 1:1, as needed to reach 65% recovery. • Lowering the cut-off grade to 0.40 g/t increased the tonnage to be placed on the pad from 6.59 to 7.51 million metric tons and decreased the average grade from 1.36 to 1.25 g/t (Figure 4-1). The ounces placed increased from 288,000 to 302,000 oz, a 5% increase. • Because no change was made to the amount of solution placed on the pad, the increase in tonnage from 6.59 to 7.51 million metric tons resulted in a decrease in solution ratio from 1.0 to 0.88. The expected recovery should have been 61% instead of 65% (Figure 4-2). • This 6% decrease in recovery exceeds the expected 5% increase in ounces placed on the pad. The total metal recovered during the year should have been expected to decrease from 187,000 to 184,000 oz. Ignoring the relationship between leach recovery and solution ratio was equivalent to ignoring a capacity constraint. The corresponding opportunity cost was ignored, and consequently the cut-off grade was underestimated. Lowering the cut-off grade to 0.40 g/t might have been justified if a cost￾effective method of increasing the recovery had been put in place. One option was to increase the volume of fresh solution placed on the pad, which would require changes in pond size, pipes, pumps, and the capacity of the carbon columns or Merrill-Crowe plant used to process the solution. Another option 45 50 55 60 65 70 Solution Ratio 0.60 0.70 0.80 0.90 1.00 1.10 Recovery 65% Recovery 61% Ratio 0.88 Ratio 1.0 Figure 4-2 Relationship between leach recovery and solution ratio Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Leach Recovery, % 56 CHAPTER FOUR was to recycle the pregnant solution on the pad, which would increase the solution-to-ore ratio without incurring some of the high costs associated with the first option. All changes to the leach plant had to take into account constraints imposed by operating permits and other conditions (technical, environmental, or legal) that would limit the options available to solve the problem. Assume that, for environmental and permitting reasons, the size of the leach plant could not be increased. Which approach should have been used to determine the optimal cut-off grade? Taking into account the low operating costs, this optimal cut-off grade was likely to be less than 0.5 g/t (which was determined on the basis of higher costs) but more than 0.4 g/t (which used the lower costs but ignored the operating constraint). An iterative approach could be used that consists of decreasing the cut-off grade by small successive increments and fully assessing the economic consequences until no further decrease is justified. 1. Assume that the cut-off grade is lowered from the current 0.50 g/t to 0.48 g/t. 2. Estimate the increase in tonnage and ounces that will be placed on the pad as a result of the lower cut-off grade. 3. Calculate the corresponding decrease in solution ratio and leach recovery. 4. Calculate the resulting change in total gold recovered, taking into account the increase in gold placed and decrease in recovery. 5. Compare the change in expected gold sold with the corresponding change in cost of operation. Differences between the cost of wasting material and placing it on the pad should be taken into account. 6. If the change in revenue from sales exceeds the change in costs, the cut-off grade can be reduced to 0.48 g/t. The analysis should then be repeated assuming a lower 0.46 g/t cut-off. The optimum cut-off is that for which the change in revenue is equal to the change in cost. C A PA C I T Y C O N S T R A I N T S A N D I N C R E A S E I N C U T- O F F GRADE All mining operations are subject to capacity constraints. These constraints can be in the mine itself, in the processing plant, in the refinery, or in the vol￾ume of material that can be sold. A consequence is that if additional material is added to the currently expected production, this material will postpone Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Capacity Constraints and Opportunity Costs 57 mining, processing, refining, or sale of this production. The net present value of the currently expected cash flow will be decreased. This decrease, which is by definition the opportunity cost of adding the new material, must be taken into account when calculating cut-off grades. Mathematical Formulation The mathematical equations that are used to determine opportunity costs were discussed in chapter 2. They can be summarized as follows. NPVi is the net present value of the currently expected cash flow, and t is the time by which this cash flow will be delayed by adding new material to the production schedule. The opportunity cost of adding this material is ( ) ( ) = + −    −  U x NPV 1 i 1 opp i t The cut-off grade is estimated by comparing the cost of adding one addi￾tional metric ton of material to the production schedule, with the value of the product recoverable from this material. A new metric ton added to the pro￾duction schedule must pay for all direct costs associated with this ton, and for the opportunity cost. When t is a very small fraction of one year, as is the case when only one metric ton of material is being considered, the opportunity cost is estimated as follows: ( ) = − ⋅ ⋅ U x i t NPV opp i The time t that is needed to mine, process, or sell one unit of material is equal to the inverse of the constrained capacity. If the constraint is in the mine, the capacity is measured in metric tons mined per year, t is the time needed to mine one metric ton of material, and the opportunity cost is mea￾sured in dollars per metric ton mined. This cost must be added to the mining costs Mo and Mw. If the constraint is in the mill, t is the time needed to pro￾cess one metric ton of material, and the opportunity cost must be added to the processing cost Po . If the constraint is in the refining capacity—which in the case of a copper mine is expressed in pounds of copper per year—t is the time needed to refine one pound of copper, and the opportunity cost must be added to the refining cost R. The following equations, defined previously, will also be used in this section: Utility of mining 1 t of ore: Uore(x) = x ∙ r ∙ (V – R) – (Mo + Po + Oo ) Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 58 CHAPTER FOUR Utility of mining 1 t of waste: Uwaste(x) = –(Mw + Pw + Ow) Mine cut-off grade: xc = [(Mo + Po + Oo ) – (Mw + Pw+ Ow)] / [r ∙ (V – R)] Mill cut-off grade: xc = (Mo + Po + Oo ) / [r ∙ (V – R)] Cut-off Grade in a Capacity-Constrained Copper Mine In this section, a copper mine example is used to illustrate the relationship between capacity constraints, opportunity cost, and cut-off grade. A prelim￾inary feasibility study was completed, which included specific assumptions concerning the mine, mill, and refinery capacities. The results of the feasibility study are summarized in Tables 4-1 and 4-2. Results of Project Feasibility Study The project is characterized by a set of operational parameters, both technical and financial, that are summarized in Table 4-1. These parameters are used to calculate a yearly cash flow, C = $115.68 million, which is assumed to be con￾stant over the ten-year life of the project. Cash flow calculation and project net present value are summarized in Table 4-2. Given a constant cash flow per year over ten years (n = 10), and a discount rate of 12% (i = 12%), the project net present value can be calculated using this formula: ( ) () () = = − +    ⋅ + = − NPV C g i,n C 1 1i 1i i 6.328 C i n The project net present value is $732.06 million. Breakeven Cut-off Grades Assuming no capacity constraint, the utility of sending one metric ton of material of grade x to the process plant is ( ) ( ) ( ) = ⋅ ⋅ − − + + = − U x x 0.84 2.95 0.75 4.50 9.00 2.20 1.848x 15.70 ore If one metric ton is sent to the waste dump, the utility is ( ) = −( ) + + = − U x 4.00 0.80 0.65 5.45 waste The mine and mill breakeven cut-off grades are calculated as follows: Mine breakeven cut-off grade: xc = 15.70/(1.848 ∙ 2,205) = 0.385% Cu Mill breakeven cut-off grade: xc = (15.70 – 5.45)/(1.848 ∙ 2,205) = 0.252% Cu Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Capacity Constraints and Opportunity Costs 59 Cut-off Grades with Mill Capacity Constraint If the mill capacity is limited to 50 million metric tons of ore per year, adding one new metric ton of material to the mill feed will postpone the currently expected cash flow by t years, where t is the inverse of the mill capacity. The corresponding opportunity cost is Uopp(x) = –i∙t∙NPVi = –0.12 ∙ 732,060,000/50,000,000 = –$1.76 per metric ton processed The mine and mill cut-off grades are calculated adding this cost to the processing cost: Mine cut-off grade: xc = (15.70 + 1.76)/(1.848 ∙ 2,205) = 0.428% Cu Mill cut-off grade: xc = (15.70 + 1.76 – 5.45)/(1.848 ∙ 2,205) = 0.295% Cu Table 4-1 Copper mine operating and financial assumptions General operating conditions Mine capacity 72,000,000 metric tons mined per year Mill capacity (flotation) 50,000,000 metric tons processed per year Refining capacity 480,000,000 pounds of copper refined per year Average grade 0.52% Cu average grade of ore processed Fixed costs (unallocated) $100,000,000 $/yr Variable costs Mw $4.00 $/waste metric ton mined Pw $0.80 $/waste metric ton mined Ow $0.65 $/waste metric ton mined Mo $4.50 $/ore metric ton mined Po $9.00 $/ore metric ton mined Oo $2.20 $/ore metric ton mined Average grade, recovery, and revenue per pound of copper x 0.55% Cu average grade of ore mined r 84% copper recovery (flotation and smelter) V $2.95 copper price per pound of recovered copper R $0.75 freight and smelting cost per pound of recovered copper Discount rate and mine life i 12% discount rate n 10 mine life (years) Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 60 CHAPTER FOUR To compensate for the mill opportunity cost, both the mine and mill cut-off grades must be increased by 0.043% Cu. Cut-Off Grades with Mine Capacity Constraint If the mine capacity is limited to 72 million metric tons of ore per year, min￾ing one new metric ton of material that was not scheduled to be mined will postpone the currently expected cash flow by t years, where t is the inverse of the mine capacity. The corresponding opportunity cost is Uopp(x) = –i∙t∙NPVi = –0.12 ∙ 732,060,000/72,000,000 = –$1.22 per metric ton mined The mine and mill cut-off grades are calculated adding this cost to the mining cost: Table 4-2 Yearly cash flow and net present value of copper project Yearly cash flow Metric tons mined 72 million metric tons per year Metric tons ore 50 million metric tons per year Metric tons waste 22 million metric tons per year Cost per metric ton ore $15.70 $/t mined and processed Cost per metric ton waste $5.45 $/t mined and wasted Cost per year ore $785.00 million $/yr Cost per year waste $119.90 million $/yr Total operating cost $904.90 million $/yr Fixed cost per year $100.00 million $/yr Total cost per year $1,004.90 million $/yr Ore grade 0.55% Cu Copper contained 606.38 million lb Cu/yr Copper recovered 509.36 million lb Cu/yr Revenue per pound sold $2.20 $/lb Revenue from sales $1,120.58 million $/yr Net cash flow $115.68 million $/yr (C) Net present value Discount rate 12% %/yr (i) Mine life 10 years (n) NPV factor g(i,n) 6.328 g(i,n) = [1 – (1 + i)–n] · (1 + i)/i Net present value $732.06 NPVi = C g(i,n) Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Capacity Constraints and Opportunity Costs 61 Mine cut-off grade: xc = (15.70 + 1.22)/(1.848 ∙ 2,205) = 0.415% Cu Mill cut-off grade: xc = [(15.70 + 1.22) – (5.45 + 1.22)]/(1.848 ∙ 2,205) = 0.252% Cu The mill breakeven cut-off grade is equal to the marginal cut-off grade. The reason is that the cost of processing one metric ton of material after it has been mined is independent of the mining cost. Cut-off Grades with Refinery Capacity Constraint If the refinery has a limited capacity of 500 million pounds of copper per year, adding one more pound of copper will postpone the currently expected copper production by t years, where t is the inverse of the refinery capacity. The corresponding opportunity cost is Uopp(x) = –i∙t∙NPVi = –0.12 ∙ 732,060,000/500,000,000 = –$0.176 per pound refined The mine and mill cut-off grades are calculated adding this cost to the cost per pound: Mine cut-off grade: xc = 15.70/[0.84 ∙ (2.95 – 0.75 – 0.176) ∙ 2,205] = 0.419% Cu Mill cut-off grade: xc = (15.70 – 5.45)/[0.84 ∙ (2.95 – 0.75 – 0.176) ∙ 2,205] = 0.273% Cu To compensate for the refinery opportunity cost, the mine and mill cut-off grades must both be increased by the same percentage: (2.95 – 0.75)/(2.95 – 0.75 – 0.176) = 8.7%. D E C R E A S I N G C U T- O F F G R A D E O V E R T I M E : A STRATEGIC OBJECTIVE In most situations, the expected net present value of future cash flows decreases over time. When the project approaches the end of its life, little material remains to be mined and the net present value of future cash flows approaches zero, or it may become negative as reclamation costs exceed gen￾erated cash flows. Consider the copper mine described previously. The yearly cash flow was expected to be constant at $115,680,000 per year. Table 4-3 shows how the Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 62 CHAPTER FOUR net present value of future cash flows should be calculated for the first three years of the mine life. In year 0, this NPV is $732.05 million; in year 1, it is $690.34 million; in year 2, it is $643.62 million. In year 9, the last year of operation, the NPV is equal to a one-year undiscounted cash flow, $115.68 million. In year 10, there is no production and the cash flow is zero. Figure 4-3 shows the relationship between the NPV of future cash flows and the year when this NPV is calculated. As the NPV of future cash flows decreases, the opportunity cost decreases and so does the cut-off grade. The relationships between NPVi , the oppor￾tunity cost i∙t∙NPVi , and the mill and mine cut-off grades are calculated in Table 4-4. The capacity constraints are assumed to be the same throughout the mine life: • Mill capacity: 50 million metric tons processed per year • Mine capacity: 72 million metric tons mined per year • Refinery capacity: 500 million lb of copper per year The cut-off grades are estimated using the equations developed earlier: • If the mill is capacity constrained, t = 1/50,000,000 yr/t and Mine cut-off grade: xc = (15.70 + i∙t∙NPVi )/(1.848 ∙ 2,205) Mill cut-off grade: xc = (15.70 + i∙t∙NPVi – 5.45)/(1.848 ∙ 2,205) Table 4-3 Calculation of remaining net present value in years 0 to 3 Project Year Year when remaining NPVi is calculated (i = 12%) 0 1 2 3 Remaining Cash Flow Discount Factor Remaining Cash Flow Discount Factor Remaining Cash Flow Discount Factor Remaining Cash Flow Discount Factor 0 115.68 1.000 1 115.68 0.893 115.68 1.000 2 115.68 0.797 115.68 0.893 115.68 1.000 3 115.68 0.712 115.68 0.797 115.68 0.893 115.68 1.000 4 115.68 0.636 115.68 0.712 115.68 0.797 115.68 0.893 5 115.68 0.567 115.68 0.636 115.68 0.712 115.68 0.797 6 115.68 0.507 115.68 0.567 115.68 0.636 115.68 0.712 7 115.68 0.452 115.68 0.507 115.68 0.567 115.68 0.636 8 115.68 0.404 115.68 0.452 115.68 0.507 115.68 0.567 9 115.68 0.361 115.68 0.404 115.68 0.452 115.68 0.507 10 0 0 0 0 NPVi $732.05 $690.34 $643.62 $591.29 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Capacity Constraints and Opportunity Costs 63 • If the mine is capacity constrained, t = 1/72,000,000 yr/t and Mine cut-off grade: xc = (15.70 + i∙t∙NPVi )/(1.848 ∙ 2,205) Mill cut-off grade: xc = (15.70 – 5.45)/(1.848 ∙ 2,205) • If the smelter is capacity constrained, t = 1/500,000,000 yr/lb and Mine cut-off grade: xc = 15.70/[0.84 ∙ (2.95 – 0.75 – i∙t∙NPVi ) ∙ 2,205] Mill cut-off grade: xc = (15.70 – 5.45)/[0.84 ∙ (2.95 – 0.75 – i∙t∙NPVi ) ∙ 2,205] Figure 4-4 shows how the mine cut-off grade changes in function of the year when it is calculated, depending on whether constraints are on mine, mill, or refinery capacity. Figure 4-5 shows the same information for the mill cut-off grade. In this example, the mill capacity constraint is that which has the highest cost, as measured by the amount of copper needed to maintain the project net present value. CUT-OFF GRADE BELOW MARGINAL CUT-OFF GRADE Situations arise when it is justified to use a cut-off grade below the marginal cut-off grade. At the end of the mine life, the net present value of future cash flows may be negative, when reclamation costs are exceeding expected 0 700 600 500 400 300 200 100 800 Project Year When NPV Is Calculated 0 1 2 3 4 5 6 7 8 9 10 Figure 4-3 Relationship between NPV of future cash flow and year when NPV is calculated Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. NPV of Future Cash Flow, $ 64 CHAPTER FOUR revenues from operations. The opportunity cost changes sign from negative to positive: postponing expenditures increases the remaining project net present value. A cut-off grade below the marginal cut-off grade may be justified, pro￾vided the loss incurred in mining and processing this material is less than the benefit of postponing reclamation expenditures. At the end of the mine life, if CR is the expected reclamation cost, the net present value of this cost is –CR. If reclamation is postponed by one year, this net present value is increased from –CR to –CR/(1 + i). The opportunity cost is [ ] () () − + = + C 1 1 1 i C i 1 i R R The cut-off grade that can be used to postpone reclamation by one year can be below the marginal cut-off grade. The only requirement is that the cash flow C generated by mining low-grade material during the last year, added to the corresponding opportunity, is positive: ( ) ( ) + + > > − + C C i 1 i 0 C C i 1 i R R The cash flow C can be negative. If the cost of reclamation is $15 million and i = 10%, the maximum loss that can be justified to postpone reclamation by one year is 15 ∙ (0.1/1.1) = $1.36 million. Table 4-4 Calculation of opportunity cost and cut-off grade per year Capacity Constraint Mill Constraint, 50 million metric tons processed Mine Constraint, 72 million metric tons mined Refinery Constraint, 500 million lb copper Year NPVi , $ million Opportunity Cost, $ Cut-off Grade, % Cu Opportunity Cost, $ Cut-off Grade, % Cu Opportunity Cost, $ Cut-off Grade, % Cu Mine Mill Mine Mill Mine Mill 0 732 1.76 0.43% 0.29% 1.22 0.42% 0.25% 0.18 0.42% 0.27% 1 690 1.66 0.43% 0.29% 1.15 0.41% 0.25% 0.17 0.42% 0.27% 2 644 1.54 0.42% 0.29% 1.07 0.41% 0.25% 0.15 0.41% 0.27% 3 591 1.42 0.42% 0.29% 0.99 0.41% 0.25% 0.14 0.41% 0.27% 4 533 1.28 0.42% 0.28% 0.89 0.41% 0.25% 0.13 0.41% 0.27% 5 467 1.12 0.41% 0.28% 0.78 0.40% 0.25% 0.11 0.41% 0.27% 6 394 0.94 0.41% 0.27% 0.66 0.40% 0.25% 0.09 0.40% 0.26% 7 311 0.75 0.40% 0.27% 0.52 0.40% 0.25% 0.07 0.40% 0.26% 8 219 0.53 0.40% 0.26% 0.36 0.39% 0.25% 0.05 0.39% 0.26% 9 116 0.28 0.39% 0.26% 0.19 0.39% 0.25% 0.03 0.39% 0.25% 10 0 0.00 0.39% 0.25% 0.00 0.39% 0.25% 0.00 0.39% 0.25% Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Capacity Constraints and Opportunity Costs 65 Postponing reclamation costs is justified if maximizing net present value is the only objective. Other financial, legal, environmental, and socioeco￾nomic factors may be taken into account that will influence the decision when to close the mine. Other circumstances occur where using a cut-off grade below the marginal cut-off grade is justified. This may be the case when the tonnage of material exceeding the marginal cut-off grade is less than the 0.38 0.43 0.42 0.41 0.40 0.39 Project Year 0 1 2 3 4 5 6 7 8 9 10 No Constraint Mine Constraint Mill Constraint Refinery Constraint Figure 4-4 Relationship between mine cut-off grade and project year 0.24 0.30 0.29 0.28 0.27 0.25 0.26 Project Year 0 1 2 3 4 5 6 7 8 9 10 No Constraint Mine Constraint Mill Constraint Refinery Constraint Figure 4-5 Relationship between mill cut-off grade and project year Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Mine Cut-off Grade, % Mill Cut-off Grade, % 66 CHAPTER FOUR plant capacity. Several options must then be considered, which have different technical and financial consequences: • Run the plant below capacity. This will increase the cost per metric ton processed. In some circumstances, higher recovery is achievable when reducing the mill throughput. How to optimize plant capacity was discussed previously. • Run the plant as a batch plant, with periodic shutdowns. The cost of shutdowns must be considered to optimize cut-off grade. • Run the plant at capacity using a cut-off grade below the marginal cut-off grade. If the tonnage needed to fill the plant is relatively small, this option is likely to be the best one. CUT-OFF GRADE OPTIMIZATION WITH O P P O R T U N I T Y COSTS: PRACTICAL CONSIDERATIONS The formula Uopp(x) = –i∙t∙NPVi is useful to determine whether cut-off grades and net present value have been optimized. The cut-off grade calculated using opportunity costs must be compared with that currently used in the mine plan. If the former is higher than the latter, consideration should be given to increasing the cut-off grade. This may be possible by increasing mine capacity or changing plant capacity. Conversely, if the planned cut-off grade is lower than the cut-off grade calculated using the opportunity cost, indications are that facilities might have been overdesigned. Constraints imposed by the geology of the deposit have a major influence on the determination of cut-off grades and optimum mine and mill capacities. This influence will be discussed in Chapter 5. Cut-off grades calculated from cash flows that have not been optimized are also not optimal. An iterative approach must be used to maximize the net present value. For example, one could first calculate a cash flow using a fixed cut-off grade such as the marginal cut-off grade. From this cash flow, cut-off grades could be reestimated using opportunity costs. But new cut-off grades imply new mine plans, new cash flows, and therefore new opportunity costs, which must be used to, once again, reestimate the cut-off grades. This lengthy process must be repeated until the net present value is optimized. Fortunately, algorithms have been developed and continue to be improved to facilitate simultaneous optimization of mine plans, production schedules, and material allocation, with implied optimization of cut-off grade strategies (see bibliography). Computer programs are available that render complex theoretical solutions accessible to mining engineers and other practitioners. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Capacity Constraints and Opportunity Costs 67 While solutions applicable to open pit environments are more advanced, underground mines are given increasing attention. Before changing the cut-off grade, all costs and benefits likely to result from this change must be carefully examined. Complex relationships occur between the space-dependent (and sometimes time-dependent) geological and geotechnical properties of the deposit, the time-dependent economic variables (capital and operating costs, metal prices, environmental and socio￾economic impact), and technical constraints, which, to some extent, can be controlled (mining and processing methods, operational capacity). These factors must all be taken into account to optimize cut-off grades, yearly production, and cash flows. No simple solutions exist for this complex multi￾dimensional optimization problem. Cut-off grades that were estimated to be optimal at the time the original mine plan was developed must be continuously reexamined. Current and expected costs and prices, mine and mill performance, and environmental and socioeconomic conditions will change over time, resulting in changes in future cash flow and opportunity costs. Maximizing net present value tends to give no value to actions where consequences will only be felt at the end of the mine life. For example, actions may have to be taken throughout the life of a project to minimize future costs of reclamation and environmental compliance. The cost of these actions may be significant from a net present value point of view, but the resulting savings, which will be incurred at the end of the mine life, may have no impact on the net present value. Similarly, stockpiling low-grade material may increase costs throughout the mine life, but revenues resulting from processing these stockpiles will only be realized at the end of the mine life. Maximizing net present value should never be the sole guide to decision making. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 69 5 As mentioned previously, constraints imposed by the geology of the deposit play a critical role in defining the optimal cut-off grade. The tonnage and grade of material that can be made available for processing is continuously changing during the life of the mine. The influence of geologic constraints is analyzed in conjunction with other constraints such as those on mine or mill capacity, and on volume of sales. CUT-OFF GRADE WHE N P R O C E S S I N G C A PA C I T Y IS FIXE D Ideally, a new mine should be designed such that mining capacity and pro￾cessing capacity are perfectly balanced and the planned cut-off grades are opti￾mum given the expected cash flow. In practice, this situation occurs only on paper, when the project is designed. As soon as operations start, imbalances invariably appear. The actual processing plant capacity exceeds or falls below that expected. The mining capacity is higher or lower than planned. Mine and mill capacities are no longer balanced, new constraints appear, and the cut￾off grade must be changed accordingly. The cut-off grade must also take into account differences between expected and actual costs, productivities, recov￾eries, and market value of product sold. When a new project is designed, mine and mill capacities and corresponding cut-off grades are chosen primarily to optimize financial objectives. Once mine and mill facilities are built, physical constraints become the main drivers and studies must be completed to deter￾mine whether removing these constraints is financially justified. Relationship Between Mine Capacity and Cut-off Grade In this section, it will be assumed that the capacity of the processing plant is fixed and cannot be changed. The only change that can be made is to the min￾ing capacity. What is the impact of a change in mining capacity on the cut-off grade and the grade of the material sent to the plant? Geologic Constraints and Opportunity Costs Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 70 CHAPTER FIVE • Consider an increase in mining capacity, defined as tonnage mined per year. • This increase requires an increase in mining capital cost, and is likely to result in an increase in total mine operating costs per year. How￾ever, it is also likely to result in decreased mining and overhead costs per metric ton mined. • Given that the processing capacity is fixed, the cut-off grade must be increased to keep the tonnage sent to the mill constant. The average grade of mill feed will increase and so will the quantity of product sold. • The mine life will decrease. • In some instances, the lower-grade material, which is not processed, will be stockpiled. Stockpiling of low-grade material was discussed previously. To decide whether an increase in mining capacity is justified, the expected net impact on the utility of the project must be assessed, taking into account the following factors: • Increased capital cost of new mining capacity • Decreased mine unit operating cost • Increased plant head grade and increased metal sales per year • Loss of low-grade material or delayed processing of some of this material • Reduced mine life and resulting socioeconomic and political impact • Reduced project life and decreased political risk, if applicable • Change in environmental impact A simple example follows. Consider a mining operation in which the plant was designed to process an average of 250,000 metric tons per month, or 3 million metric tons per year. The grade–tonnage relationship corresponding to the mineralized material expected to be mined during the coming year is shown in Figure 5-1. At the current mining capacity, the 3 million metric ton plant capacity is consistent with a cut-off grade of 0.74 g/t and a mill feed average grade of 1.56 g/t. Consideration is being given to increasing the mining capacity by 50%, and the impact such a change would have on the coming year is being investi￾gated. If the mining capacity is increased by 50%, the material currently sched￾uled to be mined in one year will be mined in eight months. Since the mill Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Geologic Constraints and Opportunity Costs 71 can process only 250,000 metric tons a month, it will only consume 2 million metric tons during this eight-month period. This tonnage corresponds to a cut-off grade of 1.03 g/t and an average mill feed head grade of 1.90 g/t (Fig￾ure 5-1). Consideration should be given to stockpiling the material between 1.03 g/t and a cut-off grade somewhat higher than 0.74 g/t. This material should be considered for re-handling and processing at a later date. The proposed 50% increase in mining capacity may or may not be optimal. To be economically justified, an increase in mining capacity must take into account financial, technical, environmental, permitting, and other constraints imposed by deposit size and shape, mining method, size of equip￾ment, safety and environmental regulations, and other parameters. Depend￾ing on the limitations imposed by these constraints, an iterative approach is best suited to mining capacity optimization. Such an approach can consist of the following steps: 1. Assume a 1 million metric ton increase in mining capacity (or some other increase that is technically achievable). 2. Calculate the resulting decrease in mine life. 3. Estimate the increase in cut-off grade and resulting higher mill head grade that is consistent with the increase in mining capacity and fixed processing capacity. 4. Estimate the increase in mine capital and yearly operating costs needed to increase the mining capacity. Calculate the corresponding 0 4 3 2 1 5 6 1.0 1.4 1.2 1.6 1.8 2.0 2.2 2.4 Cut-off Grade 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade Average 1.90 Average 1.56 Cut-off 1.03 Cut-off 0.74 Tonnage 3.0 Tonnage 2.0 Figure 5-1 Estimation of cut-off grade assuming fixed processing capacity Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade 72 CHAPTER FIVE discounted incremental mining costs (DIMC) for the remaining life of the project. 5. Estimate the increase in mill production per year (units of product sold) and calculate the corresponding discounted incremental reve￾nues (DIR). 6. If low-grade material is to be stockpiled, the net present value of this material should also be taken into account. 7. If the DIR exceed the DIMC, this analysis should be repeated assum￾ing an additional 1-million-metric-ton increase in mining capacity. 8. The optimal mining capacity is that for which DIR equals DIMC. In this previous discussion, it was assumed that the increase in mine capacity could be achieved without changing mine selectivity. The grade– tonnage curve did not change. The volumes being mined remained the same, but these volumes were mined faster. This situation will occur if more equipment of the same size is added to an open pit mine with no change to the pit design, or if more stopes are put in production simultaneously with￾out changing the underground mining method or the stope design. Some situations occur where the assumptions of constant grade–tonnage curve cannot be made. In open pit mines, larger trucks and loading equipment, increased bench height, and wider spacing between blast holes can be used to increase capacity. The result is a decrease in selectivity, resulting in a new grade–tonnage curve. Similarly, underground production can be increased by using a different mining method, which may result in lower selectivity joined with significantly lower costs per metric ton. The impact of selectivity on the grade–tonnage curve, the cut-off grade, and the mill feed average grade will be discussed at the end of Chapter 6. Increasing the mining capacity will not necessarily result in a higher head grade if this increase is realized by signifi￾cantly decreasing mine selectivity. Using Opportunity Cost as a Guide to Mine Capacity Optimization The approach to mine capacity optimization previously described can be extremely time-consuming. An alternative approach is presented here, which can be used to rapidly converge toward the desired solution, provided there is little variability in the geologic properties of the deposit being mined. Consider an open pit gold mine operating under the following conditions: • Total tonnage mined: T1 = 4.5 million t/yr • Waste mined: Tw1 = 1.5 million t/yr Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Geologic Constraints and Opportunity Costs 73 • Ore mined and processed: T+c1 = 3.0 million t/yr • Cut-off grade: xc1 = 0.74 g/t • Average grade above cut-off: x+c1 = 1.56 g/t Costs associated with waste material: • Mw = $3.50 per metric ton of waste • Pw = $0 per metric ton of waste • Ow = $0.88 per metric ton of waste Costs associated with processed material: • Mo = $3.20 per metric ton of ore • Po = $15.00 per metric ton of ore • Oo = $4.55 per metric ton of ore • Plant recovery: r = 87% • Gold price: V = $1,400 per ounce of gold (1 troy ounce = 31.1035 g) • Cost of sales: R = $50 per ounce of gold The grade–tonnage curves representing one year of production are shown in Figure 5-2. The marginal cut-off grades are as follows: Mill marginal cut-off grade: [ ] ( ) ( ) ( ) = + + − + +     − = x M P O M P O r V R 31.1035 0.49 g/t c o o o w w w Mine marginal cut-off grade: [ ] ( ) ( ) = + + − = x M P O r V R 31.1035 0.60 g/t c o o o These marginal cut-off grades are significantly lower than the 0.74 g/t cut-off grade used in the current operation. The difference reflects constraints imposed by the plant’s capacity. The total tonnage available for mining is 22.5 million metric tons. The deposit will be mined out during the five-year life of the mine. Because of low variability in the deposit geology, the yearly cash flow (YCF) is expected to be constant over a five-year mine life. This cash flow is estimated as follows: ( ) ( ) ( ) = − − + + − + + = + + + YCF T x r V R T M P O T M P O $102 million per year 1 c1 c1 c1 o o o w1 w w w Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 74 CHAPTER FIVE The net present value of this cash flow, calculated at a discount rate i = 10%, and a five-year mine life is NPVi1 = $425 million. The opportunity cost of the mill capacity constraint is i∙t∙NPVi1 = 0.10 ∙ 425/4.5 = $9.44 per metric ton processed, which is equivalent to 9.44 ∙ 31.1035/[0.87 ∙ (1,400 – 50)] = 0.25 g/t. When taking the opportunity cost into consideration, the mill cut-off grade is raised from 0.49 g/t to 0.74 g/t, which happens to be the cut-off grade used in the current mine plan. Consideration is being given to increasing mine capacity by 25%, from 4.5 to 5.625 million metric tons a year, thus reducing the mine life from five to four years. The mill capacity remains the same, at 3 million metric tons a year. The amount of waste material is increased from 1.5 million metric tons to 2.625 million metric tons, calculated as follows: • Total tonnage mined: T2 = 5.625 million t/yr • Ore mined and processed: T+c2 = 3.0 million t/yr • Waste mined: Tw2 = 2.625 million t/yr The tonnage-grade curves that represent current (4.5 million metric tons) and proposed (5.625 million metric tons) yearly production are shown in Fig￾ure 5-3. The proposed tonnage curve is obtained by multiplying the current one by 1.25. The proposed grade curve is identical to the current one. From these curves, one gets the cut-off grade to be used to fill the plant, and the average grade above cut-off grade is as follows: • Cut-off grade: xc2 = 0.90 g/t 1 4 4.5 3 3.5 2.5 2 1.5 5 1.2 1.5 1.4 1.3 1.6 1.7 1.8 1.9 2.0 2.1 Cut-off Grade 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade Average Grade 1.56 g/t Cut-off Grade 0.74 g/t 3 Million Metric Tons Figure 5-2 Grade–tonnage curves for material mined in one year at 4.5 t/yr Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade Geologic Constraints and Opportunity Costs 75 • Average grade above cut-off: x+c2 = 1.75 g/t Assuming that the costs per metric ton remain the same, the proposed yearly cash flow is estimated at ( ) ( ) ( ) = − − + + − + + = + + + YCF T x r V R T M P O T M P O $119 million per year 2 c2 c2 c2 o o o w 2 w w w As a result of a higher head grade, this yearly cash flow is significantly higher than the $102 million currently expected. The net present value of this cash flow, calculated at a discount rate i =10% and a reduced life of four years is NPVi2 = $415 million. This net present value is less than the $425 million estimated at the current mining rate. The reason is that the life of the project has been reduced from five years to four. The difference in net present value would be even more significant if, as it should be, the cost of expanding the mine capacity was taken into account. This cost would occur in year 0 and would not be discounted. The marginal cut-off grades are significantly lower than the cut-off grades used to fill the plant. This is more so when the mine capacity is increased. The net present value of both scenarios would be significantly improved if low-grade material was stockpiled and processed at the end of the mine life, after mining is completed and the higher-grade material has been pro￾cessed. More low-grade material is available for stockpiling when the mine 1 4 4.5 3 3.5 2.5 2 1.5 5 1.2 1.5 1.4 1.3 1.6 1.7 1.8 1.9 2.0 2.1 Cut-off Grade 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 1.05 1.1 Tonnage Above Cut-off Grade with 25% Increase in Mine Capacity Average Grade Above Cut-off Grade New Average Grade 1.75 g/t Current Average Grade 1.56 g/t New Cut-off Grade 0.90 g/t Current Cut-off Grade 0.74 g/t 3 Million Metric Tons Processed 3.75 Million Current Tonnage Above Cut-off Grade Figure 5-3 Current and proposed grade–tonnage curves for material mined in one year Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade 76 CHAPTER FIVE capacity is increased, and stockpiling will have a greater economic impact in this situation. Alternative mine plans should be considered, all of which should include a stockpiling option. One alternative is to lower the mine capacity and there￾fore the mill head grade, thus increasing the mine life but decreasing the yearly cash flow. Another option, if feasible, would consist of increasing the mill capacity. The optimal solution is likely to be that where declining cut-off grades are used, starting with 0.74 g/t during the current year and reaching 0.49 g/t at the end of the mine life. CUT-OFF GRADE WHEN MINING CAPACITY IS FIXE D In the previous analysis, a fixed processing capacity was assumed to be the case. Now consider the situation where the mining capacity is fixed but an increase in plant capacity is being proposed. A lower cut-off grade is needed to balance the mining capacity with the plant capacity. Increasing the mill capacity has the following impacts: • The tonnage processed per year is increased. • The tonnage mined is not changed. The cut-off grade must be decreased to keep the processing plant full. • The average grade of material sent to the mill decreases, but the metal content of this material increases. • More metal is recovered, resulting in higher revenues from sales. • The capital cost of plant expansion must be taken into account. • The plant operating costs are likely to increase per unit of time (cost per year) but should decrease per unit of production (cost per metric ton processed). The optimal plant capacity is that which maximizes the total utility of the project, taking into account financial impact (increased capital cost, decreased unit operating cost, increased revenue from sales), as well as socioeconomic, environmental, political, and other impacts. For example, consider a mining operation in which the plant was designed to process an average of 250,000 metric tons per month, or 3 million metric tons per year. The grade–tonnage relationship corresponding to the mineralized material expected to be mined at the current capacity is shown in Figure 5-4. At the current mining capacity, this plant capacity is consistent with a cut-off grade of 0.74 g/t, corresponding to a mill feed average grade of 1.56 g/t. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Geologic Constraints and Opportunity Costs 77 Management is considering increasing the size of the processing plant by 20% and investigating the impact that such a change would have on the com￾ing year. If the processing capacity is increased by 20% and the mining capac￾ity is kept constant, the cut-off grade must be decreased to 0.61 g/t to supply 3.6 million metric tons to the mill (Figure 5-4), and the average grade of mill feed will decrease to 1.41 g/t. The gold content of the material processed will increase from 2.51 million oz to 2.73 million oz. This cut-off grade calcula￾tion only takes into account capacity constraints and is independent of the economics of the project. The increase in plant capacity must be justified not only by the increase in material processed, but also taking into account capital cost requirements, possible changes (increase or decrease) in recovery, a likely decrease in operating costs, and all other direct and indirect costs and benefits. The proposed 20% increase in processing capacity may or may not be optimal. To be economically justified, an increase in plant capacity must take into account financial, technical, environmental, permitting, and other con￾straints imposed by the size of the available processing equipment, limitations on tailings dam expansion, maximum permitted dust emission, and other parameters. Depending on the limitations imposed by these constraints, an iterative approach is best suited to plant capacity optimization. This approach can consist of the following steps: 1. Assume a 1-million-metric-ton increase in processing capacity (or some other increase that is technically achievable). 0 4 3 2 1 6 5 1.0 1.4 1.2 1.6 1.8 2.0 2.2 2.4 Cut-off Grade 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Tonnage 3.6 Tonnage 3.0 Cut-off 0.61 Cut-off 0.74 Average 1.56 Average Grade 1.41 Figure 5-4 Estimation of cut-off grade assuming fixed mining capacity Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade 78 CHAPTER FIVE 2. Estimate the decrease in cut-off grade and resulting lower mill head grade that is consistent with the fixed mining capacity and higher processing capacity. 3. Estimate the increase in mill capital and yearly operating costs needed to increase the processing capacity. Calculate the corresponding dis￾counted incremental processing cost (DIPC) for the remaining life of the project. 4. Estimate the increase in mill production per year (units of product sold) and calculate the corresponding discounted incremental reve￾nue (DIR). 5. If DIR exceeds DIPC, this analysis should be repeated, assuming an additional 1-million-metric-ton increase in processing capacity. 6. The optimal processing capacity is that for which DIR equals DIPC. CUT-OFF GRADE WHEN VOLUME OF SALES IS FIXE D In this analysis, it is assumed that the volume of sales is fixed. This may be because all products are sold under contracts that specify the volume that will be bought on a yearly basis. Perhaps the market is small and the amount of product that can be sold is limited. Or it might be that management specifies the amount to be produced from a given operation for reasons external to the operation under consideration. Fixed Sales with No Mining or Processing Constraint In this example it is assumed that the recovery achieved in the processing plant is independent of tonnage processed and plant head grade. In such a situation, requiring a fixed volume of sales is equivalent to requiring a fixed quantity of metal (or other salable product) delivered by the mine to the processing plant. This quantity Q+c is equal to the tonnage delivered T+c mul￾tiplied by the average grade of plant feed x+c: + + = ⋅ + Q T c c x c Consider a gold mining operation that has been requested to supply 4 t of gold (130,000 oz) to the processing plant over a one-year period (Q+c = 4.0 t of gold). Consider three scenarios: 1. There is no constraint on either mine or plant capacity. This is usually the case only during the feasibility study. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Geologic Constraints and Opportunity Costs 79 2. The mine capacity is fixed, but the plant capacity is not. 3. The plant capacity is fixed, but the mine capacity is not. If neither the mine nor the processing plant is capacity constrained, the number of possible cut-off grades is theoretically infinite. A high cut-off grade will result in a high average grade above cut-off grade x+c. The higher the cut￾off grade, the lower the capacity T+c of the processing plant that is needed to keep sales at the required level. But in the case of an open pit mine, a higher cut-off grade will require mining more metric tons per year. In the case of an underground mine, smaller stopes may have to be designed to eliminate peripheral low-grade material, and low-grade stopes may have to be rejected. When neither mine nor plant capacity is fixed, cut-off grade optimiza￾tion requires analysis of several feasible solutions: low cut-off grade and large plant size or high cut-off grade and smaller plant size. Technical constraints, including constraints imposed by the geology of the deposit, will reduce the number of feasible options. Higher cut-off grades will result in lower capital costs for the plant and likely higher operating costs, while the impact on mine capital and operating costs will be a function of the geological properties of the deposit and the applicable mining methods. Cut-off grade optimization requires estimation of capital and operating costs and cash flow analysis for each feasible solution. Fixed Sales and Fixed Processing Rate with No Mining Constraint Cut-off grade determination becomes easier if, in addition to the constraint on the amount of metal processed, one adds a constraint on either plant or mine capacity. First assume that the plant capacity, defined as tonnage pro￾cessed per year, is fixed. With both tonnage processed T+c and metal content Q+c being fixed, the plant head grade x+c is calculated as follows: = + + + x Q T c c c If it is known what material can be mined in the coming months, one can determine the cut-off grade needed to reach the necessary average grade and the mining rate needed to reach the necessary tonnage of mill feed T+c. Consider the gold mine that was asked to supply 4 t of gold to the pro￾cessing plant during the coming year. In addition, assume that the capacity of the processing plant is fixed at 2 million t/yr. To satisfy these constraints, the head grade must be Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 80 CHAPTER FIVE ( ) ( ) = = = + + + x Q T 4,000,000 g/yr 2,000,000 t/yr 2.00 g/t c c c A preliminary mine plan was developed during which 6 million metric tons of material, both ore and waste, would be mined. The corresponding grade–tonnage relationship is shown in Figure  5-5. From this relationship, one determines that the cut-off grade needed to get an average grade of 2.0 g/t is 1.12 g/t. There are only 1.78 million metric tons of mill feed above cut-off grade in this preliminary mine plan. Since the mill capacity is 2 million t/yr, this material will be processed in 10.7 months, calculated as follows: ( ) ( ) ( ) 12 months/yr 1⋅ = .78 million metrictons 2.0 million t/yr 10.7 months Six million metric tons are scheduled to be mined in this preliminary mine plan. To mine this tonnage in 10.7 months, the mining rate must be 6.0/10.7 = 560,000 t/month or 6.7 million t/yr. In conclusion, for the mine to send 4 t of gold per year to a plant that has a capacity of 2 million t/yr, a total of 6.7 million metric tons must be mined every year and a cut-off grade of 1.12 g/t must be used. The plant head grade will be 2.0 g/t. 0 4 3 2 1 6 5 1.0 1.4 1.2 1.6 1.8 2.0 2.2 2.4 Cut-off Grade 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Average Grade 2.00 Cut-off 1.12 Tonnage 1.78 Figure 5-5 Estimation of cut-off grade and tonnage given an average grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade Geologic Constraints and Opportunity Costs 81 Fixed Sales and Fixed Mining Rate with No Processing Constraint Now consider the case where the mine capacity is constrained at 6 million t/yr and the metal content of the material to be sent to the mill is set at 4 t of gold per year. A yearly mine plan was developed in which 6 million metric tons are to be mined. The corresponding grade–tonnage relationship is shown in Fig￾ure 5-5. From the values of T+c and x+c shown in this figure, one can calculate the metal content of material above cut-off grade Q+c = T+c ∙ x+c and plot this metal content as a function of cut-off grade x (Figure 5-6). Figure 5-6 shows the relationship between cut-off grade and quantity of metal above cut-off grade, as scheduled to be mined in the current mine plan. Because the quantity of metal to be processed is Q+c = 4.0 t of gold, the cut￾off grade must be 0.97 g/t. The tonnage and average grade of material above this cut-off grade can be determined using the grade–tonnage relationship (Figure 5-7): = = + + T 2.20 million metrictons x 1.82 g/t c c Given that 6 million metric tons of material are scheduled to be mined in the coming year and that the mine must send 4 t of gold to the processing plant, a cut-off grade of 0.97 g/t must be used, resulting in 2.20 million metric tons of material being sent to the processing plant, averaging 1.82 g/t. This can only be achieved if the plant capacity is at least 2.20 million t/yr. R E L E A S I N G C A PA C I T Y C O N S T R A I N T S : A B A S E M E TA L EXAMPLE The previous analyses have shown the sometimes deterministic effect that geologic constraints imposed by nature, combined with capacity constraints, have on cut-off grades. In this section, an example is given for how to deter￾mine the economic feasibility of releasing capacity constraints under defined geologic conditions. Consider a copper mining and processing operation. Mine and mill capacities are 79 million metric tons and 39.5 million t/yr, respectively. The copper resources included in that part of the deposit scheduled to be mined in the coming year are listed in Table 5-1. The cut-off grade for mill feed is 0.25% Cu. The reserves to be mined and processed in the coming year are 39.5 million metric tons averaging 0.381% Cu and containing 150,000 metric tons of copper (332 million lb of copper). Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 82 CHAPTER FIVE Management wishes to assess the sensitivity of the project to changes in mine, mill, or smelter capacity. Four cases are to be considered: • Case 1: Assume that the mine capacity is increased by 10%, from 79 to 86.9 million metric tons, but the mill capacity remains fixed at 39.5 million t/yr. The 79 million metric tons that were scheduled to be mined in one year, including the resources shown in Table 5-1, 2.0 5.5 5.0 4.5 4.0 3.5 3.0 2.5 6.0 Cut-off Grade 0.5 1 0.9 0.8 0.7 0.6 1.1 1.2 Metal Content 4.0 Cut-off 0.97 Figure 5-6 Estimation of cut-off grade given the required metal content of mine feed 0 4 3 2 1 6 5 1.0 1.4 1.2 1.6 1.8 2.0 2.2 2.4 Cut-off Grade 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Average Grade 1.82 Cut-off 0.97 Tonnage 2.20 Figure 5-7 Estimation of tonnage and average grade above cut-off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Metal Content Above Cut-off Grade Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade Geologic Constraints and Opportunity Costs 83 will be mined in 1/1.1 = 0.91 years (10.9 months). During this period, the mill can only process 35.9 million metric tons. From Table 5-1, one sees that to send only 35.9 million metric tons to the processing plant, the cut-off grade must be increased to 0.27% Cu. The mill head grade will be 0.393% Cu. Assuming that the same aver￾age grade can be maintained over one year, 39.5 million metric tons of ore will be processed at an average grade 0.393% Cu, containing 155,000 t of copper. • Case 2: Assume that the capacity of the flotation plant is increased by 10%, from 39.5 to 43.5 million metric tons, but the mine capac￾ity is unchanged at 79 million t/yr. The resources available to feed the mill in one year remain as shown on Table 5-1. To supply 43.5 million metric tons to the mill, the cut-off grade must be lowered to 0.225% Cu. The mill head grade will average 0.367% Cu, resulting in 160,000 t of copper being processed. Table 5-1 Copper resources contained in material scheduled to be mined Cut-off Grade, % Cu Minable Tonnage, million metric tons Minable Grade, % Cu Minable Copper Content thousand metric tons Cu million lb Cu 0.15% 53.7 0.335% 180 397 0.16% 52.6 0.340% 179 395 0.17% 51.4 0.344% 177 390 0.18% 50.1 0.348% 174 384 0.19% 48.8 0.352% 172 378 0.20% 47.5 0.355% 168 372 0.21% 46.0 0.360% 165 365 0.22% 44.4 0.365% 162 357 0.23% 42.8 0.370% 159 349 0.24% 41.2 0.375% 155 341 0.25% 39.5 0.381% 150 332 0.26% 37.7 0.387% 146 322 0.27% 35.9 0.393% 141 311 0.28% 34.1 0.399% 136 300 0.29% 32.1 0.406% 131 288 0.30% 30.2 0.413% 125 275 0.31% 28.2 0.421% 119 262 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 84 CHAPTER FIVE Table 5-2 Cut-off grades, and mine and mill capacities required to satisfy specific capacity requirements Cut-off Grade, % Cu Tonnage Milled, million metric tons Average Grade, % Cu Copper Content Tonnage Mined, million metric tons thousand metric tons Cu million lb Cu Base Case Values 0.250% 39.5 0.381% 150 332 79.0 Case 1: Increase mining rate by 10%. Keep processing rate at same level. Values 0.270% 39.5 0.393% 155 342 86.9 Difference from base case 8% 0% 3% 3% 3% 10% Case 2: Increase processing rate by 10%. Keep mining rate at same level. Values 0.225% 43.5 0.367% 160 352 79.0 Difference from base case –10% 10% –4% 6% 6% 0% Case 3: Increase copper produced by 10%. Keep mining rate at same level. Values 0.210% 46.0 0.360% 165 365 79.0 Difference from base case –16% 16% –6% 10% 10% 0% Case 4: Increase copper produced by 10%. Keep milling rate at same level. Values 0.305% 39.5 0.418% 165 364 107 Difference from base case 22% 0% 10% 10% 10% 36% • Case 3: Management wishes to determine under which conditions 10% more copper could be sent to the processing plant if mine capac￾ity remains fixed at 79 million metric tons. The copper content of processed material must increase from 150,000 to 165,000 t. From Table 5-1 it can be seen that the cut-off grade must be decreased to 0.21% Cu, resulting in 46.0 million metric tons of ore being sent to the mill, averaging 0.360% Cu. If the mining rate is not changed, a 10% increase in copper processed can only be achieved by decreasing the average grade by 6% and increasing the tonnage milled by 16%. • Case 4: Management wishes to determine under which conditions 10% more copper could be sent to the processing plant if mill capacity remains fixed at 39.5 million metric tons. To increase the copper content of mill feed from 150,000 to 165,000 t, the mill head Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Geologic Constraints and Opportunity Costs 85 grade must be increased from 0.381% Cu to 165,000/39,500,000 = 0.418% Cu. Table  5-1 shows that, to reach this average grade, it is necessary to use a cut-off grade of 0.305% Cu. There are only 29.2 million metric tons above this cut-off grade. Given the mill capacity of 39.5 million metric tons, this ore will be consumed in 29.2/39.5 = 0.74 year (8.88 months). The mining rate must therefore be increased from 79 million t/yr to 79/0.74 = 107 million t/yr. If the processing rate is not changed, a 10% increase in copper processed can only be reached by increasing the average grade by 10% and increasing the tonnage mined by 36%. The results are summarized in Table 5-2. Each case is compared with the base case, in which 79 million metric tons are mined and 39.5 million metric tons are processed. This simple example shows procedures that can be used to calculate cut-off grades taking into account geologic constraints (as summarized in Table  5-1) and technical constraints, including mine, mill, or production capacities. No attempt was made to assess whether the proposed solutions were economically feasible or justified. To do so would require completing this analysis not only over one year (as done in this example) but for the life of the mine. Implementing any of the mining and processing plans summarized in Table  5-2 would require additional capital expenditures; would change operating costs; might shorten the mine life (cases 1 and 4); could justify stockpiling of low-grade material (case 4); and might require other opera￾tional changes. All these changes would result in changes in cash flow, which would have to be quantified to decide on the best option. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 87 6 While there are obvious differences, there are also many similarities between questions that must be answered when designing open pit and underground mines. These similarities are illustrated in the examples that follow. Different mining methods are considered, including open pit, selective underground, and block and panel caving. O P E N P I T M I N E : E C O N O M I C VA L U AT I O N O F A PUSHBACK In this section, a situation is analyzed where a pushback is considered to be added to an open pit mine. To make this decision and optimize the cut-off grade, two mine plans should be considered: 1. The first plan assumes that the pushback is not mined. The net pres￾ent value of future cash flows is NPVi1. 2. The second plan assumes that the pushback is mined. The cash flow generated by this plan can be divided in two parts: the cash flow cor￾responding to the material originally scheduled to be mined and the cash flow generated by the pushback. Under the second mine plan, the net present value of the material that was initially scheduled to be mined is likely to decrease from NPVi1 to NPVi2. The pushback net present value is NPV'i . The net present value of the second mine plan is NPVi2 + NPV'i . The pushback should be mined only if NPV'i exceeds the difference NPVi1 – NPVi2: NPV ' > − NPV NPV i i1 i2 The difference NPVi1 – NPVi2 represents the opportunity cost of mining the pushback. The pushback should only be mined if its net present value NPV'i is not only positive but also exceeds the opportunity cost. The reason why there is an opportunity cost to adding a pushback to an existing mine Cut-off Grade and Mine Planning Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 88 CHAPTER SIX plan is because there are capacity constraints. Mine capacity, mill capacity, or volume of sales is constrained. These capacities must be shared between the material originally scheduled to be mined and the new material coming from the pushback. To release the capacities that are required to mine the push￾back, the following changes might be considered: • Reduce the rate at which the original material is being mined, and allocate the spare mining capacity to the pushback. • Increase the cut-off grade of direct mill feed to release mill capacity for material coming from the pushback. • Delay processing of lower-grade material by stockpiling it. These actions will decrease the net present value from NPVi1 to NPVi2. Alternatively, additional capital could be invested, including that needed to increase the mine or mill capacity. This capital cost reduces the net present value NPV'i of the pushback. The net present value NPV'i of the pushback is calculated from the value of each block included in the pushback and can be expressed as follows: NPV'i = – I + Σ Ujk / (1 + i)k I = capital cost needed to mine and process the pushback Ujk = utility of mining block j in year k i = discount rate Ujk = Ujk,dir + Ujk,opp + Ujk,oth where Ujk,dir = direct utility of mining block j in year k Ujk,opp = opportunity cost of mining block j in year k Ujk,oth = other utility of mining block j in year k If there is no capacity constraint, all blocks that will generate a positive cash flow when processed (Ujk,dir > 0) should be processed when mined. The cut-off grade is independent of when the block is mined. The pushback can be mined without changing when, or whether, other material is being mined outside the pushback. The net present value of the material that was originally scheduled to be mined remains unchanged, NPVi2 = NPVi1. The pushback’s mining schedule is only a function of the sequence to be followed to mine the pushback and is independent of any other mining activity that may be taking place. The decision to mine the pushback is only a function of whether or not its net present value NPV'i is positive. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Cut-off Grade and Mine Planning 89 However, optimization of mine and mill operations implies balancing capital and operating costs, which invariably results in capacity constraints and nonzero opportunity costs: Ujk,opp > 0. Nonzero opportunity costs result in time delays, higher cut-off grades, fewer blocks being processed, and, therefore, lower pushback net present value. A pushback for which NPV'i is positive if capacity constraints are ignored may have a negative NPV'i if these constraints are taken into account. Ignoring capacity constraints may result in mining pushbacks that should not be mined and designing a pit that is larger than it should be. There are two dependent sets of constraints to be taken into account when evaluating a pushback: constraints that apply if the pushback is mined as a stand-alone operation and constraints that result from having to share capacity with the material originally scheduled to be mined. Considering a pushback as a stand-alone operation is rarely, if ever, a reasonable assumption. It may be possible to dedicate mining equipment entirely to the pushback. This capital cost must be included in NPV'i and in cut-off grade optimization. Even under these circumstances, mine schedules and cut-off grades must take into account process capacity constraints. The plant is at least in part utilized by material previously scheduled to be mined, and taking this capacity con￾straint into account will reduce the value of the pushback. When more constraints are added to the pushback, its value NPV'i decreases. Releasing these constraints can be achieved by assigning capacity to the pushback that was previously utilized elsewhere. However, this will increase the constraints on the material originally scheduled to be mined and increase the opportunity cost NPVi1 – NPVi2. The optimal capacity assignment is that where further change will result in a decrease in NPVi2 that exceeds the increase in NPV'i . Finding this optimum can be time-consuming and may require use of specialized open pit mine-design software. U N D E R G R O U N D M I N E : E C O N O M I C VA L U AT I O N O F A STOPE Situations similar to those previously described in open pit mines are also encountered in underground mines. A stope should be mined if the net present value of generated cash flow is positive. All costs and benefits must be taken into account, as well as when these costs and benefits are realized. This includes the cost of stope development (such as access drifts and cross￾cuts); the cost of waste mining, stockpiling, and re-handling; the cost of ore mining, stockpiling, re-handling, and processing; and all costs allocated to Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 90 CHAPTER SIX low-grade stockpiles, if any. Revenues include those incurred from processing ore directly sent to the mill, as well as those realized at a later date from low￾grade stockpiles. If there is no capacity constraint, all material that can generate a positive cash flow, if processed when mined, will be processed. But project optimiza￾tion invariably results in capacity constraints, such as those imposed by shaft and drift haulage capacity, ventilation, maximum speed of development, or mining method. These constraints result in nonzero opportunity costs and higher cut-off grades. When capacity constraints are taken into account, the size of some stopes is likely to be reduced, and some stopes will no longer be considered economically minable. Shifting capacity from other parts of the mine can be used to release constraints on the stope and increase the stope net present value (NPV'i). For example, one may assign shaft capacity to material produced by the new stope by slowing down mining from other stopes. However, more constraints will result, which apply to material previously scheduled to be mined. If NPVi1 was the net present value of this material before the new stope was taken into account, and NPVi2 its net present value when the new constraints are imposed, NPVi2 is less than NPVi1. The difference NPVi1 –NPVi2 is the opportunity cost of mining the new stope. Adding the new stope to the pro￾duction schedule is justified only if NPV ' > − NPV NPV i i1 i2 When more constraints are added to a stope, its value NPV'i decreases. Releasing these constraints can be achieved by assigning capacity to the stope that was previously utilized elsewhere. However, this will increase the constraints on the material originally scheduled to be mined and increase the opportunity cost NPVi1 – NPVi2. The optimal capacity assignment is that where further change will result in a decrease in NPVi2 that exceeds the increase in NPV'i . Finding this optimum can be time-consuming and may require use of specialized underground mining software. S I M I L A R I T I E S B E T W E E N O P E N P I T A N D UNDERGROUND MINE PLANNING As shown in the previous discussions, there are many similarities between questions concerning open pit and underground mines, and the approach that must be followed to answer these questions. Here are some of these questions: Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Cut-off Grade and Mine Planning 91 • How do capacity constraints influence cut-off grade and cash flow? • Which cut-off grade should be used to separate waste material, stock￾piled material, and material sent to the processing plant? • Should a pushback be mined in an open pit mine or a stope be mined in an underground mine? • Should low-grade material at the bottom of a pushback or surround￾ing a stope be mined or left in the ground? • If low-grade material must be mined, should it be wasted, stockpiled, or processed? • How should the time difference between mining, stockpiling, pro￾cessing, and selling material be taken into account in designing open pit and underground mines? BLOCK AND PANEL CAVING When a block or panel caving mining method is used, estimation of cut-off grades must take into account the limited flexibility that operators have in controlling the grade of material pulled. Cut-off grades are used to determine the location and size of a block or panel, and to decide when pulling material from a drawpoint should be stopped. Cut-off grades are not likely to play a significant role, if any, when waste or low-grade material is encountered in the block. Constraints Imposed by Block and Panel Caving Many factors must be taken into account when designing a block in addition to the geotechnical properties of the deposit and the continuity of mineral￾ization. Ideally, blocks are located in relatively high-grade areas that can be mined without significant internal or external waste dilution, the drawpoints and production levels are located in lower-grade or waste areas, and the block boundaries are located near lower-grade or waste zones. Internal and external waste or low-grade dilution will occur, which must be taken into account when locating blocks and drawpoints. When ore is drawn, waste is mixed with higher-grade material, thus eliminating the opportunity to mine waste selectively. The rate at which material is pulled from drawpoints should match the natural rate of caving. The material should be drawn in a uniform fashion across drawpoints. Production cannot be stopped in one drawpoint with￾out affecting surrounding drawpoints. If a drawpoint containing waste is Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 92 CHAPTER SIX surrounded by other high-grade drawpoints, mining waste cannot be stopped. However, if the waste drawpoint is located on the periphery of the block being mined, this drawpoint can be stopped. Production is stopped when waste indicates that the entire ore column has been pulled. Productivity is dependent on a high rate of production, which is not conducive to selective mining of ore and waste material. The capital cost of underground and surface infrastructure needed to handle waste separately from ore-grade material is likely to be high. Attempting selective mining is likely to increase the mine operating costs. For these reasons, some block cav￾ing operations have chosen to send all material mined to the processing plant, whatever the grade. Marginal Cut-off Grade and Drawpoint Management Once a block has been developed and the infrastructure is in place (including drifts, haulage facilities, drawpoints, ventilation, etc.), the utility of mining and processing one metric ton of material is ( ) ( ) = ⋅ ⋅ − ( )− + + U x x r V R M P O dir o o o where x = average grade r = recovery, or proportion of salable product recovered from the mined material V = value of one unit of salable product R = refining, transportation, and other costs of sales that are expressed in dollars per unit of salable product Mo = mining cost per metric ton processed Po = processing cost per metric ton processed Oo = overhead cost per metric ton processed The minimum grade that can be mined and processed at a profit is xc1 such that Udir(xc1) = 0: [ ] [ ] ( ) = + + ⋅ − x M P O r V R c1 o o o This cut-off grade should be used to decide whether production from a draw￾point should be stopped because of excessive lateral dilution or because the entire ore column has been mined. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Cut-off Grade and Mine Planning 93 Marginal Cut-off Grade and Block Design Incremental analysis must be used to determine the optimal size and location of a block. To decide whether a new row of drawpoints should be added along the periphery of a block, one must first estimate the tonnage T and average grade x of the material that will be pulled from these drawpoints, taking dilu￾tion into account. If one considers only operating costs and ignores the capital and opportunity cost of adding one row of drawpoints, this row should be added if the average grade x exceeds the cut-off grade xc1 calculated previously. If the average grade of the last row of drawpoints is equal to xc1, the cash flow generated from these drawpoints will not justify the capital cost of developing them. In addition, development of a larger block by addition of peripheral drawpoints will delay production from what could have been a smaller block. The cut-off grade applicable to the last row of drawpoints must take into account capital and opportunity costs. Influence of Capital Cost and Discount Rate Additional capital expenditures are needed to develop one more row of drawpoints. This capital cost I must be recovered from profits generated by the drawpoints. On an undiscounted basis, the profit made from mining and processing T metric tons of material with average grade x is T ∙ [x ∙ r ∙ (V – R) – (Mo + Po + Oo )]. This profit must be greater than or equal to the capital cost I. The cut-off grade applicable to this last row of drawpoints is determined by adding the capital cost per metric ton I/T to the operating costs M, P, and O: [ ] [ ] ( ) = + + + ⋅ − x M P O I T r V R c2 o o o where I = capital cost incurred to develop a new row of drawpoints T = tonnage to be mined from the new row of drawpoints The requirement of a minimum rate of return should be taken into account in calculating the cut-off grade. The following additional notations are used: i = minimum rate of return (discount rate) n = number of years during which material will be pulled from the new drawpoints Then make the simplifying assumption that the tonnage mined and corre￾sponding average grade will be the same every year, T/n and x, respectively. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 94 CHAPTER SIX The yearly cash flow YCF expected to be generated from the new drawpoints is ( ) ( ) ( ) = ⋅ ⋅ ⋅ − − + +     YCF T n x r V R Mo o P Oo The net present value of this cash flow (NPV'i ) is calculated as follows*:  ()() () ( ) = ⋅ + ++ + + + +     = − NPV ' YCF 1 1 1 i 1 1 i 1 1 i YCF g i,n i 2 n 1 where g(i,n) = [1– (1 + i)–n] (1 + i)/i. The minimum cut-off grade applicable to the new row of drawpoints is xc3 such that the net present value of generated cash flows NPV'i is equal to the capital investment I: [ ] ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = ⋅ ⋅ ⋅ − − + +    ⋅ = = + + + ⋅     ⋅ − NPV ' I T n x r V R M P O g i,n I x M P O n g i,n I T r V R i c3 o o o c3 o o o Opportunity Cost In addition to increasing capital costs, increasing the size of a block can delay production that could be pulled from a smaller block. Assume that a small, presumably high-grade block has been designed and that a production sched￾ule has been developed accordingly. The net present value NPVi of future cash flows expected to be generated from mining this block was calculated using the discount rate i. If t is the time by which production from the smaller block will be delayed to allow development of one more row of drawpoints, the corresponding opportunity cost is ( )= − ⋅ ⋅ U x t i NPV opp i This opportunity cost represents a decrease in net present value, which must be added to the capital cost of adding the new drawpoints. Taking this cost into account, the cut-off grade is as follows: * The notation NPV'i is used to make the distinction between the net present value of previously scheduled operations, NPVi , and the net present value of the proposed additional row of drawpoints, NPV'i . The first year of production is not discounted. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Cut-off Grade and Mine Planning 95 [ ] ( ) ( ) ( ) = + + + ⋅ + ⋅ ⋅     ⋅ − x M P O n g i,n I t i NPV T r V R c4 o o o i where Mo = mining cost per metric ton processed Po = processing cost per metric ton processed Oo = overhead cost per metric ton processed g(i,n) = [1 – (1 + i)–n] (1 + i)/i n = number of years during which material will be pulled from the new row of drawpoints i = minimum rate of return (discount rate) I = capital cost incurred to develop the new row of drawpoints t = time by which previously scheduled production will be delayed NPVi = net present value of previously scheduled production T = tonnage to be mined from the new row of drawpoints r = recovery, or proportion of salable product recovered from the mined material V = value of one unit of salable product R = refining, transportation, and other costs of sales that are expressed in dollar per unit of salable product Example: Block Caving Copper Mine Operation Consider a copper mine that is exploited by the block caving method. The deposit is scheduled to be mined using macroblocks 150 m high. The draw￾points are located on a 20-m grid. The current operating conditions are as follows: • Direct mining cost: $9.00 per metric ton of ore • Processing cost: $7.00 per metric ton of ore • General and administrative cost: $1.60 per metric ton of ore • Freight, smelting, and refining: $0.50 per pound of payable copper The mill recovery is 87% and the smelter recovery is 96%, including trans￾portation losses. The long-term copper price is assumed to be $2.30 per pound of payable copper (1 t = 2,205 lb). Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 96 CHAPTER SIX Using the above notations, this can be written as follows: Mo = $9.00 Po = $7.00 Oo = $1.60 V = $2.30 R = $0.50 r = 0.87 ∙ 0.96 = 83.52% The marginal cut-off grade for an existing drawpoint is [ ] [ ] ( ) = + + ⋅ − ⋅ = x 9.00 7.00 1.60 83.52 2.30 0.50 2,205 0.53% Cu c1 The cost of developing a new drawpoint, including access and other development costs, is I = $240,000. The specific gravity of the ore is 2.5 t/m3 , and the average tonnage contained in one column is T = 20 ∙ 20 ∙ 150 ∙ 2.5 = 150,000 t. On an undiscounted per-metric-ton basis, the cost of drawpoint development is I T = = $240,000 150,000 $1.60 per metric ton The amount of copper needed to recover this cost is [ ] ( ) − = ⋅ − ⋅ = x x $1.60 83.52 2.30 0.50 2,205 0.05% Cu c2 c1 The cut-off grade to be used to pay for operating costs and undiscounted cap￾ital cost per metric ton is = + = x 0.53 0.05 0.58% Cu c2 The life of a macroblock is estimated at n = 10 years. An i = 15% rate of return is expected for all investments. The capital recovery needed per metric ton to obtain this rate of return is n/g(i,n) ∙ (I/T): ( ) ( ) ( ) ( )( ) ( ) = − +    ⋅ + = ⋅ = ⋅ = − g i,n 1 1 0.15 1 0.15 0.15 5.772 n g i,n I T 10 5.77 1.60 $2.77 per metric ton 10 The amount of copper needed to recover this cost is [ ] ( ) − = ⋅ − ⋅ = x x $2.77 83.52 2.30 0.50 2,205 0.08% Cu c3 c1 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Cut-off Grade and Mine Planning 97 The cut-off grade to be used to pay for operating costs and discounted capital cost per metric ton is =+= x 0.53 0.08 0.61% Cu c3 The macroblock under consideration is scheduled to go into production immediately. As designed, the macroblock is composed of 15 columns for a total tonnage of 15 ∙ 150,000 = 2,250,000 t. The average grade of this material is 2.00% Cu, and the total value of the ore contained is [ ] ( ) ( ) ⋅ ⋅ ⋅ − ⋅ − + + = 2,250,000 0.02 0.8352 2.30 0.5 2,205 9.00 7.00 1.60 $109,570,896 Assuming a uniform cash flow per year over ten years, the net present value of this cash flow is ( ) ( ) = ⋅ = ⋅ = NPV 109,570,896 10 g i,n 10,957,090 5,772 $63,269,762 i Consideration is being given to adding one more drawpoint to the mac￾roblock. The time needed to develop this drawpoint is estimated at 12 days. This will delay copper production from the macroblock by 12 days at an opportunity cost: ( ) ⋅ ⋅ = ⋅ ⋅ = t i NPV 12 365 0.015 63,269,762 $311,867 i This cost is incurred today but must be recovered over the life of the project. It must be added to the capital cost I. To ensure a 15% rate of return, the amount needed per year is ( ) ( ) ( )⋅ = ⋅ = n g i,n 311,867 T 10 5.77 2.08 $3.60 per metric ton The amount of copper needed to recover this cost is [ ] ( ) − = ⋅ − ⋅ = x x $3.60 83.52 2.30 0.50 2,205 0.11% Cu c4 c3 The cut-off grade to be used to pay for operating costs, discounted capital cost per metric ton, and discounted opportunity cost is =+= x 0.61 0.11 0.72% Cu c4 This cut-off grade is significantly higher than the marginal cut-off grade xc1 = 0.53% Cu. If a lower discount rate was acceptable, such as i = 6% instead of 15%, the cut-off grade would be reduced to xc4 = 0.64% Cu. To fully take Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 98 CHAPTER SIX into account all costs and benefits associated with adding one more drawpoint to an existing macroblock, one should also look at consequences other than those considered here. These may include the opportunity cost of delaying production from other parts of the operation, lowering the head grade, and increasing the mine life. RELATIONS HI P B E T W E E N M I N E S E L E C T I V I T Y, D E P O S I T MODELING, ORE CONTROL, AND CUT-OFF GRADE In the previous examples, it was assumed that the grade–tonnage relationship which characterizes the deposit is independent of the mining capacity. How￾ever, in many instances, changes in mining capacity are accompanied by changes in mining method, size of mining equipment, bench height, stope dimensions, drill hole spacing, ore control method, and other parameters that determine mine selectivity and the shape of the grade–tonnage curve. These changes must be taken into account in establishing the likely effect that changes in mining capacity and cut-off grade will have on mill feed and reserves. As an example, consider a gold deposit for which the total resources above a zero cut-off grade are calculated at 20 million metric tons averaging 10 g/t. The geology of the deposit is such that either open pit or underground mining methods can be used. Figures 6-1 and 6-2 both show the grade– tonnage relationships corresponding to the open pit and underground mining methods. The open pit cut-off grade was estimated at 3.0 g/t. The amount of mate￾rial that could be mined above this cut-off grade was 15.2 million metric tons, averaging 12.6 g/t and containing 6.1 million oz (solid lines in Figure 6-1). If the high-selectivity model had been used to evaluate the open pit option, the reserves would have been erroneously estimated at 11.1 million metric tons, averaging 16.9 g/t and containing 6.0 million oz (dotted lines in Figure 6-1). The underground cut-off grade was estimated at 7.0 g/t. The amount of material that could be mined above this cut-off grade was 6.5 million metric tons, averaging 25.6 g/t and containing 5.3 million oz (solid lines in Figure 6-2). If the low-selectivity model had been used to determine the feasibility of the underground mining method, the reserves would have been erroneously estimated at 8.9 million metric tons, averaging 18.2 g/t and con￾taining 5.2 million oz (dotted lines in Figure 6-2). The errors made when using the open pit model to evaluate the under￾ground resources or the underground model to evaluate the open pit resources are summarized in Table 6-1. Although this table represents an extreme case, it clearly shows that changes in both mining method and cut-off grade must be Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Cut-off Grade and Mine Planning 99 evaluated jointly. Appropriate deposit models must be used, which reflect the conditions that are expected to prevail when these changes are made. When assessing the impact that changes in mining capacity may have on mill head grades, one must take into account not only changes in cut-off grades but also changes in the grade–tonnage curve. The grade–tonnage curve will remain 4 12 10 8 6 20 14 16 18 5 30 25 20 15 10 35 Cut-off Grade 1 2 3 4 5 6 7 8 9 Tonnage 15.2 Tonnage 11.1 Cut-off 3.0 Average Grade 16.9 Average Grade 12.6 Figure 6-1 Application of open pit low-selectivity cut-off grade to low- and high-selectivity models (solid lines show the low-selectivity model; dotted lines show the high-selectivity model) 4 12 10 8 6 20 14 16 18 5 30 25 20 15 10 35 Cut-off Grade 1 2 3 4 5 6 7 8 9 Tonnage 8.9 Tonnage 6.5 Cut-off 7.0 Average Grade 25.6 Average Grade 18.2 Figure 6-2 Application of the underground high-selectivity cut-off grade to high￾and low-selectivity models (solid lines show the high-selectivity model; dotted lines show the low-selectivity model) Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade Tonnage Above Cut-off Grade Average Grade Above Cut-off Grade 100 CHAPTER SIX the same only if no change is made to mining method, ore control practices, and size of mining equipment. A computer-generated deposit model is the foundation on which mine plans are developed, cut-off grades are optimized, and the tonnage and average grade of material processed are determined. For the results of a feasibility study to be meaningful, the deposit model must reflect the geological properties of the deposit. In addition, the relationship between cut-off grade, tonnage, and average grade above cut-off grade, which is implied by the deposit model, must be the same as that which will be realized when the deposit is mined. The deposit model must be developed taking into account the mining method that will be used and how selective this method will be. Different models are usually needed for open pit and underground mines, for bulk mining and selective mining, for block caving and cut-and-fill. Selectivity is a function not only of the geology of the deposit and the mining method but also of bench height and blast hole spacing, stope design, type and size of mining equipment, and ore control method. The significance of these factors must be assessed when developing the deposit model. It is not sufficient to make realistic selectivity assumptions when devel￾oping the deposit model and optimizing the cut-off grade. These assumptions must be respected when the deposit is being mined. Otherwise, the tonnage and average grade of material mined and processed will differ from that esti￾mated when the project feasibility study was completed. In practice, changes will occur during the life of the mine, which will change selectivity. Such changes may include changing mining method, using smaller or higher bench heights, designing larger or smaller stopes, changing the equipment size, and modifying ore control practices. Whenever such changes are made, one must question whether they will change the grade–tonnage curve sufficiently to require development of a new deposit model. Table 6-1 Influence of deposit model and cut-off grade on estimated mineral resources Deposit Model Open Pit Mine Cut-off, 3.0 g/t Underground Mine Cut-off, 7.0 g/t Metric Tons, millions Grade, g/t Ounces, millions Metric Tons, millions Grade, g/t Ounces, millions High selectivity 11.1 16.9 6.0 6.5 25.6 5.3 Low selectivity 15.2 12.6 6.1 8.9 18.2 5.2 Correct model 15.2 12.6 6.1 6.5 25.6 5.3 Error if correct model not used –27% 34% –2% 36% –29% –3% Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 101 7 Mining engineers face a significant challenge when determining which costs should be included in a cut-off grade calculation. Collaboration between engineers and accountants is necessary to ensure that meaningful numbers are used and that all applicable costs are included. In this chapter, some general principles concerning costs and how they should be treated in the estimation of cut-off grades are discussed, which are illustrated by specific examples. GENERAL CONSIDERATIONS Costs can be divided between fixed and variable. Fixed costs are expenses for which the total does not change in proportion to the level of activity within the relevant time period or scale of production. By contrast, variable costs change in relation to the level of activity. In cut-off grade calculations, costs incurred when drilling, sampling, blasting, loading, crushing, and grinding the ore as well as during flotation, concentrate drying, filtering and shipping, smelting and refining, and so forth, are usually considered variable costs. These costs are directly related to the production capacity. Initial capital expenditures, equipment depreciation, general administration, property taxes, marketing, public relations, government relations, and so on, are considered fixed costs. To the extent that fixed and variable costs are properly defined, cut-off grade optimization requirements only take variable costs into consideration. It is important to realize that fixed costs are fixed only within a certain range of activity or over a certain period of time. If significant changes are made to the cut-off grade that require expansion of a leach pad or tailings dam, costs related to such expansions can no longer be considered fixed. If the life of mine is extended or shortened beyond the current expected life, general and administrative (G&A) costs will change. These changes should be taken into account in the cut-off grade calculation by allocating their cost to that part of the operation (mine, mill, leach plant, concentrator, smelter, refinery, etc.) that drives the change. Which Costs Should Be Included in Cut-off Grade Calculations? Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 102 CHAPTER SEVEN Sunk costs are expenses that were incurred in the past and do not change with the level of activity. Once a mine is in full production, the costs incurred for pre-stripping, shaft sinking, plant construction, and original infrastructure are sunk. Such costs are not taken into account when deciding whether the cut-off grade should be changed. However, during a project feasibility study, all costs, including the initial capital cost, have an influence on the cut-off grade. The cut-off grade determines the tonnage, grade, and location of mate￾rial available for processing, which in turn drive mine and plant size, capital and operating costs, and financial performance. Conversely, operating costs are a critical input in the determination of the minimum cut-off grade. Dif￾ferent cut-off grade profiles, including cut-off grades that decrease over time, will require different mine plans and capital costs, and will result in better or worse financial performance. An iterative process must be used to determine the combination of cut-off grades, size of operation, and resulting capital and operating costs that will best satisfy the company’s objectives. Balancing initial and sustaining capital costs, operating costs, and cut￾off grades is a critical part of a project feasibility study. If all assumptions made during the feasibility study, including those related to the geology of the deposit, the production capacity, the cost of operations, and the value of the product sold, remained true during the entire life of the mine, the cut-off grades would remain as planned. No cut-off grade change could be justified since plans were optimized and changes would reduce the value of the proj￾ect. Decreasing the cut-off grade would require that additional lower-grade material be processed, which could not be achieved without either increasing the size of the plant or decreasing the net present value of future cash flows. Conversely, increasing the cut-off grade above that planned would result in underutilization of available capacity. In practice, operating conditions differ from those assumed during the feasibility study, the geological properties of the deposit differ from those initially expected, capacities are either not reached or exceeded, mine and mill are no longer balanced, costs and the value of products sold are better or worse than expected, and cut-off grades must be continuously reestimated. A company’s financial objectives are likely to include expectation of a minimum return on investment, which cut-off grade calculations must take into account. If the time needed to mine one metric ton of material, process it, recover a salable product, and get a return from the sale of this product exceeds one year, costs and revenues should be discounted at the company￾specified rate. While sunk costs do not influence cut-off grades, the cost of future sustaining capital expenditures must be included in the cut-off grade Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Which Costs Should Be Included in Cut-off Grade Calculations? 103 calculation to ensure that all material processed covers the capital invested, including a specified minimum return on investment. HO W S P E C I F I C C O S T S I N F L U E N C E C U T- O F F G R A D E CALCULATIONS A few examples follow that illustrate how specific costs influence the cut-off grade. It is assumed that the mining company expects a minimum 15% return (i = 15%) on all investments: • Stockpiling of low-grade material. This was discussed previously. The decision to stockpile material is more often than not a strategic deci￾sion rather than a decision based solely on expected cash flows and net present value. • Leaching operation. In leaching operations, capital and operating costs, metal recovery, and metal sales occur over several years, and the cut-off grade might have to be determined using discounted cash flows. Consider a gold leaching operation with a five-year leach cycle. The total gold recovery is expected to reach 60%. Recovery and cost per year are given in Table 7-1. The waste-leach cut-off grade is to be estimated using a gold price V = $1,150 per ounce, a cost of sales of $50 per ounce, and a discount rate of 15%. The expected yearly reve￾nue per ounce of gold contained in the material being leached is r (V – R), where r varies from year to year. This revenue is shown in Table 7-1, column 4. The last three columns in Table 7-1 show the discount factor 1/(1 + i), the discounted cost per metric ton, and the dis￾counted revenue per ounce contained. The cut-off grade is calculated as the ratio of total costs over total revenues per ounce. If cash flows are not discounted, this cut-off grade is 0.47 g/t. To obtain a 15% return on investment the cut-off grade must be increased to 0.52 g/t. • Sustaining capital. Sustaining capital is capital expenditures that must be incurred on a periodic basis to maintain production at the current level. For example, new trucks may have to be bought every eight years, leach pad expansions may be needed every four years, tails dam lifts may be added every seven years. Let I be the total cost of this investment and n its expected useful life in years. The cut-off grade should be high enough to ensure a minimum return on invest￾ment (i = 15%). This is achieved by including the cost of capital in the cut-off grade calculation. Let CI be the cost per year that must be recognized to recover the investment I over n years at the specified Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 104 CHAPTER SEVEN discount rate i. The net present value of this constant cost is CI g(i,n), where g(i,n) = [1 – (1 + i)–n] ∙ (1 + i)/i. This cost must satisfy the following equation: ( ) = C g i,n I I Therefore, the cost per year that should be included in the cut-off grade calculation is = ( ) C I g i,n I If i = 15% and n = 8, the cost of capital is CI = 0.19 I per year. If no minimum return on investment was specified, the cost of capital would be CI = I/n = 0.13 I per year. In the cut-off grade calculation, costs per year must be converted to costs per unit of production. These costs must be added to mining costs if the sustaining capital is for mining equipment, to leaching costs if it is for leach pad expan￾sion, and to milling costs if it is for a tailings dam. Incremental capital expenditures, which are optional and must be justified by a feasibility study, should not be considered as sustaining capital. • Incremental capital expenditures. Such expenditures may be required to maintain production beyond the planned life or to reach a higher level of production. The cost of these incremental capital expendi￾tures must be taken into account in deciding whether the life should be extended or production should be increased. Different cut-off grade strategies should be considered. Once the decision is made to Table 7-1 Calculation of cut-off grade in a gold leach operation Year Recovery Undiscounted Discounted Cost per Metric Ton Revenue per Ounce Contained Discount Factor Cost per Metric Ton Revenue per Ounce Contained 0 15% $6.00 $165 1.00 $6.00 $165 1 15% $1.00 $165 0.87 $0.87 $143 2 12% $1.00 $132 0.76 $0.76 $100 3 10% $1.00 $110 0.66 $0.66 $72 4 8% $1.00 $88 0.57 $0.57 $50 Total 60% $10.00 $660 — $8.85 $531 Cut-off grade oz/t — 0.0152 — — 0.0167 g/t — 0.47 — — 0.52 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Which Costs Should Be Included in Cut-off Grade Calculations? 105 make the investment, incremental capital expenditures are sunk, and only remaining capacity constraints should influence cut-off grade decisions. • Closure and reclamation costs. At the end of a mine life, one-time clo￾sure costs are incurred. These costs, CR, decrease the net present value of a project’s future cash flow and therefore the opportunity cost i∙t∙NPVi . At the beginning of the mine life, the influence of closure costs on net present value is small, but it increases when the project matures. It was shown previously that if closure can be postponed by one year by mining low-grade material, the corresponding cash flow C could be negative. The only constraint is that C > –CR i/(1 + i). • Cost to perpetuity. In some circumstances (e.g., if a water treatment plant must operate forever after closing a mining operation), a project generates a constant negative cash flow to perpetuity. It was shown previously that if C is a yearly cash flow to perpetuity, the net present value of this cash flow is NPVi = C(1 + i)/i. This net present value is constant over time and can be considered as a fixed cost that does not influence the cut-off grade. However, this cost decreases the net present value of the project and should be taken into consider￾ation when completing a feasibility study. • Overhead costs. G&A and other overhead costs must also be divided between fixed and variable costs. Variable G&A costs must be included in all cut-off grade calculations. Fixed G&A costs, usually expressed on a per-year basis, must be included if the change in cut￾off grade will change the mine life. This will be the case whenever one of the processes is capacity constrained. The fixed part of overhead costs can no longer be considered as fixed because lowering the cut￾off grade will require extending the mine life. These costs must be expressed on a per-unit-of-production basis (by dividing costs per year by production per year) and added to the unit cost of the capacity￾constrained process. O T HER COSTS AND BENEFITS Cut-off grades play a critical role in defining tonnages mined and processed, average grade of mill feed, life of mine, cash flows, and all major characteristics of a mining operation. In addition to the economically quantifiable financial impact that cut-off grade changes may have, other costs and benefits must be taken into account even though their impact is not easily quantifiable. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 106 CHAPTER SEVEN Consideration must be given not only to changes in cash flow and net present value—as measured by Udir(x) and Uopp(x)—but also to all other impacts, Uoth(x), including those of an environmental, socioeconomic, ethical, or polit￾ical nature. Costs and benefits to all stakeholders must be evaluated. For most mining operations, the following stakeholders must be taken into account: • Shareholders, who supply the capital needed for the operation and expect a return on their investment • Banks, who contribute to the supply of financial resources the min￾ing company needs to operate or expand • Analysts, who advise the investing community • Employees and their families • Users of the final product sold by the mining operation, whether it is coal, gold, copper concentrate, iron ore, processed metal, or indus￾trial minerals • Suppliers, from whom the mining operation purchases equipment, energy, consumables, supplies, services, or expertise • Local communities, including neighbors of the mining operation • The local, regional, federal, or country governments, who are respon￾sible for the welfare of their citizens and benefit from the taxes levied from the mining company. These governments must plan for new infrastructure, roads, health, education, and entertainment; increases in traffic, crime, and prostitution; and higher demand for water, food, and housing. They also have a fiduciary duty to ensure appro￾priate exploitation of national resources. • Future generations, which will live with the long-term impact, good or bad, of the mining operation • Nongovernmental organizations whose mission, self-appointed or otherwise, is to defend the interests of some of the above stakeholders Senior management decides how to balance the needs, interests, and requirements of the different stakeholders. Those in charge of mine planning must be given practical guidelines for cut-off grade determination to ensure that the projects are designed to reach the company’s objectives. Maximizing shareholder value (including minimizing shareholder liability) is often quoted as a company’s primary objective. However, a company’s objectives must include recognition of responsibilities toward all stakeholders, not only the shareholders. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Which Costs Should Be Included in Cut-off Grade Calculations? 107 Higher cut-off grades may increase short-term profitability and enhance return to shareholders and other financial stakeholders. Higher cut-off grades may shorten the payback period, thus reducing political risk of creeping or outright nationalization. But reduced mine life reduces time-dependent opportunities, such as those offered by price cycles. Conversely, lower cut-off grades may increase project life with longer economic benefit to all stakehold￾ers, including shareholders, employees, local communities, and government. Longer mine life may result in more stable employment, less socioeconomic disruption to local communities, and more stable tax revenues to government. Lower cut-off grades imply fuller consumption of mineral resources, which may present political advantages or may be required by law. All stakeholders may have to choose between higher financial returns over shorter time periods or lower returns over longer time periods. Using high but decreasing cut-off grades early in the mine life and stockpiling low-grade material for later pro￾cessing can help balance financial returns and mine life. One method of optimizing cut-off grades while taking into account unquantifiable costs and benefits consists of evaluating the project under a variety of constraints imposed on discount rate, mine or mill capacity, volume of sales, capital or operating costs, and so forth. Changes in the opportunity cost of imposing these constraints, Uopp(x), are compared with the corre￾sponding changes in other costs, Uoth(x). The optimal cut-off grade is that for which the marginal (and quantifiable) increase in opportunity cost is equal to the corresponding marginal (but subjective) decrease in other costs. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 109 8 In an increasingly complex environment, blending strategies may be critical in maximizing the profitability of an operation or set of operations. For example, a metallurgical process may require that the chemical properties of the mill feed fall within specific ranges. Sales contracts may impose limits on the quality of the product being sold and the contaminants it contains. Envi￾ronmental regulations may impose constraints on the mill feed, as needed for environmental emission control. In a gold mine where refractory ore is treated by roasting, there might be different parts of the deposit with different geochemical properties. Material with high sulfide content and low gold content may have no economic value if considered on its own. Conversely, processing material with high gold content and low sulfur content may be problematic. However, adding the calorific value of the high-sulfide material to the gold value of the high-gold material may result in a highly profitable operation. In a coal mining operation, one seam may have a low calorific value and high ash content, and another seam a high calorific value and low ash content. By appropriately mixing the two products, it may be possible to increase the tonnage of salable coal that satisfies contractual agreements on both sulfur grade and calorific value. Similarly, the product from one iron mine may have no economic value if considered on its own, but a salable product may be obtained by blending it with iron ore from another mine that exceeds con￾tractual requirements. Blending strategies applicable in relatively simple situations, such as when there are only two or three stockpiles, are discussed in this chapter. The objec￾tive is to give the reader a high-level understanding of the strategic aspect of stockpiling and blending. Although practitioners are likely to encounter more complex situations, the solution to such problems falls outside the scope of this book. Blending Strategy Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 110 CHAPTER EIGHT BLENDING TWO MATERIAL TYPES Consider a gold roasting operation that has access to two sources of con￾centrate, a 20,000 t high-sulfide, low-gold stockpile (concentrate 1) and a 40,000  t low-sulfide, high-grade stockpile (concentrate 2). The concentrate properties are summarized in Table 8-1. Table 8-2 shows that 82% of the gold is in concentrate 2. The following notations are used: Ti = tonnage of concentrate i, i = 1,2 x1i = sulfur grade of concentrate i, i = 1,2 x2i = gold grade of concentrate i, i = 1,2 To = tonnage of roaster feed x1o = sulfur grade of roaster feed x2o = gold grade of roaster feed pi = proportion of concentrate i in roaster feed, i = 1,2 t i = tonnage of concentrate i in roaster feed, i = 1,2 Blending to Maximize Gold Grade in Roaster Feed To satisfy fuel requirements, the roaster needs feed averaging at least 1.5% sulfur. An objective could be to maximize the blended gold grade while sat￾isfying the sulfur requirements. Figure 8-1, called a blending diagram, shows the relationship between sulfur grade and gold grade for the two concentrates and all possible concentrate blends. Potential roaster feeds are represented by Table 8-1 Properties of concentrates available for blending Material Source Index i Tonnage, Kt Ti Grade Metal Content Sulfur, % x1i Gold, g/t x2i Sulfur, Kt Ti x1i Gold, kg T2 x2i Concentrate 1 1 20 3.50 40 0.70 800 Concentrate 2 2 40 1.20 90 0.48 3,600 Total — 60 1.97 73.3 1.18 4,400 Table 8-2 Percentage of available tonnage and metal content in each concentrate Material Source Index i Tonnage, % Metal Content Sulfur, % Gold, % Concentrate 1 1 33 59 18 Concentrate 2 2 67 41 82 Total — 100 100 100 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Blending Strategy 111 a point on the diagram: concentrate 1 is at point A, and concentrate 2 is at point B. The only feasible grades that can be obtained by blending concen￾trates 1 and 2 are those along the line AB. If only blends that contain more than 1.5% sulfur are acceptable, the blend that maximizes the gold grade is that which exactly satisfies the sulfur constraint; it is represented by point C on the blending diagram. If p1 is the percentage of concentrate 1 in the blend, and p2 the percent￾age of concentrate 2, the optimal blend is obtained by solving the following equations: + = + = p x p x 1.5% p p 1 1 11 2 12 1 2 The concentrates should be fed to the roaster in the following proportions: ( ) ( ) =− −= = − = p 1.5% 1.2% 3.5% 1.2% 13% p 1 13% 87% 1 2 Note that p1 and p2 are equal to the ratio of the line segments on the blending diagram: p1 = CB/AB and p2 = AC/AB. After blending, the gold and sulfur grades of the concentrate delivered to the roaster are = + = = + = x p x p x 1.5% sulfur x p x p x 83.5 g/t 10 1 11 2 21 20 1 21 2 22 20 30 40 50 60 70 80 90 100 Sulfur Grade, % 0 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 B A C Concentrate 2 Selected Blend Feasible Blends Concentrate 1 Not Acceptable Acceptable Figure 8-1 Feasible blends for two concentrate stockpiles Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Gold Grade, g/t 112 CHAPTER EIGHT The total tonnage of material sent to the roaster is To , composed of t1 metric tons of concentrate 1 and t2 metric tons of concentrate 2, which must satisfy the following equations: = + = = T t t t T p t T p o 1 2 1 o 1 2 o 2 Neither t1 nor t2 can exceed the total tonnages, T1 and T2 , available in each stockpile. Therefore, the maximum value of To is the smallest value of Ti /pi . ( ) = = = = = = = ⋅ = = ⋅ = T p 20 0.13 154 thousand metrictons T p 40 0.87 46 thousand metrictons T minimum 154, 46 46 thousand metric tons t 13% 46 6 thousand metric tons t 87% 46 40 thousand metrictons 1 1 2 2 o 1 2 The gold content of the roaster feed is + = ⋅ + ⋅ = t x t x 6 40 40 90 3,840 kg, or 123 thousand ounces of gold 1 11 2 21 Blending to Maximize Gold Content in Roaster Feed In the previous example, the objective was to maximize the gold grade in the roaster feed while maintaining a sulfur grade not less than 1.5%. This resulted in 13% of stockpile 1 and 87% of stockpile 2 containing 3,840 kg of gold being sent to the roaster. This example considers the economically more meaningful objective of maximizing the amount of gold sent to the roaster while still satis￾fying the requirement that the sulfur content be not less than 1.5%. The gold and sulfur contents vary with the proportion of each stockpile sent to the roaster. Using the following formula, one can calculate the maxi￾mum tonnage To of material sent to the roaster for different proportions p1 of concentrate 1: = ( ) T minimum T p ,T p o 1 1 2 2 where p1 and p2 must be positive and their sum must be such that p1 + p2 = 1. Once the tonnage To has been calculated for a given p1 , it is possible to deduce the tonnages t1 and t2 of each stockpile sent to the roaster: Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Blending Strategy 113 = = t T p t T p 1 o 1 2 o 2 Total tonnages and tonnages originating from each stockpile are plotted in Figure 8-2. The sulfur and gold average grades of the blended material are obtained using the following equations (Figure 8-3): ( ) ( ) = + − = + − x p x 1 p x x p x 1 p x 10 1 11 1 21 20 1 21 1 22 The corresponding sulfur and gold contents, To x10 and To x2o, respectively, are shown in Figure 8-4. The blend that maximizes the gold content is com￾posed of 33% concentrate 1 and 67% concentrate 2. This results in 60,000 t of material being sent to the roaster, averaging 1.97% sulfur and 73.3 g/t gold and containing 4,400 kg of gold. Figure 8-4 shows that when the proportion of concentrate 1 decreases below 33%, the decrease in gold content is slow while the decrease in sulfur content is very sharp. These trends are reversed above 33%. An understanding of these relationships is fundamental if opti￾mal blends are to be achieved under different cost, recovery, geochemical, or operational conditions. 0 10 20 30 40 50 60 70 Percentage of Concentrate 1 in Blend, % 0 10 20 30 40 50 60 70 80 90 100 Blend with Maximum Tonnage Total Metric Tons Concentrate 1 Concentrate 2 Figure 8-2 Feasible tonnage of blended material, as a function of the percentage p1 of stockpile 1 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Metric Tons, thousands 114 CHAPTER EIGHT BLENDING T HREE MATERIAL TYPES In the previous example, the case was considered where only two stockpiles were available for blending. Now consider three stockpiles. Each stockpile is characterized by its tonnage and two grades, as shown on Table 8-3. If all stockpiles were blended to get a single product, the result would be seven 0.0 0.6 0.4 0.2 1.4 0.8 1.0 1.2 0 500 1,000 1,500 2,000 2,500 3,000 3,500 4,000 4,500 5,000 Percentage of Concentrate 1 in Blend, % 0 10 20 30 40 50 60 70 80 90 100 Blend with Maximum Tonnage Sulfur Content Gold Content Figure 8-4 Metal content of blended material, as a function of the percentage p1 of stockpile 1 0.5 2.0 1.5 1.0 4.0 2.5 3.0 3.5 30 90 80 70 60 40 50 100 Percentage of Concentrate 1 in Blend, % 0 10 20 30 40 50 60 70 80 90 100 Blend with Maximum Tonnage Sulfur Grade Gold Grade Figure 8-3 Average grade of blended material, as a function of the percentage p1 of stockpile 1 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Sulfur Content, Kt Gold Content, kg Sulfur Grade, % Gold Grade, g/t Blending Strategy 115 thousand metric tons of material with grade 1 equal to 1.03 and grade 2 equal to 12.3. To get this result, the stockpiles would have to be blended in the fol￾lowing proportions (Table 8-4): 29% from stockpile 1, 14% from stockpile 2, and 57% from stockpile 3. The blending diagram in Figure 8-5 shows the relationship between grade 1 and grade 2 for the three stockpiles. The three points, A, B, and C, represent the stockpiles. The only grade combinations that can be obtained by blending the three stockpiles are those located within the boundaries of the triangle ABC, defined as the feasible blends domain. Blending to Reach Specific Grades Consider the case where a blend needs to be obtained with specific grades: x10 for grade 1 and x20 for grade 2. To obtain these grades, the stockpiles much be blended in the following proportions, p1 , p2 , and p3 , such that + + = + + = + + = p x p x p x x p x p x p x x p pp1 1 11 2 12 3 13 10 1 21 2 22 3 23 20 1 2 3 In addition, p1 , p2 , and p3 must be positive, a condition that will be satisfied if the point with coordinates x10, x20 falls within the ABC triangle in Figure 8-5. The solution to these equations is easily calculated: Table 8-3 Stockpiles available for blending Stockpile Index i Stockpile Tonnage, Ti Stockpile Grade 1, x1i Stockpile Grade 2, x2i Contained Metal 1, Ti x1i Contained Metal 2, Ti x2i 1 1 2,000 1.00 7.0 2,000 14,000 2 2 1,000 2.00 12.0 2,000 12,000 3 3 4,000 0.80 15.0 3,200 60,000 Total 7,000 1.03 12.3 7,200 86,000 Table 8-4 Percentage of available tonnage and metal content in each stockpile Stockpile Index i Tonnage, % Metal 1, % Metal 2, % 1 1 29 28 16 2 2 14 28 14 3 3 57 44 70 Total 100 100 100 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 116 CHAPTER EIGHT ( )( )( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) ( )( ) = = − − − − − = − − − − − = = − − − − − = − − − − − = − − p A B A x x x x x x x x B x x x x x x x x p A B A x x x x x x x x B x x x x x x x x p 1 p p 1 1 1 1 10 13 22 23 20 23 12 13 1 11 13 22 23 21 23 12 13 2 2 2 2 10 13 21 23 20 23 11 13 2 12 13 21 23 22 23 11 13 3 1 2 For example, if x10 = 1.20 and x20 = 11.0, then p1 = 40.0%, p2 = 26.7%, and p3 = 33.3%. Knowing these proportions, one needs to calculate the maximum tonnage of blended material that can be produced and the tonnage of each stockpile that will be used (and therefore the tonnage that will remain in stockpiles) to reach this blended tonnage. The maximum tonnage of blended material is T0 made of t1 metric tons of stockpile 1, t2 metric tons of stockpile 2, and t3 metric tons of stockpile 3: = + + T t 0 1 t t 2 3 6 7 8 9 10 11 12 13 14 15 16 Grade 1 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 C A B Stockpile 3: Tonnage = 4,000, Grade 1 = 0.8, Grade 2 = 15 Stockpile 1: Tonnage = 2,000, Grade 1 = 1, Grade 2 = 7 Total: Tonnage = 7,000, Grade 1 = 1.03, Grade 2 = 12.3 Stockpile 2: Tonnage = 1,000, Grade 1 = 2, Grade 2 = 12 Figure 8-5 Feasible blends domain for three stockpiles Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Grade 2 Blending Strategy 117 The tonnage ti that can be taken from stockpile i cannot exceed the total tonnage Ti of material in stockpile i. Using the previous example, the maxi￾mum value of T0 is the smallest of = = = = = = T p 2,000 40.0% 5,000 t T p 1,000 26.7% 3,750 t T p 4,000 33.3% 12,000 t 1 1 2 2 3 3 Therefore, T0 = 3,750 t. The material consumed and that remaining in stock￾piles is summarized in Table 8-5. Stockpile 2 was entirely consumed, but 3,250 t remain in stockpile, of which 500 t are in stockpile 1 and 2,750 t in stockpile 3. Using Linear Algebra to Determine Stockpile Blends The following equations were used to determine how three stockpiles should be blended to obtain specific average grades: + + = + + = + + = p x p x p x x p x p x p x x p pp1 1 11 2 12 3 13 10 1 21 2 22 3 23 20 1 2 3 These equations can be written using matrix notations: Table 8-5 Material blended and processed, and material remaining in stockpile Material Type Index i Tonnage, t Ti Percentage, pi Grade 1, x1i Grade 2, x2i Material blended and processed Stockpile 1 1 1,500 40.00% 1.0 7.0 Stockpile 2 2 1,000 26.67% 2.0 12.0 Stockpile 3 3 1,250 33.33% 0.8 15.0 Total 0 3,750 100.00% 1.2 11.0 Material remaining in stockpiles Stockpile 1 1 500 — — — Stockpile 2 2 0 3.10% 2.0 12.0 Stockpile 3 3 2,750 96.90% 0.8 15.0 Total 0 3,250 100.00% 0.8 14.9 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 118 CHAPTER EIGHT                     =           x x x x x x 1 1 1 p p p x x 1 11 12 13 21 22 23 1 2 3 10 20 If X is the stockpile grade matrix, p the proportion vector, and x0 the target grade vector, the proportions p are calculated as follows: = = −− Xp x p X x 0 1 0 The vector p is calculated by multiplying the inverse of the matrix X with the vector x0 . Using the previous numerical example*:           =                 = − − − − −                 =         p − p p 1.0 2.0 0.8 7 1 1 1 2 1 1 5 1.2 11 1 0.333 0.133 2.267 0.889 0.022 1.044 0.556 0.111 0.222 1.2 11 1 40.0% 26.7% 33.3% 1 2 3 1 These equations do not guarantee that the proportions pi are all between 0 and 1. Indeed if, in the blending diagram, the point defined by the vector x0 is outside the ABC feasible area, proportions greater than 1 and less than 0 will be calculated. BLENDING TO MAXIMIZE TONNAGE There are circumstances where the objective is to maximize the tonnage of blended material subject to specified grade constraints. The tonnages that can be obtained by blending are those that correspond to points falling within the * The matrix inversion was performed using the Microsoft Excel function MINVERSE. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Blending Strategy 119 ABC feasible domain shown in Figure 8-5. To understand the relationship between grade constraints and tonnages, one must understand how tonnages vary within the feasible domain. Using the method previously described, the tonnage To can be calculated for all points within the feasible domain. The results are shown in Figure 8-6. One obtains a two-dimensional grade– tonnage surface, akin to the more familiar one-dimensional grade–tonnage curve. The constant-tonnage curves, defined as Iso-T lines, are composed of line segments that are parallel to the sides of the ABC triangle. This property of the Iso-T lines is explained in Example 7 in Appendix A. Blending to Maximize Metal Content There are circumstances, such as the one presented in the two-stockpiles example, where the objective is to maximize the metal content of the blended material, subject to specific grade constraints. For example, the objective might be to maximize the gold content, subject to the sulfur grade being within specified limits. Optimizing metal content requires understanding how this content varies within the feasible domain of blended material. Using the same example as previously mentioned, the metal content Q20 = To x20 was calculated for all points within the feasible domain, and the results were contoured as shown in Figure 8-7. A surface is obtained that represents the relationship between gold content and both sulfur and gold grades. This 6 7 8 9 10 11 12 13 14 15 16 Grade 1 of Blended Material 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 4,000 2,000 7,000 1,000 Figure 8-6 Grade–tonnage relationship and constant tonnage (Iso-T) lines of blended material Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 5,000 6,000 Grade 2 of Blended Material 2,000 3,000 2,500 1,500 2,500 3,000 4,000 5,000 120 CHAPTER EIGHT relationship can be represented by contour lines, defined as Iso-Q lines, along which the gold quantity is constant. These lines are composed of segments that are no longer parallel to the sides of the ABC triangle. Blending to Satisfy Ratio Constraints Some circumstances occur where the constraints are on variables, such as ratios, that cannot be averaged between stockpiles. For example, in a nickel laterite operation, there might be a requirement on the silica-to-magnesia ratio. The ratio of a blended product is not equal to the weighted average of the individual stockpile ratios. Using the same example as above, one can cal￾culate the ratio between the two grades (grade 2 divided by grade 1) and plot it as shown in Figure 8-8. The optimal blend might be that which maximizes one of the variables while maintaining the ratio between two other variables within specified boundaries. REMARKS CONCERNING AN INCREASINGLY COMPLEX BLENDING PROBLEM The methods described previously can be used to optimize blending of any number of stockpiles when the difference between the number of stockpiles and the number of grade or quality constraints is exactly equal to 1. It was 6 7 8 9 10 11 12 13 14 15 16 Grade 1 of Blended Material 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 60,000 86,000 14,000 12,000 Figure 8-7 Grade–metal content relationship and constant metal content (Iso-Q) lines of blended material (metal 1) Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 70,000 15,000 20,000 30,000 Grade 2 of Blended Material 60,000 50,000 40,000 30,000 20,000 Blending Strategy 121 also assumed that the objective function was to maximize tonnage or metal content, subject to specified constraints on grade or quality. When these con￾ditions are not satisfied, defining an optimal blending strategy can become increasingly complex, and mathematical programming methodologies have to be considered, which fall outside the scope of this book (see bibliography). In the previous examples, it was assumed that the stockpiles already existed. A similar situation would be that where consideration is given to blending production from mines with fixed yearly production, such that ton￾nage, grade, and quality of material available for blending are known. In such situations, the feasible domain is fully defined by the tonnages and grades/ qualities available from the stockpiles or from the different mines. When considering new mines that are not yet operating, a stockpiling strategy needs to be developed to optimize the feasibility of the project. Depending on the geological properties of the deposit, a number of stockpiling options are likely to be feasible, each stockpile being defined by both upper and lower cut-off grades. Depending on deposit geology, mine schedule, and cut-off grades, there is likely to be a very large number of feasible stockpile options. On the blending diagram, the feasible domain defined by the stockpiles must contain the target point where the requirements imposed on the blended product are satisfied. In addition, the stockpiles should be such that when blended in 6 7 8 9 10 11 12 13 14 15 16 Grade 1 of Blended Material 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 y/x = 18.75 y/x = 7 y/x = 6 Figure 8-8 Grade–metal ratio relationship and constant ratio (Iso-R) lines of blended material Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. y/x = 8 y/x = 10 y/x = 12 y/x = 14 Grade 2 of Blended Material 122 CHAPTER EIGHT the required proportions, they are consumed in their entirety. Solving this problem, known as the inverse blending problem, can be extremely complex. A number of mine planning optimization tools have been developed to assist engineers in developing stockpiling and corresponding cut-off grade strate￾gies that maximize specified objective functions during the life of a mining operation. The previous examples also assume a deterministic environment, where the stockpile properties are known and constant throughout the life of the stockpile. In practice, there is always uncertainty concerning stockpile prop￾erties. A blending solution that is optimal under deterministic assumptions is not likely to be acceptable in the presence of uncertainty. A small variation in stockpile properties may result in a blended product that does not satisfy specified requirements. In such situations, the multivariate distributions of the properties of the different sources of material must be considered to develop a stockpiling and blending strategy that is flexible and can adapt to change. This strategy might be quite different from the deterministic solution. The complexity of the blending problem also increases significantly when constraints are on variables that cannot be averaged, such as ratios between two grades. Considerable research is dedicated to finding solutions to these complex situations. Development of appropriate algorithms and related com￾puter programs is aimed at bringing practical solutions to practitioners (see the bibliography). Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 123 9 The cut-off grade determines the tonnage and average grade of material pro￾cessed and is critical in the determination of the economic feasibility of a proj￾ect. All consequences of choosing a cut-off grade must be taken into account, including technical, economic, legal, environmental, social, and political as illustrated by the following fundamental equation: () () () () = + + U x U x U x U x dir opp oth This equation represents the utility of sending material of grade x to a given destination. The optimal cut-off grade between two options is that where the utility U1 (x) of one option is equal to the utility U2 (x) of the other option. Udir(x) represents direct revenues and costs. If only this term is taken into account, the cut-off grade between two options is ( ) ( ) ( )( ) = + + − + +     − −     x M P O M P O r r V R c 1 1 1 2 2 2 1 2 Uopp(x) represents the opportunity cost of mining or processing material that was not scheduled to be mined or processed, and there are operating constraints. This cost is proportional to the net present value of future cash flows, and inversely proportional to the size of the constraints imposed on production: ( ) = − ⋅ ⋅ U x t i NPV opp i Because the net present value of future cash flows decreases over time, optimization of project economics is usually best achieved with high cut-off grades at the beginning of the mine life and lower cut-off grades at the end. Uoth(x) represents other costs that must be taken into account that are often not quantifiable in dollar terms. These costs can be highly significant Closing Remarks Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 124 CHAPTER NINE and imply that net present value is typically not the only objective to be con￾sidered when optimizing cut-off grades. Cut-off grade optimization is an iterative process. When planning a new mining operation, a cut-off grade profile must be chosen to define operating conditions, including mining rate, processing rate, volume of sales, capital and operating costs, cash flow, and net present value. Because the optimal cut-off grade is a function of operating costs and net present value, the first mine plan is not likely to be optimal. Once a cash flow has been determined under a specific set of assumptions—including assumptions related to cut-off grades—the underlying cut-off grade strategy must be reevaluated. The sensi￾tivity of the project economics must be analyzed as a function of changes in cut-off grades. As a feasibility study progresses over time, the properties of the deposit are better understood as a result of additional drilling and improved mod￾eling; mining and processing methods are better defined; constraints on production are quantified; capital and operating costs are refined; and socio￾economic, environmental, and other costs are better understood. All these changes should be taken into account when determining the cut-off grade. The determination of cut-off grades must also consider all future costs. During the feasibility study, these costs include the capital costs of building the site. When the mine and mill are operating, these costs are sunk and should no longer be taken into consideration. The cut-off grade strategy that optimizes net present value is independent of sunk costs. During the feasibil￾ity study there are no capacity constraints, but capital expenditures are taken into account to optimize cut-off grades. When the mine is operating, sunk costs should no longer be taken into account to optimize cut-off grades. How￾ever, there are capacity constraints, which are the results of decisions made during the feasibility study. If the estimates made during the feasibility study, including those related to mine and mill capacities, hold true when the mine is in operation, the optimal cut-off grade strategy remains unchanged. This strategy is the same at the end of the feasibility study when capital costs are taken into account to optimize the size of the facilities, and during operations when capacity constraints are imposed by the built facilities. But operating conditions are never identical to those assumed during the feasibility study. These conditions change continuously: • The actual mine and mill capacities differ from those which were expected before construction. • The mill recovery and optimal operating conditions are different from planned. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Closing Remarks 125 • The deposit properties differ from those in the model that was used during the feasibility study. • Costs are different from expected. • Prices of products sold are different. Consequently, cut-off grades should be continuously reviewed and changed as conditions demand. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 127 EXA M P L E 1 . N E T P R E S E N T VA L U E O F C O N S TA N T CASH FLOW: PROOF OF FORMULA Problem Statement If a project is expected to generate a constant cash flow C over n years (Ck = C for k = 0, 1, …, n – 1), the net present value of this cash flow is  ()() () = + + + + + + +     − NPV C 1 1 1 i 1 1 i 1 1 i i 2 n 1 Show that  ()() () ( ) + ++ + + + + = − 1 1 1 i 1 1 i 1 1 i g i,n 2 n 1 where g(i,n) = [1 – (1 + i)–n] ∙ (1 + i)/i. Solution Define a = 1/(1 + i). Then   1 1 1 i 1 1 i 1 1 i 1 a a a a a 2 n 1 2 3 n 2 n 1 + ++ + ()() () + + + = + + +++ + ( ) ( ) − − − If the right-hand side of this equation is multiplied by (1 – a), the following result is obtained:  ( ) + + +++ +    ⋅ − = − ( ) ( ) − − 1 a a a a a 1 a 1 a 2 3 n 2 n 1 n Therefore,  ( ) ( ) + + +++ + = − − ( ) ( ) − − 1 a a a a a 1 a 1 a 2 3 n 2 n 1 n Given that a = 1/(1 + i), this equation can be written as follows:  ()() () () ( ) + ++ + + + + =  − +   ⋅ + − − 1 1 1 i 1 1 i 1 1 i 1 1 i 1 i i 2 n 1 n Appendix A Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 128 APPENDIX A EXA M P L E 2 . N E T P R E S E N T VA L U E O F P E R P E T U I T Y CASH FLOW: PROOF OF FORMULA Problem Statement Show that if a project is expected to generate a constant cash flow C to perpe￾tuity, its net present value is NPVi = C (1 + i)/i. Solution A proof was given previously that if a project generates a constant cash flow C for n years, its net present value is NPVi = C g(i,n), where ( ) ( ) ( ) = − +    ⋅ + − g i,n 1 1 i 1 i i n If n goes to infinity, then (1 + i)n goes to infinity and (1 + i)–n goes to zero. The limit of g(i,n) is ( )= + ( ) g i,n 1 i i EXA M P L E 3 . O P P O R T U N I T Y C O S T O F M I N I N G A PERIPHERAL DEPOSIT Problem Statement Consider the open pit mine shown in Figure A-1. Two parts of the deposit remain to be mined: 1. A “main pit” that requires mining low-grade and waste material for two years, followed by eight years of mining higher-grade material. The cash flow generated during the first two years is expected to be negative at $45 million per year. During the next eight years, a posi￾tive cash flow will be generated equal to $250 million per year. 2. A “mini-pit” that can be mined in one year and is expected to gener￾ate a positive cash flow of $90 million. Under current conditions, the main pit can only be mined if the mini￾pit area can be used as a waste dump. This leaves two options: Mine only the main pit and condemn the mini-pit (option 1); or mine the mini-pit first fol￾lowed by the main pit (option 2). Assuming a 15% discount rate, what is the opportunity cost of mining the mini-pit first? Which one of the two options would you recommend? If there were other locations where waste could be dumped, a third option would be to mine the main pit first and the mini-pit last (option 3). Which option would you recommend? Repeat this analysis using a 5% discount rate. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Appendix A 129 Solution The net present values of each option are calculated in Table A-1. To show the influence that each pit has on the other, the expected main pit and mini-pit undiscounted cash flows are shown separately. The discount factors 1/(1 + i)k–1 have been calculated for each year k using a discount rate i = 15%. The net present value of each option is the sum of the product of undiscounted yearly cash flows with a discount factor. Under option 1, only the main pit is mined with net present value NPVi = $891 million. Under option 2, mining the main pit is postponed by one year. The main pit net present value is decreased to NPV'i = $775 million. The opportunity cost of mining the mini-pit first is $891 – $775 = $116 million. The net present value of the mini-pit is only $90 million, not enough to pay for the opportunity cost. The NPV of option 2 is $865 million, less than that of option 1. If the choice is between options 1 and 2, option 1 is that which should be chosen. Under option 3, the net present value of the mini-pit is $22 million, and the total net present value is $913 million. If feasible, option 3 is the best choice. If a 5% discount rate is used, the results are as shown in Table A-2. The opportunity cost of mining the mini-pit first (option 2) is $69 million. The mini-pit’s net present value is $90 million. Option 2 is better than option 1. However, option 3 is still the best option. Mini-Pit Mining: 1 year Cash Flow: $90 million/yr Main Pit - Phase 2 Mining: 8 years Cash Flow: $250 million/yr Main Pit - Phase 1 Mining: 2 years Negative Cash Flow: $45 million/yr Figure A-1 Schematic representation of main pit and mini-pit Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 130 APPENDIX A EXA M P L E 4 . S I M P L I F I E D E Q U AT I O N TO E S T I M AT E OPPORTUNITY C O S T: V E R I F I C AT I O N O F F I R S T- O R D E R APPROXIMATION Problem Statement Show that for small values of t, the difference between the exact formula [1 – (1 + i)–t] NPVi and its first-order approximation i ∙ t ∙ NPVi is less than 10%. Compare the two formulas for t varying from zero to two hundredths of a year (approximately one week) and i = 15%. Solution Figure A-2 shows the relationship between 1 – (1 + i)–t and i ∙ t as a dotted line. The maximum value of i ∙ t is 0.02 ∙ 15% = 0.003. The discount factor Table A-1 Cash flow and net present value of options 1 to 3 (discount rate i = 15%) Year k Discount Factor 1/(1 + i)k Option 1 Option 2 Option 3 Main Pit Mini-Pit Main Pit Mini-Pit Main Pit Mini-Pit 0 1.000 ($45) — $90 ($45) — 1 0.870 ($45) — ($45) — ($45) — 2 0.756 $250 — ($45) — $250 — 3 0.658 $250 — $250 — $250 — 4 0.572 $250 — $250 — $250 — 5 0.497 $250 — $250 — $250 — 6 0.432 $250 — $250 — $250 — 7 0.376 $250 — $250 — $250 — 8 0.327 $250 — $250 — $250 — 9 0.284 $250 — $250 — $250 — 10 0.247 — — $250 — $90 Net present value per pit $891 $0 $775 $90 $891 $22 Opportunity cost $0 — ($116) — $0 — Total net present value $891 $865 $913 Table A-2 Cash flow and net present value of options 1 to 3 (discount rate i = 5%) Option 1 Option 2 Option 3 Main Pit Mini-Pit Main Pit Mini-Pit Main Pit Mini-Pit Net present value per pit $1,451 $0 $1,382 $90 $1,451 $55 Opportunity cost $0 — ($69) — $0 — Total net present value $1,451 $1,472 $1,506 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Appendix A 131 calculated using the first-order approximation, i ∙ t, is less than 10% higher than that using the exact equation, 1 – (1 + i)–t. This difference decreases for smaller values of i. EXAMPLE 5. OPPORTUNITY COST OF NOT USING T HE OPTIMAL CUT-OFF GRADE IN A COPPER MINE Problem Statement Consider a copper mine in which the material can be either leached (pro￾cess 1) or sent to a flotation plant (process 2). The relevant operating condi￾tions are summarized as follows: Leach recovery: r 1 = 57% Mill recovery: r2 = 89% Leach processing cost: P01 = $2.50 per metric ton leached (including overhead) Mill processing cost: P02 = $29.00 per metric ton milled (including overhead) Copper price: V = $2.50 per pound of salable copper (after cost of sales) Demonstrate that the cut-off grade xc between leach and mill is 1.50% Cu. 0.0000 0.0030 0.0025 0.0020 0.0015 0.0010 0.0005 x = 1 – 1/(1 + i)t 0.000 0.0005 0.0010 0.0015 0.0020 0.0025 0.0030 i = 15% y = x y = 1.10x Figure A-2 Comparison of 1 – (1 + i)–t and i · t for i = 15% and t less than 0.02 years Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. y = i·t 132 APPENDIX A The mill capacity is such that it can only accept material exceeding xs = 1.60% Cu. The tonnage and metal content of material above the two cut-off grades are as follows: Above xc = 1.50% Cu: 10.842 million metric tons containing 588.84 million lb of copper Above xs = 1.60% Cu: 9.606 million metric tons containing 546.59 million lb of copper What is the opportunity cost of sending material averaging between xc and xs to the leach pad instead of the mill? Solution The cut-off grade between mill and leach is [ ] ( ) ( ) ( ) ( ) = − −     = − − ⋅ ⋅ = x P P r r V x 29.00 2.50 89% 87% 2.5 2,205 1.50% Cu c o1 o2 1 2 c A change in cut-off grade from xc = 1.50% Cu to xs = 1.60% Cu results in moving 1.236 million metric tons containing 42.25 million lb of copper from the mill to the leach pad. The result is a savings in processing costs and a loss of revenue: Cost savings: [T(xs ) – T(xc )] ∙ (Po2 – Po1) = 1.236 ∙ (29.00 – 2.50) = $32.75 million Revenue loss: [Q(xs ) – Q(xc )] ∙ (r2 – r1 ) ∙ V = 42.25 ∙ (89% – 57%) ∙ 2.5 = $33.80 million Total opportunity cost = 33.80 – 32.75 = $1.05 million For the mill to accept all material above 1.50% Cu, its capacity would have to be increased by 13%. Such an increase would be justified only if it could be achieved for less than $1.05 million. EXA M P L E 6 . S TO C K P I L I N G A N D C U T- O F F DETERMINATION IN A COPPER MINE Problem Statement Consider an open pit copper mine in which material can be sent to a flotation plant, a stockpile, or a waste dump. The parameters that apply to each option are listed in Table A-3. You are asked to do the following: Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Appendix A 133 • Calculate and plot the utility of each option as a function of the copper grade. • Calculate the cut-off grade between waste, stockpile, and mill. Inter￾pret the results. • Which cut-off grades would apply to the stockpile if the mill capacity is such that only material exceeding 0.35% Cu can be processed? Solution Table A-4 shows the mining and processing costs per year for waste dump, direct mill, and stockpile options. In this table, the $0.10 yearly cost per ton of material stockpiled applies to years 1 to 4, whereas the $1.00 cost of retrieving one ton of material from stockpile is only incurred in year 4. The $0.05 yearly Table A-3 Copper mine operating parameters Milled material Mo $1.90 Mining cost per metric ton of ore mined and shipped to the mill, including mine overhead Po $10.00 Processing cost per metric ton of ore, including process overhead r 86% Copper recovery for fresh material Wasted material Mw $2.00 Mining cost per metric ton of waste mined and placed on waste dump, including mine overhead Pw $0.05 Yearly cost per metric ton of waste dumped, including stockpile maintenance and water treatment plant operating cost Stockpiled material Mstp1 $2.50 Mining cost per metric ton of material mined and stockpiled, including overhead and capital cost per metric ton for pad preparation Pstp $0.10 Yearly cost per metric ton of material stockpiled, including stockpile maintenance and water treatment plant operating cost Mstp2 $1.00 Cost per metric ton to retrieve material from stockpile and ship it to the mill, including environmental remediation and overhead dr –3% Loss of recovery due to oxidation during life of stockpile Life of mine n 5 Life of stockpile in years, including year 0 when ore is mined and year 4 when stockpile is processed Value of recovered copper V $3.00 Value of one pound of copper recovered, net of shipping and smelting costs, and other costs of sale Discount rate i 10% Discount rate per year Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 134 APPENDIX A Table A-5 Undiscounted and discounted costs per metric ton mined, and revenues per pound processed Year Total Cost per Metric Ton Dollar per Pound of Cu Contained Discount Factor, Waste Mill Stockpile Waste Mill Stockpile 1/(1 + i)n 0 ($2.00) ($11.90) ($2.50) — $2.58 — 1.000 1 ($0.05) — ($0.10) — — — 0.909 2 ($0.05) — ($0.10) — — — 0.826 3 ($0.05) — ($0.10) — — — 0.751 4 To infinity — ($11.10) — — $2.49 0.683 Total Undiscounted Undiscounted ($7.00) ($11.90) ($13.90) $0.00 $2.58 $2.49 Discounted Discounted ($2.55) ($11.90) ($10.33) $0.00 $2.58 $1.70 cost per metric ton of waste applies to perpetuity. Table A-5 shows the total cost (mining plus processing) per year for the three processes, and the corre￾sponding value per pound contained in material processed. For direct mill feed, the value per pound is r V = 86% ∙ $3.00 = $2.58, which is recovered in year 0. For material stockpiled, this value is (r + dr) V = (86% – 3%) ∙ $3.00 = $2.49 per pound, which is recovered in year 4. Also shown in Table A-5 are cumulative costs per metric ton and value per pound, both discounted and undiscounted, for as long as the material under consideration carries costs (to perpetuity for waste, five years for stock￾piled material, one year for direct mill feed). For stockpiled material, the undiscounted value is based on the assumption that the $0.05 yearly cost will apply to 100 years, starting in year 1. The discounted value of the $0.05 cost to perpetuity is $0.05 ∙ (1 + i)/i = 11 ∙ $0.05 = $0.55. Table A-4 Mining and processing costs per year for material wasted, milled, and stockpiled Year Mining Cost per Metric Ton Processing Cost per Metric Ton Waste Mill Stockpile Waste Mill Stockpile 0 ($2.00) ($1.90) ($2.50) — ($10.00) — 1 ($0.05) — ($0.10) — — — 2 ($0.05) — ($0.10) — — — 3 ($0.05) — ($0.10) — — — 4 To infinity — ($1.10) — — ($10.00) Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Appendix A 135 The utility of sending one metric ton of material to a given process is equal to the net present value of the cash flow generated by this ton. For exam￾ple, if one metric ton of material of grade x is sent to the stockpile, the net present value of this decision is NPVstp = –$10.33 + $1.70 ∙ 2,205 ∙ x, where x is expressed in % Cu and 2,205 is the number of pounds per metric ton. The three net present values can be written as follows: For waste: Uwaste(x) = –$2.55 For direct feed: Uore(x) = –$11.90 + 2.58 ∙ 2,205 ∙ x For stockpile: U stp(x) = –$10.33 + 1.70 ∙ 2,205 ∙ x These lines are plotted in Figure A-3. The cut-off grades between two pro￾cesses are defined by the point of intersection of the lines corresponding to these processes: Waste-mill cut-off grade: ( ) ( ) = − ⋅ = x 11.90 2.55 2.58 2,205 0.16% Cu c Waste-stockpile cut-off grade: ( ) ( ) = − ⋅ = x 10.33 2.55 1.70 2,205 0.21% Cu c Mill-stockpile cut-off grade: [ ] ( ) ( ) = − − ⋅ = x 11.90 10.33 2.58 1.70 2,205 0.08% Cu c As one would expect, if there is no mill capacity constraint, the best deci￾sion is to waste all material below 0.16% Cu and to send material above 0.16% Cu directly to the mill. However, if the mill capacity is such that only material above 0.35% Cu can be processed, all material between 0.21% Cu and 0.35% Cu should be stockpiled and processed at a later date (Figure A-4). EXA M P L E 7 . P R O P E R T I E S O F C O N S TA N T- M E T R I C - TO N S CURVES IN BLENDING DIAGRAMS Problem Statement Figure A-5 shows the relationship between three stockpiles, i = 1,3. The ton￾nage of material in stockpile i is Ti with grade x1i and x2i. The first grade, x1 , is plotted on the horizontal axis, while the second grade, x2 , is plotted along the vertical axis. The stockpiles are represented by the points A, B, and C. The objective is to blend the three stockpiles to obtain a product of grade x10, x20. The proportions in which the stockpiles are blended is pi , i = 1,3. The Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 136 APPENDIX A (15) 15 10 5 0 (5) (10) Grade, % Cu 0.10 0.20 0.30 Waste-Mill Cut-off: 0.16% Cu Waste-Stockpile Cut-off: 0.21% Cu Figure A-3 Net present value of three options: best option with no mill capacity constraint (15) 15 10 5 0 (5) (10) Grade, % Cu 0.10 0.20 0.30 Mill Capacity Cut-off: 0.35% Cu Waste-Stockpile Cut-off: 0.21% Cu Figure A-4 Net present value of three options: best option with mill capacity constraint Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Stockpile Mill Profit, $/t Profit, $/t Appendix A 137 maximum tonnage of blended material is T0 . It was shown in chapter 8 that T0 is equal to the smallest value of the ratios Ti /pi . Show that, on graphs such as Figure A-5, the shapes that define blends with the same maximum tonnage (Iso-T shapes, on which T0 is a constant) are made of line segments parallel to the lines AB, AC, or AD. Solution For the tonnages T0 to be constant, the proportion pi applicable to one of the stockpiles must remain constant so that the ratio Ti /pi remains constant. If a line is drawn from A (stockpile 1), which intersects an Iso-T line in D and the line BC in E, the proportion p1 is equal to the ratio DE/AE (Figure A-5). If another line AE' is drawn, which intersects the same Iso-T line in D', the proportion p'1 is equal to the ratio D'E'/AE'. If stockpile 1 is that which lim￾its the maximum tonnage T0 , the proportions p1 and p'1 must be equal. The ratios DE/AE and DE'/AE' are equal only if the Iso-T line DD' is parallel to the line BC. 6 7 8 9 10 11 12 13 14 15 16 Grade 1 of Blended Material 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 C E' D' D E A B Figure A-5 Relationship between stockpile properties and lines with constant tonnage Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Grade 2 of Blended Material Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 139 Symbol Description C constant yearly cash flow c constant tail in recovery function CI revenues required every year during n years to get a return on investment i on a capital investment I: CI = I / g(i,n) Ck cash flow generated by a project in year k, k = 0, 1, 2, …, n CR expected end-of-life reclamation costs Cs smelter costs per metric ton of concentrate Ct cost of shipping one metric ton of concentrate to the smelter d1 metal grade deducted from recovered grade in calculation of smelter payment for metal 1 DIMC discounted incremental mining costs DIPC discounted incremental processing costs DIR discounted incremental revenues dPo (T+c)/dT+c first-order derivative of Po (T+c) with respect to T+c dQ+c/dT+c first-order derivative of Q+c with respect to T+c dr(T+c)/dT+c first-order derivative of r(T+c) with respect to T+c dU(T+c)/dT+c first-order derivative of U(T+c) with respect to T+c g(i,n) = [1 − 1/(1 + i)n ](1 + i)/i I capital cost invested i minimum rate of return (discount rate) K concentration ratio defined as number of metric tons of material that must be processed to produce one metric ton of concentrate M mining cost per metric ton processed Mo mining cost per metric ton of ore List of Symbols Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 140 LIST OF SYMBOLS Symbol Description Mo1 value of Mo for process 1 Mo2 value of Mo for process 2 Mstp current mining costs per metric ton delivered to low-grade stockpile Mw mining cost per metric ton of waste n number of years NPV net present value NPVi net present value of future cash flows, calculated using the discount rate i NSR(x1 , x2 ) net smelter return, defined as returns from selling concentrate produced from one metric ton of ore with average grades x1 , x2 , less smelting charges NSRc NSR cut-off between processing and wasting one metric ton of material Oo overhead cost per metric ton of ore Oo1 value of Oo for process 1 Oo2 value of Oo for process 2 Ostp current overhead costs associated with mining and stockpiling one metric ton of low-grade material Ow overhead cost per metric ton of waste p1 proportion of metal 1 contained in concentrate that is paid for by smelter p2 proportion of metal 2 contained in concentrate that is paid for by smelter pi in Chapter 8, proportion of stockpile or concentrate i in blended material Po processing cost per metric ton of ore Po (T+c) processing cost per metric ton of ore processed, if plant capacity is T+c Po1 processing cost for process 1 Po2 processing cost for process 2 Pstp net present value of future costs of stockpiling material that will be processed later Pw processing cost per metric ton of waste Pwaste net present value of future costs of wasting material Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. List of Symbols 141 Symbol Description Q(x) quantity of metal in material whose grade is greater than x Q+c quantity of metal contained in material above cut-off grade xc : Q+c = T+c ∙ x+c R refining, transportation, and cost of sales per unit of product sold r recovery, or proportion of valuable product recovered from the mined material r(T+c) processing plant recovery, if plant capacity is T+c r(x) process recovery for material of average grade x R1 value of R for process 1 R2 value of R for process 2 r1 value of r for process 1 r2 value of r for process 2 rc constant recovery after subtracting constant tail c Rstp refining, transportation, and cost of sales per unit of product sold from stockpile at the time product is sold rstp recovery expected at the time stockpiled material is processed s stripping ratio, defined as tons of waste per tons of ore T tonnage of ore and waste to be mined t time, measured in years T+c tonnage above cut-off grade Ti tonnage of material in stockpile or concentrate i t i tonnage of stockpile or concentrate i in blended material To tonnage of material in blended material Tw tonnage of waste to be mined U(T+c) utility of running the plant at T+c capacity for one year U(x) utility of sending one metric ton of material of grade x to a given process: U(x) = Udir(x) + Uopp(x) + Uoth(x) U1 (x) utility of sending one metric ton of material of grade x to process 1 U2 (x) utility of sending one metric ton of material of grade x to process 2 Udir(x) direct utility (profit or loss) of processing one metric ton of material of grade x Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 142 LIST OF SYMBOLS Symbol Description Ujk utility of mining block j in year k Ujk,dir direct utility of mining block j in year k Ujk,opp opportunity cost of mining block j in year k Ujk,oth other utility of mining block j in year k Umax(x) maximum utility resulting from sending material of grade x to either of two processes Uopp(x) opportunity cost or benefit of changing the processing schedule by adding one metric ton of grade x to the material flow Uore(x) utility of mining and processing one metric ton of material of grade x Uore(x1 , x2 ) utility of sending one metric ton of material with metal grades x1 , x2 to the processing plant Uoth(x) utility of other factors that must be taken into account in the calculation of cut-off grades Ustp(x) utility of stockpiling material of grade x Uwaste(x) utility of mining and wasting one metric ton of material of grade x V value of one unit of valuable product V1 value of one unit of product 1, such as one pound of copper V2 value of one unit of product 2, such as one ounce of gold Vstp dollar value of the product recovered from stockpile at the time product is sold x average grade x+c average grade above cut-off grade xc x1e grade equivalent expressed in terms of metal 1 x2e grade equivalent expressed in terms of metal 2 x1i average grade of metal 1 in stockpile or concentrate i x2i average grade of metal 2 in stockpile or concentrate i xc cut-off grade xc1 cut-off grade 1, taking only operating costs into account xc2 cut-off grade 2, taking into account operating costs and undiscounted capital cost per metric ton xc3 cut-off grade 3, taking into account operating costs and discounted capital cost per metric ton Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. List of Symbols 143 Symbol Description xc4 cut-off grade 4, taking into account operating costs, discounted capital cost per metric ton, and opportunity costs xs selected cut-off grade YCF yearly cash flow Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 145 Agricola, G. 1556. De Re Metallica (translated from the first Latin edition by Herbert Clark Hoover and Lou Henry Hoover). London: The Mining Magazine, 1912; New York: Dover Publications, 1950. Asad, M.W.A., and Dimitrakopoulos, R. 2011. Optimal production scale of open pit mining operations with uncertain metal supply and long-term stockpiles. Resourc. Policy 37(1):81–89. Asad, M.W.A., and Dimitrakopoulos, R. 2012. Implementing a parametric maximum flow algorithm for optimum open pit mine design under uncertain supply and demand. J. Oper. Res. Soc. 64(2):185–197. Asad, M.W.A., and Topal, E. 2011. Net present value maximization model for opti￾mal cut-off grade policy of open pit mining operations. J. South. Afr. Inst. Min. Metall. 111(November):741–750. Ataei, M., and Osanloo, M. 2003. Determination of optimum cutoff grade of multi￾ple metal deposits by using the Golden Section search method. J. South. Afr. Inst. Min. Metall. (October):493–499. Benito, R., Sanner, G., and Cal, W. 2011. Life of mine schedule from multiple pits: A case study for the Tenke Fungurume mining district. SME Preprint No. 11-064. Englewood, CO: SME. Brittan, M. 2008. Head grade effects on gold recovery. SME Preprint No. 08-031. Littleton, CO: SME. Bullock, R.L. 2011. Subsurface mine development. In SME Mining Engineering Handbook, 3rd ed. Englewood, CO: SME. pp. 1203–1221. Burks, S. 2013. Case studies of simultaneous mining and mineral processing optimi￾zation applied to platinum and nickel operations. J. South. Afr. Inst. Min. Metall. 113(March):221–233. Carter, P.G., Lee, D.H., and Baarsma, H. 2004. Optimisation methods for the selec￾tion of an underground mining method. In Proceedings of Orebody Modeling and Strategic Mine Planning Symposium, Perth, Australia: The Australasian Institute of Mining and Metallurgy. pp. 7–12. Cetin, E., and Dowd, P.A. 2002. The use of genetic algorithms for multiple cut-off grade optimization. In Proceedings of the 30th APCOM Symposium. Englewood, CO: SME. pp. 769–780. Bibliography Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 146 BIBLIOGRAPHY Cetin, E., and Dowd, P.A. 2013. Multi-mineral cut-off grade optimization by grid search. J. South. Afr. Inst. Min. Metall. 113:659–665. Chanda, E.K. 2004. Network linear programming optimization of an integrated mining and metallurgical complex. In Proceedings of Orebody Modeling and Stra￾tegic Mine Planning Symposium, Perth, Australia: The Australasian Institute of Mining and Metallurgy. pp. 13–19. Cook, S.E. 2013. Community management of mineral resources: The case of the Royal Bafokend Nation. J. South. Afr. Inst. Min. Metall. 113( January):61–66. Dagdelen, K., and Kawahata, K. 2008. Value creation through strategic mine plan￾ning and cutoff-grade optimization. Min. Eng. 60(1):39–45. Dimitrakopoulos, R. 2011. Stochastic optimization for strategic mine planning: A decade of developments. J. Min. Sci. 84(2):138–150. Duckworth, I., Casten, T., and Rakidjan, M. 2012. Overview of Freeport-McMoran’s proposed DMLZ mine at Grasberg. Min. Eng. 62(8):20–26. Grobler, F., Elkington, T., and Rendu, J.M. 2011. Robust decision making—Appli￾cation to mine planning under price uncertainty. In Proceedings of the 35th APCOM Symposium. Wollongong, Australia: The Australasian Institute of Min￾ing and Metallurgy. September 2011. pp. 371–380. Hall, B.E. 2003. How mining companies improve share price by destroying share￾holder value. In Proceedings CIM Mining Conference and Exhibition. Paper 1194. Montreal: Canadian Institute of Mining and Metallurgy. Hall, B.E. 2009. Short-term gain for long-term pain—How focusing on tac￾tical issues can destroy long-term value. J. South. Afr. Inst. Min. Metall. 109(March):146–156. Hoerger, S., Bachmann, J., Criss, K., and Shortridge, E. 1999. Longterm mine and process scheduling at Newmont’s Nevada operations. In Proceedings of 28th APCOM Symposium. Edited by K. Dagdelen. Golden, CO: Colorado School of Mines. pp. 739–747. Hoerger, S., Hoffman, L., and Seymour, F. 1999. Mine planning at Newmont’s Nevada operations. Min. Eng. 51(10):26–30. King, B. 2001. Optimal mine scheduling. Mineral Resource and Ore Reserve Estima￾tion. Monograph 23. Edited by A.C. Edwards. Carlton, Vic.: The Australasian Institute of Mining and Metallurgy. pp. 451–458. Lane, K.F. 1964. Choosing the optimum cut-off grade. Colo. School Mines Q. 59:811–824. Lane, K.F. 1979. Commercial aspects of choosing cutoff grades. In Proceedings of the 16th APCOM International Symposium. Littleton, CO: SME. pp. 280–285. Lane, K.F. 1988. The Economic Definition of Ore: Cut Off Grades in Theory and Prac￾tice. 1st ed., 1988; 2nd ed., 1991. London: Mining Journal Books. Lerchs, H., and Grossmann, I.F. 1965. Optimum design of open-pit mines. Trans. Can. Inst. Min. Metall. 68(633):47–54. Manchuk, J.G., and Deutsch, C.V. 2008. Optimizing stope designs and sequences in underground mines. SME Trans. 324:67–75. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Bibliography 147 Meagher, C., Dimitrakopoulos, R., and Davis, D. 2009. A new approach to con￾strained open pit pushback design using dynamic cut-off grades. In Proceedings Orebody Modelling and Strategic Mine Planning Conference, Perth, Australia, March 2009. The Australasian Institute of Mining and Metallurgy. pp. 171–175. Menabde, M., Froyland, G., Stone, P., and Yeates, G. 2007. Mining schedule optimi￾sation for conditionally simulated orebodies. In Proceedings Orebody Modelling and Strategic Mine Planning Conference, Spectrum Series 14, The Australasian Institute of Mining and Metallurgy. pp. 379–384. Minnitt, R.C.A. 2004. Cut-off grade determination for the maximum value of a small Wits-type gold mining operation. J. South. Afr. Inst. Min. Metall. 104( June):277–284. Peevers, R., and Whittle, G. 2013. Enterprise optimization for mining businesses. Presented at 2013 SME Annual Meeting & Exhibit, Denver, CO. Raiffa, H., and Schlaifer, R. 2000. Applied Statistical Decision Theory. New York: Wiley Classics Library. Rendu, J.M. 2009. Cut-off grade estimation—Old principles revisited—Application to optimisation of net present value and internal rate of return. In Proceedings Orebody Modeling and Strategic Mine Planning Conference, Perth, Australia, March 2009. The Australasian Institute of Mining and Metallurgy. Runge, I. 2011. Economic principles of decision making. In SME Mining Engineering Handbook, 3rd ed. Englewood, CO: SME. pp. 309–315. Wang, H., and Webber, T. 2012. Practical semiautomatic stope design and cutoff grade calculation method. Min. Eng. 64(10):85–91. Wharton, C. 2004. The use of extractive blending optimization for improved profit￾ability. In Proceedings Orebody Modeling and Strategic Mine Planning Conference, Perth, Australia, November 2004. The Australasian Institute of Mining and Metallurgy. pp. 69–76. Whittle, D. 2011. Open pit planning and design. In SME Mining Engineering Hand￾book, 3rd ed. Englewood, CO: SME. pp. 877–901. Whittle, J. 1999. A decade of open pit mine planning and optimization—The craft of turning algorithms into packages. In Proceedings of 28th APCOM Symposium. Edited by K. Dagdelen. Golden, CO: Colorado School of Mines. pp. 15–23. Whittle, J., and Wharton, C. 1995. Optimizing cut-offs over time. In Proceedings of the 25th International Symposium on the Application of Computers and Mathe￾matics in the Mineral Industries, Brisbane, Australia, pp. 261–265. Yellishetty, M., Mudd, G.M., Giurco, D., Mason, L., and Mohr, S. 2013. Iron Ore in Australia—too much or too hard? J. Austral. Inst. Min. Metall. No. 3:42–47. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 149 Note: f. indicates figure; n. indicates (foot)note; t. indicates table. Agricola, Georgius, on separation, 1 Blending, 109 coal of different characteristics, 109 complex cases (inverse blending problem), 120–122 gold of different characteristics, 109, 110–113, 110t., 111f., 113f., 114f. iron of different characteristics, 109 linear algebra in determining stockpile blends, 117–118 to maximize gold content, 110t., 112–113, 113f., 114f. to maximize gold grade, 110–112, 110t., 111f. to maximize metal content, 119–120, 120f. to maximize tonnage, 116f., 118–120, 119f., 120f., 121f. to reach specific grades, 115–117, 115t., 116f., 117t. to satisfy ratio constraints, 120, 121f. strategy, 16 three material types, 114–118, 115t., 116f., 117t. two material types, 110–114, 110t., 111f., 113f., 114f. See also Stockpiles and stockpiling Block caving and capital cost, 93 constraints imposed by, 91–92 and cut-off grades, 91 economic valuation example (copper mine), 95–98 marginal cut-off grade and block design, 93 marginal cut-off grade and drawpoint management, 92 and minimum rate of return, 93, 95, 96, 97 opportunity cost, 94–95 and yearly cash flow, 94 Breakeven cut-off grade, 8, 17 without capacity constraints (copper mine), 58 and polymetallic deposits, 32–38 Capacity constraints, 124 and decreasing cut-off grade and NPV over time (copper mine), 61–63, 62t., 63f., 64t., 65f. and increased cut-off grade (copper mine example), 56–61, 59t., 60t. and opportunity cost, 12–14, 53 and opportunity cost with lowered cut-off grade (gold leaching), 53–56, 54f., 55f. Index Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 150 INDEX and pushback mining, 88-89 underground mines, 23 See also Mill capacity Capital costs, 101, 102 constraints on, and changes in opportunity cost, 107 Cash flow constant over n years, 11, 12t., 127 constant to perpetuity, 11, 12t., 128 See also Net present value; Yearly cash flow Closure and reclamation costs, 105 Coal, 109 Concentrates, 9 Constant metal content lines. See Iso-Q lines Constant ratio lines. See Iso-R lines Constant tail model, 30–31 precious metal example, 31–32, 32f., 33f. Constant-tonnage curves. See Iso-T lines Copper block caving and opportunity cost, 95–98 breakeven cut-off grade without capacity constraints, 58 capacity constraints and decreasing cut-off grade and NPV over time, 61–63, 62t., 63f., 64t., 65f. capacity constraints and increased cut-off grade, 56–61, 59t., 60t. grinding circuit optimization, 44–52, 46f., 46t., 47f., 49f., 50f., 50t., 51f., 51t. mine and mill cut-off grades in open pit mines, 20–22, 22f., 23f. opportunity cost and cut-off grades with mill capacity constraint, 59–60 opportunity cost and cut-off grades with mine capacity constraint, 60–61 opportunity cost and cut-off grades with refinery capacity constraint, 61 opportunity cost of not using optimal cut-off grade, 131–132 releasing capacity constraints, 81–85, 83t., 84t. stockpiling and cut-off determination, 132–135, 133t., 134t. See also under Polymetallic deposits Costs capital, 101, 102, 107, 124 closure and reclamation, 105 and cut-off grade, 101–105 fixed, 101 future, 124 incremental capital expenditures, 104–105 leaching operation, 103, 104t. and minimum return on investment, 102–103 operating, 102, 107 other (environmental, political, etc.), 15–16 overhead, 105 to perpetuity, 105 and stakeholders, 105–107 of stockpiling of low-grade material, 103 sunk, 102, 124 sustaining capital, 103–104 variable, 101 Cut-off grades below marginal cut-off grade, 63–66 between ore and waste, 9, 17–18 and blending strategy, 16 breakeven calculations, 8, 17 and closure and reclamation costs, 105 and costs and benefits, 1–2, 12–16 decreasing, and NPV decrease over time (copper mine), 61–63, 62t., 63f., 64t., 65f. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Index 151 defined, 1, 5 and fixed mine capacity, 76–78, 77f. and fixed processing capacity, 69–76, 71f., 74f., 75f. and fixed sales volume, 78–81, 80f., 82f. fundamental equation, 6, 123–124 and grade-tonnage relationship, 6–7, 7f., 8f. and grinding circuit optimization, 44–52 and increased mining capacity with constant processing capacity, 2 and incremental capital expenditures, 104–105 and leaching operation costs, 103, 104t. and low-grade stockpiles, 40–43, 103 and net present value, 2, 3, 10 and opportunity cost, 13–14, 38–40, 39f. optimization as iterative process, 124 optimization with opportunity costs, 66–67 and overhead costs, 105 and price fluctutations, 2 and profitability, 2 and published reserves, 2–3 purposes of, 1 and stakeholders’ interests, 3 and sustaining capital, 103–104 and time-space considerations, 3 and utility functions, 17–18, 19f. See also Breakeven cut-off grade; Mine (external) cut-off grade; Mill (internal) cut-off grade; Process comparison cut-off grade; Variable recovery cut-off grade Cut-off values, 34 De Re Metallica, 1 Direct profit or loss, 8–9 precious metal example, 9–10 Discount rate. See Minimum rate of return Discounted incremental mining costs (DIMC), 71–72 Discounted incremental processing cost (DIPC), 78 Discounted incremental revenues (DIR), 72, 78 The Economic Definition of Ore: Cut-Off Grades in Theory and Practice, 3 Ex-pit cut-off grade. See Mine (external) cut-off grade External cut-off grade. See Mine (external) cut-off grade Fixed sales volume and fixed mining rate with no processing constraint, 81, 82f. and fixed processing rate with no mining constraint, 79–81, 80f. with no mining or processing constraint, 78–79 Gangue, negative effects of, 2 Geologic constraints, 66, 67 and assessment of releasing capacity constraints, 81–85, 83t., 84t. See also Mine capacity Gold capacity constraints and opportunity cost with lowered cut-off grade, 53–56, 54f., 55f. and constant tail model, 30–32, 32f., 33f. cut-off grade and specific costs, 103–104, 104t. cut-off grade between two processes, 26–28, 28f. cut-off grade in plant capacity optimization, 76–78, 77f. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 152 INDEX cut-off grade with fixed sales and no mining or processing constraint, 78–79 cut-off grade with fixed sales, fixed mining rate, and no processing constraint, 81, 82f. cut-off grade with fixed sales, fixed processing rate, and no mining constraint, 79–80, 80f. cut-off grade with variable recovery, 28–30, 29f., 30f. of different characteristics, blending, 109, 110–113, 110t., 111f., 113f., 114f., 119–120, 120f. mine selectivity, deposit modeling, ore control, and cut-off grade, 98–100, 99f., 100t. opportunity cost in mine capacity optimization, 72–76, 74f., 75f. opportunity cost of increased tonnage, 14–15 optimization of stope size, 24–26 use for high-sulfide, low-gold ore, 16 See also under Polymetallic deposits Grade-tonnage relationship, 6–7, 7f., 8f. and constant metal content (Iso-Q) lines, 119–120, 120f. and constant-tonnage curves (Iso-T lines), 119, 119f., 135–137, 137f. Grinding circuit optimization, 44 advantages of finer and coarser grinds, 44 copper mine example, 45–52, 46f., 46t., 47f., 49f., 50f., 50t., 51f., 51t. and expected grade-tonnage relationship, 45–48, 46f., 46t., 47f. formula, 44–45 and incremental utility–tonnage relationship, 50–52, 51f., 51t. and operating cost–tonnage relationship, 48, 49f., 50f., 50t. In-pit cut-off grade. See Mill (internal) cut-off grade Incremental capital expenditures, 104–105 Internal cut-off grade. See Mill (internal) cut-off grade Inverse blending problem, 120–122 Iron, 109 Iso-Q lines, 119–120, 120f. Iso-R lines, 120, 121f. Iso-T lines, 119, 119f., 135–137, 137f. Lane, F., 3 Leaching operation costs, 103, 104t. Marginal cut-off grades, 73 cut-off grade below, 63–66 Metal equivalent, 34 and NSR (calculation and reporting), 36–38 Mill capacity constraints on, and changes in opportunity cost, 107 fixed, and fixed sales volume with no mining constraint, 79–81, 80f. fixed, with increased mine capacity, 81–83, 83t., 84t. fixed, with increased mineral content, 81–82, 83t., 84–85, 84t. increased, with fixed mine capacity, 81–82, 83, 83t., 84t. and mine capacity, imbalances in, 69 unconstrained, and fixed sales volume, 78–79, 81, 82f. Mill (internal) cut-off grade, 18–19, 45 open pit mines, 21–22, 22f., 23f. underground mines, 25–26 Mine capacity assessing whether increase is justified, 70–72, 71f. changes in, and effect on cut-off grade, 69–72 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Index 153 with constant grade-tonnage curve, 70–72, 71f. without constant grade-tonnage curve, 72 constraints on, and changes in opportunity cost, 107 and cut-off grade (copper mine), 60–61 fixed, and cut-off grade, 76–78, 77f. fixed, with increased mill capacity, 81–82, 83, 83t., 84t. fixed, with increased mineral content, 81–82, 83t., 84, 84t. grade-tonnage curves, current and proposed, 73–75, 74f., 75f. increased, with fixed mill capacity, 81–83, 83t., 84t. and mill capacity, imbalances in, 69 open pit and underground methods compared for same deposit, 98–100, 99f., 100t. opportunity cost in optimization of, 72–76, 74f., 75f. unconstrained, and fixed sales volume, 78–81, 80f. Mine (external) cut-off grade with waste stripping, 19–20 without waste stripping, 19 Mine planning and block and panel caving, 91–98 open pit and underground methods compared for same deposit, 98–100, 99f., 100t. pushback mining valuation for open pit mine, 87–89 and similarities between underground and open pit mines, 90–91 stope mining valuation for underground mine, 89–90 Mines. See Mine (external) cut-off grade; Open pit mines; Stockpiles and stockpiling; Underground mines Minimum rate of return and block and panel caving, 93, 95, 96, 97 constraints on, and changes in opportunity cost, 107 Minimum return on investment, 102–103 Molybdenum. See under Polymetallic deposits Net present value (NPV), 2, 3, 10, 74 calculation example, 10, 11t. and constant cash flow over n years, 11, 12t., 127 and constant cash flow to perpetuity, 11, 12t., 128 and cut-off grade optimization, 66–67 in economic valuation of pushback, 87–89 in economic valuation of stopes, 89–90 formula, 10, 127 and opportunity cost, 12–14, 56–57 Net smelter return (NSR), 34 cut-off example (copper￾molybdenum), 35–36, 36f. and cut-off grade (formula), 34 and metal equivalent, 36–38 Open pit mines cut-off grades in, 20–22 mill cut-off grade, 21, 22f. mill cut-off grade with strip ratio, 21–22, 23f. mine cut-off grade, 21, 22f. and pushback, NPV in consideration of, 87–89 and underground method, compared for same deposit, 98–100, 99f., 100t. and underground mines, similarities in planning for, 90–91 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 154 INDEX Operating conditions, 124–125. See also Processing plant operations Operating costs, 102 and benefits, 12–15 constraints on, and changes in opportunity cost, 107 Opportunity costs and capacity constraints, 12–14, 128–129, 129f., 130t. and capacity constraints with lowered cut-off grade (gold leaching example), 53–56, 54f., 55f. and cut-off grade below marginal cut￾off grade, 63–66 in cut-off grade estimation, 13–14, 107 in cut-off grade optimization, 66–67 and cut-off grades with mill capacity constraint (copper mine), 59–60 and cut-off grades with mine capacity constraint (copper mine), 60–61 and cut-off grades with refinery capacity constraint (copper mine), 61 and decreasing cut-off grade over time (copper mine), 61–63, 62t., 63f., 64t., 65f. defined, 12, 56–57 in mine capacity optimization, 72–76, 74f., 75f. and net present value, 12–14, 56–57 of not using optimum cut-off grade, 38–40, 39f., 131–132 and peripheral deposit, 12–13, 128–129, 129f., 130t. in pushback mining, 87–88 simplified equation to estimate, 130–131, 131f. underground gold mine example, 14–15 Overhead costs, 105 Panel caving and capital cost, 93 constraints imposed by, 91–92 and cut-off grades, 91 economic valuation example (copper mine), 95–98 marginal cut-off grade and block design, 93 marginal cut-off grade and drawpoint management, 92 and minimum rate of return, 93, 95, 96, 97 opportunity cost, 94–95 and yearly cash flow, 94 Polymetallic deposits breakeven cut-off grades, 32–38 copper-gold, 32–34 copper-gold, and net smelter return (NSR), 34 copper-molybdenum, and net smelter return, 35–38 and cut-off values, 34 defined, 32 and metal equivalent, 34 metal equivalent calculation and reporting, 36–38 NSR cut-off calculation, 34–36, 36f. Process comparison cut-off grade in choosing between two processes, 26–27 precious metal example, 27–28, 28f. Processing plant operations. See Blending strategy; Grinding circuit optimization; Mill (internal) cut-off grade; Process comparison cut-off grade; Stockpiles and stockpiling Pushbacks, 91 economic valuation of, 87–89 Reserves, published, 2–3 Stakeholders’ interests and costs, 105–107 and cut-off grade, 3 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Index 155 Stockpiles and stockpiling, 1, 2, 3, 16 and cut-off estimation, 40–42, 132–135, 133t., 134t., 136f. and high-cost waste dumps, 43 and low-cost stockpiles, 42–43 low-grade, and cut-off grades, 40–43, 103 vs. wasting, 40–41 See also Blending Stopes, 91 economic valuation of, 89–90 minimum average grade, 23–24, 24f. size optimization, 24–26 Sunk costs, 102 Sustaining capital, 103–104 Symbols, list of, 139–143 Underground mines and capacity constraints, 23 cut-off grades in, 23–26 estimation of average and cut-off grades in, 23, 24f. mill cut-off grade, 25–26 mine cut-off grade, 24–25 minimum average stope grade, 23–24, 24f. and open pit method, compared for same deposit, 98–100, 99f., 100t. and open pit mines, similarities in planning for, 90–91 and opportunity costs, 23 stope size optimization, 24–26 stopes, economic valuation of, 89–90 Utility, defined, 5n. Utility functions, 5–6, 6f. and cut-off grade, 17–18, 19f. fundamental equation, 6, 123–124 Udir, 6, 8–10, 123 Uopp, 6, 12–15, 123 Uoth, 6, 15–16, 123–124 Variable costs, 101 Variable recovery cut-off grade, 28–29 constant tail model, 30–32, 32f., 33f. precious metal example, 29–30, 29f., 30f. Yearly cash flow (YCF), 73–74, 75 Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 157 About the Author Jean-Michel ( J.M.) Rendu is an independent mining consultant with forty years of experience in the mining industry. He is recognized worldwide as an expert in the estimation and public reporting of mineral resources and mineral reserves, geostatistics, and mine planning. In his former position as vice president of resources and mine planning with Newmont Mining Cor￾poration for seventeen years, he was responsible for the management of all Newmont mining engineering activities, including project reviews, staffing and support of corporate and mine-site mining groups, estimation and public reporting of mineral resources and mineral reserves, mine planning, and ore control. J.M.’s consulting activities gave him the opportunity to work on all continents except Antarctica. J.M. has authored more than fifty technical papers on deposit model￾ing, mine planning, methods and guidelines for estimation of resources and reserves, U.S. and international regulatory requirements for public reporting, and geostatistical theory and practice. He is also the author of An Introduc￾tion to Geostatistical Methods of Mineral Evaluation, a South African Institute of Mining and Metallurgy Monograph Series publication, first published in 1978 (second edition 1981). During his professional career, J.M. was assistant professor at the Uni￾versity of Wisconsin–Madison, honorary professor at the University of Queensland, adjunct associate professor at the Colorado School of Mines, Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. 158 ABOUT THE AUTHOR invited lecturer at École Polytechnique and McGill University in Montreal, and external lecturer at the University of Witwatersrand in South Africa. J.M. has taught short courses on estimation and public reporting of mineral resources and mineral reserves, cut-off grade calculation, and geostatistical theory and practice. In addition to being a Founding Registered Member of the Society for Mining, Metallurgy, and Exploration, Inc. (SME), J.M. served as SME Board member and chairman of SME’s Mining and Exploration (M&E) Division, Ethics Committee, and Resources and Reserves Committee. He was a found￾ing member and U.S. representative of the Committee for Mineral Reserves International Reporting Standards (CRIRSCO). He is a Fellow of both the Australasian Institute of Mining and Metallurgy and the South African Insti￾tute of Mining and Metallurgy. J.M. was a recipient of the Henry Krumb Lecturer Award in 1992; the Presidential Award in 1992 and 2004; the Daniel C. Jackling Award in 1994; the M&E Division Distinguished Service Award and the AIME Mineral Economics Award in 2008; the SME Distinguished Member Award and the American Mining Hall of Fame Medal of Merit Award in 2009. J.M. is an elected member of the U.S. National Academy of Engineering. Copyright © 2013 Society for Mining, Metallurgy, and Exploration, Inc. All rights reserved. GRADE ESTIMATION SECOND EDITION AN INTRODUCTION TO CUT-OFF AN INTRODUCTION TO CUT-OFF GRADEESTIMATION BY JEAN-MICHEL RENDU BLENDING CONSTRAINTS WASTE BLOCK CAVING STOCKPILES CLOSURE GRADE-TONNAGE PRICES MINE CUT-OFF LEACHING UNDERGROUND NPV SOCIOECONOMIC METAL EQUIVALENT OPEN PIT OPPORTUNITY POLYMETALLIC PROCESS PROFIT RECLAMATION SELECTIVITY STOPES UTILITY NSR VARIABLE RECOVERY NPV MILL CUT-OFF LEACHING BREAKEVEN STOCK STOCKPILES CLOSURE POLYMETALLIC UNDERGROUND NPV SOCIO-ECONOMIC METAL EQUIVALENT OPEN PIT OPPORTUNITY POLYMETALLIC PROCESS PROFIT BLENDING CONSTRAINTS WASTE BLOCK CAVING STOCKPILES CLOSURE GRADE-TONNAGE PRICES MINE CUT-OFF LEACHING UNDERGROUND NPV SOCIO￾An Introduction to Cut-off Grade Estimation examines one of the most important calculations in the mining industry. Cut-off grades are essential to determining the economic feasibility and mine life of a project. Profitability and socioeconomic impact of mining operations are influenced by the choice of cut-off grades. Cut-off grades play a key role in estimating mineral reserves that can be publicly reported. This new edition is easier to read and of greater practical interest to practitioners. The relationship between optimization of net present value, capacity constraints, and opportunity cost is explained in greater detail. A new section discusses blending strategies, which play a critical role in an increasing number of mining operations. Author Jean-Michel Rendu, an internationally recognized expert in the management, estimation, and public reporting of mineral resources, provides practical insights. As a manager in major mining companies, a consultant, and an educator, Rendu has acquired considerable experience in all aspects of mining engineering, experience that was incorporated into this publication. The Society for Mining, Metallurgy & Exploration (SME), advances the worldwide mining and minerals community through information exchange and professional development. SME is the world’s largest association of mining and minerals professionals. SECOND EDITION BY JEAN-MICHEL RENDU !SME_COGrade2013_2ndED_FullCV_F5.indd 1 12/26/13 8:21 AM AN INTRODUCTION TO CUT-OFF GRADE ESTIMATION SECOND EDITION RENDU