Strategic Mine Planning

Strategic Mine Planning 8th Edition

David Whittle Jeff Whittle Chris Wharton Geoff Hall

© Gemcom Software International Inc. 2005

Whittle Strategic Mine Planning

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Table Of Contents Strategic Mine Planning........................................................................... 1 The Process of Design .......................................................................... 31 Sensitivity Analysis in Strategic Mine Planning ..................................... 41 Introduction to Optimization................................................................... 55 Open Pit Optimization............................................................................ 63 Economic Model for Pit Optimization..................................................... 83 Generation of Nested Pit Shells ............................................................ 89 Calculating Block Values ....................................................................... 95 Producing a Practical Pit Design from an Optimal Outline ................. 103 The Effects of Underground Mining..................................................... 117 Multiple Mines ..................................................................................... 125 The Effects of Sequencing & Scheduling ........................................... 137 Economic Model for Schedule Optimization........................................ 143 Schedule Optimization......................................................................... 147 Stockpiles ............................................................................................ 155 Blending .............................................................................................. 159 Cost Models for Different Purposes..................................................... 175 Cut-Offs ............................................................................................... 192 Cut-off Optimization............................................................................. 203 The DC Model for material classification and NPV maximization........ 211 References .......................................................................................... 223


Strategic Mine Planning

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Introduction Strategic Mine Planning is the art and science of the management of businesses involved in resource exploitation. It is the convergence of business strategy on the one hand and mining optimization on the other. It therefore embraces the general nature of mining as a business, as well as the special nature of mining as an application of economic geology and engineering. The driving force behind this convergence has been information technology, in particular full value chain modeling systems such as those that have been developed around pit optimization cut-off optimization and schedule optimization. The decision-making model presented here seeks to link business strategy and optimization in a relatively sophisticated, but easy to use way. The model was first developed in a paper presented at an AusIMM conference on the subject of The Relationship between Economic Design Objectives and Reserve Estimates (Whittle 1997). The model has since been refined and presented at short courses at the University of Queensland and at training events in Canada, Chile, Brazil and other countries. In order to properly explain the model it is useful to articulate a range of concepts in a variety of fields before drawing them into the final argument. The fields are organized under the following headings:

Strategic Business Planning Situational Analysis Market Analysis Economic Evaluation

These sections leading up to the explanation of the model have been covered to a greater extent than is strictly necessary to support the decision-making behavior model. This has been done in the hope that the extra information will be interesting and useful to readers who have not received training in commerce, but who are nevertheless involved in it through their engineering or geological input to the strategic mine planning process.

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Strategic Business Planning Military Strategy Cleisthenes became the leader of Athens ~ 500 B.C., and introduced a democratic political system which included ten Strategi, each Strategus being the military leader and representative of one of the tribes of Athens. The manner in which Cleisthenes and his Strategi organized and directed the military forces is credited with the Athenians successfully defeat of the Spartans. They achieved this victory despite the fact that the Spartans were greater in number, and were considered to be individually better warriors. Strategi were leaders in the military sense, but also more broadly they were community leaders. They drew on a whole range of different resources in order to achieve a military result. Strategy is a Greek word meaning “Art of the Strategi”. Quotation from Encyclopedia Britannica on the subject of contemporary (1700 onwards) military strategy: ‘The strategist deals with many uncertainties and imponderables. Indeed the art of the strategist is the art of the “calculated risk”.’ Business Strategy 1971 - Kenneth Andrews published “Business Strategy”: • look at opportunities out in the world and match them with our capabilities. •

SWOT Analysis (Strengths, Weaknesses, Opportunities, Threats).

1980 - Michael Porter published “Competitive Strategy - Techniques for Analyzing Industries and Competitors”: •

understanding of economic and business environment is key to strategic management.

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critical success factors.

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forces driving change.

Dr Jim Landau - July 1998: • “Look to your core competencies when trying to determine your strategic direction.” •

“To be strategic, you need a window to the future.”

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“Optimize your resource usage to get a future advantage.”


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Situational Analysis Contemporary Template for Strategic Planning Situation Analysis •

Size and structure of the market – refer to the Market Analysis section below.

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Strengths – Those qualities that the company possesses which can contribute to its success.

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Weaknesses – Qualities that could contribute to the company’s success, but which the company lacks.

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Opportunities – Factors, events or circumstances in the marketplace that the company can use to its advantage.

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Threats – Factors, events or circumstances in the marketplace that could hinder the company’s success.

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Forces driving change – Factors that will influence the market in the future, including changes in competitive structures, the emergence or disappearance of substitutes or competitors in the market place, and any other factors that affect demand.

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Critical success factors – Those qualities, attributes and assets that a company needs in order to succeed in its chosen market.

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Competitor Analysis – All of the above, analyzed from the perspective of the company’s competitors, so that their actions in the market place may be anticipated.

The Strategic Plan Below is a list of common strategic plan components for any company: • Vision - a view of the future that is important to the organization. •

Mission - key statement of the purpose of the organization

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Values - wider responsibilities and guiding principles

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Sustainable Competitive Advantage – A combination of strengths and critical success factors that the company understands to be vital to future success, and which must be preserved or fostered.

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Strategic Thrust – There are three choices. Firstly a company could choose to be a cost competitor. This would be an appropriate strategic thrust where an un-differentiated high volume (and/or high value) commodity is being sold. Secondly, a company could choose to be a differentiator. Differentiation either costs money to achieve, or it exploits one of the company’s sustainable competitive advantages.

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Either way, provided it is positive an useful differentiation in the eyes or some or all of the buyers, then the seller can command a higher price. The third alternative, is to have a niche strategic thrust. This is appropriate if the product is a low volume product with few buyers, and few competitors. •

Objectives - major milestones

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Strategies - what things will be done in order achieve the objectives

Market Analysis Market analysis is an important part of Situation Analysis. There are two markets that are of interest to mining companies, the commodities market for whatever the mine produces, and the share markets where capital is raised and share value is determined. The commodities market determines the income of the business. The share market determines the shareholder wealth. Performance in the former market of course has a great impact on the performance in the latter. Commodities Markets In market theory there is the concept of a Perfect Market. This is a model of a market, which has defined characteristics and behavior. It provides a reference for discussion and analysis of real markets. Behavior of a Perfect Market The price and volume traded on a market is determined by the intersection of the supply and demand curves. Refer to Figure 1. If the market price should rise, then suppliers will be encouraged to produce greater quantities, even if their marginal cost of production increases as a result. In the long term additional suppliers will be encouraged to commence production. This potential additional supply is represented by the section of the supply curve to the right of the intersection. The section of the supply curve to the left of the intersection represents the contraction of supply that would occur should the price drop. Suppliers who can no longer make a profit will be forced out of the market, reducing the amount of the commodity available for sale on the market. The quantity of the commodity demanded by the market is determined by the demand for the utility imparted by the commodity and by the availability of substitutes.


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Figure 1: Determination of the long-term stable price in a perfect market

Characteristics of a Perfect Market Key characteristics of a Perfect Market: •

Homogeneous product (product from supplier X is much the same as product from supplier Y).

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Substitutes are available – If the price of a commodity rises too much, buyers will stop buying that commodity, and instead purchase a substitute.

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Large number of sellers – no individual seller has independent influence over the market price.

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Large number of buyers – no individual buyer has independent influence over the market price.

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Perfect Information – All buyers and sellers know what other buyers and sellers are trading, and the price at which they trade.

If these conditions are met, then it is believed that the market is operating efficiently in the long term. Sellers cannot make excessive profits, and they are forced to maintain the lowest possible cost of production. Excess demand or excess supply is remedied by a movement in either the demand curve or the supply curve, which leads to a return to equilibrium. Other Market Models Other market models include: •

Monopoly - one dominant seller. In a monopoly, the seller has a significant influence over the price at which his or her commodity is sold. Monopolies generally lead to higher prices than might prevail in

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a Perfect Market. The seller has the power to adjust his or her price in order to maximize their profits. 1 •

Duopoly - two dominant sellers. Duopolies, like monopolies, give significant market power to the sellers. It is not uncommon for the two suppliers in a duopoly to behave as if they are colluding to exploit the market. This is even the case when no actual collusion occurs.

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Oligopoly - a few dominant suppliers. Markets tend to operate like oligopolies when the percentage of supply provided by three or four suppliers reaches around 40%. Refer to Figure 2 for an illustration of this. As the market concentration increases, the sellers profits increase, because as a collective (even if no actual collusion occurs), they have more influence over the price at which products sell than in a perfect market. Monopolies, Duopolies and Oligopolies represent different degrees of seller power. Such power is advantageous to sellers, and in some cases, sellers will cooperate, rather than compete, to collectively achieve that power. An example of this happening is OPEC, a cartel of oil producers. The collective has significant more market influence than would the sum of the members, should they act independently.

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Monopsony - one dominant buyer. The market power is with this buyer.

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Oligopsony - a few dominant buyers. Buyers can cooperate through cartels to create oligopolies.

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Refer to the section Introduction to Optimization. The example optimization problem in that section is one in which the seller has considerable power in setting prices, and utilizes an optimization approach to determine the price at which his or her profit is maximized.


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Figure 2: The relationship between market concentration and profitability (Return on Capital Employed) J B Were circa 2001

How “Perfect” is the Copper Market? Characteristic

Score

Notes

Homogeneous product

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Substitutes are available

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Aluminium for electrical uses and car radiators, optical fibres in telecommunications, plastic in plumbing. 2

Large number of sellers

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400 operating mines, but much fewer companies. Some suppliers are big enough to have an influence on the market. 3

Large number of buyers

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Perfect Information

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Excellent price and trading information available through commodities markets, internet etc.

Copper is an example of a market that has many characteristics in common with a perfect market. There is however a moderate degree of market concentration, which contributes to the market tending slightly towards oligopolistic behavior (refer to Figure 2).

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Source: Crowson, P., Minerals Handbook 1996-97, Stockton Press, New York, p.115 Ibid.

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How “Perfect” is the Iron Ore Market? Characteristic

Score

Notes

Homogeneous product

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Iron is differentiated by its grade, and the grade of a number of contaminants and by its texture. Never the less,


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Aluminum, wood, plastics.


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Production is dominated by Rio Tinto, BHP Billiton and CVRD. The market has oligopolistic characteristics.


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Perfect Information

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Level of prices in general are known, but individual contract information is not. Reserves, rates of production and cost of production are generally known.

Iron ore market has characteristics of an oligopoly. Seller power in the market was illustrated by CVRD’s ability in early 2005 to negotiate a 71.5% increase in price with a significant buyer. The effect of this was to boost the fortunes of all sellers. 4

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MiningNews.net 23 February 2005 “IRON ore majors BHP Billiton and Rio Tinto led the charge on the Australian Stock Exchange this morning with iron ore explorers in their wake as Brazilian giant CVRD announced a price hike of 71.5% overnight after finalizing negotiations with long term partner, Japanese giant Nippon Steel Corporation.”


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How “Perfect” is the Gold Market? Characteristic

Score

Homogeneous product

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Perfect Information

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Notes

Platinum, palladium, silver, titanium, chromium based alloys. Dollars and other currencies, currency hedging instruments, stock.

Excellent price and trading information available through commodities markets, internet etc.

Despite all of the above, the gold market does not work anything like a perfect market. This is because gold is not only consumed in the production of goods, it is also used as a store of wealth, both in the form of jewelry and in bullion. This gold can and does find its way back into the market from these above ground stocks. This secondary market is larger than the primary market for gold. “Above ground stocks of gold are very high, and the willingness to add to, or release from, these stocks largely determines the state of the market. 5

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Source: Crowson, P., Minerals Handbook 1996-97, Stockton Press, New York, p.147

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Share Markets What factors drive the price of shares? The product is not gold, silver coal etc. The product is company shares, which need to satisfy one or more of the following buyer needs: • Potential cash flows (dividends) - Buy shares in the expectation (or hope) that the company will pay dividends in the future. •

Potential cash flows (realization of capital gains) - Buy shares in the expectation (or hope) that the market value will increase and that the increase can be realized.

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Potential unrealized capital gains - Buy shares in the expectation (or hope) that the market value will increase and use the additional value as security.

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Exposure / Diversification - Buy shares in many different companies in many different industries so as to spread the risk of investment.

How ‘perfect’ is the share market?: Characteristic

Score

Notes

Homogeneous product

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Shares are strongly differentiated.


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Because shares are so strongly differentiated, shares in one company are a substitute for shares in any other. Substitutes also include all forms of financial instruments.


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Perfect Information

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Prices and trades are public, “insider information” is not.

The primary market for shares applies to newly issued capital. A large secondary market exists for re-traded shares.


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Economic Evaluation Strategic planning cannot proceed without having a method to determine the value of the various strategic options that will be considered. If you cannot measure the value of a plan/design, you cannot know whether or not a change in it will improve it or make it worse. In a market economy, and in relation to economic matters, it is usual to represent all things in terms of their impact, or potential impact, on cash flows. Doing so gives a single-value measure of a plan/design, which allows easy comparison to all others. Simple Cash Flow Analysis Simple or undiscounted cash flow analysis involves adding up the negative and positive cashflows that are predicted for a project, yielding the net cash flow. By convention, simple cash flow analysis is performed with real dollars 6 as opposed to nominal dollars 7. The simplest possible rules that could be applied to analysis based on simple cashflows are as follows: • If the net cash flows are positive, the plan/design is profitable. If the net cashflows are negative, the plan/design is unprofitable. • In choosing between two or more plans, you should choose the one with the highest net cashflows. Some of the shortcomings of this approach are immediately obvious: • The approach does not take into account the cost of capital or the time-value of money. • There is no account made of uncertainty or risk in the predicted cashflows. • Simple cash flow analysis is a poor platform for calculating / predicting future tax liabilities as tax liabilities are principally based on accrual account calculations (see below). Despite its shortcomings, it has the advantage of being very simple, and it is commonly used for planning functions such as pit optimization. Discounted Cash Flow (DCF) Analysis DCF Analysis takes account of the fact that a dollar that we receive today is more valuable to us than a dollar that we might receive in a year’s time, and expected cash flows are discounted by an amount which increases 6

Real dollars are dollars with constant buying power. Nominal dollars are dollars at their face value, with buying power which changes over time subject to inflation or deflation. 7

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with time. The rate of increase is expressed as a discount rate, such as ten per cent per year. It is the application of the discount rate which differentiates DCF Analysis for simple cash flow analysis. The discount rate can be defined or determined in various ways, including: • Type 1 - risk adjusted discount rate consists of two components: opportunity cost and risk adjustment. Assuming it is your own money that you will invest, the opportunity cost is the rate you could earn (risk free) with the capital elsewhere. The risk adjustment is an additional amount to account for the geological, geotechnical, economic and political risks associated with the project. The risk adjustment must reflect all of the risk associated with the project. •

Type 2 – risk adjusted discount rate consists of two components: cost of capital and risk adjustment. The cost of capital is the interest rate you must pay to borrow the money. The lender may assume some risk in providing the funds, and to the extent that they do, it will be reflected in the interest rate that they set. The borrower (you) also assume the remaining risk associated with the project, and this is reflected in the risk adjustment. However the risk is shared, the total discount rate must reflect all of the risk associated with the project.

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The cost of capital (in the context of this discussion) is the rate of return investors require to supply the funds for the project where they are assuming all the risk associated with the project. It is common for a cost of capital approach to be taken when funding is provided internally, for example, from a head office to a project office.

The formula for calculating the net present value (NPV) for a project is as follows:

n

Rt NPV = ∑ t −C t = 1 (1 + k ) Where: Rt = cash flow for period t k = discount rate C = initial capital expenditure Internal rate of return (IRR) is calculated by solving the following equation for k:


0=

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n

Rt

∑ (1 + k ) t =1

t

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To explain the application of DCF Analysis, it is useful to go through an example. Consider a five year project with an expected cash flow of one million dollars per year and a discount rate of ten per cent. The cash flow calculations are shown below.

This gives a total value today, the Net Present Value (NPV), of $3,790,787. If the project had cash flows of $1.5M, $1.5M, $1M, $0.5M and $0.5M, which gives the same total undiscounted cash flow of $5M, the figures would be as shown below.

The NPV would then be $4,006,588. The increase of $215,801 (5.7%) is caused by $1M of the cash flow being brought forward from years 3 & 4 to years 1 & 2. The simplest possible rules that could be applied to analysis based on simple cashflows are as follows:

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• If the NPV is positive, the plan/design is profitable, and the decision should be to invest in the project. If the NPV is negative, the plan/design is unprofitable and the decision should be not to invest in the project. • In choosing between two or more plans (mutually exclusive alternatives), you should choose the one with the highest net cashflows. Advantages of DCF Analysis include: • DCF Analysis differentiates between projects with identical total cash flows but with different cash flow timings. •

It is very simple to calculate and understand.

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Evaluation method independent of the method for generating scenarios.

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Takes account of risk.

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Allows comparison of different projects with different risk profiles.

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Independent of inflation.

Disadvantages: • Does not take account of a project manager’s ability to adapt to future changes as they occur. Some argue accordingly that DCF analysis will underestimate ‘true’ value. •

The accounting for risk is by an extremely crude mechanism, and the process yields a single risk-weighted value, with no indication of the degree of uncertainty about the outcome (i.e. there is no probability distribution associated with the outcome).

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DCF analysis is a poor platform for calculating / predicting future tax liabilities as tax liabilities are principally based on accrual account calculations (see below).

DCF Analysis is the foundation upon which schedule optimization and cut-off optimization are based in Whittle. DCF also plays a vital role in pit optimization because despite the fact that conventional pit optimization proceeds on the basis of simple cash flow analysis, it is DCF analysis that is applied when evaluating and choosing between various pit outlines produced by the optimizer. Although DCF Analysis is a poor platform for calculating future tax liabilities, in the context of Strategic Mine Planning, it is generally sufficient to work towards maximization of pre-tax NPV, with the


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knowledge that with few exceptions, there is a positive relationship between pre-tax NPV and post-tax NPV. Accordingly maximization of pre-tax NPV, should lead indirectly to the maximization of post-tax NPV. Projected Accrual Accounts Accrual accounting is the normal method for calculating past financial performance and present financial position. It is possible to use accrual accounting, with forecast cashflows used as inputs, to calculate future profitability. The process of accrual accounting proceeds on the basis of nominal dollars. Advantages: • Better than simple cash flow analysis of discounted cash flow analysis for projecting future financial reports and tax liabilities. Disadvantages: • Deals very poorly with risk concepts. •

Subject to inflation.

In the context of Strategic Mine Planning, projected accrual accounts is best applied purely for the purpose of projecting future financial reports and tax liabilities. Option Pricing Techniques Options Pricing is a method whereby the ability to make future decisions is taken into account. It requires the determination of the probability of certain future events occurring, and also the response / decision that would be associated with those conditions in the future. The method deals with multiple future conditions/decisions. The outcome is a probability distribution of NPV. Advantages: • Can account for a project manager’s ability to adapt to future changes / future information. Disadvantages: • Requires more data. •

The application is an order of magnitude more complex that DCF Analysis.

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The evaluation technique is closely tied to the method for generating scenarios, which makes it difficult to implement dynamically with other strategic mine planning tools such as pit and schedule optimizers.

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Requires the generation of a large number of scenarios.

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Options Pricing is a subject of much interest in the mining industry, yet it is not commonly applied. Apart from its complexity compared to DCF analysis, the main reason it is not commonly applied is that because it cannot be applied dynamically with other strategic mine planning tools, what is gained in analytics, is lost in modeling precision and optimization. Monte-Carlo Analysis Monte-Carlo Analysis uses probability distributions as inputs for major economic parameters and it produces a probability distribution for value outcomes. Advantages: • Provides a probability distribution for value outcomes. Disadvantages: • Requires more data. •

Existing systems cannot incorporate pit optimization, schedule optimization and cut-off optimization.

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Monte Carlo Analysis is often implemented in spreadsheet programs with factors such as ore/waste discrimination, pit outline and schedule fixed and/or grossly simplified. As such it is generally used as an adjunct analytical process to the strategic planning function, rather than as a core function. Discrete Probability Analysis Discrete probability analysis uses simplified probability distributions for major parameters. For example, there may be three representative geological models, each assigned with a probability of it being the most representative. Refer to McRostie and Whittle (1999).


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Advantages: • Provides a probability distribution for value outcomes. •

Unlike Monte Carlo analysis, Discrete Probability Analysis can be implemented in a pit optimization / schedule optimization framework, allowing more complex interactions between major input variables.

Disadvantages: • Requires more data. •


Common Practice A survey of the evaluation practices of mineral projects in the mining industry was conducted by the Canadian Institute of Mining Management & Economics Society (CIM MES) in 2005. Refer to Figure 3 for the results of the survey. This study indicates that Discounted Cash Flow analysis is the preferred method for evaluation of feasibility studies, with NPV and IRR topping the list of evaluation metrics 8. Accounting- and market-based methods are seen to be of significantly less importance in the evaluation process. Evaluation Metric at Feasibility Stage NPV IRR Cash cost $/oz or $/lb Experience Hurdle Rate Payback NAV = NPV + cash - debt Break-Even Price Real Options Price/EBITDA ROCE EV = Market Cap + debt - cash Profitability Index Price/EBIT Price/Earnings NAV (SEC def) Price/Cash Flow Market Cap Book Value Price/Book Value

0

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0 = no importance

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6

8

10

high importance = 10

Figure 3: Evaluation Methods of Mineral Projects at the Feasibility Study Stage (CIM MES Survey of Evaluation Practices 2005)

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Source: Smith, L.D., 2005 Survey of evaluation practices in the mineral industry, CIM Management and Economics Society.

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Counterpoint We started out this section by stating that strategic planning cannot proceed without having a method to determine the value of the various alternatives that will be considered. We looked at various systems for measuring value, which cater variously for factors such as opportunity cost, cost of capital, inflation, taxation and risk. However, none of these systems are perfect, and none could be considered “complete”, where “complete” means that all decision-influencing factors are considered. Despite the apparent sophistication of some of the methods, they are only as good as the data that is entered into them and they are poor when it comes to taking into account more qualitative factors. History has provided some spectacular failures for economic analysis being applied to qualitative factors. For example, in the 1970s Ralph Nader drew the world’s attention to a leaked Ford Motor Company report which showed that the location of the fuel tank in the Ford Pinto made it prone to explode in common types of accidents. The report calculated the cost of relocating the tanks in future production (a few dollars per vehicle) and compared it to the cost of legal action against the company as a result of the deaths of future Pinto owners, if the tank was not relocated. The report concluded that the net cost to Ford would be lower if it did not change the design, so condemning scores of future Pinto occupants to death. People died as a result and when the news leaked, Ford’s good-will was seriously eroded. One could argue that the economic model Ford employed could have been corrected by taking into account (in dollar terms) the potential loss of good-will associated with the deaths, should the news have been leaked. It might have led to the fuel tank being relocated, with consequent saving of lives and of good-will. However there are perhaps some things that should be left out of cold-hearted analytical processes, including human lives.


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Decision-making behavior Decision-making behavior affects performance in terms of the commodities markets and share markets, and is influenced very much by a desire to avoid or seek exposure to risk. It is a key factor in the determination of strategies. Risk Neutral In the piloting of a corporation, the CEO is charged with the responsibility of maximizing shareholder value and it is generally accepted that the maximization of Net Present Value (NPV) is consistent with this objective. Where a decision is required as to whether or not to invest in a project, the NPV is calculated. If the NPV is positive, then this indicates that the investment should be made. In choosing the exact configuration of a mining operation for a given resource, the designer is effectively choosing between a large number of mutually exclusive projects. Stermole (1993) states that in deciding between mutually exclusive projects, you should choose the alternative with the highest NPV. In doing so, you would be exhibiting risk neutral behavior. You can afford to be risk neutral if the project under consideration is only a part of your risk exposure. If you have many projects and one fails, it is not a catastrophe. You can, in fact, diversifyaway the risk. Even if you do not have a diversified portfolio, risk neutral behavior can be appropriate. An example of this can be found in the Chairman’s Statement of the Homestake Gold Annual Report 1994, which declares that: “The company’s policy of not hedging the gold price continues in order to provide shareholders full exposure to its fluctuations.” Newmont Mining Company and many others, have similar policies. These company are making themselves attractive to potential investors by exposing themselves to the risk (and upside) associated with fluctuations in the commodity price. The advantages of making themselves more attractive to investors must outweigh any tendency of the companies to avoid risk. Risk Averse Walls (1996) performed some research on the decision-making behavior of mining CEOs and found that they do not generally make highest NPV decisions, their decisions being tempered by risk averse behavior. He found that CEOs of smaller mining companies tend to be more risk averse, and suggested that this is because the higher risk projects carried with them a greater risk of company failure for small companies. Interestingly,

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Walls also found that Australian small company CEOs were more risk averse than their U.S. counterparts but could offer no explanation for this. An alternative approach to the tempering of NPV objectives with risk averse behavior is given by Smith (1997), who suggests that rather than choose the maximum NPV production rate, this rate should be seen as the maximum rate and that the actual rate should be less than this. Part of the reasoning behind the attenuation is that a maximum NPV production rate tends to lead to a short mine life, leaving minimum time to recover from early difficulties. Sating the Market’s Thirst It is the underlying purpose of most companies to maximize shareholder wealth. For the shareholders of a company, it is advantageous to have the share market determine a high value for the shares. The market determines the value of the shares using a number of subjective and strategic criteria, but also uses a selection of quantitative measures. These measures are not necessarily the best way to measure a mining company’s value, but being based on readily available information, they are practical. The methods include 9: • Earnings per share (EPS) and Price Earnings Ratio (PER) •

Cash Flow and Price to Cash Flow Ratios (PCFR)

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Dividend Yield

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Commodity and Reserve Exposure

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Market Value

If these methods are being used to value the shares, then a strategic mine plan may be influenced by the need to sate the market’s thirst for measurable value. For instance, it may be better to structure a project so that it leads to early payment of dividends, even if this means a sacrifice of NPV. A major non-financial measure used in the valuation of shares is the Reserve. Appleyard (1997) writes: ".. the reserve is probably the major input in a company's ability to raise debt or equity finance, and, if listed, in its rating on the sharemarket." Burmeister (1997) states that: “One of the simplest yardsticks for comparing gold companies is the market capitalization per ounce of reserve or resource. Since the adoption of The AUSIMM Code for reporting on Reserves and Resources by all 9

Refer to Rudenno (1998) for a more comprehensive discussion of how investors value mining company shares.


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listed Australian gold companies, the quality of disclosure has improved to such an extent that this comparative tool can now be taken seriously. It is widely used in the USA.” If these views are truly representative of investor behavior, one could conclude that the CEO of a mining company will be inclined to quote the highest reserves that he is entitled to. Taking a Cost Position in the Market Reducing costs is always a sub-objective to maximize profit, but for strategic reasons, low costs may override the objective to maximize profit. This is the case if a company has a relative cost objective - for example, to only produce copper if it can be produced at a marginal cost in the bottom half or quartile of the marginal costs for the industry. Producers with high production costs will be the first to fall should the price fall. Because of the behavior of the market, unless massive changes occur to the demand for the commodity, low cost producers are assured of survival. Example - Copper Negatively sloping demand curve will ensure equilibrium of supply and demand is achieved if supply reduces. Contra-example - Gold A drop in primary supply will be easily taken up by supply from the secondary market. Changing the Market Structure and Behavior When one company merges with or acquires a competitor, or through project start-up gains a larger percentage share of the market, this leads to incrementally greater market concentration. In other words, the market gets a little closer to behaving like an oligopoly, a duopoly or a monopoly. With that comes the potential for increased in commodity prices and potentially beneficial changes in the price cycles (both amplitude and frequency) for all industry suppliers. It would be rare for one single project start-up or small-scale merger to have a measurable effect, never the less, it may be the company’s strategy to pursue market consolidation, and this objective might over-ride to a certain extent an NPV maximization strategy. Other Decision-making Behavior Walls & Eggert (1996) suggested reasons why mining CEOs do not follow classic highest NPV decision-making patterns. Risk aversion associated

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with potential company failure has already been discussed. Other reasons included: 1. The desire to avoid making a short term loss. The CEOs’ continued employment may be linked to their ability to consistently deliver a profit (even if this is at the expense of NPV). 2. Attempts to reconcile the interests of stakeholders other than shareholders, including CEOs themselves, employees and the communities affected by the project. In order to further delineate the basis of CEOs’ decisions, 20 Annual Reports for a variety of Australian mining companies for years 1994, 1995 and 1996 were studied (not necessarily a representative sample, but useful nevertheless). Many companies did not state objectives. Of those that did, the most common objective was to maximize shareholder value. A good example is that stated by Posgold and related companies: “…to maximize returns to shareholders in the form of dividends and capital appreciation by the creation of wealth through long term growth in earnings and assets.” Objectives stated by other companies include the following: 1. To maximize shareholder wealth through growth as a first ranking Australian resource company. 2. To achieve economies of scale through international expansion. 3. To be a low cost gold producer. 4. To operate profitably and to increase the resource base. Bad Behavior! Sometimes economic and strategic objectives are inappropriately reduced to technical objectives. For example: 5. Mining to a prescribed cut-off. 6. Mining to a prescribed stripping ratio. These are old heuristic techniques for achieving a satisfactory economic result. They are entirely superseded by modern optimization techniques discussed in this book. Representing decision-making behavior on a Reserve-tonnage / NPV curve. It is possible to conceive a set of feasible mine designs, each for a different value of a major parameter, and each of which is optimal in that all other parameters are optimized in order to maximize NPV. The major


25

parameter itself affects NPV, and, if you were to graph the NPV against the parameter values, you would have a graph which illustrates the relationship. The major design variables that we have in mind include the size of the ultimate pit, the cut-offs and the investment in processing plant (and consequently its throughput rate). The curve will typically be convex, with a single maximum. Figure 4 shows an example of such a curve for the pit tonnage of a mine.

Figure 4: The typical shape of a Pit Tonnage / NPV graph.

Such curves have been produced in countless papers. Hanson (1997) for example, like most people who produce such graphs, shows only the middle part of the curve. This is because it is the area around the maximum which is generally considered to be most interesting. Smith (1997) presents a number of similar looking graphs, but for NPV v. Production-Rate.

26


Figure 5 is an idealized example of an NPV / Ore-Tonnes-Processed curve. It is possible to create such curves for real data using one of a number of different software packages such as Whittle. These packages design optimal pits and the total ore tonnes is a function of the size of the ultimate pit and other design parameters. The total ore is equivalent to the reserve.

Figure 5: The typical shape of an Ore Tonnes (Reserve) / NPV graph.

It is useful to describe the nature of the curve in detail: • The maximum of the curve represents the optimal design from the point of view of the maximization of NPV. •

Every point on the curve represents an optimal design given the corresponding ore tonnage on the horizontal axis.

•

There are infinite feasible designs which could be represented as points on the graph, but none of these points could occur above the curve. Those designs which are represented by points under the curve are sub-optimal in one or more of the other design criteria not graphed.

•

The area under the curve but above the x-axis represents the feasible domain for pit designs which have a positive NPV and which are therefore economically viable.


27

Figure 6 is similar to Figure 5, except that we have overlaid our comments in relation to the decision-making behavior discussed in the previous section.

Figure 6: Representing decision making behaviour on an ore tonnes / NPV graph.

A risk neutral design will seek to maximize NPV. A risk averse design will not succeed in maximizing NPV because it will employ less capital than is required. This will lead to a smaller than optimal processing capacity, less efficiency and probably a less ambitious mine design. At the far right of the graph is the mine design which maximizes the reserve, while having a positive NPV. A company which is primarily motivated by the need to raise capital could present such a mine design as evidence that the reserve is economically viable. A company taking a cost position in the market may choose a design other than that which provides the highest NPV. The same can be said for a company that is seeking to effect market consolidation – that objective may lead it to choose alternatives other than those that are indicated through DCF Analysis to achieve the highest NPV. An example of such a decision-making behavior feasible domain has been creased using the Whittle package and a training data set called “Monte Bojo”. It is presented below. Several hundred life of mine simulations were performed for a range of processing plant configuration and pit size alternatives. In total, 588 life-of-mine schedules were created and analyzed. The results are shown in Figure 7. For this example, the processing plant and pit sizes were varied, but similar results could be achieved by varying any major mining parameter. One of the interesting features of this example is the flatness of the curve at its maximum. The maximum NPV (Net Present Value) design is 67 million tonnes, but the NPV curve is very flat over a range of between about 55 million and 75 million tonnes.

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In the example, the optimal design and schedule involved a mine life of only 2.3 years with an optimum processing throughput rate of 29 million tonnes per annum. We would imagine that most miners would be reluctant to engage in such a schedule. The risks are extremely high, if for example, the mining were to commence in a cyclical low price period, or if there were difficulties getting the processing plant to perform to boiler-plate recovery 10. In this case the theoretical maximum NPV is unrealistically high, and the achievable best NPV will be somewhat lower. The theoretical maximum NPV is, however, still very useful as a reference point for the economic evaluation of different practical alternatives which the engineer devises. All Production Rates 800 600 400 200 opvalue/db ($m)

0 -200 0

20,000 40,000 60,000 80,000 100,000

-400 -600 -800

Figure 7: Example of a feasible domain. The dots represent a total of about 600 complete mine designs and schedules produced by Whittle.

Application of the Decision-Making Behavior Model The purpose of the Decision-Making Behavior Model is to provide a mechanism by which the value of a broad range of strategic options can be compared and contrasted. DCF Analysis is one dimensional and quantitative. The Decision-Making behavior model provides a structure to analyze some of the qualitative or hard-to-quantify factors and objectives that influence strategic mine planning to be considered in a quantitative framework. For example, it is possible to see through the Decision-Making Behavior Model what sacrifice in NPV is being made in order to achieve Reserve Maximization or to achieve some degree of Risk Aversion. Is it better to pursue reserve maximization and make significant sacrifices in NPV, or is it better to strike a balance? 10

Smith (1997) analyses this particular issue, and in relation to production rates, concludes that the ‘optimum’ production rate should really be regarded as the maximum rate. The most sensible rate may be somewhat lower.


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If a project must be approached with Risk Averse behavior, due to the resources or structure of the operating company, might it be more valuable to (and worth selling to) a larger company, that can operate it with Risk Neutral behavior and consequently achieve a higher NPV?

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The Process of Design

31

32



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Overview of a Simple Planning Process Every ore body is different, but we will now go through the main steps involved in designing an open pit when optimization tools are available to you and where maximization of NPV is the principal objective. The steps are presented as linear for clarity. The actual process of design is iterative, with some steps or combination of steps being repeated many times, and with sensitivity analysis, what-if analysis and auditing being intermingled with the simple steps. Open pit mine planning can be broken down into a series of steps, where each leads into the next. In one such breakdown we produce each of the items shown in Figure 8. Ore-body Model Ultimate Pit Design Long-term Schedule Cut-off Schedule Short-term Schedule Figure 8: Open pit planning steps

While each item has a real effect on any following items, the reverse is not necessarily true, and they are grouped accordingly. For example, while the ore body model is rarely affected by what follows, the pit design, longterm schedule and cut-off schedule can affect each other a great deal. The aims are also different for the three boxes. In creating the ore body model, the aim is to be as accurate as possible, and to record data that will be important to the planning and subsequent mining of the ore body. In the second box, the aim is to maximize the value of the mine by deciding approximately what will be mined and processed in each year of the life of the mine. Here, the year in which a cash flow occurs can have a significant effect on the value of the pit. The emphasis is on getting the over-all scale and mining sequence right. In short-term scheduling, we aim for the smooth and efficient use of the equipment on a day-to-day basis. This affects the value of the mine by its effect on the daily cash flow. Note that, in short-term scheduling, we often have more detailed information about the ore body than is available when doing long-term scheduling. From hereon we will focus on the first two activities in the second box: Ultimate Pit Design and Long-Term Schedule. The cut-off schedule is discussed in the section titled “Cut-Off Optimization”.

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We will assume that a block model has been prepared. Each block will have a total tonnage and may have one or more “parcels”. Each parcel will have a tonnage and a quantity of each element of interest. The main steps: 1. Create a selective mining model 2. Create a sensitivity work model 3. Estimate the general size of the ultimate pit 4. Introduce pushbacks 5. Check the work with the pit design model 6. Adjust the minimum mining width 7. Do the final design and schedule The steps are discussed in greater detail below. Create a Selective Mining model There are two issues to consider with regards to selectivity: Mining Selectivity – In a selective mining model, the shape and size of the blocks is representative of the degree of selectivity that can be achieved in mining operations, most particularly that selectivity that can be achieved at the pit floor and at the pit walls. It is not critical that this be directly representative (this would in fact be difficult to achieve), as the consequences of imprecision are minor. So in practical terms, it means that blocks should be the same height as a bench, and that their width be appropriate for accurate modeling of pit slopes. As a general guideline, it is best to have a block width of no more than 1.5 times the block height, though this is not a hard and fast rule. Geostatistics block shapes and pit optimization pit shapes Note that this aspect ratio (between 1.5:1 and 1:1) may differ considerably from that used in the original generation of the model. Many geostatistical processes lead to blocks which are considerably wider than they are high. They may be longer in one direction (along strike) than the other. There are valid geostatistical reasons why this should be so, and equally valid reason why such a model must be modified to have blocks which are much closer to square, for the purposes of pit optimization. In Whittle, wide blocks can be split into smaller blocks. When this is done, the contents of the original block are distributed evenly between the new smaller blocks.


35

Processing Selectivity – The unit size to be considered when determining which material may be sent to alternate processes or to the waste dump. In pit optimization and scheduling it is the parcel which represents processing selectivity. Accordingly, if there is an ability to segregate some material within a block for channeling to a process, then this can be represented by the use of multiple parcels per block.

Multiple Parcels – Where they come from Within each block of a model there may be one or more parcels, each parcel having its own tonnage, rock type and grade characteristics. In pit optimization and later in life-of-mine (LOM) simulation and scheduling, the parcels represent processing selectivity, however many models are created in which parcels represent something else. Firstly, if a model starts out as a sub-blocked model 11 (a partial model), then it is converted to a regular block model 12, the sub-blocks may end up being represented in the new model as parcels. The sub-blocks in this case are a part of an approximate representation of a surface in the model, rather than as a representation of selectivity. Secondly, some geostatistical techniques generate multiple parcels (sometimes more than 100) in each block. Each set of parcels represent a grade distribution for the block, or a probability distribution of grade for the block. This is a different usage of parcels, which may bear little relation to processing selectivity. Whatever the original reason for the existence of these multiple parcels, it is important to remember that in pit optimization and later, in life-of-mine (LOM) simulation, each individual parcel will be assigned to a particular processing stream or assigned as waste as if it could be completely segregated from the rest of the material in the block. If the parcels in the model do not appropriately represent procession selectivity, then the result may be that when you come to actually mine the plan that was based on the optimized pit and LOM schedule, you find that you are achieving lower grades and higher tonnages than expected. The result economically will be worse than expected. One solution to this problem is to reduce the number of parcels. This can be achieved by using software such as Whittle to operate on a model to combine parcels, so that each block ends up with a smaller number of larger parcels. The user controls the combination of parcels by setting a maximum to the number of parcels that will be allowed in any block of the resultant model. A better solution is to use a model in which the parcels have been created by the geostatistician to properly represent individually selectable unit of material, for the purposes of pit optimization and LOM scheduling. 11

Sub-Block Model - A model in which some blocks are split up into smaller blocks, often to provide greater precision at contacts. 12 Regular Model - A model in which all blocks are exactly the same shape and size, with no sub-blocks.

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Create a Sensitivity Work model If the selective mining model is very large, it can lead to long processing times and slow analysis. If this is the case, then for the purposes of performing high-level analysis and sensitivity analysis, a smaller model should be produced, which will process much more quickly. This can be achieved in Whittle by “re-blocking” the selective mining model, combining blocks by a factor of 2 in each direction. Most of the analysis work can be done using this model, which makes it very quick. During this work you will home in on a design that best meets your corporate aims. You can then do a final check on the design by repeating the final simulations on the design model. It is unlikely that you will find that there is any significant change. Estimate the general size of the Ultimate Pit Set up the base case economics You need a starting point for prices, costs and recoveries. These need not be too accurate or detailed at this point. For example, if the recovery is expected to be non-linear, you can either use an average value or use a grade threshold to simulate the non-linearity approximately. In establishing costs, some assumptions will have to be made about the eventual size of the pit. Set up the slope requirements Use the average slopes, including safety berms but not haul roads at this stage, because you do not yet know where the haul roads will be. Produce a set of pit shells Rather than generating one optimal pit, generate a set of pit shells using the revenue factor parameterization technique (refer to the Pit Optimization section for details). This process provides a database of pits which are used to choose final pits and pushback designs for a broad range of economic conditions. Use revenue factors in the range 0.2 to 1.5 with 100 steps. The very low factor is required in order to generate small internal pit shells which can be used for scheduling. All the pit shells are useful for performing various kinds of sensitivity analysis and for the generation of benchmark schedules. Do Best and Worst case simulations Plot graphs of the NPVs for Best and Worst case simulations of all the pit shells and see how much the two curves differ. If they do not differ by more than two or three per cent, then the sequence of mining is unimportant, and much of what follows can be ignored. In the more normal case, where they differ by twenty or more per cent, the two curves


37

will peak at different pit sizes. Although you will not be able to mine like the Best case simulation, you can expect to get fairly close to its NPV with suitable scheduling. Choose a pit outline which lies between the two peaks, but is closer to the Best case peak. Let us call this pit shell the “working shell”. If this shell is significantly different from the pit size you assumed when setting up the base case economics, adjust the base case economics and repeat the above work. Slope sensitivity At an early stage of analysis, you may only have a rough idea of how steep the slopes can be. It is useful to know in advance just how sensitive the economics of the operation are to the pit slopes. You can check the approximate effect on NPV of changes in the pit slopes, by rerunning the pit optimization several times, each time with different pit slope settings. By rerunning the graphs for the Best and Worst case simulations and comparing the results to those for the original slope settings, you will be able to determine the sensitivity of NPV to the pit slopes. You may also have alternatives available as to pit slope angles. For example, it is sometimes possible, by using techniques such as cable bolting, to increase the pit slopes safely. You can check the approximate effect on NPV of increasing the pit slopes by, say, five degrees, increasing the mining cost by an amount commensurate with the rock-bolting costs, and re-running the above graphs. This will allow you to decide whether it is economic to use cable bolting. Roughly design the haul roads for the working shell Examine the shape of the working shell and decide where the haul roads are going to go. Set up the slopes more accurately, with allowance for the haul roads where required. Repeat the pit shell generation, simulation and the production of the graph. Again choose a working shell. Introduce pushbacks If the expected mine life is going to be more than two or three years, and the pit is physically big enough, the use of pushbacks will probably increase the pit value and allow a more constant stripping ratio. Examine the layout of the working shell and the shells within it. Choose one, two or three shells which could make reasonable intermediate pushbacks. Here the concern is mining width. Providing the separation of shells is sufficient round, say eighty per cent of their circumference, don’t worry at this point if it is too narrow elsewhere, this can be fixed later.

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Through simulations plot a graph showing NPVs for Best, Specified (with the pushbacks you have chosen) and Worst cases for several pit shells either side of the current working shell. Do this both with schedule optimization for improved NPV and for improved balance. How close are the NPV values to the Best case NPV values? Does the working shell still look the best? If it doesn’t, re-design the pushbacks for a new working shell, and produce the graph again. Experiment with different numbers of pushbacks. If increasing the number of pushbacks increases the NPV only a little, it may be wise to use the smaller number of pushbacks. This is because: • Wider pushbacks require less adjustment later, and adjustment always reduces the apparent value of the pit. • It costs money to effect a pushback, and unless you have carefully adjusted the cost models to reflect it, there will be a unmeasured differential in the values of scenarios with fewer pushbacks compared with those with more pushbacks. Some issues to consider at this time: • Is the processing capacity or mining capacity consistently underutilized? If so, there is a mismatch of capacities and you should reconsider your assumptions / designs. • Are there a few periods in which the processing capacity is not fully utilized? If so, you may be able to overcome this by increasing the mining rate temporarily (contract mining for example) or by choosing different pushbacks or by introducing buffer stockpiles. • Do the mining rates vary wildly from one period to the next? This is rarely desirable, as it means that a large mining fleet is often underutilized, and the cost of underutilization may not be adequately modeled in the simulation. The problem may be overcome by choosing different pushbacks, or by making appropriate adjustments to the cost model so that the schedule optimizer produced a more appropriate schedule. Check the work with the selective model Re-run your final version of the above work using the pit design model rather than the sensitivity work model. There will usually be very little change (1% to 2%) to any of the key figures.


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Adjust the minimum mining width Using the selective mining model and the pushbacks you have established, run the Whittle Mining Width module to adjust the mining widths before re-running again. There will be a loss in value, but it should not be anything like as big as the gain made by using pushbacks in the first place. Do the final design You now have a final pit and one or more pushbacks which you can use as guides in using the final design. This will usually be done in your GMP.

Sensitivity Analysis and What-If Analysis The need for sensitivity analysis and what-if analysis cannot be overemphasized. The more you do, the better your design and long term schedule will be. Sensitivity analysis is performed in order to determine the overall impact of variance in major input variables, such as grade, commodity price and costs. What-if analysis is used to examine alternative operational and investment scenarios. The whole design process is iterative. Do not imagine that you create a geological model, create a value model, optimize it and then do the final design. Much of what we discuss from now on will relate to sensitivity work, but, before we begin, there are two aspects of sensitivity work which are worth emphasizing. Optimization is important in sensitivity work Although an optimized outline and the corresponding detailed pit design are not the same, they do have a close relationship. Consequently, provided you are using a mathematical optimizer rather than an approximation, it is not necessary to do two detailed pit designs in order to compare the values of two optimal outlines for different conditions. Because true optimization is objective and single-valued, it is quite reasonable to take note of small value differences due to, say, changing the slopes by a few degrees or changing the price of gold by $5. This is not true when designs are done by hand, because an engineer would probably produce different designs on different days, without any change of slopes or price. Two engineers would certainly produce different designs for the same conditions.

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The main sensitivity is to economic uncertainty The size and shape of an ultimate pit outline is affected more by the economic conditions when mining ends than the economic conditions when mining starts. If the mine life is to be only a year or two, then we can probably predict the economic conditions reasonably well. Otherwise we have a problem. We can see from the formula for a block value 13: VALUE = (METAL*RECOVERY*PRICE - ORE*COSTP) ROCK*COSTM that there are three relevant economic variables. These are PRICE, COSTP and COSTM. Sensitivity to all of these must be explored. What-if Analysis There are many operational alternatives which can be checked by performing what-if analysis. Here we give just three examples. Is the processing plant the right size? You can quickly check the effect of different processing plant sizes by rerunning simulations with different throughputs and processing costs. This should always be done with schedule optimization. Of course you have to allow for the different construction costs of different sizes of processing plant. If the change looks promising, you should check if it would have an effect on the best pit size. Should we build a second processing plant later? This is very similar to the previous check. The pit is restricted by some infrastructure - should we move it? This is a bit more complicated to check because you have to produce new pit shells without the obstruction, but most of the work can be done by repeating previous runs, which is very easy. If, at this stage you find that you have to add a significant amount of waste to the pit, then the amount by which you laid back the slopes was insufficient, and you should go back and repeat some of the work above, otherwise the pit will be deeper than it should be. 13

Refer to the section “Calculating Block Values for Pit Optimization” for an explanation of the equation and terms.


Sensitivity Analysis in Strategic Mine Planning

41

42



43

Originally presented as: Whittle, D, 2000, Proteus Environment: Sensitivity Work Made Easy, in Proceedings Whittle North American Strategic Mine Planning Conference, (Breckenridge, Colorado)

Abstract There are many inputs to the process of designing an open cut mine and uncertainty in the value of any of these inputs leads to uncertainty in the economic outcomes. For example, a change in one input, such as the commodity price, can have a direct influence on the economics of resource exploitation. This paper provides general discussion of how Whittle software can be used to address the two questions: •

What is the impact of input uncertainly on the project? and,

•

What, if anything, should be done about it?

The first question is answered by performing one of several forms of sensitivity analysis. This involves varying the inputs to the design process and measuring the impact on the value of the project. This sort of analysis is easily achieved using the Whittle package and various techniques are presented. The second question is dealt with by designing and testing a number of different scenarios, which involve one of the following strategies: •

Take steps to reduce the uncertainty.

•

Design in such a way as to reduce the impact of the uncertainty.

•

Tolerate the uncertainty and accept the associated risks.

Measuring the Impact of Input Uncertainty on the Project Various forms of sensitivity analysis can be performed in order to measure the impact of input uncertainty. They include: Spider diagrams. Single or multi-variable sensitivity analysis. The McRostie/Whittle method for determining probability distributions for project Net Present Value (NPV).

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Examples of these forms of analysis are given below. Spider Diagrams A spider diagram allows the rapid testing of a variety of inputs, to determine which ones the project value is most sensitive to. The example shown in Figure 9 tests the sensitivity of a variety of inputs. The base case is the centre of the spider, and each of the legs represents a simulation in which a variable is changed plus or minus a fixed percentage. In this example, the inputs were varied plus and minus 10%. The diagram clearly shows that, in this example, the processing throughput limit to the processing plant is the input to which project value is most sensitive.

Figure 9: A spider diagram, created by Whittle. To create this graph, it is necessary to produce 15 life-of-mine schedules and graph selected results. These operations are carried out by Whittle. Once the basic operational scenario has been built, the user need only specify the parameters that are to be tested in the spider diagram.

Single or Multi-variable Sensitivity Analysis Those inputs that are deserving of further investigation can be tested in detail by running multiple scenarios, for a range of different values of the input. An example of this is shown in Figure 10.


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Figure 10: Sensitivity analysis for the recovery of sulphide rock in the processing plant. The recovery was varied in steps from 70% to 96%. For each of the 14 scenarios, Whittle produced a life of mine schedule, complete with recalculation of the cut-offs, and recalculation of discounted cashflows. The graph, which has been produced by Whittle, shows the result in terms of the total ore tonnes and NPV for the mine.

The McRostie/Whittle Determination of Probability Distributions for Project NPV McRostie and Whittle (1999) determined a method which utilized the scenario branching capabilities of Whittle to produce results that could be graphed to show the probability distribution of project values, depending on a number of uncertain inputs. The statistical concepts relied upon by McRostie and Whittle are comprehensively described in Gentry & O’Neil (1984). A summary of McRostie and Whittle (1999) is given below. This probabilistic approach to risk analysis is extremely useful as it gives not only a central NPV estimate, but also provides a range of NPV and the likelihood of the project falling within that estimate. This approach is preferable to producing a spot estimate of a project’s worth, and, using the tree structure in Whittle, risk that is associated with the assumed geological, geotechnical and operational limitations and environment can be evaluated simultaneously. To study the concept, we will use a simple example that is derived from the Whittle tutorial data set. The simple example presented here uses only

46


27 tests, and so provides very rough results. Tests of 125 or more are recommended if a reasonable estimate of the final distribution is to be made. The first step in determining the risk associated with the operation is to determine the subjective probability distribution of each major risk factor, including the geological, geotechnical and operational limitations/environment. Table 1: Subjective Probability Distribution of the Resource Model Model Name Model A

Probability 20%

Model B

60%

Model C Sum:

20% 100%

Notes Simulated by removing last 12 benches of the model. Simulated by removing last 6 benches of the model. Complete model

Table 2: Subjective Probability Distribution of Geotechnical Model Slope Name Geotechnical A Geotechnical B Geotechnical C Sum:

Probability 15% 70% 15% 100%

Notes Shallow slopes, 41° Normal slopes, 45° Steep slopes, 49°

Table 3: Subjective Probability Distribution of Operational Limitation / Environment Risk Factors

Name Operational A T Operational B

h e Operational C Sum:

Probability 25% 50% 25% 100%

Notes Low gold price (US$300/oz.) Current14 gold price (US$400/oz.) High gold price (US$500/oz.)

The probability of each result is calculated by multiplying together the subjective probabilities of each node that contributed to the result. For 14 The current gold price indicates the price at the time the Whittle tutorial data set was created, not the current gold price at the time of writing.


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example, the first branch shown in Figure 11 consists of a Block Model (20%), a Slope Set (15%) and the Pit Shell/Scenario/Analysis combination (25%) giving a probability for the result of 0.75%.

Figure 11: Probabilistic Risk Analysis Using Whittle. This screen dump shows some expanded and some collapsed branches of the analysis tree. In total there are 27 analyses (not all visible here), representing all possible combinations of the three uncertain inputs.

The probability of each result, and the NPV result itself were entered into a spreadsheet from which the weighted mean and standard deviation can be calculated. The weighted mean was calculated by summing the product of each NPV result and its associated calculated probability. The standard deviation was calculated using Equation 1. Equation 1: Standard deviation for infinite population.

σ =

∑(X

2

p( X )) − X 2

Graphing the cumulative probability versus NPV produces an unusual curve (see Figure 12) that does not look much like the classic S curve for cumulative normal distributions. The strange shape of the curve is due to

48


the small number of NPV results and because of the discrete form of the input variables.

Cumulative Probability 3x3x3 data set 1.00

0.90

0.80

0.70

Probability

0.60

0.50

0.40

0.30

0.20

0.10

0.00 -

5,000,000

10,000,000

15,000,000

20,000,000

25,000,000

NPV

Figure 12: Cumulative Probability for Simple 3x3x3 Data Set (27 Data Points)

A more advanced test was conducted which consisted of five block models, five different slope sets and five different gold prices, producing a total of 125 NPV results. This test produced a very interesting cumulative probability curve (see Figure 13) that clearly has five smaller curves combined with a skewed cumulative S curve. To highlight the underlying S curve, the data was smoothed by extracting the mid-point of each subcurve and fitting a curve to the result (see Figure 14).


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Cumulative Prob 5x5x5 Data set 1.00 95.875%

81.850%

0.80

Probability

0.60 53.200%

0.40

0.20

18.600%

3.950% 0.00 -

5,000,000

10,000,000

15,000,000

20,000,000

25,000,000

NPV

Figure 13: Cumulative Probability for More Complicated 5x5x5 Data Set (125 Data Points)

C um ulative P ro bability S m o oth ed 5x5x5 D ata set 1

1 0.95875

0.9 0.8185

0.8

0.7

Probability

0.6 0.532 0.5

0.4

0.3

0.2

0.186

0.1 0.0395 0

0 0

5000000

10000000

15000000

20000000

25000000

NPV

Figure 14: Smoothed Cumulative Probability of 5x5x5 Data Set

Once a curve like the one presented in Figure 12 has been plotted, rough estimates as to the risk involved in the project may be made. Our sample shows a mode of approximately $9m, with NPV values 25% either side of the mode of $6m and $13m. Such a range is much more useful than a spot estimate when evaluating projects.

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Strategies for Dealing with the Uncertainty Once you have determined the impact of uncertain inputs on the project value, you can determine a strategy to deal with it. The alternatives are: •

Take steps to reduce the uncertainty.

•

Design in such a way as to reduce the impact of the uncertainty.

•

Tolerate the uncertainty and accept the associated risks.

These are discussed below.

Taking Steps to Reduce the Uncertainty Figure 15 shows a model of the relationship between the uncertain inputs, the economic outcomes, and the decisions/investments that can be made to reduce input uncertainty. Details of the model are as follows: •

Ellipses represent uncertain inputs to the design process. For example, the geological model is a statistical estimate of the real mineralization, based on sample data.

•

Rectangles represent decisions or investments that can be made in order to reduce the uncertainty associated with the inputs to the design process. For example, the uncertainty of the geological model can be reduced by investing in more drilling (geological research).

•

The round-edged rectangles are intermediate and ultimate results of the various elements in the model. For example, an increase in geological research (drilling) will increase design costs.


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Figure 15: Influence diagram showing uncertain values (ellipses) and decisions (rectangles) which influence the uncertain values and lead to outcomes (rounded rectangles). It indicates, for example, that the uncertainly of the geological model can be reduced by increasing expenditure on geological research.

Four categories of uncertain inputs have been chosen for the model: •

Geological

•

Geotechnical

•

Operational, and

•

Product Price.

Apart from being generally useful categories, they also happen to map well to the design process inherent in the Whittle package. This relationship is represented in Figure 16 and Figure 17.

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Figure 16: Mapping of the uncertain inputs of the design process to the artefacts of the planning process

Figure 17: The planning process artefacts as they are represented in the Whittle

With the exception of product price, the uncertainty of all of the inputs can be influenced by the degree of research into the field. Taken to its logical extreme, it is possible to virtually eliminate the uncertainly of all inputs, except price, by investing more money in the appropriate research. Price is the exception, because it is something that does not yet exist. No matter how much money is invested into research of future prices, a great deal of its uncertainty will remain with you. With this model in mind, it is possible to test cases in Whittle whereby the impact of an uncertainty-reducing strategy can be tested. For example, you could create a hypothetical drilling campaign model. The key elements of the model for the purpose of the analysis are the cost of the campaign, and an understanding of what the campaign will achieve in terms of reducing the uncertainty of the geological model. By building the hypothetical drilling campaign into the Whittle model, it is possible to evaluate the costs and benefits of the strategy.


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Design to Reduce the Impact of the Uncertainty Design strategies can be used to reduce the impact of uncertainty. Examples are: •

The use of strategic pushbacks 15. In designing pushbacks, there are three general areas of consideration: operational, tactical and strategic. Strategic considerations include the establishment of options in the future. For example, it may be that the most efficient way to mine from an operational point of view is to bench-mine out to the ultimate pit. However, for strategic reasons, it may well be wise to build in pushbacks that, when completed, give the mine designer the option of redesigning the ultimate pit, with the benefit of more up-to-date information.

•

The use of contract mining. If mining is conducted directly by the mine owner, then the company itself will be subject to all the uncertainties and risks associated with fleet ownership and management and employment issues. Contract mining externalizes these factors and potentially reduces the uncertainty associated with future costs (depending on the structure of the contract).

•

The use of hedging. Hedging can be used to reduce the uncertainty associated with the price that will be obtained for the product of the mine.

In all cases, it is possible to simulate these scenarios using Whittle package and measure the impact on project value and risk. This is achieved using the scenario branching capabilities of Whittle.

Tolerate the Uncertainty and Accept the Associated Risks It should not be assumed that the elimination of risk (at the expense of project value) is the ultimate goal. The degree of risk the company is prepared to expose itself to will depend to a large extent on the decisionmaking behavior (Whittle 1997). Having measured the project risk using sensitivity analysis, and having determined the costs of reducing that risk, it may well be that the best strategy is to tolerate the risk.

15

Alternatively, this can be thought of as the consideration of strategic issues when choosing pushbacks.

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Introduction to Optimization

55

56



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What is Meant by Optimization? “Optimization” is a process in which something is made as effective, perfect or useful as possible. The term can be used in a general sense to mean a process through which an outcome is “optimized” through the adjustment of inputs, structures or methods. You will hear the term used in the mining domain often. For example, an optimization can be performed on a Cyclone: In such a process, the investigator seeks to determine how the Cyclone can be operated in order to facilitate best possible separation of materials, at minimum cost. Think of “Optimization” generally as a process to “make something better”. The term also has a mathematical definition: which means to find the optimal value of a function, often subject to constraints. “Optimal” in mathematical terms means one of the following: Minimal – The lowest possible value. If you were to optimize total cost, you would be seeking to minimize it. Maximal – The highest possible value. If you were to optimize total profit, you would be seeking to maximize it.

Models for Optimization A mathematical model for optimization has the following components: 1. An Objective Function – The objective function is a mathematical expression, or some other kind of mathematical model which calculates that thing which is to be optimized. If the objective function calculates profit then the optimization process will find the maximum profit through the objective function. There is only one objective function in an optimization problem. 2. Decision Variables – To optimize the objective function, an optimization process must find the appropriate settings for one or more decision variables. In algebra terms, the decision variables are independent, the objective is a dependent variable. There may be one or many decision variables in an optimization problem. 3. Constraints – The settings of the decision variables are subject to constraints, both on the decision variables themselves, and on functions of the decision variables. That’s a general description, which might seem a little esoteric. Below is simple example which will be used to illustrate the concepts.

Simple Optimization Example Many optimization problems are extremely complex. Pit optimization and schedule optimization employ advanced techniques and complex models

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to achieve a result. However, all optimization problems have some characteristics in common. To illustrate these characteristics, it is useful to work through a simple example. The example set out below uses only high-school algebra to find a solution. Consider that you are a manufacturer of something (for example, vegetable peelers), which you sell to wholesalers. The quantity you sell is dependent on the price you set. If your price goes up, the quantity you can sell will go down. At a price of $0.20 you can sell 13,000 units per week. At a price of $0.50 cents you can sell only 4,000 units per week. Let us presume that this relationship between price and volume is linear, so you can calculate the units sold with the equation Units = 19,000Price*30,000 16. Your costs are $0.01 per unit plus $1,000 per week. C(cost) = 1,000+0.01*U(units). R(revenue) = P(price)*U Pr(profit) = R–C U = 19000-30000*P The question is: What price should you set, in order to maximize weekly profit? The objective function to maximize is: Pr = R – C → Pr = PU – (1000+0.01*U) → Pr = P*(19000-30000P) – (1,000+0.01*(19000-30000P)) → Pr = -30000P2+19300P-1190 The Profit (Pr) is the thing that is to be optimized (maximized in this case). The decision variable (only one in this case) is Price (P). Constraints include such things as the fact that one cannot set a negative price, nor can you produce a negative quantity, though in this simple case, such constraints are superfluous. You may recognize the structure of the objective function as a quadratic trinomial. That is a coincidence, it happens to describe to relationship between the decision variable and the objective function in this case. Optimization problems can take many forms, but they all have an objective function, decision variables and constraints. Now, let us solve this simple optimization problem. This one is so simple, we can enumerate to work out an approximate answer. Enumeration means to try all possible values of the decision variable(s), and find the one(s) that yield the optimal result. Reading from Table 4, you can see that a price of approximately 0.30 yields the highest profit. Table 4: Enumeration of the Profit Maximization problem Price 0.00 16

Units 19000

Revenue 0

Cost 1190

Profit -1190

You can calculate this using simultaneous equations and the linear expression form Y=aX+c, where Y is the number of units sold and X is the Price.


59 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45 0.50 0.55 0.60 0.65

17500 16000 14500 13000 11500 10000 8500 7000 5500 4000 2500 1000 -500

875 1600 2175 2600 2875 3000 2975 2800 2475 2000 1375 600 -325

1175 1160 1145 1130 1115 1100 1085 1070 1055 1040 1025 1010 995

-300 440 1030 1470 1760 1900 1890 1730 1420 960 350 -410 -1320

Profit 2500 2000 1500 1000 Profit

500 0 0.00 -500

0.10

0.20

0.30

0.40

0.50

0.60

0.70

-1000 -1500

Figure 18: Profit charted against Price.

There can be more than one way to finds an optimal solution. In this simple case, we can apply some algebra to solve it, by finding the value of P at which the derivative of the objective function is equal to zero. That yields a precise answer of 0.32617. 0 = -2*30000P+19300 → P = 0.32617 The Profit (Pr) for P = 0.32617 is: Pr = -30000*(0.32617)2+19300*0.32617-1190 → Pr = 1913.47 We could also apply a step and stride routine, to try different values of P, in order to find the peak of the graph, which corresponds to the highest value of Pr. Some further observations based in part on this optimization example: 1. We made some assumptions about the behavior of sales volume (Units) with respect to price. In reality, we could probably not expect such highly predictable market behavior. The expression [U = 19000-

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30000*P] is simply a model that approximates reality. Would you really expect to sell exactly 19000 units if you set the price to zero? Would sales volume become zero at a price of exactly 0.6333? It can be said for all models – that they are simplified representations of reality. We can’t expect reality to behave exactly like a model especially when you get to extreme values. We can use models like this to our benefit, but should be ever mindful of their limitations. 2. In this example, there is only one correct answer: 0.32617. There is no other setting of P what will yield the maximum profit of 1913.47. As you can probably intuit, if you were to set the price any lower or higher than the range shown in Table 4, the profits would fall. Refer to Figure 18 which illustrates this well. In the strategic mine planning domain, pit optimization, using the Lerchs Grossmann method, is an example of an optimization problem for which there can only one correct answer. For other types of optimization problems there may be more than one optimal solution. That is, two or more settings of decision variables, that yield exactly the same optimal objective function. In the strategic mine planning domain, blend optimization is an example of an optimization problem for which it is possible for two or more settings of decision variables to yield exactly the same optimal objective function. 3. Once we have an objective function defined, we may be able to apply more than one method for finding an optimal solution. We provided examples of enumeration, utilizing a derivative and we discussed briefly step and stride. These are some of the many approaches to optimization that may be applicable to different types of optimization problems.

Conclusions We worked through a very simple optimization problem. We will consider much more complex cases in the following sections, but they will share in common the following: 1. An objective function, which calculates the variable which is to be optimized (minimized or maximized). 2. Decision variables – which are variables that influence the objective function, or to put it another way, the objective is a function of the decision variable.


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3. Constraints – Decision variables and functions of decision variables can generally only work within certain ranges. For example, you shouldn’t have negative prices and you can’t manufacture negative quantities of goods. In this case, the problem is simple and we can apply some simple techniques to find the optimal solution: 1. Enumeration 2. By solving for P when the derivative is zero 3. By a step and stride routine. We will discuss other approaches to optimization in the following sections of this book.

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Open Pit Optimization

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What is Meant by Open Pit Optimization? Let us now look in more detail at the problem of pit outline optimization. A definition of the problem is as follows: Objective Function – maximization of the profit yielded by the pit, calculated as the cumulative value of all the material (ore and waste) which is inside the pit. Decision Variables – the inclusion or exclusion of material in the pit (and consequently, the outline of the pit). Constraints – pit slopes must be obeyed. Those definitions are not mathematically precise, and they must become so before they can be used to formulate a mathematical solution. However, for the moment, these general definitions will do. Please bear them in mind as you read the following sections.

Definition of the Optimal Outline Consider Figure 19.

Figure 19: Typical Ore Body

For any orebody model there are many feasible pit outlines. In fact the number of technically feasible outlines is usually very large. Any feasible outline has a Dollar Value. In this context “feasible” means that it obeys safe slope requirements. (We will discuss haul roads later.) Dollar Value = Revenues - Costs Revenues can be calculated from ore tonnages, grades, recoveries and product price. Price is often the main unknown factor but, in order to design at all, some price must be assumed. Costs are much more complicated, and we shall spend more time on them later. For the moment, we will assume that the costs of mining and processing are known.

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The dollar value of any feasible outline can, in theory, be calculated by totalling revenues less costs for every cubic meter, or every block, within the outline. Note that, as well as the whole outline having a value, any portion of the pit has a value. The optimal outline is defined as the one with the highest dollar value. Nothing can be added to an optimal outline which will increase the value without breaking the slope constraints. Nothing can be removed from an optimal outline which will increase the value without breaking the slope constraints. In other words, we mine everything which is “worth mining”. What Affects the Optimal Outline for a Given Ore Body ? Prices and costs In general - if the product price goes up, the optimal pit gets bigger. In general - if costs go up, the optimal pit gets smaller. Slopes In general - if we use steeper slopes, the optimal pit gets deeper. Once The Above Factors Are Fixed - So is The Optimal Pit We may not know the outline, but it is fixed, and, if we exclude pit extensions of exactly zero value, there is only one optimal outline. To prove this, let us postulate that there are two optimal outlines for the same ore body. That is, there are two different outlines which satisfy the slope constraints and which both have the same total value, which is the maximum possible total value. If we can prove that this is an impossibility, then we have proven that there is only one optimal outline. There are two possible cases.

Figure 20: Disjoint pits

Figure 20 represents the first case. Here the two pits are disjoint, that is, they do not intersect. This is clearly a nonsense because, if we mine both pits, we get twice the value of either one, so that neither one could have been of maximum value (i.e. optimal).


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Figure 21: Overlapping pits

In the second case, where the two pits overlap as in Figure 21, we note that each of the three regions A, B and C have a value, and the value of the combined pit is

A+B+C If the two pits are of the same value, then

A+B=B+C or

A+B+C=B+C+C

But C must be positive, otherwise it would not be included in any optimal pit, so the combined pit has a value greater than B + C. Consequently, the supposition that two different optimal pits can have the same value is false.

A Simple Pit Optimization Example Let us consider a very simple ore body which is rectangular, of constant grade, and sits beneath a horizontal topography. Let us further assume that it is of infinite length, so that we do not have to allow for end effects, and need only consider one section. Figure 22 shows such an ore body.

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surface 1 2 3 4 5 100 tonnes waste

bench level

6 7 8 500 tonnes ore

Figure 22: Simple ore body

Using the quantities indicated in the diagram, we can calculate the tonnages for the eight possible pit outlines, and the following table shows these. Tonnages for possible outlines Pit Ore Waste Total

1 500 100 600

2 1,000 400 1,400

3 1,500 900 2,400

4 2,000 1,600 3,600

5 2,500 2,500 5,000

6 3,000 3,600 6,600

7 3,500 4,900 8,400

8 4,000 6,400 10,400

Note that, although the ore tonnage increases linearly with pit number, the waste tonnage increases as the square of the pit number. If we assume that ore is worth $2.00 per tonne and that waste costs $1.00 per tonne to mine and remove, then the following table shows the value of each pit. Pit values for ore at $2.00/T and waste at -$1.00/T Pit Value

1 900

2 1,600

3 2,100

4 2,400

5 2,500

6 2,400

7 2,100

8 1,600


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Clearly, pit 5 has the highest value, but note also that the values of pits 4 and 6 differ only by a small amount from that of pit 5. This is very important to realise, and is shown graphically in Figure 23.

Figure 23: Pit value against pit size

Note that the graph goes through a smooth maximum. Although this is an artificial example, such a smooth maximum is normal for real ore bodies. In my experience, a range of ten per cent in pit tonnage usually involves a range of less than two per cent in pit value. This has a profound effect on the process of designing pits. value

B A

tonnes

Figure 24: The small effect of design changes near the maximum

As you can see from Figure 24, small deviations from a design which is not optimal (A) can have significant effects on the pit value. Thus generations of mining engineers have experimented with small changes to try to improve their designs. This is quite unnecessary if you start from an

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optimal outline (B) where small deviations have only a second order effect on the value of the pit. Whatever value you aim to maximize, if you work near the maximum, the value you achieve becomes insensitive to the pit outline. This fact is one of the most important in the area of pit design.

What Pit Outline Optimization Methods Are Available? All currently available methods attempt to find the optimal outline in terms of a block model consisting of a regular matrix of blocks in three dimensions. The different methods try to find the list of blocks which has the maximum possible total value, while still obeying the slope constraints. The enormity of this problem is seldom appreciated. Enumeration Consider a trivial model with only one section and ten benches of ten blocks, as is shown in Figure 25.

10

10 100 Blocks Figure 25: A trivial model

If we take a very simple-minded approach, there are 100 blocks and each can either be mined or not mined. This gives 2100 or 1030 alternatives! Even if a computer could check out an alternative every microsecond, it would still take three million times the age of the universe to check them all! If we start in any position at one end and then go up one, down one, or stay at the same level, then there are about 10x39, or 200,000, alternatives. Actually it is 156,629 as the table below shows. This is because the number of alternatives at the top and bottom is not 3.


Depth 0 1 2 3 4 5 6 7 8 9 10

71 Number of different ways of reaching a particular depth in each column 1 2 3 4 5 6 7 8 9 10 1 2 5 13 35 96 267 750 2123 6046 1 3 8 22 61 171 483 1373 3923 11257 1 3 9 26 75 216 623 1800 5211 15114 1 3 9 27 80 236 694 2038 5980 17540 1 3 9 27 81 242 721 2142 6349 18782 1 3 9 27 81 243 727 2169 6453 19151 1 3 9 27 81 242 721 2142 6349 18782 1 3 9 27 80 236 694 2038 5980 17540 1 3 9 26 75 216 623 1800 5211 15114 1 3 8 22 61 171 483 1373 3923 11257 1 2 5 13 35 96 267 750 2123 6046 Total: 156629

156,629 can be handled by a computer - but if we extend the model to 10 sections, as in Figure 26, then we have about 10x299 or 1030 alternatives and three million times the age of the universe again, and this is still a very small model of only one thousand blocks.

10

10 10 1000 Blocks Figure 26: A very small model

Conclusion - Enumeration is useless. Floating Cone This is a step up from pure enumeration - a “heuristic” best described as a guided trial and error. Floating cone works by repeatedly searching for ore blocks and then checking if the blocks in the inverted slope cone which sits on the ore block has a total value which is positive. If it is positive, the cone is “mined” and the search continues. When no more positive cones can be found, the blocks “mined” constitute the optimal pit. Unfortunately, programs that use this approach can encounter problems.

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Ignoring co-operation - mining too little.

Figure 27: Ore bodies and extraction cones

With the ore and waste values shown in Figure 27, a floating cone program would examine the left-hand ore and decide that the value of the cone was 100-80-30=-10. It would decide not to mine the ore. It would then make a similar decision for the right-hand ore body and would conclude that there was nothing to mine. In fact, we can see that, if we mine both ore bodies, the mine has a total value of +10. Although all sorts of tricks are resorted to, floating cone programs cannot really get around this problem without using enumeration, and we know that that is useless. Now, totally disjoint ore bodies are not common. However, co-operation of this sort between different protuberances of an ore body will frequently make a significant difference to the outline and value of a pit. Mining too much - pulling up the waste.

Figure 28: Ore bodies and extraction cones

With the ore bodies shown in Figure 28, a floating cone program will first examine the +40 ore body and decide that it is not worth mining. It will then examine the +200 ore body and will decide that it is very well worth mining (200-80-30=+90). It will remove it from the model, together with the waste which must be removed to expose it. The situation is then as shown in Figure 29.


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Figure 29: Situation after first ore body is removed

If the program goes back to the top of the model and starts searching for ore again, then the final result will be that the +40 and +200 ore bodies will be mined and the +60 ore body will not. This would be correct. However, in computational terms, it can be very expensive to go back to the top of the model each time any ore is removed and some floating cone programs continue downwards in their search in order to save time. In this case the next ore body examined is the +60 one. The current contents of its cone are 60-70+40-20=+10, so the program mines out this cone, which is wrong. The +60 ore body should never be mined because 70 of waste has to be removed to uncover it. The main advantage of the floating cone method is that it is easy to understand. However, it can overlook co-operation and can use the value of ore to pay for the mining of waste below that ore. Two-dimensional Lerchs-Grossmann Lerchs and Grossmann published two algorithms for open pit optimization. One was a very simple method for optimizing a single section. It has two main limitations, and they are fatal. The first is that it can only handle slopes which can be produced by repeatedly moving one block up and one block across. Thus, to change the slopes, you have to change the block dimensions. In the original publication, the slopes had to be the same in both directions. Although a variant has since been published which allows one up and two across on one side, the slope modelling is still very poor, even if you have control over the block proportions. The second restriction is that the optimization works on each section independently and adjacent sections usually do not join up. This leads to considerable manual adjustment when doing a detailed design. In no sense does it arrive at an over-all optimal outline.

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Two-and-a-half Dimensional Lerchs-Grossmann Many people have extended the two-dimensional method by running it North-South and East-West and trying to join the sections up that way. This has been only partially successful. It still does not arrive at an optimal solution, and the slope modelling suffers from the same restrictions as does the original two-dimensional method. In addition, slopes are always too steep in the NW-SE and NE-SW directions. As can be seen below, one block up and one across in the North, South, East and West directions from a single valuable block produces a diamond shaped pit rather than a circular pit. If the slopes EW and N-S are 45 degrees, then the slopes NW-SE and NE-SW are about 55 degrees. This illustrated in Figure 30.

Figure 30: This diagram shows the outline of a pit generated by stepping one up and one across, starting from a single ore block and going up five benches. The numbers show the levels at which each step-out occurs.

Ernest Koenigsberg - 17th APCOM This method is similar to the two-and-a-half dimensional LerchsGrossmann method, but it works on both axes simultaneously and achieves better joins between the sections. It suffers from exactly the same slope constraints as the two-and-a-half dimensional method.


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Three-dimensional Lerchs-Grossmann and network flow In the 1960s Lerchs and Grossmann published a graph theory method, and T.B. Johnson published a network flow method for true three-dimensional optimization. Both methods guarantee to find the optimal pit in three dimensions. Both - naturally - give the same results. Both select a sub-set of blocks which gives the absolute maximum total value whilst obeying the slope requirements. Both are difficult to program for a production environment, where there are large numbers of blocks and variable slopes. The Lerchs-Grossmann method is the one most used, and it has been programmed commercially by about four different groups. The programs they produced differ in their ability to handle slope variations and differ in the machines on which they will run. Despite being difficult to program, the Lerchs-Grossmann method is relatively easy to explain and demonstrate, at least in general terms. The method works with only two types of information. These are the block values and what Lerchs and Grossmann call “arcs”. An arc is a relationship between two blocks. An arc from block A to block B indicates that, if A is to be mined, then B must be mined to expose A. The reverse is not true. If B is to be mined, block A may or may not be mined. See Figure 31.

B Arc from A to B

A Figure 31: An arc

In most optimization packages, all the slope requirements are translated into a (large) number of block relationships in the form of arcs. It might appear that we require an arc from each block to every block which is “above” it in a mining sense, but there is no need to have so many, because arcs can “chain”. That is, an arc from A to B and an arc from B to C ensure that C is mined if A is mined, despite there being no arc from A to C, as is shown in Figure 32.

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C

B

A Figure 32: Chaining of arcs

In Figure 33, three arcs used repeatedly can ensure the mining of all required blocks for the slope indicated when the bottom left-hand block is mined.

Figure 33: Required arcs for a desired slope

When this idea is extended to both sides, and to three dimensions, we typically need of the order of 50 arcs for each block, to ensure a slope accuracy of about one degree. Exact slope accuracy can never be achieved, because optimization cannot “mine” part of a block. The Lerchs-Grossmann three-dimensional optimization method achieves its aim by manipulating the block values and the arcs. It uses no other information. In other words, except for the information given by the arcs, it “knows” nothing about the positions of the blocks - nor indeed about mining. Therefore, in order to demonstrate how the method works, it is merely necessary to work with a list of blocks and a list of arcs. Whether these


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are laid out in one, two or three dimensions, and how many arcs per block are used, is immaterial. In order to demonstrate how the Lerchs-Grossmann three-dimensional method works, we will choose to work in two dimensions, because that is much easier on paper. Also for simplicity, we will use square blocks and slopes of 45 degrees, although this is not a requirement for Lerchs-Grossmann, as was shown by the previous diagram. This allows us to work with only three arcs per block, as shown in Figure 34.

Figure 34: The effect of chaining with three arcs per block

The Lerchs-Grossmann method flags each block that we currently intend to mine. During the optimization process, these flags can be turned on and off many times. A block is flagged to be mined if it currently belongs to a linked group of blocks that have a total value that is positive. These groups are called “branches”. The method repeatedly scans through the blocks looking for blocks that are flagged to be mined and that have an arc pointing to a block that is not flagged to be mined. Clearly, this is not a viable situation. The way it resolves these situations forms the core of the Lerchs-Grossmann method. The following diagrams take you through such a search.

We start with a two-dimensional model, 17 blocks long and 5 blocks high. Only three blocks contain potential ore, and they have the values shown. All other blocks are waste and have the value -1.0.

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Step 1: The first arc from a “flagged” block that we find is to a block which is not flagged.

23.9

6.9

23.9

22.9 Step 2: To resolve this, we link the two blocks together. The total value of the two-block branch is 22.9.

Step 3: We deal with the other two arcs from this block in the same way. The total value of the four-block branch is 20.9.

Step 4: We continue in the same way along the bottom bench,and then along the next bench. (Note that even waste blocks are flagged if they belong to a positive branch.)


79 Step 5: The next flagged block has an arc to a block which is also flagged. We don’t create a link for this arc or for the vertical one from the same block, because nothing has to be resolved.

Step 6: The next arc from a flagged to another flagged block is between two branches. The procedure is unchanged - we do not insert a link.

Step 7: We continue adding links until we reach the one shown. When we add this link, the branch total will become -0.1. Because of this ALL the blocks in the branch have their flags turned off.

Step 8: The next arc of interest is from a flagged block to a block which is part of a branch which is not flagged. Effectively the centre and the right-hand branches can co-operate in paying for the mining of the common waste block, which is circled.

23.9

15.9

6.9

23.9

20.8

Step 9: The Lerchs-Grossmann method includes a procedure for combining the two linked branches into one branch with only one total value.(Note that there is no requirement to always branch upwards from the root.)

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Step 10: The next arc of interest is from a flagged block to a waste block. Lerchs-Grossmann detects that this extra waste will remove the ability of the centre branch to co-operate with the right-hand branch in paying for the mining of the circled block.

Step 11: Lerchs-Grossmann includes a procedure for breaking the single branch into two branches by REMOVING a link.

23.9

6.9

23.9

-0.1

-0.1

8.9

Step 12: We continue adding links and, eventually, the total value of the left-hand branch becomes negative. The next arc after this is again between a positive and a negative branch.

Step 13: This is dealt with in the same way as before, and the left and right-hand branches are combined into one, with one total value.


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Step 14: We continue adding arcs until we reach the situation shown above. The program would then do another scan for arcs from blocks which are flagged to blocks which are not flagged. We can see that it will find none, and the optimization is complete.

Lerchs and Grossmann proved that, when no further arcs can be found that are from a flagged block to a block that is not flagged, then the flagged blocks constitute the optimal pit. In a real optimization, we would start the scan again, but visual inspection shows us that this is unnecessary because it is clear that there can be no flagged blocks with arcs to blocks which are not flagged.

In this case we have a W shaped pit that contains two ore blocks with a total value of 47.8 and 47 waste blocks with a total cost of 47. The pit is thus worth 0.8, and we can see that this is indeed the pit with the maximum possible value. Note that there are 7 extra blocks which would have to be removed to uncover the centre ore block which has a value of only 6.9. It is thus not worth mining. In real three-dimensional optimizations, there will usually be many scans of the blocks checking for arcs that have to be resolved. These continue until a scan occurs in which none have to be resolved, and we know that the optimization is complete. Since 1965, when the Lerchs and Grossmann paper was published, other algorithms have been published which achieve the same result. In general, they require a smaller number of steps to obtain the optimal outline, but the steps are more complicated. Using the Lerchs-Grossmann method with modern PCs and software, the time taken to do an optimization will frequently be as little as a minute, and should never be more than an hour or two.

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Economic Model for Pit Optimization

83

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Cash Objective If your objective is to maximize undiscounted cash-flows over the life of the mine, then the application of Lerchs-Grossmann pit optimization is perfect – it achieves the maximization of undiscounted cash-flows. Maximization if undiscounted cash-flows is relatively simple, since the exercise is completely independent of the mining schedule and mine life.

Cash Pit This is the pit outline which will yield the highest undiscounted cash value. The outcome is independent of the mining schedule.

NPV Pit The pit outline which, combined with a specific schedule, will yield the maximum NPV.

Optimizing a Pit for NPV Maximization If you objective is to maximize discounted cash-flows, the you can be said to have an NPV Objective. Optimizing pit design for an NPV objective is much more complicated than optimizing for a Cash Objective. The Lerchs-Grossmann method determines the pit outline which maximizes undiscounted value. In order to optimize for NPV one must determine not only the pit outline but also the mining schedule. This gives rise to what is known as the Circular Problem: 1. In order to optimize the pit outline for an NPV objective, you need to know the NPV of the individual blocks. 2. In order to know the NPV’s of the individual blocks, you need to know which ones are included in the pit outline and when they will be mined and processed (i.e. you need a mining schedule). You cannot produce a mining schedule without first having the pit outline. See step 1.

Discounted Pit Techniques This is a class of optimization techniques which seek to overcome the Circular Problem. It includes the DPS, DBD and NPVS techniques. Each seeks to modify the Conventional Pit Optimization Technique in order to seek an NPV Objective. They do this (essentially) by providing approximations / estimates of block NPV’s.

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DBD Technique The Discounting by Depth (DBD) approach is based on the premise that material at depth takes longer to reach, and discounting is based on the vertical position of material in the model. The options are: 1. Discounting applied on a bench-by-bench basis to the block model, assuming a fixed rate of vertical advance. Note that fixed rate of vertical advance is an extremely crude approximation of reality, as it the presumption that all block on a bench will be dealt with in the same period. 2. Discounting applied on a bench-by-bench basis, assuming a variable rate of vertical advance. This is an improvement on the fixed rate of vertical advance method, but still relies on the presumption that all blocks in a bench are dealt with in the same period. It is certainly possible to devise simple mathematical methods to calculate a vertical rate of advance, but the evidence is that in industry, the rate is determined experimentally, by trying a range of different values, and seeing which ones yield the best NPV results. The DBD technique is available in the Whittle software. NPVS Technique This is the technique applied in Earthworks NPV Scheduler. The following information has been interpreted from the help files of NPV Scheduler. The steps are: 1. Apply conventional pit optimization technique. The pit so produced provides an outer bound for the final solution. 2. Within the conventional pit and based on a user-defined mining throughput limit, calculate look-ahead values for every block. 3. Produce a block by block schedule which “mines” each block in turn, in the descending order of look-ahead value. The block by block schedule is in fact a series of nested pit, each incrementally bigger pit differing only by the addition of one block. 4. Evaluate that sequence to calculate the NPV of each nested pit. 5. Choose the pit which maximizes NPV. This technique will produce a set of nested pits, equal in number to the number of blocks in the ultimate pit.


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The approach is complicated and there are some apparent pitfalls: 1. The calculation of NPV is done on an assumed average mining rate. This is rarely the controlling factor in an open cut mine. 2. It does not in any way guarantee that the NPV values used to evaluate the blocks in the generation of the sequence (and consequently in the generation of the shape of the ultimate pit) bear relation to the actual NPV values that the blocks have once practical scheduling constraints have been applied. As a consequence, it could be that the pit produced by this method is OK if your final schedule tends towards Best Case in nature, but is inappropriate should your final schedule tend towards Worst Case in nature.

DPS Technique The Discounted Pit Shells technique has been developed by Whittle. The process involves: 1. Apply the Conventional Whittle Technique (see below), thus producing a realistic mining schedule. 2. Use the mining schedule to determine mining periods for each block in the model. 3. Based in the mining period, modify the original block model so that it contains estimates of block NPV. 4. Rerun the Conventional Whittle Technique again to generate a new set of pitshells. Note that the block NPV values calculated in step 3 are only used in the pit optimization stage. Normal discounting procedures are applies during the simulation and scheduling stage of the Whittle Technique. The DPS technique should lead to the generation of block NPV estimates that are far more accurate than those generated by the DBD technique, or the NPVS technique, as it takes into account the actual style of scheduling that is applicable to the mine (as applied by the software operator) rather than a hypothetical one. The DPS technique is available in the Whittle software.

Conventional Whittle Technique to Seek NPV Objective A technique has been developed in Whittle software over many years which has been shown to deliver good results. It applies Lerchs-

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Grossmann pit optimization in a modified framework: 1. Conventional Pit Optimization is combined with a pit parameterization technique (Revenue Factors) to produce a range of pits with different cost/price ratios. The pits nest, and are useful upon which to base push back designs, and for sensitivity analysis. Although the pit parameterization process modifies the block values, the calculation is based on undiscounted cash values. 2. Simulation and DCF Analysis are performed on a large range of benchmark and user defined schedules. There are often hundreds of different schedules performed in a single run, each with a different final pit, and a different scheduling method. From this it is easy to determine which pit out of the database provides the highest NPV for any given scheduling method. This technique effectively deals with the issue of an NPV Pit being a different size to a Cash Pit. It does so, by providing NPV evaluations of a range of Cash Pits, each one optimal for a different cost-price ratio and each with a different tonnage. The user is able to choose the pit which yields the highest NPV, for the scheduling method of choice. The technique does not deal with the issue of any difference in shape between the NPV Pit and the Cash Pit.


Generation of Nested Pit Shells

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Revenue Factor Pit Parameterization A series of value models can be prepared for a range of product prices. The outlines obtained by putting these through a pit optimizer would form a set of nested pits. In Whittle software we optimize first for infinite price. This can usually be done in one pass of the structure arcs with special program code, and thereafter we can exclude all the blocks outside this pit from further consideration. Next we optimize for the lowest price specified, which gives us the smallest pit and allows us to exclude all the blocks within it from further consideration. After these two initial optimizations, we repeatedly optimize for the price which is in the middle of the list of prices for the largest number of undivided blocks. We do this until all prices have been dealt with. For each optimization we consider only those blocks which lie between the shells for the nearest prices above and below the target price for which shells exist. We adjust the values of these blocks to allow for the new price and then do scans through the arcs which apply to these blocks until there is no further change. Figure 35 illustrates this sequence. 1

3

4

2

Figure 35: Sequence of optimization of pit shells

With this approach, each optimization is on successively fewer blocks and involves less and less adjustment to the Lerchs-Grossmann tree. Note also that, since no link in the tree crosses an existing pit shell, the links tend to become more and more parallel to the nearby shells and thus easier and easier to optimize. The combination of these effects make successive optimizations faster and faster. In practice 100 pit shells can be generated in about the time it would take to do about five simple optimizations. Note that, since we are dealing with blocks of finite size, it is entirely possible for one or more adjacent prices to produce exactly the same pit

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outline. In fact this frequently happens and the result is that fewer pit shells are generated than the list of prices supplied would suggest. In Whittle we have to deal with more than one product and thus more than one price, so we work with what we call Revenue Factors. We generate pit shells for a range of Revenue Factors which all prices are multiplied by.

Mining Phases The mining phases technique applies vertical bounding to provide a set of nested pits. The technique is available in the Whittle Pit Shells node and is described fully in the help files.

Mining Direction The mining direction technique has been developed by Whittle and generates a nested set of pit shells while imposing a horizontal mining direction. In many circumstances the nested pit shells produced by the technique are far more practical that those produced by other techniques. This technique is based on the use of user-defined expressions, to drive Pit Parameterization in a different way. Rather than generating pits on the basis of a changing price/cost ratio (the normal Revenue Factor approach), this technique includes increasingly more ore blocks in the optimization for each increment of a factor that relates to the distance from the block to an origin position. Figure 36 shows the effect of applying this technique.

Figure 36: Effect of applying the mining direction technique. The mining direction was set to emanate radially from the South East.


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The steps include the following: 1. Choose the final Using conventional techniques, choose a shell that pit that you want you want to use as your final pit. Adjust the to use economics in your Pit Shells node so that the desired final pit corresponds to a Revenue Factor of 1.00.

2. Define a User defined Expression

Copy and paste this Pit Shells node back onto the parent Pit Slopes node. This new node will be the one in which you create the Mining Direction technique. Make sure you are in Cash Flow Mode. The technique won't work if you are in Cut-off Mode. In the Expressions Tab, create an expression called "DST": D(X,0,0,0,0,0)/MX This expression will drive development straight out from the West side of the model.

3. Add Expressions to the Prices and Selling Costs

There are lots of different expressions that you can use to drive the direction. For example "D(X,Y,0,(0.5*MX),MY,0)/D((0.5*MX),MY,0,0,0,0)" drives development radially out from the centre of the North side of the model. In the Selling Tab, enter in-line function expressions for each of the Prices as follows: [price]*IF(REVFAC>=DST,1,0)/REVFAC Here, the [price] is the numerical (dollar) value given to each mineral.

4. Set the Revenue You should use: Factors Start Factor: 0.01 End Factor: 1.00 Step Size: 0.01 5. Compare the results

You should compare the results from this technique with the standard pit shells technique: (a) Is the new set of pit shells more practical to mine? (b) How much does this change the NPV? (c) Do different mining directions yield different practicality/NPV results?

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Why is this better than manually designing pushbacks with mining direction? The first advantage of this technique is that it is still strongly influenced by the economics and ore distribution of the model. That means that while you are introducing a propensity for practical development, the results are also influenced towards shapes that yield high NPV's. The second advantage - you can generate and analyze a range of different mining directions very quickly, and benchmark them against the high NPV "onion skin" pits. Further information about this technique is included in the section “Producing a Practical Pit from an Optimal Outline”.


Calculating Block Values

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This section deals exclusively with the calculation of undiscounted block values. Before we do a pit outline optimization we must first calculate the block values, and there are three basic rules to follow. The First Rule The block value must be calculated on the assumption that the block has already been uncovered. In other words, no allowance should be made for the cost of the stripping required to gain access to the block, because that is precisely what the optimizer calculates. In particular, any cut-off used to define ore should reflect the cost of processing and any extra cost of mining the block as ore rather than waste, but not the cost of stripping. If an allowance for stripping is included in the costs, the stripping will be paid for twice! The Second Rule The block value must be calculated on the assumption that the block will be mined. So, if the block contains some ore that could profitably be processed as well as some waste, the value of the ore should be added in, even if the resulting total value of the block is still negative. The optimizer will not choose to mine such a block, but if it has to mine it to get at something more valuable, the ore will help to pay for the stripping, as it would in practice. The Third Rule Any expenditure that would stop if mining stopped must be included in the cost of mining, processing or selling. Conversely, any expenditure that would not stop if mining stopped must be excluded. This is discussed in more detail below.

A Formula for a Block Value There are a number of ways of writing an expression for the value of a block. The one we use is as follows: VALUE = (METAL*RECOVERY*PRICE - ORE*COSTP) - ROCK*COSTM

where the part in parentheses is repeated for each separately minable ore parcel in the block, and where: METAL = Units of product in the ore parcel i.e. ore tonnes times grade. RECOVERY = The proportion of product recovered by processing the ore.

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PRICE ORE COSTP

= = =

ROCK COSTM

= =

The price obtainable per unit of product sold. Tonnes of ore in the ore parcel. The EXTRA cost per tonne of mining the material as ore and processing it, rather than treating it as waste. The total amount of rock (ore and waste) in the block. The cost of mining a tonne of waste.

An “ore parcel” is a region within the block of a defined rock-type, tonnage and metal content, which could be mined selectively. Its position within the block is not defined. Some generalized mining packages treat a block as a homogeneous entity. In this case, there is only one parcel per block. Note that air blocks must also be given a value so that the optimization program can differentiate them from waste. Air blocks have a value of zero. Blocks which are partially air have their block values reduced on a pro-rata basis.

Calculating Costs When preparing for a pit outline optimization, you have to calculate the expected mining, processing, rehabilitation and selling costs. However, pit outline optimization has very specific requirements with regard to the calculation of these costs, and it is important that these be fully understood. All costs must be expressed as mining cost per tonne, as processing cost per tonne, as rehabilitation cost per tonne, or as selling cost per unit of product produced. To reduce time costs to a per tonne or a per unit basis, you have to make assumptions about the production rate. If the size of the pit produced by the optimization makes these assumptions inappropriate, then the costs should be re-calculated and the optimization done again. Many people set up all their cost calculations in a computer spreadsheet. This makes recalculation much easier, and a sample is given later. What Costs to Include Incremental costs such as wages and fuel costs must obviously be included in the calculation of the cost of the activity with which they are associated. Expenditures that are related to time rather than to tonnage or production require careful thought, but, as mentioned earlier, there is a clear rule that allows you to decide which should be included: “Any expenditure that would stop if mining stopped must be included in the cost of mining, processing or selling. Conversely, any cost that would not stop if mining stopped must be excluded.” The reasoning behind this is that, when the optimizer adds a block to the pit outline, it may effectively extend the life of the mine. If it does, the extra costs that would occur as a result of this extended life must be paid


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for. Otherwise the optimizer may add blocks to the pit that reduce, rather than increase, its real value. Since the optimizer can only take note of costs expressed through the block values, it is necessary to share these time-related costs between the blocks in some way. How they should be shared depends on whether production is limited by mining, by processing or by the market. Usually it is limited by processing, and, in this case, only the mining of an ore block extends the life of the mine. The ore block values should therefore include an allowance for time costs. This is done by adding an appropriate amount to the processing cost per tonne. If production is limited by mining, as in some heap leach operations, every block that is mined extends the life of the mine, so that time costs should be added to the mining cost. A market limit means that time costs should be added to the selling cost. In each case, the amount added is the total of all the time costs per year divided by the throughput limit per year. Examples Some examples of the handling of various costs may be helpful. Processing mill Consider a processing mill that costs $6m to build and commission. If the mine were to be shut down, for whatever reason, on day 2 of operation, the mill would have a certain salvage value, say $4m. In this case $2m has gone for ever. It is an “up-front” or “sunk” cost that must be subtracted from any optimized value of the pit itself. It is not a cost for pit outline optimization purposes. We can deal with the remaining $4m in one of two ways. If we assume that there will be an ongoing program of maintenance and capital replacement that will keep the salvage value of the mill close to $4m in today's dollars, then the $4m is theoretically recoverable when the mine is closed, and so is not a cost. However the maintenance and periodic capital replacement expenses are costs for these purposes, because they would stop if mining stopped. They should be averaged and treated as a time cost. Alternatively, we can assume that only essential maintenance will be done, and that the salvage value of the mill will become progressively less, as is shown in Figure 37. In this case, the expected rate of this reduction should be treated as a time cost. Note that the rate of reduction is not necessarily the same as the depreciation rate that is used by accountants. In most cases the depreciation rate is set by taxation considerations, and may reduce the book value to zero when the salvage value is clearly not zero. Note that the life of the mill (10 years in this example) is not necessarily the same as the life of the mine.

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Re-sale value (Millions)

6

Sunk Cost

5 4 3

$0.4M/Y 2 1 0 0

2

4

Year

6

8

10

Figure 37: Calculation of rate of reduction of re-sale value

We discuss the interest on the salvage value below. Note that, if the mine is mining capacity limited, even time costs associated with the mill must be factored into the mining cost. Trucks If the expected life of the mine is shorter than the operating life of a truck, then truck purchases can be treated in the same way as the cost of the mill. If the life of the mine is much longer than the life of a truck, then trucks will have to be purchased progressively to maintain the fleet, and such purchases will stop if mining is stopped. Consequently the cost of purchasing trucks should be averaged out over the life of the mine and treated as a time cost. Unless the life of the mine is expected to be very long, some compromise between the above two approaches is usually required. Contract mining companies must take these factors into account when quoting for a job, and it is sometimes useful to think as they do when you are working out the costs for your own fleet. You should include everything that they do, except for their allowance for profit. Administration costs On-site administration costs will usually stop if mining is stopped. They must therefore be treated as a time cost. Head office administration costs may, or may not, stop if mining stops at this particular mine, and thus may, or may not, be included. Bank loans for initial costs Repayment (principal and interest) of a bank loan taken out to cover initial set-up costs will have to continue whether mining continues or not. It should therefore not be included in the costs used when calculating block values. Of course, these repayments will have to come from the cash flow of the mine. If the mine is not going to produce enough cash flow to cover them, the project should not proceed. You should not introduce these repayments as costs in an attempt to “improve” the optimization. The


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result will be quite the opposite. You will get a smaller pit with a smaller total cash flow, and we will expand on this later. Although the bank loan repayments themselves are not included, some of the items that the loan was used to pay for may be included, as you will see below. Bank loans for recoverable costs If you borrow money from the bank for day-to-day working capital or for items, such as the $4m salvage value discussed in the mill example above, then you can reasonably expect to repay the loan if mining stops. Consequently the interest paid on such a loan is a cost that stops if mining stops. It should therefore be treated as a time cost. Note that we work throughout in today’s currency, so the interest rate used should be the “real” interest rate and should not include an allowance for inflation. Grade control costs It is often necessary to do grade control work on some waste as well as ore. In this case, grade control costs apply to waste costs too. If only some of the waste is grade controlled, then the correct way to handle it is to load the cost of those particular waste blocks. However, many users make an estimate of the tonnes of such waste per tonne of ore, and load the cost of mining ore. Support – cable bolts If the permitted pit wall slope is to be increased by the use of cable bolts, the cost per tonne is related to pit size, which has to be estimated. Then a cost per square meter of wall can be transformed into a cost per tonne of waste. This is an iterative estimate, but fortunately costs per tonne are usually low. These examples do not cover all possible costs, but should indicate how to treat most costs.

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Sample cost calculation The spreadsheet in Figure 38 shows how time costs would be handled in the two different cases of an operation that could be milling or mining limited.

Time cost calculations: Time costs per year Expected yearly mill throughput Time cost per tonne milled Expected yearly mining capacity Time cost per tonne mined

980000 1000000 0.98 4000000 0.24

Incremental costs per tonne:

Bench

Mining W aste

Mining Ore

Extra for Mining Ore

Milling Ore

5 4 3 2 1

1.05 1.17 1.29 1.41 1.53

1.87 2.15 2.47 2.84 3.26

0.82 0.98 1.18 1.43 1.73

8.25 8.25 8.25 8.25 8.25

With throughput limit on milling:

Bench

Mining W aste

Milling Ore


Time costs

Milling cost for Opt.

5 4 3 2 1

1.05 1.17 1.29 1.41 1.53

8.25 8.25 8.25 8.25 8.25

0.82 0.98 1.18 1.43 1.73

0.98 0.98 0.98 0.98 0.98

10.05 10.21 10.41 10.66 10.96

With throughput limit on mining:

Bench

Mining W aste

Time costs

Mining cost for Opt.

Milling ore


Milling cost for Opt.

5 4 3 2 1

1.05 1.17 1.29 1.41 1.53

0.24 0.24 0.24 0.24 0.24

1.29 1.41 1.53 1.65 1.77

8.25 8.25 8.25 8.25 8.25

0.82 0.98 1.18 1.43 1.73

9.07 9.23 9.43 9.68 9.98

Figure 38: Cost calculation spreadsheet


Producing a Practical Pit Design from an Optimal Outline

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Introduction In this section we will consider the following issues: 1. Block model artifacts – Since an optimizer uses a block model and mines each block completely or not at all, the optimal outline is presented initially as a jagged line defined by block edges. However, it is important to remember that the blocks themselves are artifacts. The ore body itself does not consist of neat rectangular blocks. It is therefore not relevant to try to follow the jagged outline in detail, or to mine the individual blocks as though they were significant entities. 2. Choosing pit shells upon which to base Pushback designs 3. Mining width considerations – Depending on the type of equipment selected for mining, a greater or lesser amount of working space is required. The conventional term for this working space is “mining width”. Lerchs-Grossmann does not have a direct mechanism for allowing for mining width constraints, nor does pit parameterization. In both cases, the blocks in the block model are the only representation of minimum selectable mining units. The effects of mining width manifest themselves principally in the following areas: a. At the base of the final pit. b. At the bases of pushbacks. c. In the horizontal separation between one pushback and the next. 4. Mining Direction – In addition to the techniques for modifying pushbacks to allow for mining width, it is possible to employ a special pit parameterization technique which generates pushbacks which have a higher propensity for being practical to mine. 5. Working slopes – In some mines, while working pushbacks, shallow slopes will be used in intermediate phases, and steeper slopes will be used for final pit walls. The reason is as follows: a. Shallower pit slopes are easier to dress, and safer. Therefore it makes sense to use shallower pit slopes as working slopes. b. Steeper pit slopes are more difficult to dress (in order to maintain safety), but contribute to a lower stripping ratio. Therefore it makes sense to use steeper slopes for final pit walls.

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6. Haul Roads and Safety Berms – Haul roads and Safety Berms have the effect of flattening the overall slope angle.

Getting rid of the block model artifacts The precise method by which you do this depends on the tools that are available to you in your Generalized Mining Package (GMP). You may do it entirely by hand, or with varying degrees of computer assistance.

Figure 39: Putting in a smooth curve in plan

In plan, it is only necessary to draw a smooth line through the zigzag, as is shown in Figure 39.

Figure 40: Joining the centre points of the bottoms of each column of blocks in the pit

In section, the simplest method is to join the centre points of the bottom of each column of blocks that is to be mined, as is shown in Figure 40.

Choosing Pitshells Upon Which to Base Pushback Designs The principal economic reason for performing pushbacks is to maximize NPV. Elsewhere in these notes, two benchmark schedules are defined: Best Case, and Worst Case. The Best Case schedule has a great many pushbacks, and has a higher NPV. The trade-off is practicality. Specifically, too many pushbacks will or may lead to the following: 1. Excessive expense maintaining multiple working slopes.


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2. Vertical advance rates in each pushback which are difficult or impossible to achieve. 3. Operational problems associated with moving quipiment around multiple working faces. 4. Lack of adequate mining width. 5. Extra cost associated with re-working haul roads. If you experience these problems with a design that has a great many pushbacks, it could be that the NPV calculated for that particular scenario is unachievable. On the other hand, if you were to eliminate all pushbacks, you could be assured practicality, but may sacrifice NPV. The question arises as to how to reach a compromise between practicality and NPV Maximization. The answer to the question depends very much on the details of the case. A significant factor is the degree of dissemination of the ore body. Disseminated ore bodies with little stripping required, will not benefit from a very large number of pushbacks. In extreme cases, the introduction of pushbacks will make no difference to the NPV. Deposits in which target minerals are concentrated in smaller areas such as seams, will benefit more from the addition of pushbacks. The recommended method for determining the number of pushbacks is to try different options and compare the practicality and the NPV of those different options. The most efficient way to try different options is to utilize the Whittle Pushback chooser, which, given a number of pushbacks required, can choose shells to use as those pushbacks, which maximize NPV.

Ensuring the required mining width constraints are met There are two phases to allowing for minimum mining width: 1. Selection of pit-shells – In selecting pit shells to use as the basis of pushback and final pit design, consideration should be given to the Mining Width requirements as to the horizontal separation between one pushback and the next. It is common to only be able to approximately satisfy this constraint through the selection of pit shells. Final adjustments are made during the second phase (below). 2. Whittle Mining Width module – This module allows the application of mining width constraints in the base of the final pit and the pushbacks, and also applies the constraints as to the horizontal separation between one pushback and the next.

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Mining Direction Introduction This technique is based on the use of user-defined expressions in Whittle, to drive Revenue Factor Pit Parameterization in a different way. Rather than generating pits on the basis of a changing price/cost ratio (the normal Revenue Factor approach), this technique includes increasingly more ore blocks in the optimization for each increment of a factor that relates to the distance from the block to an origin position. Example An example project has been created to: a. Illustrate the technique. b. Test the expressions. It is based on the demonstration model called Marvin, but includes only the top 9 benches, leading to a broad, flat deposit. The economics and recoveries have been modified to achieve a broad pit at revenue factor 1.0, in order to illustrate this technique.

Figure 41: Overview of the example project, showing the expression validation nodes, and some example pit shells nodes.


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Figure 42: Standard pit optimization showing "onion skin" type shells (reduced number of pushbacks)

Figure 43: Same final pit, but with a Mining Direction applied (From South East Radial)

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Figure 44: Shown with Mining Direction (From West Straight)

Economic and Practical Effects of the Technique From examination of Figure 42 to Figure 44, you will see that there is quite a difference in the manner in which pits are generated, though the final pit is exactly the same in each case. The original pit shells (onion skins) should always yield the highest NPV, but with poor practicality. Introducing a mining direction increases the practicality, but sacrifices NPV. Many people complain about the onion skin pits as being “un-mine-able”. “Un-mine-able” means that the reported NPV is unachievable. Table 1 includes a comparison of NPVs for a range of different mining directions, as compared to the standard shells.


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Table 5: Comparison of different mining directions to the standard pit shells

Direction

Number of shells

NPV

Change

Standard (onion skins)

39

1,091m

-

From South East Radial (SER) From North East Radial (NER) From West Straight (WS)

36

966m

-11%

42

955m

-12%

42

996m

-9%

17

How to Apply the Technique The normal pit parameterization technique produces progressively larger shells for each incremental increase in the Revenue Factor. This is effected through the use of multiplying the Revenue Factor by the net price for each successive pit. DST Expression In the Expressions tab, you must define an expression called DST. The formulation of the expression depends on the mining direction you wish to take. The twelve examples used in the example project are shown in Table 6.

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Note that the number of pit shells is an artifact of the process. More pit shells will generally lead to a higher Best Case NPV. In order to attempt to negate that effect, the revenue factors for the standard pit shells were changed so as to produce roughly the same number of shells as the mining direction cases.

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Table 6: Example Mining Direction Expressions Name From South East

From North East

From South West

From North West From North Radial From North Straight From East Radial From East Straight From South Radial From South Straight From West Radial From West Straight

Description Radiating from the South East Corner to the North West Corner Radiating from the North East Corner to the South West Corner Radiating from the South West Corner to the North East Corner Radiating from the North West Corner to the South East Corner Radiating from the centre of the North side From North to South Radiating from the centre of the East side

Expression

From East to West

D(X,0,0,MX,0,0)/MX

Radiating from the centre of the South side From South to North Radiating from the centre of the West side From West to East

D(X,Y,0,MX,0,0)/D(MX,0,0,0,MY,0)

D(X,Y,0,MX,MY,0)/D(MX,MY,0,0,0,0)

D(X,Y,0,0,0,0)/D(0,0,0,MX,MY,0)

D(X,Y,0,0,MY,0)/D(0,MY,0,MX,0,0)

D(X,Y,0,(0.5*MX),MY,0)/D((0.5*MX),MY, 0,0,0,0) D(0,Y,0,0,MY,0)/MY D(X,Y,0,MX,(0.5*MY),0)/D(MX,(0.5*MY), 0,0,0,0)

D(X,Y,0,(0.5*MX),0,0)/D((0.5*MX),0,0,0, MY,0) D(0,Y,0,0,0,0)/MY D(X,Y,0,0,(0.5*MY),0)/D(0,(0.5*MY),0,M X,0,0) D(X,0,0,0,0,0)/MX

Price Expressions To apply the sterilization/desterilization process of the mining direction technique, we must exploit the Revenue Factor, but at the same time, stop it from changing the effective price. We do this by dividing the price by the variable REVFAC. In other words: REVFAC*PRICE/REVFAC=PRICE We then use the REVFAC to compare to a DST value which calculates the standardized distance of a block to an origin position on the model. If the REVFAC is greater than or equal to the DST value, then we multiply the


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price by 1. If the REVFAC is less than the DST value, then we multiply the price by 0. The actual price function is as follows: [price]*IF(REVFAC>=DST,1,0)/REVFAC Note that these expressions are only ever applied to the Pit Shells node, never to a Scenario node. Revenue Factors The examples in this document are calibrated to work with a maximum Revenue Factor of 1.0. You should use: Start Factor: 0.01 End Factor: 1.00 Step Size: 0.01 Note about the Technique 1. You should look to the original set of shells to see if there is any apparent “natural” direction that you can exploit. For example, if smaller pits tend to be in the west, then a mining direction working from the West will likely achieve pleasing results. 2. You should try different mining directions, and make a comparison of their NPV’s. 3. You must run in Cash Flow mode in the Pit Shells Node. The technique will not work if you have the Cut-Off mode selected in the pit shells node. 4. These expressions are only applied for the purposes of pit optimization. You must not use any of these expressions in an Economic Scenario node. 5. These expressions are generalised. They should work with any model. 6. The technique provides a final pit as being the revenue factor 1.00 pit for the specified economic conditions. If you wish to use a pit other than the revenue factor 1.0 pit, then you should do the following: a.

Multiply the DST expression by the revenue factor you wish to use. For example, if you wanted to run "From South East” and a revenue factor of 0.95, you would specify the DST expression as “D(X,Y,0,MX,0,0)/D(MX,0,0,0,MY,0*0.95”

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b.

Change the End Factor for the Revenue Factor to the desired value. 18

Allowing for Working Slopes The process involves: 1. Determining the shape of the final pit outline with the application of ultimate pit slope constraints. 2. Converting the final pit shape into a new model bounded by the new pit shape. 3. Re-optimizing the new model with working slope constraints while applying one of the pit parameterization techniques.

Figure 45: An ore body, with the final pit shown. The shape of the final pit outline is determined with the application of ultimate pit slope constraints.

Figure 46: The final pit shape converted into a new model bounded by the new pit shape. All ore and waste has been stripped out of the model beneath the pit.

18

Note that I have not tested this aspect of the technique.


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Figure 47: Re-optimizing the new model with working slope constraints by applying one of the pit parameterization techniques. The working slopes are shallower that the ultimate pit slopes.

Allowing for Haul Roads and Safety Berms If haul roads and safety berms are to be incorporated correctly, it is essential that the slopes used during pit outline optimization be laid back to allow for them. Given this, there should be no particular difficulty, as is illustrated in Figure 48.

Figure 48: Allowing for haul roads and safety berms in the average slope y°

We want to work out angle y°, which is the average slope angle for use in pit optimization. Input Variables x° is the inter-ramp slope angle. D is the width of the haul road. In this example, the one haul road intersects the wall. If two or more haul roads intersect the wall, then D is the combined width of the haul roads. C is the depth of the pit (this is an approximation – the actual value is dependent in part on the pit slopes, which of course will change as we adjust for the haul road).

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Derived Variables y° = Arctan(C/(B+D)) B=(C/Tan(x)) Substitute B y° = Arctan(C/((C/Tan(x))+D))


The Effects of Underground Mining

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Overview If it is economically viable to mine ore by either underground or open cut method, then pit optimization software can be used to determine which is the better alternative. For the purpose of pit optimization, the value attributed to blocks should be the value to the company of mining by the open pit method. If a block is worth, say, $1000 if mined by open pit method, and $800 if mined by underground method, then the value used for the purposes of pit optimization should be $200. When implementing this using Whittle, the user must specify an underground processing method, including all the costs associated with mining and processing by the underground method. The software calculates the value of the block if mined by underground method and subtracts this from its normal open pit value. The result of implementing this method is very often a smaller pit, with a lower value, but the combined value of the open pit and the underground mine will be maximized.

Which Ore should be Mined by Open Cut? Consider this existing underground mine in Figure 49. Development is represented by a vertical line (shaft) and horizontal lines (drive). Stoping is represented by oblique lines. The development and stoping shown have already been performed.

Figure 49: Existing underground mine

Further underground development is planned. This is shown as thinner lines to the East of the shaft in Figure 50. It would not be possible to mine the ore by open pit method, and this underground development will occur whether or not an open pit mine eventuates.

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Figure 50: Existing mine with planned development shown

Additional ore bodies have been identified in the East as shown in Figure 51. It would be feasible to mine the A ore body by underground method, but not the B ore body. This possible underground development is shown as dotted lines. Assume that all the ore above Drive-1 is included in Stope-1; that all the ore between Drive-2 and Drive-1 is included in Stope-2, etc. To mine each stope, the only development necessary is the digging of its associated drive. Provided that the revenue generated from mining and processing the ore in each stope is greater than the cost of the associated drive, then it is economic to mine it.

Figure 51: Newly discovered ore bodies and possible underground development.

If only the open pit method is considered, then, in this example, the pit optimizer may yield the results shown in Figure 52.


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Figure 52: This pit is optimal if only open cut methods are possible.

Parts of ore body A could be mined either by open pit, or underground method. If it is not mined by open pit method, then it will be mined by underground method. Consider any one block within the region covered by the pit and underground stopes. It will be mined, one way or another. There are therefore two possible economic results: If it is mined by open pit method: Net Value = (block revenue) - (cost to mine & process by open pit method)

Let’s say in this case the net value for the block is $1000. If it is not mined by open pit method: Net Value = (block revenue) - (cost to mine & process by underground method)

Let’s say in this case the net revenue for the block is $800. If the block is included in the pit outline, it will be worth $1000, and if it is not included in the pit outline it will be worth $800. Therefore, the value to the company of including it in the pit is $200. Pit optimizers such as Whittle can calculate the value to the company, by deducting the underground value from the open pit value. Using this system of block valuation in this example, a pit optimization is run and produces the pit shown in Figure 53.

Figure 53: The pit is smaller if the underground value of blocks is deducted from their open cut value.

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Note that the pit in Figure 53 takes in part of Stope-1. Stope-1 is only economically viable by the underground method if all of Stope-1 is mined. If the pit takes part of Stope-1, it will not be economic to mine the remainder by underground method. A new optimization should be run, which excludes the ore available to Stope-1 from consideration for underground mining. The results are shown in Figure 54.

Figure 54: Stope-1 is no longer made available for underground mining, so the final pit is larger.

The pit now takes in more of Stope-1 as expected. Stope-2 and Stope-3 remain undisturbed and should be mined by the underground method.


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When the Pit Contributes to the Underground Development In some cases, the open pit mine precedes the underground mine and it is possible to commence the underground development from the base of the pit. In this case, a deeper pit can lead to a reduction in the expense associated with underground development and it is possible to build this benefit into the block model. The method outlined below should be applied in conjunction with the method show above in the section “Which ore should be mined by open cut method?”. Step 1- Identify underground development costs You need to establish the cost per metre of vertical development in the absence of the pit. Multiply this by the height of your blocks in order to get a per-block benefit value. For example, if the cost per vertical metre of underground development is $2,000 and blocks in your block model are 10 metres high, then the perblock benefit would be $20,000. Step 2 - Modify the Block Model Run an optimization with a normal block model. Identify a set of blocks, one per bench to which you will attribute the perblock benefit. You want to influence the depth of the pit, not the horizontal position of the base. In the case of Whittle, the set of blocks should follow the bases of the shells generated for different Revenue factors. These will naturally follow the centre of the ore body or the footwall. Edit the block model, to add the per-block benefit to your chosen set of blocks. In the case of Whittle (single element) you need to create a parcel containing a quantity of metal, which, at the reference price, would yield revenue equal to your per-block benefit. For example, if the per-block benefit was $20,000, the product were Gold and the price was $10 per gram, you would create a parcel containing 2,000 grams of Gold 19. The parcel tonnage should be kept to a nominally low figure in order to reduce other biases. The rock-type should be a specially created one so that the effect of these parcels can be tracked in reports.

19

If you are using a package such as Whittle (multi-element) you can define a new element called “DOLL” for “dollars”. The price of a DOLL is always $1. This simplifies many of the calculations.

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For example: Assume that block 15 ,7, 23 originally contained one parcel. To add another parcel, you must increment the number-of-parcels value in the header line, and append the new parcel line. Column numbers: 0 1 2 3 4 123456789012345678901234567890123456789012345678

Original block record: 15 15

7 23 1 7 23 OXID

1.000 500

1.000 1000

1000

Modified block record: 15 15 15

7 23 2 7 23 OXID 7 23 UGDV

1.000 500 1

1.000 1000 2000

1000

Step 3 – Add a Special Processing Method On importation of the modifiled model, set up rock-type UGDV (UnderGround DeVelopment) with the following parameters: Rock-type mining CAF = 1.000 Rehab. cost ratio = 0.000 Processing throughput factor = 1.000 Add a processing method which is exclusively used for the UGDV rock, called DVPR (DeVelopment PRocess). Set the parameters as follows: Method Code = DVPR Rock-type Code = UGDV Processing Cost $0.01 Processing recovery fraction = 1.000 Processing Recovery Threshold = (blank) Minimum cut-off = (blank) Maximum cut-off = (blank) Step 4 - Optimization Re-optimize with the new Model file and additional Processing Method. You should find that for the same Revenue factors, that the new pits are greater than or equal to the pits generated from the original model.


Multiple Mines

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Originally presented as: Hall, G, 2004, Multi-mine Better than Multiple Mines, in Orebody Modelling and Strategic Mine Planning Symposium (Perth)

Abstract It is not uncommon for a number of open cut mines to share infrastructure in the mining value chain. This sharing offers economies of scale, and presents additional scheduling options, but also increases the complexity of design and scheduling. How do you best investigate and optimize this type of scenario in order to yield maximum economic benefit? As a senior developer in the Whittle team, the author has been involved in the creation of a modelling and optimization system which caters for multiple integrated mines. The objectives of the system design were: •

To provide a process, supported by effective tools, which enables mine planners to maximize the economic benefit of multi-mine operations.

•

To provide a modelling and optimization environment that allows multiple integrated mines to be planned and scheduled effectively, including adaptation of optimization engines to the multi-mine situation.

This paper describes the benefit the Whittle Multi-mine option can bring to such a complex operation and the features that enable that benefit.

Introduction It is not uncommon for a number of open cut mines to share infrastructure in the mining value chain. This sharing presents scale economies, and presents additional scheduling options, but also increases the complexity of design and scheduling. How do you best investigate and optimize this type of scenario in order to yield maximum economic benefit? Multiple mines could be treated in Whittle, to a certain extent, before the Whittle Multi-mine option was introduced. The simplest approach was to model each mine in isolation and then produce a schedule manually. Several people, Tom Tulp (Tulp, 1997), David Whittle (Whittle, 2001) and Chris Desoe from AMDAD developed techniques that removed some of the restrictions associated with treating multiple mines as a single model within the Whittle environment. None of these processes could entirely remove the restrictions and they all required complex setup procedures. Within their limits, however, they worked and they all enjoyed success in a restricted number of situations.

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The Multi-mine option allows the flexibility of choice of optimal pit and pushbacks for each mine independent of the other mines in the model, while still producing a schedule automatically across all mines.

Multiple mines - The background A multiple mine operation has more than one mine sufficiently close together that they share infrastructure and are planned as a single study. The past approaches to modeling this situation identified in the introduction either fail to address the benefits of producing a joint schedule, or limit the definition of the individual mines so that they do not use their optimal pit or pushbacks tailored to that mine. Planning the mining schedule of a single mine is reasonably well understood. While the process is complex, tools exist to assist the user in creating an optimal open pit shape from a model of an ore body. Tools also exist to assist in planning a mining schedule from the chosen pit. The difficulty in the multiple mine situation in particular, is in defining the best pushbacks and finding the best mining schedule. The Whittle product uses the Net Present Value (NPV) of a mine to drive both the identification of the optimal pit and the creation of the mining schedule. The optimal pit is found using the Lerchs-Grossmann (LG) algorithm (Lerchs & Grossmann, 1965). The method for determining the optimal pits is the same for a single mine or a set of mines. This, however, is just the start of the solution. The material to be removed from each mine can be extracted in any one of a number of ways, all of which can result in dramatically different NPVs for the mine. The creation of a mining sequence involves defining some useful pushbacks for those mines then mining those pushbacks in such a way that maximises the potential value of an operation.

The Whittle Multi-mine solution The model file of the Multi-mine option uses a single block model definition which identifies each mine in the block model separately. You are able to define the pushbacks and choose the optimum pit for each mine separately, then create a schedule that considers all of the mines together. This technique allows each mine to be designed to its full potential because its optimum pit is independent of any other mine. During the scheduling process, however, there are benefits from considering the mines as multiple sources of ore. The scheduler is able to decide when to choose material from the mines such that the value of the operation is maximised. The key benefits of the Multi-mine option are that it gives you independence between mines:


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•

pushbacks can be determined that are ideal for an individual mine,

•

the final pit for each mine is separate,

•

the order of processing of the mines can be changed easily, and

•

mining limits can be tailored for individual mines.

In addition, the material movements in each mine can be tracked separately and extra controls have been added to allow per-mine constraints. By using Whittle to find the theoretical maximum value of the operation, it is possible to cost the decisions that are made along the way as progress is made to a final design for the operation. Sometimes, this means that Whittle presents information that justifies a change in approach because of the increased value that is realisable when that change is implemented.

Example case study Example data The data comes from the “BlueSky” project which was originally developed as a multi-mine example by Chris Wharton and is based on the “Marvin” data developed by Norm Hanson for his students at RMIT University and used in the “Whittle Challenge”, a one day add-on to the “Optimizing With Whittle” conference in 1999. It has two mines, called NorthPark (mine 1) and SouthBorder (mine 2). SouthBorder is the standard Marvin mine with three rock-types: OX (a surface oxide), MX (a mixed ore) and PM (the primary ore). NorthPark has been modified from the original Marvin with the OX and PM rock-type tonnages being summed together (called SL) and MX being renamed to RF. The SouthBorder rock-types have their rock-type Cost Adjustment Factor (CAF) greater than one to indicate a harder rock than in the original Marvin. The slopes of both mines have been modified from the original Marvin and also made different from each other. All rock-types have gold and copper elements. Treat as single mine Optimal Pit The creation of the optimal pit for each mine proceeds as in the single mine case because the LG algorithm (in a single model file) will treat the mines as independent entities (provided the resultant pits do not touch).

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Pushbacks and Mine schedule Before we explore designs for a practical and realistic mine design and schedule, it is useful to look at the operation as a single mine case to provide some indicator values. The initial run introduces the period tonnages involved and forms a framework for developing further refinements. Key aspects of this run are: •

liberal mining rate (chosen to ensure mining limit is never, or rarely a limiting factor. In this example 80 million tonnes (mt) per annum (pa))

•

conservative processing throughput (chosen to ensure this is the limiting factor and reflects “reasonable” mine life - 20 mt pa)

With the above limits and reasonable estimates of the costs required to support the above rates, the Pit by Pit Graph node indicates that the maximum best case NPV that can be achieved is $272m. This is the pit containing 693m tonnes total with a mine life of 24 years (Table 1, line 1). We have arbitrarily chosen to develop four pushbacks. This is a decision that can be explored further when there are definite costs of starting a new pushback. The more pushbacks you have, the closer you can get to the Whittle “Best case” scenario. When the costs of a pushback are included in the analysis, you can very quickly see when the cost of adding a pushback outweighs the return. When we add four pushbacks (letting the Pushback Chooser (Whittle 2004a) decide them for us), the optimal pit is 488mt with a value of $186m over a nearly 19 year mine life (Table 1, line 2). Table 7 : Summary of results


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Three asides 1. The use of geometric values 20 in defining the revenue factor range generates a good range of pits, giving good starting pits and still keeps the overall number of pits to consider to a minimum. 2. By including the actual cost of establishing pushbacks in an operation, one can determine whether using more pushbacks would improve the value of the operation. 3. The slopes of each mine in an operation could be quite different. Since Whittle has the capability to model these without the use of the Multimine option, they will not be discussed further in this paper. Treat as multi-mine Without the Multi-mine option above, the chosen pushbacks are the same for every mine. The optimal pit is chosen by its pit number and that is also the same for every mine. Each mine is different, therefore one would expect the ideal pushbacks for each mine to be different. Using the Multi-mine model we can run the Pushback Chooser separately for each mine. This approach can be used because the Pushback Chooser only uses the relative differences between NPVs in deciding where to put the pushback boundaries. Once we have the pushbacks for each mine, we can explore schedules using input from both mines (each with its own pushbacks) and costings and limits that can be a combination of global and per mine attributes. Note that individual mine constraints are only available with the Multimine option. The user can now explore the opportunities available to vary the schedule based on the order in which the mines are considered as well as the previous variables associated with pushbacks in a single mine. At this point it useful to note which mines are the biggest contributors. This will help drive the decision as to the order in which we should mine the mines. In this example, the significantly bigger contribution comes from the NorthPark mine, so we will consider it first in the order (Fig 1). Using the Milawa algorithm will improve the NPV if an inappropriate ordering of mines is chosen, but it cannot necessarily find the best NPV.

20

“Geometric values” is a technique for defining revenue factors that produces a greater number of pits at the smaller pit end of the range than at the larger pit end (Whittle 2004b). It is useful for defining the starter pit and early pushbacks.

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Figure 55 : Cash flow contribution from each mine

With each mine having its own (four independent) pushbacks and considering NorthPark first, we end up with a schedule (Fig 2) developing an NPV of $197m from a combined tonnage of 569 mt (Table 1, line 3). This is an increase of $11m with addition of 81 mt over the previous result which is a direct result of being able to start with individually optimized mines. The following steps are not specific to the Multi-mine option when only global limits are applied, but significant gains in NPV may be available by exploring variations in the processing and mining limits. Modifying constraints When analysing the effects of constraints, you should ensure that constraints further back in the process are not making an adverse impact on downstream processes. For the illustrative purposes of this paper, selling limits are ignored (the last step in the Whittle limits) and we’ll deal with the two main limits back up the process stream: the processing capacity, then the mining capacity. Processing capacity We have started with a generous mining limit, so the impact of extending the processing capacity can be considered without a tight mining limit confusing the results.


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We consider the situation whereby spending an extra $55m we can add 10 mt pa to the processing capacity, increasing it to 30 mt pa. 21 The result of this analysis is that we can increase the NPV from the previous analysis to $280m with the same tonnage (Table 1, Line 4). The change is that the mine life is now less than 14 years as compared to over 20 years previously. The increase in NPV is due to being able to earn the money sooner. Note that at this point, mining limits have not been touched and so the mining cost (quoted per tonne) is unchanged. Mining capacity From Fig 2 we can see that there are some periods that have mined considerably more material than is required in that period. The pattern is similar after the processing capacity is increased.

Figure 56 : Two mines, independent pushbacks

Let us consider what would happen if we reduced our mining capacity to 60 mt which allows us to save $30m. 22 We see that we can add $26m to the value of the mine, increasing the NPV to $306m (Table 1, Line 5) even though we don’t fill the mill in periods 5 and 12 (by a small amount). (Fig 3)

21

In a real study, several scenarios would be considered to explore the benefits of increasing production. Some questions that would need answers are, “Should we increase production?” “If we do, by how much?” “What are the risks involved?” This example is chosen to illustrate one such scenario.

22

As with the processing capacity, this is an example of a single variation, which in practice would be one of several variations studied.

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Figure 57: Increased processing, decreased mining

Milawa algorithm The study up until now has only used fixed lead. This has been for a couple of reasons. The fixed lead approach gives results very quickly which allows us to explore many possible “what if” scenarios and gives a good feel for the performance of the mine under differing conditions. As we get closer to what we think might be a final solution, we use the Milawa algorithm 23 to see what extra benefits we can realise out of this mining operation. The result using Milawa NPV raises the value another $30m to $336m (Table 1, Line 6). Now the mill is kept full (until the end of the mine). Milawa is now changing the order of processing in the mines to achieve a greater NPV. This becomes more obvious when the mining limit is reduced even further to 50 mt pa (Fig 4).

23

The Milawa algorithm is a proprietary algorithm that modifies the selection of material available from every open pushback to produce a schedule that improves the NPV. “Milawa NPV” focuses on improving NPV, “Milawa Balanced” focuses on keeping the mining rate balanced.


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Figure 58 : Milawa NPV

The next result, from a Milawa Balanced run, shows that we can balance our mining (and keep the mill filled) at a cost of dropping the value of the operation to $246m (Table 1, Line 7; Fig 5).

Figure 59 : Milawa Balanced

From Figs 4 and 5, by inspection, it looks like the increased mining in the early years of the Milawa Balanced solution is contributing to some early costs of mining which do not occur in the Milawa NPV solution. We can now consider “tuning” the mining capacity to improve the Milawa Balanced result. In this example we can achieve a Milawa Balanced

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schedule (Fig 6) that is a significant improvement over the “un-tuned” result yielding a value of $328m (Table 1, Line 8).

Figure 60 : “Tuned” Milawa Balanced

Conclusions The optimal pits for individual mines can be determined without the Multi-mine option in Whittle. The LG algorithm will develop each mine independently. The basic approach to a multi-mine study is similar to a single mine study with all the single mine features being available in the multi-mine situation. The differences arise when the key benefits of the Multi-mine option are used: • • pushbacks can be determined that are ideal for an individual mine, •

• the final pit for each mine is separate,

•

• the order of processing of the mines can be changed easily, and

•

• mining limits can be tailored for individual mines.

The Multi-mine option can add significant value to a multiple mine operation.


The Effects of Sequencing & Scheduling

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The Effects of Sequencing and Scheduling A mining “sequence” is the order in which different parts of the pit are mined. A mining “schedule”, which tells us when things occur, can be constructed by applying the production constraints to the mining sequence. As well as determining when different parts of the pit are mined, a schedule determines when the cash flows associated with mining are produced. This is important because all dollars are not equal. The effect of time on the value of money We discussed this earlier, but now we will look at it in more detail.

A dollar that we receive today is more valuable to us than a dollar that we might receive in a year’s time. There are various reasons for this: • Inflation may reduce the value of next year’s dollar. •

If we have the money now, there is no risk of something going wrong and our not getting it.

•

If we haven’t got the money, we can’t get interest on it, or we may have to borrow money to replace it.

The standard way to allow for this is to “discount” next year’s dollar by a certain amount and to apply that idea cumulatively into the future. Thus we discount future revenues and costs by a particular discount rate and reduce them all to a “Net Present Value” or NPV. The effect of mining sequence on the optimal pit outline Let us review a very simple example shown in Figure 61. surface 1 2 3 4 5 100 tonnes waste

6 7 8 500 tonnes ore

bench level

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Figure 61: A simple ore body

Worst case mining The simplest way to mine a pit is to mine the whole of the top bench, then the whole of the next bench etc. We call this “worst case mining” because it produces the lowest NPV. However, it has the advantage of almost always being practical. If we consider the simple example again, waste at the top of the outer shells is mined early, and the cost is discounted less than the revenue from the corresponding ore, which is mined much later. The optimal pit for worst case mining is thus generally smaller than is indicated by simple pit outline optimization using today’s costs and revenues. Best case mining Each shell is mined in turn and thus the related ore and waste is mined in approximately the same time period. We call this “best case mining” because it produces the highest NPV. It is almost never practical, but sets an NPV target that we can aim for. The interaction between production rates and mining sequence

Table 8: Pit values for different mining sequences and production rates Pit Worst Mill = PRCOST However, if the processing cost is either: independent of the grades of PR1 and PR2, or varies linearly with the grades and the recovery fraction of each is independent of grade, then an “equivalent metal” approach can be used. We multiply the grade of PR2 by a factor to make its value comparable with PR1 and add the value to the grade of PR1. When mining, we apply the PR1 cut-off to the combined “equivalent metal”. We can either calculate an equivalent grade of PR1 or of PR2. Equivalent GRADE1 = GRADE1 + K2*GRADE2 where K2=(REC2*PRICE2)/(REC1*PRICE1) Equivalent GRADE2 = GRADE2 + K1*GRADE1 where K1 = (REC1*PRICE1)/(REC2*PRICE2)


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Alternatively, we can work out the factor to apply by first working out the cut-offs which would apply to each product in isolation and then dividing CUTOFF1 by CUTOFF2. Therefore, the formula for an equivalent PR1 metal is: GRADE1 + GRADE2x(CUTOFF1/CUTOFF2) Another way of looking at it is to divide each grade by the corresponding cut-off and add the resultant values together. You then apply a cut-off of 1.0 to the sum. This is shown graphically in Figure 76. Material to the right of the sloping line is processed, material to the left is not.

Figure 76: Graphical illustration of cut-off using equivalent metal approach

If the processing cost is not linear with grade, or the recovery fraction is not independent of grade, then, strictly, you should not use an equivalent metal. You should calculate the revenue and processing cost of each sample, and only process if the revenue is higher than the cost. Nevertheless, equivalent grades are often used in these circumstances for operational convenience.

Changing the Cut-offs with Time Consider a mining sequence which consists of ten 1MT increments like our earlier example, but, for simplicity, with no time costs and no mining limit, so that the marginal cut-off is 0.2. This gives a total cash flow of $27.8M and a time of 1.6 years for each increment. If we now vary the cut-off, for the first increment only, what happens to the NPV of the whole project?

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The effect can be seen in Figure 77. Why is the best cut-off for the first increment now more than 0.5, whereas we know that 0.2 gives the highest cash flow?

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NPV (Millions)

132 130 128 126 124 122 120 0.00

0.20

0.40

0.60

0.80

1.00

Cut-off for Increment 1

Figure 77: The effect of varying the cut-off used for the first increment


Cut-off Optimization

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Lane Theory of Cut-off Optimization Kenneth F. Lane explained that, in addition to the normal cash costs which must be considered in the processing cost when calculating cut-offs, there are two pseudo costs which are also important. We refer to these as the “delay cost” and the “change cost”. Both of these costs take account of the fact that any mining or processing activity takes time, and therefore delays the exploitation of the rest of the project. For example, consider the ten million tonne scenario mentioned earlier, with the cut-off that we will use to exploit the last nine million tonnes fixed at 0.20. As a consequence the cash flows for the last nine million tonnes in today's dollars is fixed. Let us now look at the effects of using three different cut-offs for the first million tonnes. Cut-off for first 1mT 0.20 0.45 0.70

Time for first 1mT 1.6Y 1.1Y 0.6Y

Cash flow for first 1mT $27.8M $25.0M $16.6M

NPV of first 1mT $25.4M $23.5M $16.0M

As the cut-off is increased, the cash flow for the first million tonnes decreases, as we would expect, since we know that the marginal cut-off of 0.20 maximizes the cash flow. The NPV of the first million tonnes also decreases, but not quite so rapidly because the discounting is reduced due to the reduced elapsed time. However, the NPV of the remaining nine million tonnes increases because its exploitation is started sooner and all its cash flows arrive earlier. Cut-off for first 1mT 0.20 0.45 0.70

Time for first 1mT 1.6Y 1.1Y 0.6Y

NPV of last 9mT $102.9M $108.9M $115.2M

Total NPV of project $128.3M $132.4M $131.2M

The NPV of the last 9mT changes by about 6 per cent with each step in cut-off, which is what you would expect with a twelve per cent discount rate, if the start time changes by half a year. Clearly, of the three cut-offs we have tried, 0.45 gives the best total NPV, and this ties in with the graph we saw earlier. Can we calculate the cut-off which maximizes the total NPV? Since the NPV of the last 9mT varies with the time taken to exploit the first 1mT, we can treat it as a time cost, here called the delay cost. The difference between starting after 0.6Y and after 1.6Y is $12.3M. If we divide this by the processing capacity, we get a delay cost of $24.6/T. It is then easy to work out the best cut-off.

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(18.00+24.6)/(100.00*0.90) = 0.47 This does not quite tie in with the graph, where the maximum is obviously greater than 0.5. The reason for this is that this simple formula makes no allowance for the discounting of the cash flow from the first 1mT. It is worth making the point here that the discounting schemes normally used in accounting, where all the cash flows for a particular calendar year are discounted by the same amount, do not reflect the real value of the cash flows, since they do not distinguish between an amount received on January 1st and one received on December 31st. These schemes were devised as convenient approximations in the days before computers, and are now thoroughly entrenched. For cut-off optimization purposes it is essential to discount on a continuous (i.e. daily) basis. This changes the slope of the cut-off/NPV curves slightly and moves the maximum. In this case it moves from 0.47 to 0.52. Note that this makes no material difference to the NPV, which is only 0.09% less with a cut-off of 0.47. This is another example of how insensitive systems become when they are near to a maximum. For the record, the discount multiplier to apply to a total of a constant daily cash flow, starting at time T1 and ending at time T2, can be calculated as follows: Discount percentage/Y:

P

Discount fraction/Y:

F = 100/(100+P)

Discount multiplier: F^T1*(F^(T2-T1)-1)/(Ln(F)*(T2-T1)) Where Ln() is log to the base “e”

We have discussed the delay cost at length, but we mentioned earlier that there are two pseudo costs. Everyone knows that cash flows are higher if we exploit our resource when the price of the product is high, and vice versa. Using the previous example, if we delay the exploitation of the last nine million tonnes to a period of lower prices, we reduce the cash flows for the nine million tonnes, and hence the NPV of this nine million tonnes. Since this effect will generally get bigger with increasing delay, we again treat it as a type of time cost, and we call it the “change cost”. It is different from all other time costs because, if the price of the product increases with time, or the costs decrease with time, it can be negative.


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Note that, if the economic circumstances are constant in today’s dollars, the change cost is, by definition, zero. If the project is mill limited, both the delay and the change costs should be divided by the mill throughput limit and added to the processing cost when calculating the cut-off. Consequently, the delay cost, which is always positive, increases the cut-off. The change cost can increase or decrease the cut-off, depending on whether the economic circumstances are deteriorating or improving, respectively. However, its general magnitude is still related to the remaining NPV. So far we have only discussed the possibility of varying the cut-off for the first increment. Why should we not vary the cut-offs also for the second and subsequent increments? We can, of course, but when we do, we change the NPV of the last 9mT, which, through the delay and change costs, changes the optimal cut-off for the first 1mT. This leads to a circular arrangement where we repeatedly go back and re-optimize earlier cut-offs until the cut-off schedule for the complete exploitation of the 10mT settles down. As the resource is used up, the NPV of the remainder of the resource tends to fall, and is zero when no further resource remains. Since both the delay and the change costs are dependent on the remaining NPV, they too tend to fall. In general, therefore, optimized cut-offs start high and progressively decrease throughout the life of the project. For the case where the economic conditions do not change, and for which, therefore, the change cost is zero, Lane proposed an approach to optimization in which you start with an estimate of the total NPV, optimize each increment using a delay cost derived from the remaining NPV and then update the remaining NPV. When all of the resource is exploited in this way, you usually find that the final remaining NPV is not zero as it should be. You then adjust the estimate for the starting NPV and try again. This process is repeated until the final remaining NPV is zero.

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If we follow this approach with the case we were looking at before, we get cut-offs as shown in Figure 78, and a total NPV of $144.3M, which is an improvement of 12.5% on the $128.3M obtained with marginal cut-offs.

0.70 0.60 Cut-off

0.50 0.40 0.30 0.20 0.10 0.00 1

2

3

4

5

6

7

8

9

10

Increment

Figure 78: Cut-off curve for simple increments using Lane method

When change costs have to be considered, Lane proposed starting two optimization loops like the one above, with the second starting one year later than the first. The optimizations would be run side by side and the difference between the two NPVs would be used to estimate the change cost. We understand that the program “OGRE”, which was developed by RTZ uses this algorithm. So far we have assumed that material below the cut-off was to be discarded. However, since the cut-offs tend to be high during the early years, it is often worthwhile stockpiling some of this material for later processing. We will address this possibility later.

Whittle Method of Cut-off Optimization When Whittle Programming started to develop a package for cut-off optimization, we decided that it had to be able to handle multiple rocktypes, multiple processing methods per rock-type, and multiple throughput limits, like Whittle. Consequently it had to calculate a number of cut-offs for each increment or time period. Since the cut-offs would normally interact with each other through the throughput limits, we had to use a multi-variable optimizing engine which could cope with a non-linear function. After testing a proprietary optimizer, which was unequal to the task, we developed our own.


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This allowed us to follow the principles of Lane optimization, in that we started from an estimated NPV, optimized the cash flow per tonne, including the pseudo costs for each increment, and then iterated the whole thing until the final NPV was zero. This can be expressed as is shown below when there are no change costs. It is much more complicated when there are. Set a starting NPV Iterate until the ending NPV is sufficiently close to zero: Set the ending NPV to the starting NPV For each increment: Find the cut-offs for this increment which maximize the cash flow per tonne including the delay cost Calculate the NPV of the increment Subtract this NPV from the ending NPV Next increment If the ending NPV is not close to zero: Adjust the starting NPV End If Next iteration This worked - mostly. However, “mostly” was not good enough for a commercial package, and we changed to the following scheme: Iterate until the total NPV stabilises: For each increment: Find the cut-offs for this increment which maximize the total NPV while keeping the cut-offs for the other increments constant Next increment Next iteration This turns out to converge reliably. In particular, it seems to handle the effects of change costs much better. It always produces an answer, but we know that with certain very odd data it can be fooled into converging on a local maximum which is not the global maximum. Because it evaluates the whole NPV rather than the NPV of a single increment when optimizing the cut-offs for an increment, it would appear to do far more computing than the Lane method. In practice, with judicious use of a store of prior calculations, it only does slightly more. This approach produces the cut-offs shown in Figure 79.

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0.60

Cut-off

0.50 0.40 0.30 0.20 0.10 0.00 1

2

3

4

5

6

7

8

9

10

Increment

Figure 79: Cut-off curve for simple increments using Whittle method

The total NPV is $145.1M, which is an improvement of $0.8M on the Lane method. What is interesting is that if we use a generalized optimizer (The Excel Solver) which “knows” nothing about cut-offs or any theory related to them, we get cut-offs which differ only in the fourth decimal place and an NPV which differs by only 36 cents. Note that the cut-off curve is convex upwards rather than downwards, as is inherent in the Lane method. Why does the Lane approach produce a result which is not as good? There are two reasons. First, Lane uses the NPV at the start of mining the increment to calculate the delay cost, whereas it is the NPV of the material after the current increment which is delayed. Second, Lane maximizes the cash flow for the increment rather than the discounted cash flow. That is, he doesn’t allow for the fact that lower cutoffs delay the exploitation of some of the current increment. When we correct either of these separately, we obtain a lower NPV. However, when we correct them both we obtain an NPV which is only 0.035% less than the best we have obtained. It is, incidentally the use of the discounted cash flow which causes the cut-off curve to be convex upwards.


The DC Model for material classification and NPV maximization

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Overview The DC Method is an approach to ore waste discrimination which utilizes a new cash-flow approach. It differs from Lane, which proceeds on the basis of grade cut-offs, and it differs for existing cash flow methods in that it can be used in Cut-Off optimization for multiple processes. Compared to Lane-based approaches: •

Far fewer decision variables need to be optimized - In a simple case (no stockpiles) Lane has one decision variable per element; per rocktype; per process, per period. For example, in a two element deposit, with three rock-types, three processes and 10 periods, there are 180 decision variables. For the same problem, the DC approach only requires 30 decision variables (one per process; per period).

•

Processes need not be ranked – The model is symmetrical – there is no need to rank processes.

•

Process balancing – The DC method provides a very solid structure for process balancing, something that the Lane-based approaches, with ranked processes, cannot do.

•

Better modeling of complex economics and recoveries - The DC method operates in cash-flow mode, which is much more flexible than the Lane approach, which operates in cut-off mode. In cut-off mode there are severe limitation on precision of selectivity in cases of complex economics and recoveries. In cashflow mode there are no known limitations.

•

No need to re-run Marginal Cut-off case – The marginal cut-off case is identical to the marginal cut-off generated in cash-flow mode by Whittle. There is no need to generate an ore selection by cut-off equivalent, and consequently no associated reconciliation problems.

•

No increase in complexity - The DC method is no more complex that the Lane approach, if anything, it is simpler, with few decision

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variables, and no ore/waste discrimination issues normally associated with cut-off-based ore/waste discrimination and grade-dependent costs and recoveries. Compared to the King Cash Flow Grade model: Can deal with multiple unranked processes with throughput limits The model implemented by Brett King deals only with ranked processes. He states “An observation from following this approach is that the Heap Leach throughput rate does not feature in the … equation. This suggests that potential exists to find a better ranking system for separating the process feeds. Answering this question is a topic for further research and investigation.” I have done a review of subsequent papers, and did not find evidence that he has published any advances on the above-mentioned topic. The DC model deals with any number of unranked processes, each with throughput limits.

Material Allocation by Cashflow We commence here with “Mill” and “Heap” as the example processes, but as the logic develops, it will be abstracted to any processes, and any number of processes. It is possible to calculate for any parcel of material the profit associated with assigning it to a process. For each parcel, you can calculate the profit for every defined process. Against those calculated values, we can apply a set of Discrimination Controls (DCs) in a structured way to allocate material to the appropriate location (a process or a waste dump). Definitions

•

Tonnes – used in this document to represent material (ore, waste) mass. It is less cumbersome than the more generic “mass units”. Substitute tons, long tons or short tons as required.

•

DC Diagram - A DC diagram is defined here to show the values for two process alternatives on the X and Y axis for a population of material parcels.

•

Increment – A grouping of material that becomes available for mining. It contains a number of Parcels, each a unit of material which can be selectively processed.

•

Material – rock coming from the mine which may be waste or ore.


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•

Mining Sequence – made up of a series of Increments. A full mining sequence represents all the material in a defined mine, in the order in which it may be accessed.

•

Parcel – A distinct package of material, with a tonnage, and various attributes such as grades, rock-types etc.

Marginal Cut-Offs

The marginal cut-off is the cut-off which maximizes the net undiscounted cashflows. Refer to Figure 80. The general rules are: 7. If the value is positive it will be processed. 8. Given 1, then the parcel will be assigned to the process which provides the highest revenue. Parcels in areas A and B are assigned to the Mill because there are no alternatives which will yield higher profits. C and D are assigned to the Heap because there are no alternatives which yield higher profits. Marginal Cut-Offs in a DC Diagram

The marginal cut-off is the cut-off which maximizes the net undiscounted cashflows (as for Whittle ore selection by cashflow. Refer to Figure 80. The general rules are: 9. If the value is positive it will be processed. 10. Given 1, then the parcel will be assigned to the process which provides the highest revenue. The -ve X and -ve Y axes and a diagonal line running between the +ve X and +ve Y axes provide the delineators for process assignment in the normal case. The normal case is equivalent to a Whittle ore selection by cash-flow case, where material is assigned to the process which makes the most profit, or if no process makes a profit, the material is assigned as waste. Parcels in areas A and B are assigned to the Mill because there are no alternatives which will yield higher profits. C and D are assigned to the Heap because there are no alternatives which yield higher profits.

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Figure 80: Process allocation in cash-flow mode


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Figure 81: DC Scatter Diagram for Marvin First Pushback

Changing Material Classification in order to maximize NPV.

The following is based on the general principals established by Lane: • For a given increment, if you reduce the time taken to process the increment, then you will increase the NPV of the sum of all subsequent increments. • The time taken to process an increment depends on the quantity of material assigned to processes which have throughput limits (tonnes/year) limits applied to them. • By reassigning material that would have been processed by the constraining processes, to other processes or to waste, the time taken to process the increment will be reduced, and the increment NPV will be reduced. The correct or “optimal” reassignment is that which maximizes the total NPV for all increments.

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Two Processes, one with a throughput limit

In the event that one process has a throughput limit on it, then a decrease in the tonnage of material sent to that process will reduce the length time required to process the increment. The reduction in time taken to process this increment, decreases the NPV of the increment, but increases the NPV of subsequent increments. Given that it might be desirable to decrease in the tonnage put through the limiting process, the question is: what type of material should the decrease be applied to? The answer: it should be applied to the material which, if reclassified, will have the lowest impact on the NPV of the increment.

Figure 82: The material that should be reallocated from Mill is that which, if reclassified, will have the lowest impact on the value of the increment.

It follows that material that should be reallocated is that which has the lowest profit per tonne, or the lowest difference in profit. Refer to Figure 82. In Figure 83, a set of red lines has been added. The position of the new red lines is defined by the original vertical axis, and the variable DCm (Shown as Dm in the figure) - Discrimination control for Mill.


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The reassignment proceeds in a manner which has the least impact on increment NPV. This means reassigning the material F and G to the next best alternative: • E is reassigned to waste. Its value as waste is $0. If it were reassigned to Heap, its value would be negative, so Waste is the better choice. •

F is reassigned to Heap. Its value assigned to Heap is lower than its value assigned to Mill, but the Heap value is greater than zero. Its value as waste is zero, so Heap is the better choice.

Figure 83: Process re-allocation that will reduce the time take for the increment to be processed (presuming a Mill throughput limit).

Two Processes, both with throughput limits If a limit is introduced on the Heap process as well, then notionally, a reduction in the tonnage assigned to Heap, could increase the sum of NPVs for this and all subsequent increments. As for the reclassification from Mill, the Heap reclassification should be done in a manner which has the minimum impact on the NPV of the increment. Figure 84 shows the result. A new set of blue lines is introduced, defined by the variables

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DCm and DCh (Shown as Dh in the figure) - Discrimination control for Heap. • G is reassigned from Heap to Waste. •

H is reassigned to Mill. This is the least cost reallocation (in fact, this reallocation actually wins back some value).

Figure 84: Process re-allocations that will reduce the time take for the increment to be processed (presuming a Mill throughput limit and a Heap throughput limit both apply).

If the Heap constraint is increased (in order to further reduce the throughput to the heap), from Dh1 to Dh2 (shown in Figure 85) then material is reassigned again.


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Figure 85: If further process allocation from Heap… The two decision variables DCm and DCh and their implementations (material classification for DCh1 and DCh2 shown).

Generalized Rule (two processes)

The processes used above are “Mill” and Heap” – familiar to miners, but unnecessary for the model to work. There is no need for the two processes to have any ranking of precedence or value. The model is entirely symmetrical. For the generalized rule we need to only refer to process 1 and process 2. =If(And((P1-DC1)>0, (P1-DC1)>=(P2-DC2)), "Pa1", If( And((P2DC2)>0, (P2-DC2)>=(P1-DC1)), "Pa2", "W")) Where: Pa1 = Process 1 assignment Pa2 = Process 2 assignment W = Waste assignment P1 = Profit through Process 1 P2 = Process through Process 2 DC1 = Discrimination control for Process 1 DC2 = Discrimination control for Process 2

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Generalized Rule (multiple processes)

For Process n of m processes =If( And((Pn-DCn)>0, (P1-DC1)>=max((P1-DC1), (P2-DC2), (P3-DC3), … (Pm-DCm)), ), "Pan",[next test]) The above test is conducted on each parcel for each of m processes. Material that is not selected for any process, is classified as waste. Where: Pan = Process n assignment Pn = Profit through Process n DCn = Discrimination control for Process n There is an order to the tests, but the order is arbitrary, as only one test can have a positive result, except in the case where the economics of two processes are identical. In this case, the parcel will be assigned to the first identical process that returns a positive result. Generalized Rule (Waste defined as an additional process)

Note that the approach described below has not been tested. It should be possible to define waste destinations as processes, each with constraints and economic models set as for regular processes. By this method, it should be possible to provide an optimal assignment of waste to one of a range of possible waste dumps. There would be a need for a default waste process to be defined, with extremely high costs associated with it to make the mathematics work. With that one qualification in mind the mathematics actually become a bit simpler. For Process n of m processes: =If((P1-DC1)>=max((P1-DC1), (P2-DC2), (P3-DC3), … (Pm-DCm)), "Pan", [next test] ) The above test is conducted on each parcel for each of m processes. Material that is not selected for any process, is classified as waste. Where: Pan = Process n assignment Pn = Profit through Process n DCn = Discrimination control for Process n


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