Orebody Modelling and Strategic Mine Planning: Integrated mineral investment and supply chain optimisation (Conference Proceedings SMP 2014)

A mining supply chain is an end-to-end supply chainincluding all value-added production operations from the

(or commodity). A typical mining supply chain in the contextof a ‘mining complex’, consists of mines, waste dumps, orestockpiles, processing plants, product warehouses and

complexes are considered, integrated mining supply chainsare formed to provide comprehensive mine-to-port supplyof materials/commodities, including mining and processingore, generating products (gold bullion, copper concentrate,iron supply, and so on) and transportation. In this work,

the near optimal decisions for a mining complex to maintain

supply and the commodity market.The mineral supply chain optimisation model proposed

herein, as shown in Figure 1, uses the data obtained fromthree program modules, ie mine scheduler, market analyserand contract manager, which are assumed already available.The spot market and contracted customers are consideredin the proposed model. The dynamic recovery rate thatdepends on the head grade of the feeding material and thethroughput of a processing plant is also considered. A salescontract designing strategy is proposed using the proposedmodel and heuristic.

The remainder of this paper is organised as follows. Inthe following section the notation and assumptions forthe optimisation model are provided. Next, a stochasticmixed integer non-linear program is formulated and thecorresponding solving heuristic is developed. A series of

numerical tests are conducted in the Model and Heuristic

heuristic. Then, a long-term sales contract design strategybased on the proposed model and heuristic is proposed.Finally, the conclusions and future work are outlined.

The mining complex’s planning horizon is T time periods, andt {1,…,T} is the period index. The uncertainty of the miningcomplex as considered here includes metal (geological)uncertainty and commodity price (market) uncertaintyrepresented by scenarios. Each scenario combines bothsources of uncertainty. Let S be the total number of scenariosthat account for the uncertainties of both mineral deposits andcommodity markets, and s {1,…,S} is the scenario index.The mining complex includes a number of mines, stockpiles,

and products (as shown in Figure 2). The detailed assumptionsfor each facility in a mining complex are described below.

A mining complex consists of I mines, each of which isindexed by i {1,…,I}. For each mine, the mining scheduleis predetermined and hence the tonnage and the grade ofthe extracted material are treated as exogenous. Because ofthe geological uncertainty, the tonnage and the grade of oreextracted in each period are stochastic. Because the mineproduction schedule is assumed to be predetermined in thiswork, the tonnage and the grade of the ore extracted in eachperiod can be simulated. The tonnage and the grade of oreextracted in each period are denoted by and

A two-stage stochastic mixed integer non-linear program is formulated for a mining complex

mineral supply chain in the context with uncertainties in both ore supply and the commoditymarket (price and demand). The endogenous spot price in the commodity market and long-termsales contracts are considered in the formulation of the mining complex’s optimisation model andan ad hoc heuristic is developed to deal with the throughput – and head-grade-dependent recoveryrate in processing plants. Numerical results indicate that the proposed heuristic is effective and

design strategy is proposed for making decisions on the contract price and strategic investments.A shadow price based method is also proposed to evaluate the existing mining schedule.

, respectively, where the superscript M indicatesthat the symbol is associated with a ‘mine’. After the materialis extracted from a mine, it can be sent to either a processingplant or a stockpile.

The mining complex has J stockpiles each of which is indexedby j {1,…,J}. A stockpile only accepts the material of aparticular type which is determined by certain parameterssuch as grade, hardness, composition, etc. A waste pile canalso be treated as a special stockpile if the waste is treatedspecial ore of which the grade is below the cutoff grade forwaste. For clarity, a binary parameter, denoted by ijts {0,1}is used to indicate if stockpile j accepts material extracted frommine i in scenario s and period t. Because a material stockpileis not homogeneous and it is usually the case that only thesurface is accessible for grade test (Holmes, 2004), the gradeof the material from a stockpile can only be tested when it isbeing moved. Thus, the grade of the material from a stockpileis treated as exogenous and stochastic. Let g be thegrade of material from stockpile j, where the superscript Hindicates that the symbol is associated with a ‘stockpile’.The tonnage of material stocked in stockpile j is denoted by

i to stockpile j is denotedby , and the cost of transporting a unit tonnage

material is denoted by *equation*. Because canalso capture other costs incurred at stockpile j, we do not usean additional parameter to denote the rehandling cost.

The mining complex has K processing plants each of whichis indexed by k {1,…,K}. For any plant k, the head gradeof the feeding material is denoted by , wherethe superscript P indicates that the symbol is associatedwith a ‘processing plant’. The head grade should satisfy

to meet the requirements of the processingmethod employed by plant k. The unit processing cost forplant k is denoted by . The throughput in plantk, denoted by , is constrained by plant k’sprocessing capacity . The recovery rate in a processing plantis dynamic depending on the throughput and the head grade.Plant k’s recovery rate function is denoted by ,which is decreasing in and increasing in based on theobservations by Hadler, Smith and Cilliers (2010) and Splaine,

ito plant k is denoted by , and

from stockpile j to plant k is denoted by , and is the unit transportation cost.

The mining complex sells its commodity to contracted buyersat contract prices or to ‘uncontracted’ buyers in the spot marketat the spot pricethe mining complex’s commodity supply. Let be plant k’scommodity supply to the spot market, and cU the expectedtransaction cost of selling a unit tonnage of commodity tothe spot market, where the superscript U indicates that thesymbol is associated with the ‘uncontracted’ spot market. Thespot price is obtained as , where and

is the spotprice’s sensitivity to the mining complex’s commodity supplyand depends on the mining complex’s market share. In theliterature, similar assumptions can be found in the articlesreviewed by Yano and Gilbert (2004). The mining complexhas L contracted buyers, each of which is indexed by l {1,…,L}. The contract price and demand for buyer l are denotedby and , respectively, where thesuperscript C indicates that the symbol is associated with a‘contract’ buyer. If buyer lby the mining complex, a penalty of is incurred for

is denoted by for buyer lfrom plant k to buyer l is denoted by , and theunit transportation cost is denoted by .

Without loss of generality, the mining complex’s

internal and outbound subsystems. The internal subsystemtransports material between mines, stockpiles and processingplants, and the outbound subsystem transport commodityfrom processing plants to buyers. The internal and outboundtransportation capacities are determined by the numberof trucks and denotethe numbers of existing internal and outbound trucks,respectively. is the transportation capacity ofeach truck. The mining complex can expand the capacity of itsinternal and/or outbound transportation system by adding

and/or trucks at a cost ofR for each.

in all planning periods. The rate of return for each planningperiod is denoted bysection is listed in Table 1.

In this section, the problem is formulated as a two-stagestochastic mixed integer non-linear program (SMINLP) and aconstructive heuristic is developed following the model.

The mining complex›s objective function is to maximise its

a

g g

,

(1)

the spot market and the contracted customers, (ii) computesthe total transportation cost, (iii) computes the total processingcost incurred in processing plants, (iv) computes the total

computes the total strategic investment considered in thiswork. As noted earlier, any scenario s contains a combinationof simulated parameters that account for the uncertainties in

The constraints of the model are formulated as follows.

1 1(2)

Equation 2 constrains that at any mine j, the total amount

j’s yield.

(3)

11 1

(4)

Equation 3, where M is a large constant, determines whetherthe material extracted from mine i is acceptable to stockpile j.Equation 4 computes the stock level in stockpile j in eachperiod.

1 1(5)

1 1 (6)

11(7)

11

1

(8)

(9)

Equations 5 and 6 compute the throughput and the headgrade of plant k, respectively. Equation 7 constrains thematerial fed to plant k to meet plant k’s requirements on headgrade. Equation 8 computes the plant k’s commodity outputbased on its material input and recovery rate function, andthe commodity is sent to contracted customers and the spotmarket. Equation 9 constrains plant k’s throughput to bewithin its processing capacity.

111111 (10)

11(11)

Equations 10 and 11 constrain the transportation to bewithin an expandable capacity. Because the transport distanceis usually in proportion to the unity transportation cost, herethe unit transportation cost is used as the transport distancewithout loss of generality.

(12)

contracted buyer in any period t and scenario s.

From the model above, it can be observed that because therecovery rate function in a processing plant, in Equation 8,depends on throughput and head grade, the mining complexhas to solve a non-convex non-linear program, which is

developed to solve the proposed program.Because our heuristic is iteration-based, for clarity, we use

a hat mark ( ) to label a constant parameter equal to optimalsolution of the corresponding variable obtained in theprevious iteration. The main idea for modifying the program

recovery rate function. Thus, in each iteration, constraints 6, 8and 9 are substituted with:

(13)

(14)

11(15)

(16)

(17)

can be obtainedas:

And withrespect to v.

The throughput in the recovery function is set to 0 andin 13 and 14, respectively, for outer linearisation. Additionalterms, 1 and 2 *, are introduced in the left-hand

grade.Because for any k, t and s, the optimal setting of in the

recovery function always makes Equation 17 bind, in ourheuristic, for all k, t and s are gradually reduced from its

maximum value, , until the slack of constraint 17 is smallenough for any k, t and s.

The proposed heuristic is tested through a series of numericalexperiments. The parameters of the mining complex’soptimisation problem are simulated as in Table 2. To test theaccuracy of the proposed heuristic, the proposed heuristic is

of solving general non-linear program, through a small-scale problem with a planning horizon of two periods. ForLingo global solver, the limit of computation time is set totwo hours. The objective value obtained by Lingo globalsolver and the proposed heuristic are displayed in Figure 3.It can be observed that within reasonable computation time,the solutions found by Lingo global solver are worse thanthe ones found by the proposed heuristic, and the proposedheuristic requires much less computation time to solve thetest problems at different scales

problem with 24 periods and the 100 scenarios. Because the

this scale, the heuristic is programmed using CPLEX OPLscript. It takes 1498.74 seconds for the proposed heuristic to

g

solve the problem. Figures 4a and 4b shows the averages ofand for all k, t and s, and Figure 4c shows

the change of the obtained objective value as the heuristicprogresses.

quantity of the commodity produced by the mining complex.For clarity, we use ‘principal buyer’ (l = 1) to refer to the buyerthat is currently signing a long-term sales contract with themining complex. In the contract negotiating phase, a commonassumption is made that the principle buyer’s preference

a practice that the contract demand increases with quantitydiscount. A good review of the operations literature onquantity discounts can be found in Viswanthan and Wang(2003). Without loss of generality, the relationship betweenthe price quote and the principal customer’s contracted

where:

1 and 1t represents the principal buyer’s willingness tobuy and demand-price sensitivity, respectively

estimated using the proposed model and heuristic if thecontract is signed at any contract price 1 . The worst-case

from the scenarios considered in the optimisation model, isalso evaluated as a consideration factor to decide if a contractoffer should be accepted. The worst-case scenario usuallyincludes parameters with low yield of ore tonnage and grade,and low market price and demand. We demonstrate theimplementation of the proposed contract design strategy bya hypothetical case with the identical settings as in the Modeland Heuristic section. Without loss of generality, the principal

buyer’s parameters are set to and for allt {1,…,24}. Figure 5 shows the mining complex’s expected

is equivalent to the case without the present contract. It can beobserved that when the contract price is over 19.5, the contract

be signed at 1 = 21. When the contract price is lower than20 but higher than 18.5, the risk-averse mining complex stillhas an incentive for signing the contract because the worst-

obtained without the present contract.

Figure 6 shows the mining complex’s optimal strategicinvestments, which are the expansions of internal andoutbound transportation capacities, at different contractprices, and the dashed lines are the mining complex strategicinvestment without the present contract. It can be observed

A B

C

that the requirement for internal transport capacity is not

because the mining complex reduces the sales quantity to spotmarket to honour contracted demand. When the contracteddemand increases to a certain level, the production capacityin the processing plant becomes the bottleneck of thesupply chain so that there is no need to continue to increasethe outbound transportation capacity and the penalty forundelivered demand is increased to reduce the mining

For a mining supply chain, the uncertainties in mineraldeposits (material types, ore grade and tonnage) and thecommodity market (price and demand) should be considered

entire supply chain. In this work, a stochastic non-linearmixed integer programming model is proposed to help amining complex to make decisions in signing a long-termsales contract to deal with the uncertainties existing in bothends of a mineral supply chain. Due to the complexity insolving the proposed non-convex and non-linear model, aheuristic is developed and tested by a number of numerical

As a suggestion for the future research, methodology inmining scheduling based on ore shadow prices (or dual price)will be studied, and the shadow-price-based mine production

the uncertainty incurred in mineral resource supply chainregardless of the complex structure of the supply chain.

Hadler, K, Smith, C and Cilliers, J, 2010. Recovery vs. mass pull: the

link to air recovery, Minerals Engineering, 23(11):994–1002.

Holmes, R J, 2004. Correct sampling and measurement the foundationof accurate metallurgical accounting, Chemometrics and IntelligentLaboratory Systems, 74(1):71–83.

Splaine, M, Browner, S and Dohm, C, 1982. The effect of head gradeJournal of The

South African Institute of Mining and Metallurgy, 82(1):6–11.

Viswanthan, S and Wang, Q, 2003. Discount pricing decisions indistribution channels with price sensitive demand, EuropeanJournal of Operational Research, 149:571–587.

Yano, C A and Gilbert, S M, 2004. Coordinated pricing andproduction/procurement decisions: a review, in ManagingBusiness Interfaces, pp 65–103 (Springer).