Heuristic Strategies for Solving Complex Interacting Stockpile Blending Problem with Chance Constraints
HHeuristic Strategies for Solving ComplexInteracting Stockpile Blending Problem withChance Constraints
Yue Xie, Aneta Neumann and Frank NeumannOptimisation and Logistics, School of Computer Science,The University of Adelaide, Adelaide, AustraliaFebruary 11, 2021
Abstract
Heuristic algorithms have shown a good ability to solve a variety of optimiza-tion problems. Stockpile blending problem as an important component of the minescheduling problem is an optimization problem with continuous search space con-taining uncertainty in the geologic input data. The objective of the optimizationprocess is to maximize the total volume of materials of the operation and subjectto resource capacities, chemical processes, and customer requirements. In thispaper, we consider the uncertainty in material grades and introduce chance con-straints that are used to ensure the constraints with high confidence. To addressthe stockpile blending problem with chance constraints, we propose a differentialevolution algorithm combining two repair operators that are used to tackle the twocomplex constraints. In the experiment section, we compare the performance ofthe approach with the deterministic model and stochastic models by consideringdifferent chance constraints and evaluate the effectiveness of different chance con-straints.
Mining is the extraction of economically valuable minerals or materials from the earth.This has raised the importance of the production scheduling process due to its signifi-cant role in the profitability and efficiency of any mining operation. Mine productionscheduling problem [14] is a well-study mining engineering problem, and it has re-ceived much attention in past decades [16] from both engineering and research. Thetask of the mine production schedule is to generate a mining sequence and ensure theproduct meets the blending resource constraints and object to maximize the net presentvalue of the mining operation.The mine production scheduling problem is commonly formulated as a Mixed-Integer Program (MIP) with binary variables [6, 3, 23]. However, since it becomes a1 a r X i v : . [ c s . N E ] F e b hallenge for the MIP when the problem deals with the blending resource constraints.Lipovetzky et al. [8] introduced a combined MIP for a mine planning problem, whichdevises a heuristic objective function in the MIP and can improve the resulting searchspace for the planner. Samavati et al. [21] proposed a heuristic approach that com-bines local branching with a new adaptive branching scheme to tackle the productionscheduling problem in open-pit mining.Stockpiles are essential components in the supply chain of the mining industry,and play a significant role in the mine scheduling problem. Jupp et al. [7] introducedthe four different reasons for stockpiling before material processing: buffering, blend-ing, storing, and grade separation. In open-pit mine production scheduling problem,stockpiles are used for blending different grades of material from the mine or keep-ing low-grade ore for possible future processing [14, 20]. Rezakhah et al. [19] useda linear-integer model to approximate the open-pit mine production scheduling withstockpiling problem which forces the stockpile to have an average grade above a spe-cific limit. Recently, some researchers present nonlinear-integer models to solve openpit mine production scheduling with stockpiles. Tabesh et al. [22] proposed a nonlin-ear model of stockpiles to optimize a comprehensive open-pit mine plan but not giveany results. Bley et al. [3] proposed a nonlinear model for mine production planning,however, they only consider one stockpile.In this paper, we study an important component of the mine production schedul-ing problem, the stockpile blending problem. This problem is challenging to addressin terms of blending material from stockpiles for parcels to match the demands ofdownstream customers. We define the stockpile blending problem as an optimizationproblem that aims to maximize the volume of valuable material from all parcels by find-ing the percentage that each stockpile provides for each parcel in the whole planning.Furthermore, the strategy has to respond to the mining schedule and the market planwhere the mine schedule provides the material mining and sending to correspondingstockpiles in each period, and the market plan provides the customer requirements.Solving the stockpile blending problem in mining optimally is critical because itis based on an uncertain supply of mineralized materials for the resource availablein the mine. This uncertainty is acknowledged in the related technical literature tobe the major reason for not meeting production expectations [2, 1]. Given its sub-stantial impact on the financial outcome of mining operations, this paper focuses ondealing with the uncertainty in metal content within a mineral deposit being mined.For the stochastic variables of the stockpile blending problem, we introduce chance-constrained programming here to tackle the uncertainty of material grades. Chance-constrained optimization problems [4, 13] whose resulting decision ensures the prob-ability of complying with the constraints and the confidence level of being feasibleto have received significant attention in the literature. Chance-constraint programminghas been widely applied in different disciplines for optimization under uncertainty [24].For example, chance-constraint programming has been applied in analog integrated cir-cuit design [11], mechanical engineering [12], and other disciplines [9, 18]. However,so far, chance-constraint programming has received little attention in the evolutionarycomputation literature [10].It is difficult for MIP to tackle such a continuous optimization problem containingthe nonlinear constraints. To address this challenge, this paper proposes two repair2perators to tackles the complex constraints. Follow the paper [25], we present thesurrogate functions of the chance constraints by using Chebyshev’s inequality. Fur-thermore, a well-known evolutionary algorithm, the Differential Evolution (DE) algo-rithm is introduced to solve the stockpile blending problem. Recently, evolutionaryalgorithms have received much attention in solving large-scale optimization problemsand multi-dimensions problem. The DE algorithm is a simple and effective evolu-tionary algorithm used to solve global optimization problems in a continuous domain[15, 17]. The DE and its variants have been successfully applied to solve numerousreal-world problems from diverse domains of science and engineering [5, 15]. Thispaper investigates the use of the DE algorithm combining the two repair operators forsolving the problem. Then we compare the impactas of different chance constraints onthe objective value.The rest of the paper is organized as follows. In the next section, we present themodel of the stockpile blending problem and a decision variable normalized operatorfor the continuous decision variables as well as a duration repair operator. After that,the chance constraints model and the surrogate functions of the chance constraintsare presented in Section 3. Following, we describe the approach we used to solvethe problem and the fitness function of the algorithm. We set up experiments andinvestigate the performance of the different fitness functions in Section 5. We concludewith Section 6. In this section, we present nonlinear formulations of the stockpiles blending problemwith a deterministic setting. In reality, some processes such as the chemical processin the concentrate production progress are highly complex to model because it is in-fluenced by many factors, some of which include the mineralogy of the ore, particlesize of milled material, temperature, and chemical reactants available in the process.The information of these variables was not available to us, therefore within this study torecovery factors of all materials from the chemical processing stage and the copper per-centage within the produced copper concentrate is assumed to be constant throughoutthe stockpiles blending and production schedule.We first introduces notation as follow, and then provide the math. We use the term”material” to include ore, i.e., rock that contains sufficient minerals including metalsthat can be economically extracted and to include waste, and we use chemical symbolrepresent the corresponding material, i.e., Cu denotes Copper, Fl denotes Flerovium.
Obj : max (cid:88) p ∈P c p = max (cid:88) p ∈P (cid:0) w p g Cup r Cup (cid:1) (1)3 ndices and sets: s ∈ S stockpiles; , . . . , Sp ∈ P parcels; , . . . , Po material; { Cu, Ag, F e, Au, U, F l, S } m ∈ M month; , . . . , m Decision variables: x ps fraction of parcel p claimed from stockpile st p produce time (duration) for parcel pw p : tonnage of parcel pθ ps : tonnage stores in stockpile s after providing material to parcel pc p : Cu tonne in parcel pg op : grade of material o in parcel p ˜ g ops : grade of material o in stockpile s when proving parcel pk p : tonne concentrate of parcel pr Cup : Cu recovery of parcel pr Flp : Fl recovery of parcel p Parameters: T mp : binary parameter, if T mp = 1 , parcel p is the first parcel need to prepare in month m , if T mp = 0 otherwise δ : discount factor for time period ˜ φ : factor in chemical processing stage φ Au : factor of Au in chemical processing stage φ U : factor of U in chemical processing stage φ Fe : factor of Fe in chemical processing stage φ Cu : factor of Cu in chemical processing stage ( γ , γ ) : factor of Cu percentage within the produced Cu concentrate µ Fl : factor of Fl recovery µ U : factor of U recovery ( µ Cu , µ Cu ) : factor of Cu recovery D m : duration of month mH ms : tonnage of material hauled to stockpile s in month mG oms : grade of material o that shipping to the stockpile s in month mK p : expected tonne concentrate of parcel pR Flp : upper threshold of Fl recovery of parcel pCu p : lower threshold of Cu grade of parcel pN m : number of planning parcels in month m s.t. (cid:88) (cid:80) m − m =1 N m +1 ≤ p ≤ (cid:80) mm =1 N m t p ≤ D m (2) (cid:88) s ∈S x ps = 1 ∀ p ∈ P (3) r Cup = µ Cu g Cup g Sp + µ Cu ∀ p ∈ P (4) g op = (cid:88) s ∈S x ps ˜ g ops ∀ p ∈ P (5) w p = δt p [ ˜ φ + ( φ Au log g Aup ) + ( φ U log g Up ) − ( φ F e log g F ep ) + ( φ Cu log g Cup )] (6)4 p = c p γ g Cup g Sp + γ (7) ˜ g ops = (cid:40) ˜ g o ( p − s · θ ( p − s + G oms · H ms θ ( p − s + H ms if T mp = 1˜ g o ( p − s otherwise (8) θ ps = (cid:40) θ ( p − s + H ms − x ps · w p if T mp = 1 θ ( p − s − x ps · w p otherwise (9) g Cup ≥ Cu p ∀ p ∈ P (10) ( K p − ≤ k p ≤ ( K p + 1) ∀ p ∈ P (11) µ F l g F lp ≤ R F lp ∀ p ∈ P (12)The objective function (1) aims to maximize the sum of Cu volume of all parcels,which is obtained by the tonnage of parcels multiply the Cu grade, and multiple theCu recovery. Constraint (2) forces the sum of duration of the parcels that planned intothe same month less than the available duration of this month. Constraint (3) ensuresthat the sum of the decision variables for the same parcel is equal to 1. Function (4)denotes the simplified calculation of Cu recovery of parcels, and function (5) calculatesthe material grades of parcels. Function (6) express the simplified calculation of parceltonne which is a component in objection function. Function (7) shows the simplifyversion of how to calculate the tonne concentrate of parcels.Constraint (8) enforces material grade balance for stockpiles when providing ma-terial to parcels. Constraint (9) enforces inventory balance when providing material toparcels. Constraint (10) forces the Cu grade of parcels to less than or equal to the givenlower bound of the Cu grade. Constraint (11) forces the value of tonne concentrate ofeach parcel is no more or less than the expected tonne concentrate by one. Constraint(12) ensures the Fl recovery of each parcel less than the bound given in advance.The stockpile blending problem is a non-linear optimization problem in the con-tinuous search space. To tackle the constraint (3), we introduce a decision variablesnormalized approach (cf. Algorithm 1) to force solutions match the constraint. Thisapproach first calculates the sum of the variables of each parcel separately, then eachvariable of the same parcel is divided by the corresponding sum. It shows the signif-icant importance of applying this approach to a generated decision vector to meet theconstraint. 5 lgorithm 1: Decision variables normalized approach
Input:
Decision vector X j = { x j , x j .., x Ij } a = (cid:80) i ∈ I x ij ; for i = 1 to I do x ij = x ij a ; return the normalized decision variables. Algorithm 2:
Duration repair operator
Input: X ∈ (0 , I · J , i ∈ { , .., I } , j ∈ { , .., J } ; parameter ζ ; availableduration D Output: parcel duration: d ∈ { , D } initialization: d = 0 , d = D , d ∈ { , D } , k = ζ · d while d ∈ { , D } and k / ∈ { K − , K + 1 } doif k > K + 1 then d := ( d + d ) / ; k := ζ · d ; if k > K + 1 then d = d ; else d = d ; else if k < K − then d := ( d + d ) / ; k := ζ · d ; if k > K + 1 then d := d ; else d := d return the duration corresponding to solution X Due to the complex constraint (11) is too tight to construct feasible solutions, wedevelop a repair operator to address this problem. As shown in function (7), the valueof tonne concentrate of parcels is related to the duration of this parcel and the mate-rial grades of the parcel. Meanwhile, referring to equation (5), material grades of theparcel are directly calculated by decision variables. Therefore, with the fixed decisionvariables of a parcel, the real tonne concentrate of this parcel is affected by the durationof this parcel. We present a duration repair operator (cf. Algorithm 2) which uses abinary search process to convert an infeasible solution into a solution without violatingconstraint (11).Since the time complexity of the binary search is log n where n denotes the lengthof the search space in the beginning. In our problem, the duration of each parcel can notexceed the total available duration of the month. The run-time of the duration repairoperator for one parcel is log d in the worst case where d denotes the total available6uration of the current month. In real-world mining engineering problems, the material grades are estimated by sometools, and in the research of mining scheduling problems, researchers treat the stochas-tic material grades as constant by using the expected values. In this paper, we are thefirst to discuss the influences of stochastic material grades on the objective value ofthe stockpile blending problem with chance constraints. Due to the complexity of theproblem, we reformulated the constraints (10) and (12) to chance-constrained. Chance-constrained programming is a competitive tool for solving optimization problems underuncertainty. The main feature is that the resulting decision ensures the probability ofcomplying with constraints, i.e. the confidence of being feasible. Thus, using chance-constrained programming the relationship between profitability and reliability can bequantified.
First, we define additional notation as follow. α Cu : confidence of Cu grade chance constraint α F l : confidence of Fl recovery chance constraintThen, the new chance constraints are
P r { g Cup ≥ Cu P } ≥ α Cu , (13)and P r { µ F l g F lp ≤ R F lp } ≥ α F l . (14)Constraints (13), (14) force the confidence of ensuring the constraint are greater thanor equal to the corresponding given bound.We use Chebyshev’s inequality to construct the available surrogate that translatesto a guarantee on the feasibility of the chance constraint imposed by the inequalities.Firstly, we use Chebyshev’s inequality to reformulate the chance constraints. The in-equality has utility for being applied to any probability distribution with known expec-tation and variance. Therefore, we assume the stochastic material grades discussed inthis paper are all estimated with given expected values and corresponding variances.Note that Chebyshev’s inequality automatically yields a two-sided tail bound, there isa one-sided version of Chebyshev’s inequality named Cantelli’s inequality. Theorem 1 (Cantelli’s inequality) . Let X be a random variable with V ar [ X ] > .Then for all λ > , P r { X ≥ E [ X ] + λ (cid:112) V ar [ X ] } ≤
11 + λ . (15)7 r { X ≤ E [ X ] − λ (cid:112) V ar [ X ] } ≤
11 + λ . (16)We assume the material grades in the stockpiles are independent of each other,each grade corresponding expectation a oms and variance σ oms . Therefore, the expectedmaterial grades of stockpiles can be denoted as E (˜ g ops ) = (cid:40) ˜ g o ( p − s · θ ( p − s + a oms · H ms θ ( p − s + H ms if T mp = 1 E (˜ g o ( p − s ) otherwise. Furthermore, the variance of the material grades are
V ar (˜ g ops ) = (cid:16) θ ( p − s θ ( p − s + H ms (cid:17) V ar (˜ g o ( p − s ) + (cid:16) H ms θ ( p − s + H ms (cid:17) σ oms if T mp = 1 V ar (˜ g o ( p − s ) otherwise. Let g Cup = (cid:80) s ∈S x ps ˜ g Cups be the Cu grade of parcel p of a given solution X = { x p , .., x ps , .., x p S } , and E [ g Cup ] = (cid:88) s ∈S x ps E (˜ g Cups ) denotes the expected Cu grade of parcel p of the solution derived by linearity of expec-tation, V ar [ g Cup ] = (cid:88) s ∈S ( x ps ) V ar (˜ g Cups ) denotes the variance of Cu grade of parcel p . To match the expression of the Cantelli’sinequality (16), we set Cu p = E [ g Cup ] − λ (cid:113) V ar [ g Cup ] and have λ = E [ g Cup ] − Cu p (cid:113) V ar [ g Cup ] for each parcel, then we have a formulation to calculate the upper bound of the chanceconstraint (13) as follows. P r { g Cup ≤ Cu p } ≤ V ar [ g Cup ] V ar [ g Cup ] + ( E [ g Cup ] − Cu p ) ≤ (1 − α Cu ) (17)Furthermore, let r F lp = µ F l (cid:80) s ∈S x ps ˜ g F lps be the FL recovery of parcel p . Let E [ r F lp ] = µ F l (cid:88) s ∈S x ps E (˜ g F lps ) V ar [ r F lp ] = (cid:88) s ∈S ( µ F l x ps ) V ar (˜ g F lps ) , is the variance of FL recovery of parcel p with solution X = { x p , .., x ps , .., x p S } . To match the expression of the Cantelli’s inequality (15), we set R F lp = µ F l E [ r F lp ] + λ (cid:113) V ar [ r F lp ] and have λ = R F lp − µ F l E [ g F lp ] (cid:113) V ar [ g F lp ] for each parcel, then we have a formulation to calculate the upper bound of the chanceconstraint (14) as follows. P r { µ F l g F lp ≥ R F lp } ≤
V ar [ g F lp ] V ar [ g F lp ]+( R F lp − µ F l E [ g F lp ]) ≤ (1 − α F l ) (18)Now, we obtain the surrogate functions of the chance constraints. In the next sec-tion, we present the approach for solving the stockpile blending problem with chanceconstraints. In this section, we present the fitness functions for the differential evolution (DE) algo-rithm which has been proved successfully used in solving the optimization problem incontinuous space.
We start by designing a fitness function for the deterministic setting model that can beused in the DE algorithm. The fitness function f for the approach needs to take allconstraints into account. The fitness function of a solution X is defined as follows. f ( X ) = ( u ( X ) , v ( X ) , w ( X ) , q ( X ) , g ( X ) , O ( X )) (19)9 ( X ) = (cid:88) p ∈P max {| K p − k p | , } v ( X ) = max { (cid:88) p ∈P t p − D m , } w ( X ) = min { (cid:88) p ∈P (cid:88) s ∈S θ ps , } q ( X ) = (cid:88) p ∈P max { Cu p − g Cup , } g ( X ) = (cid:88) p ∈P max { r F lp − R F lp , } O ( X ) = (cid:88) p ∈P c p . In this fitness function, the components u, v, q, g need to be minimize while w and O maximized, and we optimize f in lexicographic order. For the stockpile blendingproblem, any infeasible solution can at least violate one of the above constraints. Then,among solutions that meet all constraints, we aim to maximize the objective function.Formally, we have f ( X ) (cid:23) f ( Y ) iff u ( X ) < u ( Y ) or u ( X ) = u ( Y ) ∧ v ( X ) < v ( Y ) or { u, v } are equal ∧ w ( X ) > w ( Y ) or { u, v, w } are equal ∧ q ( X ) < q ( Y ) or { u, v, w, q } are equal ∧ g ( X ) < g ( Y ) or { u, v, w, q, g } are equal ∧ O ( X ) > O ( Y ) , When comparing two solutions, the feasible solution is preferred in a comparison be-tween an infeasible and a feasible solution. Between two infeasible solutions that vi-olated the same constraint, the one with a lower degree of constraint violation is pre-ferred.
Now, we design the fitness function for the stockpile blending problem with chanceconstraints. In this paper, we investigate the effectiveness of chance constraints on theobjective value. We first reformulate the components q and g of the function (19) with10 lgorithm 3: Differential evolution algorithm t ← , initialize P t = { X t , .., X tNP } randomly ; while stopping criterion not met dofor i ∈ { , .., N P } do R ← A set of randomly selected indices from { , .., N P } \ { i } ; V ti ← mutation ( P t , R, F ) ; j rand ← A randomly selected number from { , .., n } ; U ti ← crossover ( X ti , V ti , C, j rand ); for i ∈ { , .., N P } doif f ( U ti ) (cid:23) f ( X ti ) then X t +1 i ← U ti ; else X t +1 i ← X ti ; t ← t + 1 ;chance constraints (17 and 18) as follow, q (cid:48) ( X ) = (cid:88) p ∈P max (cid:8) P r { g Cup ≤ Cu p } − (1 − α Cu ) , (cid:9) (20) g (cid:48) ( X ) = (cid:88) p ∈P max (cid:8) P r { µ F l g F lp ≥ R F lp } − (1 − α F l ) , (cid:9) (21)where q (cid:48) and g (cid:48) need to be minimized.To distinguish the influence of each chance constraint, we design three fitness func-tions where the two functions consider the chance constraints separately, and the otherone uses the combination of components. f (cid:48) ( X ) = ( u ( X ) , v ( X ) , w ( X ) , q (cid:48) ( X ) , g ( X ) , O ( X )) (22) f (cid:48)(cid:48) ( X ) = ( u ( X ) , v ( X ) , w ( X ) , q ( X ) , g (cid:48) ( X ) , O ( X )) (23) f (cid:48)(cid:48)(cid:48) ( X ) = ( u ( X ) , v ( X ) , w ( X ) , q (cid:48) ( X ) , g (cid:48) ( X ) , O ( X )) (24) Algorithm (3) shows the overall procedure of the basic DE algorithm. DE is usuallyinitialized by generating a population of
N P individuals. For each i ∈ { , .., N P } , X ti is the i -th individual in the population P t . Each individual represents a d -dimensionalsolution of a problem. For each j ∈ { , .., d } , X tij is the j -th element of X ti .After the initialization of P t , the following steps are repeatedly performed until atermination condition is satisfied. For each X ti , the scale factor F > which controlsthe magnitude of the mutation, and the crossover rate Cr ∈ [0 , which controls thenumber of elements inherited from X ti to a trail vector U ti are constants and given inadvance.A set of parent indices R = { r , r } are randomly selected from { , .., n } \ { i } such that they differ from each other. For each X ti , a mutant vector V ti is generated by11pplying a mutation to X tr , X tr . There are many mutation strategies that have beenproposed in the literature [5]. Here, we use the DE/target − to − best/ strategyshown as follows, which is one of the most efficient strategies. V ti = X ti + F ( X tbest − X ti ) + F ( X tr − X tr ) , (25)where X tbest denotes the best individual in the current population.After the mutant vector V ti has been generated for each X ti , a trail vector U ti isgenerated by applying crossover to X ti and V ti . The scheme of the crossover can beoutlined as U tij = (cid:40) V tij if ( rand i,j [0 , ≤ Cr or j = j rand ) X tij otherwise (26)where rand i,j [0 , is a uniformly distributed random number, which is called a new foreach j -th element of the i -th parameter vector. j rand ∈ { , .., n } is a randomly chosenindex, which ensures that U ti gets at least one element from V ti . It is instantiated oncefor each vector per generation.After the trial vector, U ti has been generated for each parent individual, the nextstep called selection which determines whether the target or the trailing vector survivesto the next generation. The selection operation is described as X t +1 i = (cid:40) U ti if f ( U ti ) (cid:23) f ( X ti ) X tij otherwise (27)according to the fitness function. In this section, we examine the solution quality associated with different fitness func-tions. Due to business security, we are not able to investigate the proposed approach inreal-data instances. Therefore, we first design the benchmark of the stockpile blendingproblem. Afterward, we compare the results obtained by using different fitness func-tions of the instances. Furthermore, considering the complexity of the problem withchance constraints, the instances we discussed in this section only contain one monthschedule.
Table 1 lists the intervals of input parameters mentioned in Section 2.1. The threeinstances we evaluated in this paper are created by randomly generated value of pa-rameters from their intervals (see Table 1), we attach the parameters of these instancesin the appendix. The randomly generated numbers are the expected values of materialgrades, and the deviation of material grades are set equal to . multiply the expec-tation. Let α Cu = { . , . , . } and α F l = { . , . , . } . Base on this12able 1: General information about the ore and processing parameters Description Values or Value rangeNumber of parcel stockpiles; { , , } Number of Stockpile Duration of month , , Discount factor for time period ( δ ) . Factor in chemical processing stage ( ˜ φ ) [1000 , Factor of Au in chemical processing stage ( φ Au ) [200 , Factor of U in chemical processing stage ( φ U ) [300 , Factor of Fe in chemical processing stage ( φ Fe ) [560000 , Factor of Cu in chemical processing stage ( φ Cu ) [6000000 , Factor of Cu percentage within the produced Cu concentrate ( γ , γ ) ([5 , , [30 , Factor of Fl recovery ( µ Fl ) [0 . , . Factor of U recovery ( µ U ) [0 . , . Factor of Cu recovery ( µ Cu , µ Cu ) ([1 . , . , [0 , Tonnage of material hauled to stockpile [5000 , Cu grade [0 . , . Ag grade [1 . , . Fe grade [10 . , . Au grade [0 . , . U grade [30 . , . Fl grade [1200 , S grade [0 . , . Expected tonne concentrate of parcel ( K p ) [10000 , ∞ ] Threshold of Fl recovery of parcel ( R Flp ) [1300 , Threshold of Cu grade of parcel ( Cu p ) [0 . , . arrangement, we compare the performance of the DE algorithm with fitness functions(Eq. 19, 22, 23, 24) on the stockpile blending problem.We then investigate the performance of the DE algorithms with different fitnessfunctions described in Section 4 and provide the results from independent runs with generation and population for all instances. For a closer look, we report theaverage, best and worst solutions obtained by the algorithm in corresponding columns.We also evaluate the algorithm by success rate which is the percentage of success forthe algorithm in obtaining valid solutions out of runs. We benchmark our approach with the combinations from the experimental setting de-scribed above. All experiments were performed using Java of version 11.0.1 and car-ried out on a MacBook with a 2.3GHz Intel Core i5 CPU. (a) Instance 1 (b) Instance 2 (c) Instance 3
Figure 1: Bar graph for DE algorithm with single chance constraint13able 2: Fitness values obtained with single chance constraint
Deterministic Cu Chance constraint ( α Cu ) Fl Chance constraint ( α Fl ) Instance 0.999 0.99 0.9 0.999 0.99 0.91 Mean 103603035.94 99319128.52 102724900.09 103206748.35 103117715.98 103340755.20 102753876.59Best 110830487.20 100434268.90 110489368.20 111221777.51 108460913.10 110593860.12 106976825.01Worst 100025173.10 98404158.16 99426947.62 99935646.96 99979453.63 98951340.30 99444063.80Success rate 0.166666667 1 1 1 1 12 Mean 66339866.34 64280794.53 65346088.50 65440690.91 65741114.11 66128957.43 64999603.10Best 69691652.87 66062865.43 70504938.14 69307609.02 70846603.30 69265095.80 67416065.20Worst 63401302.85 62822049.78 61409848.38 62043167.43 62885220.60 63139531.31 62369471.32Success rate 0.3 1 1 1 1 13 Mean 25706739.82 25172345.85 25484058.55 25667396.95 25414602.50 25487090.60 25737554.80Best 26714591.61 25780112.93 27501228.19 26652385.99 26542657.00 26440088.30 27420748.21Worst 25042412.92 24939884.30 24517912.31 24338065.15 24301347.20 24502516.90 24675079.20Success rate 0.166666667 0.733333333 0.8 0.83333333 0.7 1
Table 3: Fitness values obtained with two chance constraints
Instance Combine Chance constraints α Cu = 0 . α Cu = 0 . α Cu = 0 . α Fl Table 2 lists the results for the three instances with using fitness function (19,22)and (23) separately. Figure 1 shows the how the chance-constrained bound α Cu or α F l affects the quality of the solutions. The bars in the graphs are corresponding tothe solutions of instances combining with the confidence of chance constraint respec-tively, and the three bars in each group corresponding to the threshold of confidence { . , . , . } . Among others, we observe that results obtained by applying thefitness function (22) are significantly affected by the value of α Cu . The results showan increasing trend as the value of α Cu decrease. However, by observing the bars in Flchance constraint group, the value of α F l does not influence the result when using theFl chance constraint.As can be seen from Table 2, the success rate shows significantly difference be-tween using the fitness functions (22) and (23) for instance and . When the confi-dence of the chance constraint (13) is tight such as . , the DE algorithm can notgenerate a pure feasible population in the last generation. While the confidence of thechance constraint (14) does not influence the success rate of the algorithm. However,for instance , which has four parcels into consideration and is the most complex in-stance in our study, the DE algorithm fails to obtain a feasible population in the lastgeneration when the value of α F l is . .Table 3 lists the results obtained by considering two chance constraints together,14he fitness function (24). For each instance, we investigate different parameters set-ting together with the different requirement on the chance constraints determined by α Cu and α F l . The results list in the columns with the same α Cu shows that there isno significant difference between the solutions obtained by applying difference α F l .Moreover, with the same α F l , the object value increase while the α Cu decrease.Now, we compare the results obtained by using single chance constraint and com-bined chance constraints. Comparing the solutions list in the column Cu Chance con-straint and
Fl Chance constraint in Table 2 against that of the combined chance con-straint in the same value of α Cu and α F l respectively. We find that for the same in-stance, the results obtained by applying a single chance constraint are better than thecombined chance constraints which happened in most cases. One interesting findingis that the value of α F l does not show significant effects on the results in the experi-ments for results in Table 2 and 3. A possible explanation for this might be that theparameters of the instances are not reliable or match the real-world situation, whichcan indicate the malfunction of the constraint. This is an important issue for featureresearch that develops approaches to create a benchmark that more reliable or moreclose to the real-world situation for the stockpile blending problem.
In this paper, we consider the stockpile blending problem which is an important com-ponent in mine scheduling with the uncertainty in the geologic input data. We modeledthe stockpile blending problem as a nonlinear optimization problem and introduced thechance constraints to tackle the stochastic material grades. We show how to incorporatea well-known probability tail, Chebyshev’s inequality, into presenting the surrogatefunctions of the chance constraints. Furthermore, we designed the four fitness func-tions with considering different chance constraints. In our experiments, which havecovered a variety of instances according to the parameters, we have observed that theconfidence of the Cu chance constraint affects the results obtained by using the fitnessfunction considering the Cu chance constraint and the fitness function with combinedchance constraints. Due to the ineffectiveness of the confidence of the Fl chance con-straint, for further studies, it could be interesting to deeply investigate the relationshipbetween chance constraints. It would be also interesting to develop benchmarks for thestockpile blending problem with chance constraints as there is no available open accessdata-set.
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Appendix
Table 4,5 and 6 list the value of input parameters of instance 1, 2 and 3 separately.Table 4: Parameters of Instance 1
Number of parcels: 3Number of stockpiles: 7Total duration: 30( δ ) 0.98( ˜ φ ) 1100( φ Au ) 270( φ U ) 340( φ Fe ) 564000( φ Cu ) 6050000 ( γ , γ ) (7 , ( µ Fl ) 0.11( µ U ) 0.79 ( µ Cu , µ Cu ) (2 . , ( R Flp ) 500( Cu p ) 0.9Ore shipping Tonage of ore Cu grade Ag grade Fe grade Au grade U grade Fl grade S grade1 480000 1.08 1.33 14.08 1.56 32.73 1263 0.212 220000 1.86 3.81 25.9 0.47 70.69 2568 0.83 970000 1.79 3.41 28.97 0.5 127.83 4500 0.744 400000 0.96 2.49 25 0.49 400 7500 0.85 3550000 1.37 2.02 14.21 0.31 44 3000 0.56 1130500 0.93 2.13 23.76 1.25 26.73 1560 0.267 5377000 1.61 2.22 16.5 0.61 31 2780 0.15Customer requirements ( K p )Parcel 1 750000Parcel 2 600000Parcel 3 420000 Number of parcels: 3Number of stockpiles: 7Total duration: 28( δ ) 0.98( ˜ φ ) 1100( φ Au ) 270( φ U ) 340( φ Fe ) 564000( φ Cu ) 6050000 ( γ , γ ) (7 , ( µ Fl ) 0.11( µ U ) 0.79 ( µ Cu , µ Cu ) (2 . , ( R Flp ) 400( Cu p ) 1Ore shipping Tonage of ore Cu grade Ag grade Fe grade Au grade U grade Fl grade S grade1 480000 1.78 1.33 14.08 1.56 32.73 1263 0.212 220000 1.86 3.81 25.9 0.47 70.69 2568 0.83 970000 1.79 3.41 28.97 0.5 127.83 4500 0.744 400000 1.16 2.49 25 0.49 400 7500 0.85 3550000 0.77 2.02 14.21 0.31 44 3000 0.56 1130500 1.23 2.13 23.76 1.25 26.73 1560 0.267 53770 1.81 2.22 16.5 0.61 31 2780 0.15Customer requirements ( K p )Parcel 1 300000Parcel 2 460000Parcel 3 330000 Table 6: Parameters of Instance 3
Number of parcels: 4Number of stockpiles: 7Total duration: 31( δ ) 0.98( ˜ φ ) 1100( φ Au ) 270( φ U ) 340( φ Fe ) 564000( φ Cu ) 6050000 ( γ , γ ) (7 , ( µ Fl ) 0.11( µ U ) 0.79 ( µ Cu , µ Cu ) (2 . , ( R Flp ) 400( Cu p ) 1Ore shipping Tonage of ore Cu grade Ag grade Fe grade Au grade U grade Fl grade S grade1 5000000 1.58 1.33 14.08 1.56 32.73 1263 0.212 4200000 1.86 3.81 25.9 0.47 70.69 2568 0.83 9700000 1.79 3.41 28.97 0.5 127.83 4500 0.744 4000000 1.16 2.49 25 0.49 400 7500 0.85 3550000 1.37 2.02 14.21 0.31 44 3000 0.56 1130500 1.13 2.13 23.76 1.25 26.73 1560 0.267 5377000 1.91 2.22 16.5 0.61 31 2780 0.15Customer requirements ( K p )Parcel 1 137000Parcel 2 94000Parcel 3 92000Parcel 4 111000)Parcel 1 137000Parcel 2 94000Parcel 3 92000Parcel 4 111000