Multi-objective approaches for the open-pit mining operational planning problem

Transcription

1 Multi-objective approaches for the open-pit mining operational planning problem V.N. Coelho a, M.J.F. Souza a, I.M. Coelho b, F.G. Guimarães c, T. Lust d and R.C. Cruz a a Federal University of Ouro Preto, Dept. of Computer Science, Ouro Preto, MG b Fluminense Federal University, Institute of Computing, Niterói, RJ / Inria Lille Nord Europe c Federal University of Minas Gerais, Department of Electrical Engineering, Belo Horizonte, MG d LIP6-CNRS, UPMC. 4 Place Jussieu Cedex 05 Paris imcoelho@ic.uff.br November 29, / 33

2 Summary Introduction Heuristic Approach Proposed Algorithms Computational Experiments and Analysis Conclusions 2 / 33

3 Open-Pit Mining Operational Planning (OPMOP) Introduction The open-pit mining operational planning problem (OPMOP) deals with: Ore and waste rocks 3 / 33

4 Open-Pit Mining Operational Planning (OPMOP) Introduction The open-pit mining operational planning problem (OPMOP) deals with: Ore and waste rocks Shovels 3 / 33

5 Open-Pit Mining Operational Planning (OPMOP) Introduction The open-pit mining operational planning problem (OPMOP) deals with: Ore and waste rocks Shovels Trucks 3 / 33

6 Open-Pit Mining Operational Planning (OPMOP) Introduction The open-pit mining operational planning problem (OPMOP) deals with: Ore and waste rocks Shovels Trucks Optimize... 3 / 33

7 Open-Pit Mining Operational Planning (OPMOP) Introduction The open-pit mining operational planning problem (OPMOP) deals with: Ore and waste rocks Shovels Trucks Optimize the allocation of mining equipment to the pits and determine the number of trips for each truck, while respecting the equipment limits, production goals and the desired mineral composition. 3 / 33

8 Introduction Heuristic Approach Proposed Algorithms Computational Experiments and Analysis Conclusions Open-Pit Mining Operational Planning Images from an open-pit mine in Brazil, situated in a region called Quadrila tero Ferrı fero 4 / 33

9 Open-Pit Mining Operational Planning (OPMOP) Characteristics Goals for ore and waste rock production 5 / 33

10 Open-Pit Mining Operational Planning (OPMOP) Characteristics Goals for ore and waste rock production Goals for ore parameters 5 / 33

11 Open-Pit Mining Operational Planning (OPMOP) Characteristics Goals for ore and waste rock production Goals for ore parameters Limited heterogeneous shovels 5 / 33

12 Open-Pit Mining Operational Planning (OPMOP) Characteristics Goals for ore and waste rock production Goals for ore parameters Limited heterogeneous shovels Limited heterogeneous trucks with compatibility issues 5 / 33

13 Open-Pit Mining Operational Planning (OPMOP) Characteristics Goals for ore and waste rock production Goals for ore parameters Limited heterogeneous shovels Limited heterogeneous trucks with compatibility issues Dynamic allocation of equipment 5 / 33

14 Open-Pit Mining Operational Planning 6 / 33

15 Open-Pit Mining Operational Planning Motivation Production costs involved are high 7 / 33

16 Open-Pit Mining Operational Planning Motivation Production costs involved are high OPMOP is a problem classified as N P-hard 7 / 33

17 Open-Pit Mining Operational Planning Motivation Production costs involved are high OPMOP is a problem classified as N P-hard The decision must be fast 7 / 33

18 Open-Pit Mining Operational Planning Motivation Production costs involved are high OPMOP is a problem classified as N P-hard The decision must be fast Successful mono-objective heuristic approaches in literature 7 / 33

19 Open-Pit Mining Operational Planning Motivation Production costs involved are high OPMOP is a problem classified as N P-hard The decision must be fast Successful mono-objective heuristic approaches in literature Novel multi-objective approach 7 / 33

20 Multi-objective Model Conflicting objectives Minimize each part of the function z(s) = (z 1 (s), z 2 (s), z 3 (s)), composed of three conflicting objectives. 8 / 33

21 Multi-objective Model Conflicting objectives Minimize each part of the function z(s) = (z 1 (s), z 2 (s), z 3 (s)), composed of three conflicting objectives. Let: T ore parameters; V vehicles; d t deviations of ore parameters from goals; P o and P w production deviations for ore and waste rock, respectively; U v utilization rate for vehicles; cost weights λ j, α, β and ω. 8 / 33

22 Multi-objective Model Conflicting objectives Minimize each part of the function z(s) = (z 1 (s), z 2 (s), z 3 (s)), composed of three conflicting objectives. Let: T ore parameters; V vehicles; d t deviations of ore parameters from goals; P o and P w production deviations for ore and waste rock, respectively; U v utilization rate for vehicles; cost weights λ j, α, β and ω. Ore quality z 1(s) = t T λ t d t + t T λ + t d + t (1) 8 / 33

23 Multi-objective Model Conflicting objectives Minimize each part of the function z(s) = (z 1 (s), z 2 (s), z 3 (s)), composed of three conflicting objectives. Let: T ore parameters; V vehicles; d t deviations of ore parameters from goals; P o and P w production deviations for ore and waste rock, respectively; U v utilization rate for vehicles; cost weights λ j, α, β and ω. Ore quality z 1(s) = t T λ t d t + t T λ + t d + t (1) Production quantity z 2(s) = α P o + α + P + o + β P w + β + P + w (2) 8 / 33

24 Multi-objective Model Conflicting objectives Minimize each part of the function z(s) = (z 1 (s), z 2 (s), z 3 (s)), composed of three conflicting objectives. Let: T ore parameters; V vehicles; d t deviations of ore parameters from goals; P o and P w production deviations for ore and waste rock, respectively; U v utilization rate for vehicles; cost weights λ j, α, β and ω. Ore quality z 1(s) = t T λ t d t + t T λ + t d + t (1) Production quantity z 2(s) = α P o + α + P + o + β P w + β + P + w (2) Number of trucks z 3(s) = v V ω l U v (3) 8 / 33

25 Heuristic Approach Representation of a Solution Table: Representation of a Solution Shovel Truck 1 Truck 2... Truck T F 1 (Shovel 1, 1) 8 X... X F 2 (Available, 0) F 3 (Shovel 8, 0) F F (Shovel 5, 1) / 33

26 Heuristic Approach Neighborhood Structures (1/2) 10 / 33

27 Heuristic Approach Neighborhood Structures (2/2) 11 / 33

28 Multi-objective Open-Pit Mining Operational Planning Proposal Heuristics Greedy Randomized Adaptative Search Procedure (GRASP) 1. Two-phase Pareto Local Search with VNS (2PPLS-VNS) 2. Multi-objective Variable Neighborhood Search (MOVNS) 3. Non-dominated Sorting Genetic Algorithm II (NSGA-II) Comparison metrics: 1. Spacing 2. Hypervolume 3. Coverage 12 / 33

29 Two-Phase Pareto Local Search Basic principle Composed of two phases: 1. In the first phase, an initial population, diversified and with a good approximation of the efficient set is generated. 2. In the second phase, the method Pareto Local Search (PLS) is applied to each individuals in the population. PLS This method is the generalization in the multiobjective case of a basic metaheuristic: the hill-climbing method. 13 / 33

30 Heuristic Approach Initial Population Generation Greedy Randomized Initial Solution Allocation of the shovels Distribution of trips 14 / 33

31 Heuristic Approach Initial Population Generation Greedy Randomized Initial Solution Allocation of the shovels Distribution of trips Waste pits Pits, the best is the one with the greatest mass Shovels, the best is the one with the greatest production Trucks, the largest one is the best. 14 / 33

32 Heuristic Approach Initial Population Generation Greedy Randomized Initial Solution Allocation of the shovels Distribution of trips Waste pits Pits, the best is the one with the greatest mass Shovels, the best is the one with the greatest production Trucks, the largest one is the best. Ore pits Pits, the best pit is the one that has the least deviation of the control parameters Shovels, the best is the one with the greatest production Trucks, the smallest one is the best. 14 / 33

33 G2PPLS-VNS Basic pseudocode Algorithm 1: G2PPLs-VNS Input: graspmax; Neighborhoods N k (x) Output: Approximation of the efficient set Xe P 0 BuildInicialSet(graspMax); Xe 2PPLs-VNS(P 0, N k (x)); return Xe 15 / 33

34 G2PPLS-VNS BuildInitialSet Algorithm 2: BuildInitialSet Input: graspmax; Output: Approximation of the efficient set Xe for i 1 until graspmax do s w BuildWasteSolution() Generate a random number γ [0, 1] s i BuildOreSolution(s w, γ) addsolution(xe, s i, f (s i )) end return Xe 16 / 33

35 G2PPLS-VNS Algorithm 3: 2PPLs com VNS Entrada: Aproximação inicial de um conjunto eficiente P 0 ; Vizinhanças N k (x); Saída: Conjunto Eficiente Xe 1 Xe P 0 ; P P 0 ; P a 2 k 1 {Tipo de estrutura de vizinhança corrente} 3 enquanto k <= r faça 4 para todo p P faça 5 para todo p N k (p) faça 6 se f (p) f (p ) então 7 addsolution(xe, p, f (p ), Added) 8 se Added = verdadeiro então 9 addsolution(p a, p, f (p )) 10 fim 11 fim 12 fim 13 fim 14 se P a então 15 P P a ; P a ; k 1 16 senão 17 k k P Xe \ {x Xe Pareto Local ótimo em relação N k (x)} 19 fim 20 fim 21 retorna Xe 17 / 33

36 Algorithm 4: GMOVNS GMOVNS Input: Neighborhoods N k (x); graspmax; levelmax Output: Approximation of the efficient set Xe Xe BuildInitialSet(graspMax) ; level 1 ; shaking 1 while stop criterion not satisfied do Select a not visited solution s Xe and check it as visited s s for i 1 until shaking do Select one neighborhood N k (.) at random s Shake(s, k) end Let k ult k ; changelevel true forall s N kult (s ) do addsolution(xe, s, f (s ), Added) if Added = true then changelevel false; end end if changelevel = true then level level + 1 else level 1; shaking 1 end if level = levelmax then level 1; shaking shaking + 1 end if all s Xe are visited then check all s Xe as non-visited solutions end end return Xe 18 / 33

37 GNSGAII-PR Basic idea The methods BuildInitialSet changes a little; Generate a population of solutions of good quality and well diversified. We do not warrant that this set is composed of non-dominated solutions; The structure of the algorithm follows the same literature. Changing only the genetic operators: Path Relinking (Ribeiro and Resende, 2012) and mutation. 19 / 33

38 GNSGAII-PR Basic pseudocode Algorithm 5: GNSGAII-PR Input: Population size N; Neighborhoods N k (x) Output: Approximation of the efficient set Xe Initial population P 0 while P 0 N do s w BuildWasteSolution() Generate a random number γ [0, 1] s i BuildOreSolution(s w, γ) P 0 s i end Q 0 SelectionPRCrossoverMutation(P 0, Neighborhoods N (k) (.)) Xe NSGA-II(P 0, Q 0, N, SelectionPRCrossoverMutation(.)) return Xe 20 / 33

39 GNSGAII-PR selectionprcrossovermutation - Genectic Operators Algorithm 6: selectionprcrossovermutation Input: mutationrate; localsearchrate Input: Population P; Neighborhoods N (k) (.) Output: Offspring population Q while Q N do Select two random individuals s 1 and s 2 P s best(path Relinking(s 1, s 2 ), Path Relinking(s 2, s 1 )) addsolution(q, s, f (s)) Generate a random number ap mutation [0, 1] if ap mutation < mutationrate then Select one neighborhood N k (.) at random s N (k) (s) else s s end Generate a random number ap localsearch [0, 1] if ap localsearch < localsearchrate then s VND(s ) addsolution(q, s, f (s )) else addsolution(q, s, f (s )) end end return Q 21 / 33

40 Experiments The code was implemented using the computational framework OptFrame 1. The battery of tests was composed of 30 runs for each algorithm with a computational time limited to 2 minutes since this runtime is suitable for real applications. The instances used for testing the algorithms were those of Souza et al. (2010) / 33

41 Spacing and Hypervolume metrics Table: G2PPLS-VNS GMOVNS GSGAII-PR: Spacing and Hypervolume Instance Spacing Hypervolume (10 6 ) G2PPLS-VNS GMOVNS GNSGAII-PR G2PPLS-VNS GMOVNS GNSGAII-PR opm opm opm opm opm opm opm opm / 33

42 Coverage metric Instance Table: G2PPLS-VNS GMOVNS: Coverage Coverage C(G2PPLS-VNS,G-MOVNS) C(G-MOVNS, G2PPLS-VNS) Best Average Std. dev. Best Average Std. dev. opm opm opm opm opm opm opm opm / 33

43 G2PPLS-VNS as a mono-objective optimization tool Table: Comparison of best results: G2PPLS-VNS GGVNS Instance Algorithm opm1 opm2 opm3 opm4 opm5 opm6 opm7 opm8 G2PPLS-VNS , GGVNS / 33

44 Obtaining the Reference Sets To obtain the Pareto Reference sets, 30 executions were performed for two hours. This test battery consisted in running GMOVNS and G2PPLS-VNS alternately. Reference sets and the solutions belonging to them are available at: 26 / 33

45 Reference Set Figure: Pareto Set Reference - opm2 instance non-dominated solutions 27 / 33

46 Conclusions The proposed multiobjective approach to OPMOP had been validated. The developed algorithms were able to generate good approximations to the Pareto fronts. For instance opm2 a reference set with 271 non-dominated solutions was found. The algorithms G2PPLS-VNS and GMOVNS, based on VNS procedures, performed better than algorithm GNSGAII-PR. The proposed algorithms seem suitable for practical applications. 28 / 33

47 Acknowledgements Federal University of Ouro Preto Brazilian agency FAPEMIG Brazilian agency CNPq 29 / 33

48 Thank you for your attention! 30 / 33

49 Questions? 31 / 33

50 References Camilo de Lelis Dias Cavalcanti. Métodos de pesquisa e lavra ii, URL lavra-mina-ceu-aberto-subterranea. Colegio Web. Brazilian mining operations, November URL conceitos-e-tipos-de-industrias. Prof. Welgeo. Comparison of a mining truck with a normal car, November URL recursos-minerais-do-brasil-metalicos.html. 32 / 33

51 G2PPLS-VNS as a mono-objective optimization tool Table: GRASP-2PPLS GRASP-MOVNS: Cardinalidade GRASP-2PPLS GRASP-MOVNS Instância Ref D 1 Ref D 1 D 2 Ref D 2 opm opm opm opm opm opm opm opm Total de Soluções / 33