A Comparative Study of Metaheuristic Algorithms forthe Fertilizer Optimization ProblemDai ChenSupervisor: Dr. Michael Hors...
MotivationHard Combinatorial Optimization ProblemHigh dimensional search spaceRuntime efficiency vs. solution qualitySolvin...
The Fertilizer Blends Optimization ProblemObjectives:Deliver nutrients to farm sites, optimize the total net profitDifficult...
The Fertilizer Blends Optimization ProblemNutrient Mixing ProcessMixture Table ABlend Delivery ProcessDelivery Rate Table ...
Metaheuristic Algorithmic SolutionsLocal Search MetaheuristicsNeighborhood movesConstructive Local Search MethodHeuristic ...
Metaheuristic Algorithmic SolutionsLocal Search MetaheuristicsNeighborhood movesConstructive Local Search MethodHeuristic ...
Local Search Metaheuristics (1)Simulated Annealing1: Initialization S ← S0, T ← T02: while Termination criterion notmet do...
Local Search Metaheuristics (1)Simulated Annealing1: Initialization S ← S0, T ← T02: while Termination criterion notmet do...
Local Search Metaheuristics (1)Simulated Annealing1: Initialization S ← S0, T ← T02: while Termination criterion notmet do...
Local Search Metaheuristics (1)Simulated Annealing1: Initialization S ← S0, T ← T02: while Termination criterion notmet do...
Local Search Metaheuristics (2)DNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3...
Local Search Metaheuristics (2)DNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3...
Local Search Metaheuristics (2)DNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3...
Local Search Metaheuristics (2)DNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3...
Constructive Local SearchGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Cons...
Constructive Local SearchGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Cons...
Constructive Local SearchGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Cons...
Empirical ExperimentsTune each algorithm to obtain its near optimal meta-parametersetting (MPS∗) for each problem instance...
Empirical Experiments - Meta-Parameter TuningSimulated Annealing1: Initialization S ← S0, T ← T02: while Termination crite...
Empirical Experiments - Meta-Parameter TuningSimulated Annealing1: Initialization S ← S0, T ← T02: while Termination crite...
Empirical Experiments - Meta-Parameter TuningDNM Local Search Algorithm1: Initialization F ← F02: while Termination criter...
Empirical Experiments - Meta-Parameter TuningDNM Local Search Algorithm1: Initialization F ← F02: while Termination criter...
Empirical Experiments - Meta-Parameter TuningDNM Local Search Algorithm1: Initialization F ← F02: while Termination criter...
Empirical Experiments - Meta-Parameter TuningGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criter...
Empirical Experiments - Meta-Parameter TuningGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criter...
Empirical Experiments - Meta-Parameter TuningGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criter...
Empirical Experiments - Meta-Parameter TuningGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criter...
Empirical Experiments - Meta-Parameter TuningGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criter...
Empirical Experiment - Performance ComparisonTune each algorithm to obtain its near optimal meta-parametersetting (MPS∗) f...
Empirical Result - Performance Comparison0.90.910.920.930.940.950.960.970.980.9910 10 20 30 40 50 60 70 80 90SolutionQuali...
ConclusionsEmpirical study of four metaheuristic algorithms for the fertilizeroptimization problemSimulated Annealing (SA)...
ConclusionsEmpirical study of four metaheuristic algorithms for the fertilizeroptimization problemSimulated Annealing (SA)...
Future WorkAutomated meta-parameter tuning and performance predictionProbabilistic models for stochastic searchAlgorithm s...
Future WorkAutomated meta-parameter tuning and performance predictionProbabilistic models for stochastic searchAlgorithm s...
Future WorkAutomated meta-parameter tuning and performance predictionProbabilistic models for stochastic searchAlgorithm s...
QuestionThank you !Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 17 / 22
Empirical Results - SA Step Size Tuning∆v / ∆a 0.001 0.005 0.011 6940 6894 68615 6974 6939 689810 6974 6954 6927Table: Loc...
Empirical Results - SA Step Size Tuning∆v / ∆a 0.001 0.005 0.011 6940 6894 68615 6974 6939 689810 6974 6954 6927Table: Loc...
Empirical Results - SA Step Size Tuning00.20.40.60.810 50000 100000 150000 200000 250000 300000SolvingProbabilityRuntime (...
Empirical Results - SA Downhill Move SchemeCooling/ kB 5 × 10−5 5 × 10−4 5 × 10−3 5 × 10−2 5 × 10−1linear 6983 7028 7057 7...
Empirical Results - SA Downhill Move Scheme0500001000001500002000002500003000001e-04 0.001 0.01 0.1AverageRuntimeCost(Loca...
Empirical Results - DNM Step Size Tuningα / δ 0.01 0.05 0.1 0.15 0.20.001 6941 6972 6980 6965 69700.005 7225 7203 7176 714...
Empirical Results - DNM Step Size Tuningα / δ 0.01 0.05 0.1 0.15 0.20.001 6941 6972 6980 6965 69700.005 7225 7203 7176 714...
Empirical Results - DNM Step Size Tuning6500660067006800690070007100720073000 20000 40000 60000 80000 100000 120000 140000...
Empirical Results - GRASP Selection Parameter Tuningα Avg.RT Best.RT AvgF.Q BestF.Q0(greedy) 101350 23367 6964 70140.2 784...
Empirical Results - GRASP Similarity Testα Cons.A Final.A Cons.V Final.V0(greedy) 6.1 1.9 193.2 111.40.2 1.3 1.1 132.7 73....
Empirical Results - GRASP Similarity Testα Cons.A Final.A Cons.V Final.V0(greedy) 6.1 1.9 193.2 111.40.2 1.3 1.1 132.7 73....
Empirical Results - GRASP Similarity Testα Cons.A Final.A Cons.V Final.V0(greedy) 6.1 1.9 193.2 111.40.2 1.3 1.1 132.7 73....
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Metaheuristic Algorithms for Fertilizer Blends Optimization

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A comparative study of metaheuristic algorithms for fertilizer blends optimization

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Transcript of "Metaheuristic Algorithms for Fertilizer Blends Optimization"

  1. 1. A Comparative Study of Metaheuristic Algorithms forthe Fertilizer Optimization ProblemDai ChenSupervisor: Dr. Michael HorschDepartment of Computer ScienceUniversity of SaskatchewanM.Sc. Thesis Defense, 2006.8Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 1 / 22
  2. 2. MotivationHard Combinatorial Optimization ProblemHigh dimensional search spaceRuntime efficiency vs. solution qualitySolving Problems by Search: MetaheuristicsIncomplete search & non-deterministicHigh level strategy for search space explorationDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 2 / 22
  3. 3. The Fertilizer Blends Optimization ProblemObjectives:Deliver nutrients to farm sites, optimize the total net profitDifficulties:Large number of farm sitesSites have different nutrient requirementsSolutions:Compound blends for all sites, use different delivery rates atdifferent sites.Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 3 / 22
  4. 4. The Fertilizer Blends Optimization ProblemNutrient Mixing ProcessMixture Table ABlend Delivery ProcessDelivery Rate Table VDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 4 / 22
  5. 5. Metaheuristic Algorithmic SolutionsLocal Search MetaheuristicsNeighborhood movesConstructive Local Search MethodHeuristic space sampling & local searchDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 5 / 22
  6. 6. Metaheuristic Algorithmic SolutionsLocal Search MetaheuristicsNeighborhood movesConstructive Local Search MethodHeuristic space sampling & local searchDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 5 / 22
  7. 7. Local Search Metaheuristics (1)Simulated Annealing1: Initialization S ← S0, T ← T02: while Termination criterion notmet do3: Alter S to obtain neighbor S4: if Acceptance criterion metthen5: S = S6: end if7: Update T using the coolingscheme8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return SbestDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 6 / 22
  8. 8. Local Search Metaheuristics (1)Simulated Annealing1: Initialization S ← S0, T ← T02: while Termination criterion notmet do3: Alter S to obtain neighbor S4: if Acceptance criterion metthen5: S = S6: end if7: Update T using the coolingscheme8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return SbestA + ∆aV + ∆vSSDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 6 / 22
  9. 9. Local Search Metaheuristics (1)Simulated Annealing1: Initialization S ← S0, T ← T02: while Termination criterion notmet do3: Alter S to obtain neighbor S4: if Acceptance criterion metthen5: S = S6: end if7: Update T using the coolingscheme8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return Sbestf(s)SSAcceptedf (s)Accepted with pSSp = e− ∆EkBTDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 6 / 22
  10. 10. Local Search Metaheuristics (1)Simulated Annealing1: Initialization S ← S0, T ← T02: while Termination criterion notmet do3: Alter S to obtain neighbor S4: if Acceptance criterion metthen5: S = S6: end if7: Update T using the coolingscheme8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return Sbestf(s)SSAcceptedf (s)Accepted with pSSp = e− ∆EkBTDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 6 / 22
  11. 11. Local Search Metaheuristics (2)DNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3: Flow augmentation, update F4: if Overdose effect happensthen5: Micro tuning, update F6: end if7: Compute solution S from F8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return Sbestm1Market Nutrientsm2m3k1Farm Sitesk2k3k4Site YieldNutrientsDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 7 / 22
  12. 12. Local Search Metaheuristics (2)DNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3: Flow augmentation, update F4: if Overdose effect happensthen5: Micro tuning, update F6: end if7: Compute solution S from F8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return Sbestm1Market Nutrientsm2m2m3k1Farm Sitesk2k3k4Site YieldNutrientsDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 7 / 22
  13. 13. Local Search Metaheuristics (2)DNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3: Flow augmentation, update F4: if Overdose effect happensthen5: Micro tuning, update F6: end if7: Compute solution S from F8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return Sbestm1Market Nutrientsm2m3k1Farm Sitesk2k3k3k4Site YieldNutrientsDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 7 / 22
  14. 14. Local Search Metaheuristics (2)DNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3: Flow augmentation, update F4: if Overdose effect happensthen5: Micro tuning, update F6: end if7: Compute solution S from F8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return Sbestm1Market Nutrientsm2m3k1Farm Sitesk2k3k3k4Site YieldNutrientsDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 7 / 22
  15. 15. Constructive Local SearchGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Construct initial solution S03: Perform local search on S0 toobtain S14: if f(S1) > f(Sbest) then5: Sbest = S16: end if7: end while8: return SbestDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 8 / 22
  16. 16. Constructive Local SearchGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Construct initial solution S03: Perform local search on S0 toobtain S14: if f(S1) > f(Sbest) then5: Sbest = S16: end if7: end while8: return SbestCandidate ListRestricted Candidate ListA VDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 8 / 22
  17. 17. Constructive Local SearchGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Construct initial solution S03: Perform local search on S0 toobtain S14: if f(S1) > f(Sbest) then5: Sbest = S16: end if7: end while8: return SbestCandidate ListRestricted Candidate ListA VDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 8 / 22
  18. 18. Empirical ExperimentsTune each algorithm to obtain its near optimal meta-parametersetting (MPS∗) for each problem instance.Performance comparison among algorithmsProblem Instance40-site Canola Data100-site Wheat Data40-site Wheat DataMetaheuristic AlgorithmsLocal Search MetaheuristicsSimulated AnnealingDNM Local SearchIterative ImprovementGreedy Randomized AdaptiveSearch Procedure (GRASP)Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 9 / 22
  19. 19. Empirical Experiments - Meta-Parameter TuningSimulated Annealing1: Initialization S ← S0, T ← T02: while Termination criterion notmet do3: Alter S to obtain neighbor S4: if Acceptance criterion satisfiedthen5: S = S6: end if7: Update T using the coolingscheme8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return SbestMeta-ParametersStep sizes (∆a, ∆v)Cooling scheduleBolztman constant kB∆a : [0.001, 0.005]∆v : [5, 10]Linear Cooling &kB = 0.005Exponential Cooling& kB = 0.05Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 10 / 22
  20. 20. Empirical Experiments - Meta-Parameter TuningSimulated Annealing1: Initialization S ← S0, T ← T02: while Termination criterion notmet do3: Alter S to obtain neighbor S4: if Acceptance criterion satisfiedthen5: S = S6: end if7: Update T using the coolingscheme8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return SbestMeta-ParametersStep sizes (∆a, ∆v)Cooling scheduleBolztman constant kB∆a : [0.001, 0.005]∆v : [5, 10]Linear Cooling &kB = 0.005Exponential Cooling& kB = 0.05Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 10 / 22
  21. 21. Empirical Experiments - Meta-Parameter TuningDNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3: Flow augmentation, update F4: if Overdose effect happensthen5: Micro tuning, update F6: end if7: Compute solution S from F8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return SbestMeta-ParametersFlow augmentationsize δMicro tuning size αδ : [0.01, 0.05]α : 0.005Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 11 / 22
  22. 22. Empirical Experiments - Meta-Parameter TuningDNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3: Flow augmentation, update F4: if Overdose effect happensthen5: Micro tuning, update F6: end if7: Compute solution S from F8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return SbestMeta-ParametersFlow augmentationsize δMicro tuning size αδ : [0.01, 0.05]α : 0.005Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 11 / 22
  23. 23. Empirical Experiments - Meta-Parameter TuningDNM Local Search Algorithm1: Initialization F ← F02: while Termination criterion notmet do3: Flow augmentation, update F4: if Overdose effect happensthen5: Micro tuning, update F6: end if7: Compute solution S from F8: if f(Sbest) < f(S) then9: Sbest = S10: end if11: end while12: return SbestMeta-ParametersFlow augmentationsize δMicro tuning size αδ : [0.01, 0.05]α : 0.005Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 11 / 22
  24. 24. Empirical Experiments - Meta-Parameter TuningGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Construct initial solution S03: Perform local search on S0 toobtain S14: if f(S1) > f(Sbest) then5: Sbest = S16: end if7: end while8: return SbestMeta-ParametersRCL selectionparameter α{0, 0.2, 0.4, 0.6, 0.8, 1}Structure SimilarityTestα = 0 :Cons.S > Final.Sα = 1 :Final.S > Cons.SDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 12 / 22
  25. 25. Empirical Experiments - Meta-Parameter TuningGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Construct initial solution S03: Perform local search on S0 toobtain S14: if f(S1) > f(Sbest) then5: Sbest = S16: end if7: end while8: return SbestMeta-ParametersRCL selectionparameter α{0, 0.2, 0.4, 0.6, 0.8, 1}Structure SimilarityTestα = 0 :Cons.S > Final.Sα = 1 :Final.S > Cons.SDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 12 / 22
  26. 26. Empirical Experiments - Meta-Parameter TuningGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Construct initial solution S03: Perform local search on S0 toobtain S14: if f(S1) > f(Sbest) then5: Sbest = S16: end if7: end while8: return SbestMeta-ParametersRCL selectionparameter α{0, 0.2, 0.4, 0.6, 0.8, 1}Structure SimilarityTestα = 0 :Cons.S > Final.Sα = 1 :Final.S > Cons.SDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 12 / 22
  27. 27. Empirical Experiments - Meta-Parameter TuningGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Construct initial solution S03: Perform local search on S0 toobtain S14: if f(S1) > f(Sbest) then5: Sbest = S16: end if7: end while8: return SbestMeta-ParametersRCL selectionparameter α{0, 0.2, 0.4, 0.6, 0.8, 1}Structure SimilarityTestα = 0 :Cons.S > Final.Sα = 1 :Final.S > Cons.SDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 12 / 22
  28. 28. Empirical Experiments - Meta-Parameter TuningGreedy Randomized AdaptiveSearch Procedure (GRASP)1: while Termination criterion notmet do2: Construct initial solution S03: Perform local search on S0 toobtain S14: if f(S1) > f(Sbest) then5: Sbest = S16: end if7: end while8: return SbestMeta-ParametersRCL selectionparameter α{0, 0.2, 0.4, 0.6, 0.8, 1}Structure SimilarityTestα = 0 :Cons.S > Final.Sα = 1 :Final.S > Cons.SDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 12 / 22
  29. 29. Empirical Experiment - Performance ComparisonTune each algorithm to obtain its near optimal meta-parametersetting (MPS∗) for the problem.Performance comparison among algorithmsDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 13 / 22
  30. 30. Empirical Result - Performance Comparison0.90.910.920.930.940.950.960.970.980.9910 10 20 30 40 50 60 70 80 90SolutionQuality(PercentageofBest-known)Runtime Cost (CPU seconds)SADNMIIAGRASPIIAFigure: Runtime Quality ComparisonDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 14 / 22
  31. 31. ConclusionsEmpirical study of four metaheuristic algorithms for the fertilizeroptimization problemSimulated Annealing (SA)Iterated Improvement (IIA)DNM Local Search (DNM)Greedy Randomized Adaptive Search Procedure (GRASPIIA)Meta-parameter Tuning of the algorithmsLocal search step size → Landscape GranularityDownhill move & Strict uphill moveDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 15 / 22
  32. 32. ConclusionsEmpirical study of four metaheuristic algorithms for the fertilizeroptimization problemSimulated Annealing (SA)Iterated Improvement (IIA)DNM Local Search (DNM)Greedy Randomized Adaptive Search Procedure (GRASPIIA)Meta-parameter Tuning of the algorithmsLocal search step size → Landscape GranularityDownhill move & Strict uphill moveDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 15 / 22
  33. 33. Future WorkAutomated meta-parameter tuning and performance predictionProbabilistic models for stochastic searchAlgorithm selection problem - no free lunch theorem for searchDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 16 / 22
  34. 34. Future WorkAutomated meta-parameter tuning and performance predictionProbabilistic models for stochastic searchAlgorithm selection problem - no free lunch theorem for searchDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 16 / 22
  35. 35. Future WorkAutomated meta-parameter tuning and performance predictionProbabilistic models for stochastic searchAlgorithm selection problem - no free lunch theorem for searchDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 16 / 22
  36. 36. QuestionThank you !Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 17 / 22
  37. 37. Empirical Results - SA Step Size Tuning∆v / ∆a 0.001 0.005 0.011 6940 6894 68615 6974 6939 689810 6974 6954 6927Table: Local search step size tuning matrix. Each entry gives the averagefound solution quality (total net profit) of SA under different (∆v, ∆a) settings.Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 18 / 22
  38. 38. Empirical Results - SA Step Size Tuning∆v / ∆a 0.001 0.005 0.011 6940 6894 68615 6974 6939 689810 6974 6954 6927Table: Local search step size tuning matrix. Each entry gives the averagefound solution quality (total net profit) of SA under different (∆v, ∆a) settings.Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 18 / 22
  39. 39. Empirical Results - SA Step Size Tuning00.20.40.60.810 50000 100000 150000 200000 250000 300000SolvingProbabilityRuntime (Local Search Moves)∆a = 0.001, ∆v = 1∆a = 0.005, ∆v = 5∆a = 0.01, ∆v = 10Figure: Runtime Distribution MeasurementDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 18 / 22
  40. 40. Empirical Results - SA Downhill Move SchemeCooling/ kB 5 × 10−5 5 × 10−4 5 × 10−3 5 × 10−2 5 × 10−1linear 6983 7028 7057 7041 6749exponential 6973 7006 7059 7064 6982Table: Solution quality comparision. Each entry gives the average best-foundsolution quality (total net profit) of SA under different downhill move schemes.Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 19 / 22
  41. 41. Empirical Results - SA Downhill Move Scheme0500001000001500002000002500003000001e-04 0.001 0.01 0.1AverageRuntimeCost(LocalSearchSteps)Bolztman ConstantLinear CoolingExponential CoolingFigure: Average Runtime MeasurementDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 19 / 22
  42. 42. Empirical Results - DNM Step Size Tuningα / δ 0.01 0.05 0.1 0.15 0.20.001 6941 6972 6980 6965 69700.005 7225 7203 7176 7148 71510.01 7136 7128 7100 7076 70620.015 7117 7106 7092 7082 70680.02 7101 7089 7078 7059 7050Table: A parameter tuning matrix for DNM local search. Each entry gives theaverage achieved solution quality (total net profit) under one (α, δ) setting.Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 20 / 22
  43. 43. Empirical Results - DNM Step Size Tuningα / δ 0.01 0.05 0.1 0.15 0.20.001 6941 6972 6980 6965 69700.005 7225 7203 7176 7148 71510.01 7136 7128 7100 7076 70620.015 7117 7106 7092 7082 70680.02 7101 7089 7078 7059 7050Table: A parameter tuning matrix for DNM local search. Each entry gives theaverage achieved solution quality (total net profit) under one (α, δ) setting.Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 20 / 22
  44. 44. Empirical Results - DNM Step Size Tuning6500660067006800690070007100720073000 20000 40000 60000 80000 100000 120000 140000 160000AverageSolutionQuality(TotalNetProfit)Runtime (Local Search Moves)δ = 0.01, α = 0.001δ = 0.01, α = 0.005δ = 0.01, α = 0.010δ = 0.01, α = 0.015Figure: Runtime Quality MeasurementDai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 20 / 22
  45. 45. Empirical Results - GRASP Selection Parameter Tuningα Avg.RT Best.RT AvgF.Q BestF.Q0(greedy) 101350 23367 6964 70140.2 78410 16917 6984 70390.4 76057 16933 6980 70250.6 72937 14862 6977 70200.8 65289 18384 6983 70271(random) 59227 18636 6985 7034Table: Selection parameter α tuning. Measurements include: average andbest runtime (local search steps) to 98% of best-known solution quality(Avg.RT and Best.RT); average and best-found final solution quality (AvgF.Qand BestF.Q).Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 21 / 22
  46. 46. Empirical Results - GRASP Similarity Testα Cons.A Final.A Cons.V Final.V0(greedy) 6.1 1.9 193.2 111.40.2 1.3 1.1 132.7 73.30.4 0.6 1.2 77.0 57.00.6 0.3 1.4 52.8 52.70.8 0.4 1.5 50.4 55.01(random) 0.4 1.5 50.0 56.1Table: Structure similarity test. Measurements include: structure similarity ofconstructed solutions set (Cons.A and Cons.V); structure similarity of finalsolutions set (Final.A and Final.V).Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 22 / 22
  47. 47. Empirical Results - GRASP Similarity Testα Cons.A Final.A Cons.V Final.V0(greedy) 6.1 1.9 193.2 111.40.2 1.3 1.1 132.7 73.30.4 0.6 1.2 77.0 57.00.6 0.3 1.4 52.8 52.70.8 0.4 1.5 50.4 55.01(random) 0.4 1.5 50.0 56.1Table: Structure similarity test. Measurements include: structure similarity ofconstructed solutions set (Cons.A and Cons.V); structure similarity of finalsolutions set (Final.A and Final.V).Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 22 / 22
  48. 48. Empirical Results - GRASP Similarity Testα Cons.A Final.A Cons.V Final.V0(greedy) 6.1 1.9 193.2 111.40.2 1.3 1.1 132.7 73.30.4 0.6 1.2 77.0 57.00.6 0.3 1.4 52.8 52.70.8 0.4 1.5 50.4 55.01(random) 0.4 1.5 50.0 56.1Table: Structure similarity test. Measurements include: structure similarity ofconstructed solutions set (Cons.A and Cons.V); structure similarity of finalsolutions set (Final.A and Final.V).Dai Chen (USASK) Metaheuristics for the Fertilizer Problem M.Sc. Thesis Defense 22 / 22

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