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FEUP | PDEEC | Decision Support

Metaheuristics: GRASP. Group 1: Clara Gouveia Daniel Oliveira Fabrício Sperandio Filipe Sousa [Presenter]. FEUP | PDEEC | Decision Support. January 3 rd , 2011. Metaheuristics: GRASP. Outline. Part One: Introduction to GRASP. GRASP Overview Construction Phase Local Search Phase

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FEUP | PDEEC | Decision Support

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  1. Metaheuristics: GRASP Group 1: Clara Gouveia Daniel Oliveira FabrícioSperandio Filipe Sousa [Presenter] FEUP | PDEEC | Decision Support January 3rd, 2011

  2. Metaheuristics: GRASP Outline Part One: Introduction to GRASP • GRASP Overview • Construction Phase • Local Search Phase • GRASP Example • GRASP Variations Part Two: Paper Presentation • 3D BPP Definition • 2D BPP Definition • Hybrid GRASP/VND for BPP • Preprocess Phase • Construction Phase • Improvement Phase • Improvement Procedures • Combined Strategies • Diversification Phase • Computational Results • Comparison with other Algorithms • Conclusions FEUP | PDEEC | Decision Support

  3. Metaheuristics: GRASP GRASP Overview G reedy R andomized A daptive S earch FEUP | PDEEC | Decision Support P rocedure

  4. Metaheuristics: GRASP GRASP Overview • GRASP is a metaheuristic: • A metaheuristic is a method that works with local improvement procedures and other higher level strategiesto create processes capable of escaping from local optima, and performing a robust search of a solution space. • GRASP was first described in 1989: • Published by Thomas A. Feo and Mauricio G. C. Resende. • In this first publication GRASP was applied to the set covering problem. FEUP | PDEEC | Decision Support References: M. Gendreau, and J. Potvin. Handbook of Metaheuristics. 2nd edition, Springer, 2010. T. A. Feo, and M. G. C. Resende. A Probabilistic Heuristic for a Computationally Difficult Set Covering Problem. Operations Research Letters, no. 8, pp. 67-71, 1989.

  5. Metaheuristics: GRASP GRASP Overview • GRASP can be divided in two phases: • Construction: where a feasible solution is built. • Local Search: from the solution obtained a neighborhood is built and the search is then performed. For N iterations FEUP | PDEEC | Decision Support Construction Phase Local Search Phase Best Solution Improved Solution References: T. A. Feo, and M. G. C. Resende. Greedy Randomized Adaptive Search Procedures. Journal of Global Optimization, no. 6, pp. 109-133, 1995. M. G. C. Resende. Greedy Randomized Adaptive Search Procedures (GRASP). AT&T Labs Research Technical Report, 98.41.1, Dec. 1998. M. G. C. Resende, and Celso C. Ribeiro. Greedy Randomized Search Procedures. AT&T Labs Research Technical Report TD-53RSJY, version 2, Aug. 2002

  6. Metaheuristics: GRASP GRASP Overview • GRASP is easy to implement: • Few parameters need to be set and tuned. • Effort can be transferred to the implementation of efficient data structures. • GRASPis also easily implemented in parallel. For N iterations Construction Phase Local Search Phase FEUP | PDEEC | Decision Support compare Improved Solution Best Solution Construction Phase Local Search Phase Improved Solution References: T. A. Feo, and M. G. C. Resende. Greedy Randomized Adaptive Search Procedures. Journal of Global Optimization, no. 6, pp. 109-133, 1995. M. G. C. Resende. Greedy Randomized Adaptive Search Procedures (GRASP). AT&T Labs Research Technical Report, 98.41.1, Dec. 1998. M. G. C. Resende, and Celso C. Ribeiro. Greedy Randomized Search Procedures. AT&T Labs Research Technical Report TD-53RSJY, version 2, Aug. 2002

  7. Metaheuristics: GRASP Construction Phase • The RCL (Restricted Candidate List) is built based on the α parameter: • α is variable between 0 and 1: • 0 makes the construction too random. • 1 makes the construction too greedy. • If βis the best free element and Σ represents the free elements, RCL is composed by: • RCL U {Σ≥ α.β} Repeat until element list is empty FEUP | PDEEC | Decision Support Initialize elements Build RCL Contructed Solution Update Candidate List Update Solution Randomly choose an element References: T. A. Feo, and M. G. C. Resende. Greedy Randomized Adaptive Search Procedures. Journal of Global Optimization, no. 6, pp. 109-133, 1995. M. G. C. Resende. Greedy Randomized Adaptive Search Procedures (GRASP). AT&T Labs Research Technical Report, 98.41.1, Dec. 1998. M. G. C. Resende, and Celso C. Ribeiro. Greedy Randomized Search Procedures. AT&T Labs Research Technical Report TD-53RSJY, version 2, Aug. 2002

  8. Metaheuristics: GRASP Construction Phase • The randomness of GRASP is present when an element is picked by chance from the RCL. • After the element is added to the current solution the cost of each free element is updated: • This is the adaptive part of the GRASP metaheuristic. Repeat until element list is empty FEUP | PDEEC | Decision Support Initialize elements Build RCL Update Candidate List Contructed Solution Update Solution Randomly choose an element References: T. A. Feo, and M. G. C. Resende. Greedy Randomized Adaptive Search Procedures. Journal of Global Optimization, no. 6, pp. 109-133, 1995. M. G. C. Resende. Greedy Randomized Adaptive Search Procedures (GRASP). AT&T Labs Research Technical Report, 98.41.1, Dec. 1998. M. G. C. Resende, and Celso C. Ribeiro. Greedy Randomized Search Procedures. AT&T Labs Research Technical Report TD-53RSJY, version 2, Aug. 2002

  9. Metaheuristics: GRASP Local Search Phase • The CreateNeighborhoodcan be implement in several different ways: • This is problemdependent. • The stopping criteriavaries with the implemented method. Repeat until stopping criteria is satisfied Contructed Solution Create Neighborhood Compare with Existing Best Improved Solution Select Best Solution FEUP | PDEEC | Decision Support References: T. A. Feo, and M. G. C. Resende. Greedy Randomized Adaptive Search Procedures. Journal of Global Optimization, no. 6, pp. 109-133, 1995. M. G. C. Resende. Greedy Randomized Adaptive Search Procedures (GRASP). AT&T Labs Research Technical Report, 98.41.1, Dec. 1998. F. Parreño, R. Alvarez-Valdes, J. F. Oliveira, and J. M. Tamarit. A hybrid GRASP/VND algorithm for two- and three-dimensional bin packing. Annals of Operations Research, vol. 179, no. 1, pp. 203-220, Oct. 2008.

  10. Metaheuristics: GRASP GRASP Example • Set Covering Problem: • Having α = 40% implies a RCL = {P1, P4, P5, P6, P7}. • Randomly selecting P5 translates into: • Covering elements 3, 4, and 5. • The next GRASP step consists in the candidate list update. FEUP | PDEEC | Decision Support References: T. A. Feo, and M. G. C. Resende. Greedy Randomized Adaptive Search Procedures. Journal of Global Optimization, no. 6, pp. 109-133, 1995.

  11. Metaheuristics: GRASP GRASP Example • Set Covering Problem: • RCL = {P3, P4, P6, P7}. • Randomly choosing P3 leaves P6 as the only option: • Translates into solution{P5, P3, P6}. FEUP | PDEEC | Decision Support References: T. A. Feo, and M. G. C. Resende. Greedy Randomized Adaptive Search Procedures. Journal of Global Optimization, no. 6, pp. 109-133, 1995.

  12. Metaheuristics: GRASP GRASP Example • Set Covering Problem: • Once again, having α = 40% implies a RCL = {P1, P4, P5, P6, P7}. • However, if P6 had been randomly chosen, and then P4 we would have reached an optimal solution: • {P6, P4}. FEUP | PDEEC | Decision Support References: T. A. Feo, and M. G. C. Resende. Greedy Randomized Adaptive Search Procedures. Journal of Global Optimization, no. 6, pp. 109-133, 1995.

  13. Metaheuristics: GRASP GRASP Variations • Reactive GRASP: • RCL size is adjusted according to the quality of the solutions previously found. • GRASP using GA (Genetic Algorithm) methodology : • Introduces a mutation in the local search phase. • GRASP with cost perturbation: • The cost associated with an element is modified in some way to cause a perturbation in the greedy function. • Other variations exist... FEUP | PDEEC | Decision Support References: T. A. Feo, and M. G. C. Resende. Greedy Randomized Adaptive Search Procedures. Journal of Global Optimization, no. 6, pp. 109-133, 1995. M. G. C. Resende, and Celso C. Ribeiro. Greedy Randomized Search Procedures. AT&T Labs Research Technical Report TD-53RSJY, version 2, Aug. 2002

  14. Metaheuristics: GRASP Outline Part One: Introduction to GRASP • GRASP Overview • Construction Phase • Local Search Phase • GRASP Example • GRASP Variations Part Two: Paper Presentation • 3D BPP Definition • 2D BPP Definition • Hybrid GRASP/VND for BPP • Preprocess Phase • Construction Phase • Improvement Phase • Improvement Procedures • Combined Strategies • Diversification Phase • Computational Results • Comparison with other Algorithms • Conclusions FEUP | PDEEC | Decision Support

  15. Metaheuristics: GRASP Part Two: Paper Presentation A hybrid GRASP/VND algorithm for two- and three-dimensional bin packing. F. Parreño, R. Alvarez-Valdes, J.F. Oliveira, and J.M. Tamarit Annals of Operations Research Volume 179, Number 1, Pages 203-220, 25 October 2008. FEUP | PDEEC | Decision Support

  16. Metaheuristics: GRASP 3D BPP Definition • The three-dimensional bin packing problem (3BP): • NP-hardproblem. • Useful for industrial applications (loading cargo into pallets, containers or vehicles, or packaging design). • Assumptions: • Known dimensions: • Bin (W,H,D) . • Boxes (wi,hi, di ), (i = 1, . . . , n). • wi≤ W, hi ≤ H, and di ≤ D. • The items cannot be rotated. FEUP | PDEEC | Decision Support

  17. Metaheuristics: GRASP 2D BPP Definition • The two-dimensional bin packing problem (2BP): • Special case of 3BP where di =D , i=1, . . . , n. • Assumptions: • Known dimensions: • Bin (W,H). • Boxes (wi,hi), (i = 1, . . . , n). • wi ≤ W, and hi≤ H. • The items cannot be rotated. FEUP | PDEEC | Decision Support

  18. Metaheuristics: GRASP Hybrid GRASP/VND for BPP k=1; f=[] B={b1,...,bn} • Preprocess: • Simplify the problem. • Constructive Phase: • Develop a feasible solution. • Improvement Phase: • Improves the solution. • Inputs: • Items to pack. • Number of iterations. • Alpha. Preprocess B={b1,...,bm} S={E} Constructive Phase k=k+1 Target: n-1 FEUP | PDEEC | Decision Support YES NO NO or YES YES NO Improvement Phase f=[]

  19. Metaheuristics: GRASP Hybrid GRASP/VND for BPP Constructive Phase • Vectors with boxes left to pack: • B={b1,...,bm}. • Set of empty maximal spaces: • S={E}. (0) Initialization (1) Choose Maximal Space – S* (2) Choose Boxes to Pack (3) Update List S FEUP | PDEEC | Decision Support and

  20. Metaheuristics: GRASP Hybrid GRASP/VND for BPP Constructive Phase: Step 1 (0) Initialization • Choose Maximal Space: • Maximal Space (S*): maximal space in S closer to a corner. • The volume can be used as a tie breaker. (1) Choose Maximal Space – S* • Objective: • First fill the corners. • Then the sides. • Later the innerspace. (2) Choose Boxes to Pack FEUP | PDEEC | Decision Support (3) Update List S

  21. Metaheuristics: GRASP Hybrid GRASP/VND for BPP Constructive Phase: Step 2 (0) Initialization • Choose Boxes to Pack: • Boxes fitting in S* can be packed: • Individually. • Layer: several boxes with the same dimensions. • Using one of the following criteria: • Best volume: order by the box that produces the largest increase in the bin volume. • Best fit: order by the box which fits best in the maximal space. • The box to be moved is selected randomly among the (1-α)*100%blocks. • α is the RCL GRASP parameter. (1) Choose Maximal Space – S* (2) Choose Boxes to Pack FEUP | PDEEC | Decision Support (3) Update List S

  22. Metaheuristics: GRASP Hybrid GRASP/VND for BPP Constructive Phase: Step 2 (0) Initialization • α parameter: • Responsible for the random selection of the boxes to pack. • The authors used the reactive-GRASP principle to determine α: • α is initially chosen randomly from the discrete set {0.1, 0.2,...,0.9}. • After some iterations the probability distribution of αis adjusted taking into account the relative quality of the solutions. • This incorporates a learning mechanism in GRASP, which is memory less in its constructive phase. (1) Choose Maximal Space – S* (2) Choose Boxes to Pack FEUP | PDEEC | Decision Support (3) Update List S

  23. Metaheuristics: GRASP Hybrid GRASP/VND for BPP Constructive Phase: Step 3 (0) Initialization • Update List S: • Update S: • If the box intercepts other maximal space. • If the box does not fit exactly in the space S*. • If the unpacked boxes are larger than the remaining maximal spaces. • Update B: • The placed boxes must be removed from the list. • If and then return to step 1: • Otherwise the constructive phase ends with n bins used. • For the next GRASP iteration, the target of the constructive phase is n-1. (1) Choose Maximal Space – S* (2) Choose Boxes to Pack FEUP | PDEEC | Decision Support (3) Update List S

  24. Metaheuristics: GRASP • Initial Procedure • Improvement • Procedures • Repacking • Objects • Combined • Procedures Hybrid GRASP/VND for BPP Improvement Phase Order Solution by Non-increasing Volume Procedure 1 Procedure 2 Procedure 3 Procedure 4 Local Search Best Fit Best Volume FEUP | PDEEC | Decision Support All Moves 1→ 4 VND (N1, N2, N3, N4) VNDseq (N1, N2, N3, N4)

  25. Metaheuristics: GRASP Improvement Phase • Procedure 1: • Eliminates the last k% items in the solution. • k is a random value between 30 and 90%. • Procedure 2: • Removes the last k% pieces packed from each bin, whose occupied volume is less than the average. • Procedure 3: • Select all the bins in which the occupied volume is lower than the overall average occupancy. • Split each bin into two parts randomly selecting how to cut: vertically or horizontally. • Choose randomly one side of the bin: [up/down left/right]: • Remove all boxes that are mostly in the selected side. Improvement Procedures FEUP | PDEEC | Decision Support

  26. Metaheuristics: GRASP Improvement Phase Improvement Procedures • Procedure 4 – Local Search: • Neighborhood: consists in all pairs of bins in which at least one have an occupancy bellow the average. • Search method: • For each pair in the neighborhood unpack all the items. • The first box to pack must be one of the boxes that was not packed (initial solution). • The moves are chosen considering one of the following criteria: • First Improve: select the first pair which improves the current solution. • Best Improve: Examine all the neighborhood and select the best improvement. • The move improves the solution if the total volume of boxes packed into the bins increases. FEUP | PDEEC | Decision Support

  27. Metaheuristics: GRASP Improvement Phase Combined Strategies • All moves: • The improvement procedures are applied: 1→ 4. • VND (Variable Neighborhood Descent): • Neighborhood (N1 to N4): generated by the four improvement procedures. • Start with the solution given by the constructive heuristic (x): • set p ← 1 • while p <= 4 • Exploration of the neighborhood to find the best x’ in the neighbor Np(x). • Move if x’ is better then x. Return to p ← 1. Otherwise set p ← p +1. • VNDseq (Sequential Variable Neighborhood Descent): • Similar to VND but: • In step 2.a) instead of setting p=1, the algorithm proceeds sequentially to the (p+1)th. FEUP | PDEEC | Decision Support

  28. Metaheuristics: GRASP Computational Results • Computation and Data Sets: • The algorithm was coded in C++ and run on a standard laptop. • 2D and 3D data sets were used in order to test the proposed algorithm: • For the 3D simulation, a standard benchmark generated by Martello et al was used. This data set was also used by Faroe et al. • For the 2D simulation, more than one data set was used. To compare with already published results of other heuristics. FEUP | PDEEC | Decision Support

  29. Metaheuristics: GRASP Computational Results Experiments • Choosing the best strategy: • Constructive Phase: • Number of corners to consider: • [1, 2, 3, 4] for the 2D BPP. • [1, 2, 3, 4, 8] for the 3D BPP. • Deterministic or random constructive phase. • Improvement Phase: • Compare the performance of the four improvement methods individually. • Compare the performance of the combined improvement methods. • Comparison with other algorithms to solve 2D and 3D BPP. FEUP | PDEEC | Decision Support

  30. Metaheuristics: GRASP Computational Results Constructive The results obtained for each dataset showed that using the randomizedconstructive algorithm proved to be better than the determinist. • Applying the constructive phase with randomization: • Each improvement methods were tested with both objective functions: Best Volume and Best Fit. • At each iteration one of the four strategies to choose the corner is randomly selected. • The algorithm run for 5000 iterations. FEUP | PDEEC | Decision Support

  31. Metaheuristics: GRASP Computational Results Improvements • 2D: • Method 4 was the best independently of the objective function used. • Method 2 with best fit approach also produced good results. • 3D • For both objective functions, method 4 performed better followed by method 2. FEUP | PDEEC | Decision Support

  32. Metaheuristics: GRASP Computational Results Combined Improvement • The diversification algorithm is applied after 500 iterations without improvementin the following 100 iterations . • The analysis of the three strategies for the combination of improvements shows: • 2D/3D: • “All moves” is the worst strategy. • Small difference among the others. FEUP | PDEEC | Decision Support • The method chosen for the final implementation was: • VNDseq + Diversification.

  33. Metaheuristics: GRASP Comparison with other Algorithms • 2D statistical analysis show: • GRASPis betterthan TS3 and it favors GRASP against the others algorithms. • 3D statistical analysis show: • GRASPand SCH are significantly better than TS3, GLS, and HBP. • GRASPand SCHachieved optimal solutions: • Indicates a good performance of the proposed algorithm. FEUP | PDEEC | Decision Support

  34. Metaheuristics: GRASP Conclusions • Regarding the GRASP algorithm: • Grasp consists in a constructive phase followed by a local search procedure. • It is easy to implement  due to the reduced number of parameters • Parameter α defines the randomness of Grasp • Basic version of GRASP does not have memory. It can be combined with other procedures such as reactive  GRASP that adjust  α. • Hybrid GRASP/VND for BPP: • Combination of GRASP with VND allowed  the algorithm to obtain high quality solutions. • The quality of the solutions were similar to well known algorithms to solve 2D and 3D BPP. • Reinforces the fact that GRASP is flexible and could be adapted easily to accommodate constraints or other conditions. FEUP | PDEEC | Decision Support

  35. Metaheuristics: GRASP Thank you for your attention!!! FEUP | PDEEC | Decision Support Questions?

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