1 / 35

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

Télécharger la présentation

FEUP | PDEEC | Decision Support

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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?

More Related