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A GRASP Heuristic to the Extended Car Sequencing Problem

A GRASP Heuristic to the Extended Car Sequencing Problem. Lucas Rizzo. Sebastián Urrutia. Federal University of Minas Gerais. { surrutia , rizzo }@dcc.ufmg.br. Outline. The Extended Car Sequencing Problem Heuristic Approach Results Conclusions. Introduction.

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A GRASP Heuristic to the Extended Car Sequencing Problem

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  1. A GRASP Heuristic to the Extended Car Sequencing Problem Lucas Rizzo SebastiánUrrutia Federal Universityof Minas Gerais {surrutia,rizzo}@dcc.ufmg.br

  2. Outline • TheExtendedCarSequencingProblem • Heuristic Approach • Results • Conclusions

  3. Introduction • The car sequencing problem (CSP) involves the schedule of different kinds of cars along an assembly line • A group of options need to be installed in each car along this assembly line • Each type of option is installed by a station • A limited number of cars can be simultaneously handled by each station

  4. The Car Sequencing Problem Definition • The CSP is defined by the set ofentries • C = {c1, c2, ... , cn} • V = {v1, v2, ..., vk} • Vectorspandqassociated to eachoptionvi, suchthat in everysequenceofq(vi) carsatmostp(vi) ofthenrequirevi • A functiont: C x V → {0,1}, that for eachcarci andeachoptionvjreturns 1 ifvjmustbeinstalled in ciand 0 otherwise.

  5. The Car Sequencing Problem Definition • The CSP is a NP-Complete [2] feasibilityproblem • It consists in finding an arrangement of the cars in a sequence such that the number of upper violations in the objective function is zero where

  6. The Extended Car Sequencing Problem • This work deals with the extended version of CSP (xCSP) [1] • Now is alsoconsideredlower limitations on the number of cars that a station can simultaneously deal with • This new limitations aim a more homogeneous distribution of work in the option installation stations

  7. The Extended Car Sequencing Problem • The xCSP is defined by thesame set (C,V,t,p,q) ofentriesshown for the CSP • Differentlywehaveonly • Vectorsrandsassociated to eachoptionvi, suchthat in everysequenceofs(vi)carsatleastr(vi) ofthenrequirevi

  8. The Extended Car Sequencing Problem • The xCSPisanoptimizationproblemwithanobjectivefunctionthatconsidersbothupperandlowerviolations • Upperviolations are consideredinfinite times more importantthanthelowerones

  9. The Extended Car Sequencing Problem • For upperviolations, subsequences with size smaller than q(vi) are considered at the beginning of the assembly line • The reason for this is that in the optimization problem, a subsequence that exceeds the limit of a station by a larger number of cars is considered worse than a subsequence that exceeds by a smaller number of cars

  10. Example • Suppose a optionvasuchthatp(va) = 2 andq(va) = 4. Alsosupposer(va) = 1 ands(va) = 4. X _ _ _ _ XXX _ XXX _ _ _ X X _ _ _ _ _ XX AssemblyLine 1 AssemblyLine 2 AssemblyLine 3

  11. Example • Suppose a optionvasuchthatp(va) = 2 andq(va) = 4. Alsosupposer(va) = 1 ands(va) = 4. X _ _ _ _ XXX _ XXX _ _ _ X X _ _ _ _ _ XX AssemblyLine 1 AssemblyLine 2 AssemblyLine 3

  12. Example • Suppose a optionvasuchthatp(va) = 2 andq(va) = 4. Alsosupposer(va) = 1 ands(va) = 4. X _ _ _ _ XXX _ XXX _ _ _ X X _ _ _ _ _ XX AssemblyLine 1 AssemblyLine 2 AssemblyLine 3

  13. Example • Suppose a optionvasuchthatp(va) = 2 andq(va) = 4. Alsosupposer(va) = 1 ands(va) = 4. X _ _ _ _ XXX _ XXX _ _ _ X X _ _ _ _ _ XX AssemblyLine 1 AssemblyLine 2 AssemblyLine 3

  14. Example • Suppose a optionvasuchthatp(va) = 2 andq(va) = 4. Alsosupposer(va) = 1 ands(va) = 4. X _ _ _ _ XXX _ XXX _ _ _ X X _ _ _ _ _ XX AssemblyLine 1 AssemblyLine 2 AssemblyLine 3

  15. Example • Suppose a optionvasuchthatp(va) = 2 andq(va) = 4. Alsosupposer(va) = 1 ands(va) = 4. X _ _ _ _ XXX _ XXX _ _ _ X X _ _ _ _ _ XX AssemblyLine 1 AssemblyLine 2 AssemblyLine 3

  16. Example • Suppose a optionvasuchthatp(va) = 2 andq(va) = 4. Alsosupposer(va) = 1 ands(va) = 4. X _ _ _ _ XXX _ XXX _ _ _ X X _ _ _ _ _ XX AssemblyLine 1 AssemblyLine 2 AssemblyLine 3

  17. Heuristic Approach • To heuristically solve the problem we implemented an algorithm based on the GRASP metaheuristic 1. Final Solution ← 2. For k=1 to MAX ITERATIONS 3. Solution ← RandomizedGreedyAlgorithm 4. Solution ← Local Search (Solution) 5. IfSolutionbetterthanFinal SolutionThen 6. Final Solution ← Solution 7. EndIf 8. End For 9. ReturnFinal Solution

  18. Heuristic Approach • To heuristically solve the problem we implemented an algorithm based on the GRASP metaheuristic 1. Final Solution ← 2. For k=1 to MAX ITERATIONS 3. Solution ← RandomizedGreedyAlgorithm 4. Solution ← Local Search (Solution) 5. IfSolutionbetterthanFinal SolutionThen 6. Final Solution ← Solution 7. EndIf 8. End For 9. ReturnFinal Solution

  19. Heuristic Approach • To heuristically solve the problem we implemented an algorithm based on the GRASP metaheuristic 1. Final Solution ← 2. For k=1 to MAX ITERATIONS 3. Solution ← RandomizedGreedyAlgorithm 4. Solution ← Local Search (Solution) 5. IfSolutionbetterthanFinal SolutionThen 6. Final Solution ← Solution 7. EndIf 8. End For 9. ReturnFinal Solution

  20. RandomizedGreedyAlgorithm • Wecan build a initial solution byiterativelychoosingthecarthat minimize thecostofthe new sequence • Thiscar is added to thelastpositionofthesequence • Ties in this criterion are not only possible but frequent, for this case we present tie-breaking criterion

  21. Tie-Breaking Criterion • Choosethecarthat minimizes thelowerviolations • Iftiespersist, create a restricted candidate listbyrandomlychoosingtwoofthecarstied. • Choosethecarthathasthelargerdynamichighestutilization rate [3].

  22. Tie-Breaking Criteria • The dynamichighestutilization (DHU) for anoptionvi is theratiobetweenthenumberofcarsrequiringtheoptionvi in thecurrentsequence, andthenumberofcarsrequiringtheoptionthatcouldbe schedule in thecurrentsequencewithoutviolations • Thecarwiththelargerdynamichighestutilization rate is theonewhichhastheoptionswiththehighestDHU.

  23. Local Search • Onlythe swap operatorwasimplemented. It consists in exchangingtwogivencars in a givensequence • All feasible solutions are reachable from any other solution by a finite number of swap movements. Therefore the neighborhood is connected. • Because of this property, the swap operator when well implemented allow us to construct an efficient local search

  24. Local Search • To compare neighborssolutionsweconsiderfirstofallupperviolationsandonlyifthey are tied in thiscriterium, we compute lowerviolations. This is becauseupperviolationshaveinfinite times more importancethanlowerviolations • Furthermorethisstrategyreducestheevaluation time, resulting in a more efficient local search.

  25. Swap Operator • Givencarsciandcj, the swap willonlychangethevalueoftheobjectivefunctionifexistanoptionvi Vsuchthatt(ci,vi) ≠ t(cj,vi) • The costfunctionmayonlyvary in subsequencesaffectedbythemovement • Auxiliaryvectors for eachoptionvistoresthenumberofcarsthathavevi in theirsubsequences. Thisallow us, in O(1), to decide when a carwillgenerate a violationifplaced in a sequence.

  26. Swap Operator • Thatimplies a complexityof O(|V|*qmax + |V|*smax) for a possibleexchange • Withthislowcomplexityallpossibleexchangescanbetested in a veryefficientway.

  27. Results • A total of 74 instances, provided by CSPlib [7], were used in the experiments • Four instances have 100 cars while the other 70 instances have 200 cars. Each instance considers five possible options to be or not installed in each car

  28. Results • Instances are divided in eight groups according to the percentage of use of each option. These vary between 60% and 90%. The first group contains four “classical” instances while the rest of the groups contain 10 instances each. • In order to adapt the instances to the xCSP we build the restrictions of minimum as s(vi) = q(vi) and r(vi) = p(vi) – 1 • Results are compared with [1]. As far as we know this is the only work dealing with the xCSP.

  29. Results • Both instances and machine are identical to the ones used in [1] • The average time to each GRASP iteration was 0.3 seconds for the proposed algorithm while in [1] an average time of 1 second per GRASP iteration is reported

  30. Results

  31. Results

  32. Results

  33. Results

  34. Results

  35. Conclusions • A heuristicbasedonthe GRASP metaheuristicwasimplemented for theExtendedCarSequencingProblem • Thisheuristicpresents a randomizedgreedyalgorithmand a simplebutefficient local search • Resultswerecomparedwiththeliteratureusingthesameinstancesand machine

  36. Conclusions • A careful implementation of the local search procedure in terms of the cost evaluation of neighbors solutions allowed us to decrease the time of execution of every GRASP iteration in comparison with [1] • In consequence our algorithm found much better solutions in the same execution time

  37. Conclusions • Further research includes the usage of other neighborhoods in the local search procedure and the developing of new heuristics based on other metaheuristics • We intend to perform a fine tuning of the algorithm, for example for the size of the restricted candidate list. Preliminary experiments show that better results may be obtained.

  38. A GRASP Heuristic to the Extended Car Sequencing Problem Lucas Rizzo SebastiánUrrutia Federal Universityof Minas Gerais {surrutia,rizzo}@dcc.ufmg.br

  39. References • [1] Bautista, J., Pereira, J. and Adenso-Díaz, B. A GRASP approach for the extended car sequencing problem, Journal of Scheduling, v.11, p.3-16, 2008. • [2] Gent, I. Two results on car-sequencing problems (Technical report APES-02-1998). • [3] Gottlieb, J., Puchta, M. and Solnon, C. A study of greedy, local search and ant colony optimization approaches for car sequencing problems. In Lecture notes in computer science: Vol. 2611, 246-257, 2003. • [4] Feo T. A. and Resende M. G. C. A probabilistic heuristic for a computationally difficult set covering problem. Operations Research Letters, 8, 67-71, 1989

  40. References • [5] Regin, J. C. and Puget, J. F. A filtering algorithm for global sequencing constraints. In Lecture notes in computer science: Vol. 1330, 32-46, 1997. • [6] Resende, M.G.C. and Ribeiro, C.C. Greedy randomized adaptive search procedures: Advances, hybridizations, and applications, Handbook of Metaheuristics (M. Gendreau e J.- Y. Potvin, editors), 283-319, Springer, 2nd edition, 2010. • [7] http://www.csplib.org/

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