Your SlideShare is downloading. ×
icaps 2009 Talk
Upcoming SlideShare
Loading in...5
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Text the download link to your phone
Standard text messaging rates apply

icaps 2009 Talk

281

Published on

Talk given at ICAPS 2009 (Thessaloniki, Greece) on local search for fuzzy job shop.

Talk given at ICAPS 2009 (Thessaloniki, Greece) on local search for fuzzy job shop.

Published in: Education
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total Views
281
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
Downloads
0
Comments
0
Likes
0
Embeds 0
No embeds

Report content
Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
No notes for slide

Transcript

  • 1. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Improved Local Search for Job Shop Scheduling with Uncertain Durations Inés González-Rodríguez1 Camino R. Vela2 Jorge Puente2 Alejandro Hernández-Arauzo2 1 Dept. of Mathematics, Statistics and Computing, University of Cantabria (Spain) 2 Dept. of Computer Science and A.I. Centre, University of Oviedo (Spain) ICAPS 2009.
  • 2. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Starting Point Previous work on job shop scheduling with uncertain durations: using local search very competitive makespan values :) but with a high computational load :( We would like to. . . keep makespan quality while improving efficiency.
  • 3. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Outline 1 The Fuzzy Job Shop Scheduling Problem 2 Improved Local Search Existing Approach Makespan Calculation for Neighbours Makespan Lower Bound Reduced Neighbourhood 3 Experimental Results 4 Conclusions
  • 4. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Job Shop Scheduling Problem J1 J2 J3
  • 5. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Job Shop Scheduling Problem J1 J2 J3
  • 6. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Job Shop Scheduling Problem J1 J2 J3
  • 7. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Job Shop Scheduling Problem J1 J2 J3
  • 8. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Motivation For Uncertainty Classical JSP assumes complete knowledge! about processing times. But. . . what can we do if we only have incomplete knowledge? (Un)fortunately, THE REAL WORLD IS FULL OF UNCERTAINTY !!!
  • 9. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Triangular Fuzzy Numbers Assume we only know: an interval of possible values for the duration [a1 , a3 ], a1 a3
  • 10. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Triangular Fuzzy Numbers Assume we only know: an interval of possible values for the duration [a1 , a3 ], the most likely duration a2 in this interval. a1 a2 a3
  • 11. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Triangular Fuzzy Numbers This knowledge is represented by a triangular fuzzy number, TFN, A = (a1 , a2 , a3 ): a1 a2 a3
  • 12. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Working With Triangular Fuzzy Numbers addition: A + B = (a1 + b1 , a2 + b2 , a3 + b3 ) maximum: max(A, B) ≈ (max{a1 , b1 }, max{a2 , b2 }, max{a3 , b3 }) expected value: 1 1 E[A] = (a + 2a2 + a3 ) 4
  • 13. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Solution Graph With TFNs A solution is represented by an acyclic graph: (3, 4, 7) 1 2 (3, 4, 7) (1, 2, 3) (4, 5, 6) (2, 3, 4) 0 3 4 7 (4, 5, 6) (1, 2, 6) (2, 3, 4) (1, 2, 4) 5 6 (1, 2, 6) or by a task processing order: π =(1 3 5 4 2 6).
  • 14. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Fuzzy? Schedule What is fuzzy in this schedule? Task starting and completion times and the makespan are fuzzy. But, the task processing order is precise.
  • 15. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Optimal Makespan We want to minimise a fuzzy makespan. But. . . Problem It is not clear when a TFN is less than another:
  • 16. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Optimal Makespan We want to minimise a fuzzy makespan. But. . . Problem It is not clear when a TFN is less than another: Solution The expected value defines a total ordering; A ≤E B iff E[A] ≤ E[B]:
  • 17. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Optimal Makespan We want to minimise a fuzzy makespan. But. . . Problem It is not clear when a TFN is less than another: Solution The expected value defines a total ordering; A ≤E B iff E[A] ≤ E[B]: Objective for FJSP Minimise the expected makespan E[Cmax ]
  • 18. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Criticality With Exact Durations 3 1 2 3 1 4 2 0 3 4 7 4 1 2 1 5 6 1 Critical path: (3, 4, 6); critical arcs: (4, 6); critical blocks: (3), (4, 6).
  • 19. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Criticality With TFNs: Parallel Graphs Solution Graph G: (3, 4, 7) 1 2 (3, 4, 7) (1, 2, 3) (4, 5, 6) (2, 3, 4) 0 3 4 7 (4, 5, 6) (1, 2, 6) (2, 3, 4) (1, 2, 4) 5 6 (1, 2, 6)
  • 20. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Criticality With TFNs: Parallel Graphs Parallel Solution Graph G1 : (3,4,7) 1 2 (3,4,7) (1,2,3) (4,5,6) (2,3,4) 0 3 4 7 (4,5,6) (1,2,6) (2,3,4) (1,2,4) 5 6 (1,2,6) 1 Critical path: 3,4,6; Cmax = 7
  • 21. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Criticality With TFNs: Parallel Graphs Solution Graph G: (3, 4, 7) 1 2 (3, 4, 7) (1, 2, 3) (4, 5, 6) (2, 3, 4) 0 3 4 7 (4, 5, 6) (1, 2, 6) (2, 3, 4) (1, 2, 4) 5 6 (1, 2, 6) Critical paths: [3,4,6], [3,5,6]; Cmax = (7, 10, 16)
  • 22. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Previous Neighbourhood Idea: obtain neighbours by reversing every resource critical arc. Definition Take a task processing order π a resource arc v = (x, y ) in the associated solution graph π(v ) , the order obtained from π after reversing arc v . The neighbourhood structure obtained from π is: H(π) = {π(v ) : v ∈ R(π) is critical}.
  • 23. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Previous Neighbourhood Solution Graph G(π), π=(1 3 5 4 2 6) (3, 4, 7) 1 2 (3, 4, 7) (1, 2, 3) (4, 5, 6) (2, 3, 4) 0 3 4 7 (4, 5, 6) (1, 2, 6) (2, 3, 4) (1, 2, 4) 5 6 (1, 2, 6) Critical paths: [3,4,6], [3,5,6]; Cmax = (7, 10, 16)
  • 24. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Previous Neighbourhood Neighbour Solution Graph G(σ), σ=(1 5 3 4 2 6) (3, 4, 7) 1 2 (3, 4, 7) (1, 2, 3) (4, 5, 6) (4, 5, 6) (2, 3, 4) 0 3 4 7 (2, 3, 4) (1, 2, 4) (1, 2, 6) 5 6 (1, 2, 6)
  • 25. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Previous Neighbourhood: Properties Good News :) All neighbours in H are feasible solutions. The reversal of a non-critical arc can never reduce makespan.
  • 26. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Previous Neighbourhood: Properties Good News :) All neighbours in H are feasible solutions. The reversal of a non-critical arc can never reduce makespan. Bad News :( In the fuzzy case, the neighbourhood has a considerable size, with the consequent increase in computational cost.
  • 27. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Proposed Improvements How can we improve the neighbourhood structure? Ideas: 1 More efficient makespan calculation for neighbours. 2 Use inexpensive makespan lower bound to discard neighbours. 3 Reduce neighbourhood.
  • 28. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Neighbour’s Makespan Calculation Idea: avoid unnecesary calculations using available information Makespan (and more) known for initial solution π; Neighbour σ = πv differs from π “only” in the reversal of an arc.
  • 29. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Heads and Tails Definition For a solution graph G(π) and a task x with processing time px : The head of x is the starting time of x, denoted rx : rx = max{rPJx + pPJx , rPνx + pPνx } The tail of x is the time lag between x’s completion time and the completion time of the last task, denoted qx : qx = max{qSJx + pSJx , qSνx + pSνx }
  • 30. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Heads and Tails in the Neighbour Pνx π Pνx σ = π(x,y ) PJx x SJx PJx x SJx PJy y SJy PJy y SJy Sνy Sνy To calculate heads (tails) in σ, we need only re-calculate the heads (tails) of tasks from x onwards (backwards) in G(σ).
  • 31. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Neighbour’s Makespan Lower Bound Idea: We are only interested in neighbours with improving makespan. Use makespan lower bound to discard non-improving neighbours. How can we calculate an inexpensive lower bound?
  • 32. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Makespan After a Move Notation for a processing order π and tasks x and y : D[P] a TFN where D i [P] is the length of the longest path in a set of paths P of Gi , i = 1, 2, 3. Pπ (x ∨ y ): set of all paths in the solution graph G(π) containing x or y ; Pπ (x ∧ y ): set of all paths in G(π) containing both x and y ; Pπ (¬x): set of all paths in G(π) not containing x. Proposition For σ = π(v ) , where v = (x, y ) is an arc in G(π) Cmax (σ) = max{D[Pπ (¬x)], D[Pσ (x ∨ y )]}
  • 33. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Lower Bound for Makespan After a Move Consequence: D[Pσ (x ∨ y )] ≤E Cmax (σ). Using heads and tails: Corollary Let rx (qx ) be the head (tail) of x in σ = π(x,y ) . The TFN LBσ = max{rx + px + qx , ry + py + qy } provides a lower bound for Cmax (σ). If Cmax (π) ≤E LBσ , we already know the neighbour is not improving.
  • 34. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Idea to Reduce Neighbourhood In the crisp case, reversing arcs inside a critical block does not improve the makespan. Can we prove the same for the fuzzy job shop?
  • 35. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Reduced Neighbourhood Definition If V(π) is the set of all arcs in the extreme of a critical block, the reduced neighbourhood structure for π is: HR (π) = {π(v ) : v ∈ V(π)}. Properties HR (π) ⊂ H(π) HR (π) contains only feasible schedules
  • 36. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Reduced Neighbourhood The discarded neighbours from H(π) − HR (π) never improve the makespan. Theorem For a feasible processing order π, ∀σ ∈ H(π) − HR (π), E[Cmax (π)] ≤ E[Cmax (σ)]
  • 37. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Reduced Neighbourhood Sufficient condition for optimality: Proposition If the neighbourhood HR (π) = ∅, then π is an optimal processing order. If the local search procedure stops because the neighbourhood is empty, it has found a global optimum.
  • 38. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Experiment Setup Goal Evaluate empirically the contribution of our proposals to improving local search efficiency. Problem Instances 1 Take 12 crisp hard job shop benchmark problems; 2 Generate 10 fuzzy versions of each instance (durations: TFNs with a2 = original value). Total of 120 fuzzy job shop instances.
  • 39. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Experiment Setup Solving Method Based on memetic algorithm GVPV08 proposed in ICAPS 2008 [2] combining: GA: Genetic algorithm LS: Hill-climbing with H. We plug 3 new proposals in the local search.
  • 40. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Experiment Setup GVPV08 compared favourably with state-of-the-art methods (SA [1] and GA [3]) in terms of makespan. New improvements do not affect makespan, only computational load of LS. We know results will be “good” re. makespan; our aim is to evaluate efficiency improvement.
  • 41. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Scheduling Neighbours With Heads and Tails Problem Size GVPV08 (s) MA (s) Red% FT10 10 × 10 801.2 588.2 26.59% FT20 20 × 5 1693.9 682.1 59.73% La21 15 × 10 1769.4 1072.8 39.37% La24 15 × 10 1562.4 950.1 39.19% La25 15 × 10 1722.8 993.7 42.32% La27 20 × 10 4137.8 2242.8 45.80% La29 20 × 10 3936.0 2071.7 47.37% La38 15 × 15 3037,6 2556.7 15.83% La40 15 × 15 3220.4 2652.2 17.64% ABZ7 20 × 15 7396.1 5294.7 28.41% ABZ8 20 × 15 8098.5 5780.9 28.62% ABZ9 20 × 15 7308.0 5652.1 22.66% Average CPU time reduction of 34.5% using MA
  • 42. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Makespan Estimate And Reduced Neighbourhood Incorporate proposed improvements to MA (using heads and tails): 1 MA(R)= MA using HR instead of H. 2 MA(LB) = MA using LB to estimate makespan. 3 MA(LB+R) = MA using both HR and LB. We compare: number of evaluated neighbours, CPU time in seconds (averaged across 30 executions and 10 fuzzy instances of each problem).
  • 43. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Number Of Evaluated Neighbours Problem MA MA(R) MA(LB) MA(R+LB) FT10 2.83E+07 1.44E+07 6.25E+06 3.61E+06 FT20 6.78E+07 1.14E+07 2.25E+07 5.16E+06 La21 4.46E+07 1.93E+07 1.07E+07 5.76E+06 La24 3.87E+07 2.04E+07 9.00E+06 5.59E+06 La25 4.28E+07 1.78E+07 1.03E+07 5.75E+06 La27 8.37E+07 2.77E+07 2.53E+07 9.79E+06 La29 7.85E+07 2.44E+07 2.40E+07 8.90E+06 La38 5.16E+07 3.10E+07 1.11E+07 8.02E+06 La40 5.42E+07 3.03E+07 1.16E+07 8.42E+06 ABZ7 9.74E+07 4.07E+07 2.57E+07 1.30E+07 ABZ8 1.08E+08 4.60E+07 2.87E+07 1.48E+07 ABZ9 9.65E+07 4.43E+07 2.42E+07 1.33E+07 Av. Red. – 57% 75% 87%
  • 44. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References CPU Time In Seconds Problem MA MA(R) MA(LB) MA(R+LB) FT10 588.2 372.90 179.3 166.5 FT20 682.1 283.1 295.4 211.0 La21 1072.8 610.9 327.8 292.2 La24 950.1 636.4 304.5 284.0 La25 993.7 555.3 322.4 289.5 La27 2242.8 1072.5 641.9 511.6 La29 2071.7 962.8 609.8 485.8 La38 2556.7 1732.8 546.7 514.0 La40 2652.2 1643.2 565.1 531.5 ABZ7 5294.7 2671.0 1073.8 898.4 ABZ8 5780.9 2936.5 1168.0 975.2 ABZ9 5652.1 3107.7 1078.5 925.8 Av. Red. – 45% 73% 76%
  • 45. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Conclusions Previous work: good makespan but high computational cost. Proposed improvements: 1 Simple but effective neighbour scheduling algorithm. 2 Makespan estimate: evaluate only 25% of neighbours. 3 Reduced neighbourhood structure: < 1/2 of original neighbourhood. Combined Use Evaluate 13% of original neighbours in 23% of CPU time!!!
  • 46. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References Future Work Better neighbourhood structures to improve makespan. Alternative meta-heuristics (taboo search, PSO,. . . ). Alternative uncertainty models.
  • 47. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References References I Fortemps, P. (1997) Jobshop scheduling with imprecise durations: a fuzzy approach. IEEE Transactions of Fuzzy Systems 7:557–569. González Rodríguez, I.; Vela, C. R.; Puente, J.; and Varela, R. (2008) A new local search for the job shop problem with uncertain durations. In Proc. of ICAPS08, 124–131. Sidney: AAAI Press Sakawa, M., and Kubota, R. (2000) Fuzzy programming for multiobjective job shop scheduling with fuzzy processing time and fuzzy duedate through genetic algorithms. European Journal of Operational Research 120:393–407.
  • 48. The Fuzzy Job Shop Scheduling Problem Improved Local Search Experimental Results Conclusions References References II Taillard, E. D. (1994) Parallel taboo search techniques for the job shop scheduling problem. ORSA Journal on Computing 6(2):108–117. Van Laarhoven, P.; Aarts, E.; and Lenstra, K. (1992) Job shop scheduling by simulated annealing. Operations Research 40:113–125.

×