The two-machine flowshop total completion time problem: A branch-and-bound based on network-flow formulation

Transcription

1 The two-machine flowshop total completion time problem: A branch-and-bound based on network-flow formulation Boris Detienne 1, Ruslan Sadykov 1, Shunji Tanaka 2 1 : Team Inria RealOpt, University of Bordeaux, France 2 : Department of Electrical Engineering, Kyoto University, Japan Multidisciplinary International Scheduling conference: Theory and Application MISTA 2015 August 25th, 2015 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 1 / 43

2 Outline 1 Introduction Problem description Literature Contribution 2 Lower bounds Network flow formulation Extended network flow formulation 3 Branch-and-bound 4 Numerical results B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 2 / 43

3 1 Introduction Problem description Literature Contribution 2 Lower bounds 3 Branch-and-bound 4 Numerical results B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 3 / 43

4 Two-machine flow-shop problem F2 ST SI C i Input data: A set I of n jobs composed of 2 operations The first operation is processed on machine 1, the second on machine 2 For all i I, s 2 i is the sequence-independent setup time on machine 2 Assumption: data are integer and deterministic Constraints Each machine can process only one operation at a time Operations of a same job cannot be processed simultaneously Objective Find a schedule that minimizes the sum of the completion times of the jobs on the second machine. B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 4 / 43

5 Example i p 1 i p 2 i s 2 i B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 5 / 43

6 Example i p 1 i p 2 i s 2 i B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 5 / 43

7 Example i p 1 i p 2 i s 2 i B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 5 / 43

8 Example i p 1 i p 2 i s 2 i B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 5 / 43

9 Example i p 1 i p 2 i s 2 i Cost of the schedule: = 40 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 5 / 43

10 Properties of the problem Complexity Strongly NP-hard [Conway et al., 1967] Dominating solutions There is a least one optimal schedule that is: active (operations are performed as soon as possible, no unforced idle time) such that the sequences of the jobs on both machines are the same (permutation schedule) [Conway et al., 1967, Allahverdi et al., 1999] The problem comes to find one optimal sequence of jobs. B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 6 / 43

11 Literature Lower bounds and exact algorithms L.B.: Single machine problems [Ignall and Schrage, 1965], [Ahmadi and Bagchi, 1990], [Della Croce et al., 1996], [Allahverdi, 2000] Branch-and-bound, up to 10, 15 and 30 jobs (p i 20), 20 jobs (p i 100) L.B.: Lagrangian relaxation of precedence constraints [van de Velde, 1990], [Della Croce et al, 2002], [Gharbi et al., 2013] Branch-and-bound, up to 20 and 45 jobs (p i 10) L.B.: linear relaxation of a positional/assignment model [Akkan and Karabati, 2004], [Hoogeven et al., 2006], [Haouari and Kharbeche, 2013], [Gharbi et al., 2013] : 35 jobs (p i 100) L.B.: Lagrangian relaxation of the job cardinality ctr., flow model [Akkan and Karabati, 2004] Branch-and-bound, up to 60 jobs (p i 10), 45 jobs (p i 100) B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 7 / 43

12 Contribution Branch-and-bound based on the network flow model of [Akkan and Karabati, 2004] Improvements Stronger lower bound by using a larger size network Advantages Stronger Lagrangian relaxation bound Allows integration of dominance rules inside the network Disadvantages (Too) high memory and CPU time requirements Reduction of the size of the network using Lagrangian cost variable fixing Extension to sequence-independent setup times B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 8 / 43

13 1 Introduction 2 Lower bounds Network flow formulation Extended network flow formulation 3 Branch-and-bound 4 Numerical results B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 9 / 43

14 Lag-based models [Akkan and Karabati], [Gharbi et al.] Lag variables L c k = C2 [k] C1 : time lag elapsed between the completion of the job in [k] position k on machine 1 and on machine 2 Total completion time B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 10 / 43

15 Lag-based models [Akkan and Karabati], [Gharbi et al.] Lag variables L c k = C2 [k] C1 : time lag elapsed between the completion of the job in [k] position k on machine 1 and on machine 2 Total completion time B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 10 / 43

16 Lag-based models [Akkan and Karabati], [Gharbi et al.] Lag variables L c k = C2 [k] C1 : time lag elapsed between the completion of the job in [k] position k on machine 1 and on machine 2 Total completion time B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 10 / 43

17 Lag-based models [Akkan and Karabati], [Gharbi et al.] Lag variables L c k = C2 [k] C1 : time lag elapsed between the completion of the job in [k] position k on machine 1 and on machine 2 Total completion time B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 10 / 43

18 Lag-based models [Akkan and Karabati], [Gharbi et al.] Lag variables L c k = C2 [k] C1 : time lag elapsed between the completion of the job in [k] position k on machine 1 and on machine 2 Total completion time 3p Lc 1 2p Lc 2 p Lc 3 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 10 / 43

19 Lag-based models [Akkan and Karabati], [Gharbi et al.] Lag variables L c k = C2 [k] C1 : time lag elapsed between the completion of the job in [k] position k on machine 1 and on machine 2 Total completion time 3p Lc 1 2p Lc 2 p Lc 3 n k=1 C2 [k] = n k=1 (n k + 1)p 1 [k] + Lc k B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 11 / 43

20 Lag-based models [Akkan and Karabati], [Gharbi et al.] Lag variables Recursive formula for lag: L c k 0, = max L c k 1 + s2 [k] p1 + p 2 [k] [k] Total completion time 3p Lc 1 2p Lc 2 p Lc 3 n k=1 C2 [k] = n k=1 (n k + 1)p 1 [k] + Lc k B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 11 / 43

21 Network flow formulation [Akkan et Karabati, 2004] Lag-based models The contribution of a job to the objective function only depends on: Its position in the sequence Its lag, which is directly deduced from the lag of the preceding job Structure of the network One node a pair (position, lag) One arc the processing of a job initial node determines the position terminal node determines the lag The cost of an arc is the corresponding contribution to the objective function B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 12 / 43

22 Network flow formulation [Akkan et Karabati, 2004]: G 1 p 1 = (10, 7); p 2 = (7, 3); p 3 = (1, 3) l = 3 cost= = 24 l = 3 l = 3 1 l = 5 cost=6 cost=37 l = 7 l = 5 2 l = 7 2 l = 7 l = 9 l = 9 l = 11 k = 0 k = 1 k = 2 k = 3 k = 4 Shortest path + Each job is processed exactly once B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 13 / 43

23 Network flow formulation [Akkan et Karabati, 2004]: G 1 p 1 = (10, 7); p 2 = (7, 3); p 3 = (1, 3) l = 3 cost= = 24 l = 3 l = 3 1 l = 5 cost=6 cost=37 l = 7 l = 5 2 l = 7 2 l = 7 l = 9 l = 9 l = 11 k = 0 k = 1 k = 2 k = 3 k = 4 Shortest path + Each job is processed once L.B. by Lagrangian relaxation B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 13 / 43

24 Network flow formulation [Akkan et Karabati, 2004]: G 1 Disadvantage: many infeasible paths "weak" lower bound l = 3 l = 3 l = 3 1 l = 5 l = 7 l = 5 l = l = 7 l = 9 l = 9 l = 11 k = 0 k = 1 k = 2 k = 3 k = 4 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 14 / 43

25 Extended network flow formulation: G 2 Structure of the network One node a triplet (position, lag, job) One arc the processing of a job initial node determines the position and the job terminal node determines the lag and the next job The cost of an arc is the corresponding contribution to the objective function B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 15 / 43

26 Extended network G 2 l = 7 l = 7 l = 7 l = 3 J3 l = 3 l = 3 k = 0 k = 1 k = 2 k = 3 k = 4 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 16 / 43

27 Extended network G 2 - Example of reduction l = 7 l = 7 l = 7 l = 3 J3 l = 3 l = 3 k = 0 k = 1 k = 2 k = 3 k = 4 Jobs cannot be processed twice consecutively B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 17 / 43

28 Extended network G 2 - Example of reduction l = 7 l = 7 l = 3 l = 3 l = 3 k = 0 k = 1 k = 2 k = 3 k = 4 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 18 / 43

29 Extended network G 2 - Example of reduction l = 7 l = 7 l = 3 l = 3 l = 3 k = 0 k = 1 k = 2 k = 3 k = 4 If p 1 i + s2 j p 1 j + s2 i, p2 i + s2 i p 2 j + s2 j, and p2 j p 2 i, then i j [Allahverdi, 2000] B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 19 / 43

30 Extended network G 2 - Example of reduction l = 7 l = 3 l = 3 k = 0 k = 1 k = 2 k = 3 k = 4 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 20 / 43

31 Extended network G 2 - Example of reduction Given a position k, a lag l and a sub-sequence σ: f (k, l, σ): cost of scheduling σ at (k, l) L(k, l, σ): lag of the last job of σ scheduled at (k, l) Dominance Sub-sequence σ is dominated at (k, l) by sub-sequence σ if: The set of jobs in σ and σ is the same f (k, l, σ) > f (k, l, σ ) The partial schedule up to the end of σ will be less costly L(k, l, σ) L(k, l, σ ) The partial schedule after σ will not be more costly B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 21 / 43

32 Extended network G 2 - Example of reduction f (k = 1,, σ = (, )) = 37 L(k = 1,, σ = (, )) = 9 l = 7 l = 3 J 4 l = 9 J 4 l = 7 l = 3 f (k = 1,, σ = (, )) = 22 L(k = 1,, σ = (, )) = 7 k = 0 k = 1 k = 2 k = 3 k = 4 Example: σ = 2 allows us to remove some arcs B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 22 / 43

33 Lagrangian cost variable fixing Additional input data An upper bound UB of the optimum is known Principle Assume that one dominant optimal solution satisfies hypothesis h The optimal path goes through a given arc Compute a (Lagrangian) lower bound LB h under h If LB h > UB, then h is not satisfied in any optimal dominant solution The arc can be removed from the graph B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 23 / 43

34 Lagrangian cost variable fixing (1) Removing arcs from the network Hypothesis: the path goes through e [Ibaraki and Nakamura, 1994] Given Lagrangian multipliers π v J i e w J j SP(w, ) SP(0, v) SP(e) = SP(, v) + cost(e) + SP(w, ) If SP(e) j π j > UB, then e is part of no optimal solution. Computing SP(e) for all e E is done in O( E )-time B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 24 / 43

35 Lagrangian cost variable fixing (2) Removing arcs from the network [Detienne et al., 2012] Given Lagrangian multipliers π and a job J i SP i (0, v) v J i e w J j SP i (w, ) e represents J i SP i (a, b) : SP from a to b going through no arc representing J i SP i (e) = SP i (, v) + cost(e) + SP i (w, ) If SP i (e) j π j > UB, then e is part of no optimal solution. Computing SP i (e) for all e E and i I is done in O(n E )-time B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 25 / 43

36 Lagrangian cost variable fixing (2) Removing arcs from the network [Detienne et al., 2012] Given Lagrangian multipliers π and a job J i SP i (0, v) SP i (0, v) v J k e w J j SP i (w, ) SP i (w, ) e does not represent J i SP i (a, b) : SP from a to b going through exactly one arc representing J i SP i (e) = min{sp i (, v) + cost(e) + SP i (w, ), SP i (, v) + cost(e) + SP i (w, )} If SP i (e) j π j > UB, then e is part of no optimal solution. Computing SP i (e) for all e E and i I is done in O(n E )-time B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 26 / 43

37 1 Introduction 2 Lower bounds 3 Branch-and-bound 4 Numerical results B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 27 / 43

38 Preprocessing Initial upper bound A good feasible solution is obtained by a local search procedure Dynasearch [Tanaka, 2010] Pre-computation of lower bounds Construction of network G 1 Lagrangian cost variable fixing (subgradient procedure) Construction of the extended network G 2 from G 1 Lagrangian cost variable fixing (subgradient procedure) For the best Lagrangian multipliers, SP i (v, ) and SP i (v, ) are stored for each i I and v V B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 28 / 43

39 Branching scheme Solution space explored Feasible sequences of jobs Feasible constrained paths in G 2 Depth-First Search, starting from start node Branching Current sequence σ ( path) is extended with job J i iff: There is a corresponding arc in G 2 All predecessors of J i are in σ and J i is not in σ The sequence of the last 5 jobs obtained would not be dominated by one of its permutations The sequence is not dominated by a previously explored sequence (Memory Dominance Rule, [Baptiste et al., 2004], [T Kindt et al., 2004], [Kao et al., 2008]) B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 29 / 43

40 Lower bound for σ path ending at v in G 2 Lower bound coming from jobs not sequenced yet LB 1 = cost(σ) + max SP i (v, ) π i i/ σ i/ σ Lower bound coming from sequenced jobs LB 2 = cost(σ) + max SP i(v, ) i σ Computing max{lb 1, LB 2 } is done in (n)-time. i/ σ π i B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 30 / 43

41 Tentative upper bound Weakness of the approach If the initial upper bound is too large, variable fixing is not efficient. Overall procedure 1 Build and filter G 2 using the initial upper bound (dynasearch) 2 If G 2 is sufficiently small, run the Branch-and-Bound, STOP 3 Build and filter G 2 using a tentative upper bound 4 Run the Branch-and-Bound 5 If a feasible solution is found, it is optimal, STOP 6 Otherwise, increase the tentative upper bound and go to 3 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 31 / 43

42 1 Introduction 2 Lower bounds 3 Branch-and-bound 4 Numerical results B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 32 / 43

43 No setup times - F 2 C i Coded in C++ (MS VS 2012) MS Windows 8 laptop with 16GB RAM and Intel Core Instances Randomly generated [Akkan and Karabati, 2004], [Haouari and Kharbeche 2013] Up to 100 jobs, p 1 i and p 2 i are drawn from [1, 100] Results for 100 job instances (40 instances) Avg. time: 216 s., Max. time: 602 s. Tentative upper bound is useless Root gap Variable fixing reduces the number of arcs by a factor 5 Avg.: 166K nodes, 1.4M arcs, Max.: 239K nodes, 2.9M arcs B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 33 / 43

44 Sequence-independent setup times - F 2 ST SI C i Instances Subset of the testbed of [Gharbi et al., 2013] Up to 100 jobs, p 1 i, p2 i and s 2 i are drawn from [1, 100] Results for 100 job instances (200 instances) Avg. time: 935 s., Max. time: 6443 s. Tentative upper bound is critical Reduces the number of arcs from 18.5M to 2.2M at the root node Lagrangian Variable fixing + Tentative upper bound reduce the number of arcs by a factor 17 Avg.: 237K nodes, 2.2M arcs, Max.: 440K nodes, 4.9M arcs B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 34 / 43

45 Conclusion Contributions New lower bound for F2 C i and F2 ST SI C i Efficient management of the size of the extended network Dominance rules are embedded in the structure of the network The lower bound is used with success in an exact solving approach All 100-job instances of our test bed are solved in less than two hours 98% are solved in less than one hour Future directions Use Successive Sublimation Dynamic Programming instead of Branch-and-Bound Adapt for other min-sum objective functions? Adapt for more than two machines permutation flowshop? B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 35 / 43

46 Thank you for your attention B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 36 / 43

47 Network flow formulation [Akkan et Karabati, 2004]: G 1 V 1, A 1 : sets of nodes and arcs x v,w,j : amount of flow on the arc representing j between nodes v and w min c v,w,j x v,w,j (v,w,j) A 1 s.t. x v,w,j = x w,v,j v V 1 {(0, 0), (n + 1, 0)} (v,w,j) A 1 (w,v,j) A 1 x v,w,j = 1 j = 1,..., n (v,w,j) A 1 x 0,w,j = 1 (0,w,j) A 1 x v,w,j {0, 1} (v, w, j) E 1 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 37 / 43

48 Lower bound by Lagrangian relaxation V 1, A 1 : sets of nodes and arcs x v,w,j : amount of flow on the arc representing j between nodes v and w n L(π) = min c v,w,j x v,w,j + π j x v,w,j 1 (v,w,j) A 1 j=1 (v,w):(v,w,j) A 1 s.t. x v,w,j = x w,v,j v V 1 {(0, 0), (n + 1, 0)} (v,w,j) A 1 (w,v,j) A 1 x v,w,j = 1 j = 1,..., n (v,w,j) A 1 x 0,w,j = 1 (0,w,j) A 1 x v,w,j {0, 1} (v, w, j) A 1 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 38 / 43

49 Lower bound by Lagrangian relaxation V 1, A 1 : sets of nodes and arcs x v,w,j : amount of flow on the arc representing j between nodes v and w L(π) = min s.t. (v,w,j) A 1 cv,w,j +π j xv,w,j n j=1 π j x v,w,j = x w,v,j v V 1 {(0, 0), (n + 1, 0)} (v,w,j) A 1 (w,v,j) A 1 x v,w,j = 1 j = 1,..., n (v,w,j) A 1 x 0,w,j = 1 (0,w,j) A 1 x v,w,j {0, 1} (v, w, j) A 1 Subproblem: shortest path in the network B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 39 / 43

50 Lag-based models [Akkan and Karabati], [Gharbi et al.] Lag variables C m : completion time of the job in position k on machine m [k] L c k : time lag elapsed between the completion of the job in position k on machines 1 and 2 L c k 0, = C2 [k] C1 [k] = max L c k 1 + s2 [k] p1 + p 2 [k] [k] L c 1 + s2 [2] p1 [2] Lc 2 = p2 [2] B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 40 / 43

51 Lag-based models [Akkan and Karabati], [Gharbi et al.] Lag variables C m : completion time of the job in position k on machine m [k] L c k : time lag elapsed between the completion of the job in position k on machines 1 and 2 L c k 0, = C2 [k] C1 [k] = max L c k 1 + s2 [k] p1 + p 2 [k] [k] L c 2 + s2 [3] > p1 [3] Lc 3 = Lc 2 + s2 [3] p1 [3] + p2 [3] L c 1 + s2 [2] p1 [2] Lc 2 = p2 [2] B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 40 / 43

52 Lag-based models Formulating the objective function Minimizing the sum of completion times: n n C 2 [k] = C 1 [k] + Lc k k=1 = = k=1 n k p 1 [r] + Lc k k=1 n k=1 r=1 (n k + 1)p 1 [k] + Lc k B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 41 / 43

53 Example p 1 = (3, 5); p 2 = (7, 4); p 3 = (2, 7) = = Cost of the schedule: ( ) + ( ) + ( ) = 43 B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 42 / 43

54 Lower bound for σ path ending at v in G 2 Lower bound coming from jobs not sequenced yet LB 1 = cost(σ) + max SP i (v, ) π i i/ σ i/ σ Lower bound coming from sequenced jobs LB 2 = cost(σ) + max SP i(v, ) i σ Computing max{lb 1, LB 2 } is done in (n)-time. i/ σ π i B. Detienne, R. Sadykov, S. Tanaka Two-machine flow-shop 43 / 43