Event-based MIP models for the resource constrained project scheduling problem

Transcription

1 Event-based MIP models for the resource constrained project scheduling problem Oumar Koné, Christian Artigues, Pierre Lopez LAAS-CNRS, Université de Toulouse, France Marcel Mongeau IMT, Université de Toulouse, France July 7, 2009 O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

2 1 The resource constrained project scheduling problem 2 Overview of state-of-the-art methods 3 Compact MIP formulations for the RCPSP 4 Benchmark instances and instance indicators 5 Experimental comparison of MIP formulations 6 New event-based MIP formulations 7 Computational evaluation of the new formulations O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

3 The resource constrained project scheduling problem (1/3) Data A = {1,...,n} set of activities p i duration of activity i A R = {1,...,m} set of resources B k number of available units of resource k R b ik number of units of resource k R required by activity i A H = {0,...,T 1} discrete set of time periods. E set of precedence constraints A RCPSP example with m =2and n =10 i p i b 1i b 2i Successors 3 6,7 4, , O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

4 The resource constrained project scheduling problem (2/3) Feasible schedule S i H start time of activity i A in schedule S A(S, t) ={i A S i t S i + p i 1} set of activities in progress at time t H for schedule S. S set of feasible schedule where S S verifies S j H, S j S i + p i, (i, j) E and i A(S,t) b ik B k, k R RCPSP min S S max i A (S i + p i ) O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

5 The resource constrained project scheduling problem (3/3) Feasible schedule A RCPSP example with m =2and n =10 i p i b 1i b 2i Successors 3 6,7 4, , O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

6 Overview of state-of-the-art methods Heuristics [Kolish and Hartmann 2006] MIP search can be used to solve subproblems in a large neighborhood framework [Palpant et al 2004] Lower bounds [Néron et al 2006] Best bounds combine constraint propagation and linear programming relaxations Exact methods [Demeulemeester and Herroelen 1997] [Sprecher 2000] [Laborie 2005] Best methods based on problem structure, MIP results not reported on the standard benchmark instances What is the performance of today s MIP solvers for solving exactly the RCPSP (through compact formulations)? O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

7 Compact MIP formulations for the RCPSP 1/3 Basic Discrete time formulation (DT) [Pritsker et al. 1969] min tx n+1,t x t H tx jt tx it + p i (i, j) E t H t H n t b ik x iτ B k t H, k R i=1 τ=t p i +1 x it =1 i A {0, n +1} t H x it {0, 1} i A {0, n +1}, t H O and n +1dummy start and end activities. (n +2)T binary variables O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

8 Compact MIP formulations for the RCPSP 2/3 Disaggregated Discrete time formulation (DDT) [Christofides et al. 1987] min tx n+1,t x t H t+p T i 1 x iτ + x jτ 1, t H, (i, j) E τ=t τ=0 n t b ik x iτ B k t H, k R i=1 τ=t p i +1 x it =1 i A {0, n +1} t H x it {0, 1} i A {0, n +1}, t H (n +2)T binary variables O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

9 Compact MIP formulations for the RCPSP 3/3 Flow-based continuous-time formulation (FCT) [Artigues et al. 2003] min S,f,x S n+1 S j S i M +(p i + M)x ij (i, j) (A {0, n +1}) 2 f ijk min (b ik, b jk )x ij (i, j) (A {0} A {n +1}), k R f ijk = b ik i A {0, n +1}, k R j A {0,n+1} f ijk = b jk i A {0, n +1}, k R i A {0,n+1} x ij =1 (i, j) E f ijk 0 (i, j) (A {0, n +1}) 2, k R f n+1,0k = B k k R S i 0 i A {0, n +1} x ij {0, 1} (i, j) (A {0, n +1}) 2 (n +2) 2 binary variables, m(n +2) 2 + n +2continuous variables O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

10 Benchmark instances and instance indicators (1/2) Benchmark instances KSD30, 30 activities, 4 resources BL: Baptiste and Le Pape 2000, 20 and 25 activities, 3 resources Pack: Carlier and Néron 2001, 17 to 35 activities, 3 resources KSD15 d: keep 15 first activities of KSD30 and multiply some p i by 15. Pack d: Pack instances where some p i are miltiplied by 50 Instance indicators OS: Order strength NC: Network complexity RF: Resource factor RS: Resource Strength DR Disjunction Ratio PR Processing time Range O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

11 Benchmark instances and instance indicators (2/2) KSD30 BL Pack KSD15 d Pack d V R T OS NC RF RS DR PR O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

12 Experimental comparison of MIP formulations known result : LP relaxation DDT DT FCT Comparison of exact solving through branch and bound (CPLEX 11, 500s on a XEON 5110 biprocessor Dell PC clocked at 1.6Ghz with 4GB RAM) (gap from best known solutions) Instances Formulations Opt. Sol. Non-opt. Sol. Total Sol. No sol. % Time % %Gap % % KSD30 DT DDT FCT Pack DT DDT FCT BL DT DDT FCT KSD15 d DT DDT FCT Pack d DT DDT FCT O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

13 New Event-Based MIP Formulations 1/5 Can we solve highly cumulative instances with a high processing time range with MIP a solver? Idea : reducing the number of variables through the concept of event (cf machine scheduling [Lasserre and Queyranne 1992] [Dauzère-Pérès and Lasserre 1995] [Pinto and Grossmann 1995]) Can we adapt these formulations to discrete resources? O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

14 New Event-Based MIP Formulations 2/5 How many events (E set of events)? i A either S i =0or j A, S i = S j + p j = E n +1 Start/End Event-based formulation (SEE) Variable x ie {0, 1} : activity i starts at event e. Variable y ie {0, 1} : activity i ends at event e. t e time of event e 2n 2 +2n binary variables, (n +1)continuous variables On/Off Event-based formulation (OOE) Variable z ie {0, 1} : z ie is set to 1 if activity i starts at event e or if it still being processed immediately after event e activity i starts at event e or if it still being processed immediately after event e. n 2 binary variables, (n +1)continuous variables O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

15 New Event-Based MIP Formulations 3/5 Example e x 6e y 6e z 6e x 7e y 7e z 7e t x 6t x 7t O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

16 New Event-Based MIP Formulations 4/5 Start/End Event-based formulation (SEE) min t n t 0 =0 t f t e + p i x ie p i (1 y if ) t e+1 t e x ie =1, y ie =1 e E e E n e 1 y ie + x je 1 e =e e =0 (e, f ) E 2, f > e, i A e E, e < n i A (i, j) E, e E r 0k = i A b ik x i0 k K r ek = r (e 1)k + b ik x ie b ik y ie i A i A e E, e 1, k R r ek B k e E, k R x ie {0, 1}, y ie {0, 1} i A {0, n +1}, e E t e 0, r ek 0 e E, k R. O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

17 New Event-Based MIP Formulations 5/5 On/Off Event-based formulation (OOE) min C max C max t e +(z ie z i(e 1) )p i t 0 =0, t e+1 t e t f t e +((z i e z i(e 1) ) (z if z i(f 1) ) 1)p i e 1 e =0 z ie e(1 (z ie z i(e 1) )), z ie 1 e E e z ie + z je 1+(1 z ie )e e =0 n 1 b ik z ie B k i=0 t e 0 z ie {0, 1} n 1 e =e z ie e(1 + (z ie z i(e 1) )) e E, i A e n 1 E (e, f, i) E 2 A, f > e 0 e 0 E i A e E, (i, j) E e E, k R e E i A, e E O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

18 Computational evaluation of the new formulations poor LP relaxations Comparison of exact solving through branch and bound (CPLEX 11, 500s on a XEON 5110 biprocessor Dell PC clocked at 1.6Ghz with 4GB RAM) Instances Formulations Opt. Sol. Non-opt. Sol. Total Sol. No sol. % Time % %Gap % % KSD30 DDT SEE OOE Pack DDT SEE OOE BL DDT SEE OOE KSD15 d FCT SEE OOE Pack d FCT SEE OOE O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19

19 Concluding remarks and further research No dominant MIP formulation in terms of direct exact solving through CPLEX Guidelines for the choice of the accurate MIP formulation Small horizon and integer durations : use DDT Large horizon or non integer durations Disjunctive problems : use FCT Cumulative problems : use OOE Further research : find a dominant formulation, improve LP relaxation, column generation, play with the number of events... Research report: O. Koné, C. Artigues, P. Lopez, and M. Mongeau. Event-based MILP models for resource-constrained project scheduling problems. LAAS report 09102, Université de Toulouse, LAAS-CNRS, Toulouse, France, March ( O. Koné, C. Artigues, P. Lopez, M. Mongeau ()Event-based MIP models for the RCPSP July 7, / 19