CHSP: extension of an evolutionary approach to multifunction tanks

1. CHSP: extension of anevolutionary approach to multifunction tanks Maria MAHALEAN, Hervé MANIER, Marie-Ange MANIER, Christelle BLOCH Systems and Transport Laboratory University of Technology of Belfort-Montbéliard

2. Outlines A scheduling problem : the Hoist Scheduling Problem Tackled problem : the Cyclic Hoist Scheduling Problem Evolutionary approaches to solve the CHSP Creating an initial population of feasible solutions in the monofunction case Extension to multifunction tanks Conclusion 2

3. hoists track carriers No storage no wait no preemption tanks TheHoist Scheduling Problem (HSP) Goal : Determine a scheduling of soaking operations that maximizes productivity, while ensuring quality of parts • Spatial constraints : • risks of collision between hoists • Temporal constraints : • moving times as large • as processing times • bounded processing times • Resource constraints • limited capacity • Potential constraints : • processing sequence

4. Simple line (dissociated load - unload) Associated load - unload Multifunction tanks Duplicated tanks Multihoist Different kinds of lines 4

5. CHSP Min Cmax T >1 =1 =1 =1 >1 >1 no no no yes yes no =1 =1 >1 =1 =1 =1 =1 no no no no no yes yes yes yes yes Tmin Tmin varied Tmin Tmin Tmin var. Tmin Tmin Tmin var. Tackled problem : the CHSP (Cyclic Hoist Scheduling Problem) Objective : aims at finding a repetitive (cyclic)sequence of processing operations mh ct circ nps recrc g Song et al., 1995 Calvez et al., 1997 Hanen and Munier, 1993 Shapiro and Nuttle, 1988 Manier-Lacoste, 1994 Lei and Wang, 1991 Baptiste et al., 1994 Hanen, 1994 Phillips and Unger, 1976 Lei and Wang, 1989 Lei and Wang, 1994 Ng, 1995 Ng, 1996 Bracker and Chapman, 1985 Baptiste et al., 1992 Baptiste et al., 1993 Bacheluet al., 1997 Lei, 1993 Armstrong et al., 1994 Lim, 1997 Chen et al., 1998 Ng and Leung, 1997 Manier et al., 2003 Ptuskin, 1995 Lei et al., 1993 Baptiste et al., 1996 Varnier et al.,1997 Manieret al., 2000 Armstrong et al., 1996 Manier and Bloch, 2003 CHSP | mt / / ass | /nop, recrc | Tmin

6. the beginning of a processing operation the end of a transportoperation the beginning of a transportoperation the end of a processing operation Hoist point of view : transport unloaded move the end of a transportoperation the beginning of an unloaded move Time Scheduling processing operations Scheduling transport operations Scheduling unloaded moves Tackled problem : the CHSP (Cyclic Hoist Scheduling Problem) Product point of view : Time soaking assumption

7. Evolutionary approaches to solve the CHSP Chromosomes defined by Lim (1997): * based on the representation of transportoperations : a gene = a tank, beginning of a transport operation * same sizen (= number of tanks) for all the chromosomes * example: the chromosome 6 5 2 3 4 represents the following sequence of transport operations: {(1,2), (6,1), (5,6), (2,3), (3,4), (4,5)} Chromosomes defined by Manier, Manier and Bloch (2003): * based on the representation of unloaded moves : a couple of successive genes i j = an unloaded move from tank i to tank j * the size of chromosomes varies: between 0 (“empty” chromosome) and n. * previous example: the chromosome that corresponds to the scheduling described above is: 1 5 / 2 6 it represents the unloaded moves:{(1,5), (5,1), (2,6), (6,2)} 7

8. Constraints of the problem (1) feasible solution sequence of unloaded moves, satisfies the constraints : - number of hoists a single hoist - unit capacity of tanks each tank processes one carrier at once -1-cyclic in each cycle, each treatment is performed exactly once, whatever the corresponding tank is 8

9. Creation of an initial population * based on the fact that each tank is unloaded exactly once during a cycle possible choices already unloaded during the cycle The current tank represents the position of the hoist. Current tank unloaded moves (2,4) (1,3) (4,2) (3,1) chromosome 13 / 24

10. transport unloaded move hypothesis: tanks numbered according to the order of operations in the processing sequence transport operations:from tank i to tank i+1 transport unloaded move i i i+1 i+1 j j j+1 j+1 hypothesis : treatments numbered according to the order of operations in the proc. sequence transport operations:from treatment i to treatment i+1 Extension to multifunction tanks The case of monofunction tanks The case of multifunction tanks a couple of successive genes i j = an unloaded move from treatment i to treatment j

11. j-1 i i+1 j+1 j Constraints of the problem (2) A. Constraints linked to the unit capacity of tanks If i and j are two treatments performed in the same multifunction tank, and treatment i is in progress, neither unloaded move to treatment j-1 nor unloaded move to treatment j can be introduced .

12. the tank remains empty till the loading of treatment j i i+1 j-1 j+1 j Constraints of the problem (3) B. Constraints on the multifunction tank, linked to the first passage When treatment i has been unloaded, the multifunctiontank is empty until the loading oftreatment j, consequently no unloaded move to treatment j can be introduced, before this loading.

13. There must be at least one empty tank here !  j-1 i+1 i j+1 j treatment i OR treatment j is in progress in the multifunction tank Constraints of the problem (4) C. Constraints on the other tanks, linked to the firts passage in a multifunction tank (1) If i and j are two successive treatments performed in the same multifunction tank, and one of them is in progress, then there is at least one empty tank among those associated with the treatments that are between i and j. 13

14. There is no empty tank  !  j-1 i+1 i j+1 j treatment i in progress in the multifunction tank. Constraints of the problem (5) C. Constraints on the other tanks, linked to the firts passage in a multifunction tank (2) Proof (1): Deadlock: no move is possible! 14

15. There is no empty tank  !   j-1 i+1 i j+1 j treatment j in progress in the multifunction tank. Constraints of the problem (6) C. Constraints on the other tanks, linked to the firts passage in a multifunction tank (3) Proof(2): treatment j is unloaded  the multifunction tank becomes free  2 possibilities: - either treatment i is loaded  deadlock (previous case) - or treatment j-1 is unloaded  treatment j is performed twice 15

16. j-1 i+1 i j+1 j Successive treatments in a multifunction tank Constraints of the problem (7) C-c1. Constraints on the other tanks, linked to the firts passage in a multifunction tank - consequences (1) Consequence 1: Land i et j be two successivetreatments performed in the same multifunction tank. If this tank has been unloaded after treatment j, then there is at least one empty tank among those associated with the treatments that are between i and j. There must be at least one empty tank here !  16

17. All the tanks are busy !  j-1 i-1 i+1 i j+1 j Successive treatments in a multifunction tank Constraints of the problem (8) C-c2. Constraints on the other tanks, linked to the firts passage in a multifunction tank - consequences (2) Consequence 2: Let i et j be two successivetreatments performed in the same multifunction tank. If this tank has been unloaded after treatment i and if all the tanks associated with the treatments that are between i and j are busy, then treatment i-1 can not be unloaded. Deadlock ! 17

18. Tank 3 is multifunction and the treatments 3 and 5 are two successive treatments in this tank. 5 1 2 3 4 1 2 3 4 5 6 Creation of an initial population and feasibility check of a chromosome (1) example with 5 tanks and the following processing sequence: tank (i) 1 2 3 4 3 5 treatment (t) 1 2 3 4 5 6 tanks treatments processing sequence 18

19. 1 2 3 4 3 5 1 2 3 4 3 5 1 2 3 4 3 5 1 2 3 4 3 5 1 2 3 4 3 5 1 2 3 4 3 5 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Creation of an initial population and feasibility check of a chromosome(2) possible choices already unloaded during the cycle forbidden moves The current tank represents the position of the hoist. B A A A A i = 3 empty till 2nd passage t = 5 in i = 3 first passage Current tank The sequence of unloaded moves from the point of view of tanks(2,4) (3,5) (1,3) (5,2) (3,3) (4,1) from the point of view of treatments(2,4) (5,6) (1,5) (6,2) (3,3) (4,1) the chromosome{15624} 19

20. Conclusion • Implementation and tests (benchmark Phillips and Unger) in progress • First results: very few solutions satisfy all the constraints • (in particular bounded processing times) •  A multiobjective algorithm to handle this constraint without penalty function • Extension to other cases, in particular multihoist case • the same genotype whatever the number of hoists is This approach Lim 1997 Vertex i transport from tank i to tank i+1 an unloaded move from tank i Arc (i,j) sequence of transports performed unloaded move (i,j) Graphe G 1 1 2 2 Chromosome Chromosome 3 3 15/26 6 6 65234 ó 4 4 5 5 useful to both find the optimal number of hoists and maximize throughput 20

21. Creation of an initial population * Each solution is built while satisfying the constraints previously described by using the following variables: * D(t) indicates whether the treatment t has been unloaded or not D(t)=1 treatment t has been unloaded D(t)=0 otherwise * p(i), defined for multifunction tanks, indicates whether there has been a first passage in the multifunction tank i or not p(i)=1 there has been a first passage p(i)=0 otherwise * T(i, t) indicates whether there is treatment in progress in tank i or not T(i, t)=1 if treatment t is in progress in tank i T(i, t)=0 otherwise 21

22. 1 2 3 4 3 5 1 2 3 4 3 5 1 2 3 4 3 5 1 2 3 4 3 5 1 2 3 4 3 5 1 2 3 4 3 5 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 Creation of an initial population and feasibility check of a chromosome(2) possible choices already unloaded during the cycle forbidden moves The current tank represents the position of the hoist. B A A A A Current tank The sequence of unloaded moves from the point of view of tanks(2,4) (3,5) (1,3) (5,2) (3,3) (4,1) from the point of view of treatments(2,4) (5,6) (1,5) (6,2) (3,3) (4,1) the chromosome{15624} 22