1. Shop Scheduling Reformulation of Vehicle Routing Evgeny Selensky Dept of Computing Science Glasgow University

2. Overview Why is reformulation important? Examples of reformulation Vehicle Routing Problem and its instances Job shop scheduling problem and its instances Tools to study reformulation issues Models and search procedures used in study Some preliminary results Outline of research (big picture) Future work

3. Why Reformulate? Better problem-solving (using representation benefits) Solution process improvement (speed of solvers, memory use) Human interface improvements (better understanding of problems) Software reuse (generic types of constraints, heuristics, etc)

4. Examples of reformulation N-queens  n 2 0/1 variables  n variables with domains of size n  B.Nadel (reformulation of n-queens, case study, IEEE Expert, June 1990) Potential search space of each representation Benefits of each? One might be more pruningful than another Better heuristics might be available May reduce symmetries Graph colouring  n-variables, k-colours, e edges, go colour them  k set variables, partition the set of n nodes such that … Crossword puzzles construction Stable marriage (research frontier) Scheduling and vehicle routing

5. N identical vehicles of capacity C M customers with demands D i >0, i = 1..M Each vehicle serves subset of customers Side constraints may be present (e.g.,time windows, precedence constraints) Find tours for subset of vehicles such that: all customers served, each once one tour per vehicle total distance minimal Vehicle routing problem (delivery)

6. VRP instances Repair/install equipment Pick up money from banks Deliver prisoners from jail to court etc Street cleaning, garbage collection Automated guided vehicles in a factory Ambulance routing Drilling circuit boards Robot arm movements Computer networks

7. Job shop scheduling problem M machines, i = 1..M, M  2 N jobs each of S operations, j = 1..S, of duration d ij  j : O ij < O ij+1 (chain-type precedence constraints)  i  j  tr_cost ij  0  j : O ij requires specific resource No preemption Minimise makespan = LatestEnd - EasliestStart Open shop relaxation  j : start(O ij ) start(O ij+1 ) very hard, no polynomial time approximation within 5/4 from optimal solution Multipurpose machines  j : O ij requires alternative resource time Earliest start time Latest end time

8. JSSP instances engineering job shop  making engines: forge and machine pistons cast block treat surfaces drill, machine, heat treat, etc. high speed communications networks navigation less like construction and assembly

9. Tools ILOG Solver 5.0 general constraint programming problems offers enhanced search facilities ILOG Scheduler 5.0 scheduling and resource allocation ILOG Dispatcher 3.0 advanced local search algorithms for routing

10. ILOG Scheduler models Real-world processes represented by resources and activities Activity starting time, processing time, completion time, demand performed by resource Resource capacity or state Capacitated resources unary ( capacity  1 ) - useful when dealing with transition costs discrete ( capacity  1 ) - allows one to take into account capacity constraints

11. An OSSP models of a TSP and VRP Vehicles/salesman are machines on the shop floor the visits are operations (aka activities) the visits (activities) pass through the vehicles (machines)! Relativity of representation

12. A JSSP model of a TSP N+1 activities as visits to N cities + 1 additional visit earliest and latest time on each activity 1 unary resource as salesman Transition costs as distances between cities First city picked out arbitrarily,  i  [2.. N+1] end(act 1 )  start(act i ) Last visit in tour to first city,  i  [1.. N] start(act N+1 )  end(act i ) Search goal minimise(  tr_cost)

13. An OSSP model of a VRP N+2*M activities as visits to N customers and base for M vehicles Pair of resources as vehicle Transition costs as distances M additional activities (starts of tours)  i  [1.. M] act i is setup  i  [1.. M] act i requires vehicle i N actual visits  i  [M+1.. N+M] act i requires vehicle 0  vehicle 1  …  vehicle M-1 M additional activities (ends of tours)  i  [N+M+1.. N+2*M] act i is teardown  i  [N+M+1.. N+2*M] act i requires vehicle i Search goal minimise(   tr_cost)

14. An OSSP model of a VRP Why bother?  Maybe that scheduling heuristics work for vrp  maybe scheduling propagation works for vrp edge finding, energetic reasoning, … maybe as vrp becomes urban it looks like jssp urban, low transition costs? Can you see the symmetric argument?  Ossp modeled as a vrp

15. Search Search facilities constraint propagation, heuristics goal should be specified performed by Solver engine Limited Discrepancy Search William D Harvey, Matthew L Ginsberg, August 1995, IJCAI Depth Bounded Discrepancy Search Toby Walsh, August 1997, IJCAI

16. LDS trace Discrepancy = 0 Discrepancy = 2 Discrepancy = 1 Discrepancy = 3

17. Results. JSSP representation of TSP instances nearest neighbour heuristic (schedule activity with min earliest start time) Intel Pentium III 933 MHz, 1Gb RAM Number of cities CPU Time, s Distance Optimal Distance Number of Branching Points Number of Backtracks 14 1.046 3323 3323 1592 935 16 28.703 6859 6859 127150 100470 17 2.969 2085 2085 10009 4923 21 2.766 2707 2707 5683 3239 22 6195.86 7013 7013 19657050 15299594 24 1031.734 1272 1272 2311096 1640314 26 78.438 937 937 167029 71832

18. Results. An OSSP representation of VRP instances VRP instances with time windows (CVRPTW) harder to solve approx. 2 times worse in distance than with local search The best known worst-case performance ratio for 3-machine dense OSSP schedules 3/2 Time, s Number of Customers, CVRP Capacitated VRP instances (CVRP) 9 - 12 customers served by 2 vehicles 13-14 customers served by 3 vehicles Intel Pentium III 933 MHz, 1Gb RAM Resource selection heuristics minimal capacity maximal capacity

19. Interesting, but so what? … An early step in our research

20. A sketch of our planned research in the field VRP model, ILOG Dispatcher  model ossp as a vrp  use vrp heuristics  see what happens as we vary the vrp properties of the problem: setups few operations per job increase alternative resources OSSP model, ILOG Scheduler  model vrp as ossp  use ossp heuristics  see what happens as we vary ossp properties: setups vs durations many operations per job few alternative resources

21. Conclusion. Future work TSP and VRP empirically represented as JSSP Search for solutions Complete and quasi-complete embedded in binary chop Different resource and activity selection criteria So no conclusion yet… In the future A VRP representation of OSSP Other problems … (SM, SMTI, Car Sequencing, …)