A hybrid heuristic for an inventory routing problem

1. A hybrid heuristic for an inventory routing problem C.Archetti, L.Bertazzi, M.G. Speranza University of Brescia, Italy A.Hertz Ecole Polytechnique and GERAD, Montréal, Canada DOMinant 2009, Molde, September 20-23

2. The literature • Surveys • Federgruen, Simchi-Levi (1995), in ‘Handbooks in • Operations Research and Management Science’ Campbell et al (1998), in ‘Fleet Management and Logistics’ Cordeau et al (2007) in ‘Handbooks in Operations Research and Management Science: Transportation’ Bertazzi, Savelsbergh, Speranza (2008) in ‘The vehicle routing problem...’, Golden, Raghavan, Wasil (eds) • Pioneering papers • Bell et al (1983), Interfaces • Federgruen, Zipkin (1984), Operations Research • Golden, Assad, Dahl (1984), Large Scale Systems • Blumenfeld et al (1985), Transportation Research B • Dror, Ball, Golden (1985), Annals of Operations Research • ……

3. Deterministic product usage - no inventory holding costs in the objective function Jaillet et al (2002), Transp. Sci. Campbell, Savelsbergh (2004), Transp. Sci. Gaur, Fisher (2004), Operations Research Savelsbergh, Song (2006), Computers and Operations Research …… Deterministic product usage - inventory holding costs in the objective function Anily, Federgruen (1990), Management Sci. Speranza, Ukovich (1994), Operations Research Chan, Simchi-Levi (1998), Management Sci. Bertazzi, Paletta, Speranza (2002), Transp. Sci. Archetti et al (2007), Transp. Sci. …… The literature

4. Deterministic product usage - inventory holding costs in the objective function – production decision Bertazzi, Paletta, Speranza (2005), Journal of Heuristics Archetti, Bertazzi, Paletta, Speranza , forthcoming Boudia, Louly, Prins (2007), Computers and Operations Research Boudia, Prins (2007), EJOR Boudia, Louly, Prins (2008), Production Planning and Control The literature

5. The problem Data n customers H time units 1 vehicle 1 Availability at t Demand of s at t 0 2 Capacity of s 3 + initial inventory + travelling costs + inventory costs + vehicle capacity How much to deliver to s at time t to minimize routing costs + inventory costs No stock-out No lost sales

6. Replenishment policies 1 0 2 3 • Order-Up-to Level (OU) • Maximum Level (ML) Constraints on the quantities to deliver

7. Order-Up-to level policy (OU) Inventory at customers Maximum level Us Initial level Time

8. The Maximum Level policy (ML) Every time a customer is visited, the shipping quantity is such that at most themaximum level is reached Us

9. Basic decision variables

10. Problem formulation • Inventory definition at the supplier • Stock-out constraints at the supplier

11. Inventory definition at the customers • Stock-out constraints at the customers • Capacity constraints

12. Order-up-to level constraints The quantity shipped to s at time t is Maximum level constraints

13. Routing constraints

14. Known algorithms Local search Very fast Error? • A heuristic • (for the OU policy) • Bertazzi, Paletta, Speranza (2002), Transp. Science Instances up to H=3, n=50 H=6, n=30 • A branch-and-cut algorithm • (for the OU and for the ML policies) • Archetti, Bertazzi, Laporte, Speranza (2007), Transp. Science

15. History of the hybrid heuristic design Exact approach allowed us to compute errors generated by the local search Design of a tabu search Design of a hybrid heuristic (tabu search +MILP models) Often large errors, rarely optimal Sometimes large errors, sometimes optimal Excellent results

16. HAIR (Hybrid Algorithm for Inventory Routing) Initialize generates initial solution A Tabu search is run Whenever a new best solution is found Improvements is run Every JumpIter iterations without improvements Jump is run

17. OU policy - Initialize Each customer is served as late as possible Initial solution may be infeasible (violation of vehicle capacity or stock-out at the supplier)

18. OU policy – Tabu search Search space: feasible solutions infeasible solutions (violation of vehicle capacity or stock-out at the supplier) Solution value: total cost + two penalty terms Moves for each customer: Removal of a day Move of a day Insertion of a day Swap with another customer After the moves: Reduce infeasibility Reduce costs

19. Route assignment Goal: to find an optimal assignment of routes to days optimizing the quantities delivered at the same time. Removal of customers is allowed. OU policy – Improvements – MILP 1 Optimal solution of a MILP model

20. The route assignment model OU policy – Improvements – MILP 1 Binary variables: assignment of route r to time t removal of customer s from route r Continuous variables: quantity to customer s at time t inventory level of customer s at time t inventory level of the supplier at time t

21. The route assignment model OU policy – Improvements – MILP 1 Min inventory costs – saving for removals s.t. Stock-out constraints OU policy defining constraints Vehicle capacity constraints Each route can be assigned to one day at most Technical constraints on possibility to serve or remove a customer # of binary variables: (n+H)*(# of routes)+n*H NP-hard

22. OU policy – Improvements – MILP 1 Day 5 Day 6 Day 1 Day 2 Day 3 Day 4 Node removed Incumbent solution Unused The optimal route assignment

23. Customer assignment Objective: to improve the incumbent solution by merging a pair of consecutive routes. Removal of customers from routes, insertion of customers into routes and quantities delivered are optimized. OU policy – Improvements – MILP 2 Optimal solution of a MILP model For each merging and possible assignment day of the merged route a MILP is solved

24. The customer assignment model OU policy – Improvements – MILP 2 Binary variables: removal of customer s from time t insertion of customer s into time t Continuous variables: quantity to customer s at time t inventory level of customer s at time t inventory level of the supplier at time t

25. The customer assignment model OU policy – Improvements – MILP 2 Min inventory costs + insertion costs – saving for removals s.t. Stock-out constraints OU policy defining constraints Vehicle capacity constraints Each route can be assigned to one day at most Technical constraints on possibility to insert or remove a customer # of binary variables: n*H NP-hard

26. OU policy - Jump After a certain number of iterations without improvements a jump is made. Jump: move customers from days where they are typically visited to days where they are typically not visited. (In our experiments a jump is made only once)

27. The hybrid heuristic for the ML policy The ML policy is more flexible than the OU policy The entire hybrid heuristic has been adapted

28. 160 benchmark instances from Archetti et al (2007), TS Known optimal solution H = 3, n = 5,10, …., 50 H = 6, n = 5,10, …., 30 Inventory costs low, high Tested instances

29. Summary of results

30. Summary of results

31. Summary of results

32. HAIR has been slightly changed: the improvement procedure is called only if at least 20 iterations were performed since its last application; the swap move is not considered. Large instances Optimal solution unknown • H = 6 • n = 50, 100, 200 • Inventory costs: low, high • 10 instances for each size for a total of 60 instances

33. Summary of results – large instances Running time for OU: 1 hour Running time for ML: 30 min Errors taken with respect to the best solution found Running time BPS: always less than 3 min

34. Summary of results – large instances Running time for OU: 1 hour Running time for ML: 30 min Errors taken with respect to the best solution found Running time BPS: always less than 3 min

35. Hybrid vs tabu – OU policy

36. Hybrid vs tabu – ML policy

37. Tabu search combined with MILP models very successful Use of the power of CPLEX Ad hoc designed MILP models used to explore in depth parts of the solution space It is crucial to find the appropriate MILP models (models that are needed, models that explore promising parts of the solution space, trade-off between size of search and complexity) Conclusions