# Solving the Vehicle Routing Problem with Multiple Trips by Adaptive Memory Programming

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1 Solving the Vehicle Routing Problem with Multiple Trips by Adaptive Memory Programming Alfredo Olivera and Omar Viera Universidad de la República Montevideo, Uruguay ICIL 05, Montevideo, Uruguay, February 2005

2 Contents 1. Motivation and problem description. 2. Adaptive memory procedure. 3. Experimental results. 4. Conclusions and future work.

3 Motivation and problem description

4 The classic Vehicle Routing Problem (VRP) Given: A set of n customers and one depot. Travelling costs c ij. Customer demands q i and vehicle capacity Q. Find a set of routes such that: Each route starts and ends at the depot. Every customer belongs to exactly one route. Vehicle capacity is not exceeded in any route. The total cost is minimized.

5 A sample VRP instance

6 The optimal VRP solution Total cost 525 (5 routes).

7 The VRP solution in practice - finite fleet Total cost 525 (5 routes). Problem: 3 vehicles available. Solution:?

8 The VRP solution in practice - finite fleet Total cost 525 (5 routes). Problem: 3 vehicles available. Solution: Assign more than one route to the same vehicle.

9 The VRP solution in practice - time horizon Total cost 525 (5 routes). Problem: must finish the work before time 200. Solution:?

10 The VRP solution in practice - time horizon Total cost 554 (6 routes). Problem: must finish the work before time 200. Solution: re-calculate!

11 VRP limitations The classic VRP model does not consider: a finite number of vehicles, a time horizon, the assignment of routes to vehicles. It is not always possible to account for these constraints at a second stage.

12 The VRP with Multiple Trips (VRPMT) Besides the VRP constraints, the VRPMT considers: A finite number of vehicles (m). A time horizon (T ). Solutions include an assignment of routes to vehicles. More than one route may be assigned to the same vehicle. The duration for each vehicle should not be greater than T.

13 Some remarks about the VRPMT The problem incorporates realistic constraints. Finding a feasible solution is NP-Hard (by reduction of the Bin Packing Problem). Little work reported in the OR literature.

15 The Adaptive Memory Procedure (AMP) Proposed by Rochat and Taillard in 1995 for VRP and VRPTW. Probabilistic intensification and diversification. Enhancement of local search algorithms. Gives accurate solutions when used to solve other VRPs.

16 AMP - The algorithm A memory M with components of visited solutions is kept. A generic template: 1. Initialize M. 2. Repeat for AMP iter iterations: (a) Build a solution s using M. (b) Apply tabu search to s. (c) Update M using the obtained solution.

17 AMP - The Memory The memory M contains individual routes. Routes belonging to better solutions are stored first. The size of the memory is bounded.

18 AMP - Initialization Initially, the memory is empty. The following is repeated a fixed number of times: 1. Build a solution using a fast randomized heuristic. 2. Apply the tabu search to that solution. 3. Add the routes of the new solution to M.

19 AMP - Solution building Probabilistically select non-overlapping routes from M. Higher probability is assigned to the first routes. The selected routes are assigned to the vehicles, using a simple greedy heuristic.

20 Tabu Search - neighborhood definition For each solution s, a set of neighbors N(s) is defined: Original routes Customer insert Customer swap Original assignment Route insert Route swap

21 Tabu Search - exploration strategy Infeasible solutions may be visited and penalized in the objective function. Repeat for T S iter iterations: 1. Make the best non-tabu move. 2. Declare tabu the inverse of the move for δ iterations.

22 Experimental results

23 Benchmark problems The AMP was tested over a set of 104 benchmark problems. Highly constrained instances (finding a feasible solution is the main issue). Results were compared with: Brandão and Mercer (BM), Taillard, Laporte and Gendreau (TLG), Petch and Salhi (PS).

24 Summarized results AMP BM TLG PS Feasible solutions Average LT R (%) Average Overtime Average GAP with respect to a VRP lower bound: 3.09%. The GAP standard deviation average was 0.76%. The AMP solved 6 previously unsolved instances.

25 Conclusions and future work

26 Conclusions We solved a hard and practical vehicle routing problem. The AMP was effective (once more). The algorithm seems to be flexible to handle more constraints.

27 Future work Add constraints to the VRPMT: heterogeneous fleet, multiple depots, time windows, control of depot congestion.

28 Questions

29

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