Skip to main content
Log in

The min-max multi-depot vehicle routing problem: heuristics and computational results

  • General Paper
  • Published:
Journal of the Operational Research Society

Abstract

In the multi-depot vehicle routing problem (MDVRP), there are several depots where vehicles can start and end their routes. The objective is to minimize the total distance travelled by all vehicles across all depots. The min-max multi-depot vehicle routing problem (Min-Max MDVRP) is a variant of the standard MDVRP. The primary objective is to minimize the length of the longest route. We develop a heuristic (denoted by MD) for the Min-Max MDVRP that has three stages: (1) simplify the multi-depot problem into a single depot problem and solve the simplified problem; (2) improve the maximal route; (3) improve all routes by exchanging customers between routes. MD is compared with two alternative heuristics that we also develop and an existing method from the literature on a set of 20 test instances. MD produces 15 best solutions and is the top performer. Additional computational experiments on instances with uniform and non-uniform distributions of customers and varying customer-to-vehicle ratios and with real-world data further demonstrate MD’s effectiveness in producing high-quality results.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Figure 1
Figure 2

Similar content being viewed by others

References

  • Carlsson J, Ge D, Subramaniam A, Wu A and Ye Y (2009). Min-max multi-depot vehicle routing problem. In: Lectures on Global Optimization (Volume 55 in the series Fields Institute Communications). American Mathematical Society: Providence, RI.

  • Chen S, Golden B and Wasil E (2007). The split delivery vehicle routing problem: Applications, algorithms, test problems, and computational results. Networks 49 (4): 318–329.

    Article  Google Scholar 

  • Codenotti B, Manzini G, Margara L and Resta G (1996). Perturbation: An efficient technique for the solution of very large instances of the Euclidean TSP. INFORMS Journal on Computing 8 (2): 125–133.

    Article  Google Scholar 

  • Cook W (2003). Concorde TSP solver. http://www.math.uwaterloo.ca/tsp/concorde/index.html, accessed 15 January 2014.

  • Dantzig GB and Ramser JH (1959). The truck dispatching problem. Management Science 6 (1): 80–91.

    Article  Google Scholar 

  • Golden BL, Magnanti TL and Nguyen HQ (1977). Implementing vehicle routing algorithms. Networks 7 (2): 113–148.

    Article  Google Scholar 

  • Groër C, Golden B and Wasil E (2010). A library of local search heuristics for the vehicle routing problem. Mathematical Programming Computation 2 (2): 79–101.

    Article  Google Scholar 

  • Gulczynski D, Golden B and Wasil E (2011). The multi-depot split delivery vehicle routing problem: An integer programming-based heuristic, new test problems, and computational results. Computers & Industrial Engineering 61 (3): 794–804.

    Article  Google Scholar 

  • Gurobi Optimization, Inc (2014). Gurobi optimizer reference manual, http://www.gurobi.com, accessed 15 March 2014.

  • Hansen P and Mladenović N (2001). Variable neighborhood search: Principles and applications. European Journal of Operational Research 130 (3): 449–467.

    Article  Google Scholar 

  • Helsgaun K (2012). LKH TSP solver. http://www.akira.ruc.dk/~keld/research/LKH, accessed 24 June 2013.

  • Narasimha KV, Kivelevitch E, Sharma B and Kumar M (2013). An ant colony optimization technique for solving min-max multi-depot vehicle routing problem. Swarm and Evolutionary Computation 13: 63–73.

    Article  Google Scholar 

  • Ren C (2011). Solving min-max vehicle routing problem. Journal of Software 6 (9): 1851–1856.

    Article  Google Scholar 

  • Wottawa M (2004). VRPLIB—A library of capacitated vehicle routing problems, http://elib.zib.de/pub/Packages/mp-testdata/vehicle-rout/vrplib/, accessed 29 April 2014.

  • Yakici E and Karasakal O (2013). A min-max vehicle routing problem with split delivery and heterogeneous demand. Optimization Letters 7 (7): 1611–1625.

    Article  Google Scholar 

  • Ye Y (2006). LP-based load balancing implementation using MATLAB. http://www.stanford.edu/class/msande310/LP-TSP.zip, accessed 11 July 2012.

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Edward Wasil.

Appendix

Appendix

A.1. Illustration

To illustrate the MD algorithm, consider an instance with three depots, with each depot having one vehicle and 10 customers. (We denote this problem by MM1. The coordinates of the customers and depots are given in the online appendix.) The locations of customers and depots are shown in Figure A1. Customers are represented by circles and depots are represented by stars.

Figure A1
figure 3

Locations of customers and depots.

A.1.1. Initialization

Customers 1, 9, 5, and 2 are assigned to depot 1. Customers 7, 3, and 4 are assigned to depot 2. Customers 10, 6, and 8 are assigned to depot 3. The LKH solver generates a TSP on each route. This gives the initial solution that is shown in Figure A2.

Figure A2
figure 4

Initial solution.

The number next to a line segment is the length between neighbouring nodes. The number in parentheses next to a depot is the total length of the route passing through that depot (170.778 for depot 1, 143.072 for depot 2, and 219.02 for depot 3).

A.1.2. Local search

The maximal route passes through depot 3 and has a length of 219.02. There are three customers on this route. The savings produced by removing a customer from this route are given in Table A1.

Table A1 Savings if a customer is removed from route 3

The customer with the largest savings (customer 10) is removed from the maximal route. It may be inserted onto the route passing through depot 1 or onto the route passing through depot 2. The increase in route length is estimated by inserting customer 10 between every pair of adjacent nodes on the route. These costs are given in Table A2.

Table A2 Cost of inserting customer 10 onto routes 1 and 2

Customer 10 is inserted in the least-cost way (27.837) between customer 7 and customer 3 on route 2 that passes through depot 2. The TSP solution on routes 2 and 3 are calculated again using the LKH solver. The following solution is produced after iteration 1 (see Figure A3). Route 1 passes through depot 1 with customers 1, 9, 5, and 2 and a length of 170.778. Route 2 passes through depot 2 with customers 7, 10, 3, and 4 and a length of 170.909. Route 3 passes through depot 3 with customers 8 and 6 and a length of 127.098.

Figure A3
figure 5

Iteration 1.

The objective function value decreased from 219.02 to 170.909, so the procedure continues. The maximal route is the second route with four customers. The savings from removing a customer are given in Table A3.

Table A3 Savings if a customer is removed from route 2

Customer 10 is removed and inserted onto route 1 or route 3. The insertion costs are given in Tables A2 and A4.

Table A4 Cost of inserting customer 10 onto route 3

If customer 10 is inserted onto route 3, the estimated increase in route length is 91.922. If it is inserted onto route 1, the estimated increase in route length is 88.507 (see Table A2). So we try to insert customer 10 onto route 1. After solving the TSP on routes 1 and 2 using the LKH solver, the new objective function value is 238.877, which is greater than 170.909. There is no improvement if customer 10 is removed from the maximal route (ie, route 2).

We go back to Table A3 and consider removing customer 7. The increase in route length is estimated by inserting customer 7 between every pair of adjacent nodes on routes 1 and 3. These costs are given in Table A5.

Table A5 Cost of inserting customer 7 onto routes 1 and 3

Customer 7 is inserted in the least-cost way between customers 9 and 5 on route 1. After solving the TSP on routes 1 and 2 using the LKH solver, the new objective function value is 203.438, which is greater than 170.909. The solution is not updated.

We go back to Table A3 and consider removing customers 3 and 4. Neither removal produces a smaller objective function value, so the solution with the objective function value of 170.909 (Figure A3) is retained.

A.1.2.1. Improvement by perturbation

For each of the three depots, we compute the average of the distances to its preceding and succeeding customers. This is the radius of the first perturbation. The angle of the first perturbation is generated randomly. The new positions of the depots after perturbation are given in Table A6. The sequence of nodes on each route is unchanged and the feasible solution to the new problem is shown in Figure A4. The objective function value is 240.507.

Table A6 Depot perturbation
Figure A4
figure 6

A feasible solution to the perturbed problem.

We apply our local improvement procedure to the problem with the perturbed positions of the depots. When the local improvement procedure ends, we have a solution with an objective function value of 175.466. This solution is shown in Figure A5.

Figure A5
figure 7

Solution to the perturbed problem after local search.

The depots are then set to their original positions with the sequence of nodes on each route given by the perturbed solution (see Figure A6). This solution is feasible and has an objective function value 227.925. We apply the local improvement procedure to this solution and obtain another feasible solution with objective function value 226.275 in Figure A7. The first perturbation does not result in a better solution, and we continue with the second to the fifth with the angles of perturbations shown in Table A7. None of these improves the current solution, so MD stops. The final solution has an objective function value of 170.909.

Table A7 Angles of each perturbation
Figure A6
figure 8

A feasible solution to the original problem.

Figure A7
figure 9

The feasible solution generated after one perturbation.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Wang, X., Golden, B. & Wasil, E. The min-max multi-depot vehicle routing problem: heuristics and computational results. J Oper Res Soc 66, 1430–1441 (2015). https://doi.org/10.1057/jors.2014.108

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1057/jors.2014.108

Keywords

Navigation