An ant colony optimization technique for solving min-max Multi-Depot Vehicle Routing Problem

Abstract The Multi-Depot Vehicle Routing Problem (MDVRP) involves minimizing the total distance traveled by vehicles originating from multiple depots so that the vehicles together visit the specified customer locations (or cities) exactly once. This problem belongs to a class of Nondeterministic Polynomial Hard (NP Hard) problems and has been used in literature as a benchmark for development of optimization schemes. This article deals with a variant of MDVRP, called min–max MDVRP, where the objective is to minimize the tour-length of the vehicle traveling the longest distance in MDVRP. Markedly different from the traditional MDVRP, min–max MDVRP is of specific significance for time-critical applications such as emergency response, where one wants to minimize the time taken to attend any customer. This article presents an extension of an existing ant-colony technique for solving the Single Depot Vehicle Routing Problem (SDVRP) to solve the multiple depots and min–max variants of the problem. First, the article presents the algorithm that solves the min–max version of SDVRP. Then, the article extends the algorithm for min–max MDVRP using an equitable region partitioning approach aimed at assigning customer locations to depots so that MDVRP is reduced to multiple SDVRPs. The proposed method has been implemented in MATLAB for obtaining the solution for the min–max MDVRP with any number of vehicles and customer locations. A comparative study is carried out to evaluate the proposed algorithm's performance with respect to a currently available Linear Programming (LP) based algorithm in literature in terms of the optimality of solution. Based on simulation studies and statistical evaluations, it has been demonstrated that the ant colony optimization technique proposed in this article leads to more optimal results as compared to the existing LP based method.

[1]  Francisco Herrera,et al.  A practical tutorial on the use of nonparametric statistical tests as a methodology for comparing evolutionary and swarm intelligence algorithms , 2011, Swarm Evol. Comput..

[2]  Xin Yao,et al.  Memetic Algorithm With Extended Neighborhood Search for Capacitated Arc Routing Problems , 2009, IEEE Transactions on Evolutionary Computation.

[3]  J. Tukey,et al.  Generalized “sandwich” theorems , 1942 .

[4]  Paolo Toth,et al.  The Vehicle Routing Problem , 2002, SIAM monographs on discrete mathematics and applications.

[5]  Ann Melissa Campbell,et al.  Routing for Relief Efforts , 2008, Transp. Sci..

[6]  Giovanni Storchi,et al.  Multiperiod integrated routing and scheduling of World Food Programme cargo planes in Angola , 2007, Comput. Oper. Res..

[7]  Gilbert Laporte,et al.  An adaptive memory heuristic for a class of vehicle routing problems with minmax objective , 1997, Comput. Oper. Res..

[8]  John Gunnar Carlsson,et al.  Solving Min-Max Multi-Depot Vehicle Routing Problem ⁄ , 2007 .

[9]  Benita M. Beamon,et al.  Last Mile Distribution in Humanitarian Relief , 2008, J. Intell. Transp. Syst..

[10]  Jan Karel Lenstra,et al.  A Computational Study of Local Search Algorithms for Job Shop Scheduling , 1994, INFORMS J. Comput..

[11]  Marco Dorigo,et al.  Ant system: optimization by a colony of cooperating agents , 1996, IEEE Trans. Syst. Man Cybern. Part B.

[12]  Richard F. Hartl,et al.  An improved Ant System algorithm for theVehicle Routing Problem , 1999, Ann. Oper. Res..

[13]  G. Laporte,et al.  A tabu search heuristic for periodic and multi-depot vehicle routing problems , 1997, Networks.

[14]  B. Gillett,et al.  Multi-terminal vehicle-dispatch algorithm , 1976 .

[15]  Alain Hertz,et al.  Ants can colour graphs , 1997 .

[16]  Michel Gendreau,et al.  Vehicle Routing Problem with Time Windows, Part I: Route Construction and Local Search Algorithms , 2005, Transp. Sci..

[17]  Richard F. Hartl,et al.  D-Ants: Savings Based Ants divide and conquer the vehicle routing problem , 2004, Comput. Oper. Res..

[18]  Yinyu Ye,et al.  Finding equitable convex partitions of points in a polygon efficiently , 2010, TALG.

[19]  Richard M. Karp,et al.  Reducibility Among Combinatorial Problems , 1972, 50 Years of Integer Programming.

[20]  Manish Kumar,et al.  Ant colony optimization technique to solve the min-max Single Depot Vehicle Routing Problem , 2011, Proceedings of the 2011 American Control Conference.

[21]  Brian W. Kernighan,et al.  An Effective Heuristic Algorithm for the Traveling-Salesman Problem , 1973, Oper. Res..

[22]  J. Beardwood,et al.  The shortest path through many points , 1959, Mathematical Proceedings of the Cambridge Philosophical Society.

[23]  Gilbert Laporte,et al.  Solving a Family of Multi-Depot Vehicle Routing and Location-Routing Problems , 1988, Transp. Sci..

[24]  Patrick R. McMullen,et al.  Ant colony optimization techniques for the vehicle routing problem , 2004, Adv. Eng. Informatics.

[25]  Marco Dorigo,et al.  Ant system for Job-shop Scheduling , 1994 .

[26]  George B. Dantzig,et al.  The Truck Dispatching Problem , 1959 .

[27]  F. Tillman The Multiple Terminal Delivery Problem with Probabilistic Demands , 1969 .

[28]  Joseph ORourke,et al.  Computational Geometry in C Second Edition , 1998 .

[29]  Sergey Bereg,et al.  Generalizing Ham Sandwich Cuts to Equitable Subdivisions , 1999, SCG '99.

[30]  Andrew Lim,et al.  Multi-depot vehicle routing problem: a one-stage approach , 2005, IEEE Transactions on Automation Science and Engineering.

[31]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[32]  Bruce L. Golden,et al.  A new heuristic for the multi-depot vehicle routing problem that improves upon best-known solutions , 1993 .

[33]  Roberto Montemanni,et al.  Ant colony optimization for real-world vehicle routing problems , 2007, Swarm Intelligence.

[34]  Luca Maria Gambardella,et al.  Ant colony system: a cooperative learning approach to the traveling salesman problem , 1997, IEEE Trans. Evol. Comput..

[35]  John R. Current,et al.  An improved ant colony optimization based algorithm for the capacitated arc routing problem , 2010 .

[36]  M Dorigo,et al.  Ant colonies for the travelling salesman problem. , 1997, Bio Systems.

[37]  Jalel Euchi,et al.  The urban bus routing problem in the Tunisian case by the hybrid artificial ant colony algorithm , 2012, Swarm Evol. Comput..

[38]  E Bonabeau,et al.  Swarm Intelligence: A Whole New Way to Think about Business , 2001 .

[39]  William J. Cook,et al.  Solution of a Min-Max Vehicle Routing Problem , 2002, INFORMS Journal on Computing.

[40]  Vittorio Maniezzo,et al.  The Ant System Applied to the Quadratic Assignment Problem , 1999, IEEE Trans. Knowl. Data Eng..

[41]  Chunyu Ren,et al.  Solving Min-Max Vehicle Routing Problem , 2011, J. Softw..