Analysis of Mixed-Integer Linear Programming Formulations for a Fuel-Constrained Multiple Vehicle Routing Problem

This paper addresses a fuel-constrained, multiple vehicle routing problem (FCMVRP) in the presence of multiple refueling stations. We are given a set of targets, a set of refueling stations, and a depot where m vehicles are stationed. The vehicles are allowed to refuel at any refueling station, and the objective of the problem is to determine a route for each vehicle starting and terminating at the depot, such that each target is visited by at least one vehicle, the vehicles never run out of fuel while traversing their routes, and the total travel cost of all the routes is a minimum. We present four new mixed-integer linear programming (MILP) formulations for the problem. These formulations are compared both analytically and empirically, and a branch-and-cut algorithm is developed to compute an optimal solution. Extensive computational results on a large class of test instances that corroborate the effectiveness of the algorithm are also presented.

[1]  Laurence A. Wolsey,et al.  Technical Note - Facets and Strong Valid Inequalities for Integer Programs , 1976, Oper. Res..

[2]  Elise Miller-Hooks,et al.  A Green Vehicle Routing Problem , 2012 .

[3]  Mark S. Daskin,et al.  AN INTEGRATED MODEL OF FACILITY LOCATION AND TRANSPORTATION NETWORK DESIGN , 2001 .

[4]  Manfred W. Padberg,et al.  On the facial structure of set packing polyhedra , 1973, Math. Program..

[5]  Gilbert Laporte,et al.  Two exact algorithms for the distance-constrained vehicle routing problem , 1984, Networks.

[6]  Martin W. P. Savelsbergh,et al.  Sequence Independent Lifting in Mixed Integer Programming , 2000, J. Comb. Optim..

[7]  Kaarthik Sundar,et al.  Formulations and algorithms for the multiple depot, fuel-constrained, multiple vehicle routing problem , 2015, 2016 American Control Conference (ACC).

[8]  Roberto Musmanno,et al.  Tabu Search Heuristics for the Arc Routing Problem with Intermediate Facilities under Capacity and Length Restrictions , 2004, J. Math. Model. Algorithms.

[9]  Kaarthik Sundar,et al.  Route planning algorithms for unmanned aerial vehicles with refueling constraints , 2012, 2012 American Control Conference (ACC).

[10]  J. Desrosiers,et al.  A Primer in Column Generation , 2005 .

[11]  A. Atamtūrk,et al.  Sequence Independent Lifting for Mixed-Integer Programming , 2004 .

[12]  Gilbert Laporte,et al.  Improvements and extensions to the Miller-Tucker-Zemlin subtour elimination constraints , 1991, Oper. Res. Lett..

[13]  E. Balas,et al.  Facets of the Knapsack Polytope From Minimal Covers , 1978 .

[14]  Dominik Goeke,et al.  The Electric Vehicle-Routing Problem with Time Windows and Recharging Stations , 2014, Transp. Sci..

[15]  Kaarthik Sundar,et al.  Algorithms for Routing an Unmanned Aerial Vehicle in the Presence of Refueling Depots , 2013, IEEE Transactions on Automation Science and Engineering.

[16]  R. A. Zemlin,et al.  Integer Programming Formulation of Traveling Salesman Problems , 1960, JACM.

[17]  Jean-Claude Latombe,et al.  Robot motion planning , 1970, The Kluwer international series in engineering and computer science.

[18]  Richard F. Hartl,et al.  A variable neighborhood search for the capacitated arc routing problem with intermediate facilities , 2008, J. Heuristics.

[19]  Matteo Fischetti,et al.  Solving the Orienteering Problem through Branch-and-Cut , 1998, INFORMS J. Comput..

[20]  R. Gomory.  Some polyhedra related to combinatorial problems , 1969 .

[21]  R. Ravi,et al.  Approximation algorithms for distance constrained vehicle routing problems , 2012, Networks.

[22]  Chung-Lun Li,et al.  On the Distance Constrained Vehicle Routing Problem , 1992, Oper. Res..

[23]  Richard F. Hartl,et al.  The Electric Fleet Size and Mix Vehicle Routing Problem with Time Windows and Recharging Stations , 2013, Eur. J. Oper. Res..

[24]  K. Sundar,et al.  Heuristics for Routing Heterogeneous Unmanned Vehicles with Fuel Constraints , 2014 .

[25]  Dirk Van Oudheusden,et al.  The orienteering problem: A survey , 2011, Eur. J. Oper. Res..

[26]  Kaarthik Sundar,et al.  Generalized multiple depot traveling salesmen problem - Polyhedral study and exact algorithm , 2015, Comput. Oper. Res..