The Stochastic Vehicle Routing Problem for Minimum Unmet Demand

In this chapter, we are interested in routing vehicles to minimize unmet demand with uncertain demand and travel time parameters. Such a problem arises in situations with large demand or tight deadlines so that routes that satisfy all demand points are difficult or impossible to obtain. An important application is the distribution of medical supplies to respond to large-scale emergencies, such as natural disasters or terrorist attacks. We present a chance constrained formulation of the problem that is equivalent to a deterministic problem with modified demand and travel time parameters under mild assumptions on the distribution of stochastic parameters and relate it to a robust optimization approach. A tabu heuristic is proposed to solve this MIP and simulations are conducted to evaluate the quality of routes generated from both deterministic and chance constrained formulations. We observe that chance constrained routes can reduce the unmet demand by around 2%-6% for moderately tight deadline and total supply constraints.

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

[2]  Dimitris Bertsimas,et al.  A Stochastic and Dynamic Vehicle Routing Problem in the Euclidean Plane , 1991, Oper. Res..

[3]  Michel Gendreau,et al.  A Tabu Search Heuristic for the Vehicle Routing Problem with Soft Time Windows , 1997, Transp. Sci..

[4]  Gilbert Laporte,et al.  The vehicle routing problem: An overview of exact and approximate algorithms , 1992 .

[5]  Gilbert Laporte,et al.  The Vehicle Routing Problem with Stochastic Travel Times , 1992, Transp. Sci..

[6]  Gilbert Laporte,et al.  STOCHASTIC VEHICLE ROUTING. , 1996 .

[7]  Antoine Jézéquel,et al.  Probabilistic vehicle routing problems , 1985 .

[8]  Edward P. C. Kao,et al.  A Preference Order Dynamic Program for a Stochastic Traveling Salesman Problem , 1978, Oper. Res..

[9]  Dimitris Bertsimas,et al.  A Vehicle Routing Problem with Stochastic Demand , 1992, Oper. Res..

[10]  Gilbert Laporte,et al.  An Exact Algorithm for the Vehicle Routing Problem with Stochastic Demands and Customers , 1995, Transp. Sci..

[11]  Alain Hertz,et al.  A Tabu Search Algorithm for the Split Delivery Vehicle Routing Problem , 2003, Transp. Sci..

[12]  James P. Kelly,et al.  A Network Flow-Based Tabu Search Heuristic for the Vehicle Routing Problem , 1996, Transp. Sci..

[13]  Petros A. Ioannou,et al.  Truck route planning in nonstationary stochastic networks with time windows at customer locations , 2006, IEEE Transactions on Intelligent Transportation Systems.

[14]  Paolo Toth,et al.  The Granular Tabu Search and Its Application to the Vehicle-Routing Problem , 2003, INFORMS J. Comput..

[15]  Giuseppe Carlo Calafiore,et al.  Uncertain convex programs: randomized solutions and confidence levels , 2005, Math. Program..

[16]  G. Clarke,et al.  Scheduling of Vehicles from a Central Depot to a Number of Delivery Points , 1964 .

[17]  Moshe Dror,et al.  Vehicle Routing with Stochastic Demands: Properties and Solution Frameworks , 1989, Transp. Sci..

[18]  Gilbert Laporte,et al.  Models and exact solutions for a class of stochastic location-routing problems , 1987 .

[19]  Eleni Hadjiconstantinou,et al.  Routing under Uncertainty: An Application in the Scheduling of Field Service Engineers , 2002, The Vehicle Routing Problem.

[20]  Melvyn Sim,et al.  Robust discrete optimization and network flows , 2003, Math. Program..

[21]  Giovanni Righini,et al.  Heuristic algorithms for the vehicle routing problem with simultaneous pick-up and delivery , 2007, Comput. Oper. Res..

[22]  Thomas L. Morin,et al.  Generalized Dynamic Programming for Stochastic Combinatorial Optimization , 1989, Oper. Res..

[23]  M. Dror.  Modeling vehicle routing with uncertain demands as a stochastic program: Properties of the corresponding solution , 1993 .

[24]  Nicola Secomandi,et al.  A Rollout Policy for the Vehicle Routing Problem with Stochastic Demands , 2001, Oper. Res..

[25]  A R Odoni,et al.  THE PROBABILISTIC VEHICLE ROUTING PROBLEM. VEHICLE ROUTING: METHODS AND STUDIES. STUDIES IN MANAGEMENT SCIENCE AND SYSTEMS - VOLUME 16 , 1988 .

[26]  Michael D. Metzger,et al.  Emergency Response for Homeland Security: Lessons Learned and the Need for Analysis , 2004 .

[27]  Garud Iyengar,et al.  Ambiguous chance constrained problems and robust optimization , 2006, Math. Program..

[28]  Arkadi Nemirovski,et al.  Robust Convex Optimization , 1998, Math. Oper. Res..

[29]  Jason D. Papastavrou,et al.  A stochastic and dynamic model for the single-vehicle pick-up and delivery problem , 1999, Eur. J. Oper. Res..

[30]  Peng Sun,et al.  A Robust Optimization Perspective on Stochastic Programming , 2007, Oper. Res..

[31]  Moshe Dror,et al.  Stochastic vehicle routing with modified savings algorithm , 1986 .

[32]  Dimitris Bertsimas,et al.  Stochastic and Dynamic Vehicle Routing in the Euclidean Plane with Multiple Capacitated Vehicles , 1993, Oper. Res..

[33]  Ibrahim H. Osman,et al.  Metastrategy simulated annealing and tabu search algorithms for the vehicle routing problem , 1993, Ann. Oper. Res..

[34]  Jürgen Branke,et al.  Waiting Strategies for Dynamic Vehicle Routing , 2005, Transp. Sci..

[35]  J. Papastavrou.  A stochastic and dynamic routing policy using branching processes with state dependent immigration , 1996 .

[36]  Michel Gendreau,et al.  A Tabu Search Heuristic for the Vehicle Routing Problem with Stochastic Demands and Customers , 1996, Oper. Res..

[37]  David Simchi-Levi,et al.  A New Generation of Vehicle Routing Research: Robust Algorithms, Addressing Uncertainty , 1996, Oper. Res..

[38]  Russell Bent,et al.  Scenario-Based Planning for Partially Dynamic Vehicle Routing with Stochastic Customers , 2004, Oper. Res..

[39]  Dritan Nace,et al.  A robust approach to the chance-constrained knapsack problem , 2008, Oper. Res. Lett..

[40]  F. Ordóñez,et al.  A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty , 2008 .

[41]  Gilbert Laporte,et al.  A Tabu Search Heuristic for the Vehicle Routing Problem , 1991 .

[42]  Gilbert Laporte,et al.  Designing collection routes through bank branches , 1991, Comput. Oper. Res..

[43]  Bruce L. Golden,et al.  Stochastic vehicle routing: A comprehensive approach , 1983 .