A Stochastic and Dynamic Vehicle Routing Problem in the Euclidean Plane

We propose and analyze a generic mathematical model for dynamic, stochastic vehicle routing problems, the dynamic traveling repairman problem (DTRP). The model is motivated by applications in which the objective is to minimize the wait for service in a stochastic and dynamically changing environment. This is a departure from classical vehicle routing problems where one seeks to minimize total travel time in a static, deterministic environment. Potential areas of application include repair, inventory, emergency service and scheduling problems. The DTRP is defined as follows: Demands for service arrive in time according to a Poisson process, are independent and uniformly distributed in a Euclidean service region, and require an independent and identically distributed amount of on-site service by a vehicle. The problem is to find a policy for routing the service vehicle that minimizes the average time demands spent in the system. We propose and analyze several policies for the DTRP. We find a provably optima...

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

[2]  J. Kingman Some inequalities for the queue GI/G/1 , 1962 .

[3]  William L. Maxwell,et al.  Theory of scheduling , 1967 .

[4]  Leonard Kleinrock,et al.  Queueing Systems - Vol. 1: Theory , 1975 .

[5]  Teofilo F. Gonzalez,et al.  P-Complete Approximation Problems , 1976, J. ACM.

[6]  Richard M. Karp,et al.  Probabilistic Analysis of Partitioning Algorithms for the Traveling-Salesman Problem in the Plane , 1977, Math. Oper. Res..

[7]  Averill M. Law,et al.  A Sequential Procedure for Determining the Length of a Steady-State Simulation , 1979, Oper. Res..

[8]  Richard C. Larson,et al.  Urban Operations Research , 1981 .

[9]  J. Steele Subadditive Euclidean Functionals and Nonlinear Growth in Geometric Probability , 1981 .

[10]  Sheldon M. Ross,et al.  Stochastic Processes , 2018, Gauge Integral Structures for Stochastic Calculus and Quantum Electrodynamics.

[11]  Averill M. Law,et al.  Simulation Modeling and Analysis , 1982 .

[12]  Michael J. Ferguson,et al.  Exact Results for Nonsymmetric Token Ring Systems , 1985, IEEE Trans. Commun..

[13]  Eugene L. Lawler,et al.  A Guided Tour of Combinatorial Optimization , 1985 .

[14]  N. Biggs THE TRAVELING SALESMAN PROBLEM A Guided Tour of Combinatorial Optimization , 1986 .

[15]  Dimitri P. Bertsekas,et al.  Data Networks , 1986 .

[16]  D. Bertsimas Probabilistic combinatorial optimization problems , 1988 .

[17]  L. Platzman,et al.  Heuristics Based on Spacefilling Curves for Combinatorial Problems in Euclidean Space , 1988 .

[18]  Amedeo R. Odoni,et al.  A single-server priority queueing-location model , 1988, Networks.

[19]  Patrick Jaillet,et al.  A Priori Solution of a Traveling Salesman Problem in Which a Random Subset of the Customers Are Visited , 1988, Oper. Res..

[20]  Dimitris Bertsimas,et al.  The probabilistic vehicle routing problem , 1988 .

[21]  Thomas L. Magnanti,et al.  Extremum Properties of Hexagonal Partitioning and the Uniform Distribution in Euclidean Location , 1988, SIAM J. Discret. Math..

[22]  John J. Bartholdi,et al.  Spacefilling curves and the planar travelling salesman problem , 1989, JACM.

[23]  E. Minieka The delivery man problem on a tree network , 1990 .

[24]  Patrick Jaillet,et al.  A Priori Optimization , 1990, Oper. Res..

[25]  Dimitris Bertsimas,et al.  Stochastic Dynamic Vehicle Routing in the Euclidean Plane: The Multiple-Server, Capacitated Vehicle Case , 1990 .

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