A dynamic scheme for stochastic vehicle routing

This paper considers stochastic load-constrained vehicle routing systems in which a fleet of homogeneous vehicles is dispatched from a central depot to serve the demands of surrounding customers. Customer locations and load sizes are known only in distribution before operations. Variants of this problem are often treated with very simple uncoordinated operating schemes, which are mathematically tractable but not very flexible. This research proposes a more flexible scheme denoted threshold global sharing (TGS) that utilizes a real-time reoptimization control to significantly reduce costs. The TGS scheme jointly replans the operations of the entire vehicle fleet at a single decision epoch during the operating period. Exact models for determining an optimal configuration for such a scheme are intractable; thus, a continuous approximation approach is employed to generate near-optimal configurations for large-scale problems with many customers and vehicles. TGS is shown to improve substantially on current schemes. 1 Load-capacitated vehicle routing with uncertain demand This paper considers single-period, single-depot vehicle routing systems in which a fleet of homogeneous vehicles is dispatched from a central depot to serve the demands of surrounding customers. All customers are served during the period, and vehicles return to the depot after their tasks are completed. Such a system is load-capacitated if the set of tasks each vehicle can complete is constrained primarily by the size of the maximum load that each vehicle can transport. In stochastic load-capacitated vehicle routing systems, customer locations and/or shipment demands are not known with certainty during planning. Many real-world collection and distribution routing problems are best modeled as stochastic load-constrained systems (see Erera (2000) for application examples). Operators of such systems must design an operating scheme for the vehicle fleet that serves all customers with minimum long-run average cost. Transportation cost is typically modeled as a non-decreasing function of the number of vehicles dispatched and the total distance traveled. This paper considers a collection system where customer locations and load sizes are known only in distribution before operations. Variants of this problem are often treated with very simple uncoordinated operating schemes, which are mathematically tractable but not very flexible. This research proposes a more flexible scheme denoted threshold global sharing (TGS) that utilizes a real-time reoptimization control to significantly reduce costs. The TGS scheme jointly replans the operations of the entire vehicle fleet at a single decision epoch during the operating period. Exact models for determining an optimal configuration for such a scheme are intractable; thus, a continuous approximation approach is employed to generate near-optimal configurations for largescale problems with many customers and vehicles. TGS is shown to improve substantially on current schemes. The remaining sections of this paper are organized as follows. Section 2 classifies operating schemes for stochastic vehicle routing systems, and reviews relevant research. Section 3 then defines the TGS operating scheme. Section 4 defines the scheme configuration problem, and develops an idealized problem setting for the subsequent analysis. Section 5 develops an approximation model for the standard detour-to-depot scheme, and Section 6 for the TGS scheme. Section 7 briefly discusses a simulation model used to validate the approximation model. Finally, section 8 compares results for the two schemes on randomized problems. 2 Classification of operating schemes Determining an effective fleet operating scheme is typically very difficult for stochastic vehicle routing systems. The characteristics of these systems that create challenges include: 1. Customer service: Service constraints that guarantee that all customers are completely served during the operating period can often only be enforced probabilistically when demands are uncertain.