A Study of Proxies for Shapley Allocations of Transport Costs

TL;DR

This paper explores methods to approximate the Shapley value for transport cost allocation in vehicle routing, demonstrating that some proxies are computationally feasible and closely match the ideal allocations in real-world and synthetic scenarios.

Contribution

It introduces six proxies for the Shapley value in transport cost allocation and proves the NP-hardness of exact approximation, providing practical solutions for operational use.

Findings

Several proxies closely approximate the Shapley value.

Approximation within a constant factor is NP-hard.

Proxies perform well in real-world and synthetic tests.

Abstract

We propose and evaluate a number of solutions to the problem of calculating the cost to serve each location in a single-vehicle transport setting. Such cost to serve analysis has application both strategically and operationally in transportation. The problem is formally given by the traveling salesperson game (TSG), a cooperative total utility game in which agents correspond to locations in a traveling salesperson problem (TSP). The cost to serve a location is an allocated portion of the cost of an optimal tour. The Shapley value is one of the most important normative division schemes in cooperative games, giving a principled and fair allocation both for the TSG and more generally. We consider a number of direct and sampling-based procedures for calculating the Shapley value, and present the first proof that approximating the Shapley value of the TSG within a constant factor is NP-hard. Treating the Shapley value as an ideal baseline allocation, we then develop six proxies for that value which are relatively easy to compute. We perform an experimental evaluation using Synthetic Euclidean games as well as games derived from real-world tours calculated for fast-moving consumer goods scenarios. Our experiments show that several computationally tractable allocation techniques correspond to good proxies for the Shapley value.