Closed form distance formula for the balanced multiple travelling salesmen

TL;DR

This paper introduces a closed-form formula for the mTSP distance based on the number of vehicles and customers, and compares heuristics for balancing routes, providing both theoretical and empirical insights.

Contribution

It presents a novel closed-form formula for mTSP distances and compares heuristics, enhancing understanding of route balancing and distance estimation.

Findings

Nearest node heuristic is about 1% better in solution quality.

Distance grows approximately linearly with the number of vehicles.

Distance is nearly proportional to the square root of the number of customers.

Abstract

As a first contribution the mTSP is solved using an exact method and two heuristics, where the number of nodes per route is balanced. The first heuristic uses a nearest node approach and the second assigns the closest vehicle (salesman). A comparison of heuristics with test-instances being in the Euclidean plane showed similar solution quality and runtime. On average, the nearest node solutions are approximately one percent better. The closest vehicle heuristic is especially important when the nodes (customers) are not known in advance, e.g. for online routing. Whilst the nearest node is preferable when one vehicle has to be used multiple times to service all customers. The second contribution is a closed form formula that describes the mTSP distance dependent on the number of vehicles and customers. Increasing the number of salesman results in an approximately linear distance growth for uniformly distributed nodes in a Euclidean grid plane. The distance growth is almost proportional to the square root of number of customers (nodes). These two insights are combined in a single formula. The minimum distance of a node to $n$ uniformly distributed random (real and integer) points was derived and expressed as functional relationship dependent on the number of vehicles. This gives theoretical underpinnings and is in agreement with the distances found via the previous mTSP heuristics. Hence, this allows to compute all expected mTSP distances without the need of running the heuristics.