A Unified Maximum Entropy Principle Approach for a Large Class of Routing Problems

Abstract

We present a novel modeling and algorithmic approach, a Maximum Entropy Principle (MEP) heuristic for Routing and Scheduling, for a large class of problems including the Traveling Salesman Problem (TSP), multiple Traveling Salesmen Problem (mTSP), the Vehicle Routing Problem (VRP) and the Close-Enough Traveling Salesman Problem (CETSP). Our approach models these routing and scheduling problems as ‘equivalent’ facility location problems with side-constraints, and then employs tools from statistical physics for assigning resources (routes/vehicles) to each node (city) such that the resource allocation results in feasible, sub-optimal routes. The approach is very flexible and can incorporate side-constraints such as minimum tour-lengths, capacity constraints, schedule constraints, and reachability constraints (like CETSP). Analytically, our model results in a second-order non-linear system of complex implicit equations; and we show that an iterative approach effectively solves these equations, is equivalent to a gradient descent and converges to a local minimum. While the optimization model is non-linear, the algorithm converges to an integer optimal solution. Computationally, we compare our approach to the Simulated Annealing (SA) heuristic, the CMT-14 benchmark instances for the VRP and randomly generated instances for the CETSP. Our approach consistently outperforms SA for all constrained routing problems. On the CMT-14 benchmark instances, our approach finds the optimal (when verifiable) number of vehicles, with a cumulative tour distance within 5.7% and in comparable computation times of the best-known solutions (over all approaches for each instance). We also demonstrate the efficacy of our approach on randomly generated instances of the CETSP and discuss our results.