The Profitable Single Truck and Trailer Routing Problem with Time Windows: Formulation, valid inequalities and branch-and-cut algorithms

Abstract

The Profitable Single Truck and Trailer Routing Problem with Time Windows (PSTTRPTW) is defined by a fleet of trucks and trailers and a set of customers that reward a profit if their supply is collected within a time window. Trailers can be attached to trucks in order to raise capacity, at the expense of increasing travel time and cost. Trailers can be decoupled from trucks at satellite depots, after which the truck performs subtours, visiting customers inaccessible by trailers. A route may include any number of subtours, as long as the vehicle’s capacity is respected. The goal is to find a route for a single choice of truck and trailer maximizing the difference between the collected profit and the route cost. We introduce an Integer Programming formulation, preprocessing tests, valid inequalities and investigate two PSTTRPTW Branch-and-cut (BC) algorithms, one of them based on the basic formulation and the other, on the strengthened model. Our numerical results show that these preprocessing tests reduced the number of variables by about 30%. In addition, the Linear Programming relaxations provided by the strongest model are, on average, 25% sharper than those given by the basic model. As a result, the reinforced BC algorithm has the edge: it is faster, solves all instances solved by its competitor and 13 additional ones, and provides optimality gaps 14% smaller when optimality is not certified.