The Capacitated Canadian Postman Problem

Abstract

Routing problems can be classified according to where the demand is located on the network: on nodes, on arcs or on both. The travelling salesman problem (tsp) is an example in which a vehicle must visit every node of the network and return to the original node at minimum cost. If a capacity restraint is added to the vehicle and the total demand exceeds the capacity of one vehicle, a vehicle routing problem (vrp) is obtained. When the demand to be satisfied lies on the arcs of the network, the equivalent to the tsp is the Chinese postman problem (cpp). This problem consists of covering, at minimum cost, all the arcs of a network with a route starting and ending at a specified originating point. If the generation of more than one route to serve the demand is necessary because of a capacity constraint, the result is termed the capacitated Chinese postmand problem (ccpp). This paper adapts the mathematical formulation of the ccpp to the context of the Canada post corporation. No capacity constraints on the route are present in this context, but there is a time limit on each route. Furthermore, this time limit includes the travel time between a central depot and the beginning and end of the route. A general heuristic is presented to solve the problem under certain hypotheses. To solve the sequencing problem, a euler's tour problem is defined with penalties assigned to each change of arc; this penalty corresponds to street crossing and the change of streets when passing from one arc to another. This problem has many other real-life applications, including mail distribution, snow removal and salt application, garbage collection, rural school transportation and parking meter collection.