On some generalized routing problems

The paper examines applications of combinatorial optimization in logistics and transport, and considers some mathematical optimization problems from the perspective of Operational Research (OR). It also lists some of the numerous applications of these problems in economic reality, describes how such problems can be mathematically modelled, proposes algorithms for their solution, and presents the results of extensive computational experiments with implementations of the proposed algorithms. Routing problems consider (primary) objects which are able to move on or over or along other (secondary) objects. That is, the primary objects can be subjected to spatial-temporal transformations, and they are themselves able to perform such transformations on the secondary objects in order to fulfill given tasks to achieve certain objectives. This may not be done in an arbitrary manner, but only in compliance with given general conditions, within a general framework or set-up. The problem consists in finding the best possible manner of movements, respectively, of spatial-temporal transformations of the primary objects, with regard to the given objectives, while respecting the general conditions. Routing problems are mathematical optimization problems (arithmetic problems), i.e., they consist of decision variables, a (not necessarily scalar) objective function, and side constraints. Routing problems are modelled and formulated on weighted graphs (networks). The solutions to routing problems consist completely or partly of undirected or directed walks in the underlying networks. Some well-known routing problems are the shortest path problem (SPP), the travelling salesman problem (TSP), the vehicle routing problem with time windows (VRPTW), and the Chinese postman problem (CPP). The paper considers the following three generalized routing problems: (i) the generalized directed rural postman problem (GDRPP) (which generalizes the CPP) The GDRPP is a straightforward generalization of the directed rural postman problem (DRPP). The DRPP consists in finding an optimal postman tour in a digraph, i.e., a least-cost cycle traversing each arc of a specified subset of the digraph’s arcs at least once. The GDRPP is to the DRPP what the generalized travelling salesman problem (GTSP) is to the travelling salesman problem (TSP): In the GDRPP, there are several subsets (groups, classes, clusters) of arcs and the requirement is to find a least-cost cycle traversing at least one arc from each subset at least once. (ii) the vehicle routing problem with trailers and transshipments (VRPTT) (which generalizes the VRPTW) In the VRPTT, the vehicle fleet consists of autonomous vehicles able to move on their own, and of non-autonomous vehicles which must be accompanied by an autonomous vehicle to be able to move (e.g., lorries and trailers). Moreover, both autonomous and non-autonomous vehicles are either collection or support vehicles. Collection vehicles are used to pick up the supplies of the customers. Support vehicles are used as mobile depots by the collection vehicles. (iii) the truck-and-trailer routing problem (TTRP) (also a generalization of the VRPTW and a special case of the VRPTT) In the TTRP, there is a fixed assignment of each non-autonomous vehicle to an autonomous vehicle. This means that no support vehicles are used.