Using an exact bi-objective decoder in a memetic algorithm for arc-routing (and other decoder-expressible) problems

Abstract

We address the bi-objective Capacitated Arc Routing Problem (CARP) by considering two levels of solution interpretation: implicit and explicit solutions. An algorithm that translates implicit solutions into explicit solutions is called a decoder. In this work, the decoder takes as input a permutation of the required edges and generates a Pareto frontier of CARP solutions. While bi-objective CARP was our main focus and starting point, we could also use the proposed framework to solve a bi-objective version of the traveling salesman problem by plugging-in a different decoder. Recall that bi-objective CARP asks to service (the demands of) a set of required edges using a fleet of vehicles of limited capacity so as to minimize: (i) the total travelled distance and (ii) the length of the longest route. Any permutation s of the required edges constitutes an implicit CARP solution. The decoder constructs all non-dominated explicit solutions that service the edges in the order indicated by s, i.e., the decoder is an exact algorithm that returns the optimal Pareto frontier subject to the service order s. To achieve competitive CARP results it is also important to reinforce the decoder using a local search operator that acts on explicit routes (and that may change the service order s). For nine instances, the resulting algorithm was even able to find a new total-cost upper bound, improving upon the best solutions reported in the (considerably larger) mono-objective CARP literature. This shows that (some of) the proposed ideas can also be useful in single objective optimization: the second objective can be seen as a guide for the mono-objective search process.