Using dual feasible functions to construct fast lower bounds for routing and location problems

In cutting and packing problems, Dual Feasible Functions (DFFs) represent a well established tool for deriving high quality lower bounds in very short times. A well-known DFF lower bounding approach consists of using DFFs to rapidly generate feasible values for the dual variables of a Column Generation (CG) program (i.e., of an extended formulation commonly associated to CG). This paper presents a method for extending this approach to problems such as Capacitated p-Median, Distance Constrained General Routing and related variants (e.g., Capacitated Arc-Routing). In contrast to classical DFF bounds for cutting and packing, our extended DFF bounds deal with non-equal column costs, i.e., the column costs (primal objective function coefficients) have a more complex structure depending on some sums of distances. The general idea is to use a (reformulated) classical CG model in which a feasible dual solution is expressed as a linear combination of both DFF and non-DFF terms. In fact, one of the proposed approaches (the mixed CG-DFF bound for Capacitated p-Median in Section 2.2) still requires optimizing a restricted CG program, but with fewer variables and easier pricing sub-problems. The most refined bound version is the “DFF warm-started CG” from Section 2.2.2: it takes dual constraints generated while solving the above restricted CG program are re-uses them to warm-start a full CG phase. In the best cases, this can yield a speed-up between 2 and 3 relative to the pure CG. We present numerical experiments on Capacitated p-Median, Capacitated Arc-Routing (with fixed costs) and Distance Constrained Arc Routing; the numerical comparisons with the CG bound concern both the quality and the running time.