Large-scale constrained clustering for rationalizing pickup and delivery operations

The paper presents a three-phase procedure for clustering a large number of data points subject to both configuration and resource constraints. Motivated by the desire of a shipping carrier to reduce its fixed costs, the problem is to construct a set of compact work areas for regional pickup and delivery operations. In general terms, the objective is to find the minimum number of clusters (homogeneous vehicles) that satisfy volume, time and contiguity constraints. The problem is placed in context by formulating it as a mixed-integer goal program. Because practical instances contain anywhere from 6000 to 50,000 data points and can only be described in probabilistic terms, it is not possible to obtain provably optimal solutions to the proposed model. Instead, a novel solution methodology is developed that makes use of metaheuristic and mathematical programming techniques. In the preprocessing phase, a fraction of the data points are aggregated to reduce the problem size. This is shown to substantially decrease the computational burden without compromising solution quality. In the main step, an efficient adaptive search procedure is used to form the clusters. Randomness is introduced at each inner iteration to ensure a full exploration of the feasible region, and an incremental slicing scheme is used to overcome local optimality. In metaheuristic terms, these two refinements are equivalent to diversification and intensification search strategies. To improve the results, a set covering problem is solved in the final phase. The individual clusters obtained from the heuristic provide the structure for this model. To test the methodology, six data sets provided by the sponsoring company were analyzed. All runs for the first two phases took less than 4min, and in all but one case produced a tangible improvement over the current service area configurations. The set covering solution provided further improvement, which collectively averaged 11.2%.