GRASP with path relinking for commercial districting

• A real-world territory design problem is addressed.
• A new more robust measure of dispersion is studied.
• A GRASP with path relinking heuristic is proposed.
• GRASP-based construction phase enhances performance of existing mechanisms.
• Both static and dynamic path relinking strategies improve algorithmic performance.

The problem of grouping basic units into larger geographic territories subject to dispersion, connectivity, and balance requirements is addressed. The problem is motivated by a real-world application from the bottled beverage distribution industry. Typically, a dispersion function is minimized as compact territories are sought. Existing literature reveals that practically all the works on commercial districting use center-based dispersion functions. These center-based functions yield mixed-integer programming models with some nice properties; however, they have the disadvantage of being very costly to be properly evaluated when used within heuristic frameworks. This is due to the center updating operations frequently needed through the heuristic search. In this work, a more robust dispersion measure based on the diameter of the formed territories is studied. This allows a more efficient heuristic search computation. For solving this particular territory design problem, a greedy randomized adaptive search procedure (GRASP) that incorporates a novel construction procedure where territories are formed simultaneously in two main stages using different criteria is proposed. This also differs from previous literature where GRASP was used to build one territory at a time. The GRASP is further enhanced with two variants of forward-backward path relinking, namely static and dynamic. Path relinking is a sophisticated and very successful search mechanism. This idea is novel in any districting or territory design application to the best of our knowledge. The proposed algorithm and its components have been extensively evaluated over a wide set of data instances. Experimental results reveal that the construction mechanism produces feasible solutions of acceptable quality, which are improved by an effective local search procedure. In addition, empirical evidence indicate that the two path relinking strategies have a significant impact on solution quality when incorporated within the GRASP framework. The ideas and components of the developed method can be further extended to other districting problems under balancing and connectivity constraints.