This paper introduces a novel class of generalized assignment problems with location/allocation considerations that arises in several applications including retail shelf space allocation. We consider a set of items where each item may represent a family of products and a set of variable-sized knapsacks that may represent shelves which comprise contiguous segments having distinct attractiveness. The decision-maker seeks to assign the set of items to these knapsacks, specify their segment assignments within knapsacks, and determine their total allocated space within predetermined lower/upper bounds in a fashion that maximizes a reward-based objective function. We develop an effective optimization-based Very Large-Scale Neighborhood Search (VLSN) that enables the derivation of improved solutions in a neighborhood by re-optimizing selected subsets of knapsacks using a proposed mixed-integer formulation. The VLSN approach greatly outperforms the best solution identified by CPLEX within one CPU hour, whereas general-purpose solver heuristics (e.g., feasibility pump, relaxation induced neighborhood search, or local branching) failed to provide feasible solutions to most of the larger instances within a time limit comparable to VLSN algorithm run times. To provide further evidence of the near-optimality of the VLSN solutions, we derived tighter upper bounds by solving a 0-1 MIP relaxation of the problem using an enhanced column generation approach. Our computational study was carried out on randomly generated computationally challenging instances with up to 210 items and 42 knapsacks. Our results demonstrate that the proposed approach delivers high-quality solutions that exhibit an average deviation below 0.4%.
We investigate modeling approaches and exact solution methods for a generalized assignment problem with location/allocation (GAPLA) considerations. In contrast with classical generalized assignment problems, each knapsack in GAPLA is discretized into consecutive segments having different levels of attractiveness. To maximize a total reward function, the decision maker decides not only about item knapsack assignments, but also the specific location of items within their assigned knapsacks and their total space allocation within prespecified lower and upper bounds. Mathematical programming formulations are developed for single and multiple knapsack variants of this problem along with valid inequalities, preprocessing routines, and model enhancements. Further, a branch-and-price algorithm is devised for a set partitioning reformulation of GAPLA, and is demonstrated to yield substantial computational savings over solving the original formulation using branch-and-bound/cut solvers such as CPLEX over challenging problem instances.
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