Exact Solution Methods for a Generalized Assignment Problem with Location/Allocation Considerations

Abstract

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.

Solving a Special Case of the Generalized Assignment Problem Using the Modified Differential Evolution Algorithms: A Case Study in Sugarcane Harvesting

“…The mutation process needs to be executed in order to get mutant vectors. Mutant vectors can be obtained by using Equation (12), which randomly selects three vectors out of all target vectors. Let r 1 , r 2 , and r 3 denote randomly selected vectors from target vectors, while F is the scaling factor.…”

“…S h,G−1 is the score of the set of vectors h in the last iteration. S h,G−1 is set at 5 when the set of vectors h is used in Equation (12) in the last iteration, and it is found as the new optimal solution. S h,G−1 is set at 3 when the set of vectors h is used in Equation 12in the last iteration, and it is found as the new personal best solution.…”

“…S h,G−1 is set at 3 when the set of vectors h is used in Equation 12in the last iteration, and it is found as the new personal best solution. S h,G−1 is set at 1 when the set of vectors h is used in Equation (12) in the last iteration, and it gives the worst solution. For example, if the current attractiveness of TV, BV, and RV is 100, 120, and 90, respectively, in the particular iteration G. The calculated new score is shown in Table 8.…”

“…Alfares [10] tried to optimize the number of workers, while Elshafei and Alfares [11] tried to minimize the labor cost. Recently, location/allocation has been considered while making decisions in GAP [12]. The proposed problem that is closest to the GAP type is the multilevel GAP, which was introduced by Laguna et al [13].…”

Using Product Network Analysis to Optimize Product-to-Shelf Assignment Problems

“…The reward resulting from assigning a product category to a shelf depends on the attractiveness of the shelf segments it is assigned to and its aggregate gross profit. Ghoniem et al [15] studied the unclustered variant of the problem, where product categories are taken individually, as a generalized assignment problem with location/allocation considerations for which they develop preprocessing schemes, valid inequalities, and a branch-and-price algorithm that significantly outperforms CPLEX for small-and mid-sized instances. Zhao et al [16] developed integrated optimization models with inventory replenishment, shelf display location, and shelf space allocation.…”