The Linear Multiple Choice Knapsack Problem with Two Criteria: Profit and Equity

We study an extension of the Linear Multiple Choice Knapsack (LMCK) Problem that considers two criteria. The problem can be used to find the optimal allocation of an available resource to a group of disjoint sets of activities, while also ensuring that a certain balance on the resource amounts allocated to the activity sets is attained. The first criterion maximizes the profit incurred by the implementation of the considered activities. The second criterion minimizes the maximum difference between the resource amounts allocated to any two sets of activities. We present the mathematical formulation and explore the fundamental properties of the problem. Based on these properties, we develop an efficient algorithm that obtains the entire frontier of nondominated solutions. The algorithm is very efficient compared to generic multiple objective linear programming (MOLP) algorithms. We present theoretical findings which provide insight into the behavior of the algorithm, and report computational results which demonstrate its efficiency.