A Simple and Efficient Technique to Generate Bounded Solutions for the Multidimensional Knapsack Problem: a Guide for OR Practitioners

The 0-1 Multidimensional Knapsack Problem (MKP) is a NP-Hard problem that has important applications in business and industry. Approximate solution approaches for the MKP in the literature typically provide no guarantee on how close generated solutions are to the optimum. This article demonstrates how general-purpose integer programming software (Gurobi) is iteratively used to generate solutions for the 270 MKP test problems in Beasley’s OR-Library such that, on average, the solutions are guaranteed to be within 0.094% of the optimums and execute in 88 seconds on a standard PC. This methodology, called the simple sequential increasing tolerance (SSIT) matheuristic, uses a sequence of increasing tolerances in Gurobi to generate a solution that is guaranteed to be close to the optimum in a short time. This solution strategy generates bounded solutions in a timely manner without requiring the coding of a problem-specific algorithm. The SSIT results (although guaranteed within 0.094% of the optimums) when compared to known optimums deviated only 0.006% from the optimums—far better than any published results for these 270 MKP test instances.

Optimization Method to Address Psychosocial Risks through Adaptation of the Multidimensional Knapsack Problem

This paper presents a methodological scheme to obtain the maximum benefit in occupational health by attending to psychosocial risk factors in a company. This scheme is based on selecting an optimal subset of psychosocial risk factors, considering the departments’ budget in a company as problem constraints. This methodology can be summarized in three steps: First, psychosocial risk factors in the company are identified and weighted, applying several instruments recommended by business regulations. Next, a mathematical model is built using the identified psychosocial risk factors information and the company budget for risk factors attention. This model represents the psychosocial risk optimization problem as a Multidimensional Knapsack Problem (MKP). Finally, since Multidimensional Knapsack Problem is NP-hard, one simulated annealing algorithm is applied to find a near-optimal subset of factors maximizing the psychosocial risk care level. This subset is according to the budgets assigned for each of the company’s departments. The proposed methodology is detailed using a case of study, and thirty instances of the Multidimensional Knapsack Problem are tested, and the results are interpreted under psychosocial risk problems to evaluate the simulated annealing algorithm’s performance (efficiency and efficacy) in solving these optimization problems. This evaluation shows that the proposed methodology can be used for the attention of psychosocial risk factors in real companies’ cases.

Simulated Annealing Algorithm for the Multiple Choice Multidimensional Knapsack Problem

The multiple choice multidimensional knapsack problem (MCMK) is a harder version of the 0/1 knapsack problem, and is ever more complex than the 0/1 multidimensional knapsack problem. In MCMK, there are several groups of items. The objective is to maximize the value (profit) by choosing exactly 1 item from each group such that all the constraints are satisfied. It is difficult and NP-hard even to find a solution that does not violate all constraints. In this work, we present a simulated annealing based algorithm with open source C++ code to find good solutions to the multidimensional multiple choice knapsack problem. In all of the benchmark instances we used, the algorithm is able to find optimum (or close) solutions, thereby proving that the algorithm is suitable for solving larger instances for which optimal solutions are unknown.