Development of an Exact Method for Zero-One Linear Programming Model

Abstract

The paper presents a new method for solving the 0–1 linear programming problems (LPs). The general 0–1 LPs are believed to be NP-hard and a consistent, efficient general-purpose algorithm for these models has not been found so far. Cutting planes and branch and bound approaches were the earliest exact methods for the 0–1 LP. Unfortunately, these methods on their own failed to solve the 0–1 LP model consistently and efficiently. The hybrids that are a combination of heuristics, cuts, branch and bound and pricing have been used successfully for some 0–1 models. The main challenge with these hybrids is that these hybrids cannot completely eliminate the threat of combinatorial explosion for very large practical 0–1 LPs. In this paper, a technique to reduce the complexity of 0–1 LPs is proposed. The given problem is used to generate a simpler version of the problem, which is then solved in stages in such a way that the solution obtained is tested for feasibility and improved at every stage until an optimal solution is found. The new problem generated has a coefficient matrix of 0 s and 1 s only. From this study, it can be concluded that for every 0–1 LP with a feasible optimal solution, there exists another 0–1 LP (called a double in this paper) with exactly the same optimal solution but different constraints. The constraints of the double are made up of only 0 s and 1 s. It is not easy to determine this double 0–1 LP by mere inspection but can be obtained in stages as given in the numerical illustration presented in this paper. The 0–1 integer programming models have applications in so many areas of business. These include large economic/financial models, marketing strategy models, production scheduling and labor force planning models, computer design and networking models, military operations, agriculture, wild fire fighting, vehicle routing and health care and medical models