A linearization method for quadratic minimum spanning tree problem

The crisp and fuzzy quadratic minimum spanning tree (Q-MST) problem can be formulated as a linear model, and thus, the global optimum can be obtained by the proposed method. Conventionally, the Q-MST problem, which contains a quadratic term in the objective function, is solved by genetic algorithm and other heuristic methods. However, these methods cannot guarantee to obtain a global optimal solution. To address this issue, the proposed method transforms the quadratic term into linear formulations for crisp and fuzzy Q-MST problems, and yields the global optimum solutions by linear integer programming. Two examples are given to demonstrate the proposed method in greater detail. decision environment, thus the fuzzy Q-MST problem was formulated. The three models, with simulation-based genetic algorithms, are more efficient than the approach of Zhou and Gen, which cannot guarantee achieving the global optimum. Whether a crisp or fuzzy Q-MST problem, all must be solved by genetic or other heuristic algorithms because of the nonlinear issues. Thus, the proposed method adopts Chang and Chang’s concept [1], which is to linearize the nonlinear terms of Gao and Lu’s models [3]. Both crisp and fuzzy Q-MST problems can be reformulated into a linear integer-programming problem, and thus, the global optimal solution can be easily obtained. The remainder of this paper is organized as follows. Section 2 introduces the crisp and fuzzy Q-MST problem. Section 3 presents Chang’s linearization method [1]. Section 4 discusses the proposed method. Section 5 provides two examples to illustrate the proposed method. Section 6 is the conclusion.