An expected value model of quadratic clique problem on a graph with fuzzy parameters

A clique is a subset of vertices of a graph, such that there is an edge between every pair of vertices of the subset. The maximum weight clique problem, which has wide applications in the fields of communication, and distribution networks, asks for a clique of maximum weight. Here we have considered the quadratic version of the maximum weight clique problem. Costs of all the vertices are considered as fuzzy numbers. The direct cost associated with the vertices may represent the construction or running cost, whereas, the penalty cost, which is caused by simultaneous selection of particular pair of vertices, may explain two interactional activities. Using a generalization of credibility measure, the problem is formulated as an expected value model. Then the crisp equivalents are derived when the fuzzy costs are characterized by intervals or triangular fuzzy numbers. Furthermore, a fuzzy simulation based hybrid genetic algorithm is designed to solve the proposed model.