Rough-fuzzy quadratic minimum spanning tree problem

A quadratic minimum spanning tree problem determines a minimum spanning tree of a network whose edges are associated with linear and quadratic weights. Linear weights represent the edge costs whereas the quadratic weights are the interaction costs between a pair of edges of the graph. In this study, a bi-objective rough-fuzzy quadratic minimum spanning tree problem has been proposed for a connected graph, where the linear and the quadratic weights are represented as rough-fuzzy variables. The proposed model is formulated by using rough-fuzzy chance-constrained programming technique. Subsequently, three related theorems are also proposed for the crisp transformation of the proposed model. The crisp equivalent models are solved with a classical multi-objective solution technique, the epsilon-constraint method and two multi-objective evolutionary algorithms: (a) nondominated sorting genetic algorithm II (NSGA-II) and (b) multi-objective cross-generational elitist selection, heterogeneous recombination, and cataclysmic mutation (MOCHC) algorithm. A numerical example is provided to illustrate the proposed model when solved with different methodologies. A sensitivity analysis of the example is also performed at different confidence levels. The performance of NSGA-II and MOCHC are analysed on five randomly generated instances of the proposed model. Finally, a numerical illustration of an application of the proposed model is also presented in this study.