Rough‐fuzzy quadratic minimum spanning tree problem

A quadratic minimum spanning tree (QMST) problem is to determine a minimum spanning tree of a connected graph having edges which are associated with linear and quadratic weights. The linear weights are the edge costs which are associated with every edge whereas the quadratic weights are the interaction costs between a pair of edges of the graph. In this paper, a bi-objective ({\alpha},\b{eta}) rough fuzzy quadratic minimum spanning tree problem (b-({\alpha},\b{eta})RFQMSTP) has been considered for a connected graph whose linear and quadratic weights are expressed as rough fuzzy variable. {\alpha} and \b{eta} determine the predefined confidence levels of the model. The b-({\alpha},\b{eta}) RFQMSTP is transformed into crisp form using chance-constrained programming technique. The crisp equivalent of b-({\alpha},\b{eta}) RFQMSTP model is solved using epsilon-constraint method and two multi-objective evolutionary algorithms: nondominated sorting genetic algorithm II (NSGA-II) and 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 techniques. A sensitivity analysis of the example is performed at different confidence levels of {\alpha} and \b{eta}. Finally a performance analysis of NSGA-II and MOCHC are performed on 5 randomly generated bi-objective ({\alpha},\b{eta}) rough fuzzy quadratic minimum spanning tree instances. The result of our proposed model when solved by epsilon-constraint method is obtained by using the standard optimization software, LINGO. Whereas the results of evolutionary algorithms for our model are obtained by using an open source optimization package, jMETAL.