Imprecise constrained covering solid travelling salesman problem with credibility

In this article, we model an “Imprecise Constrained Covering Solid Travelling Salesman Problem with Credibility” (ICCSTSPC), a generalization of Covering Salesman Problem (CSP), in fuzzy environment. A salesman begins from an initial node, visits a subset of nodes exactly once using any one of appropriate vehicles available at each step, so that unvisited nodes are within a predetermined distance from the visited nodes, and returns to the initial node within a restricted time. Here the travelling costs and travelling times between any two nodes and the covering distance all are considered as fuzzy. Thus the problem reduces to find the optimal tour for a set of nodes with the proper conveyances so that total travelling cost is minimum within a restricted time. The ICCSTSPC is reduced to a set of Imprecise Constrained Covering Solid Travelling Salesman Problems by solving Unicost Set Cover Problem (USCP) using Random Insertion-Deletion (RID). These reduced Constrained Solid Travelling Salesman Problems (CSTSPs) are solved by an Improved Genetic Algorithm (IGA), which consists of probabilistic selection, order crossover, proposed generation dependent inverse mutation. A random mutation for vehicles is proposed to get a better cost at each generation of IGA by choosing an alternative vehicle for each node. Hence the ICCSTSPC is solved by a random insertion-deletion (RID) for covering set and IGA, i.e., RID-IGA. To justify the performance of the RID-IGA, some test problems are solved. The model is illustrated with some randomly generated crisp and fuzzy data.