In this paper, Capacitated vehicle routing problem(CVRP) with fuzzy stochastic demands have been presented. Discrete fuzzy random variables have been used to represent the demands of the customers. The objective of CVRP with fuzzy stochastic demands is to obtain a set of routes which originates as well as terminates at the source node and while traversing the route, the demands of all the customers present in the network are satisfied. The task here is to carry out all these operations with minimum cost. CVRP in imprecise and random environment has been considered here, and an a priori route construction technique has been adopted for which branch and bound algorithm has been used. The recourse policy used in this work is reactive, i.e. recourse to depot is only done only upon the occurrence of the failure. The delivery policy considered here is full delivery. Demands of the customers are the only source of impreciseness and randomness in the problem under construction. Parametric Graded Mean Integration Representation(PGMIR) method has been used for the comparison purposes, whenever required. A numerical example with 4 customers have been solved to present the proposed methodology.

This paper considers Capacitated Vehicle Routing Problem(CVRP) in an imprecise and random environment. The deterministic version of the problem deals with finding a set of routes in such a way that the demand of all the customers present in the network are satisfied and the cost incurred in performing these operations comes out to be a minimum. In practical life situations, problems are not always defined in crisp form. Phenomenons like randomness and impreciseness are quite natural to arise in real life. This work presents CVRP in such a mixed environment. In this work, the demands of the customers are assumed to be stochastic in nature and are revealed only upon the arrival of the salesman. Moreover, the edge weights are representing the time required to traverse the edge and hence are both imprecise and random in nature. Different traffic conditions, weather conditions and many other factors corresponds to the random nature of edge weights and varying speed of the vehicle corresponds to the impreciseness. Thus, in this work, edge weights are represented by discrete fuzzy random variables. In this paper, an expectation based approach has been used to deal with randomness. A procedure based on Branch and Bound algorithm has been used to find routes with minimum cost. A numerical example has also been presented to explain the working of the method proposed.

In this paper, an approach for solving the Intuitionistic Fuzzy Multi-Objective Linear Fractional Programming Problem (IFMOLFPP) has been presented. The coefficients of objective functions, resources, and all the technological coefficients are taken to be a triangular intuitionistic fuzzy number (TIFN). In this paper, a component-wise optimization method has been adopted to handle the impreciseness. First, each intuitionistic fuzzy fractional objective is transformed into an equivalent crisp multiple fractional objectives, and then using the fuzzy programming approach, an optimal solution is obtained for every intuitionistic fuzzy fractional objective. Then, the optimal solution of each individual intuitionistic fuzzy fractional objective is used to transform it into a corresponding intuitionistic fuzzy linear objective and thus the original problem now transforms to Multi-Objective Intuitionistic fuzzy LPP which is further transformed into an Intuitionistic Fuzzy LPP with a single objective and the problem is then solved by using component-wise optimization and fuzzy goal programming approach. A numerical example is also presented to explain the methodology of the approach defined in this paper.