Fully fuzzy multi-item two-stage fixed charge four-dimensional transportation problems with flexible constraints

In real-life transportation problems (TPs), the demands, availabilities, transportation capacities, transportation costs, and fixed charges are uncertain. For this reason, the transported amounts from sources to destinations become uncertain. When the parameters and the decision variables of a problem are fuzzy in nature, the environment is termed ‘fully fuzzy’. Nowadays, with the development of infrastructure for transportation, TPs are developed with several conveyances and routes between sources and destinations. Till now, this type of TPs, i.e., four-dimensional TPs (4DTPs), did not receive much attention. Considering this type of practical problem, a fully fuzzy multi-item two-stage fixed charge 4DTP (FF-MITSFC-4DTP) with breakability during the transportation is considered. The problem consists of the models without and with flexible constraints. Two different methods, the modified graded mean integrated value method (MGMIVM) and an algorithm based on the order relation of fuzzy numbers, are, respectively, used to convert the fully fuzzy models without and with flexibility into respective equivalent deterministic problems. The defuzzified deterministic problems are solved using the generalized reduced gradient (GRG) method through Lingo (18.0) software. The efficiency of the methods is illustrated by solving a real-life problem numerically and comparing the results with an existing model. Results of particular TPs of different dimensions and without breakability are obtained and compared. Some managerial insights are discussed. A sensitivity analysis is presented. The novelty of this investigation is that, for the first time, real-life multi-item two-stage fully fuzzy 4DTPs with flexible constraints for minimum cost and their appropriate solution methodologies are presented.