Three Optimization Methods for Solving the Fully Rough Interval Multilevel Solid Transportation Problem

The paper introduces a multilevel model for the solid transportation problem (STP) involving rough interval parameters. When multiple decision makers are available to model STP in any hierarchical system, multilevel programming addresses the issue. Rough interval environments provide a comparatively new and broader alternative by giving the decision-makers more flexible ways to handle the uncertain information during the decision-making process. Our literature review indicates that no algorithm has been developed for multilevel solid transportation problems employing rough interval numbers as parameters. First, we present a decomposition technique for addressing the multilevel STP, referred to as FRMSTP, wherein all parameters and decision variables exist within rough interval settings. This technique decomposes the FRMSTP into four distinct linear multilevel STPs, with an additional bounded variable constraint for the crisp problems, wherein the optimization variables of the lower problems are treated as parameters in the upper problems. Secondly, we enhance three methodologies—the constraint technique, the interactive approach, and the fuzzy approach—by employing imprecise interval data to derive a compromised optimal solution for the FRMSTP. A numerical example is provided to highlight the advantages of utilizing computational techniques. For the comparison, the results from the example indicated that all methods found a very similar rough preferred solution for the same objective functions.