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This paper considers a multi-objective fixed-charge transportation problem (MOFCTP) in which the parameters of the objective functions are random rough variables, while the supply and the demand parameters are rough variables. In real-life situations, the parameters of a multi-objective fixed-charge transportation problem may not be defined precisely, because of globalization of the market, uncontrollable factors, etc. As such, the multi-objective fixed-charge transportation problem is proposed under rough and random rough environments. To tackle uncertain (rough and random rough) parameters, the proposed model employs an expected value operator. Furthermore, a procedure is developed for converting the uncertain multi-objective fixed-charge transportation problem into a deterministic form and then solving the deterministic model. Three different methods, namely, the fuzzy programming, global criterion, and ϵ-constrained methods, are used to derive the optimal compromise solutions of the suggested model. To provide the preferable optimal solution of the formulated problem, a comparison is drawn among the optimal solutions that are extracted from different methods. Herein, the ϵ-constrained method derives a set of optimal solutions and generates an exact Paretofront. Finally, in order to show the applicability and feasibility of the proposed model, the paper includes a real-life example of a multi-objective fixed-charge transportation problem. The main contribution of the paper is that it deals with MOFCTP using two types of uncertainties, thus making the decision making process more flexible.

In this paper, we analyze the interval programming using interval and rough interval (RI). Interval programming is one of the tools to tackle uncertainty in mathematical programming. In the real world situations, we often encounter the cases where the data cannot be determined with certainty. So, the value of the data is assessed using an interval. Here, we consider that the parameters of the Fixed-Charge Transportation Problem (FCTP) are imprecise. In our proposed problem, uncertainties are considered in terms of interval and RI. We present a study of the FCTP using the interval programming. An example of the FCTP is included to show the effectiveness and usefulness of our proposed study. Finally, concluding remarks and future scopes of our paper are discussed.