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In this paper, we introduce a new concept 'safety factor' in a transportation problem. When items are transported from plants to destinations through different conveyances, there are some difficulties/risks to transport the items due to bad road, insurgency, land slide, etc. in some routes. Due to these, a desired total safety factor is being introduced, and depending upon the nature of the safety factor, we develop five models. In this paper, a solid transportation problem (STP) with imprecise unit costs is considered. The sources' availabilities, destinations' demands, and capacities of conveyances are also represented by fuzzy numbers like trapezoidal and triangular numbers. The transportation problem has been formulated with and without a safety factor. To reduce the different models into its crisp equivalent, we introduce different methods as chance-constraint programming, an approach using interval approximation of fuzzy numbers and the application of the expected value model. Generalized reduced gradient technique is used to find the optimal solutions for a set of given numerical data. To illustrate the model, a numerical example has been presented and solved using LINGO.12 software. The effect of safety factors on transported amount is illustrated.

To maximization the total profit and minimization the total cost we investigate two transportation models with interval transportation parameters and we present a solution of our newly developed models using appropriate methods. The solid transportation problem has been formulated with and without interval entropy function defined by Shannon's measure of entropy. Also an additional constraint on the total budget at each destination is imposed. When we transport the goods from plants to destination then few items should be damage due to bad condition of route and vehicles, for this reason we introduce the breakability in our respective models. We employ two different approaches (Hu and Wang's Approach and Mahato and Bhunia's Approach) based on interval ranking and made a comparison on it. To convert the multi-objective programming problem into single-objective problem we apply weighted sum method. The models are illustrated with numerical examples and solved using the LINGO.13 optimization software. The results of the models with and without entropy are also obtained and compared.