Soft Computing

pp 1–25

# Uncertain interval programming model for multi-objective multi-item fixed charge solid transportation problem with budget constraint and safety measure

Methodologies and Application

## Abstract

This paper presents uncertain interval programming models for multi-objective multi-item fixed charge solid transportation problem with budget constraint and safety measure (MOMIFCSTPBCSM). The human languages usually involve imperfect or unknown information and are in the lack of certainty, and often, it is impossible to exactly describe an existing state or a future outcome. In using the probability theory, we must have enough historical information to estimate the probability distributions and in the case of fuzzy theory, we must have a trustworthy membership function, which is not easy to do. Thus, we often estimate the degree of belief with some hesitation that each condition may occur. To deal with such a situation, the uncertain interval theory may be very useful. Based on these facts, the parameters of the formulated problem are chosen as uncertain intervals. We consider unit transportation costs, fixed charges, transportation times, deterioration of items, supplies at origins, demands at destinations, conveyance capacities, budget at each destination, selling prices and purchasing costs, and we assume the safety factor and the desired safety measure are interval uncertain parameters. To formulate the proposed MOMIFCSTPBCSM, we use interval theory and uncertain programming techniques to develop two different models: an Expected Value Model and a Chance-Constrained Model. The equivalent deterministic models are formulated and solved using a linear weighted method, a fuzzy programming method and the goal programming method.

## Keywords

Multi-objective multi-item fixed charge solid transportation problem Budget constraint Safety measure Deterioration of item Theories of interval

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