Abstract
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.
Highlights
A transportation model plays a vital role in ensuring the efficient movement and intime availability of raw materials and finished goods from sources to destinations
Real-life problems are modeled with multi-objective functions which are measured in different respects, and they are noncommensurable and conflicting in nature
In order to describe a quantity with both fuzziness and randomness, we introduce the concept of hybrid variable as follows: Definition 7
Summary
A transportation model plays a vital role in ensuring the efficient movement and intime availability of raw materials and finished goods from sources to destinations. Our aim is to formulate and solve solid transportation problems with safety constraints with different types of uncertain (fuzzy, random, and hybrid) parameters. In order to describe a quantity with both fuzziness and randomness, we introduce the concept of hybrid variable as follows: Definition 7. Model 1: solid transportation problem with hybrid penalties, fuzzy resources, demands, conveyance capacities, and without safety factor. To formulate the model, we assume that there are no risks to transport the commodities from plants to destination by different conveyances, i.e., all routes are totally safe for the transport of goods, and unit transportation cost is a hybrid variable. Þ; ð8Þ xijk ≥ 0; ∀i; j; k: Model 2: solid transportation problem with crisp penalties, fuzzy resources, demands, conveyance capacities, and desired safety measure as crisp. 1 0 for xijk > 0 otherwise and B is the desired safety measure for the whole transportation system
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