A solid transportation problem with safety factor under different uncertainty environments

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.

Computation of Some Stochastic Transportation Problems Using Essen Inequality

“…In another article, they [9] presented a stochastic transportation problem where the supply and demand parameters follow extream value distribution and some of the cost parameters are multi-choice type. Baidya et al [10] presented a new concept of the safety factor in a solid transportation problem and developed five different models in which they considered the availability of source parameters, demands, and capacities of conveyances as fuzzy numbers namely, trapezoidal and triangular fuzzy numbers. Kundu et al [11] studied threedimensional multi-objective transportation problems in which they considered penalties as random, fuzzy and hybrid variables.…”

A multi-objective solid transportation problem with reliability for damageable items in random fuzzy environment

“…Samanta and Roy (2005) proposed an algorithm for solving multi-objective entropy transportation problem under fuzzy environment. Also, Baidya et al (2013) and Chakraborty et al (2014) have developed a STP in uncertainty environments.…”

A bi-objective model for uncertain multi-modal shortest path problems

“…Constraints (12) and (13) mean that the total travel time and travel cost will be less than f 1 and f 2 with confidence levels α and β, respectively. Constraints (14), (15), and (16) ensure that a feasible path can be generated in the network. Constraint (17) implies the total transfer times will not exceed the given limit N. It is noted that the travel time and travel cost on arcs are imprecise due to the complexity of the actual network.…”

“…In this case, uncertainty theory was founded to rationally deal with belief degrees, which enlightened a new approach to describe non-deterministic phenomena. Up to now, the uncertainty theory has been applied to many classical optimization problems, such as solid transportation problem [14], project scheduling problem [15], maximum flow problem [16], optimal assignment problem [17], uncertain graph [18], etc. As for the shortest path problem, Liu [19] introduced three concepts of uncertain path according to different decision criteria, including the expected shortest path, α-shortest path, and the most shortest path.…”