Abstract
In this paper, based on the concepts of credibility measure and expectation theory, we derive the expectation formulae for the three reductions of a type-2 trapezoidal fuzzy variable (T2TrFV), which are attained by adopting the critical value (CV) reduction methods. We minimize the total transportation cost and the total transportation time over a single layered distribution system consisting of vendors and customers represented as a multi-objective solid transportationproblem. To portray the uncertainty in a real life choice environment, we consider the unit cost of transportation, demands, availabilities, conveyance capacities, unit transportation time and unit loading and unloading time as T2TrFVs. The corresponding deterministic model, which is obtained by the application of expectation formulas deduced earlier, is converted to a single objective optimization problem using goal programming technique and weighted sum method via the soft computing technique—generalized reduced gradient (LINGO-14.0). A numerical experiment is finally illustrated and corresponding graphical representations are provided.
Highlights
The thought of type-2 fuzzy set (T2FS) was first introduced by Zadeh [1] as an extension of the concept of an ordinary fuzzy set or a type-1 fuzzy set (T1FS)
Zadeh [7] gave the definition of type-1 fuzzy sets (T1FSs), and later in Zadeh [1] elaborated on the fact that in fuzzy logic everything is allowed to be a matter of degree which led to the introduction of T2FS
Dutta and Jana Journal of Uncertainty Analysis and Applications (2017) 5:3 to deal with a type-2 fuzzy variable (T2FV) more than a type-1 fuzzy variable (T1FV) since T2FV has the possibility of taking a crisp value and the possibility is again a fuzzy number in [ 0, 1], i.e, due to the fuzzy membership function of a type-2 fuzzy number, the computation complexity is very high in practical applications
Summary
The thought of type-2 fuzzy set (T2FS) was first introduced by Zadeh [1] as an extension of the concept of an ordinary fuzzy set or a type-1 fuzzy set (T1FS). We have for the first time, solved a STP under type-2 trapezoidal fuzzy (T2TrF) environment using CV reduction method and expectation formulas of the reductions via Goal Programming Technique (GPT). Which is the expectation formula of the reduction of type-2 trapezoidal fuzzy variable ξ ̃ = (r1, r2, r3, r4, θl, θr) obtained by the optimistic critical value reduction method. Which is the expectation formula of the reduction of type-2 triangular fuzzy variable ξ ̃ = (r1, r2, r4, θl, θr) obtained by the optimistic critical value reduction method. 2θl which is the expectation formula of the reduction of type-2 trapezoidal fuzzy variable ξ ̃ = (r1, r2, r3, r4, θl, θr) obtained by the pessimistic critical value reduction method. Using weighted sum method, the problem has been solved using LINGO-14.0 and the optimum results has been tabulated in Tables 5, 6, and 7, by which decision maker choose their expected results by choosing different weights
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