A New Approach for Solving Type-2-Fuzzy Transportation Problem

In this paper, a new approach is introduced to solve transportation problem with type-2-fuzzy variables. In most of the real-life situations, the available data do not happen to be crisp in nature. It gives rise to the fuzzy transportation problem (FTP). This proposed approach concentrates on the problem when the vertical slices of type-2-fuzzy sets (T2FSs) are trapezoidal fuzzy numbers (TFNs). The original problem reduces to three different linear programming problems (LPPs) which are solved using the simplex algorithm. Then the effectiveness of this paper is discussed with numerical example. In conclusion, the significance of the paper and the scope of future study are discussed.


Introduction
The transportation problem (TP) is one of the important linear programming problem which arises in many real-life situations. Hence it has drawn considerable attention in the literature. The core idea of TP is to distribute commodities from certain origins to some destinations in such a way that the total transportation cost gets minimized. The transportation problem has many real-life applications, such as scheduling, production, investment, deciding plant location and inventory control etc. The classical TP has some limitations for designing real-life problem of TP through single objective function where this limitation may be removed by multi objective transportation problem. Many researchers have considered transportation problem with multiple choices and multiple objectives (Mahapatra et al., 2010;Mahapatra et al., 2013;Maity and Roy, 2016;, Das et al., 2019Saxena, 2019;Maity et al. 2019). In classical transportation problem, the given data are crisp in nature. However, in real-life situations, most of the cases given data are of imprecise nature. This imprecise nature are measured by fuzzy number in our formulated model. Zadeh (1975) first introduced the concept of type-2-fuzzy set (T2FS) along with extension principle to extend the basic operations to the set of fuzzy numbers. He also introduced -cut representation which happens to be equivalent with the concept of extension principle. In order to make use of T2FSs in transportation problem, it is needed to introduce meaningful operations for fuzzy sets. Extension principle and -cut representation are the connection between crisp binary operations, such as addition, multiplication and fuzzy binary operations. Xie and Lee (2017) worked on extended type-reduction method for general type-2-fuzzy sets.
Trapezoidal membership functions are commonly used membership functions to represent fuzzy numbers in real-life applications. Triangular membership function can be viewed as a particular type of trapezoidal membership function etc. There are some other types of popular membership functions, such as Gaussian membership function, power membership function etc. Triangular, trapezoidal, power, and Gaussian fuzzy numbers are all particular type of LR flat fuzzy numbers. Thus we introduce a new method for modelling and solving FTPs in which all the parameters are T2FSs in nature.
Being a particular type of linear programming problem, fuzzy linear programming approach can be used in solving fuzzy transportation problem. Verdegay (1984) investigated the application of Zimmermann's (2001) fuzzy programming approach and presented the additive fuzzy programming model. Fang et al. (1999) proposed a new method for solving LPPs with fuzzy coefficients in the constraints. Fan et al. (2013) developed a direct transformation algorithm for solving the fuzzy linear programming (FLP) model with uncertainties expressed as fuzzy sets on the left-hand and right-hand sides of the constraints and the objective function. The solution of a fuzzy linear programming should be fuzzy in nature. But, the mentioned techniques come up with crisp solutions which are not expected. Crisp and fuzzy solutions have also been derived for FLP (Gupta and Kumar, 2012;Kumar and Ram, 2018) but no fuzzy solution for the fuzzy transportation problem has been found. As a particular type of LPP, there have been some special types of solution approaches to TP.
Though several researchers have worked on transportation problem under fuzzy environment (Chanas and Kuchta, 1996;Cadenas and Verdegay, 2006;Ebrahimnejad, 2016;Maity and Roy, 2017) but a few of them has worked in type-2-fuzzy environment. In some cases, data cannot be handled with general type-1-fuzzy approaches to transportation problem due to higher order of fuzziness involved with data. As for example, transportation cost may vary in different places, which may depend upon time. Hence type-2-fuzzy set is needed to be introduced for tackling such data. Based on this consideration, a new approach is introduced to handle transportation problem with type-2-fuzzy data. The contributions of this paper are as follows:

Basic Definitions
Here some basic definitions are defined for development of the paper. In this regard, the domains of the fuzzy sets are considered to be the universe of the corresponding fuzzy set, as fuzzy sets are treated as functions in the paper.
Fuzzy Set: A fuzzy subset of a universe is a function : → [0,1].

Crisp Set:
A crisp subset of a universe is a function : → {0,1} where ( ) = 1 if belongs to and ( ) = 0 otherwise. We can realize crisp set to be a particular type of fuzzy set.
-Cut: -cut ( ∈ [0,1]) of a fuzzy set (in the universe ) is a crisp set , is given by, can be represented in terms of -cut as follows:

Basic Operations of Fuzzy Sets
Here we now define some binary operations of fuzzy set in the universe of real numbers using extension principle. Throughout the paper trapezoidal fuzzy number ( , , , ) is considered to be positive, i.e., ≥ 0.
Addition: Let and be two fuzzy sets in the universe of real numbers. Then the addition of two fuzzy sets is defined by, ⊕ ( ) = sup + = min{ ( ), ( )}, where ⊕ is the fuzzy addition.
Though the multiplication of two trapezoidal fuzzy numbers belonging to the set [0,1] is also an element of [0,1], but the same is not true for addition. Hence, we modify the definition of fuzzy addition of trapezoidal fuzzy numbers as follows: The addition of trapezoidal fuzzy numbers = ( 1 , 2 , 3 , 4 ) is given as: Now, some order relations in the set of trapezoidal fuzzy numbers belonging to [0,1] are needed to be introduced for formulating the proposed model. Our objective is to maximize the length where the trapezoidal fuzzy number takes the value 1. Hence we define the order as follows: Let = (0, 1 , 2 , 1) and = (0, 1 , 2 , 1) then ≥ if 1 ≤ 1 and 2 ≥ 2 .

Model Formulation
Classical transportation problem was first introduced by Hitchcock (1941) where decision maker is certain about the transportation cost, supply and demand, is formulated as follows: Where goods are transported from origins to destinations, the transportation cost of a unit measure of goods from ℎ origin to ℎ destination is , the supply at ℎ origin is 1 and the demand at ℎ destination is 2 .
In this transportation problem, the decision maker has the privilege of having crisp data for transportation cost, supply and demand. But in real-life situation, the decision maker may not be provided with crisp data. In this model, all the given data and variables are considered to be type-2-fuzzy variables. And the type-2-fuzzy variables are of the form ( , ( , ) ), where , , are real variables and ( , ) corresponds to the trapezoidal fuzzy number (0, , , 1). Imposing type-2fuzziness in the classical transportation model, type-2-fuzzy transportation model is designed as follows:
The boundary conditions on and are given in order to maintain the consistency in the solutions. If we split the above fuzzy problem in coordinate wise, then we get three mutually different crisp LPPs. Considering the first coordinate we obtain the LPP, is defined in Model 2 as follows: ≤ 1 ( = 1,2, … , ), ∑ =1 ≥ 2 ( = 1,2, … , ), ≥ 0, ∀ .
It have been already mentioned that our objective would be to maximize ( 2 , 3 ) . In order to do that we have to minimize 2 and maximize 3 . Considering the second coordinate we obtain the LPP, given by Model 3.
corresponds to the minimum point where the trapezoidal fuzzy variable takes the functional value 1.

Model 3
Considering the third coordinate we get the LPP, considered in Model 4. corresponds to the maximum point where the trapezoidal fuzzy variable takes the functional value 1.

Conclusions
In contrast to classical transportation problem, we have considered all the constraints and variables to be type-2-fuzzy variables which has given rise to fuzzy transportation problem. In most of the existing methods, in order to solve the problems, the original problem is transformed into a single crisp problem and which comes up with crisp solution. And in that process we lose significant amount of information which results in deterioration in accuracy of solution. But in our approach, we have considered transportation problem in type-2-fuzzy environment and it comes up with type-2-fuzzy solution. Here the original problem has been converted into three crisp transportation problems and combining the three optimal solutions we have obtained the optimal solution of the original problem.
However, in order to convert into crisp transportation problems, the multiplication of trapezoidal fuzzy number is approximated with a trapezoidal fuzzy number. The approximation is done in a convenient manner to make the computations simplified. In future this flaw can be addressed with some better approximation methods, such as area minimizing method and sup-norm minimizing method etc.