Goal programming approach to fully fuzzy fractional transportation problem

In this paper, authors devoted to study a fully fuzzy fractional multi-objective transportation problem by using goal programming approach. Also trapezoidal membership functions are applied to each objective function and constraints to describe a each fuzzy goal. A numerical example is provided to illustrate the efficiency of the multi-objective proposed approach.


Introduction
Transportation problem is a special type of linear programming problem (LPP), which deals with shipping commodities from sources to destinations. The basic transportation problem was originally developed by Hitchcock [1]. Depending on the nature of the cost function the transportation problem can be categorized into linear and non-linear transportation problems. In real life situations, all the transportation problems are not single objective. A special type of LPP in which constraints are of equality type and all the objectives are conflicting with each other is called multi-objective transportation problem (MOTP). Buckly and Feuring, in [2], considered the fully fuzzified linear programming problem (FFLP) by establishing all the coefficients and variables of a linear programme as being quantities. They transform the fully fuzzified programming problem into a multi-objective deterministic problem which, treated in the general case, is non-linear.
Fractional programming is important to our daily life, because various optimization problems from engineering, social life and economy consider the minimization of a ratio between physical and/or economical functions, for example cost/time, cost/volume, cost/profit, or other quantities that measure the efficiency of a system is minimized. Fractional linear programmes have a richer set of objective functions. In contrast a linear fractional programming is used to achieve the highest ratio of outcome to cost, the ratio representing the highest efficiency.
Transportation problem comprise a special class of linear fractional programming. In a typical problem of this type of trucking company may be interested in finding the least expensive way of transporting each unit of large quantities of a product from a number of warehouses to a number of stores. The main reason for interest in fractional programming from the fact that linear fractional objective functions occur frequently as a measure of performance in a variety of circumstances such as when satisfying objectives under uncertainty.
The general format of linear fractional problem (LFP) may be written as Transportation problem with fractional objective function are widely used as performance measures in many real life situations. Goal programming (GP) is a branch of multi-objective optimization, which in turn is a branch of multi-criteria decision analysis (MCDA) also known as multi-criteria decision making. This is an optimization programme. It can be thought of as an extension or generalization of linear programming to multiple, normally conflicting objective measures. Each of these measures is given a GP variant used. A form of linear programming that allows for consideration of multiple goals. GP can be used to determine the optimal solution to a multi-objective decision making problem. GP requires the decision maker to set an aspiration level for each goal which can be a very difficult task as there are several of uncertainties in nature must be considered. Lee and Moore [3] applied GP to find a solution of MOTP. Anukokila and Radhakrishnan [4] studied a GP approach for solving multi-objective fractional transportation problem by representing the parameters (γ , δ) in terms of interval valued fuzzy numbers.
On the other hand, GP method has the ability to deal with piecewise linear function with appropriate linearization of the constraints. The point that optimizes a single objective GP determines the point that best satisfies the set of goals in the decision problem. GP attempts to minimize the deviations from the goals. Thus, the objective function contains mainly the deviational variables representing each goal or subgoal. The deviational variable is represented in two dimensions (a positive and a negative deviation from each sub-goal and/or constraint) in the objective function. GP has been widely applied to solve different real world problems which involve multiple objectives [5][6][7][8].
Motivated by Pop and Stancu-Minasian [9], in this paper the author's extended to solve the fully fuzzy fractional transportation problem (FFFTP) with GP approach and there by the primary objective is thus to introduce the concept and a computational method for solving FFFTP, by using Kerre's method to evaluate a fuzzy constraint. The method of variation change on the under-and-over deviational variables of the membership goals associated with the fuzzy goals of the model is introduced to solve the problem efficiently by using a linear goal programming methodology. Moreover, the achievement of FFFTP, all the parameters and variables are considered here as trapezoidal fuzzy numbers. Lingo [10] software package is used to solve the optimization problem.
The rest of the paper is organized as follows. Section 2 discuss about the review of the proposed problem. Section 3 provides problem formulation of MOTP and GP formulations. Section 4 presents the preliminaries of trapezoidal fuzzy numbers. In Section 5, FFFTP technique are developed to deriving fuzzy goal programming (FGP) approach also Kerre's method to apply fuzzy inequalities for fuzzy numbers. An illustrative example provided and the optimal solution is compared with the proposed approach in Section 6.

Literature review
Fractional programming has been studied by many researchers Chang [11], Pal et al. [7] and Stanojevic and Stancu-Minasian [12]. To come into a conclusion how important this area of research is we further investigated and noticed that there are entire books and chapters devoted to this subject by researchers by Craven [13], Stancu-Minasian [14]. Stancu-Minasian's text book [15] contains the state of the art theory and practice of fractional programming. FGP approach studied by Mohamed [6] is an important technique in dealing with conflicting objectives of decision makers for satisfying decision for overall benefit of the organization. Pramanik [16] and Gang [5] for solving MOTP with FGP approach. Pop and Stancu-Minasian [17] proposed a method to solve the fully fuzzified linear fractional programming problem, where all the variables and parameters are represented by triangular fuzzy number. Fuzzy set theory was proposed by Zadeh [18] and has been found extensive in various fields. Ammar [19] and Ebrahimnejad [20] studied multi-objective transportation with fuzzy numbers. Amarpreetkaur and Amit Kumar [21] discussed new approach for solving transportation problem using generalized trapezoidal fuzzy number. Recently Kumar Das and Mandal [22] proposed a new approach for solving fully fuzzy linear fractional programming problems by using multi-objective programming problems. Gupta et al. [23] discussed the exact fuzzy optimal solution of unbalanced fully fuzzy MOTPs using Mehar's method.

Problem formulation
The proposed fuzzy mathematical model programming is based on the following assumptions.
Index Set • U k − Upper tolerance limit for the kth fuzzy goal.
• L k − Lower tolerance limit for the kth fuzzy goal.

Objective function
• Multi-objective transportation problem is • Multi-objective fractional fuzzy transportation problem is • Goal programming objective transportation problem is

Constraints
• Constraints on supply available for each source i: • Constraints on demand for each destination j: • Constraints on GP transportation problem: • Constraints on weighted FGP: • Non-negative constraints on decision variables:

Trapezoidal fuzzy numbers
Definition 4.1: A trapezoidal fuzzy numberZ is a structure (z 1 , z 2 , z 3 , z 4 ) ∈ R 4 . The membership function of the trapezoidal numberZ is defined in terms of the real numbers (z 1 , z 2 , z 3 , z 4 ) as follows:

Definition 4.2 (The principle of extension):
Let us consider the function f : X 1 × X 2 × · · · × X n → Y defined on a Cartesian product of non-fuzzy sets by values in a non-fuzzy set. Considering the fuzzy sets where ⊆ means the fuzzy inclusion relation.
The principle of extension was formulated by Zadeh [18] in order to extend the known models implying fuzzy elements to the case of fuzzy entities. Applying this principle of extension, the following definitions of the operations with trapezoidal fuzzy number follow. Given two trapezoidal fuzzy numbersĀ = (a 1 , a 2 , a 3 , a 4 ), (i) the addition of two trapezoidal fuzzy numbers (ii) the symmetry of a trapezoidal fuzzy number −Ā = (−a 4 , −a 3 , −a 2 , −a 1 ); (iii) the multiplication of two trapezoidal fuzzy num-bersĀ (iv) the division of two trapezoidal fuzzy numbers

Fully fuzzified fractional transportation problem
Let us consider the fully fuzzified fractional transportation problem where C k ij , α k and D k ij , β k represents the coefficients of the fractional transportation objective function. A k i , B k j represents the right-hand side of the linear constraints, respectively. The most often used fuzzy numbers are socalled trapezoidal fuzzy numbers, we denote by A major difficulty of solving fractional programming is that it is a highly non-linear fractional programming problem. In order to solve the problem, the membership function μ k for the kth fuzzy goal F k (x) ≥ G k 1 can be expressed as follows: where L k is the lower tolerance limit for the kth fuzzy goal (G k 1 − L k ) is the tolerance which is subjectively chosen. Again the membership function μ k for the kth fuzzy goal F k (x) ≤ G k1 can be expressed as follows: where U k is the upper tolerance limit for the fuzzy goal and (U k − G k 2 ) is the tolerance which is subjectively chosen. In fuzzy programming technique represent the under-and-over deviations, respectively, from the aspired level. In this approach, only the underdeviational variable D − k is required to be minimized to achieve the aspired levels of the fuzzy goals.
In order to solve the problem, we assume that L k ≤ F k (x) ≤ U k , where L k and U k are upper and lower bounded of F k (x), respectively. Introducing FGP technique the achievement of highest membership value of goal can be represented as follows: Equation (3) can be expressed as the following FGP problem: where D − k and D + k are respectively. In conventional programming, the under and/or over deviational variables are included in the achievement function minimizing then and that depend upon the type of the objective functions to be optimized.

Kerre's method for fuzzy numbers
Kerre's method [24] defines a fuzzy-max which is problem dependent. Kerre's provides the result M 1 > M 2 . Kerre's method would favour a fuzzy set with smaller area measurement, regardless of its relative location on the x-axis. It gives a ranking order M 3 > M 2 > M 1 . Kerre's method seems better than Yager's method and this method not logically sound either. Kerre's method would result in a smaller Hamming distance between M 1 and the fuzzy-max. Therefore M 1 > M 2 , which is against the obvious fact that M 2 > M 1 . The main concept of comparison of fuzzy numbers is based on the comparison of areas determined by membership functions. We shall operate a system of constraints which has to be satisfied by the components of a trapezoidal fuzzy numbers in order to be considered negative (in the fuzzy Kerre meaning). We apply this way of defining an inequality between fuzzy numbers to be trapezoidal fuzzy numbers M and0. Now we describe the inequality of trapezoidal fuzzy numbers through a system of deterministic disjunctive constraints. Applying Kerre's method to (2), the given problem is reduced to the following deterministic multiobjective fractional transportation problem with disjunctive constraints.
In the next section, we use the above considerations to solve the fully fuzzified fractional transportation problems.

Solving method for FFFTP
First we transform problem (2) into a fully fuzzified fractional transportation problem using the Charnes-Cooper [25] transformation ( m i=1 n j=1 D k ij x ij + β k = 1/T k and x ij T k = Y k ij ) and obtain the following problem: After aggregating the fuzzy quantities we change the objective function, which is described by a trapezoidal fuzzy number, and obtain the following deterministic multiple objective transportation problem with fuzzy constraints: Solutions (y 1 ij , y 2 ij , y 3 ij , y 4 ij ) i,j=1,2,...,m,n and (t 1 , t 2 , t 3 , t 4 ) are obtained, namely the trapezoidal fuzzy number (Y k ij ) i,j=1,2,...,m,n and T k . Then the optimal solution of problem (6) is (x ij = Y k ij /T k , i, j = 1, 2, . . . , m, n). Using a symmetrical definition for trapezoidal fuzzy numbers M = (m − 1, m, m + 1, m + 2).
By applying Proposition 5.2 to the inequality of index i in Equation (7), we obtain the following disjunctive system of constraints By applying Proposition 5.3 to the equality in (8), we obtain the disjunctive system of constraints Consequently, problem (2) is reduced to the following deterministic multiple objective fractional transportation problem, subject to a conjunctive system of disjunctive non-linear constraints: According to method described in Patkar and Stancu-Minasian [26], we shall consider the variable (δ to eliminate the disjunctive and to obtain (9) as system of conjunctive constraints. Solving problem (7)-(9) will allow us to obtain solution (y 1 ij , y 2 ij , y 3 ij , y 4 ij ) j=1,2,...,n , ..,m , namely the trapezoidal fuzzy numbers (y ij ) j=1,2,...,n i=1,2,...,m and (t 1 , t 2 , t 3 , t 4 ), which represents the solution of problem (6).

Solving FFFTP using GP approach
Applying the minmax form of GP to the fuzzy model of MOTP with the membership function leads the following model: Now, let the tolerance limits of the two fuzzy objective goals be (−1, −2). The membership functions of the goals are obtained as Then the membership goals can be expressed as where

Example
To illustrate the FGP approach, consider the following fractional MOTP as and Using fuzzy programming technique the optimal solution of the problem is obtained by using the following steps Step 1: Step 2: (2)).
We attach now to this problem to a fully fuzzy fractional transportation problem. Consider its real number coefficient m as being symmetric trapezoidal fuzzy number M of spread 2, having the following form: Using (2), we get the fully fuzzified fractional transportation problem as still having fuzzy constraints using (14) Similarly we solve F 2 (x) by the same procedure. By evaluating the fuzzy constraints with Kerre's method, described in Section 5, we obtain the following equivalent system of disjunctive constraints: In order to obtain a synthesis function of the form objective function from (16) and applying to it the results presented in Stancu-Manasian [27] we use the importance coefficients π 1 = 0.1, π 2 = 0.3 respectively. Now obtain the optimum values Step 3. Using (9), the membership goal can be restated as minimize φ sub to 3y 1 11 ij ≥ 0, i, j = 1, 2, 3, 4.
The problem was solved by the linear interactive global optimization (LINGO) software the compromise solution is presented as follows:

Conclusion
In fuzzy programming technique, decision maker can obtain a satisfying solution. The proposed approach allows using of the fully fuzzy fractional transportation problem and Kerre's method in the process of obtaining the preferred solution with the decision making process. The main contribution of this proposed paper is the GP approach for solving multi-objective fractional transportation problem with trapezoidal fuzzy number. The FGP solution procedure provides the most satisfactory solution for all the decision makers at both the levels by reaching the aspired levels of the membership goals. The method can be easily implemented to solve any non-linear multi-objective programming problem.