Development of Unrestricted Fuzzy Linear Fractional Programming Problems Applied in Real Case

Purpose: We formulate a linear fractional programming (LFP) problem in which costs of the objective functions and constraints all are taken to be triangular fuzzy numbers. Methodology: The fuzzy LFP problem is transformed into an equivalent crisp line fractional programming (CLFP) problem by using the centroid ranking function. This proposed method is based on crisp LFP and has a simple structure. Findings: To show the efficiency of our proposed method a real life problem has been illustrated. The discussion of the practical problem will help decision makers to realise the usefulness of the CLFP problem. Value: Using centroid ranking function, we overcome the all limitations of our day to day real life problem. Finally, a result analysis is also established for applicability of our method.


Introduction
The decision-making problems can be derived as the minimised of several fractional terms and this is the well-known fractional programming (FP) problem. If these fractional terms are in linear terms, appearing in objective function subject to a linear constraint, then these types of problem are called linear FP (LFP) problems. LFP problems have attracted the interest of many researchers due to its application in decision-making such as production planning, marketing and media selection, university planning and student admissions, financial and corporate planning, health care and hospital planning, etc. see [1][2][3][4][5] and references therein. In the literature, many researchers have been recommended to solve LFP problems. Isbell and Marlow [6] first identified an example of LFP problem and solved it by a sequence of linear programming problems. Charnes and Cooper [5] considered the variable transformation method to solve LFP problems. Bitran and Novaes [7] considered the updated objective functions method to solve LFP problems by solving a sequence of linear programmes. Martos [8], Swarup [9], Pandy and Punnen [10], Das and Mandal [11] solved the LFP problem by various types of solution procedures based on the simplex method; see also [12][13][14][15]. These methods are interesting; however, in daily life circumstances, due to ambiguous information supplied by decision-makers, the parameters are often illusory and it is very hard challenge for decision-maker to make a decision. In such a case, it is more appropriate to interpret the ambiguous coefficients and the vague aspiration parameters by means of the fuzzy set (FS) theory. The concept of fuzzy set and fuzzy numbers was first introduced [16] and applied efficiently for linear optimisation; see [17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32] and references therein. Furthermore, several researchers have investigated linear fractional programming problems in the fuzzy framework. For example, Sakawa and Yano [33] proposed a method to solve a multi-objective LFP (MOLFP) problem under a fuzzy environment. Dutta et al. [34] established the sensitivity analysis in the fuzzy LFP (FLFP) problems. Some authors solved the FLFP problems by the fuzzy goal programming approach [35][36][37][38][39]. De and Deb [40] considered a fuzzy linear fractional programming problem using the sign distance ranking method where all the terms are triangular fuzzy numbers. Youness et al. [41] imported a design to find a bi-level multi-objective fractional integer programming problem that consists of fuzzy numbers in the right-hand side of the constraints. Pop and Minasian [42] proposed a method for solving fully falsified linear fractional programming problems where all the parameters and variables are triangular fuzzy numbers. In [43,44], they considered the same problem of [42] for solving fully fuzzy linear fractional programming problems. Veeramani and Sumathi [45] proposed a solution procedure for solving fuzzy linear fractional programming problem by using the fuzzy mathematical programming approach. Very recently, a number of papers have exhibited their interest to solve the FLFP problems [46][47][48][49][50][51]. Recently, several researchers had focused for solving linear programming by using multi-objective and transformation technique under fuzzy circumstances, see refer [52][53][54][55][56]. The main works and features of this paper are that we consider a new type of fuzzy arithmetic for triangular numbers in which the coefficients of the objective function and the constraints were represented by triangular fuzzy numbers with inequality constraints utilised in daily life problems. The proposed technique is very easy and involves mathematical calculation. The rest of our work is organised as follows: In Section2, we review some concept and arithmetic between two triangular fuzzy numbers. In Section 3, formulation of FFLFP problems and use of ranking function are discussed. The new method for solving FFLFP problems is affirmed in Section 4. In Section 5, the numerical examples and the obtained results are given for illustrating the new method. In Section 6, advantages of the proposed method are discussed. Finally, the conclusion is given in Section 7.

Preliminaries
In this section, the basic definitions, involving fuzzy sets, fuzzy numbers and operations on fuzzy numbers, are outlined. For detailed information on the fuzzy set theory, we refer the interested reader to [20,57].

Definition 2.5: A convenient method for comparing of the fuzzy numbers is by the use of ranking function. So, we define a ranking function : F( ) → which maps for each fuzzy number in to real line. The centroid of centroid ranking of triangular fuzzy numbers is
here w = 1.

a) be any triangular fuzzy number andB = (e, f , d) be a non-negative triangular fuzzy number, theñ
Note: It is clear from

Linear Fractional Programming (LFP) Problem
In this section, the general form of LFP problem is discussed. Furthermore, Charnes and Cooper's linear transformation is summarised.
For some values of x, G(x) may be equal to zero. To avoid such cases, one requires that either For convenience, assume that LFP problem satisfies the condition that (2)

Theorem 3.1 ([3]):
Assume that no point (z, 0) with z ≥ 0 is feasible for the following linear programming problem.
Then, with the condition of relation (2), the LFP problem (1) is equivalent to the linear programming problem model (3). Now, consider the two related problems and, where model (4) is obtained from model (1) by the transformation t = 1/G(x), z = tx and model (5) differs from model (4) by replacing the equality constraints tG(z/t) = 1 by an inequality constraint tG(z/t) ≤ 1. (1) is a standard concave-convex programming problem which reaches a maximum at a point x * , then the corresponding transformed problem model (5) attains the same maximum value at a point (t * , z * ) where z * /t * = x * . Moreover, model (5) has a concave objective function and a convex feasible set. Suppose that:

Theorem 3.2 ([3]): If model
where F(x) is concave and negative for each x ∈ S and G(x) is concave and positive on S, then where −F(x) is convex and positive. Therefore, the problem (6) is converted into a standard concave-convex programming problem transformed to the following linear programming problem:

Fuzzy Linear Fractional Programming Problem and Its Solution
Consider the following fuzzy linear fractional programming (FLFP) problem. We are going to approach m fuzzy equality constraints and n fuzzy variables where all the terms are triangular fuzzy numbers. (i)ỹ is a non-negative fuzzy number, then,ỹ is also an exact optimal solution of the problem (8) and is called a substitute optimal solution.
Step 2: If all the terms represent the triangular fuzzy numbers, then write the FFLFP problem as follows: Step 3: By utilising Definition 2.5, the new centroid ranking function of the problem should be transformed into a crisp LFP problem. The model can be written as: Step 4: The above problems are crisp linear fractional programming problems, which can be changed into crisp LP problems by using a Charnes-Cooper transformation model, as discussed in Section 3.
Step 5: Solved the crisp LP problem by utilising any technique.
Step 6: Write the solution of FLFP problems in the form of x and obtain the optimal solution asz.
Step 7: Finally, by using Definition 2.7, compare the results.

Application of Our Proposed Method
In this section, we take some real-life problems and proved the ability of our proposed method: Example 5.1: In TATA Hospital Jamshedpur, India has two nutritional experiments (Vitamin A and Calcium) with two products Milk (glass) and Salad (500 mg) with profit around 6 dollars and around 2 dollars per unit, respectively. However, the cost for each unit of the above product is around 1 dollar. Consider that a fixed cost of around 2 dollars is added to the cost function. Determine the maximum profit of these two products.
Here, the environmental coefficients, such as profit (due to market situations), cost (due to market conditions), vitamin A and calcium (due to the presence of the suppliers), are imprecise numbers with triangular possibility distributions over the planning horizon due to incomplete information. For example, the profit of the product A is (4,6,8) dollars. Similarly, the other parameters and variables are assumed to be triangular fuzzy numbers. Hence, the above problem can be formulated as the following FFLFP problem (Table 1).
Then, we obtain the optimal solution as x 1 = 4.097, and x 2 = 3.064 And the optimal value of the problem as Z = 3.33. In Veeramani and Sumathi methods [51] the optimal values of the problems are  Figure 1. Membership function of the proposed method vs. existing methods [51,58]. 14, 4.4, 22). In Stanojevic and Stancu method [58], the fuzzy optimal values are: 3.2, 4.5, 28) By comparing the proposed method results with those of the existing method [51,58], based on Definition 2.7, we conclude that our method is more efficient than other existing methods.
3.33 = (z)proposed method= (z) methodof [51] < (z) methodof [58] = 3.2 In Figure 1, we compare the membership function for the proposed method and the existing methods [51,58]. Graph (Figure 1) shows that the modified technique yields better values of most of the membership functions and individual objective functions in comparison to the existing methods [51,58]. It is clear that both the approaches are closer, but the modified methodology is efficient and requires less computation than earlier technique in terms of considering the solution preferences by the decision-maker at each level. In Figure 1, Z is an objective function and Z π is a membership function.

Result Analysis
This section provides a comparative study of the proposed method with the existing method for fuzzy linear fractional programming problems.
• In our proposed model, our results are better than the existing results. In our model, we introduced a new centroid ranking function method for solving the fuzzy LFP problem and transformed into a crisp LFP problem. • In fig-1, we have compared our proposed technique with other existing technique, we have found that the objective value of our proposed method is more than that of the existing method [51,58]. • Our model is very simple and efficient compared to the existing method [51,58].
• Our model is applied in a real-life problem and also in a large-scale problem.

Conclusion
In the past few years, growing interests are shown in fuzzy linear fractional programming and currently there are several methods for solving FLFP. However, to the best of our knowledge, a few efficient optimal solutions were found in fuzzy linear fractional programming (FLFP). In this paper, we proposed a new efficient method for solving FLFP problems, in order to obtain the fuzzy optimal solution. Furthermore, the limitations of other existing methods have been pointed out. To show the efficiency of the proposed method, some numerical examples are illustrated. We concluded that our proposed model is very easy to handle, efficient and shows better outcomes than other models.