A new method for solving fuzzy linear fractional programming problems

Abstract

Fuzzy Linear Fractional Programming problem has been used as an important planning tool for the different disciplines such as engineering, business, finance, economics, etc. In this paper, a new algorithm is developed to solve the Fuzzy Linear Fractional programming Problem (FLFPP) where the cost of the objective function, the resources and the technological coefficients are triangular fuzzy numbers. For this, the FLFPP is transformed into an equivalent deterministic Multi Objective Linear Fractional Programming Problem (MOLFPP) and solved them each objective function. From the obtained solutions, we define an imprecise and aspiration level for each objectives. Hence, the objectives are transformed as fuzzy goal. Then the goal programming approach is used to achieve the highest degree of each of the membership goals by minimizing their deviational variables. Finally, two real examples and one case study problem will be used to illustrate our algorithm and compare it with the existing method.

Keywords

Triangular fuzzy number linear programming problem multi objective linear programming problem goal programming

Nomenclature

FGP

Fuzzy Goal Programming

FLPP

Fuzzy Linear Programming Problem

FLFPP

Fuzzy Linear Fractional Programming Problem

FMOLPP

Fuzzy Multi-Objective Linear Programming Problem

Goal Programming

LFPP

Linear Fractional Programming Problem

LPP

Linear Programming Problem

MOLPP

Multi-Objective Linear Programming Problem

MOLFPP

Multi-Objective Linear Fractional Programming Problem

1 Introduction

Mathematical models taking objective function as a ratio of two linear functions have many applications. For instance, it seen many times models used in finance corporate planning, bank balance sheet management, water resources, and health care are of fractional type modeling. Indeed, in such situations, it is often a question of optimizing a ratio debt/equity, output/employee, actual cost/standard cost, profit/cost, inventory/sales, risk-assets/capital, student/cost, doctor/patient, and so on subject to some constraints. The above problems can be modeled efficiently through Linear Fractional Programming problems (LFPP).

In the literature, several methods (see [25, 26]) have been recommended to solve LFPP. Isbell and Marlow [13] first identified an example of LFPP and solved it by a sequence of linear programming problems. Charnes and Cooper [7] considered variable transformation method to solve LFPP and the updated objective function method were developed by Bitran and Novaes [4]. Gilmore and Gomory [10], Martos [17], Swarup [28], Wagner and Yuan [33], and Sharma et al. [24] solved the LFPP by various types of solution procedures based on the simplex method developed by Dantzig [8]. Chadha [5] has proposed a procedure for solving a fractional programming problem with absolute-value functions. Later, Chan [6] proposed a fuzzy goal programming approach for solving fractional programming with absolute-value functions. Tantawy [30, 31] proposed two different approaches namely; a feasible direction approach and a duality approach to solve the LFPP. Mojtaba et al. [19] solved the LFPP with interval coefficients in objective function which is based on Charnes and Cooper technique.

In LFPPs, the coefficients are assumed to be exactly known. In practice, the coefficients (some or all) are not exact due to the errors of measurement or vary with market conditions etc. These situations can be modeled efficiently through Fuzzy Linear Fractional Programming (FLFP). In fuzzy decision making problems, the idea of maximizing decision was anticipated by Bellman and Zadeh [3]. The theory of fuzzy linear programming on general level was initially proposed by Tanaka et al. [29]. Hashemi et al. [11] proposed a two-phase approach for solving the FFLPPs. Pop and Stancu Minasian [22], Stanojevi’c and Stancu-Minasianb [27] used deterministic multiple objective linear programming problem by quadratic constraints to work out FFLPPs.

Abousina and Baky [1] proposed fuzzy goal programming procedure to bilevel multi-objective linear fractional programming problems. Baky [2] solved multi-level multi-objective linear programming problems through fuzzy goal programming approach. Dutta et al. [9] discussed the efficient of tolerance in fuzzy linear programming problem. Li and Chen [15] developed the method for solving fuzzy linear fractional programming through fuzzy programming approach. Luhandjula [16] introduced the concept of fuzzy approaches for multiple objective LFPPs. Pal et al. studied a goal programming procedure for fuzzy multi-objective linear fractional programming problem. Mehra et al. [18] proposed a method to compute an (α, β) acceptable optimal solution where α ∈ [0, 1] and β ∈ [0, 1] is the grade of satisfaction associated with the fuzzy objective function and with the fuzzy constraints, respectively. Pop and Stancu Minasian [22], analysed a method to solve the fully fuzzified LFPPs. Recently, Veeramani and Sumathi [32] proposed solution procedure for solving fuzzy linear fractional programming problem by using fuzzy mathematical programming approach.

Most of the work listed above deals with fuzziness either in the constraint inequalities and/or in the aspiration levels of the decision makers. In this paper, we consider the FLFPP with cost and coefficient of the decision variables are triangular fuzzy numbers. First the FLFPP is transformed into a deterministic MOLFPP. This transformation is obtained by using Zadeh extension principle. In the proposed approach goal programming model for achievement of the highest membership value of each of fuzzy goals defined for the fractional objectives is formulated. In the solution process, the method of variable change on under and lover deviational variables are associated with fuzzy goals off the model is introduced and solve the problems by using linear goal programming approach. An illustrative real life examples and case study problem demonstrates the feasibility and effectiveness of the proposed method.

The rest of our work is organized as follows: In Section 2, we review some concepts of fuzzy numbers. In Section 3, the method of converting LFPP into LP problem and multi objective linear fractional goal programming approach is discussed. In Section 4, solution procedure for FLFPP is developed. In Section 5, advantages of the developed method is discussed. The proposed procedure illustrated through a real time example in Section 6.

2 Some basic notions

In this section, the basic definitions involving fuzzy sets, fuzzy numbers and operations on fuzzy numbers are outlined.

Definition 2.1. If X is a collection of objects denoted generically by x, then a fuzzy set $\tilde{A}$ in X is a set of ordered pairs $\tilde{A} = {(x, μ_{\tilde{A}} (x)) | x \in X}$ where $μ_{\tilde{A}} (x)$ is called membership function or grade of membership(also degree of compatibility or degree of truth) of $x \in \tilde{A}$ that maps X to [0, 1].

Definition 2.2. The support of a fuzzy set $\tilde{A}$ is the set of all points x in X such that $μ_{\tilde{A}} (x) > 0$

Definition 2.3. A fuzzy set $\tilde{A}$ is said to be normal if $μ_{\tilde{A}} (x) = 1$ for atleast one x ∈ X.

Definition 2.4. A fuzzy set $\tilde{A}$ on X is convex if and only if $μ_{\tilde{A}} (λ x_{1} + (1 - λ) x_{2}) \geq \min (μ_{\tilde{A}} (x_{1}), μ_{\tilde{A}} (x_{2}))$ for all x₁, x₂ ∈ X and for all λ ∈ [0, 1] where min denotes the minimum operator.

Definition 2.5. A fuzzy subset $\tilde{A}$ of the real line R with membership function $μ_{\tilde{A}} : R \to [0, 1]$ is called fuzzy number if

$\tilde{A}$ is normal and convex fuzzy set.

Support of $\tilde{A}$ must be bounded.

Definition 2.6. A triangular fuzzy number $\tilde{A}$ is denoted by $\tilde{A} = (a^{(1)}, a^{(2)}, a^{(3)})$ with a⁽¹⁾ < a⁽²⁾ < a⁽³⁾ is a fuzzy set where the membership function can be defined as $μ_{\tilde{A}} (x) = {\begin{matrix} \frac{x - a^{(1)}}{a^{(2)} - a^{(1)}}, & a^{(1)} \leq x \leq a^{(2)}, \\ \frac{a^{(3)} - x}{a^{(3)} - a^{(2)}}, & a^{(2)} \leq x \leq a^{(3)}, \\ 0 & otherwise, \end{matrix}$

Definition 2.7. Let $\tilde{A} = (a^{(1)}, a^{(2)}, a^{(3)})$ and $\tilde{B} = (b^{(1)}, b^{(2)}, b^{(3)})$ be two positive triangular fuzzy numbers, where a⁽¹⁾, a⁽²⁾, a⁽³⁾, b⁽¹⁾, b⁽²⁾, b⁽³⁾ ∈ R. Then the arithmetic operations and scalar multiplications are defined by the extension principle and can be equivalently represented as follows:

$\tilde{A} + \tilde{B}$ = (a⁽¹⁾ + b⁽¹⁾, a⁽²⁾ + b⁽²⁾, a⁽³⁾ + b⁽³⁾)

$\tilde{A} - \tilde{B}$ = (a⁽¹⁾ − b⁽³⁾, a⁽²⁾ − b⁽²⁾, a⁽³⁾ − b⁽¹⁾)

$k \tilde{A} = {\begin{matrix} ({ka}^{(1)}, {ka}^{(2)}, {ka}^{(3)}), & if k is positive, \\ ({ka}^{(1)}, {ka}^{(3)}, {ka}^{(2)}), & if k is negative \end{matrix}$

Remark 2.1. Let $\tilde{A} = (a^{(1)}, a^{(2)}, a^{(3)})$ and $\tilde{B} = (b^{(1)}, b^{(2)}, b^{(3)})$ be any two positive triangular fuzzy numbers. If $\tilde{A} \leq \tilde{B}$ then a⁽²⁾ ≤ b⁽²⁾, a⁽¹⁾ ≤ b⁽¹⁾ and a⁽³⁾ ≤ b⁽³⁾.

3 Linear Fractional Programming Problem

In this section, the general form of LFPP is discussed. Also, Charnes and Cooper’s [7] linear transformation is summarized.

Consider the standard form of LFPP as follows: $\begin{matrix} Max Z (x) = \frac{\sum c_{j} x_{j} + p}{\sum d_{j} x_{j} + q} = \frac{N (x)}{D (x)} \\ subject to \end{matrix}$ (1) $x \in S = {x \in R^{n} : Ax (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b, x \geq 0}$ where j = 1, 2 . . . n, A ∈ R^m×n, b ∈ R^m, c, d ∈ Rⁿ and p, q ∈ R. For some values of x, D (x) may be equal to zero. To avoid such cases, one requires

that either ${x \geq 0, Ax (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b \Rightarrow D (x) > 0}$ or ${x \geq 0, Ax (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b \Rightarrow D (x) < 0}$ . For convenience, assume that LFP satisfies the condition that $x \geq 0, Ax (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b \Rightarrow D (x) > 0 .$ (2)

Remark 3.1. The problem (1) is said to be standard concave-convex programming problem, if N (x) is concave on S with N (ζ) ≥0 for some ζ ∈ S and D (x) is convex and positive on S.

Definition 3.1. [6] The two mathematical programming problem (i) Max F (x), subject to x ∈ S, (ii) Max G (x), subject to x ∈ U will be said to be equivalent iff there is a one to one map f of the feasible set of (i), onto the feasible set of (ii), such that F (x) = G (f (x)) for all x ∈ S

Consider the following problem $\begin{matrix} Max tN (z / t) \\ subject to \end{matrix}$ (3) $A (z / t) - b (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) 0, tD (z / t) \leq 1, t > 0, z \geq 0$ where (3) is obtained from (1) by the transformation t = 1/D (x) , z = tx and the LFPP (1) is equivalent to LPP (3).

Theorem 3.1. [23] Let for someζ ∈ S, N (ζ) ≥0, if (1) reaches a (global) maximum atx = x^*, then (3) reaches a (global) maximum at a point (t, z) = (t^*, z^*), wherez^*/t^* = x^*and the objective functions at these points are equal.

Theorem 3.2. [23] If (1) is a standard concave-convex programming problem which reaches a (global) maximum at a pointx^*, then the corresponding transformed problem (3) attains the same maximum value at a point (t^*, z^*) wherez^*/t^* = x^*. Moreover (3) has a concave objective function and a convex feasible set.

Suppose that $\begin{matrix} Max Z (x) = \frac{N (x)}{D (x)} \\ subject to \end{matrix}$ (4) $x \in S = {x \in R^{n} : Ax (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b, x \geq 0}$ where N (x) is concave and negative for each x ∈ S, D (x) is concave and positive on S, then $\begin{matrix} {Max}_{x \in S} \frac{N (x)}{D (x)} \Leftrightarrow {Min}_{x \in S} \frac{- N (x)}{D (x)} \\ \Leftrightarrow {Max}_{x \in S} \frac{D (x)}{- N (x)} \end{matrix}$ here −N (x) is convex and positive. Therefore, the problem (4) is converted into standard concave-convex programming problem. Hence, the problem (4) is transformed to the following linear programming problem: $\begin{matrix} Max tD (z / t) \\ subject to \end{matrix}$ (5) $A (z / t) - b (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) 0, - tN (z / t) \leq 1, t > 0, z \geq 0$

3.1 Multi-Objective Linear Fractional Programming Problem

The general format of a classical MOLFPP may be written as $\begin{matrix} Max Z (x) = [Z_{1} (x), Z_{2} (x), . . ., Z_{K} (x)] \\ subject to \end{matrix}$ (6) $x \in S = {x \in R^{n} : Ax (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b, x \geq 0}$ with b ∈ R^m,A ∈ R^m×n and $Z_{k} (x) = \frac{\sum c_{k} x + p_{k}}{\sum d_{k} x + q_{k}} = \frac{N_{k} (x)}{D_{k} (x)}$ , where k = 1, 2,. . . , K, c_k, d_k ∈ Rⁿ and p_k, q_k ∈ R.

3.2 Fuzzy linear fractional goal programming approach

To solve MOLFPP, we will assign an imprecise aspiration level for each objective function. These fuzzy objectives are called fuzzy goals [21]. Let L_k be the aspiration level assigned to the k^th objective Z_k (X). Then the fuzzy goals appear as (a) Z_k (X) ≽ L_k (for maximizing) and (b) Z_k (X) ⪯ L_k (for minimizing), where ≽ and ⪯ indicate the fuzziness of the aspiration levels, and is to be understood as essentially less than in the sense of Zimmermannn [34].

Hence, the fuzzy linear fractional goal programming can be stated as follows: $\begin{matrix} Find X \\ soastosatisfy \\ Z_{k} (X) ≽ L_{k}, k = 1, 2, . . ., k_{1} \\ Z_{k} (X) ⪯ L_{k}, k = k_{1} + 1, . . ., K \\ subject to \\ x \in S = {x \in R^{n} : Ax (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b, x \geq 0} \end{matrix}$ In fuzzy programming, the fuzzy goals are characterized by their associated membership functions. The membership function μ_k for the k^th fuzzy goal Z_k (X) ≽ L_k can be expressed algebraically according [29] as $μ_{k} (X) = {\begin{matrix} 1, & if Z_{k} (X) \geq L_{k} \\ \frac{Z_{k} (X) - L_{k}}{U_{k} - L_{k}}, & if L_{k} \leq Z_{k} (X) \leq U_{k}, \\ 0 & if Z_{k} (X) \leq L_{k}, \end{matrix}$ (7) On the other hand the membership function μ_k for the k^th goal Z_k (X) ⪯ L_k can be defined as $μ_{k} (X) = {\begin{matrix} 1, & if Z_{k} (X) \leq L_{k} \\ \frac{U_{k} - Z_{k} (X)}{U_{k} - L_{k}}, & if L_{k} \leq Z_{k} (X) \leq U_{k}, \\ 0 & if Z_{k} (X) \geq U_{k}, \end{matrix}$ (8) In fuzzy decision environment, the achievement of the objective goals to their aspired levels to the extent possible is actually represented by the possible achievement of their respective membership values to the highest degree.

4 MOLFPP formulation of FLFPP

In this section, a procedure for solving FLFPP through MOLFPP is developed. Here, we consider the cost of the objective function, the resources and the technological coefficients are triangular fuzzy numbers.

Let us consider the FLFPP $\begin{matrix} Max \tilde{Z} = \frac{\sum \tilde{c_{j}} x_{j} + \tilde{p}}{\sum \tilde{d_{j}} x_{j} + \tilde{q}} \\ subject to \end{matrix}$ (9) $\begin{matrix} \sum \tilde{a_{ij}} x_{j} (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) \tilde{b_{i}}, i = 1, 2, . . ., m \\ x_{j} \geq 0, j = 1, 2, . . ., n \end{matrix}$ we assume that, $\tilde{c_{j}}$ , $\tilde{p}$ , $\tilde{d_{j}}$ , $\tilde{q}$ , $\tilde{b_{i}}$ , $\tilde{a_{ij}}$ are triangular fuzzy numbers for each i = 1, . . . , m and j = 1,. . . , n. Therefore, the problem (9) can be written as $\begin{matrix} Max \tilde{Z} = \frac{\sum (c_{j}^{(1)}, c_{j}^{(2)}, c_{j}^{(3)}) x_{j} + (p^{(1)}, p^{(2)}, p^{(3)})}{\sum (d_{j}^{(1)}, d_{j}^{(2)}, d_{j}^{(3)}) x_{j} + (q^{(1)}, q^{(2)}, q^{(3)})} \\ subject to \end{matrix}$ (10) $\begin{matrix} \sum (a_{ij}^{(1)}, a_{ij}^{(2)}, a_{ij}^{(3)}) x_{j} (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) (b_{i}^{(1)}, b_{i}^{(2)}, b_{i}^{3}), \\ i = 1, 2, . . ., m x_{j} \geq 0, j = 1, 2, . . ., n \end{matrix}$ By using Zadeh’s extension principle of fuzzy numbers, the problem (10) reduce to an equivalent MOLP problem as follows: $\begin{matrix} Max {Z_{1} = \frac{\sum c_{j}^{(1)} x_{j} + p^{(1)}}{\sum d_{j}^{(3)} x_{j} + q^{(3)}}, Z_{2} = \frac{\sum c_{j}^{(2)} x_{j} + p^{(2)}}{\sum d_{j}^{(2)} x_{j} + q^{(2)}} \\ Z_{3} = \frac{\sum c_{j}^{(3)} x_{j} + p^{(3)}}{\sum d_{j}^{(1)} x_{j} + q^{(1)}}} \\ subject to \end{matrix}$ (11) $\begin{matrix} \sum a_{ij}^{(1)} x_{j} (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b_{i}^{(1)}, \sum a_{ij}^{(2)} x_{j} (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b_{i}^{(2)} \\ \sum a_{ij}^{(3)} x_{j} (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b_{i}^{(3)}, i = 1, 2, . . ., m \\ x_{j} \geq 0, j = 1, 2, . . ., n \end{matrix}$ which is a MOLFPP.

Theorem 4.1. If x ^* is an optimal solution of the problem (11), then x ^* is an optimal solution of the problem (10).

Proof. Let y be a feasible solution of (10). Clearly, y be the feasible solution of (11). Since, x^* be an optimal solution of (11), then Z₁ (x^*) ≥ Z₁ (y), Z₂ (x^*) ≥ Z₂ (y) and Z₃ (x^*) ≥ Z₃ (y). This implies that $\tilde{Z} (x) > \tilde{Z} (y)$ . Hence the proof.

Remark 4.1. In the case of FLFP problem with the cost of the objective function, the resources and the technological coefficients are trapezoidal fuzzy numbers, we decompose it into four objective crisp LFP problem.

4.1 Construction of membership functions

Let us assume that Z₁, Z₂ and Z₃ ≥ 0 for the feasible region. Hence, the MOLFPP can be solved by using the above procedure (as discussed in section 3) for each objective function. Calculate the individual minimum and maximum values of all the objective functions from the obtained solutions. Set the maximum value is U_k, (k = 1, 2, 3) and minimum value is L_k, (k = 1, 2, 3). Let L_k and U_k are the aspired level of achievement and the highest acceptable level of achievement for the k^th objective function, respectively. Define the membership function μ_k (k = 1, 2, 3) in the following manner: $μ_{1} = \frac{\frac{\sum c_{j}^{(1)} x_{j} + p^{(1)}}{\sum d_{j}^{(3)} x_{j} + q^{(3)}} - L_{1}}{U_{1} - L_{1}}$ (12) $μ_{2} = \frac{\frac{\sum c_{j}^{(2)} x_{j} + p^{(2)}}{\sum d_{j}^{(2)} x_{j} + q^{(2)}} - L_{2}}{U_{2} - L_{2}}$ (13) $μ_{3} = \frac{\frac{\sum c_{j}^{(3)} x_{j} + p^{(3)}}{\sum d_{j}^{(1)} x_{j} + q^{(1)}} - L_{3}}{U_{3} - L_{3}}$ (14)

4.2 Construction of membership goals

In MOLFPP, the aim is to achieve the highest membership value of the associated fuzzy goal in order to obtain the absolute satisfactory solution. In practice, achievement of all membership values to the highest degree is not possible due to conflicting objectives. Therefore, the decision policy for minimizing the regrets of all the levels should be taken into consideration. Hence, each objective function should try to maximize their membership function as close as possible to unity by minimizing its negative deviational variables. Therefore, in effect, we are simultaneously optimizing all the objective functions. So, for the defined membership functions in Equations (12), (13), and (14), the flexible membership goals having the aspired level unity can be represented as follows: $\frac{\frac{\sum c_{j}^{(1)} x_{j} + p^{(1)}}{\sum d_{j}^{(3)} x_{j} + q^{(3)}} - L_{1}}{U_{1} - L_{1}} + d_{1}^{-} - d_{1}^{+} = 1$ (15) $\frac{\frac{\sum c_{j}^{(2)} x_{j} + p^{(2)}}{\sum d_{j}^{(2)} x_{j} + q^{(2)}} - L_{2}}{U_{2} - L_{2}} + d_{2}^{-} - d_{2}^{+} = 1$ (16) $\frac{\frac{\sum c_{j}^{(3)} x_{j} + p^{(3)}}{\sum d_{j}^{(1)} x_{j} + q^{(1)}} - L_{3}}{U_{3} - L_{3}} + d_{3}^{-} - d_{3}^{+} = 1$ (17) where $d_{k}^{-} (\geq 0)$ and $d_{k}^{+} (\geq 0)$ with $d_{k}^{-} d_{k}^{+} = 0$ (k = 1, 2, 3) represent under and over deviations, respectively from the aspired levels.

In conventional GP, the under and or over deviational variables are included in the achievement function for minimizing them and that depend upon the type of the objective functions to be optimized.

In this approach, only the under deviational variable $d_{k}^{-}$ is required to be minimized to achieve the aspired levels of the fuzzy goals. It may be noted that any over deviation from a fuzzy goal indicates the full achievement of the membership value.

Now it can be easily realized that the membership goals (15), (16) and (17) are inherently non linear in nature and this may create computational difficulties in the solution process. To avoid such problems, a linearization process is presented in the following section.

4.3 Linearization of membership goals

Algebraic procedures such as pivoting are so powerful for manipulating linear qualities and inequalities that many nonlinear programming algorithms replace the given problem by an approximating linear problems. Here the membership goals are nonlinear. Hence, to convert nonlinear membership goals into linear in the following manner. The first membership goal (15) can be presented as $\begin{matrix} \frac{\sum c_{j}^{(1)} x_{j} + p^{(1)}}{\sum d_{j}^{(3)} x_{j} + q^{(3)}} - L_{1} + d_{1}^{-} (U_{1} - L_{1}) \\ - d_{1}^{+} (U_{1} - L_{1}) = U_{1} - L_{1} \\ \sum (c_{j}^{(1)} + U_{1} d_{j}^{(3)}) x_{j} + d_{1}^{-} (U_{1} - L_{1}) \\ (\sum d_{j}^{(3)} x_{j} + q^{(3)}) - d_{1}^{+} (U_{1} - L_{1}) \\ (\sum d_{j}^{(3)} x_{j} + q^{(3)}) = U_{1} q^{(3)} - p^{(1)} \end{matrix}$ (18) Similarly, the second (16) and third (17)membership goal can be presented as follows: $\begin{matrix} \sum (c_{j}^{(2)} + U_{1} d_{j}^{(2)}) x_{j} + d_{2}^{-} (U_{2} - L_{2}) \\ (\sum d_{j}^{(2)} x_{j} + q^{(2)}) - d_{2}^{+} (U_{2} - L_{2}) \\ (\sum d_{j}^{(2)} x_{j} + q^{(2)}) = U_{2} q^{(2)} - p^{(2)} \end{matrix}$ (19) $\begin{matrix} \sum (c_{j}^{(3)} + U_{3} d_{j}^{(1)}) x_{j} + d_{3}^{-} (U_{3} - L_{3}) \\ (\sum d_{j}^{(1)} x_{j} + q^{(1)}) - d_{3}^{+} (U_{3} - L_{3}) \\ (\sum d_{j}^{(1)} x_{j} + q^{(1)}) = U_{3} q^{(1)} - p^{(3)} \end{matrix}$ (20) Now, using the method of variable change as presented by [14], the goal expression (18), (19) and (20) can be linearized as follows: Letting $D_{1}^{-} = d_{1}^{-} (U_{1} - L_{1}) (\sum d_{j}^{(3)} x_{j} + q^{(3)})$ and $D_{1}^{+} = d_{1}^{+} (U_{1} - L_{1}) (\sum d_{j}^{(3)} x_{j} + q^{(3)})$ , the linear form of the expression in (18) is obtained as

$\begin{matrix} \sum (c_{j}^{(1)} + U_{1} d_{j}^{(3)}) x_{j} + D_{1}^{-} - D_{1}^{+} \\ = U_{1} q^{(3)} - p^{(1)} \end{matrix}$ (21) with $D_{1}^{-}, D_{1}^{+} \geq 0$ and $D_{1}^{-} D_{1}^{+} = 0$ since $d_{1}^{-}, d_{1}^{+} \geq 0$ and $\sum d_{j}^{(3)} x_{j} + q^{(3)} > 0 .$ Now, in making decision, minimization of $d_{1}^{-}$ means minimization of $D_{1}^{-} / ((U_{1} - L_{1}) (\sum d_{j}^{(3)} x_{j} + q^{(3)}))$ , which is also a non linear one. It may be noted that when a membership goal is fully achieved, $d_{1}^{-} = 0$ and when its achievement is zero, $d_{1}^{-} = 1$ are found in the solution. Hence, involvement of $d_{1}^{-} \leq 1$ in the solution leads to impose the following constraint to the model of the problem:\\ $\frac{D_{1}^{-}}{(U_{1} - L_{1}) (\sum d_{j}^{(3)} x_{j} + q^{(3)})} \leq 1$ . That is, $D_{1}^{-} - (U_{1} - L_{1}) \sum d_{j}^{(3)} x_{j} \leq (U_{1} - L_{1}) q^{(3)}$ Similarly, Equations (19) and (20) can be decompose into the following manner $\begin{matrix} \sum (c_{j}^{(2)} + U_{2} d_{j}^{(2)}) x_{j} + D_{2}^{-} - D_{2}^{+} \\ = U_{2} q^{(2)} - p^{(2)} \end{matrix}$ (22) $\begin{matrix} D_{2}^{-} - (U_{2} - L_{2}) \sum d_{j}^{(2)} x_{j} \\ \leq (U_{2} - L_{2}) q^{(2)} \end{matrix}$ (23) $\begin{matrix} \sum (c_{j}^{(3)} + U_{3} d_{j}^{(1)}) x_{j} + D_{3}^{-} - D_{3}^{+} \\ = U_{3} q^{(1)} - p^{(3)} \end{matrix}$ (24) $\begin{matrix} D_{3}^{-} - (U_{3} - L_{3}) \sum d_{j}^{(1)} x_{j} \\ \leq (U_{3} - L_{3}) q^{(1)} \end{matrix}$ (25) Here, on the basis of the previous discussion, it may be pointed out that any such constraint corresponding to $d_{k}^{+}, (k = 1, 2, 3)$ does not arise in the model formulation. Therefore, the equivalent GP problem for the FLFPP is

$\begin{matrix} Min Z = w_{1} D_{1}^{-} + w_{2} D_{2}^{-} + w_{3} D_{3}^{-} \\ and satisfy \\ \sum (c_{j}^{(1)} + U_{1} d_{j}^{(3)}) x_{j} + D_{1}^{-} - D_{1}^{+} = U_{1} q^{(3)} - p^{(1)} \\ \sum (c_{j}^{(2)} + U_{2} d_{j}^{(2)}) x_{j} + D_{2}^{-} - D_{2}^{+} = U_{2} q^{(2)} - p^{(2)} \\ \sum (c_{j}^{(3)} + U_{3} d_{j}^{(1)}) x_{j} + D_{3}^{-} - D_{3}^{+} = U_{3} q^{(1)} - p^{(3)} \\ subject to \\ D_{1}^{-} - (U_{1} - L_{1}) \sum d_{j}^{(3)} x_{j} \leq (U_{1} - L_{1}) q^{(3)} \\ D_{2}^{-} - (U_{2} - L_{2}) \sum d_{j}^{(2)} x_{j} \leq (U_{2} - L_{2}) q^{(2)} \\ D_{3}^{-} - (U_{3} - L_{3}) \sum d_{j}^{(1)} x_{j} \leq (U_{3} - L_{3}) q^{(1)} \end{matrix}$

$\begin{matrix} \sum a_{ij}^{(1)} x_{j} (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b_{i}^{(1)}, \sum a_{ij}^{(2)} x_{j} (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b_{i}^{(2)} \\ \sum a_{ij}^{(3)} x_{j} (\begin{matrix} \leq \\ = \\ \geq \end{matrix}) b_{i}^{(3)}, i = 1, 2, . . ., m \\ x_{j} \geq 0, j = 1, 2, . . ., n \end{matrix}$ (26) where Z represents the fuzzy achievement function consisting of the weighted under deviational variables, where the numerical weights w_k (≥0) , k = 1, 2, 3 represent the relative importance of achieving the aspired levels of the respective fuzzy goal subject to the constraints set in the decision situation. To assess the relative importance of the fuzzy goals properly, the weighting scheme suggested by [17] can be used to assign the values w_k, k = 1, 2, 3. In the present formulation w_k, k = 1, 2, 3. In the present formula w_k is determined as $w_{k} = \frac{1}{U_{k} - L_{k}}$ (27) The minsum GP method [12] can be used to solve the problem (27).

4.4 Algorithm

The proposed approach for solving FLFPP can be summarized as follows:

Step 1. Formulate the given real life problems as a FLFPPPs.

Step 2. Convert FLFPP into MOLFPP by using Zadeh’s extension principle of fuzzy numbers.

Step 3. Solve each objective function one by one, subject to the given set of constraints in the following manner:

If N_k (x) , (k = 1, 2, 3) is concave and positive, D_k (x) , (k = 1, 2, 3) is convex and positive then formulate the model (3) and solve.

If N_k (x) , (k = 1, 2, 3) is concave and negative, D_k (x) , (k = 1, 2, 3) is concave and positive then formulate the model (5) and solve.

Step 4. Calculate the individual minimum and maximum values of all the objective functions from the obtained solutions in step 2.

Step 5. From step 3, find for each objective the best U_k and worst L_k values corresponding to the set of solutions. Set L_k and U_k are aspired level of achievement and highest acceptable level of achievement for the k^thobjective function respectively.

Step 6. Define the membership functions μ_k, (k = 1, 2, 3) for each fractional objective function.

Step 7. Construct the fuzzy membership goals for each factional objective function.

Step 8. Convert non linear membership goals into linear.

Step 9. Evaluate the weights $w_{k} = \frac{1}{U_{k} - L_{k}}, (k = 1, 2, 3)$ .

Step 10. Formulate the LPP for the FLFPP.

Step 11. Solve the Model (27) to get the satisfactory solution of the FLFPP.

The flow chart (as shown in Fig. 1) is showing the procedure of the proposed method.

The advantages of the proposed method and disadvantages of the existing methods are discussed in the following section.

5 Advantages of the proposed method

The existing FLFP/MOLFP models [1– 3 , 21] cannot be consider the cost of the objective function, technological coefficients and resources are fuzzy numbers. They consider the fuzziness either in the constraint inequalities and/or in the aspiration levels of the of the objective function while the proposed work has been studied on FLFP model with fuzzy coefficients. M.K. Mehlawat and S. Kumar [17] presented a vertex-following solution method using a linear ranking function for a FLP Problem with fuzzy numbers. In this method the FLFPP defuzzified into LFP problem using linear ranking function and then solved simplex method. The shortcoming of the method is the fuzziness is neglected initially. Moreover, in literature there are many ranking methods are used. Each one have own merits and demerits. Hence, the selecting a suitable ranking function is a difficult one. Also, to solve fuzzy optimization problems using ranking function does not provide efficient solution. In the proposed approach does not use any ranking function to defuzzify the fuzzy numbers. Veeramani and Sumathi [32] proposed fuzzy mathematical programming approach for solving fuzzy linear factional Programming Problem. The shortcoming of it is that the optimal solution in not efficient. But, the proposed method provide the better solution. To show the efficiency of the proposed method compared with the existing method [32]. Hence the solution is improved in the proposed method.

6 Numerical examples

In this section, we illustrate the efficiency of the proposed algorithm using two real life example and one case study problem are discussed. Also, compare our results with mathematical programming approach [32]. A mathematical solver Lingo will be used to solve the mathematical programs.

Example 6.1. A company manufactures two kinds of products A and B with profit 10 and 20 dollar per unit respectively and fixed profit is 10 dollars. However the cost for each one unit of the above products is 3 and 4 dollars respectively. Assume that a fixed cost of 20 dollar is added to the cost function due to expected duration through the process of production. Suppose the raw material needed for manufacturing product A and B is 1 units per pound and 3 units per pound respectively, the supply for this raw material is restricted to 50 pounds. Man-hours per unit for the product A is 3 hours and product B is 2 hours per unit for manufacturing but total Man-hour available 80 hours daily. Determine how many Products A and B should be manufactured in order to maximize the total profit.

Here, the environmental coefficients such as profit (due to market situations), cost (due to market conditions), man-hour (due to presents of the workers, efficiency of the workers), raw materials (due to wastage) are imprecise numbers with triangular possibility distributions over the planning horizon due to incomplete information. For example, profit of product A is (8,10,12) dollars. Man-hours per unit of the product A is (2,3,4) hours. In this case, let x₁ and x₂ to be the amount of units of A and B to be produced. Then the above problem can be formulated to the following FLFPP:

$\begin{matrix} Max Z = \frac{\tilde{10} x_{1} + \tilde{20} x_{2} + \tilde{10}}{\tilde{3} x_{1} + \tilde{4} x_{2} + \tilde{20}} \\ subject to \\ \tilde{1} x_{1} + \tilde{3} x_{2} \leq \tilde{50}, \tilde{3} x_{1} + \tilde{2} x_{2} \leq \tilde{80}, x_{1}, x_{2} \geq 0 . \end{matrix}$ Let us assume that $\tilde{10} = (8, 10, 12)$ , $\tilde{20} = (17, 20, 23)$ , $\tilde{10} = (8, 10, 11)$ , $\tilde{3} = (2, 3, 4)$ , $\tilde{4} = (3, 4, 6)$ , $\tilde{20} = (16, 20, 22)$ , $\tilde{1} = (1, 1, 2)$ , $\tilde{3} = (2, 3, 4)$ , $\tilde{50} = (45, 50, 55)$ , $\tilde{3} = (1, 3, 5)$ , $\tilde{2} = (1, 2, 3)$ and $\tilde{80} = (70, 80, 90)$ .

The above FLFPP is equivalent to the following MOLFPP

$\begin{matrix} Max {Z_{1} = \frac{8 x_{1} + 17 x_{2} + 8}{4 x_{1} + 6 x_{2} + 22}, \\ Z_{2} = \frac{10 x_{1} + 20 x_{2} + 10}{3 x_{1} + 4 x_{2} + 20} \\ Z_{3} = \frac{12 x_{1} + 23 x_{2} + 11}{2 x_{1} + 3 x_{2} + 16}} \\ subject to \\ x_{1} + 2 x_{2} \leq 45, x_{1} + 3 x_{2} \leq 50, 2 x_{1} + 4 x_{2} \leq 55, \\ x_{1} + x_{2} \leq 70, 3 x_{1} + 2 x_{2} \leq 80, 5 x_{1} + 3 x_{2} \leq 90, \\ x_{1}, x_{2} \geq 0 . \end{matrix}$ (28) Here, it is observed that Z₁ (x) , Z₂ (x) and Z₃ (x) ≥0 for the feasible region. Therefore, the MOLFPP is solved for each of the objective function one by one, we obtain X₁ = (x₁ = 0, x₂ = 13); X₂ = (x₁ = 0, x₂ = 18) and X₃ = (x₁ = 0, x₂ = 12).

The objective function values are: Z₁ (X₁) =2.29, Z₁ (X₂) =2.42, Z₁ (X₃) =2.23, Z₂ (X₁) =3.75, Z₂ (X₂) =4.02, Z₂ (X₃) =3.68, Z₃ (X₁) =5.64,Z₃ (X₂) =6.07 and Z₃ (X₃) =5.52.

The upper and lower bounds of each objective function can be written as follows: 2.23 ≤ Z₁ ≤ 2.42, 3.68 ≤ Z₂ ≤ 4.02 and 5.52 ≤ Z₃ ≤ 6.07. That is L₁ = 2.23, L₂ = 3.68, L₃ = 5.52, U₁ = 2.42, U₂ = 4.02, U₃ = 6.07.

Using the linear membership function as defined in (7), an equivalent crisp model can be formulated as: $\begin{matrix} Min Z = 5.26 D_{1}^{-} + 2.94 D_{2}^{-} + 1.82 D_{3}^{-} \\ and satisfy \\ - 1.68 x_{1} + 2.48 x_{2} + D_{1}^{-} - D_{1}^{+} = 45.24 \\ - 2.06 x_{1} + 3.92 x_{2} + D_{2}^{-} - D_{2}^{+} = 70.4 \\ - . 14 x_{1} + 4.79 x_{2} + D_{3}^{-} - D_{3}^{+} = 86.12 \\ subject to \\ - 0.76 x_{1} - 1.14 x_{2} + D_{1}^{-} \leq 4.18 \\ - 1.02 x_{1} - 1.36 x_{2} + D_{2}^{-} \leq 6.8 \\ - 1.1 x_{1} - 1.65 x_{2} + D_{3}^{-} \leq 8.8 \\ x_{1} + 2 x_{2} \leq 45, x_{1} + 3 x_{2} \leq 50 \\ 2 x_{1} + 4 x_{2} \leq 55, x_{1} + x_{2} \leq 70 \\ 3 x_{1} + 2 x_{2} \leq 80, 5 x_{1} + 3 x_{2} \leq 90 \\ x_{1}, x_{2} \geq 0 . \end{matrix}$ The problem is solved the results are x₁ = 0, x₂ = 13.75 and Z = (2.861, 3.8, 4.81). The above problem can be solved fuzzy mathematical programming approach [32], the solution is x₁ = 0, x₂ = 13 and Z = (2.29, 3.75, 5.363). Figure 2 depicts that the membership values for the optimal solution of the Proposed method and Fuzzy mathematical programming method. It is noted from Fig. 2 that as proposed method is the better core value of the existing one. Moreover, the proposed solution is better than the existing solution. Hence, the solution is improved by the proposed method.

Example 6.2. Consider the Grand Stand Oil Company, which produces regular-grade and premium-grade gasoline products by blending three petroleum components. The gasoline are sold at different prices, and the petroleum components have different costs. Petroleum components profit (dollar) and cost (dollar) per gallon shown in the Tables 1 and 2. The available petroleum supplies are component 1, 2 and 3 is 5000, 10000 and 10000 gallons respectively. The product specifications are shown in the Table 3. The requirement for at least 10000 gallons of regular grade gasoline. The Grand Strand blending problem is to determine how many gallons of each component should be used in the regular blend and in the premium blend. The optimum solution should be maximize the firm’s profit/cost subject to the constraints.

The environmental coefficients generally imprecise numbers with triangular possibility distributions over the planning horizon due to incomplete or unobtainable information. For example, the profit of the regular gasoline component 1 is (60,70,80), the cost of the regular gasoline component 1 is (5,6,7) and the product specification regular gasoline component 1 is (0.2,0.3,0.4), etc.

Let x_ij = gallons of component i used in gasoline j, where i = 1, 2, or 3 for component 1,2, or 3 and j = r if regular or j = p if premium. The six decision variables become $\begin{matrix} x_{1 r} & = & gallons of component 1 in regular gasoline \\ x_{2 r} & = & gallons of component 2 in regular gasoline \\ x_{3 r} & = & gallons of component 3 in regular gasoline \\ x_{1 p} & = & gallons of component 1 in premium gasoline \\ x_{2 p} & = & gallons of component 2 in premium gasoline \\ x_{3 p} & = & gallons of component 3 in premium gasoline \end{matrix}$ With the notations for the six decision variables just defined, the total gallons of each type of gasoline produced can be expressed as the sum of the gallons of the blended components $\begin{matrix} Regular gasoline & = & x_{1 r} + x_{2 r} + x_{3 r}, \\ Premium gasoline & = & x_{1 p} + x_{2 p} + x_{3 p} \end{matrix}$ Similarly, the total gallons of each component used can be expressed as these sums: $\begin{matrix} Component 1 & = & x_{1 r} + x_{1 p}, \\ Component 2 & = & x_{2 r} + x_{2 p} \\ Component 3 & = & x_{3 r} + x_{3 p} \end{matrix}$ The total profit gained by the firm may be expressed in the following form: $\begin{matrix} N (X) & = & (60, 75, 85) x_{1 r} + (50, 60, 70) x_{2 r} \\ + (45, 50, 56) x_{3 r} + (70, 80, 90) x_{1 p} \\ + (65, 70, 75) x_{2 p} + (80, 90, 100) x_{3 p} . \end{matrix}$ Obviously, total cost is $\begin{matrix} D (X) & = & (5, 6, 7) x_{1 r} + (4, 5, 6) x_{2 r} \\ + (3.5, 4, 5) x_{3 r} + (4, 5, 5.5) x_{1 p} \\ + (6, 7, 7.5) x_{2 p} + (7, 8, 9) x_{3 p} . \end{matrix}$ Limitations on the availability of the three components can be expressed by the three constraints: $\begin{matrix} x_{1 r} + x_{1 p} & \leq & 5000 (Component 1), \\ x_{2 r} + x_{2 p} & \leq & 10000 (Component 2), \\ x_{3 r} + x_{3 p} & \leq & 10000 (Component 3) \end{matrix}$ The constraint for at least 10000 gallons of the regular grade gasoline is written x_1r + x_2r + x_3r ≥ 10000. The first specification states that component 1 must account for at most 30 percent of the total gallons of regular gasoline produced. That is, $\frac{x_{1 r}}{x_{1 r} + x_{2 r} + x_{3 r}} \leq (0.2, 0.3, 0.4)$ When this constraint is rewritten with the variables on the left side of the equation and a constant on the right side, the first product-specification constraint becomes $\begin{matrix} (0.6, 0.7, 0.8) x_{1 r} + (- 0.4, - 0.3, - 0.2) x_{2 r} \\ + (- 0.4, - 0.3, - 0.2) x_{3 r} \leq 0 . \end{matrix}$ Similarly, the remaining 5 blending specifications can be written as 5 constraints. Thus, the complete FLFP model with six decision variables and 10 constraints can be written as follows: $\begin{matrix} Max Z \\ = \frac{60 x_{1 r} + 50 x_{2 r} + 45 x_{3 r} + 70 x_{1 p} + 65 x_{2 p} + 80 x_{3 p}}{7 x_{1 r} + 6 x_{2 r} + 5 x_{3 r} + 5.5 x_{1 p} + 7.5 x_{2 p} + 9 x_{3 p}}, \\ \frac{75 x_{1 r} + 60 x_{2 r} + 50 x_{3 r} + 80 x_{1 p} + 70 x_{2 p} + 90 x_{3 p}}{6 x_{1 r} + 5 x_{2 r} + 4 x_{3 r} + 5 x_{1 p} + 7 x_{2 p} + 8 x_{3 p}}, \\ \frac{80 x_{1 r} + 70 x_{2 r} + 56 x_{3 r} + 90 x_{1 p} + 75 x_{2 p} + 100 x_{3 p}}{5 x_{1 r} + 4 x_{2 r} + 3.5 x_{3 r} + 4 x_{1 p} + 6 x_{2 p} + 7 x_{3 p}} \end{matrix}$ $\begin{matrix} subject to \\ x_{1 r} + x_{1 p} \leq 5000, x_{2 r} + x_{2 p} \leq 10000 \\ x_{3 r} + x_{3 p} \leq 10000, x_{1 r} + x_{2 r} + x_{3 r} \geq 1000 \\ 0.6 x_{1 r} - 0.4 x_{2 r} - 0.4 x_{3 r} \leq 0 \\ 0.7 x_{1 r} - 0.3 x_{2 r} - 0.3 x_{3 r} \leq 0 \\ 0.8 x_{1 r} - 0.2 x_{2 r} - 0.2 x_{3 r} \leq 0 \\ - 0.5 x_{1 r} + 0.5 x_{2 r} - 0.5 x_{3 r} \geq 0 \\ - 0.4 x_{1 r} + 0.6 x_{2 r} - 0.4 x_{3 r} \geq 0 \\ - 0.3 x_{1 r} + 0.7 x_{2 r} - 0.3 x_{3 r} \geq 0 \\ - 0.25 x_{1 r} - 0.25 x_{2 r} + 0.75 x_{3 r} \leq 0 \\ - 0.2 x_{1 r} - 0.2 x_{2 r} + 0.8 x_{3 r} \leq 0 \\ - 0.1 x_{1 r} - 0.1 x_{2 r} + 0.9 x_{3 r} \leq 0 \\ 0.7 x_{1 p} - 0.3 x_{2 p} - 0.3 x_{3 p} \geq 0 \\ 0.75 x_{1 p} - 0.25 x_{2 p} - 0.25 x_{3 p} \geq 0 \\ 0.8 x_{1 p} - 0.2 x_{2 p} - 0.2 x_{3 p} \geq 0 \\ - 0.5 x_{1 p} + 0.5 x_{2 p} - 0.5 x_{3 p} \leq 0 \\ - 0.4 x_{1 p} + 0.6 x_{2 p} - 0.4 x_{3 p} \leq 0 \\ - 0.3 x_{1 p} + 0.7 x_{2 p} - 0.3 x_{3 p} \leq 0 \\ - 0.4 x_{1 p} - 0.4 x_{2 p} + 0.6 x_{3 p} \geq 0 \\ - 0.3 x_{1 p} - 0.3 x_{2 p} + 0.7 x_{3 p} \geq 0 \\ - 0.2 x_{1 p} - 0.2 x_{2 p} + 0.8 x_{3 p} \geq 0 \\ x_{1 r}, x_{2 r}, x_{3 r}, x_{1 p}, x_{2 p}, x_{3 p} \geq 0 \end{matrix}$ Here, it is observed that Z₁ (x) , Z₂ (x) and Z₃ (x) ≥0 for the feasible region. Therefore, the MOLFPP is solved for each of the objective function one by one, we obtain X₁ = (x_1r = 0, x_2r = 8583, x_3r = 954, x_1p = 4768, x_2p = 0, x_3p = 3179); X₂ = (x_1r = 0, x_2r = 8584, x_3r = 954, x_1p = 4769, x_2p = 0, x_3p = 3179) and X₃ = (x_1r = 0, x_2r = 9456, x_3r = 0, x_1p = 4728, x_2p = 0, x_3p = 3152).

The objective function values are: Z₁ (X₁) =9.54, Z₁ (X₂) =9.55, Z₁ (X₃) =9.5, Z₂ (X₁) =12.81,Z₂ (X₂) =12.82, Z₂ (X₃) =12.78, Z₃ (X₁) =17.73, Z₃ (X₂) =17.74 and Z₃ (X₃) =17.8.

The upper and lower bounds of each objective function can be written as follows: 9.5 ≤ Z₁ ≤ 9.55, 12.78 ≤ Z₂ ≤ 12.82 and 17.73 ≤ Z₃ ≤ 17.8. That is L₁ = 9.5, L₂ = 12.78, L₃ = 17.73, U₁ = 9.55, U₂ = 12.82, U₃ = 17.8.

Using the linear membership function as defined in (7), an equivalent crisp model can be formulated as: $\begin{matrix} Min Z = 20 D_{1}^{-} + 25 D_{2}^{-} + 14.29 D_{3}^{-} \\ and satisfy \\ - 6.85 x_{1 r} - 7.3 x_{2 r} - 2.75 x_{3 r} + 17.47 x_{1 p} \\ - 6.63 x_{2 p} - 5.95 x 3 p + D_{1}^{-} - D_{1}^{+} = 0 \\ - 1.92 x_{1 r} - 4.1 x_{2 r} - 1.28 x_{3 r} + 15.9 x_{1 p} \\ - 19.74 x_{2 p} - 12.56 x 3 p + D_{2}^{-} - D_{2}^{+} = 0 \\ - 9 x_{1 r} - 1.2 x_{2 r} - 6.3 x_{3 r} + 18.8 x_{1 p} \\ - 3.18 x_{2 p} - 24.6 x 3 p + D_{3}^{-} - D_{3}^{+} = 0 \\ subject to \\ 0.35 x_{1 r} + 0.3 x_{2 r} + 0.25 x_{3 r} + 0.28 x_{1 p} \\ + 0.38 x_{2 p} + 0.45 x 3 p + D_{1}^{-} - \geq 0 \\ 0.24 x_{1 r} + 0.2 x_{2 r} + 0.16 x_{3 r} + 0.2 x_{1 p} \\ + 0.28 x_{2 p} + 0.32 x 3 p + D_{2}^{-} - \geq 0 \\ 0.35 x_{1 r} + 0.28 x_{2 r} + 0.24 x_{3 r} + 0.28 x_{1 p} \\ + 0.42 x_{2 p} + 0.49 x 3 p + D_{3}^{-} - \geq 0 \end{matrix}$ and add the twenty two constraints as defined in the original problem. Now, the problem is solved the results are x_1r = 0, x_2r = 9000, x_3r = 1000, x_1p = 5000, x_2p = 0, x_3p = 3333.33 and Z = (9.54, 12.814, 17.74). The above problem can be solved fuzzy mathematical programming approach [32], the solution is x_1r = 0, x_2r = 8626, x_3r = 912, x_1p = 4764, x_2p = 0, x_3p = 3176 and Z = (9.50, 12.78, 17.90). Figure 3 represents the membership values for the optimal solution of the Proposed method and Fuzzy mathematical programming method. It shows that as proposed method is the better core value of the existing one. Also, the discussed method provides the better decision variables values (ie. the available resources are efficiently used in the presented method). Therefore, the proposed solution is better than the existing solution.

7 Concluding remarks

In the presented paper a new method has been suggested to solve fuzzy linear fractional programming problems with cost of the objective function, technological coefficients and resources are triangular fuzzy numbers. Based on Zadeh’s extension principle, the FLFPP is transformed into equivalent MOLFPP. In the proposed approach fuzzy goal programming model is formulated for achievement of the highest membership value of each fractional objectives. In the solution process, the method of variable change on under and lover deviational variables are associated with fuzzy goals off the model is introduced and solve the problems by using linear goal programming approach. By using the real time problems the proposed method compared with the fuzzy mathematical programming approach [32]. The need for having such methodology is evident and hence this technique can contribute significantly to the literature of fractional programming.

Acknowledgments

The authors would like to thank the Editor-in-Chief and anonymous referees for the various suggestions which have led to an improvement in both the quality and the clarity of this paper.

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