Implementing deviation degree of two closed intervals to decode fully fuzzy multiobjective linear programming problem

Abstract

In this paper, a new concept of deviation distance and deviation degree between two closed intervals is being introduced. A new algorithm has been proposed to solve a Fully Fuzzy Multiobjective Linear Programming (FFMOLP) problem with all the constraints as fuzzy inequalities. In the algorithm, the FFMOLP problem is transformed into Crisp Nonlinear Programming (CNLP) problem by using the proposed concept of deviation degree, fuzzy nearest interval approximation, and goal programming (GP) technique. Then, it is proved that the δ-optimal solution of the CNLP problem is the fuzzy Pareto optimal solution of FFMOLP problem. Finally, to illustrate our method and to show its effectiveness, numerical examples are solved and are compared with the existing methods.

Keywords

Fully fuzzy multiobjective linear programming problem deviation distance deviation degree goalprogramming

1 Introduction

Multiobjective linear programming problem plays a vital role in many areas of engineering and management as it is helpful for Decision Maker (DM) to consider simultaneously several objectives in order to obtain a set of adequate solutions.

Moreover, in most of the situations, the coefficients of multiobjective models are not known precisely due to inexistent or scarce nature of relevant data, inadequate information, etc. To deal with such imprecise situations, Zimmermann [7] has applied fuzzy set theory concepts in multiobjective linear programming problem in 1978. In the past four decades, many researchers [6 , 13] have proposed different methods based on the concept of deviation degree, GP, membership function, etc, for solving various types of fuzzy multiobjective linear programming problems. In recent years, some authors have shifted their focus from partially fuzzy linear programming problems, in which either decision variables or decision parameters are fuzzy but not all of them are fuzzy in nature, to Fully Fuzzy Linear Programming (FFLP) problem in which decision variables and decision parameters both are fuzzy in nature. Allahviranloo et al. [19] have introduced a new method for solving the FFLP problem using ranking function. Lotfi et al. [5] defined the FFLP problem with all the parameters and variables as triangular fuzzy numbers. Kumar et al. [1] have given a new method, in which, there is no restriction on the elements of coefficient matrix for solving FFLP problem. Kumar and Kaur [10] have introduced a new method called Mehar’s method for solving FFLP problem with trapezoidal fuzzy numbers. Ezzati et al. [16] have given a new lexicographical ordering for triangular fuzzy numbers and converted the FFLP problem into crisp multiobjective linear programming problem. Then, they have found the exact optimal solution of FFLP problem. Recently, Jayalakshmi and Pandian [12] have found the proper efficient solution of FFMOLP problem using Total objective-segregationmethod.

In most of the methods fuzzy numbers are replaced by real numbers due to which most of the information is lost. The aim of this paper is to convert the FFMOLP problem into Multiobjective Interval Linear Programming (MOILP) problem by using fuzzy nearest interval approximation instead of converting fuzzy problem directly into crisp problem in order to avoid the pitfall of fuzziness. The paper is organized as follows: in Section 2, we have given some basic definitions related to closed intervals and fuzzy set theory; a concept of deviation distance and deviation degree is introduced in Section 3; Section 4 provides a new method for solving FFMOLP problem and finding its fuzzy Pareto optimal solution; and in Section 5, a comparison is made between the existing method and proposed method through numerical examples; Finally, the conclusion is drawn inSection 6.

2 Preliminaries

In this section, some basic definitions related to closed interval and fuzzy set theory have been reviewed.

Definition 2.1. [3] Let A = [a_L, a_U] be a closed interval then the center and width of A is defined as $m (A) = \frac{a_{L} + a_{U}}{2}$ and $w (A) = \frac{a_{U} - a_{L}}{2}$ respectively.

Remark 2.1. [8] Let A = [a_L, a_U] be a closed interval. The closed interval is also denoted by its center and width as A = 〈m (A) , w (A) 〉.

Definition 2.2. [8] Let A = [a_L, a_U] and B = [b_L, b_U] are the closed interval then the following order relations are defined between two closed intervals as follows:

A ≤ _mw B iff m (A) ≤ m (B) and w (A) ≥ w (B),

A < _mw B iff m (A) < m (B) and w (A) > w (B).

Definition 2.3. [1] The characteristic function μ_A of a crisp set A ⊆ X assigns a value either 0 or 1 to each member in X. This function can be generalized to a function $μ_{\tilde{A}}$ such that the value assigned to the element of the universal set X fall within a specified range i.e. $μ_{\tilde{A}} : X \to [0, 1]$ . The assigned value indicates the membership grade of the element in the set A. The function $μ_{\tilde{A}}$ is called the membership function and the set $\tilde{A} = {(x, μ_{\tilde{A}} (x)) : x \in X}$ defined by $μ_{\tilde{A}} (x)$ for each x ∈ X is called a fuzzy set.

Definition 2.4. [15] A fuzzy subset $\tilde{A}$ of the real line R with membership function $μ_{\tilde{A}} : X \to [0, 1]$ is called a fuzzy number if

$\tilde{A}$ is normal, i.e. there exist an element x such that $μ_{\tilde{A}} (x) = 1$ ;

$\tilde{A}$ is fuzzy convex, i.e. $μ_{\tilde{A}} (λ x + (1 - λ) y) \geq μ_{\tilde{A}} (x) \land μ_{\tilde{A}} (y)$ ∀ x, y ∈ R, ∀ λ ∈ [0, 1];

$μ_{\tilde{A}}$ is upper semi-continuous;

$supp \tilde{A}$ is bounded, where $supp (\tilde{A}) = \bar{{x \in R : μ_{\tilde{A}} (x) > 0}}$ .

Remark 2.2.F (R) denotes the set of all fuzzy numbers.

Definition 2.5. [1] A fuzzy number $\tilde{A} = (a, b, c)$ is said to be a triangular fuzzy number if its membership function is given by $\begin{matrix} μ_{\tilde{A}} (x) = {\begin{matrix} \frac{x - a}{b - a} & if a \leq x \leq b \\ \frac{x - c}{b - c} & if b \leq x \leq c \\ 0 & otherwise \end{matrix} \end{matrix}$

Remark 2.3.TF(R) denotes the set of all triangular fuzzy numbers.

Definition 2.6. [1] A triangular fuzzy number $\tilde{A} = (a, b, c)$ is said to be non-negative fuzzy number if and only if a ≥ 0.

Definition 2.7. [1] Let $\tilde{A} = (a, b, c)$ and $\tilde{B} = (e, f, g)$ be two triangular fuzzy numbers then arithmetic operations on triangular fuzzy numbers are defined as

$\tilde{A} \oplus \tilde{B} = (a, b, c) \oplus (e, f, g) = (a + e, b + f, c + g)$

$- \tilde{A} = - (a, b, c) = (- c, - b, - a)$

$\tilde{A} ⊖ \tilde{B} = (a, b, c) ⊖ (e, f, g) = (a - g, b - f, c - e)$

If $\tilde{B} = (x, y, z)$ be a non-negative triangular fuzzy number, then $\begin{matrix} \tilde{A} \otimes \tilde{B} ≅ {\begin{matrix} (ax, by, cz), & a \geq 0, \\ (az, by, cz), & a < 0, c \geq 0, \\ (az, by, cx), & c < 0, \end{matrix} \end{matrix}$

Definition 2.8. [1] A ranking function is a function $R : F (R) \to R$ , which maps each fuzzy number into the real line, where a natural order exists.

In particular, $R : TF (R) \to R$ is defined by $R (\tilde{A}) = \frac{a + 2 b + c}{4}$ , for all $\tilde{A} = (a, b, c) \in TF (R)$ .

Definition 2.9. [2] Let $\tilde{A}$ and $\tilde{B}$ be two triangular fuzzy numbers then

$\tilde{A} ≻ \tilde{B}$ iff $R (\tilde{A}) > R (\tilde{B})$

$\tilde{A} \approx \tilde{B}$ iff $R (\tilde{A}) = R (\tilde{B})$

$\tilde{A} ≺ \tilde{B}$ iff $R (\tilde{A}) < R (\tilde{B})$

Definition 2.10. [15] Let $\tilde{A}$ and $\tilde{B}$ be two fuzzy numbers with α-cut [A_L (α) , A_U (α)] and [B_L (α) , B_U (α)], respectively, then the metric d on F (R) is defined as $\begin{matrix} d (\tilde{A}, \tilde{B}) = \sqrt{(\begin{matrix} \int_{0}^{1} (A_{L} (α) - B_{L} (α))^{2} d α \\ + \int_{0}^{1} (A_{U} (α) - B_{U} (α))^{2} d α \end{matrix})} . \end{matrix}$

Definition 2.11. [15] The nearest interval approximation of fuzzy numbers $\tilde{A}$ with respect to metric d is defined as $\begin{matrix} C_{d} (\tilde{A}) & = [\int_{0}^{1} {\tilde{A}}_{L} (α) d α, \int_{0}^{1} {\tilde{A}}_{U} (α) d α] \\ = [(C_{d} (\tilde{A}))_{L}, (C_{d} (\tilde{A}))_{U}] . \end{matrix}$ If $\tilde{A} = (a, b, c)$ be a triangular fuzzy number then $C_{d} (\tilde{A}) = [\frac{a + b}{2}, \frac{b + c}{2}]$ .

We can easily observe the following fuzzy order relation on F (R) using Definitions 2.2 and 2.11:

Let $\tilde{A}, \tilde{B} \in F (R)$ then

$\tilde{A} ⪯_{mw} \tilde{B}$ iff $m (C_{d} (\tilde{A})) \leq m (C_{d} (\tilde{B}))$ and $w (C_{d} (\tilde{A})) \geq w (C_{d} (\tilde{B}))$ ,

$\tilde{A} ≺_{mw} \tilde{B}$ iff $m (C_{d} (\tilde{A})) < m (C_{d} (\tilde{B}))$ and $w (C_{d} (\tilde{A})) > w (C_{d} (\tilde{B}))$ .

$\tilde{A} ⪰_{mw} \tilde{B}$ iff $m (C (\tilde{A})) \geq m (C_{d} (\tilde{B}))$ and $w (C_{d} (\tilde{A})) \leq w (C_{d} (\tilde{B}))$ ,

$\tilde{A} ≻_{mw} \tilde{B}$ iff $m (C_{d} (\tilde{A})) > m (C_{d} (\tilde{B}))$ and $w (C_{d} (\tilde{A})) < w (C_{d} (\tilde{B}))$ .

Both of the above order relations ⪯_mw and ⪰_mw are reflexive, antisymmetric and transitive and hence, define partial ordering between fuzzy numbers.

3 Deviation distance and deviation degree between two closed intervals

Ishibuchi and Tanaka [8] have proposed two different concepts of ordering two closed intervals on the basis of their end points, and their center and width. Various approaches [3 , 19] have been adopted by many researchers to solve the interval programming problem using the closed interval ordering concept.

Cheng et al. [6] have extended the concept of distance measure between two triangular fuzzy numbers to deviation degree between two triangular fuzzy numbers. In this section, we define a new concept of deviation degree between two closed intervals with the help of their center and width as follows:

Definition 3.1. Let A = [a_L, a_U] and B = [b_L, b_U] are two closed intervals then

the positive deviation distance of A from B is $\begin{matrix} d_{B} (A^{+}) & = max (m (A) - m (B), 0) \\ + max (w (A) - w (B), 0), \end{matrix}$

the negative deviation distance of A from B is $\begin{matrix} d_{B} (A^{-}) & = - min (m (A) - m (B), 0) \\ - min (w (A) - w (B), 0) . \end{matrix}$

In an interval programming problem, since different constraints may have different scale, the deviation distance of two closed intervals may also have different scale. Therefore, it is appropriate to normalize all the deviation distance to the same scale. For comparing two kinds of inequality constraints, we define two kinds of deviation degree as follows:

Definition 3.2. Let A = [a_L, a_U] and B = [b_L, b_U] are two closed intervals then

the positive deviation degree of A from B is $\begin{matrix} D_{B} (A^{+}) = \frac{d_{B} (A^{+})}{2 (M - m)}, \end{matrix}$

the negative deviation distance of A from B is $\begin{matrix} D_{B} (A^{-}) = \frac{d_{B} (A^{-})}{2 (M - m)}, \end{matrix}$

where M = max {a_L, b_L} and m = min {a_L, b_L}.

4 Fully fuzzy multiobjective linear programming problem

In the literature two type of approaches are used to solve the fuzzy linear programming problem. One of them is using different ranking functions, each fuzzy number is converted into the single point of the real line and corresponding to that we obtain the crisp linear programming problem. The other approach ranks fuzzy numbers by means of a fuzzy relationship. In this approach, DM takes different degree of satisfaction of constraints for the desirable optimal solution, which allows DM to look for a non dominated solution. In this section, we propose a new algorithm to solve the FFMOLP problem and find the fuzzy Pareto optimal solution to it. This is based on the second approach which provides cushion to DM to decide how much DM wants to compromise with the feasibility for the desire objective function value.

FFMOLP problem with k objectives, m inequality constraints and n variables can be formulated as follows: $\begin{matrix} (P 1) & Max {\tilde{Z}}_{1} (\tilde{X}) = \sum_{j = 1}^{n} {\tilde{c}}_{1 j} \otimes {\tilde{x}}_{j} \\ ⋮ \\ Max {\tilde{Z}}_{k} (\tilde{X}) = \sum_{j = 1}^{n} {\tilde{c}}_{kj} \otimes {\tilde{x}}_{j} \\ subject to \\ \sum_{j = 1}^{n} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j} ⪯ {\tilde{b}}_{i}, i = 1, 2, \dots, m_{1}, \\ \sum_{j = 1}^{n} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j} ⪰ {\tilde{b}}_{i}, i = m_{1} + 1, m_{1} + 2, \dots, m, \\ \tilde{X} \geq \tilde{0}, \end{matrix}$ where ${\tilde{C}}_{q}$ = $[{\tilde{c}}_{qj}]_{n \times 1}$ , $\tilde{X}$ = $[{\tilde{x}}_{j}]_{n \times 1}$ , ${\tilde{x}}_{j}$ = (x_j, y_j, z_j), ${\tilde{a}}_{ij}$ , ${\tilde{b}}_{j}$ , ${\tilde{c}}_{qj} \in TF (R)$ , q = 1, 2, …, k, j = 1, 2, …, n and i = 1, 2, …, m. ⪯ and ⪰ are fuzzy less than equal to and fuzzy greater than equal to inequalities respectively.

In the multiobjective programming problem, it is unlikely that all the objective functions will attain their optimal values at the same time due to the conflicting nature of the objective functions; hence, it is very difficult to maximize all the objective functions simultaneously. Hence, Pareto optimal solution provides a better compromise solution.

In fuzzy environment, fuzzy Pareto optimal solution has been defined by various authors [6 , 17] using the deviation degree, membership function and magnitude of fuzzy number for different kind of fuzzy multiobjective linear programming problems, in which fuzzy objective functions are replaced by a single numbers. Therefore, critical information is lost in the process. In order to avoid such situations, we are comparing fuzzy objective functions on the basis of their center value and width of the fuzzy objective functions, respectively. Now, we will define a fuzzy Pareto optimal solution based on the concept of fuzzy order relations ⪯_mw and ≺_mw for FFMOLP problem as follows:

Definition 4.1. A vector ${\tilde{X}}^{*} = [{\tilde{x}}_{j}^{*}]_{n \times 1}$ where ${\tilde{x}}_{j}^{*} = (x_{j}^{*}, y_{j}^{*}, z_{j}^{*})$ , j = 1, 2, …, n, is said to be fuzzy Pareto optimal solution of FFMOLP problem if it satisfies the following conditions

${\tilde{X}}^{*}$ satisfying constraints of (P1),

There does not exist any $\tilde{X} = [{\tilde{x}}_{j}]_{n \times 1}$ where ${\tilde{x}}_{j} = (x_{j}, y_{j}, z_{j})$ , j = 1, 2, …, n, satisfying constraints of (P1) such that $\begin{matrix} {\tilde{Z}}_{q} ({\tilde{X}}^{*}) & ⪯_{mw} {\tilde{Z}}_{q} (\tilde{X}) for all q = 1, 2, \dots, k \\ and \\ {\tilde{Z}}_{r} ({\tilde{X}}^{*}) & ≺_{mw} {\tilde{Z}}_{r} (\tilde{X}) for at least one r = 1, 2, \dots, k . \end{matrix}$

4.1 Procedure to find the fuzzy Pareto optimal solution of FFMOLP problem

In this subsection, a new algorithm is proposed to find the fuzzy Pareto optimal solution of (P1). The steps of the proposed algorithm are given as follows:

Step 1: Using Definition 2.7 and assuming $\sum_{j = 1}^{n} {\tilde{c}}_{qj} \otimes {\tilde{x}}_{j} = (e_{q}, f_{q}, g_{q})$ , $\sum_{j = 1}^{n} {\tilde{a}}_{ij} \otimes {\tilde{x}}_{j} = (u_{i}$ , v_i, w_i) and ${\tilde{b}}_{i} = (b_{i 1}, b_{i 2}, b_{i 3})$ , rewrite (P1) as: $\begin{matrix} (P 2) & Max {\tilde{Z}}_{1} (\tilde{X}) = (e_{1}, f_{1}, g_{1}) \\ ⋮ \\ Max {\tilde{Z}}_{k} (\tilde{X}) = (e_{k}, f_{k}, g_{k}) \\ subject to \\ (u_{i}, v_{i}, w_{i}) ⪯ (b_{i 1}, b_{i 2}, b_{i 3}), i = 1, 2, \dots, m_{1}, \\ (u_{i}, v_{i}, w_{i}) ⪰ (b_{i 1}, b_{i 2}, b_{i 3}), \\ i = m_{1} + 1, m_{1} + 2, \dots, m, \\ \tilde{X} \geq \tilde{0} . \end{matrix}$

Step 2: Using nearest interval approximation of fuzzy numbers (Definition 2.11) with respect to metric d, convert each of the fuzzy numbers involved in objectives and constraints of (P2) obtained in Step 1, to the following: $\begin{matrix} {\tilde{Z}}_{q} (\tilde{X}) & = (e_{q}, f_{q}, g_{q}) \approx C_{d} ({\tilde{Z}}_{q} (\tilde{X})) \\ = [\frac{e_{q} + f_{q}}{2}, \frac{f_{q} + g_{q}}{2}] \\ for all q = 1, 2, \dots, k, \\ (u_{i}, v_{i}, w_{i}) & \approx C_{d} (u_{i}, v_{i}, w_{i}) \\ = [\frac{u_{i} + v_{i}}{2}, \frac{v_{i} + w_{i}}{2}] \\ for all i = 1, 2, \dots, m, \\ (b_{i 1}, b_{i 2}, b_{i 3}) & \approx C_{d} (b_{i 1}, b_{i 2}, b_{i 3}) \\ = [\frac{b_{i 1} + b_{i 2}}{2}, \frac{b_{i 2} + b_{i 3}}{2}] \\ for all i = 1, 2, \dots, m . \end{matrix}$

Step 3: Calculate the center and width (Remark 2.1) of the intervals $C_{d} ({\tilde{Z}}_{q} (\tilde{X}))$ , C_d (u_i, v_i, w_i) and C_d (b_i1, b_i2, b_i3). Then denote these intervals as follows: $\begin{matrix} C_{d} ({\tilde{Z}}_{q} (\tilde{X})) & = 〈 \frac{e_{q} + 2 f_{q} + g_{q}}{4}, \frac{g_{q} - e_{q}}{4} 〉, \\ for all q = 1, 2, \dots, k, \\ C_{d} (u_{i}, v_{i}, w_{i}) & = 〈 \frac{u_{i} + 2 v_{i} + w_{i}}{4}, \frac{w_{i} - v_{i}}{4} 〉, \\ for all i = 1, 2, \dots, m, \end{matrix}$ $\begin{matrix} C_{d} (b_{i 1}, b_{i 2}, b_{i 3}) & = 〈 \frac{b_{i 1} + 2 b_{i 2} + b_{i 3}}{4}, \frac{b_{i 3} - b_{i 1}}{4} 〉, \\ for all i = 1, 2, \dots, m . \end{matrix}$

Step 4: Convert (P2) obtained in Step 1 to the following problem: $\begin{matrix} (P 3) & Max C_{d} ({\tilde{Z}}_{1} (\tilde{X})) = [\frac{e_{1} + f_{1}}{2}, \frac{f_{1} + g_{1}}{2}] \\ ⋮ \\ Max C_{d} ({\tilde{Z}}_{k} (\tilde{X})) = [\frac{e_{k} + f_{k}}{2}, \frac{f_{k} + g_{k}}{2}] \\ subject to \\ [u_{i} + v_{i}, v_{i} + w_{i}] \leq_{mw} [b_{i 1} + b_{i 2}, b_{i 2} + b_{i 3}], \\ i = 1, 2, \dots, m_{1}, \\ [u_{i} + v_{i}, v_{i} + w_{i}] \geq_{mw} [b_{i 1} + b_{i 2}, b_{i 2} + b_{i 3}], \\ = m_{1} + 1, m_{1} + 2, \dots, m, \\ x_{j}, y_{j} - x_{j}, z_{j} - y_{j} \geq 0, j = 1, 2, \dots, n . \end{matrix}$

Step 5: Converting the interval constraints of (P3) obtained in Step 4, into crisp nonlinear constraints with the help of Definition 3.2, then (P3) reduces to the following problem: $\begin{matrix} (P 4) & Max C_{d} ({\tilde{Z}}_{1} (\tilde{X})) = [\frac{e_{1} + f_{1}}{2}, \frac{f_{1} + g_{1}}{2}] \\ ⋮ \\ Max C_{d} ({\tilde{Z}}_{k} (\tilde{X})) = [\frac{e_{k} + f_{k}}{2}, \frac{f_{k} + g_{k}}{2}] \\ subject to \\ D_{V_{i}} (U_{i}^{+}) \leq δ_{i}, i = 1, 2, \dots, m_{1}, \\ D_{V_{i}} (U_{i}^{-}) \leq δ_{i}, i = m_{1} + 1, m_{1} + 2, \dots, m, \\ x_{j}, y_{j} - x_{j}, z_{j} - y_{j} \geq 0, j = 1, 2, \dots, n, \end{matrix}$ where δ = (δ₁, δ₂, …, δ_m) and δ_i is the deviation degree of the i-th constraint specified by DM, U_i = [u_i + v_i, v_i + w_i] and V_i = [b_i1 + b_i2, b_i2 + b_i3], i = 1, 2, …, m.

Step 6: To find the maximum value of k interval objective functions of (P4), first find the aspiration level of their centers and widths and then use GP technique as follows:

Step 6(a): Solve 2k CNLP problem, obtained by maximizing and minimizing the centers and widths of the k objective functions of (P4) by solving them individually with the constraints of (P4) respectively, as follows: $\begin{matrix} Max m (C_{d} ({\tilde{Z}}_{q} (\tilde{X}))), q = 1, 2, \dots, k \\ subject to \\ D_{V_{i}} (U_{i}^{+}) \leq δ_{i}, i = 1, 2, \dots, m_{1}, \\ D_{V_{i}} (U_{i}^{-}) \leq δ_{i}, i = m_{1} + 1, m_{1} + 2, \dots, m, \\ x_{j}, y_{j} - x_{j}, z_{j} - y_{j} \geq 0, j = 1, 2, \dots, n . \end{matrix}$ $\begin{matrix} Min w (C_{d} ({\tilde{Z}}_{q} (\tilde{X}))), q = 1, 2, \dots, k \\ subject to \\ D_{V_{i}} (U_{i}^{+}) \leq δ_{i}, i = 1, 2, \dots, m_{1}, \\ D_{V_{i}} (U_{i}^{-}) \leq δ_{i}, i = m_{1} + 1, m_{1} + 2, \dots, m, \\ x_{j}, y_{j} - x_{j}, z_{j} - y_{j} \geq 0, j = 1, 2, \dots, n . \end{matrix}$

Let $Z_{q}^{m}$ and $Z_{q}^{w}$ be the aspiration level of the centers and the widths of the k objective function of (P4), which are the optimal solutions of 2k CNLP problems respectively.

Step 6(b): Using GP technique, (P4) can be written as: $\begin{matrix} (P 5) & Min \sum_{q = 1}^{k} n_{q}^{m} + \sum_{q = 1}^{k} p_{q}^{w} \\ subject to \\ m (C_{d} ({\tilde{Z}}_{q} (\tilde{X}))) + n_{q}^{m} - p_{q}^{m} = Z_{q}^{m}, \\ q = 1, 2, \dots, k, \\ w (C_{d} ({\tilde{Z}}_{q} (\tilde{X}))) + n_{q}^{w} - p_{q}^{w} = Z_{q}^{w}, \\ q = 1, 2, \dots, k, \\ D_{V_{i}} (U_{i}^{+}) \leq δ_{i}, i = 1, 2, \dots, m_{1}, \\ D_{V_{i}} (U_{i}^{-}) \leq δ_{i}, i = m_{1} + 1, m_{1} + 2, \dots, m, \\ x_{j}, y_{j} - x_{j}, z_{j} - y_{j} \geq 0, j = 1, 2, \end{matrix}$ where $n_{q}^{m}$ and $p_{q}^{m}$ are negative and positive deviation degree of $m (C_{d} ({\tilde{Z}}_{q} (\tilde{X})))$ , $n_{q}^{w}$ and $p_{q}^{w}$ are negative and positive deviation degree of $w (C_{d} ({\tilde{Z}}_{q} (\tilde{X})))$ respectively.

Step 7: Solve (P5) by LINGO 14.0 and find the optimal values of (x_j, y_j, z_j), j = 1, 2, …, n.

Remark 4.1. Till now we have not found any software which tell us about the alternate optimal solutions of (P5) and we are still working on that.

Definition 4.2. A vector (x_j, y_j, z_j, $n_{q}^{m}$ , $p_{q}^{m}$ , $n_{q}^{w}$ , $p_{q}^{w}) (δ)$ , j = 1, 2, …, n, q = 1, 2, …, k is δ-feasible solution of (P5) within the deviation degree δ = (δ₁, δ₂, …, δ_m) if it satisfies constraints of problem (P5). Let X (δ) denotes the set of all δ-feasible solutions of problem (P5).

Definition 4.3. A δ-feasible solution $(x_{j}^{*}$ , $y_{j}^{*}$ , $z_{j}^{*}$ , $n_{q}^{m^{*}}$ , $p_{q}^{m^{*}}$ , $n_{q}^{w^{*}}$ , $p_{q}^{w^{*}}) (δ) \in X (δ)$ , j = 1, 2, …, n, q = 1, 2, …, k is the δ-optimal solution of (P5) if for all δ-feasible solution (x_j, y_j, z_j, $n_{q}^{c}$ , $p_{q}^{c}$ , $n_{q}^{w}$ , $p_{q}^{w}) (δ) \in X (δ)$ , j = 1, 2, …, n, q = 1, 2, …, k we have $\begin{matrix} \sum_{q = 1}^{k} n_{q}^{m^{*}} + \sum_{q = 1}^{k} p_{q}^{w^{*}} \leq \sum_{q = 1}^{k} n_{q}^{m} + \sum_{q = 1}^{k} p_{q}^{w} \end{matrix}$ Now, we are going to show that the δ-optimal solution of (P5) is the fuzzy Pareto optimal solution of the FFMOLP problem.

Theorem 4.1. If $(x_{j}^{*}, y_{j}^{*}, z_{j}^{*}, n_{q}^{m^{*}}, p_{q}^{m^{*}}, n_{q}^{w^{*}}, p_{q}^{w^{*}}) (δ)$ , j = 1, 2, …, n, q = 1, 2, …, k, is the δ-optimal solution of (P5), then ${\tilde{X}}^{*} = [{\tilde{x}}_{j}^{*}]_{n \times 1}$ where ${\tilde{x}}_{j}^{*} = (x_{j}^{*}, y_{j}^{*}, z_{j}^{*})$ , j = 1, 2, …, n, is the fuzzy Pareto optimal solution of (P1).

Proof. Let if possible ${\tilde{X}}^{*}$ is not a fuzzy Pareto optimal solution of (P1), then there exist ${\tilde{X}}^{\circ} = [{\tilde{x}}_{j}^{\circ}]_{n \times 1}$ where ${\tilde{x}}_{j}^{\circ} = (x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ})$ , j = 1, 2, …, n, satisfying constraints of (P1) such that

$\Rightarrow \begin{matrix} {\tilde{Z}}_{q} ({\tilde{X}}^{*}) ⪯_{mw} {\tilde{Z}}_{q} ({\tilde{X}}^{\circ}) for all q = 1, 2, \dots, k, \\ and \\ {\tilde{Z}}_{r} ({\tilde{X}}^{*}) ≺_{mw} {\tilde{Z}}_{r} ({\tilde{X}}^{\circ}) \\ for at least one r = 1, 2, \dots, k . \end{matrix}}$ (1)

$\Rightarrow \begin{matrix} m (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{*}))) \leq m (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{\circ}))) \\ for all q = 1, 2, \dots, k, \\ and \\ w (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{*}))) \geq w (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{\circ}))) \\ for all q = 1, 2, \dots, k, \\ and \\ m (C_{d} ({\tilde{Z}}_{r} ({\tilde{X}}^{*}))) > m (C_{d} ({\tilde{Z}}_{r} ({\tilde{X}}^{\circ}))) \\ for at least one r = 1, 2, \dots, k, \\ and \\ w (C_{d} ({\tilde{Z}}_{r} ({\tilde{X}}^{*}))) > w (C_{d} ({\tilde{Z}}_{r} ({\tilde{X}}^{\circ}))) \\ for at least one r = 1, 2, \dots, k . \end{matrix}}$ (2) Since (P2) is obtained from (P1) using Definition 2.7. Hence, ${\tilde{X}}^{\circ} = [{\tilde{x}}_{j}^{\circ}]_{n \times 1}$ is also the feasible solution of (P2).

Let S and S′ be the set of all the feasible solutions of (P2) and (P3) respectively.

We claim ${\tilde{X}}^{\circ} \in S$ if and only if $(x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ}) \in S^{'}$ .

Let ${\tilde{X}}^{\circ} \in S$ $\begin{matrix} \Leftrightarrow & {\tilde{X}}^{\circ} satisfy (u_{i}, v_{i}, w_{i}) {⪯, ⪰} (b_{i 1}, b_{i 2}, b_{i 3}) . \\ \Leftrightarrow & (x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ}) satisfy \\ C_{d} (u_{i}, v_{i}, w_{i}) {\leq_{mw}, \geq_{mw}} C_{d} (b_{i 1}, b_{i 2}, b_{i 3}) . \\ \Leftrightarrow & (x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ}) \in S^{'} . \end{matrix}$ Hence, $(x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ})$ is a feasible solution of (P3).

Using Definition 3.2, we can easily observe that $(x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ}) (δ)$ is a feasible solution of (P4), corresponding to the feasible solution $(x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ})$ of (P3).

Since (P4) and 2k CNLP problems have same constraints, therefore $(x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ}) (δ)$ is also a feasible solution of 2k CNLP problems. Hence,

$\begin{matrix} Z_{q}^{m} \geq m (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{\circ}))) \\ and \\ Z_{q}^{w} \leq w (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{\circ}))) . \end{matrix}}$ (3) Corresponding to the feasible solution $(x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ}) (δ)$ of (P4), there exist $n_{q}^{m^{\circ}}, p_{q}^{m^{\circ}}, n_{q}^{w^{\circ}}$ and $p_{q}^{w^{\circ}}$ such that $(x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ}, n_{q}^{m^{\circ}}, p_{q}^{m^{\circ}}, n_{q}^{w^{\circ}}, p_{q}^{w^{\circ}}) (δ)$ is the feasible solution of (P5).

Since $(x_{j}^{*}, y_{j}^{*}, z_{j}^{*}) (δ)$ also satisfy the constraints of (P4), therefore $(x_{j}^{*}, y_{j}^{*}, z_{j}^{*}) (δ)$ is the feasible solution of 2k CNLP problems. Therefore,

$\begin{matrix} Z_{q}^{m} \geq m (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{*}))) \\ and \\ Z_{q}^{w} \leq w (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{*}))) . \end{matrix}}$ (4) Both $(x_{j}^{\circ}, y_{j}^{\circ}, z_{j}^{\circ}, n_{q}^{m^{\circ}}, p_{q}^{m^{\circ}}, n_{q}^{w^{\circ}}, p_{q}^{w^{\circ}}) (δ)$ and $(x_{j}^{*}, y_{j}^{*}$ , $z_{j}^{*}, n_{q}^{m^{*}}, p_{q}^{m^{*}}, n_{q}^{w^{*}}, p_{q}^{w^{*}}) (δ)$ are feasible solution of (P5). Then from (2), (3) and (4) we have

$\Rightarrow \begin{matrix} Z_{q}^{m} \geq m (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{\circ}))) \geq m (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{*}))) \\ for all q = 1, 2, \dots, k, \\ and \\ Z_{q}^{w} \leq w (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{\circ}))) \leq w (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{*}))) \\ for all q = 1, 2, \dots, k, \\ and \\ Z_{r}^{m} \geq m (C_{d} ({\tilde{Z}}_{r} ({\tilde{X}}^{\circ}))) > m (C_{d} ({\tilde{Z}}_{r} ({\tilde{X}}^{*}))) \\ for at least one r = 1, 2, \dots, k, \\ and \\ Z_{r}^{w} \leq w (C_{d} ({\tilde{Z}}_{r} ({\tilde{X}}^{\circ}))) < w (C_{d} ({\tilde{Z}}_{r} ({\tilde{X}}^{*}))) \\ for at least one r = 1, 2, \dots, k . \end{matrix}}$ (5)

$\Rightarrow \begin{matrix} Z_{q}^{m} - m (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{\circ}))) \leq Z_{q}^{m} - m (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{*}))) \\ for all q = 1, 2, \dots, k, \\ and \\ w (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{\circ}))) - Z_{q}^{w} \leq w (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{*}))) - Z_{q}^{w} \\ for all q = 1, 2, \dots, k, \\ and \\ Z_{r}^{m} - m (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{\circ}))) < Z_{r}^{m} - m (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{*}))) \\ for at least one r = 1, 2, \dots, k \\ and \\ w (C_{d} ({\tilde{Z}}_{q} (X^{\circ}))) - Z_{r}^{w} < w (C_{d} ({\tilde{Z}}_{q} ({\tilde{X}}^{*}))) - Z_{r}^{w} \\ for at least one r = 1, 2, \dots, k . \end{matrix}}$ (6)

Using the constraints of (P5), we get

$\Rightarrow \begin{matrix} n_{q}^{m^{\circ}} - p_{q}^{m^{\circ}} \leq n_{q}^{m^{*}} - p_{q}^{m^{*}} \\ for all q = 1, 2, \dots, k, \\ and \\ - n_{q}^{w^{\circ}} + p_{q}^{w^{\circ}} \leq - n_{q}^{w^{*}} + p_{q}^{w^{*}} \\ for all q = 1, 2, \dots, k, \\ and \\ n_{r}^{m^{\circ}} - p_{r}^{m^{\circ}} < n_{r}^{m^{*}} - p_{r}^{m^{*}} \\ for at least one r = 1, 2, \dots, k, \\ and \\ - n_{r}^{w^{\circ}} + p_{r}^{w^{\circ}} < - n_{r}^{w^{*}} + p_{r}^{w^{*}} \\ for at least one r = 1, 2, \dots, k . \end{matrix}}$ (7) From (3), (4) and $p_{q}^{m^{*}} n_{q}^{m^{*}}$ = $p_{q}^{w^{*}} n_{q}^{w^{*}}$ = $p_{q}^{m^{\circ}} n_{q}^{m^{\circ}}$ = $p_{q}^{w^{\circ}} n_{q}^{w^{\circ}}$ = 0, we get $p_{q}^{m^{*}}$ = $p_{q}^{m^{\circ}} = n_{q}^{w^{\circ}}$ = $n_{q}^{w^{\circ}}$ = 0. Therefore

$\Rightarrow \begin{matrix} n_{q}^{m^{\circ}} \leq n_{q}^{m^{*}} for all q = 1, 2, \dots, k, \\ and \\ p_{q}^{w^{\circ}} \leq p_{q}^{w^{*}} for all q = 1, 2, \dots, k, \\ and \\ n_{r}^{m^{\circ}} < n_{r}^{m^{*}} for at least one r = 1, 2, \dots, k, \\ and \\ p_{r}^{w^{\circ}} < p_{r}^{w^{*}} for at least one r = 1, 2, \dots, k . \end{matrix}}$ (8)

From (8), we get $\begin{matrix} \sum_{q = 1}^{k} n_{q}^{m^{\circ}} + \sum_{q = 1}^{k} p_{q}^{m^{\circ}} < \sum_{q = 1}^{k} n_{q}^{m^{*}} + \sum_{q = 1}^{k} p_{q}^{m^{*}} . \end{matrix}$ This is a contradiction as $(x_{j}^{*}, y_{j}^{*}, z_{j}^{*}, n_{q}^{m^{*}}, p_{q}^{m^{*}}, n_{q}^{w^{*}}$ , $p_{q}^{w^{*}}) (δ)$ is the optimal solution of (P5). Hence, ${\tilde{X}}^{*} = [{\tilde{x}}_{j}^{*}]_{n \times 1}$ where ${\tilde{x}}_{j}^{*} = (x_{j}^{*}, y_{j}^{*}, z_{j}^{*})$ , j = 1, 2, …, n, is the fuzzy Pareto optimal solution of (P1). □

5 Numerical examples

In this section, the proposed method is illustrated with the help of two examples and compares the proposed method with the Total objective-segregation method [12].

Example 5.1. $\begin{matrix} \begin{matrix} Max {\tilde{Z}}_{1} (\tilde{X}) = (1, 2, 3) \otimes {\tilde{x}}_{1} \oplus (3, 4, 5) \otimes {\tilde{x}}_{2} \\ Max {\tilde{Z}}_{2} (\tilde{X}) = (2, 3, 4) \otimes {\tilde{x}}_{1} \oplus (1, 3, 5) \otimes {\tilde{x}}_{2} \\ subject to \\ (- 2, 2, 6) \otimes {\tilde{x}}_{1} \oplus (1, 2, 3) \otimes {\tilde{x}}_{2} ⪯ (- 3, 8, 24), \\ (2, 4, 6) \otimes {\tilde{x}}_{1} \oplus (1, 3, 5) \otimes {\tilde{x}}_{2} ⪰ (1, 13, 32), \\ {\tilde{x}}_{1}, {\tilde{x}}_{2} ⪰ \tilde{0} . \end{matrix}} \end{matrix}$

Using Definitions 2.7 and 2.11, convert aboveproblem into MOILP problem as follows: $\begin{matrix} Max C_{d} ({\tilde{Z}}_{1} (\tilde{X})) = [\frac{x_{1} + 3 x_{2} + 2 y_{1} + 4 y_{2}}{2}, \\ \frac{2 y_{1} + 4 y_{2} + 3 z_{1} + 5 z_{2}}{2}] \\ Max C_{d} ({\tilde{Z}}_{2} (\tilde{X})) = [\frac{2 x_{1} + x_{2} + 3 y_{1} + 3 y_{2}}{2}, \\ \frac{3 y_{1} + 3 y_{2} + 4 z_{1} + 5 z_{2}}{2}] \\ subject to \\ [- 2 z_{1} + x_{2} + 2 y_{1} + 2 y_{2}, 2 y_{1} + 2 y_{2} + 6 z_{1} + 3 z_{2}] \\ \leq [5, 32], \\ [2 x_{1} + x_{2} + 4 y_{1} + 3 y_{2}, 4 y_{1} + 3 y_{2} + 6 z_{1} + 5 z_{2}] \\ \geq [14, 45], \\ x_{j}, y_{j} - x_{j}, z_{j} - y_{j} \geq 0, j = 1, 2 . \end{matrix}}$ (9)

Using Steps 3–6, $max {a, b} = \frac{a + b}{2} + | \frac{a - b}{2} |$ and $min {a, b} = \frac{a + b}{2} - | \frac{a - b}{2} |$ , (10) can be written as

$\begin{matrix} Min n_{1}^{m} + p_{1}^{w} + n_{2}^{m} + p_{2}^{w} \\ subject to \\ \frac{x_{1} + 3 x_{2} + 4 y_{1} + 8 y_{2} + 3 z_{1} + 5 z_{2}}{4} + n_{1}^{m} - p_{1}^{m} = Z_{1 δ}^{m}, \\ \frac{3 z_{1} + 5 z_{2} - x_{1} - 3 x_{2}}{4} + n_{1}^{w} - p_{1}^{w} = Z_{1 δ}^{w}, \\ \frac{2 x_{1} + x_{2} + 6 y_{1} + 6 y_{2} + 4 z_{1} + 5 z_{2}}{4} + n_{2}^{m} - p_{2}^{m} = Z_{2 δ}^{m}, \\ \frac{4 z_{1} + 5 z_{2} - 2 x_{1} - x_{2}}{4} + n_{2}^{w} - p_{2}^{w} = Z_{2 δ}^{w}, \\ P \leq Q, \\ R \leq S, \\ x_{j}, y_{j} - x_{j}, z_{j} - y_{j} \geq 0, j = 1, 2, \end{matrix}}$ (10)

where, $\begin{matrix} P & = 2 (2 y_{1} + 2 y_{2} + 6 z_{1} + 3 z_{2} - 32) \\ + | x_{2} + 4 y_{1} + 4 y_{2} + 4 z_{1} + 3 z_{2} - 37 | \\ + | 8 z_{1} + 3 z_{2} - x_{2} - 27 |, \\ Q & = 2 δ (8 z_{1} + 3 z_{2} - x_{2} + 27 \\ + | 2 y_{1} + 2 y_{2} + 6 z_{1} + 3 z_{2} - 32 | \\ + | - 2 z_{1} + x_{2} + 2 y_{1} + 2 y_{2} - 5 |), \end{matrix}$

$\begin{matrix} R & = 2 (45 - 4 y_{1} - 3 y_{2} - 6 z_{1} - 5 z_{2}) \\ + | 2 x_{1} + x_{2} + 8 y_{1} + 6 y_{2} + 6 z_{1} + 5 z_{2} - 59 | \\ + | 6 z_{1} + 5 z_{2} - 2 x_{1} - x_{2} - 31 |, \end{matrix}$ $\begin{matrix} S & = 2 δ (6 z_{1} + 5 z_{2} - 2 x_{1} - x_{2} + 31 \\ + | 4 y_{1} + 3 y_{2} + 6 z_{1} + 5 z_{2} - 45 | \\ + | 2 x_{1} + x_{2} + 4 y_{1} + 3 y_{2} - 14 |), \end{matrix}$

$Z_{q δ}^{m}$ and $Z_{q δ}^{w}$ , q = 1, 2 are the aspiration level of center and width of the q-th objective function of (10) corresponding to the different values of deviation degree δ and their values are given in Table 1 below.

Solving (11) by LINGO 14.0 and using Theorem 4.1, we obtain the fuzzy Parteo optimal solution and the fuzzy objective functions values of problem (9) for different values of δ which are listed in Table 2.

Now, using Total objective-segregation method [12], the fuzzy optimal solution, fuzzy objective functions values and the real values corresponding to fuzzy objective functions of problem (9) are listed in Table 3.

Using the Definition 2.9, comparison between the fuzzy objective functions value is drawn as follows:

For δ = 0.001 to 0.0003, 0.0005 to 0.0008and 0.001, we can easily see from Table 2 and 3 that $\begin{matrix} R ({\tilde{Z}}_{1} (\tilde{X}))_{M} > R ({\tilde{Z}}_{1} (\tilde{X}))_{N} \\ and \\ R ({\tilde{Z}}_{2} (\tilde{X}))_{M} > R ({\tilde{Z}}_{2} (\tilde{X}))_{N} \end{matrix}$

For δ = 0.0004 and 0.0009, $\begin{matrix} R ({\tilde{Z}}_{1} (\tilde{X}))_{M} > R ({\tilde{Z}}_{1} (\tilde{X}))_{N} \\ and \\ R ({\tilde{Z}}_{2} (\tilde{X}))_{M} < R ({\tilde{Z}}_{2} (\tilde{X}))_{N} \end{matrix}$

where M = Proposed method and N = Total objective-segregation method.

Therefore, from the above observations, we can easily conclude that the proposed method gives better solution than the Total objective-segregation method [12].

The following example problem is taken from Jayalakshmi and Pandian [12].

Example 5.2

$\begin{matrix} Max {\tilde{Z}}_{1} (\tilde{X}) = (1, 2, 3) \otimes {\tilde{x}}_{1} \oplus (2, 4, 5) \otimes {\tilde{x}}_{2} \\ Max {\tilde{Z}}_{2} (\tilde{X}) = (2, 3, 4) \otimes {\tilde{x}}_{1} \oplus (3, 4, 5) \otimes {\tilde{x}}_{2} \\ subject to \\ (0, 1, 2) \otimes {\tilde{x}}_{1} \oplus (1, 2, 3) \otimes {\tilde{x}}_{2} ⪯ (1, 10, 27), \\ (1, 2, 3) \otimes {\tilde{x}}_{1} \oplus (0, 1, 2) \otimes {\tilde{x}}_{2} ⪯ (2, 11, 28), \\ {\tilde{x}}_{1}, {\tilde{x}}_{2} ⪰ \tilde{0} . \end{matrix}}$ (11)

Using Definitions 2.7 and 2.11, convert problem (12) into MOILP problem as follows:

$\begin{matrix} Max C_{d} ({\tilde{Z}}_{1} (\tilde{X})) = [\frac{x_{1} + 2 x_{2} + 2 y_{1} + 4 y_{2}}{2}, \\ \frac{2 y_{1} + 4 y_{2} + 3 z_{1} + 5 z_{2}}{2}] \\ Max C_{d} ({\tilde{Z}}_{2} (\tilde{X})) = [\frac{2 x_{1} + 3 x_{2} + 3 y_{1} + 4 y_{2}}{2}, \\ \frac{3 y_{1} + 4 y_{2} + 4 z_{1} + 5 z_{2}}{2}] \\ subject to \\ [x_{2} + y_{1} + 2 y_{2}, y_{1} + 2 y_{2} + 2 z_{1} + 3 z_{2}] \leq [11, 37], \\ [x_{1} + 2 y_{1} + y_{2}, 2 y_{1} + y_{2} + 3 z_{1} + 2 z_{2}] \leq [13, 39], \\ x_{j}, y_{j} - x_{j}, z_{j} - y_{j} \geq 0, j = 1, 2 . \end{matrix}}$ (12) Using Steps 3–7, $max {a, b} = \frac{a + b}{2} + | \frac{a - b}{2} |$ and $min {a, b} = \frac{a + b}{2} - | \frac{a - b}{2} |$ , (13) can be written as

$\begin{matrix} Min n_{1}^{m} + p_{1}^{w} + n_{2}^{m} + p_{2}^{w} \\ subject to \\ \frac{x_{1} + 2 x_{2} + 2 y_{1} + 4 y_{2} + 3 z_{1} + 5 z_{2}}{4} + n_{1}^{m} - p_{1}^{m} = Z_{1 δ}^{m}, \\ \frac{3 z_{1} + 5 z_{2} - x_{1} - 2 x_{2}}{4} + n_{1}^{w} - p_{1}^{w} = Z_{1 δ}^{w}, \\ \frac{2 x_{1} + 3 x_{2} + 6 y_{1} + 8 y_{2} + 4 z_{1} + 5 z_{2}}{4} + n_{2}^{m} - p_{2}^{m} = Z_{2 δ}^{m}, \\ \frac{4 z_{1} + 5 z_{2} - 2 x_{1} - 3 x_{2}}{4} + n_{2}^{w} - p_{2}^{w} = Z_{2 δ}^{w}, \\ A \leq B, \\ C \leq D, \\ x_{j}, y_{j} - x_{j}, z_{j} - y_{j} \geq 0, j = 1, 2, \end{matrix}}$ (13)

where, $\begin{matrix} A & = 2 (y_{1} + 2 y_{2} + 2 z_{1} + 3 z_{2} - 37) \\ + | x_{2} + 2 y_{1} + 4 y_{2} + 2 z_{1} + 3 z_{2} - 48 | \\ + | 2 z_{1} + 3 z_{2} - x_{2} - 26 |, \\ B & = 2 δ (2 z_{1} + 3 z_{2} - x_{2} + 26 \\ + | y_{1} + 2 y_{2} + 2 z_{1} + 3 z_{2} - 37 | \\ + | x_{2} + y_{1} + 2 y_{2} - 11 |), \\ C & = 2 (2 y_{1} + y_{2} + 3 z_{1} + 2 z_{2} - 39) \\ + | x_{1} + 4 y_{1} + 2 y_{2} + 3 z_{1} + 2 z_{2} - 52 | \\ + | 3 z_{1} + 2 z_{2} - x_{1} - 26 |, \\ D & = 2 δ (3 z_{1} + 2 z_{2} - x_{1} + 26 \\ + | 2 y_{1} + y_{2} + 3 z_{1} + 2 z_{2} - 39 | \\ + | x_{1} + 2 y_{1} + y_{2} - 13 |), \end{matrix}$ $Z_{q δ}^{m}$ and $Z_{q δ}^{w}$ , q = 1, 2, are the aspiration level of center and width of the q-th objective function of (13) corresponding to the different value of the deviation degree δ and their values are given below inTable 4.

Solving (14) by LINGO 14.0 and using Theorem 4.1, we obtain the fuzzy Parteo optimal solution and the fuzzy objective functions value of problem (12) and and the real values corresponding to the fuzzy objective functions of problem (12) (Definition 2.8) for different values of δ are listed in Table 5.

Next, we have compared the fuzzy objective functions value obtain by proposed method and Total objective-segregation method [12] with the help of Definition 2.9 as follows:

for all values of δ = 0.001 to 0.0001 in Tables 5 and 6, we can easily observe that $\begin{matrix} R ({\tilde{Z}}_{1} (\tilde{X}))_{M} > R ({\tilde{Z}}_{1} (\tilde{X}))_{N} \\ and \\ R ({\tilde{Z}}_{2} (\tilde{X}))_{M} > R ({\tilde{Z}}_{2} (\tilde{X}))_{N} \end{matrix}$ Hence, from the above observations, we can easily conclude that our result is more reliable.

Next, we will solve the following linear programming with fuzzy parameters which is a particular case of our problem, taken from Jimenez et al. [14]

Example 5.3.

$\begin{matrix} Min \tilde{Z} (X) = (19, 20, 21) x_{1} + (29, 30, 31) x_{2} \\ subject to \\ (4.5, 5, 5.5) x_{1} + (2.5, 3, 4) x_{2} \geq (194, 200, 206), \\ (3, 4, 5) x_{1} + (6.5, 7, 7.5) x_{2} \geq (230, 240, 250), \\ x_{1}, x_{2} \geq 0 . \end{matrix}}$ (14) Using Definitions 2.7 and 2.11, convert problem (15) into interval linear programming problem asfollows:

$\begin{matrix} Min C_{d} (\tilde{Z} (X)) = [\frac{39 x_{1} + 59 x_{2}}{2}, \frac{41 x_{1} + 61 x_{2}}{2}] \\ subject to \\ \geq [394, 406], \\ [7 x_{1} + 13.5 x_{2}, 9 x_{1} + 14.5 x_{2}] \geq [470, 490], \\ x_{1}, x_{2} \geq 0 . \end{matrix}}$ (15) Using Steps 3–7, $max {a, b} = \frac{a + b}{2} + | \frac{a - b}{2} |$ and $min {a, b} = \frac{a + b}{2} - | \frac{a - b}{2} |$ , (16) can be written as

$\begin{matrix} Min n_{1}^{m} + p_{1}^{w} \\ subject to \\ \frac{80 x_{1} + 120 x_{2}}{4} + n_{1}^{m} - p_{1}^{m} = Z_{δ}^{m}, \\ \frac{3 x_{1} + 3 x_{2}}{4} + n_{1}^{w} - p_{1}^{w} = Z_{δ}^{w}, \\ E \leq F, \\ G \leq H, \\ x_{1}, x_{2} \geq 0 . \end{matrix}}$ (16) where, $\begin{matrix} E & = 2 (406 - 10.5 x_{1} - 7 x_{2}) + | 20 x_{1} + 12.5 x_{2} - 800 | \\ + | x_{1} + 1.5 x_{2} - 12 |, \end{matrix}$ $\begin{matrix} F & = 2 δ (X_{1} + 1.5 X_{2} + 12 + | 10.5 X_{1} + 7 x_{2} - 406 | \\ + | 9.5 x_{1} + 5.5 x_{2} - 960 |), \\ G & = 2 (490 - 9 x_{1} - 14.5 x_{2}) + | 16 x_{1} + 28 x_{2} - 960 | \\ + | 2 x_{1} + x_{2} - 20 |, \\ H & = 2 δ (2 x_{1} + x_{2} + 20 + | 9 x_{1} + 14.5 x_{2} - 490 | \\ + | 7 x_{1} + 13.5 x_{2} - 470 |), \end{matrix}$ $Z_{δ}^{m}$ and $Z_{δ}^{w}$ , are the aspiration level of center and width of the objective function of (17) corresponding to the different value of the deviation degree δ and their values are given below in Table 7.

Solving (17) by LINGO 14.0. we obtain the fuzzy optimal solution and the fuzzy objective function value of problem (15) for different values of δ, which are listed in Table 8.

The optimal value and the fuzzy objective function value for different value of α obtain by Jimenez et al. method [14] and the real value corresponding to the fuzzy objective function are listed in Table 9.

With the help of Definition 2.9, we can easily observe from Table 8 and 9 that the proposed method give better result than Jimenez et al. method [14].

6 Conclusion

In this paper, a new method has been proposed in which FFMOLP problem is first converted into MOILP problem by nearest interval approximations in order to avoid pitfalls of the essential information. Then with the help of deviation degree and GP approach, MOILP can be transformed into CNLP problem and fuzzy Pareto optimal solution for FFMOLP problem can be obtained by solving CNLP problem using LINGO 14.0 software. Our method can also solve the FFLP problem. This method gives the flexibility to the DM to choose the value of deviation degree δ between the two closed intervals in each constraint. In the end, a comparison is made with the existing method. The proposed method has the following advantages:

captures basic features of the original fuzzy quantities and avoid pitfalls of fuzziness.

the optimal solution is a fuzzy pareto optimal solution.

DM need to know only about arithmetic operations of TrFN and near interval approximation of TrFN. Hence, it is very easy to apply and understand.

Without any difficulty it can be easily implemented into a programming language.

Footnotes

Acknowledgments

The second author is thankful to the Council of Scientific and Industrial Research (CSIR), New Delhi, India, for the financial support (09/045(1153)/2012-EMR-I).

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