An approach for finding fuzzy optimal and approximate fuzzy optimal solution of fully fuzzy linear programming problems with mixed constraints

Abstract

Kumar and Kaur [A. Kumar and J. Kaur Fuzzy optimal solution of fully fuzzy linear programming problems using ranking function. Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology,26(1)(2014), 337–344.] proposed a method to find the fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems with mixed constraints. They claimed that the FFLP problem with mixed constraints in which decision variables are represented by nonnegative triangular fuzzy numbers and the remaining parameters are represented by any type of triangular fuzzy numbers cannot be solved by any of the existing methods while all such FFLP problems can be solved by using their proposed method. In this paper, it is showed that the infeasibility case of the FFLP cannot be handled by the Kumar and Kaur’s method. We have provided an extension to the method of Kumar and Kaur [A. Kumar and J. Kaur Fuzzy optimal solution of fully fuzzy linear programming problems using ranking function. Journal of Intelligent & Fuzzy Systems: Applications in Engineering and Technology,26(1)(2014), 337–344.] to find fuzzy optimal solution to the feasible case and approximate fuzzy optimal solution to the infeasible case of FFLP problem with mixed constraints. To the best of our knowledge, there does not exist any study which deals with the infeasibility case of FFLP problem.

Keywords

Fully fuzzy linear programming problem (FFLP)triangular fuzzy numbers ranking function infeasibility

1 Introduction

Linear Programming (LP) is an optimization problem of very important area of mathematics called “optimization theory” which deals with finding optimal solutions under set of linear inequality or linear equality constraints.

LP is called as fuzzy linear programming (FLP) which has at least one fuzzy parameter. FFLP is defined in which all parameters i.e. coefficients as well as variables in the objective function and constraints are taken as fuzzy numbers.

Kumar and Kaur [22] declared that the FFLP problem with mixed constraints in which decision variables are represented by nonnegative triangular fuzzy numbers and the remaining parameters are represented by any type of triangular fuzzy numbers cannot be solved by any of the existing methods while their proposed method can be found the fuzzy optimal solution of this type of FFLP. In this paper, it is showed that the infeasibility case of the FFLP cannot be handled by the Kumar and Kaur’s method. We have provided an extension to the method of Kumar and Kaur [22] to find fuzzy optimal solution to the feasible case and approximate fuzzy optimal solution to the infeasible case of FFLP problem with mixed constraints.

This paper is organized as follows. In Section 3, brief information about the triangular fuzzy numbers and some basic definitions are presented. FFLP problem and related definitions are given in Section 4. Section 5 presents an extension to the method of Kumar and Kaur [22] to find fuzzy optimal solution to the feasible case and approximate fuzzy optimal solution to the infeasible case of FFLP problem with mixed constraints. Numerical Experiments are given in section 6 to illustrate the extension of Kumar and Kaur [22]’s method and the conclusion is given in Section 7.

2 Literature review

In the last several years, FLP and FFLP problems has gained great importance and there is a vast literature on the investigation of optimal solutions of these problems. Zimmermann [35] presented the application of FLP approaches to the linear vector maximum problem. Werners [33] introduced an interactive system which supports a decision maker in solving programming models with crisp or fuzzy constraints and crisp or fuzzy goals. Campos and Verdegay [6] considered LP problems with fuzzy constraints and fuzzy coefficients in both matrix and right hand side of the constraint set. Inuiguchi et al. [14] proposed a technique to solve the FLP using a standard LP when membership functions are strictly quasiconcave and the minimum operator is adopted for aggregating fuzzy goals. Rommelfanger [31] presented a survey on methods for solving FLPs. Cadenas and Verdegay [5] considered a LP problem in which all its elements are defined as fuzzy sets. Fang et al. [8] presented a method for solving LP problems with fuzzy coefficients in constraints and showed that such problems can be reduced to a linear semi-infinite programming problem. Buckley and Feuring [4] presented a method to find solutions to the fully fuzzified LP where all the parameters and variables are fuzzy numbers and an evolutionary algorithm is designed to solve the fuzzy flexible program. Maleki et al. [25] proposed a new method for solving LP problems with fuzzy variables using the concept of comparison of fuzzy numbers. Zhang et al. [34] concerned the solution of FLP problems which involve fuzzy numbers in coefficients of objective functions and they converted the FLP to a multi-objective optimization problem with four-objective functions. Nehi et al. [28] introduced the lexicographic ranking function to order fuzzy numbers using the concepts of value, ambiguity and fuzziness for a fuzzy number and they applied the lexicographic ranking function method to solve fuzzy number LP problems. Ramik [30] introduced a class of FLP problems based on fuzzy relations and a concept of duality for FLP. Ganesan and Veeramani [9] dealed with a kind of FLP problem involving symmetric trapezoidal fuzzy numbers and they derived solution of FLP problems without converting them to crisp LP problems. Hashemi et al. [12] proposed a two-phase approach to find the optimal solutions of a FFLP. Mahdavi-Amiri and Nasseri [24] developed a dual algorithm for solving the LP problems with trapezoidal fuzzy variables by using linear ranking function to order trapezoidal fuzzy numbers. Jimenez et al. [15] proposed a method for solving LP problems where all the coefficients are, in general, fuzzy numbers. Allahviranloo et al. [1] proposed a method to solve the FFLP problem by using linear ranking function for defuzzifying it. Nasseri [27] proposed a new method for solving the FLP problems by solving the classical LP problems without using any ranking function. Lotfi et al. [23] discussed FFLP problems of which all parameters and variable are symmetric triangular fuzzy numbers. Kumar et al. [17] proposed a method to find the fuzzy optimal solution of FFLP problems with inequality constraints by transforming crisp LP form. Kumar et al. [18] proposed to find the fuzzy optimal solution of FFLP problems with inequality constraints by using trapezoidal fuzzy numbers. Kumar et al. [19] proposed to find the fuzzy optimal solution of FFLP problems with equality constraints. Kumar and Kaur [20] proposed Mehar’s method to find the fuzzy optimal solution of FLP problems. Kumar and Kaur [21] pointed out the shortcomings of existing general form of FFLP problems and proposed a new general form of FFLP problems. Kaur and Kumar [16] proposed the product of unrestricted L-R flat fuzzy numbers and Mehar’s method for solving FFLP problems by using their proposed product. Kumar and Kaur [22] proposed a method to find the fuzzy optimal solution of FFLP problems with mixed constraints where parameters are triangular fuzzy numbers. Otadi [29] presented a method to find the fuzzy optimal solution of FFLP problems. Hatami and Kazemipoor [13] adopted Mehar’s Method for solving FFLP problem involving symmetric trapezoidal fuzzy numbers including both equality and inequality constraints. Ezzati et al. [7] proposed a model based on a new lexicographic ordering on triangular fuzzy numbers to solve the FFLP problem by converting it to its equivalent multi-objective LP problem and solved by the lexicographic method. Bhardwaj and Kumar [3] proved that the FFLP problems with inequality constraints cannot be transformed into FFLP problems with equality constraints and hence, the algorithm, proposed by Ezzati et al. [7] cannot be used for finding the fuzzy optimal solution of FFLP problems with inequality constraints. Mottaghi et al. [26] presented a method for solving the FLP problems in which the coefficients of the objective function and the values of the right-hand side are represented by fuzzy numbers, while the elements of the coefficient matrix are represented by real numbers and developed Karush–Kuhn–Tucker (KKT) optimality conditions for FLP problems. Shamooshaki et al. [32] proposed a model by using the lexicography method to solve FFLP problem with L-R fuzzy number and find the fuzzy optimal solution of it. Baykasoglu and Subulan [2] claimed that their study is the first study in the literature which presents fuzzy efficient solutions and analysis for a fully fuzzy reverse logistics network design problem with fuzzy decision variables which is a type of FFLP. Gri et al. [10] proposed some approaches to solve the fully fuzzy fixed charge multi-item solid transportation problems (FFFCMISTPs), in which direct costs, fixed charges, supplies, demands, conveyance capacities and transported quantities(decision variables) are fuzzy in nature. Gupta et al. [11] pointed out that Giri et al. [10] have used some mathematical incorrect assumptions in their proposed approach and hence the claim of Giri et al. is not valid.

Our study differs from the FFLP problem literature in that it deals with infeasibility case of FFLP problem.

3 Preliminaries

In this section, brief information about the triangular fuzzy numbers and some basic definitions are presented.

3.1 Basic definitions

Definition 1. [36] If X is a collection of objects denoted 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 the membership function or grade of membership of x in $\tilde{A}$ which maps X to the membership space M(M contains [0, 1]).

Definition 2. A fuzzy number $\tilde{A} = (a, b, c)$ is said to be an arbitrary triangular fuzzy number if its membership function is given by $μ_{\tilde{A}} (x) = {\begin{matrix} \frac{x - a}{b - a} & , a \leq x < b \\ \frac{x - c}{b - c} & , b \leq x \leq c \\ 0 & , Otherwise \end{matrix}$

where a, b, c ∈ R and a ≤ b ≤ c.

Definition 3. An arbitrary triangular fuzzy number $\tilde{A} = (a, b, c)$ is called positive (negative), denoted by $\tilde{A} > 0 (\tilde{A} < 0)$ if its membership function $μ_{\tilde{A}} (x) = 0, \forall x < 0 (\forall x > 0)$ .

Definition 4. Two arbitrary triangular fuzzy numbers $\tilde{A} = (a, b, c)$ and $\tilde{B} = (d, e, f)$ are said to be equal if and only if a = d, b = e and c = f.

Definition 5. [22] A ranking function R is a mapping from a fuzzy set F (R) to real numbers where a natural order exists. Let $\tilde{A} = (a, b, c)$ be a triangular fuzzy number then $R (\tilde{A}) = \frac{a + 2 b + c}{4}$ .

Definition 6. Let $\tilde{A} = (a, b, c)$ and $\tilde{B} = (d, e, f)$ be arbitrary triangular fuzzy numbers then $\begin{matrix} (i) \tilde{A} ≺ \tilde{B} iff R (\tilde{A}) < R (\tilde{B}) \\ (ii) \tilde{A} ≻ \tilde{B} iff R (\tilde{A}) > R (\tilde{B}) \\ (iii) \tilde{A} ≅ \tilde{B} \Rightarrow R (\tilde{A}) = R (\tilde{B}) \end{matrix}$

3.2 Arithmetic operations

Some algebraic operations on triangular fuzzy numbers are defined as follows:

Let $\tilde{A} = (a, b, c)$ and $\tilde{B} = (d, e, f)$ be arbitrary two triangular fuzzy numbers, $\begin{matrix} (i) \tilde{A} + \tilde{B} = (a + d, b + e, c + f) \\ (ii) - \tilde{A} = (- c, - b, - a) \\ (iii) \tilde{A} - \tilde{B} = (a - f, b - e, c - d) . \end{matrix}$

(iv) The fuzzy multiplication based on the extension principle is performed via the following equation: $\tilde{A} * \tilde{B} = (l, m, u)$ with $\begin{matrix} m = be \\ l = min {ad, af, cd, cf} \\ u = max {ad, af, cd, cf} . \end{matrix}$

Definition 7. Suppose $\tilde{A}, \tilde{B} \in F (R)$ then $R (\tilde{A} + \tilde{B}) = R (\tilde{A}) + R (\tilde{B})$ .

This property can be generalized for n fuzzy numbers ${\tilde{A}}_{1}, {\tilde{A}}_{2}, \dots, {\tilde{A}}_{n}$ i.e. $R (\sum_{j = 1}^{n} {\tilde{A}}_{j}) = \sum_{j = 1}^{n} R ({\tilde{A}}_{j})$

4 Fully fuzzy linear programming problems

In the real life problems, it is hard to estimate parameters of problems imprecisely. Fuzzy set theory is a powerful tool to estimate and describe the vagueness and ambiguity in these parameters.

Here, we handle FFLP problem with mixed constraints in which decision variables are represented by nonnegative triangular fuzzy numbers and the remaining parameters are represented by any type of triangular fuzzy numbers.

FFLP problems with m mixed constraints and n fuzzy variables can be formulated as follows:

$\begin{matrix} Max (or Min) ({\tilde{c}}_{1} \otimes {\tilde{x}}_{1}) \oplus \dots \oplus ({\tilde{c}}_{n} \otimes {\tilde{x}}_{n}) \\ subject to \\ ({\tilde{a}}_{11} \otimes {\tilde{x}}_{1}) \oplus \dots \oplus ({\tilde{a}}_{1 n} \otimes {\tilde{x}}_{n}) \underline{≺}, =, \underline{≻} {\tilde{b}}_{1} \\ ({\tilde{a}}_{21} \otimes {\tilde{x}}_{1}) \oplus \dots \oplus ({\tilde{a}}_{2 n} \otimes {\tilde{x}}_{n}) \underline{≺}, =, \underline{≻} {\tilde{b}}_{2} \\ ⋮ ⋱ ⋮ \\ ({\tilde{a}}_{m 1} \otimes {\tilde{x}}_{1}) \oplus \dots \oplus ({\tilde{a}}_{mn} \otimes {\tilde{x}}_{n}) \underline{≺}, =, \underline{≻} {\tilde{b}}_{m} \\ x_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n} \geq 0 . \end{matrix}$ (1) Using matrix notation we get $\begin{matrix} Max (or Min) {\tilde{C}}^{T} \otimes \tilde{X} \\ \tilde{A} \otimes \tilde{X} \underline{≺}, =, \underline{≻} \tilde{b} \end{matrix}$ where ${\tilde{C}}^{T} = {[{\tilde{c}}_{j}]}_{1 xn}, \tilde{X} = {[{\tilde{x}}_{j}]}_{nx 1}, \tilde{A} = {[{\tilde{a}}_{ij}]}_{mxn}, \tilde{b} = {[{\tilde{b}}_{i}]}_{mx 1}$ and ${\tilde{a}}_{ij}, {\tilde{c}}_{j}, {\tilde{b}}_{i} \in F (R)$ and ${\tilde{x}}_{j}$ is non-negative triangular fuzzy number.

Definition 8. The fuzzy feasible solution of FFLP problem (P1) will be ${\tilde{X}}^{*} = {[{\tilde{x}}_{j}^{*}]}_{nx 1}$ if it satisfies the following characteristics:

${\tilde{x}}_{j}^{*}$ is a non-negative fuzzy number

$\tilde{A} \otimes {\tilde{X}}^{*} \underline{≺}, =, \underline{≻} \tilde{b}$

A FFLP is infeasible if it has no feasible solutions, i.e. the feasible region is empty.

Definition 9. [22] The fuzzy optimal solution of FFLP problem (P1) will be ${\tilde{X}}^{*} = {[{\tilde{x}}_{j}^{*}]}_{nx 1}$ if it satisfies the following characteristics:

${\tilde{x}}_{j}^{*}$ is a non-negative fuzzy number

$\tilde{A} \otimes {\tilde{X}}^{*} \underline{≺}, =, \underline{≻} \tilde{b}$

If there exist any non-negative $\tilde{X} = {[{\tilde{x}}_{j}]}_{nx 1}$ such that $\tilde{A} \otimes {\tilde{X}}^{*} \underline{≺}, =, \underline{≻} \tilde{b}$ then $R ({\tilde{C}}^{T} \otimes$ ${\tilde{X}}^{*}) \geq R ({\tilde{C}}^{T} \otimes \tilde{X})$ (in case of maximization problem) and $R ({\tilde{C}}^{T} \otimes {\tilde{X}}^{*}) \leq R ({\tilde{C}}^{T} \otimes \tilde{X})$ (in case of minimization problem).

Definition 10. [22] Let ${\tilde{X}}^{*} = {[{\tilde{x}}_{j}^{*}]}_{nx 1}$ be a fuzzy optimal solution of FFLP problem (P1) if there exist ${\tilde{Y}}^{*} = {[{\tilde{y}}_{j}^{*}]}_{nx 1}$ such that

${\tilde{y}}_{j}^{*}$ is a non-negative fuzzy number

$\tilde{A} \otimes {\tilde{Y}}^{*} \underline{≺}, =, \underline{≻} \tilde{b}$

$R ({\tilde{C}}^{T} \otimes {\tilde{X}}^{*}) = R ({\tilde{C}}^{T} \otimes {\tilde{Y}}^{*})$

then ${\tilde{Y}}^{*} = {[{\tilde{y}}_{j}^{*}]}_{nx 1}$ is said to be an alternative fuzzy optimal solution of (P1).

Definition 11. For equality constraints in (P1), let the fuzzy number $\sum_{j = 1}^{n} {\tilde{a}}_{ij} {\tilde{x}}_{j}$ is denoted by $(t_{i}^{1}, t_{i}^{2}, t_{i}^{3})$ and ${\tilde{b}}_{i} = (l_{i}, m_{i}, u_{i})$ . $\tilde{x} = ({\tilde{x}}_{1}, {\tilde{x}}_{2}, \dots, {\tilde{x}}_{n})$ is said to be an approximate fuzzy optimal solution of (P1) if it satisfies the following conditions:

$\tilde{x}$ is not a feasible fuzzy solution,

$l_{i} \leq t_{i}^{3}$ and $u_{i} \geq t_{i}^{1}$ for ∀i∈ { 1, 2, …, m }.

Here, the condition (i) implies that the fuzzy number obtained in the left hand side of at least one equality constraint is not equal to its right hand side fuzzy number.

5 Extension of the method of Kumar and Kaur [22] to find the fuzzy optimal solution of FFLP problems

Kumar and Kaur [22] proposed a method to solve FFLP problems with mixed constraints in which decision variables are represented by non-negative triangular fuzzy numbers and other parameters are represented by any type of triangular fuzzy numbers. Their method includes Step 1 → Step 2 (Case i–iii) → Step 3 → Step 5 → Step 7 → Step 9.

However, the infeasibility case of the FFLP cannot be handled by the proposed method. Therefore, we have provided an extension of this method to find fuzzy optimal solution to the feasible case and approximate fuzzy optimal solution to the infeasible case of FFLP problem with mixed constraints.

The steps of the our extended method are as follows (See Fig. 1):

Step 1. The FFLP problem (P1) can be written as: $\begin{matrix} Max (or Min) {\tilde{C}}^{T} \otimes \tilde{X} \\ \tilde{A} \otimes \tilde{X} \underline{≺}, =, \underline{≻} \tilde{b} \end{matrix}$ where ${\tilde{C}}^{T} = {[{\tilde{c}}_{j}]}_{1 xn}, \tilde{X} = {[{\tilde{x}}_{j}]}_{nx 1}, \tilde{A} = {[{\tilde{a}}_{ij}]}_{mxn},$ $\tilde{b} = {[{\tilde{b}}_{i}]}_{mx 1}$ and ${\tilde{a}}_{ij}, {\tilde{c}}_{j}, {\tilde{b}}_{i} \in F (R)$ and ${\tilde{x}}_{j}$ is non-negative triangular fuzzy number and ${\tilde{a}}_{ij}, {\tilde{c}}_{j}$ and ${\tilde{b}}_{i}$ are arbitrary fuzzy numbers.

Step 2. Check the type of all the constraints, i.e.

Case (i)

If $\sum_{j = 1}^{n} {\tilde{a}}_{pj} \otimes {\tilde{x}}_{j} \underline{≺} {\tilde{b}}_{p}$ for some p ∈ { 1, 2, …, m } then convert such type of inequality constraints into equality constraints by introducing fuzzy variables ${\tilde{S}}_{p}$ to the left side ${\tilde{S}}_{p}^{'}$ to the right side of the constraints, i.e.

$\sum_{j = 1}^{n} {\tilde{a}}_{pj} \otimes {\tilde{x}}_{j} \oplus {\tilde{S}}_{p} = {\tilde{b}}_{p} \oplus {\tilde{S}}_{p}^{'}$ where

$R ({\tilde{S}}_{p}) - R ({\tilde{S}}_{p}^{'}) \geq 0 .$ Go to Step 3.

Case (ii)

If $\sum_{j = 1}^{n} {\tilde{a}}_{qj} \otimes {\tilde{x}}_{j} \underline{≻} {\tilde{b}}_{q}$ for some q∈ { 1, 2, …, m } then convert such type of inequality constraints into equality constraints by introducing fuzzy variables ${\tilde{S}}_{q}$ to the left side ${\tilde{S}}_{q}^{'}$ to the right side of the constraints, i.e.

$\sum_{j = 1}^{n} {\tilde{a}}_{qj} \otimes {\tilde{x}}_{j} \oplus {\tilde{S}}_{q} = {\tilde{b}}_{q} \oplus {\tilde{S}}_{q}^{'}$ where

$R ({\tilde{S}}_{q}) - R ({\tilde{S}}_{q}^{'}) \leq 0 .$ Go to Step 3.

Case (iii)

If $\sum_{j = 1}^{n} {\tilde{a}}_{rj} \otimes {\tilde{x}}_{j} = {\tilde{b}}_{r}$ for some r∈ { 1, 2, …, m } then there will be no change in that constraints, i.e. $\sum_{j = 1}^{n} {\tilde{a}}_{rj} \otimes {\tilde{x}}_{j} = {\tilde{b}}_{r}$ . Go to Step 3.

Case (iv)

If $\sum_{j = 1}^{n} {\tilde{a}}_{rj} \otimes {\tilde{x}}_{j} = {\tilde{b}}_{r}$ for some r∈ { 1, 2, …, m } then rearrange equality constraints by introducing fuzzy variables ${\tilde{S}}_{r}$ to the left side ${\tilde{S}}_{r}^{'}$ to the right side of the constraints i.e.

$\sum_{j = 1}^{n} {\tilde{a}}_{rj} \otimes {\tilde{x}}_{j} \oplus {\tilde{S}}_{r} = {\tilde{b}}_{r} \oplus {\tilde{S}}_{r}^{'}$ . Go to Step 4.

Step 3. The FFLP problem obtained from Step 2, can be written as follows: $\begin{matrix} Max (or Min) (\sum_{j = 1}^{n} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}) \\ subject to \\ \sum_{j = 1}^{n} {\tilde{a}}_{pj} \otimes {\tilde{x}}_{j} \oplus {\tilde{S}}_{p} = {\tilde{b}}_{p} \oplus {\tilde{S}}_{p}^{'} \\ \sum_{j = 1}^{n} {\tilde{a}}_{qj} \otimes {\tilde{x}}_{j} \oplus {\tilde{S}}_{q} = {\tilde{b}}_{p} \oplus {\tilde{S}}_{q}^{'} \\ \sum_{j = 1}^{n} {\tilde{a}}_{rj} \otimes {\tilde{x}}_{j} = {\tilde{b}}_{r} \\ R ({\tilde{S}}_{p}) - R ({\tilde{S}}_{p}^{'}) \geq 0 \\ R ({\tilde{S}}_{q}) - R ({\tilde{S}}_{q}^{'}) \leq 0 \end{matrix}$ ${\tilde{x}}_{j}$ is non-negative triangular and ${\tilde{a}}_{ij}, {\tilde{c}}_{j}, {\tilde{b}}_{i},$ ${\tilde{S}}_{p}, {\tilde{S}}_{p}^{'}, {\tilde{S}}_{q}$ and ${\tilde{S}}_{q}^{'}$ are arbitrary fuzzy numbers. Go to Step 5.

Step 4. The FFLP problem obtained from Step 2, can be written as follows: $\begin{matrix} Max (or Min) (\sum_{j = 1}^{n} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}) \\ subject to \\ \sum_{j = 1}^{n} {\tilde{a}}_{pj} \otimes {\tilde{x}}_{j} \oplus {\tilde{S}}_{p} = {\tilde{b}}_{p} \oplus {\tilde{S}}_{p}^{'} \\ \sum_{j = 1}^{n} {\tilde{a}}_{qj} \otimes {\tilde{x}}_{j} \oplus {\tilde{S}}_{q} = {\tilde{b}}_{p} \oplus {\tilde{S}}_{q}^{'} \\ \sum_{j = 1}^{n} {\tilde{a}}_{rj} \otimes {\tilde{x}}_{j} \oplus {\tilde{S}}_{r} = {\tilde{b}}_{r} \oplus {\tilde{S}}_{r}^{'} \\ R ({\tilde{S}}_{p}) - R ({\tilde{S}}_{p}^{'}) \geq 0 \\ R ({\tilde{S}}_{q}) - R ({\tilde{S}}_{q}^{'}) \leq 0 \end{matrix}$ ${\tilde{x}}_{j}$ is non-negative triangular and ${\tilde{a}}_{ij}, {\tilde{c}}_{j},$ ${\tilde{b}}_{i}, {\tilde{S}}_{p}, {\tilde{S}}_{p}^{'}, {\tilde{S}}_{q}, {\tilde{S}}_{q}^{'}, {\tilde{S}}_{r}$ and ${\tilde{S}}_{r}^{'}$ are arbitrary triangular fuzzy numbers. Go to Step 6.

Step 5. If all parameters are arbitrary triangular fuzzy numbers ${\tilde{c}}_{j} = (p_{j}, q_{j}, r_{j})$ , ${\tilde{a}}_{ij} = (a_{ij}, b_{ij}, c_{ij})$ , ${\tilde{b}}_{i} = (b_{i}, g_{i}, h_{i})$ , ${\tilde{S}}_{i} = (s_{i}, t_{i}, u_{i})$ , ${\tilde{S}}_{i}^{'} = (s_{i}^{'}, t_{i}^{'}, u_{i}^{'})$ and decision variables ${\tilde{x}}_{j} = (x_{j}, y_{j}, z_{j})$ are non-negative triangular fuzzy numbers, then FFLP problem obtained in Step 3 can be written as: $\begin{matrix} Max (or Min) \sum_{j = 1}^{n} (p_{j}, q_{j}, r_{j}) \otimes (x_{j}, y_{j}, z_{j}) \\ subject to \\ \sum_{j = 1}^{n} (a_{pj}, b_{pj}, c_{pj}) \otimes (x_{j}, y_{j}, z_{j}) \oplus (s_{p}, t_{p}, u_{p}) \\ = (b_{p}, g_{p}, h_{p}) \oplus (s_{p}^{'}, t_{p}^{'}, u_{p}^{'}) \\ \sum_{j = 1}^{n} (a_{qj}, b_{qj}, c_{qj}) \otimes (x_{j}, y_{j}, z_{j}) \oplus (s_{q}, t_{q}, u_{q}) \\ = (b_{q}, g_{q}, h_{q}) \oplus (s_{q}^{'}, t_{q}^{'}, u_{q}^{'}) \\ \sum_{j = 1}^{n} (a_{rj}, b_{rj}, c_{rj}) \otimes (x_{j}, y_{j}, z_{j}) = (b_{r}, g_{r}, h_{r}) \\ R (s_{q}, t_{q}, u_{q}) - R (s_{q}^{'}, t_{q}^{'}, u_{q}^{'}) \geq 0 \\ R (s_{q}, t_{q}, u_{q}) - R (s_{q}^{'}, t_{q}^{'}, u_{q}^{'}) \leq 0 . Go to Step 7 . \end{matrix}$ Step 6. If all parameters are arbitrary triangular fuzzy numbers ${\tilde{c}}_{j} = (p_{j}, q_{j}, r_{j})$ , ${\tilde{a}}_{ij} = (a_{ij}, b_{ij}, c_{ij})$ , ${\tilde{b}}_{i} = (b_{i}, g_{i}, h_{i})$ , ${\tilde{S}}_{i} = (s_{i}, t_{i}, u_{i})$ , ${\tilde{S}}_{i}^{'} = (s_{i}^{'}, t_{i}^{'}, u_{i}^{'})$ and decision variables ${\tilde{x}}_{j} = (x_{j}, y_{j}, z_{j})$ are non-negative triangular fuzzy numbers then FFLP problem, obtained in Step 3, can be written as: $\begin{matrix} Max (or Min) \sum_{j = 1}^{n} (p_{j}, q_{j}, r_{j}) \otimes (x_{j}, y_{j}, z_{j}) \\ subject to \\ \sum_{j = 1}^{n} (a_{pj}, b_{pj}, c_{pj}) \otimes (x_{j}, y_{j}, z_{j}) \oplus (s_{p}, t_{p}, u_{p}) \\ = (b_{p}, g_{p}, h_{p}) \oplus (s_{p}^{'}, t_{p}^{'}, u_{p}^{'}) \\ \sum_{j = 1}^{n} (a_{qj}, b_{qj}, c_{qj}) \otimes (x_{j}, y_{j}, z_{j}) \oplus (s_{q}, t_{q}, u_{q}) \\ = (b_{q}, g_{q}, h_{q}) \oplus (s_{q}^{'}, t_{q}^{'}, u_{q}^{'}) \\ \sum_{j = 1}^{n} (a_{rj}, b_{rj}, c_{rj}) \otimes (x_{j}, y_{j}, z_{j}) \oplus (s_{r}, t_{r}, u_{r}) \\ = (b_{r}, g_{r}, h_{r}) \oplus (s_{r}^{'}, t_{r}^{'}, u_{r}^{'}) \\ R (s_{p}, t_{p}, u_{p}) - R (s_{p}^{'}, t_{p}^{'}, u_{p}^{'}) \geq 0 \\ R (s_{q}, t_{q}, u_{q}) - R (s_{q}^{'}, t_{q}^{'}, u_{q}^{'}) \leq 0 . Go to Step 8 . \end{matrix}$ Step 7. Assuming (a_ij, b_ij, c_ij) ⊗ (x_j, y_j, z_j) = (m_ij, n_ij, o_ij) the FFLP problem can be converted into the following Crisp LP problem by using arithmetic operations defined in Section 2. $\begin{matrix} Max (or Min) R (\sum_{j = 1}^{n} (p_{j}, q_{j}, r_{j}) \otimes (x_{j}, y_{j}, z_{j})) \\ subject to \\ \sum_{j = 1}^{n} m_{pj} \oplus s_{p} = b_{p} \oplus s_{p}^{'} \\ \sum_{j = 1}^{n} m_{qj} \oplus s_{q} = b_{q} \oplus s_{q}^{'} \\ \sum_{j = 1}^{n} m_{rj} = b_{r} \\ \sum_{j = 1}^{n} n_{pj} \oplus t_{p} = g_{p} \oplus t_{p}^{'} \\ \sum_{j = 1}^{n} n_{qj} \oplus t_{q} = g_{q} \oplus t_{q}^{'} \\ \sum_{j = 1}^{n} n_{rj} = g_{r} \\ \sum_{j = 1}^{n} o_{pj} \oplus u_{p} = h_{p} \oplus u_{p}^{'} \\ \sum_{j = 1}^{n} o_{qj} \oplus u_{q} = h_{q} \oplus u_{q}^{'} \\ \sum_{j = 1}^{n} o_{rj} = h_{r} \\ (s_{p} + 2 t_{p} + u_{p}) - (s_{p}^{'} + 2 t_{p}^{'} + u_{p}^{'}) \geq 0 \\ (s_{q} + 2 t_{q} + u_{q}) - (s_{q}^{'} + 2 t_{q}^{'} + u_{q}^{'}) \leq 0 \\ x_{j} \geq 0, y_{j} - x_{j} \geq 0, z_{j} - y_{j} \geq 0 \\ t_{p} - s_{p} \geq 0, u_{p} - t_{p} \geq 0 \\ t_{q} - s_{q} \geq 0, u_{q} - t_{q} \geq 0 \\ t_{p}^{'} - s_{p}^{'} \geq 0, u_{p}^{'} - t_{p}^{'} \geq 0 \\ t_{q}^{'} - s_{q}^{'} \geq 0, u_{q}^{'} - t_{q}^{'} \geq 0 . Go to Step 9 . \end{matrix}$ Step 8. Assuming (a_ij, b_ij, c_ij) ⊗ (x_j, y_j, z_j) = (m_ij, n_ij, o_ij) the FFLP problem can be converted into the following Crisp LP problem by using arithmetic operations defined in Section 2. $\begin{matrix} Max (or Min) R (\sum_{j = 1}^{n} (p_{j}, q_{j}, r_{j}) \otimes (x_{j}, y_{j}, z_{j})) \\ subject to \\ \sum_{j = 1}^{n} m_{pj} \oplus s_{p} = b_{p} \oplus s_{p}^{'} \\ \sum_{j = 1}^{n} m_{qj} \oplus s_{q} = b_{q} \oplus s_{q}^{'} \\ \sum_{j = 1}^{n} m_{rj} \oplus s_{r} = b_{r} \oplus s_{r}^{'} \\ \sum_{j = 1}^{n} n_{pj} \oplus t_{p} = g_{p} \oplus t_{p}^{'} \\ \sum_{j = 1}^{n} n_{qj} \oplus t_{q} = g_{q} \oplus t_{q}^{'} \\ \sum_{j = 1}^{n} n_{rj} \oplus t_{r} = g_{r} \oplus t_{r}^{'} \\ \sum_{j = 1}^{n} o_{pj} \oplus u_{p} = h_{p} \oplus u_{p}^{'} \\ \sum_{j = 1}^{n} o_{qj} \oplus u_{q} = h_{q} \oplus u_{q}^{'} \\ \sum_{j = 1}^{n} o_{rj} \oplus u_{r} = h_{r} \oplus u_{r}^{'} \\ (s_{p} + 2 t_{p} + u_{p}) - (s_{p}^{'} + 2 t_{p}^{'} + u_{p}^{'}) \geq 0 \\ (s_{q} + 2 t_{q} + u_{q}) - (s_{q}^{'} + 2 t_{q}^{'} + u_{q}^{'}) \leq 0 \\ x_{j} \geq 0, y_{j} - x_{j} \geq 0, z_{j} - y_{j} \geq 0 \\ t_{p} - s_{p} \geq 0, u_{p} - t_{p} \geq 0 \\ t_{q} - s_{q} \geq 0, u_{q} - t_{q} \geq 0 \\ t_{r} - s_{r} \geq 0, u_{r} - t_{r} \geq 0 \\ t_{p}^{'} - s_{p}^{'} \geq 0, u_{p}^{'} - t_{p}^{'} \geq 0 \\ t_{q}^{'} - s_{q}^{'} \geq 0, u_{q}^{'} - t_{q}^{'} \geq 0 \\ t_{r}^{'} - s_{r}^{'} \geq 0, u_{r}^{'} - t_{r}^{'} \geq 0 . Go to Step 10 . \end{matrix}$ Step 9. Find the optimal solution x_j, y_j and z_j by solving the Crisp LP problem obtained in Step 7, and find the fuzzy optimal solution by putting the values of x_j, y_j and z_j in ${\tilde{x}}_{j} = (x_{j}, y_{j}, z_{j})$ . Determine the fuzzy optimal value by putting ${\tilde{x}}_{j}$ in $\sum_{j = 1}^{n} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j}$ . If the problem does not have any feasible solution then go to Step 2 Case (iv).

Step 10. Find the optimal solution x_j, y_j and z_j by solving the Crisp LP problem obtained in Step 8, and find the approximate fuzzy optimal solution by putting the values of x_j, y_j and z_j in ${\tilde{x}}_{j} = (x_{j}, y_{j}, z_{j})$ . Determine the fuzzy approximate optimal value by putting ${\tilde{x}}_{j}$ in $\sum_{j = 1}^{n} {\tilde{c}}_{j} \otimes {\tilde{x}}_{j} .$ This solution is the most acceptable solution under the limitation of the available resources in real life problems for the decision makers.

6 Numerical experiments

In this section, we present two numerical experiments to illustrate the extended solution procedure.

Example 1. [22] $\begin{matrix} Maximize ((1, 2, 3) \otimes {\tilde{x}}_{1} \oplus (3, 4, 5) \otimes {\tilde{x}}_{2}) \\ subject to \\ (- 2, 2, 6) \otimes {\tilde{x}}_{1} \oplus (1, 2, 3) \otimes {\tilde{x}}_{2} \underline{≺} (- 3, 8, 24) \\ (2, 4, 6) \otimes {\tilde{x}}_{1} \oplus (1, 3, 5) \otimes {\tilde{x}}_{2} \underline{≻} (1, 13, 32) \\ (- 1, 0, 1) \otimes {\tilde{x}}_{1} \oplus (- 3, - 2, - 1) \otimes {\tilde{x}}_{2} = (- 14, - 6, 1) \end{matrix}$ ${\tilde{x}}_{1}$ and ${\tilde{x}}_{2}$ are non-negative triangular fuzzy numbers.

By solving the Crisp LP problem constructed in Step 7, the feasible solution is obtained. Using Step 9, the fuzzy optimal solution of the FFLP problem is ${\tilde{x}}_{1} = (1.5, 1.5, 1.5)$ and ${\tilde{x}}_{2} = (0.5, 3, 4.1666)$ and the optimal value is (3, 15, 25.3333).

Example 2. $\begin{matrix} Maximize ((1, 2, 3) \otimes {\tilde{x}}_{1} \oplus (3, 4, 5) \otimes {\tilde{x}}_{2}) \\ subject to \\ (- 2, 2, 6) \otimes {\tilde{x}}_{1} \oplus (1, 2, 3) \otimes {\tilde{x}}_{2} \underline{≺} (- 3, 8, 24) \\ (2, 4, 6) \otimes {\tilde{x}}_{1} \oplus (1, 3, 5) \otimes {\tilde{x}}_{2} \underline{≻} (1, 13, 32) \\ (- 1, 0, 1) \otimes {\tilde{x}}_{1} \oplus (- 3, - 2, - 1) \otimes {\tilde{x}}_{2} = (- 14, - 6, 1) \\ (- 3, - 1, 2) \otimes {\tilde{x}}_{1} \oplus (1, 2, 4) \otimes {\tilde{x}}_{2} = (- 3, - 2, - 1) \\ (- 4, - 3, - 2) \otimes {\tilde{x}}_{1} \oplus (1, 3, 6) \otimes {\tilde{x}}_{2} = (1, 4, 5) \end{matrix}$

By solving constructed Crisp LP problem in Step 7, the infeasible case is occurred. By using Step 2 Case (iv) the reconstructed Crisp LP problem is solved and the fuzzy approximate optimal solution of the FFLP problem is ${\tilde{x}}_{1} = (0.875, 0.875, 0.875)$ and ${\tilde{x}}_{2} = (3.7500, 3.7500, 3.7500)$ and the optimal value is (12.1250, 16.7500, 21.3750).

7 Conclusion

LP is an optimization problem of a very important area of mathematics called “optimization theory” which deals with finding optimal solutions under set of linear inequality or linear equality constraints. Here, we dealt with FFLP defined in which all parameters i.e coefficients as well as variables in the objective function and constraints are taken as arbitrary triangular fuzzy numbers. Kumar and Kaur [22] proposed a method to find the fuzzy optimal solution of FFLP problems with mixed constraints. However, their method cannot be used for finding the fuzzy optimal solution of FFLP with mixed constraints when infeasibility case may occurred. In this paper, we have provided an extension to the method of Kumar and Kaur [22] to find fuzzy optimal solution to the feasible case and approximate fuzzy optimal solution to the infeasible case of FFLP problem with mixed constraints. To our knowledge, there is no scientific paper related with the infeasibility case of FFLP problem. It is important to provide an approximate fuzzy optimal solution of FFLP which is the most acceptable solution under the limitation of the available resources in real life problems for the decision makers. For future work, we try our method to solve FFLP problems with trapezoidal fuzzy numbers.

Footnotes

Acknowledgments

This research has been supported by Yıldız Technical University Scientific Research Projects Coordination Department. Project Number: 2014-01-03-GEP01.

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