[Retracted] Optimization of <span class="nowrap"><svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-0.2379999pt" id="M1" height="11.491pt" version="1.1" viewBox="-0.0657574 -11.253 20.0945 11.491" width="20.0945pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g190-77" d="M495 163C480 117 462 85 444 65C421 39 387 34 332 34C290 34 256 36 236 47C218 57 213 77 213 131V526C213 612 222 616 301 622V650H40V622C122 616 128 611 128 526V126C128 41 120 34 36 28V0H489C498 31 519 126 525 157L495 163Z"/></g><g transform="matrix(.017,0,0,-0.017,9.232,0)"><path id="g190-83" d="M631 18C609 24 585 35 559 65C534 91 514 117 478 169C448 214 406 281 389 313C462 346 516 399 516 485C516 545 490 590 449 616C412 641 363 650 290 650H42V622C120 615 128 612 128 527V125C128 40 120 34 38 28V0H300V28C221 34 212 40 212 125V284H244C295 284 312 272 329 244C359 195 395 133 430 84C475 19 516 -3 592 -7C603 -8 615 -8 627 -8L631 18ZM212 316V563C212 591 215 602 223 607C231 613 248 617 277 617C352 617 423 577 423 469C423 415 407 375 368 345C343 324 310 316 260 316H212Z"/></g></svg>-</span>Type Fully Bipolar Fuzzy Linear Programming Problems

Mehmood, Muhammad Athar; Akram, Muhammad; Alharbi, Majed G.; Bashir, Shahida

doi:https://doi.org/10.1155/2021/1199336

Mathematical Problems in Engineering

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Volume 2021 | Article ID 1199336 | https://doi.org/10.1155/2021/1199336

[Retracted] Optimization of -Type Fully Bipolar Fuzzy Linear Programming Problems

Muhammad Athar Mehmood,¹Muhammad Akram,²Majed G. Alharbi,³and Shahida Bashir¹

Academic Editor: Naeem Jan

Received29 Apr 2021

Revised28 May 2021

Accepted10 Jun 2021

Published02 Jul 2021

Abstract

In this study, we present a technique to solve -type fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints. We define -type bipolar fuzzy numbers and their arithmetic operations. We discuss multiplication of -type bipolar fuzzy numbers. Furthermore, we develop a method to solve -type FBFLPPs with equality constraints involving -type bipolar fuzzy numbers as parameters and variables. Moreover, we define ranking for -type bipolar fuzzy numbers which transform the -type FBFLPP into a crisp linear programming problem. Finally, we consider numerical examples to illustrate the proposed method.

1. Introduction

Zadeh [1–3]’s fuzzy set (FS) theory has been shown to be a useful tool to describe situations in which the data are imprecise or vague. FSs handle such situations by attributing a degree to which a certain object belongs to a set. In 1994, Zhang [4] initiated the concept of bipolar fuzzy sets (BFSs) as an extension of FSs. The BFS representation is useful when irrelevant elements and contrary elements are needed to be discriminated. Furthermore, Zhang [5] introduced NPN fuzzy sets and NPN qualitative algebra. On the other hand, Akram and Arshad [6] defined bipolar fuzzy numbers and proposed a novel trapezoidal bipolar fuzzy TOPSIS method for group decision making, and Singh [7] discussed two schemes based on the properties of next neighbors and Euclidean distance in a bipolar fuzzy environment. Chakraborty et al. [8] introduced pentagonal neutrosophic numbers and further analyzed their properties. They studied the definitions of score function and accuracy function which transform pentagonal neutrosophic numbers into crisp numbers and also presented transportation models in a neutrosophic environment.

In mathematical programming models, the simplest model is a linear programming problem. Linear programming is a mathematical modeling technique in which a linear function is maximized or minimized when subjected to various constraints. Uncertainty is also introduced in the linear programming problem, which is widely studied by many scholars. Bellman and Zadeh [9] studied about objectives and human decision making. Zimmerman [10] presented a scheme to solve the fuzzy linear programming (FLP) problem by using multiobjective function. Tanaka et al. [11] suggested a method and obtained solution of FLP problems. Later, several methods have been developed in [12–17] to solve FLP problems and fuzzy systems of linear equations. Recently, certain methods have been developed in [18–20] to solve bipolar fuzzy linear system (BFLS) of equations.The notion of -FN was introduced by Dubois and Prade [21]. Dehghan et al. [22] presented a technique and obtained solution of fully fuzzy linear system in which the coefficient matrix and right hand column vector contain -fuzzy numbers. Kaur and Kumar [15] defined arithmetic operations of -FNs and suggested Mehar’s method to solve fully FLP problems by using these numbers as variables and parameters. Buckley [23] gave the idea of fuzzy complex set and fuzzy complex number. Akram et al. [18] proposed a technique to solve -BFLS, -complex BFLS with real coefficients, and -complex BFLS with complex coefficients of equations. Recently, Akram et al. [24, 25] presented methods to solve Pythagorean FLP problems. Mehmood et al. [26] proposed a method for solving fully BFL programming problems. In this research article, we present a technique to solve -type fully bipolar fuzzy linear programming problems (FBFLPPs) with equality constraints. We define -type bipolar fuzzy numbers and their arithmetic operations. We discuss multiplication of -type bipolar fuzzy numbers. Furthermore, we develop a method to solve -type FBFLPPs with equality constraints involving -type bipolar fuzzy numbers as parameters and variables. Moreover, we define ranking for -type bipolar fuzzy numbers which transform the -type FBFLPP into a crisp linear programming problem. Finally, we consider numerical examples to illustrate the proposed method.

We have organized the research article as follows: In Section 2, some preliminary concepts are presented. In Section 3, arithmetic operations are introduced. Section 4 presents a method to solve -type FBFLPPs with equality constraints in which variables and parameters are -type BFNs. In Section 5, the numerical example and model are illustrated. Conclusion is given in Section 6. The list of acronyms used in the research article is given in Table 1.

2. Preliminaries

Definition 1. [4]). Let . A BFS in is an object having the formwhich is characterized by the functions and , known as the truth membership function and falsity membership function, respectively. Here, truth or positive membership degree indicates the satisfaction degree of those elements which fulfill a certain property relating to BFS , and negative membership degree indicates the satisfaction degree of those elements which fulfill some counter property relating to BFS .

Definition 2. [18]). A BFN, is a BFS of the mapping , with satisfaction degree and dissatisfaction degree such that

Definition 3. [6]). Let be a BFN. Then, its cut is defined aswhere .

Definition 4. [18]). A BFN is said to be an -bipolar fuzzy number of the form , where are real numbers and and , if its membership function and nonmembership function , are represented bywhere , and are called the mean value and left and right spreads of the positive side of , respectively, while , and are called the mean value and left and right spreads of negative side of , respectively. Also, and are continuous and decreasing functions from to , while and are continuous and increasing functions from to such that(1)(2)(3)(4)

Remark 1. If we putin Definition 4, then converts to an -type bipolar fuzzy number.

Definition 5. An -type BFN is said to be nonnegative if and and is said to be nonpositive if and .

Definition 6. An -type BFN is positive if and and is negative if and .

Definition 7. An -type BFN is said to be unrestricted if and are real numbers.

Definition 8. An -type BFNs is said to be zero if and only if , and .

Definition 9. Two -type BFNs and are equal if .

Definition 10. Let be an -type BFN; then, their and are given as follows:where and .

Definition 11. Let be an -type BFN; then, ranking of , represented by , is defined aswhere and .
Let and be two -type BFNs; then,(i), if (ii), if (iii), if For other concepts and applications, refer to [27–32].

3. Arithmetic Operations

In this section, we study about arithmetic operations for -type BFNs.

Theorem 1. Let and be two -type BFNs; then,

Proof. Let and be two -type BFNs; then, their and , , and are given as follows:So,By setting in equation (170), we getBy setting in equation (170), we getNow,By setting in equation (181), we getBy setting in equation (181), we getOn combining the equations (173), (178), (185), and (15), the result follows.

Theorem 2. Let and be two -type BFNs; then,

Proof. Let and be two -type BFNs; then, their and , , and are given as follows:So,By setting in equation (18), we getBy setting in equation (18), we getNow,By setting in equation (21), we getBy setting in equation (21), we getOn combining equations (19), (20), (22), and (23), the result follows.

Theorem 3. Let be an -type BFN and be an arbitrary real number; then,

Proof. Let be an -type BFN and be an arbitrary real number; then, their and , , and are given as follows:Now, if , thenBy setting in equation (26), we getBy setting in equation (26), we getAlso,By setting in equation (29), we getBy setting in equation (29), we getOn combining the equations (27), (28), (30), and (31), the case follows.
If , thenBy setting in equation (32), we getBy setting in equation (32), we getAlso,By setting in equation (35), we getBy setting in equation (35), we getOn combining equations (33), (34), (36), and (37), the case follows. Thus result is as required.

Theorem 4. Let and be two nonnegative -type BFNs; then,

Proof. Let and be two nonnegative -type BFNs; then, their and , , and are given as follows:So,By setting in equation (40), we getBy setting in equation (40), we getAlso,By setting in equation (44), we getBy setting in equation (44), we getOn combining the equations (41), (43), (45), and (47), the result follows.

Theorem 5. Let be a nonnegative -type BFN and be a non-positive -type BFN, then

Proof. Let be a nonnegative -type BFN and be a nonpositive -type BFN; then, their and , , and are given as follows:So,By setting in equation (50), we getBy setting in equation (50), we getNow,By setting in equation (54), we getBy setting in equation (54), we getOn combining equations (51), (53), (55), and (57), the result follows.

Theorem 6. Let be a nonpositive -type BFN and be a nonnegative -type BFN; then,

Proof. Let be a nonpositive -type BFN and be a nonnegative -type BFN; then, their and , , and are given as follows:So,By setting in equation (60), we getBy setting in equation (60), we getNow,By setting in equation (64), we getBy setting in equation (64), we getOn combining the equations (61), (63), (65), and (67), the result follows.

Theorem 7. Let and be two nonpositive -type BFNs; then,

Proof. Let and be two nonpositive -type BFNs; then, their and , , and are given as follows:So,By setting in equation (70), we getBy setting in equation (70), we getNow,By setting in equation (74), we getBy setting in equation (74), we getOn combining the equations (191), (73), (75), and (77), the result follows.

Theorem 8. Let be an -type BFN in which , and be an unrestricted -type BFN; then,

Proof. Let be an -type BFN in which , and be an unrestricted -type BFN; then, their and , , and are given as follows:So,By setting in equation (80), we getBy setting in equation (80), we getNow,By setting in equation (84), we getBy setting in equation (84), we getOn combining the equations (81), (83), (85), and (87), the result follows.

Theorem 9. Let be an -type BFN in which , , and be an unrestricted -type BFN; then,

Proof. Let be an -type BFN in which , , and be an unrestricted -type BFN; then, their and , , and are given as follows:So,By setting in equation (90), we getBy setting in equation (90), we getNow,By setting in equation (94), we getBy setting in equation (94), we getOn combining the equations (91), (93), (95), and (97), the result follows.

Theorem 10. Let be an -type BFN in which , and be an unrestricted -type BFN; then,

Proof. Let be an -type BFN in which , and be an unrestricted -type BFN; then, their and , , and are given as follows:So,By setting in equation (100), we getBy setting in equation (100), we getNow,By setting in equation (104), we getBy setting in equation (104), we getOn combining equations (101), (103), (105), and (107), the result follows.

Theorem 11. Let be an -type BFN in which , , and be an unrestricted -type BFN; then,

Proof. Let be an -type BFN in which , , and be an unrestricted -type BFN; then, their and , , and are given as follows:So,By setting in equation (110), we getBy setting in equation (110), we getNow,By setting in equation (114), we getBy setting in equation (114), we getOn combining the equations (111), (113), (115), and (117), the result follows.

Theorem 12. Let be an -type BFN in which , , , and be an unrestricted -type BFN; then,

Proof. Let be an -type BFN in which , , , and be an unrestricted -type BFN; then, their and , , and are given as follows:So,By setting in equation (120), we getBy setting in equation (120), we getNow,By setting in equation (124), we getBy setting in equation (124), we getOn combining the equations (121), (123), (125), and (127), the result follows.

Theorem 13. Let be an -type BFN in which , , and be an unrestricted -type BFN; then,

Proof. Let be an -type BFN in which , , and be an unrestricted -type BFN; then, their and , , and are given as follows:So,By setting in equation (130), we getBy setting in equation (130), we getNow,By setting in equation (134), we getBy setting in equation (135), we getOn combining the equations (131), (133), (135), and (137), the result follows.

Theorem 14. Let be an -type BFN in which , and be an unrestricted -type BFN; then,

Proof. Let be an -type BFN in which , and be an unrestricted -type BFN; then, their and , , and are given as follows:So,By setting in equation (140), we getBy setting in equation (140), we getNow,By setting in equation (144), we getBy setting in equation (144), we getOn combining equations (141), (143), (145), and (147), the result follows.

Theorem 15. Let be an -type BFN in which , , and be an unrestricted -type BFN; then,

Proof. Let be an -type BFN in which , , and be an unrestricted -type BFN; then, their and , , and are given as follows:So,By setting in equation (150), we getBy setting in equation (150), we getNow,By setting in equation (154), we getBy setting in equation (154), we getOn combining equations (151), (153), (155), and (157), the result follows.

Theorem 16. Let be an -type BFN in which , and be an unrestricted -type BFN; then,

Proof. Let be an -type BFN in which , and be an unrestricted -type BFN; then, their and , , and are given as follows:So,By setting in equation (160), we getBy setting in equation (160), we getNow,By setting in equation (164), we getBy setting in equation (164), we getOn combining equations (161), (163), (165), and (167), the result follows.

4. LR-Type FBFLPP

In this section, we study about the -type fully bipolar fuzzy linear programming problem with equality constraints in which all the variables are represented by -type BFNs. Consider an -type FBFLPP with -type BFNssubject towhere , , and are -type BFNs.

Definition 12. A bipolar fuzzy optimal solution of -type FBFLPP (168) will be -type BFN if(1) are -type BFNs(2)(3)If there exists any -type BFN satisfying the constraints, then In case of a maximization problem, In case of a minimization problem,

4.1. Methodology

subject towhere , and -type BFNs.

Step 1. assume thatThe -type FBFLPP (170) can be transformed into the following problem:subject towhere is an -type BFN, .

Step 2. By using the product of -type BFNs given in Section 3, we suppose thatThe -type FBFLPP (173) transforms to a problem as follows:subject towhere is an -type BFN, .

Step 3. By using arithmetic operations, given in Section 3, the -type FBFLPP (176) transforms into a problem as follows:subject to.

Step 4. By using ranking function, the -type FBFLPP converts into a crisp mathematical problem as follows:subject to.

Step 5. Consideringthe crisp mathematical problem (181) can be transformed into the following problem:subject to.

Step 6. By applying the linearity property , here let be a BFN, the crisp mathematical problem (185) can be transformed into the following problem:subject to.

Step 7. By applying ranking for -type BFNs (2.11), the crisp mathematical problem (188) becomessubject to.

Step 8. By solving the crisp mathematical problem (190), we get the optimal solution , .

Step 9. find the exact -type bipolar fuzzy optimal solution of -type FBFLPP by assigning the values of , and in . .

Step 10. find the -type bipolar fuzzy optimal value by putting the values of in , .
Thus, we state the existence condition for the optimal solution of bipolar fuzzy LPP in the following theorem.

Theorem 17. The solution of -type FBFLPP.
Maximize/Minimize, and are -type BFNs, which exist with the solution of the associated crisp mathematical problem.
Maximize/Minimizesubject to exists. Otherwise, there is no guarantee that the -type bipolar fuzzy optimal solution exists.

Proof. The proof is straightforward.

5. Numerical Examples

In this section, the methodology presented in Section 4 is illustrated by solving a numerical example and model.

Example 1. subject towhere and are -type BFNs and . Step 1: Let and be -type BFNs; then, subject to where and are -type BFNs Step 2: by using the product of -type BFNs given in Section 3, the -type FBFLPP converts as Maximize subject to Step 3: using arithmetic operations (12), the -type FBFLPP converts as Maximize subject to Step 4: by applying ranking function and using (2.8), the -type FBFLPP converts as Maximize Step 5: using ranking (2.11) for -type BFNs, the problem converts to crisp LPP as Maximize subject to Step 6: by using software Maple, an optimal solution of the crisp LPP is Step 7: the exact -type BFOS is Step 8: the bipolar fuzzy optimal value of -type FBFLPP is

Example 2. (Fitness Problem). A chef purchases beef and mutton for the players of a football team. Both types of meat possess plenty of proteins and minerals. The cost of beef and mutton per kilograms is Rs. and Rs. , respectively. Each player must have to take grams of proteins and grams of minerals daily to maintain the physical fitness standards, and further details are presented in Table 2. How many units of beef and mutton should be used to fulfill the demand of each player at the minimum cost?
We apply the proposed method to solve this problem.
Let and units be taken of beef and mutton; then, the given problem converts to an -type FBFLPP assubject towhere and are -type BFNs and . Step 1: let and be -type BFNs; then, subject to where and are -type BFNs Step 2: by using the product of -type BFNs given in Section 3, -type FBFLPP converts as Minimize subject to Step 3: using arithmetic operations (12), the -type FBFLPP converts as Minimize subject to Step 4: by applying ranking function and using and , the -type FBFLPP converts as Minimize subject to Step 5: using ranking (2.11) for -type BFNs, the -type FBFLPP converts to a crisp non-LPP as Minimize subject to Step 6: minimize subject to Step 7: by using software Maple, Optimal solution of the crisp non-LPP is Step 8: the exact -type BFOS is Step 9: the bipolar fuzzy optimal value of the -type fully bipolar fuzzy linear programming problem is Thus, units of mutton and units of beef should be given to each player at a minimum cost of Rs. .

6. Conclusions

Linear programming is applied to optimize an objective function subject to constraints. It has vast application in many fields such as science, marketing, industry, business, agriculture, and telecommunication. In this research article, we have defined -type BFNs and their arithmetic operations, and particularly by considering different cases, multiplication of -type BFNs is discussed. We have introduced the ranking for -type BFNs that transform the -type FBFLPP into a crisp linear programming problem. The proposed technique is applied to solve -type FBFLPPs with equality constraints involving variables and parameters as -type BFNs. The given method has been interpreted with a numerical example and a model. The obtained optimal solution satisfies all the constraints of the -type FBFLPP and justifies that the proposed scheme is accurately designed. In future, our scheme can be extended to a complex bipolar fuzzy LPP.

Data Availability

No data were used to support this study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2021 Muhammad Athar Mehmood et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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