Methods for Solving LR-Type Pythagorean Fuzzy Linear Programming Problems with Mixed Constraints

A Pythagorean fuzzy set is the superset of fuzzy and intuitionistic fuzzy sets, respectively. Yager proposed the concept of Pythagorean fuzzy sets in which he relaxed the condition that sum of square of both membership degree and nonmembership degree of an element of a set must not be greater than 1. $is paper introduces two new techniques to solve LR-type fully Pythagorean fuzzy linear programming problems with mixed constraints having unrestricted LR-type Pythagorean fuzzy numbers as variables and parameters by introducing unknown variables and using a ranking function. Furthermore, we show the equivalence of both the proposed methods and compare the solutions obtained by the two techniques. Besides this, we solve an already existing practical model using proposed techniques and compare the result.


Introduction
e origin of linear programming is the 1940s (World War II). Linear programming is a technique in which a function (called objective function) is an optimized subject to a given set of restrictions (called constraints). It is mostly used in a situation where there is some quantity to be optimized within available resources. e nature of the linear programming model is trivial and easily applicable to various real-life applications, including transportation problems, supplier selections, assignment problems, production planning problems, and supply chain management. Linear programming in a fuzzy environment is a very interesting field in which many researchers showed interest around the globe. It can be used very effectively in the situations, where the data is fuzzy, vague, or uncertain, where crisp theory fails to cope with. Hence, in these situations, fuzzy linear programming technique is very effective in making decisions.
Zadeh [1,2] introduced the concepts of fuzzy sets and fuzzy numbers. Bellman and Zadeh [3] first introduced the concept of decision-making in a fuzzy environment. Zimmermann [4] studied the fuzzy programming technique to solve the multiobjective linear programming problem under a fuzzy environment. e fuzzy optimization technique is based on the maximization of the marginal satisfaction (membership functions and degree of belongingness) of each element into the fuzzy decision set. Tanaka et al. [5] also discussed mathematical linear programming in a fuzzy environment. Allahviranloo [6] presented the Adomian decomposition method for a fuzzy system of linear equations. Allahviranloo et al. [7] discovered a method for solving fully fuzzy linear programming problems by the ranking function. Lotfi et al. [8] presented a method for solving a full fuzzy linear programming using lexicography method and fuzzy approximate solution. Kumar et al. [9] proposed a new method for solving fully fuzzy linear programming problems. Kumar and Kaur [10] studied a method for exact fuzzy optimal solution of fully fuzzy linear programming problems with unrestricted fuzzy variables. Kaur and Kumar [11] presented Mehar's method for solving fully fuzzy linear programming problems with LR fuzzy parameters. Moloudzadeh et al. [12] introduced a new method for solving an arbitrary fully fuzzy linear system. Behera et al. [13] studied new methods for solving imprecisely defined linear programming problems under trapezoidal fuzzy uncertainty. Najafi and Edalatpanah [14] introduced a new method for solving fully fuzzy linear programming problems. Pe �rez-Ca� nedo et al. [15] gave a revised version of a lexicographical-based method for solving fully fuzzy linear programming problems with inequality constraints.
Later on, it was realized that only the membership degrees are not well enough to represent the marginal attainment of the element into the fuzzy decision set. To extend or explore the fuzzy set, Atanassov [16] introduced the concept of fuzzy set to intuitionistic fuzzy set in which there is a nonmembership function along with the membership function. In an intuitionistic fuzzy set, the sum of membership degree and nonmembership degree of an element should not be greater than 1. Angelov [17] first considered the intuitionistic fuzzy optimization techniques based on intuitionistic fuzzy decision set in decision-making problems. Dubey and Mehra [18] presented linear programming with triangular intuitionistic fuzzy numbers. Parvathi and Malathi [19] proposed a method to solve intuitionistic fuzzy linear programming problems. Nagoorgani and Ponnalagu [20] revealed a new approach on solving intuitionistic fuzzy linear programming problems. Parvathi and Malathi [21] studied intuitionistic fuzzy linear optimization. Parvathi et al. [22] gave intuitionistic fuzzy linear regression analysis. Garg et al. [23] presented an intuitionistic fuzzy optimization technique for solving multiobjective reliability optimization problems in an interval environment. Suresh et al. [24] gave a method of solving intuitionistic fuzzy linear programming problems by ranking function. Nagoorgani et al. [25] presented the knowledge of expert opinion in intuitionistic fuzzy linear programming problems. Singh and Yadav [26] proposed optimization of unrestricted LR-type intuitionistic fuzzy mathematical programming problems. Singh and Yadav [27] proposed intuitionistic fuzzy multiobjective linear programming problems with various membership functions. Malathi and Umadevi [28] gave a new procedure for solving linear programming problems in an intuitionistic fuzzy environment. Abhishekh and Nishad [29] proposed a novel ranking approach for solving fully LR-intuitionistic fuzzy transportation problem. Bharati and Singh [30] studied a method for the solution of multiobjective linear programming problems in intervalvalued intuitionistic fuzzy environments. Kabiraj et al. [31] proposed another method for intuitionistic fuzzy linear programming problems. Perez-Canedo and Concepcion-Morales [15] presented a method for unique optimal values of LR-type fully intuitionistic fuzzy linear programming with inequality constraints.
Unfortunately, intuitionistic fuzzy sets fail to deal with the situations where the sum of membership degree and nonmembership degree of an element exceeds 1. To overcome this difficulty, Yager [32] introduced the concept of Pythagorean fuzzy set in which he relaxed the condition that sum of square of both membership degree and nonmembership degree of an element of a set must not be greater than 1. Yager and Abbasov [33] presented Pythagorean membership grades, complex numbers, and decision-making. Yager [34] introduced Pythagorean membership grades in multicriteria decision-making. Zhang and Xu [35] extended the TOPSIS to multiple-criteria decision-making with Pythagorean fuzzy sets. Peng et al. [36] studied Pythagorean fuzzy information measures and their applications. Wan et al. [37] gave a Pythagorean fuzzy mathematical programming method for multiattribute group decisionmaking with Pythagorean fuzzy truth degrees. Kumar et al. [38] proposed a Pythagorean fuzzy approach to the transportation problem. Luqman et al. [39] presented a digraph and matrix approach for risk evaluations under Pythagorean fuzzy information. Wan et al. [40] gave a new order relation for Pythagorean fuzzy numbers and its application to multiattribute group decision-making. On the contrary, Ahmad et al. [41] studied spherical fuzzy linear programming problems and Akram et al. [42] developed methods to solve fully Pythagorean fuzzy linear programming problems with equality constraints. Wei et al. [43] studied green supplier selection based on the CODAS method in a probabilistic uncertain linguistic environment. Wei et al. [44] extended COPRAS model for multiple attribute group decision-making based on single-valued neutrosophic 2tuple linguistic environment. Zhang et al. [45] studied the TODIM method based on cumulative prospect theory for multiple attribute group decision-making under a 2-tuple linguistic Pythagorean fuzzy environment. Recently, Akram et al. [46] have introduced a method to solve linear programming problems in which all the variables and parameters are LR-type PFNs having equality constraints. As a continuation of this work, we study two methods for solving FPFLPPs with mixed constraints in which all the variables and parameters are unrestricted LR-type Pythagorean fuzzy numbers (PFNs). e main contribution of this research paper is as follows: (1) e first method is presented to solve FPFLPPs with mixed constraints, in which all the variables and parameters are LR-type PFNs (2) e second method is presented to handle inequality constraints in FPFLPPs using a ranking function (3) A comparison of the proposed methods with the existing methods is given (4) e results of the methods are shown graphically e article is organized as follows. Section 2 presents some preliminaries. Section 3 explains the proposed methods to solve FPFLPPs with unrestricted LR-type PFNs with mixed constraints. Section 4 presents the equivalence of the proposed techniques. Section 5 is devoted for numerical examples to explain the proposed methods. Section 6 includes comparative analysis and some discussions. In Section 7, merits of the proposed methods are given. In Section 8, we conclude the paper.

Preliminaries
In this section, we review elementary concepts that are useful for this article.

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Definition 1 (see [34]). A PFS Ω on X is an object of the form: where μ A : X ⟶ [0, 1] and ] A : X ⟶ [0, 1] are membership function and nonmembership function of Ω, respectively, such that 0 ≤ μ 2 is called a Pythagorean fuzzy index or degree of hesitancy of x in Ω. For computational convenience, Ω � (μ Ω , ] Ω ) is called a PFN [35].
Definition 2 (see [46]). A PFN A � (a; η, θ; η ′ , θ ′ ) LR is defined as an LR-type PFN if its membership (μ A ) and nonmembership (] A ) functions are given as where η ≤ η′, θ ≤ θ ′ and 0 ≤ μ 2 A + ] 2 A ≤ 1. L and R are continuous, decreasing functions on [0, ∞) and L ′ and R ′ are continuous increasing functions on [0, ∞) such that a is called the mean value of A, η and θ are the left and right spreads of μ A , and η ′ and θ ′ are left and right spreads of ] A , respectively.

Mathematical Problems in Engineering
Similarly, for some scalar c, R(cA 1 ) � cR(A 1 ).
Hence, ranking function, as defined in Definition 7, is a linear function.

Methodology to Solve LR-Type Fully Pythagorean Fuzzy Linear Programming Problems
We state here our proposed FPFLPP with LR-type PFNs as which subject to where A ij , X j , B i , and C j are LR-type PFNs.

Method 1: FPFLPP Using Unknown Variables.
Here, we state a criterion for the optimal solution of FPFLPP (13).
Definition 8. An LR-type Pythagorean fuzzy optimal solution of FPFLPP (13) with LR-type PFNs will be LR-type e statement of our proposed problem is given in equation (13). We now present steps to solve proposed FPFLPP (13).
Step 1: separating all the constraints into three categories, where X j are LR-type PFNs and M 1 � i: Step 2: introduce the variable S l on left side and S l ′ on right side of the inequality constraint n j�1 A lj ⊗ X j ≼ B l , ∀l ∈ M 1 , to convert it into equality constraint as below: where R(S l ) − R(S l ′ ) ≥ 0. Introduce the variable S s on left side and S s ′ on right side of the inequality constraint n j�1 A sj ⊗ X j ≽ B s , ∀s ∈ M 3 , to convert it into equality constraint as below: where R(S s ) − R(S s ′ ) ≤ 0. e FPFLPP (15) can be written as which subject to where X j , S l , S l ′ , S s , and S s ′ are LR-type PFNs.

Step 3: by assuming
Mathematical Problems in Engineering (19) can be rewritten as Step 4: by using the product as discussed in Section 2 and taking (a ij ; (21) can be written as where Step 5: using arithmetic operations as discussed in Section 2 and using Definition 6, the FPFLPP (23) takes the form which subject to Mathematical Problems in Engineering where Step 6: now, we have to find LR-type Pythagorean fuzzy feasible solution out of all LR-type Pythagorean fuzzy feasible solutions corresponding to which the ranking of the objective is optimum. By applying ranking, the FPFLPP (24) can be written as which subject to 8 Mathematical Problems in Engineering . , m and ∀j � 1, 2, . . . , n.
Step 8: by using the linearity property of ranking function, problem (28) takes the form which subject to Mathematical Problems in Engineering 9 . . , m and ∀j � 1, 2, . . . , n.
Step 11: find the LR-type Pythagorean fuzzy optimal solution X * j of the FPFLPP (13) by substituting the values of x * j , α * j , β * j , α ′ * j , and β ′ * Step 12: find the LR-type Pythagorean fuzzy optimal value of the FPFLPP (13) by substituting the values of X * j , as calculated in Step (11), in n j�1 C j ⊗ X j .

Method 2: FPFLPP Using Ranking
Function. Now, we present another method to solve FPFLPP (13). We present a criterion for the optimal solution.
Definition 9. An LR-type Pythagorean fuzzy optimal solution of FPFLPP (13) with LR-type PFNs will be LR-type If there exist any LR-type PFNs X j ′ satisfying step 2, then R( n j�1 C j ⊗ X j ) > R( n j�1 C j ⊗ X j ′ ) in maximization problem and R( n j�1 C j ⊗ X j ) < R ( n j�1 C j ⊗ X j ′ ) in minimization problem (15) can be rewritten as
Step 2: by using the product as discussed in Section 2 and taking (a ij ; ρ ij , where (x j ; α j , β j ; α j ′ , β j ′ ) LR is an LR-type PFN.
Step 3: using arithmetic operations as discussed in Section 2 and using Definition 6, the FPFLPP (36) can be rewritten as which subject to where (x j ; α j , β j ; α j ′ , β j ′ ) LR is an LR-type PFN.
Step 9: find the LR-type Pythagorean fuzzy optimal solution X * j of the FPFLPP (13) by substituting the values of Step 10: find the LR-type Pythagorean fuzzy optimal value of the FPFLPP (13) by substituting the values of X * j , as calculated in Step (9), in n j�1 C j ⊗ X j .

Equivalence of the Proposed Methods
Here, we confirm that the two techniques proposed in Section 3.1 and Section 3.2 give the same solution.
If A 1 and A 2 are any two PFNs such that A 1 � A 2 , then R(A) � R(B).
us, the 1 st and 2 nd constraints [ n (19) can be written as Since the ranking function as discussed in Definition 7 is linear, so equations (47) and (48) can be written as Mathematical Problems in Engineering us, equations (49) and (50) can be written as Now, by using the 4 th and 5 th constraints (19), equations (51) and (52) convert to Hence, the proposed techniques (method 1 and method 2) are equivalent. Both the techniques give almost the same solution. However, there is a little bit of difference that when solving the translated crisp problem, one of them may give an answer more faster than the other one. So, depending on the initial guess for the solver, technique which gives faster optimal solution is not known in advance.

Numerical Examples
Let X 1 be the number of X plants and X 2 be the number of Y plants that farmer should grow.
en, the problem converts to the following LR-type FPFLPP: which subject to where X j are LR-type PFNs for j � 1, 2 and L(x) � R(x) � max 0, 1 − x 3 and L ′ (x) � R ′ (x) � min 1, x 2 . Now, we solve Example 1 by using method 1 as discussed in Section 3.1.
Step 3: by using arithmetic operations discussed in Section 2 and Definition 6, the FPFLPP, obtained in Step 2, can be rewritten as which subject to Step 4: using Step 4 of the proposed method 2, the FPFLPP, obtained in Step 3, can be rewritten as which subject to Step Step 4, can be written as which subject to 5x 1 + 5x 2 + x 3 � 155, ≥ 0, f 2 ′ ≥ 0, g 2 ′ − e 2 ′ ≥ 0, and h 2 ′ − f 2 ′ ≥ 0.

Comparison with Existing Linear Programming Model
Pe � rez-Ca� nedo et al. [15] developed a method to solve LR-type fully intuitionistic fuzzy linear programming model. We have proposed two methods to solve LR-type FPFLPP with mixed constraints. By applying the proposed methods to Example 1, we have obtained the solution.
Results of Example 1 are given in Table 1 and are shown graphically in Figure 1. Furthermore, we have solved the practical model [15] by using L(x) � R(x) � max 0, 1 − x { } and L ′ (x) � R ′ (x) � min 1, x { }, and results are given in Table 2. Solution with existing method [15] with permutation (S, A, M, D, E) and solution with method 1 as discussed in Section 3.1 are compared in Figure 2. We observe the following facts: (1) Our proposed methods are equivalent in terms of handling the inequality constraints of FPFLPP with LR-type PFNs as variables and parameters. (2) Example 1 is solved by using our proposed methods.
We see from Table 1 and Figure 1 that both methods produce the optimal solution which is almost the same. (3) e solutions of both the methods (method 1 and method 2) are obtained by solving ultimately a crisp linear programming problem, which is mostly done using any software. e iterations needed for the solution of the crisp problem may vary problem to problem and may also depend on one of the methods used. (4) We compare the solutions of our proposed method 1 with the existing method [15]. We see from Table 2 and Figure 2 that both the solutions are consistent to a large extent.

Merits of the Proposed Methods
e proposed mathematical model is based on the Pythagorean fuzzy environment. e advantages of the proposed method as compared to the existing method are as follows: (1) ere is no method to solve FPFLPP in which all the variables and parameters are unrestricted LR-type PFNs. us, this contribution is new and very helpful for the decision makers. (2) A Pythagorean fuzzy model is more powerful than an intuitionistic fuzzy model since the intuitionistic fuzzy model cannot handle the situation where sum of membership degree μ and nonmembership degree ] of an element exceed 1. So, these techniques are more general and can be used in an intuitionistic fuzzy environment or fuzzy environment. (3) e proposed techniques give almost the same results, so these techniques can be used depending on the interest of the decision maker.
Apart from all the benefits, the presented methods have some limitations. Our proposed methods fail where the condition μ 2 + ] 2 > 1.

Conclusions and Future Directions
In mathematical programming models, linear programming problems are the simplest and most extensively used model. e linear programming model is easily applicable to various real-life applications. In this article, we have studied two techniques to solve LR-type FPFLPP with mixed constraints. We have shown the equivalence of both the presented methods. We have compared the results obtained from both the proposed techniques which come out to be almost the same. Furthermore, we have compared with the existing method [15]. In the future, our research work can be extended for nonlinear programming problems, fractional programming problems, and transportation problems.

Data Availability
No data were used to support this study.

Ethical Approval
is article does not contain any studies with human participants or animals performed by any of the authors.  Figure 2: Comparison of optimal value using the existing method [15] is in blue and using method 1, as discussed in Section 3.1, is in red.

Conflicts of Interest
Mathematical Problems in Engineering 27