# Methods for SolvingLR-Type Pythagorean Fuzzy Linear Programming Problems with Mixed Constraints

Authors: Muhammad Akram; Inayat Ullah; Majed G. Alharbi
Journal: Mathematical Problems in Engineering
 (2021)

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## Abstract

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. This paper introduces two new techniques to solveLR-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.


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## Body

## 1. Introduction

The 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. The 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. The 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-Canˇedo 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 interval-valued 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 decision-making 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 2-tuple 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).The main contribution of this research paper is as follows:(1)
The first method is presented to solve FPFLPPs with mixed constraints, in which all the variables and parameters areLR-type PFNs(2)
The 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)
The results of the methods are shown graphicallyThe article is organized as follows. Section2 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.For further information, the reader can refer to [47–57].


## 2. Preliminaries

In this section, we review elementary concepts that are useful for this article.Definition 1.
(see [34]). A PFS Ω on X is an object of the form:(1)Ω=x,μΩx,νΩx|x∈X,where μA:X⟶0,1 and νA:X⟶0,1 are membership function and nonmembership function of Ω, respectively, such that 0≤μΩ2x+νΩ2x≤1,∀x∈X. Moreover, for all x∈X, πΩx=1−μΩ2x−νΩ2x is called a Pythagorean fuzzy index or degree of hesitancy of x in Ω. For computational convenience, Ω=μΩ,νΩ is called a PFN [35].Definition 2 (see [46]).
A PFNA=a;η,θ;η′,θ′LR is defined as an LR-type PFN if its membership μA and nonmembership νA functions are given as(2)μAx=La−xη,x≤a,η>0,Rx−aθ,x≥a,θ>0,νAx=L′a−xη′,x≤a,η′>0,R′x−aθ′,x≥a,θ′>0,where η≤η′,θ≤θ′ and 0≤μA2+νA2≤1. L and R are continuous, decreasing functions on 0,∞ and L′ and R′ are continuous increasing functions on 0,∞ such that(1)
L0=R0=1(2)
limx⟶∞Rx=limx⟶∞Lx=0(3)
L′0=R′0=0(4)
limx⟶∞R′x=limx⟶∞L′x=1
a is called the mean value of A, η and θ are the left and right spreads of μA, and η′and θ′ are left and right spreads of νA, respectively.Remark 1 (see [46]).
(1)
If we setL′x=1−Lx and R′x=1−Rx in Definition 2, then A=a;η,θ;η′,θ′LR becomes LR-type intuitionistic fuzzy number [26].(2)
If we take(3)Lx=Rx=1−x,0≤x≤1,0,otherwise,L′x=R′x=x,0≤x≤1,1,otherwise,in Definition 2, then A=a;η,θ;η′,θ′LR becomes triangular PFN.Definition 3.
(see [46]). An LR-type PFN A=a;η,θ;η′,θ′LR is nonnegative (respectively, nonpositive), denoted as A≥0 (respectively, A≤0), if a−η′≥0 (respectively a+θ′≤0) and A is unrestricted if a belongs to real numbers.Definition 4.
(see [46]). An LR-type PFN A=a;η,θ;η′,θ′LR is positive if a−η′>0 and negative if a+θ′<0.Definition 5.
(see [46]). An LR-type PFN A=a;η,θ;η′,θ′LR is zero if and only if a=0, η=0, θ=0, η′=0, and θ′=0.Definition 6.
(see [46]). Two LR-type PFNs A1=a1;η1,θ1;η1′,θ1′LR and A2=a2;η2,θ2;η2′,θ2′LR are equal if a1=a2, η1=η2, θ1=θ2, η1′=η2′,andθ1′=θ2′.Theorem 1 (see [46]).
LetA1=a1;η1,θ1;η1′,θ1′LR and A2=a2;η2,θ2;η2′,θ2′LR be two LR-type PFNs; then, A1⊕A2=a1+a2;η1+η2,θ1+θ2;η1′+η2′,θ1′+θ2′LR.Theorem 2 (see [46]).
LetA1=a1;η1,θ1;η1′,θ1′LR and A2=a2;η2,θ2;η2′,θ2′LR be two LR-type PFNs; then, A1⊖A2=a1−a2;η1+θ2,θ1+η2;η1′+θ2′,θ1′+η2′LR.Theorem 3 (see [46]).
LetA=a;η,θ;η′,θ′LR be an LR-type PFN and c be any real number; then,(4)cA=ca;cη,cθ;cη′,cθ′LR,c≥0,ca;−cθ,−cη;−cθ′,−cη′LR,c<0.Theorem 4 (see [46]).
LetA1=a1;η1,θ1;η1′,θ1′LR and A2=a2;η2,θ2;η2′,θ2′LR be two nonnegative LR-type PFNs; then, A1⊗A2=a1a2;a1η2+a2η1−η1η2,a1θ2+a2θ1+θ1θ2;a1η2′+a2η1′−η1′η2′,a1θ2′+a2θ1′+θ1′θ2′LR.Theorem 5 (see [46]).
LetA1=a1;η1,θ1;η1′,θ1′LR be nonnegative LR-type PFN and A2=a2;η2,θ2;η2′,θ2′LR be nonpositive LR-type PFN; then, A1⊗A2=a1a2;a1η2−a2θ1+η2θ1,a1θ2−a2η1−η1θ2;a1η2′−a2θ1′+η2′θ1′,a1θ2′−a2η1′−η1′θ2′LR.Theorem 6 (see [46]).
LetA1=a1;η1,θ1;η1′,θ1′LR be an LR-type PFN in which a1−η1′<0, a1−η1≥0 and A2=a2;η2,θ2;η2′,θ2′LR be an unrestricted LR-type PFN; then, A1⊗A2=a;η,θ;η′,θ′LR, where a=a1a2:(5)η=a1a2−mina1a2−η2a1−η1a2+η1η2,a1a2−η2a1+θ1a2−η2θ1,θ=maxa1a2+θ2a1+θ1a2+θ1θ2,a1a2+θ2a1−η1a2−η1θ2−a1a2,η′=a1a2−mina1a2−η1′a2+θ2′a1−η1′θ2′,a1a2+θ1′a2−η2′a1−η2′θ1′,θ′=maxa1a2−η1′a2−η2′a1+η1′η2′,a1a2+θ1′a2+θ2′a1+θ1′θ2′−a1a2.Theorem 7 (see [46]).
LetA1=a1;η1,θ1;η1′,θ1′LR be an LR-type PFN in which a1−η1<0, a1≥0 and A2=a2;η2,θ2;η2′,θ2′LR be an unrestricted LR-type PFN; then, A1⊗A2=a;η,θ;η′,θ′LR, where a=a1a2:(6)η=a1a2−mina1a2−η1a2+θ2a1−η1θ2,a1a2+θ1a2−η2a1−η2θ1,θ=maxa1a2−η1a2−η2a1+η1η2,a1a2+θ1a2+θ2a1+θ1θ2−a1a2,η′=a1a2−mina1a2−η1′a2+θ2′a1−η1′θ2′,a1a2+θ1′a2−η2′a1−η2′θ1′,θ′=maxa1a2−η1′a2−η2′a1+η1′η2′,a1a2+θ1′a2+θ2′a1+θ1′θ2′−a1a2.Theorem 8 (see [46]).
LetA1=a1;η1,θ1;η1′,θ1′LR be an LR-type PFN in which a1<0, a1+θ1≥0 and A2=a2;η2,θ2;η2′,θ2′LR be an unrestricted LR-type PFN; then, A1⊗A2=a;η,θ;η′,θ′LR, where a=a1a2:(7)η=a1a2−mina1a2−η1a2+θ2a1−η1θ2,a1a2+θ1a2−η2a1−η2θ1,θ=maxa1a2−η1a2−η2a1+η1η2,a1a2+θ1a2+θ2a1+θ1θ2−a1a2,η′=a1a2−mina1a2−η1′a2+θ2′a1−η1′θ2′,a1a2+θ1′a2−η2′a1−η2′θ1′,θ′=maxa1a2−η1′a2−η2′a1+η1′η2′,a1a2+θ1′a2+θ2′a1+θ1′θ2′−a1a2.Theorem 9 (see [46]).
LetA1=a1;η1,θ1;η1′,θ1′LR be an LR-type PFN in which a1+θ1<0, a1+θ1′≥0 and A2=a2;η2,θ2;η2′,θ2′LR be an unrestricted LR-type PFN; then, A1⊗A2=a;η,θ;η′,θ′LR, where a=a1a2:(8)η=a1a2−mina1a2−η1a2+θ2a1−η1θ2,a1a2+θ2a1+θ1a2+θ1θ2,θ=maxa1a2+θ1a2−η2a1−η2θ1,a1a2−η2a1−η1a2+η1η2−a1a2,η′=a1a2−mina1a2−η1′a2+θ2′a1−η1′θ2′,a1a2+θ1′a2−η2′a1−η2′θ1′,θ′=maxa1a2−η1′a2−η2′a1+η1′η2′,a1a2+θ1′a2+θ2′a1+θ1′θ2′−a1a2.Theorem 10 (see [46]).
LetA1=a1;η1,θ1;η1′,θ1′LR be an LR-type PFN in which a1+θ1′<0 and A2=a2;η2,θ2;η2′,θ2′LR be an unrestricted LR-type PFN; then, A1⊗A2=a;η,θ;η′,θ′LR, where a=a1a2:(9)η=a1a2−mina1a2−η1a2+θ2a1−η1θ2,a1a2+θ2a1+θ1a2+θ1θ2,θ=maxa1a2+θ1a2−η2a1−η2θ1,a1a2−η2a1−η1a2+η1η2−a1a2,η′=a1a2−mina1a2−η1′a2+θ2′a1−η1′θ2′,a1a2+θ2′a1+θ1′a2+θ1′θ2′,θ′=maxa1a2+θ1′a2−η2′a1−η2′θ1′,a1a2−η1′a2−η2′a1+η1′η2′−a1a2.Theorem 11 (see [46]).
LetA1=a1;η1,θ1;η1′,θ1′LR be an LR-type PFN in which a1−η1′≥0 and A2=a2;η2,θ2;η2′θ2′LR be an unrestricted LR-type PFN; then, A1⊗A2=a;η,θ;η′,θ′LR, where a=a1a2:(10)η=a1a2−mina1a2−η2a1−η1a2+η1η2,a1a2−η2a1+θ1a2−η2θ1,θ=maxa1a2+θ2a1+θ1a2+θ1θ2,a1a2+θ2a1−η1a2−η1θ2−a1a2,η′=a1a2−mina1a2−η2′a1−η1′a2+η1′η2′,a1a2−η2′a1+θ1′a2−η2′θ1′,θ′=maxa1a2+θ2′a1+θ1′a2+θ1′θ2′,a1a2+θ2′a1−η1′a2−η1′θ2′−a1a2.Definition 7.
(see [46]). Let A=a;η,θ;η′,θ′LR be an LR-type PFN; then, ranking of A, denoted ℜA, can be defined as(11)ℜA=14∫01a−ηL−1αdα+∫01a+θR−1αdα+∫01a−η′L′−1αdα+∫01a+θ′R′−1αdα.
LetA1 and A2 be two LR-type PFN; then, we see that(i)
A1≼A2 if ℜA1<ℜA2(ii)
A1≽A2 if ℜA1>ℜA2(iii)
A1≈A2 if ℜA1=ℜA2Remark 2.
(see [46]). Ranking function, as defined in Definition 7, is a linear function.Proof.
LetA1=a1;η1,θ1;η1′,θ1′LR and A2=a2;η2,θ2;η2′,θ2′LR be two LR-type PFNs; then, A1⊕A2=a1+a2;η1+η2,θ1+θ2;η1′+η2′,θ1′+θ2′LR.
Now, ifL=R,(12)ℜA1⊕A2=Rea1+a2;η1+η2,θ1+θ2;η1′+η2′,θ1′+θ2′LR=14∫01a1+a2−η1+η2L−1αdα+∫01a1+a2+θ1+θ2R−1αdα+∫01a1+a2−η1′+η2′L′−1αdα+∫01a1+a2+θ1′+θ2′R′−1αdα=14∫01a1−η1L−1αdα+∫01a1+θ1R−1αdα+∫01a1−η1′L′−1αdα+∫01a1+θ1′R′−1αdα+14∫01a2−η2L−1αdα+∫01a2+θ2R−1αdα+∫01a2−η2′L′−1αdα+∫01a2+θ2′R′−1αdα=ℜa1;η1,θ1;η1′,θ1′LR+ℜa2;η2,θ2;η2′,θ2′LR=ℜA1+ℜA2.
Similarly, for some scalarc, ℜcA1=cℜA1.
Hence, ranking function, as defined in Definition7, is a linear function.


## 3. Methodology to SolveLR-Type Fully Pythagorean Fuzzy Linear Programming Problems

We state here our proposed FPFLPP withLR-type PFNs as(13)Max/Min∑j=1nCj⊗Xj,which subject to(14)∑j=1nAij⊗Xj≼¯,=,≽¯Bi,∀i=1,2,…,m,where Aij,Xj,Bi, and Cj are LR-type PFNs.


### 3.1. Method 1: FPFLPP Using Unknown Variables

Here, we state a criterion for the optimal solution of FPFLPP (13).Definition 8.
AnLR-type Pythagorean fuzzy optimal solution of FPFLPP (13) with LR-type PFNs will be LR-type PFNs Xj if(1)
Xj are LR-type PFNs(2)
∑j=1nAij⊗Xj⊕Si=Bi⊕Si′, ∀i=1,2,…,m, such that ∑j=1nAij⊗Xj≽¯Bi for some PFNs Si and Si′ satisfying ℜSi−ℜSi′≥0; ∑j=1nAij⊗Xj⊕Si=Bi⊕Si′, ∀i=1,2,…,m, such that ∑j=1nAij⊗Xj≽¯Bi for some PFNs Si and Si′ satisfying ℜSi−ℜSi′≤0; and ∑j=1nAij⊗Xj=Bi, ∀i=1,2,…,m, such that ∑j=1nAij⊗Xj=Bi(3)
If there exist anyLR-type PFNs Xj′ satisfying step 2, then ℜ∑j=1nCj⊗Xj>ℜ∑j=1nCj⊗Xj′ in maximization problem and ℜ∑j=1nCj⊗Xj<ℜ∑j=1nCj⊗Xj′ in minimization problemThe 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, ∑j=1nAlj⊗Xj≼¯Bl,∀l∈M1, ∑j=1nArj⊗Xj=Br,∀r∈M2 and ∑j=1nAsj⊗Xj≽¯Bs,∀s∈M3, the FPFLPP (13) can be rewritten as(15)Max/Min∑j=1nCj⊗Xj,which subject to(16)∑j=1nAlj⊗Xj≼¯Bl,∀l∈M1,∑j=1nArj⊗Xj=Br,∀r∈M2,∑j=1nAsj⊗Xj≽¯Bs,∀s∈M3,whereXj are LR-type PFNs and M1=i:1≤i≤m,∑j=1nAij⊗Xj≼¯Bi, M2=i:1≤i≤m,∑j=1nAij⊗Xj=Bi, and M3=i:1≤i≤m,∑j=1nAij⊗Xj≽¯Bi.Step 2: introduce the variable Sl on left side and Sl′ on right side of the inequality constraint ∑j=1nAlj⊗Xj≼¯Bl,∀l∈M1, to convert it into equality constraint as below:(17)∑j=1nAlj⊗Xj⊕Sl=Bl⊕Sl′,∀l∈M1,whereℜSl−ℜSl′≥0.Introduce the variableSs on left side and Ss′ on right side of the inequality constraint ∑j=1nAsj⊗Xj≽¯Bs,∀s∈M3, to convert it into equality constraint as below:(18)∑j=1nAsj⊗Xj⊕Ss=Bs⊕Ss′,∀s∈M3,whereℜSs−ℜSs′≤0.The FPFLPP (15) can be written as(19)Max/Min∑j=1nCj⊗Xj,which subject to(20)∑j=1nAlj⊗Xj⊕Sl=Bl⊕Sl′,∀l∈M1,∑j=1nAsj⊗Xj⊕Ss=Bs⊕Ss′,∀s∈M3,∑j=1nArj⊗Xj=Br,∀r∈M2,ℜSl−ℜSl′≥0,∀l∈M1,ℜSs−ℜSs′≤0,∀s∈M3,whereXj,Sl,Sl′,Ss, and Ss′ are LR-type PFNs.Step 3: by assuming Aij=aij;ρij,τij;ρij′,τij′LR, Xj=xj;αj,βj;αj′,βj′LR, Bi=bi;θi,ϕi;θi′,ϕi′LR, Cj=cj;ζj,ηj;ζj′,ηj′LR, Si=mi;ei,fi;gi,hiLR, and Si′=mi′;ei′,fi′;gi′,hi′LR, the FPFLPP (19) can be rewritten as(21)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(22)∑j=1nalj;ρlj,τlj;ρlj′,τlj′LR⊗xj;αj,βj;αj′,βj′LR⊕ml;el,fl;gl,hlLR=bl;θl,ϕl;θl′,ϕl′LR⊕ml′;el′,fl′;gl′,hl′LR,∀l∈M1,∑j=1nasj;ρsj,τsj;ρsj′,τsj′LR⊗xj;αj,βj;αj′,βj′LR⊕ms;es,fs;gs,hsLR=bs;θs,ϕs;θs′,ϕs′LR⊕ms′;es′,fs′;gs′,hs′LR,∀s∈M3,∑j=1narj;ρrj,τrj;ρrj′,τrj′LR⊗xj;αj,βj;αj′,βj′LR=br;θr,ϕr;θr′,ϕr′LR,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,wherexj;αj,βj;αj′,βj′LR, ml;el,fl;gl,hlLR, ml′;el′,fl′;gl′,hl′LR, ms;es,fs;gs,hsLR, and ms′;es′,fs′;gs′,hs′LR are LR-type PFN.Step 4: by using the product as discussed in Section 2 and taking aij;ρij,τij;ρij′,τij′LR⊗xj;αj,βj;αj′βj′LR=λij;δij,γij;δij′,γij′LR, the FPFLPP (21) can be written as(23)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,∑j=1nλlj;δlj,γlj;δlj′,γlj′LR⊕ml;el,fl;gl,hlLR=bl;θl,ϕl;θl′,ϕl′LR⊕ml′;el′,fl′;gl′,hl′LR,∀l∈M1,∑j=1nλsj;δsj,γsj;δsj′,γsj′LR⊕ms;es,fs;gs,hsLR=bs;θs,ϕs;θs′,ϕs′LR⊕ms′;es′,fs′;gs′,hs′LR,∀s∈M3,∑j=1nλrj;δrj,γrj;δrj′,γrj′LR=br;θr,ϕr;θr′,ϕr′LR,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,wherexj;αj,βj;αj′,βj′LR, ml;el,fl;gl,hlLR, ml′;el′,fl′;gl′,hl′LR, ms;es,fs;gs,hsLR, and ms′;es′,fs′;gs′,hs′LR are LR-type PFN.Step 5: using arithmetic operations as discussed in Section 2 and using Definition 6, the FPFLPP (23) takes the form(24)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(25)∑j=1nλlj+ml=b1+m1′,∀l∈M1,∑j=1nδlj+el=θl+el′,∀l∈M1,∑j=1nγlj+fl=ϕl+fl′,∀l∈M1,∑j=1nδlj′+gl′=θl′+gl′,∀l∈M1,∑j=1nγlj′+hl=ϕl′+hl′,∀l∈M1,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nλsj+ms=bs+ms′,∀s∈M3,∑j=1nδsj+es=θs+es′,∀s∈M3,∑j=1nγsj+fs=ϕs+fs′,∀s∈M3,∑j=1nδsj′+gs′=θs′+gs′,∀s∈M3,∑j=1nγsj′+hs=ϕs′+hs′,∀s∈M3,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,wherexj;αj,βj;αj′,βj′LR, ml;el,fl;gl,hlLR, ml′;el′,fl′;gl′,hl′LR, ms;es,fs;gs,hsLR, and ms′;es′,fs′;gs′,hs′LR are LR-type PFN.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(26)Max/Minℜ∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(27)∑j=1nλlj+ml=b1+m1′,∀l∈M1,∑j=1nδlj+el=θl+el′,∀l∈M1,∑j=1nγlj+fl=ϕl+fl′,∀l∈M1,∑j=1nδlj′+gl′=θl′+gl′,∀l∈M1,∑j=1nγlj′+hl=ϕl′+hl′,∀l∈M1,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nλsj+ms=bs+ms′,∀s∈M3,∑j=1nδsj+es=θs+es′,∀s∈M3,∑j=1nγsj+fs=ϕs+fs′,∀s∈M3,∑j=1nδsj′+gs′=θs′+gs′,∀s∈M3,∑j=1nγsj′+hs=ϕs′+hs′,∀s∈M3,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,αj≥0, βj≥0, αj′−αj≥0, βj′−βj≥0, ei≥0, fi≥0, ei′−ei≥0, and fi′−fi≥0, ∀i=1,2,…,m and ∀j=1,2,…,n.Step 7: by taking cj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR=ωj;σj,κj;σj′,κj′LR, problem (26) can be written as(28)Max/Minℜ∑j=1nωj;σj,κj;σj′,κj′LR,which subject to(29)∑j=1nλlj+ml=b1+m1′,∀l∈M1,∑j=1nδlj+el=θl+el′,∀l∈M1,∑j=1nγlj+fl=ϕl+fl′,∀l∈M1,∑j=1nδlj′+gl′=θl′+gl′,∀l∈M1,∑j=1nγlj′+hl=ϕl′+hl′,∀l∈M1,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nλsj+ms=bs+ms′,∀s∈M3,∑j=1nδsj+es=θs+es′,∀s∈M3,∑j=1nγsj+fs=ϕs+fs′,∀s∈M3,∑j=1nδsj′+gs′=θs′+gs′,∀s∈M3,∑j=1nγsj′+hs=ϕs′+hs′,∀s∈M3,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,αj≥0, βj≥0, αj′−αj≥0, βj′−βj≥0, ei≥0, fi≥0, ei′−ei≥0, and fi′−fi≥0, ∀i=1,2,…,m and ∀j=1,2,…,n.Step 8: by using the linearity property of ranking function, problem (28) takes the form(30)Max/Min∑j=1nℜωj;σj,κj;σj′,κj′LR,which subject to(31)∑j=1nλlj+ml=b1+m1′,∀l∈M1,∑j=1nδlj+el=θl+el′,∀l∈M1,∑j=1nγlj+fl=ϕl+fl′,∀l∈M1,∑j=1nδlj′+gl′=θl′+gl′,∀l∈M1,∑j=1nγlj′+hl=ϕl′+hl′,∀l∈M1,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nλsj+ms=bs+ms′,∀s∈M3,∑j=1nδsj+es=θs+es′,∀s∈M3,∑j=1nγsj+fs=ϕs+fs′,∀s∈M3,∑j=1nδsj′+gs′=θs′+gs′,∀s∈M3,∑j=1nγsj′+hs=ϕs′+hs′,∀s∈M3,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,αj≥0, βj≥0, αj′−αj≥0, βj′−βj≥0, ei≥0, fi≥0, ei′−ei≥0, and fi′−fi≥0, ∀i=1,2,…,m and ∀j=1,2,…,n.Step 9: by using Definition 7, problem (30) can be converted into(32)Max/Min∑j=1n14∫01ωj−σjL−1αdα+∫01ωj+κjR−1αdα+∫01ωj−σj′L′−1αdα+∫01ωj+κj′R′−1αdα,which subject to(33)∑j=1nλlj+ml=b1+m1′,∀l∈M1,∑j=1nδlj+el=θl+el′,∀l∈M1,∑j=1nγlj+fl=ϕl+fl′,∀l∈M1,∑j=1nδlj′+gl′=θl′+gl′,∀l∈M1,∑j=1nγlj′+hl=ϕl′+hl′,∀l∈M1,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nλsj+ms=bs+ms′,∀s∈M3,∑j=1nδsj+es=θs+es′,∀s∈M3,∑j=1nγsj+fs=ϕs+fs′,∀s∈M3,∑j=1nδsj′+gs′=θs′+gs′,∀s∈M3,∑j=1nγsj′+hs=ϕs′+hs′,∀s∈M3,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,14∫01ml−elL−1αdα+∫01ml+flR−1αdα+∫01ml−glL′−1αdα+∫01ml+hlR′−1αdα−14∫01ml′−el′L−1αdα+∫01ml′+fl′R−1αdα+∫01ml′−gl′L′−1αdα+∫01ml′+hl′R′−1αdα≥0,∀l∈M1,14∫01ms−esL−1αdα+∫01ms+fsR−1αdα+∫01ms−gsL′−1αdα+∫01ms+hsR′−1αdα−14∫01ms′−es′L−1αdα+∫01ms′+fs′R−1αdα+∫01ms′−gs′L′−1αdα+∫01ms′+hs′R′−1αdα≤0,∀l∈M3,αj≥0, βj≥0, αj′−αj≥0, βj′−βj≥0, ei≥0, fi≥0, ei′−ei≥0, and fi′−fi≥0, ∀i=1,2,…,m and ∀j=1,2,…,n.Step 10: now, solve the crisp linear/nonlinear programming problem (32) by any existing method to find the optimal solution xj∗,αj∗,βj∗,αj′∗,βj′∗.Step 11: find the LR-type Pythagorean fuzzy optimal solution Xj∗ of the FPFLPP (13) by substituting the values of xj∗, αj∗, βj∗, αj′∗, and βj′∗ in Xj∗=xj∗;αj∗,βj∗;αj′∗,βj′∗LR.Step 12: find the LR-type Pythagorean fuzzy optimal value of the FPFLPP (13) by substituting the values of Xj∗, as calculated in Step (11), in ∑j=1nCj⊗Xj.


### 3.2. 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.
AnLR-type Pythagorean fuzzy optimal solution of FPFLPP (13) with LR-type PFNs will be LR-type PFNs Xj if(1)
Xj are LR-type PFNs(2)
ℜ∑j=1nAij⊗Xj≤,=,≥ℜBi, for all i=1,2,…,m(3)
If there exist anyLR-type PFNs Xj′ satisfying step 2, then ℜ∑j=1nCj⊗Xj>ℜ∑j=1nCj⊗Xj′ in maximization problem and ℜ∑j=1nCj⊗Xj<ℜ∑j=1nCj⊗Xj′ in minimization problemStep 1: by assuming Aij=aij;ρij,τij;ρij′,τij′LR, Xj=xj;αj,βj;αj′,βj′LR, Bi=bi;θi,ϕi;θi′,ϕi′LR, and Cj=cj;ζj,ηj;ζj′,ηj′LR, the FPFLPP (15) can be rewritten as(34)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(35)∑j=1nalj;ρlj,τlj;ρlj′,τlj′LR⊗xj;αj,βj;αj′,βj′LR≼¯bl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,∑j=1nasj;ρsj,τsj;ρsj′,τsj′LR⊗xj;αj,βj;αj′,βj′LR±bs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1narj;ρrj,τrj;ρrj′,τrj′LR⊗xj;αj,βj;αj′,βj′LR=br;θr,ϕr;θr′,ϕr′LR,∀r∈M2,wherexj;αj,βj;αj′,βj′LR is an LR-type PFN.Step 2: by using the product as discussed in Section 2 and taking aij;ρij,τij;ρij′,τij′LR⊗xj;αj,βjαj′,βj′LR=λij;δij,γij;δij′,γij′LR, the FPFLPP (34) can be written as(36)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,∑j=1nλlj;δlj,γlj;δlj′,γlj′LR≼¯bl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,∑j=1nλsj;δsj,γsj;δsj′,γsj′LR≽¯bs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1nλrj;δrj,γrj;δrj′,γrj′LR=br;θr,ϕr;θr′,ϕr′LR,∀r∈M2,wherexj;αj,βj;αj′,βj′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(37)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(38)∑j=1nλlj;δlj,γlj;δlj′,γlj′LR≼¯bl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,∑j=1nλsj;δsj,γsj;δsj′,γsj′LR≽¯bs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,wherexj;αj,βj;αj′,βj′LR is an LR-type PFN.Step 4: by applying ranking, the FPFLPP (37) takes the form(39)Max/Minℜ∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(40)ℜ∑j=1nλlj;δlj,γlj;δlj′,γlj′LR≤ℜbl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,ℜ∑j=1nλsj;δsj,γsj;δsj′,γsj′LR≥ℜbs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,αj≥0, βj≥0, αj′−αj≥0, and βj′−βj≥0, ∀j=1,2,…,n.Step 5: by taking cj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR=ωj;σj,κj;σj′,κj′LR, problem (39) can be rewritten as(41)Max/Minℜ∑j=1nωj;σj,κj;σj′,κj′LR,which subject to(42)ℜ∑j=1nλlj;δlj,γlj;δlj′,γlj′LR≤ℜbl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,ℜ∑j=1nλsj;δsj,γsj;δsj′,γsj′LR≥ℜbs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,αj≥0, βj≥0, αj′−αj≥0, and βj′−βj≥0, ∀j=1,2,…,n.Step 6: by using the linearity of ranking function, problem (41) can be written as(43)Max/Min∑j=1nℜωj;σj,κj;σj′,κj′LR,which subject to(44)∑j=1nℜλlj;δlj,γlj;δlj′,γlj′LR≤ℜbl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,∑j=1nℜλsj;δsj,γsj;δsj′,γsj′LR≥ℜbs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,αj≥0, βj≥0, αj′−αj≥0, and βj′−βj≥0, ∀j=1,2,…,n.Step 7: by using Definition 7, problem (43) can be converted into problem (45):(45)Max/Min∑j=1n14∫01ωj−σjL−1αdα+∫01ωj+κjR−1αdα+∫01ωj−σj′L′−1αdα+∫01ωj+κj′R′−1αdα,which subject to(46)∑j=1n14∫01λlj−δljL−1αdα+∫01λlj+γljR−1αdα+∫01λlj−δlj′L′−1αdα+∫01λlj+γlj′R′−1αdα≤14∫01bl−θlL−1αdα+∫01bl+ϕlR−1αdα+∫01bl−θl′L′−1αdα+∫01bl+ϕlR′−1αdα,∀l∈M1,∑j=1n14∫01λsj−δsjL−1αdα+∫01λsj+γsjR−1αdα+∫01λsj−δsj′L′−1αdα+∫01λsj+γsj′R′−1αdα≥14∫01bs−θsL−1αdα+∫01bs+ϕsR−1αdα+∫01bs−θs′L′−1αdα+∫01bs+ϕsR′−1αdα,∀s∈M3,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,αj≥0, βj≥0, αj′−αj≥0, and βj′−βj≥0, ∀j=1,2,…,n.Step 8: solve the crisp linear/nonlinear programming problem (45) by any existing method to find the optimal solution xj∗,αj∗,βj∗,αj′∗,βj′∗.Step 9: find the LR-type Pythagorean fuzzy optimal solution Xj∗ of the FPFLPP (13) by substituting the values of xj∗, αj∗, βj∗, αj′∗, and βj′∗ in Xj∗=xj∗;αj∗,βj∗;αj′∗,βj′∗LR.Step 10: find the LR-type Pythagorean fuzzy optimal value of the FPFLPP (13) by substituting the values of Xj∗, as calculated in Step (9), in ∑j=1nCj⊗Xj.


## 3.1. Method 1: FPFLPP Using Unknown Variables

Here, we state a criterion for the optimal solution of FPFLPP (13).Definition 8.
AnLR-type Pythagorean fuzzy optimal solution of FPFLPP (13) with LR-type PFNs will be LR-type PFNs Xj if(1)
Xj are LR-type PFNs(2)
∑j=1nAij⊗Xj⊕Si=Bi⊕Si′, ∀i=1,2,…,m, such that ∑j=1nAij⊗Xj≽¯Bi for some PFNs Si and Si′ satisfying ℜSi−ℜSi′≥0; ∑j=1nAij⊗Xj⊕Si=Bi⊕Si′, ∀i=1,2,…,m, such that ∑j=1nAij⊗Xj≽¯Bi for some PFNs Si and Si′ satisfying ℜSi−ℜSi′≤0; and ∑j=1nAij⊗Xj=Bi, ∀i=1,2,…,m, such that ∑j=1nAij⊗Xj=Bi(3)
If there exist anyLR-type PFNs Xj′ satisfying step 2, then ℜ∑j=1nCj⊗Xj>ℜ∑j=1nCj⊗Xj′ in maximization problem and ℜ∑j=1nCj⊗Xj<ℜ∑j=1nCj⊗Xj′ in minimization problemThe 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, ∑j=1nAlj⊗Xj≼¯Bl,∀l∈M1, ∑j=1nArj⊗Xj=Br,∀r∈M2 and ∑j=1nAsj⊗Xj≽¯Bs,∀s∈M3, the FPFLPP (13) can be rewritten as(15)Max/Min∑j=1nCj⊗Xj,which subject to(16)∑j=1nAlj⊗Xj≼¯Bl,∀l∈M1,∑j=1nArj⊗Xj=Br,∀r∈M2,∑j=1nAsj⊗Xj≽¯Bs,∀s∈M3,whereXj are LR-type PFNs and M1=i:1≤i≤m,∑j=1nAij⊗Xj≼¯Bi, M2=i:1≤i≤m,∑j=1nAij⊗Xj=Bi, and M3=i:1≤i≤m,∑j=1nAij⊗Xj≽¯Bi.Step 2: introduce the variable Sl on left side and Sl′ on right side of the inequality constraint ∑j=1nAlj⊗Xj≼¯Bl,∀l∈M1, to convert it into equality constraint as below:(17)∑j=1nAlj⊗Xj⊕Sl=Bl⊕Sl′,∀l∈M1,whereℜSl−ℜSl′≥0.Introduce the variableSs on left side and Ss′ on right side of the inequality constraint ∑j=1nAsj⊗Xj≽¯Bs,∀s∈M3, to convert it into equality constraint as below:(18)∑j=1nAsj⊗Xj⊕Ss=Bs⊕Ss′,∀s∈M3,whereℜSs−ℜSs′≤0.The FPFLPP (15) can be written as(19)Max/Min∑j=1nCj⊗Xj,which subject to(20)∑j=1nAlj⊗Xj⊕Sl=Bl⊕Sl′,∀l∈M1,∑j=1nAsj⊗Xj⊕Ss=Bs⊕Ss′,∀s∈M3,∑j=1nArj⊗Xj=Br,∀r∈M2,ℜSl−ℜSl′≥0,∀l∈M1,ℜSs−ℜSs′≤0,∀s∈M3,whereXj,Sl,Sl′,Ss, and Ss′ are LR-type PFNs.Step 3: by assuming Aij=aij;ρij,τij;ρij′,τij′LR, Xj=xj;αj,βj;αj′,βj′LR, Bi=bi;θi,ϕi;θi′,ϕi′LR, Cj=cj;ζj,ηj;ζj′,ηj′LR, Si=mi;ei,fi;gi,hiLR, and Si′=mi′;ei′,fi′;gi′,hi′LR, the FPFLPP (19) can be rewritten as(21)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(22)∑j=1nalj;ρlj,τlj;ρlj′,τlj′LR⊗xj;αj,βj;αj′,βj′LR⊕ml;el,fl;gl,hlLR=bl;θl,ϕl;θl′,ϕl′LR⊕ml′;el′,fl′;gl′,hl′LR,∀l∈M1,∑j=1nasj;ρsj,τsj;ρsj′,τsj′LR⊗xj;αj,βj;αj′,βj′LR⊕ms;es,fs;gs,hsLR=bs;θs,ϕs;θs′,ϕs′LR⊕ms′;es′,fs′;gs′,hs′LR,∀s∈M3,∑j=1narj;ρrj,τrj;ρrj′,τrj′LR⊗xj;αj,βj;αj′,βj′LR=br;θr,ϕr;θr′,ϕr′LR,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,wherexj;αj,βj;αj′,βj′LR, ml;el,fl;gl,hlLR, ml′;el′,fl′;gl′,hl′LR, ms;es,fs;gs,hsLR, and ms′;es′,fs′;gs′,hs′LR are LR-type PFN.Step 4: by using the product as discussed in Section 2 and taking aij;ρij,τij;ρij′,τij′LR⊗xj;αj,βj;αj′βj′LR=λij;δij,γij;δij′,γij′LR, the FPFLPP (21) can be written as(23)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,∑j=1nλlj;δlj,γlj;δlj′,γlj′LR⊕ml;el,fl;gl,hlLR=bl;θl,ϕl;θl′,ϕl′LR⊕ml′;el′,fl′;gl′,hl′LR,∀l∈M1,∑j=1nλsj;δsj,γsj;δsj′,γsj′LR⊕ms;es,fs;gs,hsLR=bs;θs,ϕs;θs′,ϕs′LR⊕ms′;es′,fs′;gs′,hs′LR,∀s∈M3,∑j=1nλrj;δrj,γrj;δrj′,γrj′LR=br;θr,ϕr;θr′,ϕr′LR,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,wherexj;αj,βj;αj′,βj′LR, ml;el,fl;gl,hlLR, ml′;el′,fl′;gl′,hl′LR, ms;es,fs;gs,hsLR, and ms′;es′,fs′;gs′,hs′LR are LR-type PFN.Step 5: using arithmetic operations as discussed in Section 2 and using Definition 6, the FPFLPP (23) takes the form(24)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(25)∑j=1nλlj+ml=b1+m1′,∀l∈M1,∑j=1nδlj+el=θl+el′,∀l∈M1,∑j=1nγlj+fl=ϕl+fl′,∀l∈M1,∑j=1nδlj′+gl′=θl′+gl′,∀l∈M1,∑j=1nγlj′+hl=ϕl′+hl′,∀l∈M1,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nλsj+ms=bs+ms′,∀s∈M3,∑j=1nδsj+es=θs+es′,∀s∈M3,∑j=1nγsj+fs=ϕs+fs′,∀s∈M3,∑j=1nδsj′+gs′=θs′+gs′,∀s∈M3,∑j=1nγsj′+hs=ϕs′+hs′,∀s∈M3,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,wherexj;αj,βj;αj′,βj′LR, ml;el,fl;gl,hlLR, ml′;el′,fl′;gl′,hl′LR, ms;es,fs;gs,hsLR, and ms′;es′,fs′;gs′,hs′LR are LR-type PFN.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(26)Max/Minℜ∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(27)∑j=1nλlj+ml=b1+m1′,∀l∈M1,∑j=1nδlj+el=θl+el′,∀l∈M1,∑j=1nγlj+fl=ϕl+fl′,∀l∈M1,∑j=1nδlj′+gl′=θl′+gl′,∀l∈M1,∑j=1nγlj′+hl=ϕl′+hl′,∀l∈M1,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nλsj+ms=bs+ms′,∀s∈M3,∑j=1nδsj+es=θs+es′,∀s∈M3,∑j=1nγsj+fs=ϕs+fs′,∀s∈M3,∑j=1nδsj′+gs′=θs′+gs′,∀s∈M3,∑j=1nγsj′+hs=ϕs′+hs′,∀s∈M3,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,αj≥0, βj≥0, αj′−αj≥0, βj′−βj≥0, ei≥0, fi≥0, ei′−ei≥0, and fi′−fi≥0, ∀i=1,2,…,m and ∀j=1,2,…,n.Step 7: by taking cj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR=ωj;σj,κj;σj′,κj′LR, problem (26) can be written as(28)Max/Minℜ∑j=1nωj;σj,κj;σj′,κj′LR,which subject to(29)∑j=1nλlj+ml=b1+m1′,∀l∈M1,∑j=1nδlj+el=θl+el′,∀l∈M1,∑j=1nγlj+fl=ϕl+fl′,∀l∈M1,∑j=1nδlj′+gl′=θl′+gl′,∀l∈M1,∑j=1nγlj′+hl=ϕl′+hl′,∀l∈M1,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nλsj+ms=bs+ms′,∀s∈M3,∑j=1nδsj+es=θs+es′,∀s∈M3,∑j=1nγsj+fs=ϕs+fs′,∀s∈M3,∑j=1nδsj′+gs′=θs′+gs′,∀s∈M3,∑j=1nγsj′+hs=ϕs′+hs′,∀s∈M3,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,αj≥0, βj≥0, αj′−αj≥0, βj′−βj≥0, ei≥0, fi≥0, ei′−ei≥0, and fi′−fi≥0, ∀i=1,2,…,m and ∀j=1,2,…,n.Step 8: by using the linearity property of ranking function, problem (28) takes the form(30)Max/Min∑j=1nℜωj;σj,κj;σj′,κj′LR,which subject to(31)∑j=1nλlj+ml=b1+m1′,∀l∈M1,∑j=1nδlj+el=θl+el′,∀l∈M1,∑j=1nγlj+fl=ϕl+fl′,∀l∈M1,∑j=1nδlj′+gl′=θl′+gl′,∀l∈M1,∑j=1nγlj′+hl=ϕl′+hl′,∀l∈M1,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nλsj+ms=bs+ms′,∀s∈M3,∑j=1nδsj+es=θs+es′,∀s∈M3,∑j=1nγsj+fs=ϕs+fs′,∀s∈M3,∑j=1nδsj′+gs′=θs′+gs′,∀s∈M3,∑j=1nγsj′+hs=ϕs′+hs′,∀s∈M3,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,ℜml;el,fl;gl,hlLR−ℜml′;el′,fl′;gl′,hl′LR≥0,∀l∈M1,ℜms;es,fs;gs,hsLR−ℜms′;es′,fs′;gs′,hs′LR≤0,∀s∈M3,αj≥0, βj≥0, αj′−αj≥0, βj′−βj≥0, ei≥0, fi≥0, ei′−ei≥0, and fi′−fi≥0, ∀i=1,2,…,m and ∀j=1,2,…,n.Step 9: by using Definition 7, problem (30) can be converted into(32)Max/Min∑j=1n14∫01ωj−σjL−1αdα+∫01ωj+κjR−1αdα+∫01ωj−σj′L′−1αdα+∫01ωj+κj′R′−1αdα,which subject to(33)∑j=1nλlj+ml=b1+m1′,∀l∈M1,∑j=1nδlj+el=θl+el′,∀l∈M1,∑j=1nγlj+fl=ϕl+fl′,∀l∈M1,∑j=1nδlj′+gl′=θl′+gl′,∀l∈M1,∑j=1nγlj′+hl=ϕl′+hl′,∀l∈M1,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nλsj+ms=bs+ms′,∀s∈M3,∑j=1nδsj+es=θs+es′,∀s∈M3,∑j=1nγsj+fs=ϕs+fs′,∀s∈M3,∑j=1nδsj′+gs′=θs′+gs′,∀s∈M3,∑j=1nγsj′+hs=ϕs′+hs′,∀s∈M3,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,14∫01ml−elL−1αdα+∫01ml+flR−1αdα+∫01ml−glL′−1αdα+∫01ml+hlR′−1αdα−14∫01ml′−el′L−1αdα+∫01ml′+fl′R−1αdα+∫01ml′−gl′L′−1αdα+∫01ml′+hl′R′−1αdα≥0,∀l∈M1,14∫01ms−esL−1αdα+∫01ms+fsR−1αdα+∫01ms−gsL′−1αdα+∫01ms+hsR′−1αdα−14∫01ms′−es′L−1αdα+∫01ms′+fs′R−1αdα+∫01ms′−gs′L′−1αdα+∫01ms′+hs′R′−1αdα≤0,∀l∈M3,αj≥0, βj≥0, αj′−αj≥0, βj′−βj≥0, ei≥0, fi≥0, ei′−ei≥0, and fi′−fi≥0, ∀i=1,2,…,m and ∀j=1,2,…,n.Step 10: now, solve the crisp linear/nonlinear programming problem (32) by any existing method to find the optimal solution xj∗,αj∗,βj∗,αj′∗,βj′∗.Step 11: find the LR-type Pythagorean fuzzy optimal solution Xj∗ of the FPFLPP (13) by substituting the values of xj∗, αj∗, βj∗, αj′∗, and βj′∗ in Xj∗=xj∗;αj∗,βj∗;αj′∗,βj′∗LR.Step 12: find the LR-type Pythagorean fuzzy optimal value of the FPFLPP (13) by substituting the values of Xj∗, as calculated in Step (11), in ∑j=1nCj⊗Xj.


## 3.2. 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.
AnLR-type Pythagorean fuzzy optimal solution of FPFLPP (13) with LR-type PFNs will be LR-type PFNs Xj if(1)
Xj are LR-type PFNs(2)
ℜ∑j=1nAij⊗Xj≤,=,≥ℜBi, for all i=1,2,…,m(3)
If there exist anyLR-type PFNs Xj′ satisfying step 2, then ℜ∑j=1nCj⊗Xj>ℜ∑j=1nCj⊗Xj′ in maximization problem and ℜ∑j=1nCj⊗Xj<ℜ∑j=1nCj⊗Xj′ in minimization problemStep 1: by assuming Aij=aij;ρij,τij;ρij′,τij′LR, Xj=xj;αj,βj;αj′,βj′LR, Bi=bi;θi,ϕi;θi′,ϕi′LR, and Cj=cj;ζj,ηj;ζj′,ηj′LR, the FPFLPP (15) can be rewritten as(34)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(35)∑j=1nalj;ρlj,τlj;ρlj′,τlj′LR⊗xj;αj,βj;αj′,βj′LR≼¯bl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,∑j=1nasj;ρsj,τsj;ρsj′,τsj′LR⊗xj;αj,βj;αj′,βj′LR±bs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1narj;ρrj,τrj;ρrj′,τrj′LR⊗xj;αj,βj;αj′,βj′LR=br;θr,ϕr;θr′,ϕr′LR,∀r∈M2,wherexj;αj,βj;αj′,βj′LR is an LR-type PFN.Step 2: by using the product as discussed in Section 2 and taking aij;ρij,τij;ρij′,τij′LR⊗xj;αj,βjαj′,βj′LR=λij;δij,γij;δij′,γij′LR, the FPFLPP (34) can be written as(36)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,∑j=1nλlj;δlj,γlj;δlj′,γlj′LR≼¯bl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,∑j=1nλsj;δsj,γsj;δsj′,γsj′LR≽¯bs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1nλrj;δrj,γrj;δrj′,γrj′LR=br;θr,ϕr;θr′,ϕr′LR,∀r∈M2,wherexj;αj,βj;αj′,βj′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(37)Max/Min∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(38)∑j=1nλlj;δlj,γlj;δlj′,γlj′LR≼¯bl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,∑j=1nλsj;δsj,γsj;δsj′,γsj′LR≽¯bs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,wherexj;αj,βj;αj′,βj′LR is an LR-type PFN.Step 4: by applying ranking, the FPFLPP (37) takes the form(39)Max/Minℜ∑j=1ncj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR,which subject to(40)ℜ∑j=1nλlj;δlj,γlj;δlj′,γlj′LR≤ℜbl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,ℜ∑j=1nλsj;δsj,γsj;δsj′,γsj′LR≥ℜbs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,αj≥0, βj≥0, αj′−αj≥0, and βj′−βj≥0, ∀j=1,2,…,n.Step 5: by taking cj;ζj,ηj;ζj′,ηj′LR⊗xj;αj,βj;αj′,βj′LR=ωj;σj,κj;σj′,κj′LR, problem (39) can be rewritten as(41)Max/Minℜ∑j=1nωj;σj,κj;σj′,κj′LR,which subject to(42)ℜ∑j=1nλlj;δlj,γlj;δlj′,γlj′LR≤ℜbl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,ℜ∑j=1nλsj;δsj,γsj;δsj′,γsj′LR≥ℜbs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,αj≥0, βj≥0, αj′−αj≥0, and βj′−βj≥0, ∀j=1,2,…,n.Step 6: by using the linearity of ranking function, problem (41) can be written as(43)Max/Min∑j=1nℜωj;σj,κj;σj′,κj′LR,which subject to(44)∑j=1nℜλlj;δlj,γlj;δlj′,γlj′LR≤ℜbl;θl,ϕl;θl′,ϕl′LR,∀l∈M1,∑j=1nℜλsj;δsj,γsj;δsj′,γsj′LR≥ℜbs;θs,ϕs;θs′,ϕs′LR,∀s∈M3,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,αj≥0, βj≥0, αj′−αj≥0, and βj′−βj≥0, ∀j=1,2,…,n.Step 7: by using Definition 7, problem (43) can be converted into problem (45):(45)Max/Min∑j=1n14∫01ωj−σjL−1αdα+∫01ωj+κjR−1αdα+∫01ωj−σj′L′−1αdα+∫01ωj+κj′R′−1αdα,which subject to(46)∑j=1n14∫01λlj−δljL−1αdα+∫01λlj+γljR−1αdα+∫01λlj−δlj′L′−1αdα+∫01λlj+γlj′R′−1αdα≤14∫01bl−θlL−1αdα+∫01bl+ϕlR−1αdα+∫01bl−θl′L′−1αdα+∫01bl+ϕlR′−1αdα,∀l∈M1,∑j=1n14∫01λsj−δsjL−1αdα+∫01λsj+γsjR−1αdα+∫01λsj−δsj′L′−1αdα+∫01λsj+γsj′R′−1αdα≥14∫01bs−θsL−1αdα+∫01bs+ϕsR−1αdα+∫01bs−θs′L′−1αdα+∫01bs+ϕsR′−1αdα,∀s∈M3,∑j=1nλrj=br,∀r∈M2,∑j=1nδrj=θr,∀r∈M2,∑j=1nγrj′=ϕr′,∀r∈M2,∑j=1nγrj=ϕr,∀r∈M2,∑j=1nδrj′=θr′,∀r∈M2,αj≥0, βj≥0, αj′−αj≥0, and βj′−βj≥0, ∀j=1,2,…,n.Step 8: solve the crisp linear/nonlinear programming problem (45) by any existing method to find the optimal solution xj∗,αj∗,βj∗,αj′∗,βj′∗.Step 9: find the LR-type Pythagorean fuzzy optimal solution Xj∗ of the FPFLPP (13) by substituting the values of xj∗, αj∗, βj∗, αj′∗, and βj′∗ in Xj∗=xj∗;αj∗,βj∗;αj′∗,βj′∗LR.Step 10: find the LR-type Pythagorean fuzzy optimal value of the FPFLPP (13) by substituting the values of Xj∗, as calculated in Step (9), in ∑j=1nCj⊗Xj.


## 4. Equivalence of the Proposed Methods

Here, we confirm that the two techniques proposed inSection 3.1 and Section 3.2 give the same solution.IfA1 and A2 are any two PFNs such that A1=A2, then ℜA=ℜB. Thus, the 1st and 2nd constraints [∑j=1nAlj⊗Xj⊕Sl=Bl⊕Sl′,∀l∈M1, ∑j=1nAsj⊗Xj⊕Ss=Bs⊕Ss′,∀s∈M3] of Problem (19) can be written as(47)ℜ∑j=1nAlj⊗Xj⊕Sl=ℜBl⊕Sl′,∀l∈M1,(48)ℜ∑j=1nAsj⊗Xj⊕Ss=ℜBs⊕Ss′,∀s∈M3.Since the ranking function as discussed in Definition7 is linear, so equations (47) and (48) can be written as(49)ℜ∑j=1nAlj⊗Xj+ℜSl=ℜBl+ℜSl′,∀l∈M1,(50)ℜ∑j=1nAsj⊗Xj+ℜSs=ℜBsℜSs′,∀s∈M3.Thus, equations (49) and (50) can be written as(51)ℜBl−ℜ∑j=1nAlj⊗Xj=ℜSl−ℜSl′,∀l∈M1,(52)ℜBs−ℜ∑j=1nAsj⊗Xj=ℜSs−ℜSs′,∀s∈M3.Now, by using the4th and 5th constraints [ℜSl−ℜSl′≥0,ℜSs−ℜSs′≤0] of problem (19), equations (51) and (52) convert to(53)ℜBl−ℜ∑j=1nAlj⊗Xj≥0,∀l∈M1,ℜBs−ℜ∑j=1nAsj⊗Xj≤0,∀s∈M3,or(54)ℜ∑j=1nAlj⊗Xj≤ℜBl,∀l∈M1,ℜ∑j=1nAsj⊗Xj≥ℜBs,∀s∈M3.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.


## 5. Numerical Examples

Example 1.
A farmer has45;41,80;44,138LR square-feet land. He wants to grow two types of plants, namely, X and Y. Each X plant needs 5;3,2;3,4LR square-feet of land and 2;1,1;2,3LR man-per-hour labor. Each Y plant needs 6;5,4;5,6LR square-feet of land and 3;2,3;3,3LR man-per-hour labor. Maximum labor which is available is 21;18,48;21,74LR man-per-hour. Profit for each X plant is 4;1,2;3,4LR and for each Y plant is 3;2,3;3,5LR. Farmer wants to maximize his profit subject to give available resources with Lx=Rx=max0,1−x3 and L′x=R′x=min1,x2.
LetX1 be the number of X plants and X2 be the number of Y plants that farmer should grow. Then, the problem converts to the following LR-type FPFLPP:(55)Max4;1,2;3,4LR⊗X1⊕3;2,3;3,5LR⊗X2, which subject to(56)2;1,1;2,3LR⊗X1⊕3;2,3;3,3LR⊗X2≼¯21;18,48;21,74LR,5;3,2;3,4LR⊗X1⊕6;5,4;5,6LR⊗X2=45;41,80;44,138LR,where Xj are LR-type PFNs for j=1,2 and Lx=Rx=max0,1−x3 and L′x=R′x=min1,x2.
Now, we solve Example1 by using method 1 as discussed in Section 3.1.Step 1: by applying Step 2 of the presented method 1 inSection 3.1, the problem becomes(57)Max4;1,2;3,4LR⊗X1⊕3;2,3;3,5LR⊗X2,which subject to(58)2;1,1;2,3LR⊗X1⊕3;2,3;3,3LR⊗X2⊕S=21;18,48;21,74LR⊕S′,5;3,2;3,4LR⊗X1⊕6;5,4;5,6LR⊗X2=45;41,80;44,138LR,ℜS−ℜS′≥0, where Xj,S, and S′ are LR-type PFNs for j=1,2 and Lx=Rx=max0,1−x3 and L′x=R′x=min1,x2.Step 2: letX1=x1;α1,β1;α1′,β1′LR, X2=x2;α2,β2;α2′,β2′LR, S=m;e,f;g,hLR, and S′=m′;e′,f′;g′,h′LR, and the problem obtained in Step (1) can be written as(59)Max4;1,2;3,4LR⊗x1;α1,β1;α1′,β1′LR⊕3;2,3;3,5LR⊗x2;α2,β2;α2′,β2′LR,which subject to(60)2;1,1;2,3LR⊗x1;α1,β1;α1′,β1′LR⊕3;2,3;3,3LR⊗x2;α2,β2;α2′,β2′LR⊕m;e,f;g,hLR=21;18,48;21,74LR⊕m′;e′,f′;g′,h′LR,5;3,2;3,4LR⊗x1;α1,β1;α1′,β1′LR⊕6;5,4;5,6LR⊗x2;α2,β2;α2′,β2′LR=45;41,80;44,138LR,ℜm;e,f;g,hLR−ℜm′;e′,f′;g′,h′LR≥0,wherex1;α1,β1;α1′,β1′LR, x2;α2,β2;α2′,β2′LR, m;e,f;g,hLR, and m′;e′,f′;g′,h′LR are LR-type PFNs.Step 3: using the product as discussed inSection 2, the FPFLPP, obtained in Step 2, can be written as(61)Max4x1;4x1−min3x1−3α1,6x1−6α1,max6x1+6β1,3x1+3β1−4x1;4x1−minx1−α1′,8x1−8α1′,max8x1+8β1′,x1+β1′−4x1LR⊕3x2;3x2−minx2−α2,6x2−6α2,max6x2+6β2,x2+β2−3x2;3x2−min0,8x2−8α2′,max8x2+8β2′,0−3x2LR,which subject to(62)2x1;2x1−minx1−α1,3x1−3α1,max3x1+3β1,x1+β1−2x1;2x1−min0,5x1−5α1′,max5x1+5β1′,0−2x1LR⊕3x2;3x2−minx2−α2,6x2−6α2,max6x2+6β2,x2+β2−3x2;3x2−min0,6x2−6α2′,max6x2+6β2′,0−3x2LR⊕m;e,f;g,hLR=21;18,48;21,74LR⊕m′;e′,f′;g′,h′LR,5x1;5x1−min2x1−2α1,7x1−7α1,max7x1+7β1,2x1+2β1−5x1;5x1−min2x1−2α1′,9x1−9α1′,max9x1+9β1′,2x1+2β1′−5x1LR⊕6x2;6x2−minx2−α2,10x2−10α2,max10x2+10β2,x2+β2−6x2;6x2−minx2−α2′,12x2−12α2′,max12x2+12β2′,x2+β2′−6x2LR=45;41,80;44,138LR,ℜm;e,f;g,hLR−ℜm′;e′,f′;g′,h′LR≥0,wherex1;α1,β1;α1′,β1′LR, x2;α2,β2;α2′,β2′LR, m;e,f;g,hLR, and m′;e′,f′;g′,h′LR are LR-type PFNs.Step 4: by using arithmetic operations discussed inSection 2 and using Definition 6, the FPFLPP, obtained in Step 3, can be rewritten as(63)Max4x1+3x2;4x1−min3x1−3α1,6x1−6α1+3x2−minx2−α2,6x2−6α2,max6x1+6β1,3x1+3β1−4x1+max6x2+6β2,x2+β2−3x2;4x1−minx1−α1′,8x1−8α1′+3x2−min0,8x2−8α2′,max8x1+8β1′,x1+β1′−4x1+max8x2+8β2′,0−3x2LR,which subject to(64)2x1+3x2+m=21+m′,5x1+6x2=45,2x1−minx1−α1,3x1−3α1+3x2−minx2−α2,6x2−6α2+e=18+e′,max3x1+3β1,x1+β1−2x1+max6x2+6β2,x2+β2−3x2+f=48+f′,2x1−min0,5x1−5α1′+3x2−min0,6x2−6α2′+g=21+g′,max5x1+5β1′,0−2x1+max6x2+6β2′,0−3x2+h=74+h′,5x1−min2x1−2α1,7x1−7α1+6x2−minx2−α2,10x2−10α2=41,max7x1+7β1,2x1+2β1−5x1+max10x2+10β2,x2+β2−6x2=80,5x1−min2x1−2α1′,9x1−9α1′+6x2−minx2−α2′,12x2−12α2′=44,max9x1+9β1′,2x1+2β1′−5x1+max12x2+12β2′,x2+β2′−6x2=138,ℜm;e,f;g,hLR−ℜm′;e′,f′;g′,h′LR≥0, where x1;α1,β1;α1′,β1′LR, x2;α2,β2;α2′,β2′LR, m;e,f;g,hLR, and m′;e′,f′;g′,h′LR are LR-type PFNs.Step 5: using Step 6 of the proposed method 1, the FPFLPP, obtained in Step 4, can be rewritten as(65)Maxℜ4x1+3x2;4x1−min3x1−3α1,6x1−6α1+3x2−minx2−α2,6x2−6α2,max6x1+6β1,3x1+3β1−4x1+max6x2+6β2,x2+β2−3x2;4x1−minx1−α1′,8x1−8α1′+3x2−min0,8x2−8α2′,max8x1+8β1′,x1+β1′−4x1+max8x2+8β2′,0−3x2LR,which subject to(66)2x1+3x2+m=21+m′,5x1+6x2=45,2x1−minx1−α1,3x1−3α1+3x2−minx2−α2,6x2−6α2+e=18+e′,max3x1+3β1,x1+β1−2x1+max6x2+6β2,x2+β2−3x2+f=48+f′,2x1−min0,5x1−5α1′+3x2−min0,6x2−6α2′+g=21+g′,max5x1+5β1′,0−2x1+max6x2+6β2′,0−3x2+h=74+h′,5x1−min2x1−2α1,7x1−7α1+6x2−minx2−α2,10x2−10α2=41,max7x1+7β1,2x1+2β1−5x1+max10x2+10β2,x2+β2−6x2=80,5x1−min2x1−2α1′,9x1−9α1′+6x2−minx2−α2′,12x2−12α2′=44,max9x1+9β1′,2x1+2β1′−5x1+max12x2+12β2′,x2+β2′−6x2=138,ℜm;e,f;g,hLR−ℜm′;e′,f′;g′,h′LR≥0,α1≥0,β1≥0,α1′−α1≥0,β1′−β1≥0,α2≥0,β2≥0,α2′−α2≥0,β2′−β2≥0,e≥0,f≥0,g−e≥0,h−f≥0,e′≥0,f′≥0,g′−e′≥0,andh′−f′≥0.Step 6: usingmina,b=a+b/2−a−b/2 and maxa,b=a+b/2+a−b/2 and Steps 8 and 9 of method 1, presented in Section 3.1, the FPFLPP, obtained in Step 5, can be written as(67)Max6512x1+6112x2−278α1−98x1−α1−218α2−158x2−α2+278β1+98x1+β1+218β2+158x2+β2−3α1′−73x1−α1′−83α2′−83x2−α2′+3β1′+73x1+β1′+83β2′+83x2+β2′,which subject to(68)2x1+3x2+m=21+m′,5x1+6x2=45,2α1+x1−α1−12x2+72α2+52x2−α2+e=18+e′,2β1+x1+β1+12x2+72β2+52x2+β2+f=48+f′,−12x1+52α1′+52x1−α1′+3α2′+3x2−α2′+g=21+g′,12x1+52β1′+52x1+β1′+3β2′+3x2+β2′+h=74+h′,12x1+92α1+52x1−α1+12x2+112α2+92x2−α2=41,−12x1+92β1+52x1+β1−12x2+112β2+92x2+β2=80,−12x1+112α1′+72x1−α1′−12x2+132α2′+112x2−α2′=44,12x1+112β1′+72x1+β1′+12x2+132β2′+112x2+β2′=138,8m−e+f−g+h−8m′+e′−f′+g′−h′≥0,α1≥0,β1≥0,α1′−α1≥0,β1′−β1≥0,α2≥0,β2≥0,α2′−α2≥0,β2′−β2≥0,e≥0,f≥0, g−e≥0,h−f≥0,e′≥0,f′≥0,g′−e′≥0,andh′−f′≥0.Step 7: the optimal solution of the crisp nonlinear programming problem, obtained in Step 5(using: MATLAB R2014a, solver “fmincon,” algorithm “interior point,” TolFun = eps, TolX = eps, TolCon = 0.000001) isx1=9.0001, α1=0.001, β1=8.857, α1′=0.1011, β1′=11.3333, x2=−0.0001, α2=1.3996, β2=0.0001, α2′=1.3997, and β2′=0.0002.Step 8: substituting the values ofx1,α1,β1,α1′,β1′,x2,α2,β2,α2′, and β2′ in X1=x1;α1,β1;α1′,β1′LR and X2=x2;α2,β2;α2′,β2′LR, the exact LR-type Pythagorean fuzzy optimal solution is X1=9.0001;0.001,8.8570.1011,11.3333LR, and X2=−0.0001;1.3996,0.0001;1.3997,0.0002LR.Step 9: substituting the values ofX1 and X2, obtained in Step 8, into the objective function, the LR-type Pythagorean fuzzy optimal value is 36.0001;9.0034,75.3404;38.2995,126.6415LR. So, the farmer should grow 9.0001;0.001,8.857;0.1011,11.3333LR number of plants of X and −0.0001;1.3996,0.0001;1.3997,0.0002LR number of plants of Y to get a maximum profit of 36.0001;9.0034,75.3404;38.2995,126.6415LR.
We now solve Example1 by using method 2 as discussed in Section 3.2:Step 1: letX1=x1;α1,β1;α1′,β1′LR and X2=x2;α2,β2;α2′,β2′LR; then, problem of Example 1 can be written as(69)Max4;1,2;3,4LR⊗x1;α1,β1;α1′,β1′LR⊕3;2,3;3,5LR⊗x2;α2,β2;α2′,β2′LR,which subject to(70)2;1,1;2,3LR⊗x1;α1,β1;α1′,β1′LR⊕3;2,3;3,3LR⊗x2;α2,β2;α2′,β2′LR≼¯21;18,48;21,74LR,5;3,2;3,4LR⊗x1;α1,β1;α1′,β1′LR⊕6;5,4;5,6LR⊗x2;α2,β2;α2′,β2′LR=45;41,80;44,138LR,wherex1;α1,β1;α1′,β1′LR and x2;α2,β2;α2′,β2′LR are LR-type PFNs.Step 2: using the product as discussed inSection 2, the FPFLPP, obtained in Step 1, can be written as(71)Max4x1;4x1−min3x1−3α1,6x1−6α1,max6x1+6β1,3x1+3β1−4x1;4x1−minx1−α1′,8x1−8α1′,max8x1+8β1′,x1+β1′−4x1LR⊕3x2;3x2−minx2−α2,6x2−6α2,max6x2+6β2,x2+β2−3x2;3x2−min0,8x2−8α2′,max8x2+8β2′,0−3x2LR,which subject to(72)2x1;2x1−minx1−α1,3x1−3α1,max3x1+3β1,x1+β1−2x1;2x1−min0,5x1−5α1′,max5x1+5β1′,0−2x1LR⊕3x2;3x2−minx2−α2,6x2−6α2,max6x2+6β2,x2+β2−3x2;3x2−min0,6x2−6α2′,max6x2+6β2′,0−3x2LR≼¯21;18,48;21,74LR,5x1;5x1−min2x1−2α1,7x1−7α1,max7x1+7β1,2x1+2β1−5x1;5x1−min2x1−2α1′,9x1−9α1′,max9x1+9β1′,2x1+2β1′−5x1LR⊕6x2;6x2−minx2−α2,10x2−10α2,max10x2+10β2,x2+β2−6x2;6x2−minx2−α2′,12x2−12α2′,max12x2+12β2′,x2+β2′−6x2LR=45;41,80;44,138LR,wherex1;α1,β1;α1′,β1′LR and x2;α2,β2;α2′,β2′LR are LR-type PFNs.Step 3: by using arithmetic operations discussed inSection 2 and Definition 6, the FPFLPP, obtained in Step 2, can be rewritten as(73)Max4x1+3x2;4x1−min3x1−3α1,6x1−6α1+3x2−minx2−α2,6x2−6α2,max6x1+6β1,3x1+3β1−4x1+max6x2+6β2,x2+β2−3x2;4x1−minx1−α1′,8x1−8α1′+3x2−min0,8x2−8α2′,max8x1+8β1′,x1+β1′−4x1+max8x2+8β2′,0−3x2LR,which subject to(74)2x1+3x2;2x1−minx1−α1,3x1−3α1+3x2−minx2−α2,6x2−6α2,max3x1+3β1,x1+β1−2x1+max6x2+6β2,x2+β2−3x2;2x1−min0,5x1−5α1′+3x2−min0,6x2−6α2′,max5x1+5β1′,0−2x1+max6x2+6β2′,0−3x2LR≼¯21;18,48;21,74LR,5x1−min2x1−2α1,7x1−7α1+6x2−minx2−α2,10x2−10α2=41,max7x1+7β1,2x1+2β1−5x1+max10x2+10β2,x2+β2−6x2=80,5x1−min2x1−2α1′,9x1−9α1′+6x2−minx2−α2′,12x2−12α2′=44,max9x1+9β1′,2x1+2β1′−5x1+max12x2+12β2′,x2+β2′−6x2=138,5x1+6x2=45, where x1;α1,β1;α1′,β1′LR and x2;α2,β2;α2′,β2′LR are LR-type PFNs.Step 4: using Step 4 of the proposed method 2, the FPFLPP, obtained in Step 3, can be rewritten as(75)Maxℜ4x1+3x2;4x1−min3x1−3α1,6x1−6α1+3x2−minx2−α2,6x2−6α2,max6x1+6β1,3x1+3β1−4x1+max6x2+6β2,x2+β2−3x2,4x1−minx1−α1′,8x1−8α1′+3x2−min0,8x2−8α2′,max8x1+8β1′,x1+β1′−4x1+max8x2+8β2′,0−3x2LR,which subject to(76)ℜ2x1+3x2;2x1−minx1−α1,3x1−3α1+3x2−minx2−α2,6x2−6α2,max3x1+3β1,x1+β1−2x1+max6x2+6β2,x2+β2−3x2;2x1−min0,5x1−5α1′+3x2−min0,6x2−6α2′,max5x1+5β1′,0−2x1+max6x2+6β2′,0−3x2LR≤ℜ21;18,48;21,74LR,5x1−min2x1−2α1,7x1−7α1+6x2−minx2−α2,10x2−10α2=41,max7x1+7β1,2x1+2β1−5x1+max10x2+10β2,x2+β2−6x2=80,5x1−min2x1−2α1′,9x1−9α1′+6x2−minx2−α2′,12x2−12α2′=44,max9x1+9β1′,2x1+2β1′−5x1+max12x2+12β2′,x2+β2′−6x2=138,5x1+6x2=45,α1≥0,β1≥0,α1′−α1≥0,β1′−β1≥0,α2≥0,β2≥0,α2′−α2≥0,andβ2′−β2≥0.Step 5: using mina,b=a+b/2−a−b/2 and maxa,b=a+b/2+a−b/2 and Steps 6 and 7 of the presented method 2, the FPFLPP, obtained in Step 4, can be written as(77)Max6512x1+6112x2−278α1−98x1−α1−218α2−158x2−α2+278β1+98x1+β1+218β2+158x2+β2−3α1′−73x1−α1′−83α2′−83x2−α2′+3β1′+73x1+β1′+83β2′+83x2+β2′,which subject to(78)83x1+154x2−32α1−34x1−α1−218α2−158x2−α2+32β1+34x1+β1+218β2+158x2+β2−53α1′−53x1−α1′−2α2′−2x2−α2′+53β1′+53x1+β1′+2β2′+2x2+β2′≤35.458312x1+92α1+52x1−α1+12x2+112α2+92x2−α2=41,−12x1+92β1+52x1+β1−12x2+112β2+92x2+β2=80,−12x1+112α1′+72x1−α1′−12x2+132α2′+112x2−α2′=44,12x1+112β1′+72|x1+β1′|+12x2+132β2′+112|x2+β2′|=138,5x1+6x2=45,α1≥0,β1≥0,α1′−α1≥0,β1′−β1≥0,α2≥0,β2≥0,α2′−α2≥0,andβ2′−β2≥0.Step 6: the optimal solution of the crisp nonlinear programming problem, obtained in Step 5(using MATLAB R2014a, solver “fmincon,” algorithm “interior point,” TolFun = 1, TolX = eps, TolCon = 1), is x1=9, α1=0, β1=8.8, α1′=0, β1′=11.33, x2=0, α2=1.3, β2=0, α2′=1.3, and β2′=0.Step 7: substituting the values of x1,α1,β1,α1′,β1′,x2,α2,β2,α2′, and β2′ in X1=x1;α1,β1;α1′,β1′LR and X2=x2;α2,β2;α2′,β2′LR, the exact LR-type Pythagorean fuzzy optimal solution is X1=9;0,8.8;0,11.33LR and X2=0;1.3,0;1.3,0LR.Step 8: substituting the values of X1 and X2, obtained in Step 7 into the objective function, the LR-type Pythagorean fuzzy optimal value is 36;16.8,70.8;37.4,126.64LR.
So, according to this technique, the farmer should grow9;0,8.8;0,11.33LR number of plants of X and 0;1.3,0;1.3,0LR number of plants of Y to get a maximum profit of 36;16.8,70.8;37.4,126.64LR.Example 2.
Let us solve the practical model, discussed in [15], by method 1 as discussed in Section 3.1, using Lx=Rx=max0,1−x and L′x=R′x=min1,x:(79)Max8;2,2;52,52LR⊗X1⊕12;2,2;52,52LR⊗X2⊕1;14,14;12,12LR⊗X3,which subject to(80)5;12,12;34,34LR⊗X1⊕5;12,12;34,34LR⊗X2⊕X3=155;45,52;60,70LR,6;14,14;12,12LR⊗X1⊕2;14,14;12,12LR⊗X2≼¯125;18,22;30,35LR,1;12,14;34,12LR⊗X1⊕4;14,12;12,14LR⊗X2≼¯110;42,38;65,60LR,where Xj are LR-type PFNs for j=1,2,3.
LetX1=x1;α1,β1;α1′,β1′LR, X2=x2;α2,β2;α2′,β2′LR,X3=x3;α3,β3;α3′,β3′LR, S1=m1;e1,f1;g1,h1LR, S2=m2;e2,f2;g2,h2LR, S1′=m1′;e1′,f1′;g1′,h1′LR, and S2′=m2′;e2′,f2′;g2′,h2′LR. By solving step by step as discussed in Section 3.1, we obtain the crisp problem as given under(81)Max8x1+12x2+x3−α1−14x1−α1−32α2−14x2−α2−18α3−132x3−α3+β1+14x1+β1+32β2+14x2+β2+18β3+132x3+β3−α1′−516x1−α1′−32α2′−516x2−α2′−18α3′−116x3−α3′+β1′+516x1+β1′+32β2′+516x2+β2′+12β3′+116x3+β3′,which subject to(82)5x1+5x2+x3=155,6x1+2x2+m1=125+m1′,x1+4x2+m2=110+m2′,5α1+12x1−α1+5α2+12x2−α2+α3=45,5β1+12x1+β1+5β2+12x2+β2+β3=52,5α1′+34x1−α1′+5α2′+34x2−α2′+α3′=60,5β1′+34x1+β1′+5β2′+34x2+β2′+β3′=70,6α1+14x1−α1+2α2+14x2−α2+e1=18+e1′,6β1+14x1+β1+2β2+14x2+β2+f1=22+f1′,6α1′+12x1−α1′+2α2′+12x2−α2′+g1=30+g1′,6β1′+12x1+β1′+2β2′+12x2+β2′+h1=35+h1′,18x1+78α1+38x1−α1−18x2+338α2+38x2−α2+e2=42+e2′,−18x1+78β1+38x1+β1+18x2+338β2+38x2+β2+f2=38+f2′,18x1+78α1′+58x1α1′−18x2+338α2′+58x2−α2′+g2=65+g2′,−18x1+78β1′+58x1+β1′+18x2+338β2′+58x2+β2′+h2=60+h2′,8m1−e1+f1−g1+h1−8m1′+e1′−f1′+g1′−h1′≥0, 8m2−e2+f2−g2+h2−8m2′+e2′−f2′+g2′−h2′≥0, α1≥0,β1≥0,α1′−α1≥0,β1′−β1≥0,α2≥0,β2≥0,α2′−α2≥0,β2′−β2≥0,e1≥0, f1≥0,g1−e1≥0,h1−f1≥0,e1′≥0,f1′≥0,g1′−e1′≥0,h1′−f1′≥0,e2≥0,f2≥0, g2−e2≥0,h2−f2≥0,e2′≥0,f2′≥0,g2′−e2′≥0,andh2′−f2′≥0.
The optimal solution of this nonlinear problem (using MATLAB R2014a, solver “fmincon,” algorithm “interior point,” TolFun = 0.1, TolX = eps, TolCon = 0.1) isX1=12.9809,0.0332,0.0866;0.0749,0.1206LR, X2=23.8686;4.4142,3.0225;5.8714,4.3456LR, and X3=−29.2479;6.5785,16.4640;7.0914,16.6821LR. The optimal value of this problem is 368.27;117.03,109.26;159.43,146.96LR.


## 6. Comparison with Existing Linear Programming Model

Pe´rez-Canˇedo 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 Lx=Rx=max0,1−x and L′x=R′x=min1,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 withLR-type PFNs as variables and parameters.(2)
Example1 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)
The 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. The 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.Table 1
Solution of Example1.
X1X2Optimal valueUsing proposed method 19.0001;0.0010,8.8570;0.1011,11.3333LR−0.0001;1.3996,0.0001;1.3997,0.0002LR36.0001;9.0034,75.3404;38.2995,126.6415LRUsing proposed method 29;0,8.8;0,11.33LR0;1.3,0;1.3,0LR36;16.8,70.8;37.4,126.64LRFigure 1
Comparison of the Pythagorean fuzzy optimal value using proposed methods (method 1 as discussed inSection 3.1 = red and method 2 as discussed in Section 3.2 = blue).Table 2
Solution of Example2.
X1X2X3Optimal valueUsing proposed method 112.9809;0.0332,0.0866;0.0749,0.1206LR23.8686;4.4142,3.0225;5.8714,4.3456LR−29.2479;6.5785,16.4640;7.0914,16.6821LR361.08;133.5743,136.536;173.5292,169.364LRUsing existing method [15]13.03;0,0;0,0LR24.87;0,0;0,1.21LR−34.58;26.04,33.04;31.56,34.58LR368.27;117.03,109.26;159.43,146.96LRFigure 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.


## 7. Merits of the Proposed Methods

The proposed mathematical model is based on the Pythagorean fuzzy environment. The advantages of the proposed method as compared to the existing method are as follows:(1)
There is no method to solve FPFLPP in which all the variables and parameters are unrestrictedLR-type PFNs. Thus, 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 degreeμ and nonmembership degree ν of an element exceed 1. So, these techniques are more general and can be used in an intuitionistic fuzzy environment or fuzzy environment.(3)
The 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 conditionμ2+ν2>1.


## 8. Conclusions and Future Directions

In mathematical programming models, linear programming problems are the simplest and most extensively used model. The linear programming model is easily applicable to various real-life applications. In this article, we have studied two techniques to solveLR-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.



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