LR -type fully Pythagorean fuzzy linear programming problems with equality constraints

1 Introduction

Fuzzy set theory [57] deals with the uncertain or vague information. A fuzzy set is characterized by membership function which assigns grades to the elements of fuzzy set. Atanassov [10] proposed the concept of intuitionistic fuzzy set(IFS) to deal with uncertain information. There is a non-membership grade along with the membership grade in IFS but sum of both membership and non-membership grades must not exceed 1. Thus, IFS restricts the decision maker to some extent. A Pythagorean fuzzy set(PFS), discussed by Yager [54], is a generalization of IFS and allows the decision maker to assign membership grades such that the sum of squares of both membership and non-membership must not greater than 1. This shows that PFS is more flexible towards the uncertain information and is more powerful to deal with such kind of information.

Linear programming is a technique in which a function (called objective function) is optimized subject to a given set of restrictions (called constraints). Nowadays, linear programming has become a part and parcel of every industry. The planar is interested to maximize their profit and minimize the cost subject to availability of the resources. Thus, linear programming has become a very vital theory in many real life problems like airline scheduling, telecommunication network, industry and many more. Fuzzy linear programming became an interesting field for many researchers around the globe. The concept of decision making in fuzzy environment was introduced by Bellman and Zadeh [12]. Tanaka et al. [47] considered a method to solve fuzzy mathematical programming problems. Zimmerman [61] developed a method to solve linear programming problem with multi objective function. There are number of methods available in the literature to solve fuzzy linear programming problems in which some or all of the parameters and/or variables are fuzzy numbers or LR-type fuzzy numbers. Kaur and Kumar [21] presented a method to find exact fuzzy optimal solution of fully fuzzy linear programming problems with unrestricted fuzzy variables. Kumar et al. [23] studied a new method for solving fully fuzzy linear programming problems. Kaur and Kumar [22] discussed Mehar’s method for solving fully fuzzy linear programming problems with LR fuzzy parameters. Najafi and Edalatpanah [33] introduced a new method for solving fully fuzzy linear programming problems. Najafi et al. [34] presented a nonlinear model for fully fuzzy linear programming with fully unrestricted variables and parameters. Pérez-Ca n ˇ edo et al. [41] described a revised version of a lexicographical-based method for solving fully fuzzy linear programming problems with inequality constraints. For further terminologies, one may consult [11, 30].

To extend or explore the fuzzy set, Atanassov [10] introduced the concept of fuzzy set to intuitionistic fuzzy set in which there is a non-membership function along with the membership function. Serval researchers have studied certain techniques to solve linear programming problems in an intuitionistic fuzzy environment. Angelov [9] proposed a technique for optimization in an intuitionistic fuzzy environment. Nagoorgani and Ponnalagu [31] discussed an approach for solving intuitionistic fuzzy linear programming problems. Nagoorgani et al. [32] presented a method for the knowledge of expert opinion in intuitionistic fuzzy linear programming problems. Parvathi and Malathi [35] considered another method for intuitionistic fuzzy linear programming problems. Suresh et al. [46] suggested a new method for solving intuitionistic fuzzy linear programming problems by ranking function. Singh and Yadav [45] presented a method to solve LR-type intuitionistic fuzzy linear programming problems. Perez-Canedo and Concepcion-Morales [40] extended the method presented by Singh and Yadav [45] to inequality constraints. Abhishekh and Nishad [1] discussed a novel approach for solving fully LR intuitionistic fuzzy transportation problem. Recently, Akram et al. [5] introduced a method to solve fully Pythagorean fuzzy linear programming problems with equality constraints. Gou et al. [18] presented the properties of continuous Pythagorean fuzzy information. Ren [42] discussed Pythagorean fuzzy TODIM approach to multi-criteria decision making. Wu et al. [50] presented an integrated approach to green supplier selection based on the interval type-2 fuzzy best-worst and extended VIKOR methods. Wu et al. [51] considered a linguistic distribution behavioral multi-criteria group decision making model integrating extended generalized TODIM and quantum decision theory. Wu et al. [52] presented an enhancing multiple attribute group decision making flexibility based on information fusion technique and hesitant Pythagorean fuzzy sets. Zhang et al. [58] described consensus and opinion evolution-based failure mode and effect analysis approach for reliability management in social network and uncertainty contexts. Zhang et al. [59] gave an overview on feedback mechanisms with minimum adjustment or cost in consensus reaching in group decision making: research paradigms and challenges. Kumar et al. [24] presented a Pythagorean fuzzy transportation problem. Wan et al. [48] discussed a Pythagorean fuzzy mathematical programming method for multi-attribute group decision making. Chen [14] considered a Pythagorean fuzzy linear programming technique for multi-dimensional analysis of preference using a squared-distance-based approach for multiple criteria decision analysis.

All the existing methods are applicable to those linear programming problems in which some/all the parameters and variables are non-negative/unrestricted crisp numbers, fuzzy numbers, LR fuzzy numbers, intuitionistic fuzzy numbers and LR intuitionistic fuzzy numbers. The method discussed in this paper deals with the problems in which all the parameters and variables are unrestricted LR-type Pythagorean fuzzy numbers(PFNs) and can also be used to solve almost all the problems which can be solved by existing methods.

In this paper, we define an LR-type PFN and propose a technique to solve LR-type fully Pythagorean fuzzy linear programming problems (FPFLPP) with equality constraints using ranking function. The motivation of this article depends on the following facts:

We present the concept of LR-type PFN and arithmetic operations of LR-type PFNs using α-cut and β-cut.

The list of acronyms used in research paper is given in Table 1.

For further information, the readers are referred to [2–4, 55].

Definition 2.1. [56] Let X be a fixed set. A PFS A on X is an object of the form: A = { 〈 x , μ A ( x ) , ν A ( x ) 〉 | x ∈ X } , characterized by a membership function ( μ A ) and a non-membership function ( ν A ) , where μ A : X → [ 0 , 1 ] , x ∈ X → μ A ( x ) ∈ [ 0 , 1 ] , ν A : X → [ 0 , 1 ] , x ∈ X → ν A ( x ) ∈ [ 0 , 1 ] such that 0 ≤ μ A 2 ( x ) + ν A 2 ( x ) ≤ 1 for all x ∈ X . Moreover, for all x ∈ X, π A ( x ) = 1 - μ A 2 ( x ) - ν A 2 ( x ) is called a Pythagorean fuzzy index or degree of hesitancy of x in A . For computational convenience, A = ( μ A , ν A ) is called a PFN [60]. Now we define basic concepts based on [45].

Definition 2.2. Let A be a PFS in X, then its α-cut and β-cut are defined as A ^α  = {x ∈ X : μ _A  (x) ≥ α} and A ^β  = {x ∈ X : ν _A  (x) ≤ β} , ∀α, β ∈ [0, 1] .

Definition 2.3. A PFN A = (a ; l, r ; l′, r′)  _LR is defined as an LR-type PFN, if its membership (μ _A ) and non-membership (ν _A ) functions are given as: μ A ( x ) = { L ( a - x l ) , x ≤ a , l > 0 , R ( x - a r ) , x ≥ a , r > 0 , and ν A ( x ) = { L ′ ( a - x l ′ ) , x ≤ a , l ′ > 0 , R ′ ( x - a r ′ ) , x ≥ a , r ′ > 0 , where l ≤ l′,r ≤ r′ and 0 ≤ μ A 2 + ν A 2 ≤ 1 . L and R are monotone, continuous, decreasing functions from [0, ∞) to [0, 1] and L′ and R′ are monotone, continuous, increasing functions from [0, ∞) to [0, 1] such that

L (0) = R (0) =1,

Note: In this paper, we will discuss only those LR-type PFNs in which lim x → 1 R ( x ) = lim x → 1 L ( x ) = 0 , and lim x → 1 R ′ ( x ) = lim x → 1 L ′ ( x ) = 1 .

Example 2.1. Consider an LR-type PFN (50 ; 20, 25 ; 30, 35)  _LR with L (x) = R (x) = max {0, 1 - x³}, L′ (x) = R′ (x) = min {1, x²} . This LR-type PFN is shown graphically in Fig. 1.

OPEN IN VIEWER

Fig. 1

LR-type PFN with L (x) = R (x) = max {0, 1 - x³}, L′ (x) = R′ (x) = min {1, x²} .

Remark 2.1. (1) If we set L′ (x) =1 - L (x) and R′ (x) =1 - R (x) in Definition 2.3, then A = (a ; l, r ; l′, r′)  _LR becomes LR-type intuitionistic fuzzy number [45].

(2) If we take L ( x ) = R ( x ) = { 1 - x , 0 ≤ x ≤ 1 , 0 , otherwise , and L ′ ( x ) = R ′ ( x ) = { x , 0 ≤ x ≤ 1 , 1 , otherwise , in Definition 2.3, then A = (a ; l, r ; l′, r′)  _LR becomes triangular PFN.

Definition 2.4. An LR-type PFN A = (a ; l, r ; l′, r′)  _LR is non-negative(respectively non-positive), denoted as A ≥ 0(respectively A ≤ 0), if a - l′≥0(respectively a + r′≤0) and A is unrestricted if a belongs to real numbers.

Definition 2.5. An LR-type PFN A = (a ; l, r ; l′, r′)  _LR is positive if a - l′>0 and negative if a + r′<0.

Definition 2.6. An LR-type PFN A = (a ; l, r ; l′, r′)  _LR is zero if and only if a = 0, l = 0, r = 0, l′=0 and r′=0.

Definition 2.7. Two LR-type PFNs A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR are equal if a₁ = a₂, l₁ = l₂, r₁ = r₂, l₁′ = l₂′,r₁′ = r₂′ .

Theorem 2.1. Let A = (a ; l, r ; l′, r′)  _LR be an LR-type PFN, then its α - cut and β - cut are A ^α  = [a - lL^-1 (α) , a + rR^-1 (α)] and A ^β  = [a - lL^′-1 (β) , a + rR^′-1 (β)], ∀α, β ∈ [0, 1].

Proof. By using the Definition 2.2, the theorem can be proved easily. □

Definition 2.8. Let A = (a ; l, r ; l′, r′)  _LR be an LR-type PFN, then ranking of A, denoted R ( A ) , can be defined as

R ( A ) = 1 4 [ ( ∫ 0 1 ( a - lL - 1 ( α ) ) d α + ∫ 0 1 ( a + rR - 1 ( α ) ) d α ) + ( ∫ 0 1 ( a - l ′ L ′ - 1 ( α ) ) d α + ∫ 0 1 ( a + r ′ R ′ - 1 ( α ) ) d α ) ] .

Let A₁ and A₂ be two LR-type PFNs, then:

A₁ ≺ A₂ if R ( A 1 ) < R ( A 2 ) ,

Remark 2.2. Ranking function, as in Definition 2.8, is a linear function.

Theorem 2.2. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be two LR-type PFNs, then A₁ ⊕ A₂ = (a₁ + a₂ ; l₁ + l₂, r₁ + r₂ ; l₁′ + l₂′, r₁′ + r₂′)  _LR  .

Proof. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be two LR-type PFNs, then their α - cut and β - cut for all α, β ∈ [0, 1] , are given as: A 1 α = [ a 1 - l 1 L - 1 ( α ) , a 1 + r 1 R - 1 ( α ) ] , A 2 α = [ a 2 - l 2 L - 1 ( α ) , a 2 + r 2 R - 1 ( α ) ] , A 1 β = [ a 1 - l 1 ′ L ′ - 1 ( β ) , a 1 + r 1 ′ R ′ - 1 ( β ) ] , A 2 β = [ a 2 - l 2 ′ L ′ - 1 ( β ) , a 2 + r 2 ′ R ′ - 1 ( β ) ] . Thus, A 1 α + A 2 α = [ a 1 - l 1 L - 1 ( α ) + a 2 - l 2 L - 1 ( α ) , a 1 + r 1 R - 1 ( α ) + a 2 + r 2 R - 1 ( α ) ] .(1) By taking α = 1 in Equation 1, we have ( A 1 + A 2 ) α = 1 = a 1 + a 2 .(2) By taking α = 0 in Equation 1, we have

( A 1 + A 2 ) α = 0 = [ a 1 + a 2 - l 1 - l 2 , a 1 + a 2 + r 1 + r 2 ] .(3) Now

A 1 β + A 2 β = [ a 1 - l 1 ′ L ′ - 1 ( β ) + a 2 - l 2 ′ L ′ - 1 ( β ) , a 1 + r 1 ′ R ′ - 1 ( β ) + a 2 + r 2 ′ R ′ - 1 ( β ) ] .(4) By taking β = 0 in Equation 4, we have ( A 1 + A 2 ) β = 0 = a 1 + a 2 .(5) By taking β = 1 in Equation 4, we have

( A 1 + A 2 ) β = 1 = [ a 1 + a 2 - l 1 ′ - l 2 ′ , a 1 + a 2 + r 1 ′ + r 2 ′ ] .(6) By combining the Equations 2,3,5,6, the result follows. □

Theorem 2.3. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be two LR-type PFNs, then A₁ ⊖ A₂ = (a₁ - a₂ ; l₁ + r₂, r₁ + l₂ ; l₁′ + r₂′, r₁′ + l₂′)  _LR  .

Proof. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be two LR-type PFNs, then their α - cut and β - cut for all α, β ∈ [0, 1] , are given as: A 1 α = [ a 1 - l 1 L - 1 ( α ) , a 1 + r 1 R - 1 ( α ) ] , A 2 α = [ a 2 - l 2 L - 1 ( α ) , a 2 + r 2 R - 1 ( α ) ] , A 1 β = [ a 1 - l 1 ′ L ′ - 1 ( β ) , a 1 + r 1 ′ R ′ - 1 ( β ) ] , A 2 β = [ a 2 - l 2 ′ L ′ - 1 ( β ) , a 2 + r 2 ′ R ′ - 1 ( β ) ] .

Thus,

A 1 α - A 2 α = [ a 1 - l 1 L - 1 ( α ) - a 2 - r 2 R - 1 ( α ) , a 1 + r 1 R - 1 ( α ) - a 2 + l 2 L - 1 ( α ) ] .(7) By taking α = 1 in Equation 7, we have ( A 1 - A 2 ) α = 1 = a 1 - a 2 .(8) By taking α = 0 in Equation 1, we have

( A 1 - A 2 ) α = 0 = [ a 1 - a 2 - l 1 - r 2 , a 1 - a 2 + l 2 + r 1 ] .(9) Also,

A 1 β - A 2 β = [ a 1 - l 1 ′ L ′ - 1 ( β ) - a 2 - r 2 ′ R ′ - 1 ( β ) , a 1 + r 1 ′ R ′ - 1 ( β ) - a 2 + l 2 ′ L ′ - 1 ( β ) ] .(10) By taking β = 0 in Equation 4, we have ( A 1 - A 2 ) β = 0 = a 1 - a 2 .(11) By taking β = 1 in Equation 4, we have

( A 1 - A 2 ) β = 1 = [ a 1 - a 2 - l 1 ′ - r 2 ′ , a 1 - a 2 + l 2 ′ + r 1 ′ ] .(12) By combining the Equations 8, 9, 11, 12, the result follows. □

Theorem 2.4. Let A = (a ; l, r ; l′, r′)  _LR be an LR-type PFN and c be any real number, then cA = { ( ca ; cl , cr ; cl ′ , cr ′ ) LR , c ≥ 0 , ( ca ; - cr , - cl ; - cr ′ , - cl ′ ) LR , c < 0 .

Proof. Let A = (a ; l, r ; l′, r′)  _LR be an LR-type PFN and c be any real number, then A ^α  = [a - lL^-1 (α) , a + rR^-1 (α)], A ^β  = [a - l′L^′-1 (β) , a + r′R^′-1 (β)] .

Now, if c ≥ 0, then

cA α = [ ca - clL - 1 ( α ) , ca + crR - 1 ( α ) ] .(13) By taking α = 1 in Equation 7, we have cA α = 1 = ca .(14) By taking α = 0 in Equation 7, we have cA α = 0 = [ ca - cl , ca + cr ] .(15) Also,

cA β = [ ca - cl ′ L ′ - 1 ( β ) , ca + cr ′ R ′ - 1 ( β ) ] .(16) By taking β = 0 in Equation 10, we have cA β = 0 = ca .(17) By taking β = 1 in Equation 10, we have cA β = 1 = [ ca - cl ′ , ca + cr ′ ] .(18) By combining the Equations 14, 15, 17, 18, the case c ≥ 0 follows.

If c < 0, then

cA α = [ ca + crR - 1 ( α ) , ca - clL - 1 ( α ) ] .(19) By taking α = 1 in Equation 13, we have cA α = 1 = ca .(20) By taking α = 0 in Equation 13, we have cA α = 0 = [ ca + cr , ca - cl ] .(21) Also,

cA β = [ ca + cr ′ R ′ - 1 ( β ) , ca - cl ′ L ′ - 1 ( β ) ] .(22) By taking β = 0 in Equation 22, we have cA β = 0 = ca .(23) By taking β = 1 in Equation 22, we have cA β = 1 = [ ca + cr ′ , ca - cl ′ ] .(24) By combining the Equations 20, 21, 23,24, the case c < 0 follows. This completes the proof. □

Theorem 2.5. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be two non-negative LR-type PFNs, then A₁ ⊗ A₂ = (a₁a₂ ; a₁l₂ + a₂l₁ - l₁l₂, a₁r₂ + a₂r₁ + r₁r₂ ; a₁l₂′ + a₂l₁′ - l₁′l₂′, a₁r₂′ + a₂r₁′ + r₁′r₂′)  _LR  .

Proof. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be two non-negative LR-type PFNs, then their α - cut and β - cut for all α, β ∈ [0, 1] , are given as: A 1 α = [ a 1 - l 1 L - 1 ( α ) , a 1 + r 1 R - 1 ( α ) ] , A 2 α = [ a 2 - l 2 L - 1 ( α ) , a 2 + r 2 R - 1 ( α ) ] , A 1 β = [ a 1 - l 1 ′ L ′ - 1 ( β ) , a 1 + r 1 ′ R ′ - 1 ( β ) ] , A 2 β = [ a 2 - l 2 ′ L ′ - 1 ( β ) , a 2 + r 2 ′ R ′ - 1 ( β ) ] . Thus,

A 1 α A 2 α = [ ( a 1 - l 1 L - 1 ( α ) ) ( a 2 - l 2 L - 1 ( α ) ) , ( a 1 + r 1 R - 1 ( α ) ) ( a 2 + r 2 R - 1 ( α ) ) ] .(25) By taking α = 1 in Equation 25, we have ( A 1 A 2 ) α = 1 = a 1 a 2 .(26) By taking α = 0 in Equation 25, we have ( A 1 A 2 ) α = 0 = [ a 1 a 2 - a 1 l 2 - a 2 l 1 + l 1 l 2 , a 1 a 2 + a 1 r 2 + a 2 r 1 + r 1 r 2 ] .(27) Also

A 1 β A 2 β = [ ( a 1 - l 1 ′ L ′ - 1 ( β ) ) ( a 2 - l 2 ′ L ′ - 1 ( β ) ) , ( a 1 + r 1 ′ R ′ - 1 ( β ) ) ( a 2 + r 2 ′ R ′ - 1 ( β ) ) ] .(28) By taking β = 0 in Equation 28, we have ( A 1 A 2 ) β = 0 = a 1 a 2 .(29) By taking β = 1 in Equation 28, we have ( A 1 A 2 ) β = 1 = [ a 1 a 2 - a 1 l 2 ′ - a 2 l 1 ′ + l 1 ′ l 2 ′ , a 1 a 2 + a 1 r 2 ′ + a 2 r 1 ′ + r 1 ′ r 2 ′ ] .(30) By combining the Equations 26, 27, 29, 30, the result follows. □

Theorem 2.6. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR be non-negative LR-type PFN, and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be non-positive LR-type PFN, then A₁ ⊗ A₂ = (a₁a₂ ; a₁l₂ - a₂r₁ + l₂r₁, a₁r₂ - a₂l₁ - l₁r₂ ; a₁l₂′ - a₂r₁′ + l₂′r₁′, a₁r₂′ - a₂l₁′ - l₁′r₂′)  _LR  .

Proof. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR be non-negative LR-type PFN and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be non-positive LR-type PFN, then their α - cut and β - cut for all α, β ∈ [0, 1] , are given as: A 1 α = [ a 1 - l 1 L - 1 ( α ) , a 1 + r 1 R - 1 ( α ) ] , A 2 α = [ a 2 - l 2 L - 1 ( α ) , a 2 + r 2 R - 1 ( α ) ] , A 1 β = [ a 1 - l 1 ′ L ′ - 1 ( β ) , a 1 + r 1 ′ R ′ - 1 ( β ) ] , A 2 β = [ a 2 - l 2 ′ L ′ - 1 ( β ) , a 2 + r 2 ′ R ′ - 1 ( β ) ] . Thus,

A 1 α A 2 α = [ ( a 1 + r 1 R - 1 ( α ) ) ( a 2 - l 2 L - 1 ( α ) ) , ( a 1 - l 1 L - 1 ( α ) ) ( a 2 + r 2 R - 1 ( α ) ) ] .(31) By taking α = 1 in Equation 31, we have ( A 1 A 2 ) α = 1 = a 1 a 2 .(32) By taking α = 0 in Equation 31, we have ( A 1 A 2 ) α = 0 = [ a 1 a 2 - a 1 l 2 + a 2 r 1 - r 1 l 2 , a 1 a 2 + a 1 r 2 - a 2 l 1 - l 1 r 2 ] .(33) Also

A 1 β A 2 β = [ ( a 1 + r 1 ′ R ′ - 1 ( β ) ) ( a 2 - l 2 ′ L ′ - 1 ( β ) ) , ( a 1 - l 1 ′ L ′ - 1 ( β ) ) ( a 2 + r 2 ′ R ′ - 1 ( β ) ) ] .(34) By taking β = 0 in Equation 34, we have ( A 1 A 2 ) β = 0 = a 1 a 2 .(35) By taking β = 1 in Equation 34, we have ( A 1 A 2 ) β = 1 = [ a 1 a 2 - a 1 l 2 ′ + a 2 r 1 ′ - r 1 ′ l 2 ′ , a 1 a 2 + a 1 r 2 ′ - a 2 l 1 ′ - l 1 ′ r 2 ′ ] .(36) By combining the Equations 32, 33, 35, 36, the result follows. □

Theorem 2.7. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR be an LR-type PFN in which a₁ - l′₁ < 0, a₁ - l₁ ≥ 0 and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be an unrestricted LR-type PFN, then A₁ ⊗ A₂ = (a ; l, r ; l′, r′)  _LR , where a = a₁a₂, l = a₁a₂ - min {a₁a₂ - l₂a₁ - l₁a₂ + l₁l₂, a₁a₂ - l₂a₁ + r₁a₂ - l₂r₁}, r = max {a₁a₂ + r₂a₁ + r₁a₂ + r₁r₂, a₁a₂ + r₂a₁ - l₁a₂ - l₁r₂} - a₁a₂, l′ = a₁a₂ - min {a₁a₂ - l₁′a₂ + r₂′a₁ - l₁′r₂′, a₁a₂ + r₁′a₂ - l₂′a₁ - l₂′r₁′} and r′ = max {a₁a₂ - l₁′a₂ - l₂′a₁ + l₁′l₂′, a₁a₂ + r₁′a₂ + r₂′a₁ + r₁′r₂′} - a₁a₂ .

Proof. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR be an LR-type PFN such that a₁ - l₁′<0, a₁ - l₁ ≥ 0 and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be an unrestricted LR-type PFN, and their α - cut and β - cut for all α, β ∈ [0, 1] , are given as: A 1 α = [ a 1 - l 1 L - 1 ( α ) , a 1 + r 1 R - 1 ( α ) ] , A 2 α = [ a 2 - l 2 L - 1 ( α ) , a 2 + r 2 R - 1 ( α ) ] , A 1 β = [ a 1 - l 1 ′ L ′ - 1 ( β ) , a 1 + r 1 ′ R ′ - 1 ( β ) ] , A 2 β = [ a 2 - l 2 ′ L ′ - 1 ( β ) , a 2 + r 2 ′ R ′ - 1 ( β ) ] . Thus, A 1 α A 2 α = [ min { ( a 1 - l 1 L - 1 ( α ) ) ( a 2 - l 2 L - 1 ( α ) ) , ( a 1 + r 1 R - 1 ( α ) ) ( a 2 - l 2 L - 1 ( α ) ) } ,

max { ( a 1 + r 1 R - 1 ( α ) ) ( a 2 + r 2 R - 1 ( α ) ) , ( a 1 - l 1 L - 1 ( α ) ) ( a 2 + r 2 R - 1 ( α ) ) } ] .(37) By taking α = 1 in Equation 97, we have ( A 1 A 2 ) α = 1 = a 1 a 2 .(38) By taking α = 0 in Equation 97, we have ( A 1 A 2 ) α = 0 = [ min { a 1 a 2 - a 1 l 2 - a 2 l 1 + l 1 l 2 , a 1 a 2 - a 1 l 2 + a 2 r 1 - l 2 r 1 } , max { a 1 a 2 + a 1 r 2 + a 2 r 1 + r 1 r 2 , a 1 a 2 + a 1 r 2 - a 2 l 1 - l 1 r 2 } ] .(39) Also A 1 β A 2 β = [ min { ( a 1 - l 1 ′ L ′ - 1 ( β ) ) ( a 2 + r 2 ′ R ′ - 1 ( β ) ) , ( a 1 + r 1 ′ R ′ - 1 ( β ) ) ( a 2 - l 2 ′ L ′ - 1 ( β ) ) } ,

max { ( a 1 - l 1 ′ L ′ - 1 ( β ) ) ( a 2 - l 2 ′ L ′ - 1 ( β ) ) , ( a 1 + r 1 ′ R ′ - 1 ( β ) ) ( a 2 + r 2 ′ R ′ - 1 ( β ) ) } ] .(40) By taking β = 0 in Equation 40, we have ( A 1 A 2 ) β = 0 = a 1 a 2 .(41) By taking β = 1 in Equation 41, we have ( A 1 A 2 ) β = 1 = [ min { a 1 a 2 + a 1 r 2 ′ - a 2 l 1 ′ - l 1 ′ r 2 ′ , a 1 a 2 - a 1 l 2 ′ + a 2 r 1 ′ - l 2 ′ r 1 ′ } , max { a 1 a 2 - a 1 l 2 ′ - a 2 l 1 ′ + l 1 ′ l 2 ′ , a 1 a 2 + a 1 r 2 ′ + a 2 r 1 ′ + r 1 ′ r 2 ′ } ] .(42) By combining the Equations 98, 99, 41, 42, the result follows. □

By using similar arguments as used in the Theorem (2.7), the following theorems can be proved easily.

Theorem 2.8 Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR be an LR-type PFN in which a₁ - l₁ < 0, a₁ ≥ 0 and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be an unrestricted LR-type PFN, then A₁ ⊗ A₂ = (a ; l, r ; l′, r′)  _LR , where a = a₁a₂, l = a₁a₂ - min {a₁a₂ - l₁a₂ + r₂a₁ - l₁r₂, a₁a₂ + r₁a₂ - l₂a₁ - l₂r₁}, r = max {a₁a₂ - l₁a₂ - l₂a₁ + l₁l₂, a₁a₂ + r₁a₂ + r₂a₁ + r₁r₂} - a₁a₂, l′ = a₁a₂ - min {a₁a₂ - l₁′a₂ + r₂′a₁ - l₁′r₂′, a₁a₂ + r₁′a₂ - l₂′a₁ - l₂′r₁′}, r′ = max {a₁a₂ - l₁′a₂ - l₂′a₁ + l₁′l₂′, a₁a₂ + r₁′a₂ + r₂′a₁ + r₁′r₂′} - a₁a₂ .

Theorem 2.9. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR be an LR-type PFN in which a₁ < 0, a₁ + r₁ ≥ 0 and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be an unrestricted LR-type PFN, then A₁ ⊗ A₂ = (a ; l, r ; l′, r′)  _LR , where a = a₁a₂, l = a₁a₂ - min {a₁a₂ - l₁a₂ + r₂a₁ - l₁r₂, a₁a₂ + r₁a₂ - l₂a₁ - l₂r₁}, r = max {a₁a₂ - l₁a₂ - l₂a₁ + l₁l₂, a₁a₂ + r₁a₂ + r₂a₁ + r₁r₂} - a₁a₂, l′ = a₁a₂ - min {a₁a₂ - l₁′a₂ + r₂′a₁ - l₁′r₂′, a₁a₂ + r₁′a₂ - l₂′a₁ - l₂′r₁′}, r′ = max {a₁a₂ - l₁′a₂ - l₂′a₁ + l₁′l₂′, a₁a₂ + r₁′a₂ + r₂′a₁ + r₁′r₂′} - a₁a₂ .

Theorem 2.10. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR be an LR-type PFN in which a₁ + r₁ < 0, a₁ + r₁′≥0 and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be an unrestricted LR-type PFN, then A₁ ⊗ A₂ = (a ; l, r ; l′, r′)  _LR , where a = a₁a₂, l = a₁a₂ - min {a₁a₂ - l₁a₂ + r₂a₁ - l₁r₂, a₁a₂ + r₂a₁ + r₁a₂ + r₁r₂}, r = max {a₁a₂ + r₁a₂ - l₂a₁ - l₂r₁, a₁a₂ - l₂a₁ - l₁a₂ + l₁l₂} - a₁a₂, l′ = a₁a₂ - min {a₁a₂ - l₁′a₂ + r₂′a₁ - l₁′r₂′, a₁a₂ + r₁′a₂ - l₂′a₁ - l₂′r₁′}, r′ = max {a₁a₂ - l₁′a₂ - l₂′a₁ + l₁′l₂′, a₁a₂ + r₁′a₂ + r₂′a₁ + r₁′r₂′} - a₁a₂ .

Theorem 2.11. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR be an LR-type PFN in which a₁ + r₁′<0 and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be an unrestricted LR-type PFN, then A₁ ⊗ A₂ = (a ; l, r ; l′, r′)  _LR , where a = a₁a₂, l = a₁a₂ - min {a₁a₂ - l₁a₂ + r₂a₁ - l₁r₂, a₁a₂ + r₂a₁ + r₁a₂ + r₁r₂}, r = max {a₁a₂ + r₁a₂ - l₂a₁ - l₂r₁, a₁a₂ - l₂a₁ - l₁a₂ + l₁l₂} - a₁a₂, l′ = a₁a₂ - min {a₁a₂ - l₁′a₂ + r₂′a₁ - l₁′r₂′, a₁a₂ + r₂′a₁ + r₁′a₂ + r₁′r₂′}, r′ = max {a₁a₂ + r₁′a₂ - l₂′a₁ - l₂′r₁′, a₁a₂ - l₁′a₂ - l₂′a₁ + l₁′l₂′} - a₁a₂ .

Theorem 2.12. Let A₁ = (a₁ ; l₁, r₁ ; l′₁, r′₁)  _LR be an LR-type PFN in which a₁ - l′₁ ≥ 0, and A₂ = (a₂ ; l₂, r₂ ; l′₂, r′₂)  _LR be an unrestricted LR-type PFN, then A₁ ⊗ A₂ = (a ; l, r ; l′, r′)  _LR , where a = a₁a₂, l = a₁a₂ - min {a₁a₂ - l₂a₁ - l₁a₂ + l₁l₂, a₁a₂ - l₂a₁ + r₁a₂ - l₂r₁}, r = max {a₁a₂ + r₂a₁ + r₁a₂ + r₁r₂, a₁a₂ + r₂a₁ - l₁a₂ - l₁r₂} - a₁a₂, l′ = a₁a₂ - min {a₁a₂ - l₂′a₁ - l₁′a₂ + l₁′l₂′, a₁a₂ - l₂′a₁ + r₁′a₂ - l₂′r₁′}, r′ = max {a₁a₂ + r₂′a₁ + r₁′a₂ + r₁′r₂′, a₁a₂ + r₂′a₁ - l₁′a₂ - l₁′r₂′} - a₁a₂ .

Let us consider FPFLPP with LR-type PFNs.

Max / Min ∑ j = 1 n C j ⊗ X j(43) subject to ∑ j = 1 n A ij ⊗ X j = B i ∀ i = 1 , 2 , ⋯ , m where A _ij ,X _j ,B _i , and C _j are LR-type PFNs.

Definition 3.1. An LR-type Pythagorean fuzzy optimal solution of FPFLPP 43 with LR-type PFNs will be LR-type PFNs X _j if:

X _j are LR-type PFNs.

We now present procedure to solve FPFLPP 43 with LR-type PFNs as follows:

Step 1: Assuming A = (a _ij  ; l _ij , r _ij  ; l′ _ij , r′ _ij )  _LR , X = (x _j  ; y _j , z _j  ; y′ _j , z′ _j )  _LR , B = (b _i  ; p _i , q _i  ; p′ _i , q′ _i )  _LR , and C = (c _j  ; m _j , n _j  ; m′ _j , n′ _j )  _LR , the FPFLPP 44 can be written as:

Max / Min ∑ j = 1 n ( c j ; m j , n j ; m ′ j , n ′ j ) LR ⊗ ( x j ; y j , z j ; y ′ j , z ′ j ) LR(44) subject to ∑ j = 1 n ( a ij ; l ij , r ij ; l ′ ij , r ′ ij ) LR ⊗ ( x j ; y j , z j ; y ′ j , z ′ j ) LR = ( b i ; p i , q i ; p ′ i , q ′ i ) LR , ∀i = 1, 2, ⋯ , m where (x _j  ; y _j , z _j  ; y′ _j , z′ _j )  _LR is LR-type PFN.

Step 2: By using the product as proposed in Section 2 and letting (a _ij  ; l _ij , r _ij  ; l′ _ij , r′ _ij )  _LR  ⊗ (x _j  ; y _j , z _j  ; y′ _j , z′ _j )  _LR  = (a_ij1 ; l_ij1, r_ij1 ; l′_ij1, r′_ij1)  _LR , the FPFLPP 44 can be written as:

Max / Min ∑ j = 1 n ( c j ; m j , n j ; m ′ j , n ′ j ) LR ⊗ ( x j ; y j , z j ; y ′ j , z ′ j ) LR(45) subject to ∑ j = 1 n ( a ij 1 ; l ij 1 , r ij 1 ; l ′ ij 1 , r ′ ij 1 ) LR = ( b i ; p i , q i ; p ′ i , q ′ i ) LR , ∀i = 1, 2, ⋯ , m,

where (x _j  ; y _j , z _j  ; y′ _j , z′ _j )  _LR is LR-type PFN.

Step 3: By using arithmetic operations as defined in Section 2 and using Definition 2.7, the FPFLPP 45 can be written as:

Max / Min ∑ j = 1 n ( c j ; m j , n j ; m ′ j , n ′ j ) LR ⊗ ( x j ; y j , z j ; y ′ j , z ′ j ) LR(46) subject to ∑ j = 1 n a ij 1 = b i , ∑ j = 1 n l ij 1 = p i , ∑ j = 1 n r ij 1 = q i , ∑ j = 1 n l ′ ij 1 = p ′ i , ∑ j = 1 n r ′ ij 1 = q ′ i , y _j  ≥ 0, z _j  ≥ 0, y _j ′ - y _j  ≥ 0, z _j ′ - z _j  ≥ 0, ∀i = 1, 2, ⋯ , m, ∀j = 1, 2, ⋯ , n .

Step 4: Now we have to find LR-type Pythagorean fuzzy feasible solution X t = ( x j t ; y j t , z j t ; y j ′ t , z j ′ t ) LR such that R ( X t ) ≥ R ( X ) for all feasible solutions X. Applying ranking, the FPFLPP 46 can be solved by solving

Max / Min R ( ∑ j = 1 n ( c j ; m j , n j ; m ′ j , n ′ j ) LR ⊗ ( x j ; y j , z j ; y ′ j , z ′ j ) LR )(47) subject to ∑ j = 1 n a ij 1 = b i , ∑ j = 1 n l ij 1 = p i , ∑ j = 1 n r ij 1 = q i , ∑ j = 1 n l ′ ij 1 = p ′ i , ∑ j = 1 n r ′ ij 1 = q ′ i , y _j  ≥ 0, z _j  ≥ 0, y _j ′ - y _j  ≥ 0, z _j ′ - z _j  ≥ 0, ∀i = 1, 2, ⋯ , m, ∀j = 1, 2, ⋯ , n .

Step 5: Assuming ( c j ; m j , n j ; m ′ j , n ′ j ) LR ⊗ ( x j ; y j , z j ; y ′ j , z ′ j ) LR = ( x j s ; y j s , z j s ; y j ′ s , z j ′ s ) LR , the Problem 47 can be written as:

Max / Min R ( ∑ j = 1 n ( x j s ; y j s , z j s ; y j ′ s , z j ′ s ) LR )(48) subject to ∑ j = 1 n a ij 1 = b i , ∑ j = 1 n l ij 1 = p i , ∑ j = 1 n r ij 1 = q i , ∑ j = 1 n l ′ ij 1 = p ′ i , ∑ j = 1 n r ′ ij 1 = q ′ i , y j s ≥ 0 , z j s ≥ 0 , y j ′ s - y j s ≥ 0 , z j ′ s - z j s ≥ 0 , ∀i = 1, 2, ⋯ , m, ∀j = 1, 2, ⋯ , n .

Step 6: As ranking function is linear, thus, Problem 48 can be written as:

Max / Min ∑ j = 1 n R ( x j s ; y j s , z j s ; y j ′ s , z j ′ s ) LR(49) subject to ∑ j = 1 n a ij 1 = b i , ∑ j = 1 n l ij 1 = p i , ∑ j = 1 n r ij 1 = q i , ∑ j = 1 n l ′ ij 1 = p ′ i , ∑ j = 1 n r ′ ij 1 = q ′ i , y j s ≥ 0 , z j s ≥ 0 , y j ′ s - y j s ≥ 0 , z j ′ s - z j s ≥ 0 , ∀i = 1, 2, ⋯ , m, ∀j = 1, 2, ⋯ , n .

Step 7: Using the Definition 2.8, Problem 49 can be converted into Problem 50: Max / Min ∑ j = 1 n [ 1 4 { ( ∫ 0 1 ( x j s - y j s L - 1 ( α ) ) d α + ∫ 0 1 ( x j s + z j s R - 1 ( α ) ) d α )

+ ( ∫ 0 1 ( x j s - y j ′ s L ′ - 1 ( α ) ) d α + ∫ 0 1 ( x j s + z j ′ s R ′ - 1 ( α ) ) d α ) } ](50) subject to ∑ j = 1 n a ij 1 = b i , ∑ j = 1 n l ij 1 = p i , ∑ j = 1 n r ij 1 = q i , ∑ j = 1 n l ′ ij 1 = p ′ i , ∑ j = 1 n r ′ ij 1 = q ′ i , y j s ≥ 0 , z j s ≥ 0 , y j ′ s - y j s ≥ 0 , z j ′ s - z j s ≥ 0 , ∀i = 1, 2, ⋯ , m, ∀j = 1, 2, ⋯ , n .