A new fuzzy decision support system approach; analysis and applications

1.   Introduction

The idea of a fuzzy set was initiated by Zadeh ^[1] to measure ambiguity and vagueness. Fuzzy sets use only membership values for the estimation of uncertainty. But it's difficult for experts to use crisp values for their decision purpose thus Zadeh ^[8,9,10,11] proposed the idea of interval-valued fuzzy sets. Atanassov ^[2,3] provided the further generalization of fuzzy sets by adding non-membership grades with the membership grades known as intuitionistic fuzzy sets. Yager ^{[4,5,6,7,33,34,35]} further extended the idea of intuitionistic fuzzy sets to Pythagorean fuzzy sets which are further extended to Q-rung orthopair fuzzy sets. The idea of interval-valued intuitionistic fuzzy sets ^[12,13,14] and interval-valued Pythagorean fuzzy sets ^{[15,16,17,18]} was also a very valuable addition. More details can be seen in ^[33,34,35] about applications of Pythagorean fuzzy sets. The idea of cubic sets (combination of interval-valued fuzzy sets and fuzzy sets) given by Jun ^[19,20,21] uses interval-valued fuzzy data as well as the crisp term of fuzzy set. Detailed about the applications of cubic sets can be seen in ^{[36,37,38,39,40,41,42,43,44]}. On the same lines, the idea of cubic Pythagorean fuzzy sets ^[22] and neutrosophic cubic set ^[23,24,25] was developed. For the applications of neutrosophic cubic sets we refer the reader ^[45,46,47]. Chang's extent analysis of the analytical hierarchy process, ^{[27,28,29,30]} used hierarchy form of data in terms of a fuzzy environment instead of crisp data. For more applications of AHP we refer the reader ^[50]. Jun et al. ^[31] extends the idea of Zadeh's fuzzy sets and introduced a new approach which is called negative-valued function, and constructed N-structures. Further, they applied N-structure theory to subtraction algebra and BCK/BCI-algebra and study their related properties. They also discussed N-ideals of subtraction algebra and used N-fuzzy sets which is the extension of fuzzy sets where they used [-1, 0] instead of [0, 1]. In 2013, ^[32] William and Saeid further gave the concept of generalized N-ideals of subtraction algebras. Rashid et al. ^[26] applied N-structure on cubic fuzzy set as N-cubic fuzzy set. Applications of N-structures can be seen in ^[48,49]. The following are some examples of applications that deal with the negative aspects of objects or their effects.

(1) Due to the lack of any approved treatment, health workers propose some possible treatment (Clinical Management Protocol 2020) to cure the unexpected virus infection, where the choice of drugs has a significant impact on the patients' recovery rate. A few researchers have experimented with the selection of drugs for COVID-19 affected patients, according to the literature. The recommended drugs for treating COVID-19 patients have a variety of functions, including effectiveness, adverse effects, and some unknown consequences. As a result, we use these sets to look for drugs that have a detrimental influence on COVID-19 or other disorders.

(2) Another example is that this set might be used to assess the drawbacks of utilizing social media, such as Facebook, which can lead to addiction and privacy issues.

It has already been studied that after crisp sets, the need for fuzzy sets was felt where true membership was discussed due to its imprecise and vague properties. Moreover, sets for false membership were defined and investigated. Many other fuzzy sets were defined in a way rather than a negative side to describe the positive behavior that is used in many real-life problems, but another thing is that these sets only illustrate positive behavior of things to show only positive features. Obviously, in exact decision-making, we observe the limitations of these things and keep them in mind.

So how do make any decision based on only the positive side?

Is it possible for anything that has the specialty of good quality features only?

These things show negative features as well as a positive behavior.

To notice this side, we define N-cubic Pythagorean fuzzy set (NCPFNs), which apply to real-life problems to describe the drawbacks that are precisely used in decision making in a better way. This paper is organizes as follows: In Section 1, we discussed brief history of different sets and motivation about our work. In Section 2, we recall some basic definitions. In Section 3, we introduce the novel concept of N-cubic Pythagorean fuzzy set (NCPFNs) with some interesting properties. In Section 4, we develop triangular N-cubic Pythagorean fuzzy set. Further application of the proposed work is discussed in Section 5 using AHP method. Section 6, discussed comparison analysis and conclusions are providing in Section 7.

2.   Preliminaries

We recall some basic definitions as:

Definition: ^[4] Suppose G be the non-empty universal set, then Pythagorean fuzzy set is defined as:

$\{{ < \mathrm{g}, \mathtt{μ} }_{G}\left(\mathrm{g}\right), {\eta }_{G}\left(\mathrm{g}\right) > \mathrm{g}\in \mathrm{G}\}$, For ${\mathtt{μ} }_{G}\left(\mathrm{g}\right):\mathrm{G}\to \left[\mathrm{0, 1}\right]$ and ${\mathtt{η}}_{G}\left(\mathrm{g}\right):\mathrm{G}\to \left[\mathrm{0, 1}\right]$ be the membership and non-membership values of g in G.

Also, $0 \le {{\mathtt{μ} }}_{G}(\mathrm{g}{)}^{2}+{\mathtt{η}}_{G}\left(\mathrm{g}{)}^{2}\right\}\le 1$ and ${\pi }_{P}\left(\mathrm{g}\right)\mathrm{ } = \sqrt{1-{{\mathtt{μ} }_{G}\left(g\right)}^{2}-{\eta }_{G}{\left(g\right)}^{2}}$ is the degree of indeterminacy.

Definition: ^[20] Suppose X be the set; cubic sets are the structure given as:

where ${\mathtt{μ} }_{X}\left(x\right)$be the interval valued fuzzy set and ${\eta }_{X}\left(\mathrm{x}\right)$ be the fuzzy set.

Definition: ^[22] Let X be a fixed non-empty set. By a cubic Pythagorean fuzzy set, we mean a structure of the form

where ${\mathtt{μ} }_{C}\left(x\right)$ is an interval valued Pythagorean fuzzy set in X and ${\eta }_{C}\left(\mathrm{x}\right)$ is Pythagorean fuzzy set in X. Let ${\rm{ \mathsf{ π} }}_{CP}\left(\mathrm{x}\right) = < \left[{\pi }_{CP}^{-}\left(\mathrm{x}\right), {\pi }_{C\left(P\right)}^{+}\left(\mathrm{x}\right)\right], {\rm{ \mathsf{ π} }}_{CP}\left(\mathrm{x}\right) >$, then ${\rm{ \mathsf{ π} }}_{CP}\left(\mathrm{x}\right)$ is said to be cubic Pythagorean fuzzy index of element x ∈ X to set C(P), where

and also, $\left[\mathrm{0, 0}\right]\le [{\mathtt{μ} }_{1}{\left(x\right)}^{-}, {\mathtt{μ} }_{1}{\left(x\right)}^{+}{]}^{2}+[{\mathtt{μ} }_{2}{\left(x\right)}^{-}, {\mathtt{μ} }_{2}{\left(x\right)}^{+}{]}^{2}\le \left[\mathrm{1, 1}\right], 0\le {{\eta }_{1}\left(x\right)}^{2}+{{\eta }_{2}\left(x\right)}^{2}\le 1$.

3.   N-Cubic Pythagorean fuzzy sets

In this section we initiated the study of N-cubic Pythagorean fuzzy sets (NCPFN). As cubic sets consist of two sets, i.e., interval valued fuzzy sets and fuzzy sets. For achieving NCPFN, we must first define N-Pythagorean fuzzy sets (NPFS) and N-interval valued Pythagorean fuzzy sets (NIVPFS). We discuss some basic operations and properties of N-cubic Pythagorean fuzzy sets (NCPFN).

(Note that the superscript 2(2) denotes the even power of membership values and non-membership value, while 2(2)+1 be the odd power of (-1) that are used in the whole manuscript.)

3.1.   N-Pythagorean fuzzy sets

Definition: Let A be a fixed set. Then N-Pythagorean fuzzy set is a structure of the form

where ${\mathtt{μ} }_{X}\left(\mathrm{x}\right):\mathrm{X}\to [-\mathrm{1, 0}]$ and ${\eta }_{X}\left(\mathrm{x}\right):\mathrm{X}\to [-\mathrm{1, 0}]$ be the N-valued fuzzy membership and N-valued fuzzy non-membership of x ∈ X with the condition that

and

are the N- Pythagorean degree of indeterminacy. The spacing graph of N-Pythagorean fuzzy set can be demonstrated by Figure 1.

Remark: In general, for N-q rung ortho pair fuzzy set -1 ≤ (-1${)}^{2q+1}${µ_X (x${)}^{2q}$ + η_X(x${)}^{2q}$} ≤ 0 for q ≥ 1 and π_p(x) = (-1${)}^{2q+1}\sqrt[q]{{{\mathtt{μ} }}_{X}{\left(\mathrm{x}\right)}^{2q}+{\mathtt{η}}_{X}{\left(x\right)}^{2q}-{{{\mathtt{μ} }}_{X}\left(\mathrm{x}\right)}^{2q}{\mathtt{η}}_{X}{\left(x\right)}^{2q}}$ for q ≥ 3.

3.2.   N-Interval valued Pythagorean fuzzy sets

Definition: Let X be a fixed set. Then N-interval valued Pythagorean fuzzy set is a structure of the form

where ${\mathtt{μ} }_{A}\left(\mathrm{x}\right)\in D\left[-\mathrm{1, 0}\right], {\mathtt{η}}_{A}\left(\mathrm{x}\right)\in D\left[-\mathrm{1, 0}\right]$ be the N-interval valued fuzzy membership and N-interval valued fuzzy non-membership of x ∈ X with the condition that

Also,

where

be the N- interval valued Pythagorean degree of indeterminacy. The spacing graph of N-interval valued Pythagorean fuzzy set can be demonstrated by Figure 2.

3.3.   N-Cubic Pythagorean fuzzy sets

Definition: Let X is a fixed nonempty set. By an N-cubic Pythagorean fuzzy set we mean a structure of the form

Where ${\mathtt{μ} }_{NP}\left(x\right)$ is an N-interval valued Pythagorean fuzzy set in X and ${\eta }_{NP}\left(\mathrm{x}\right)$ is N-Pythagorean fuzzy set in X.

Let ${\rm{ \mathsf{ π} }}_{NP}\left(\mathrm{x}\right) = < \left[{\pi }_{NP}^{-}\left(\mathrm{x}\right), {\pi }_{NP}^{+}\left(\mathrm{x}\right)\right], {\rm{ \mathsf{ π} }}_{NP}\left(\mathrm{x}\right) >$.

Then ${\rm{ \mathsf{ π} }}_{NP}\left(\mathrm{x}\right)$ is said to be N-cubic Pythagorean fuzzy index of element x ∈ X to set N_C^P(x), where

Also,

where,

and,

We denote N-cubic Pythagorean fuzzy set as NCPFs = < N-IVPFs, N-PFs > . The spacing graph of N-cubic Pythagorean fuzzy set can be demonstrated by Figure 3.

Some arithmetic operations on N-cubic Pythagorean fuzzy sets

Definition: Let A and B be two N-cubic Pythagorean fuzzy sets then, we define:

(2)

(3)

(4)

(5)

3.4.   N-Cubic Pythagorean fuzzy numbers (NCPFN's)

In this section we define the concept of N-cubic Pythagorean fuzzy number. We define some basic properties of proposed NCPFN's.

Definition: The set NCPFs = < N-IVPFs, N-PFs > is said to be N-cubic Pythagorean fuzzy number, if following conditions fulfilled.

(1) N-Cubic Pythagorean subset of real lines.

(2) Normal, their exist x ∈ R such that $< {{μ} }_{NP}\left(\mathrm{x}\right) > = \left[-1, -1\right], < {\eta }_{NP}\left(x\right) > = 0$.

(3) Concave for the membership, i.e.,

(4) Convex for non-membership, i.e.,

Theorem: For any two NCPFN's N_A and N_B, and ∂, ∂₁, ∂₂ > 0 then

$(1)\; {N}_{A}+{N}_{B} = {N}_{B}+{N}_{A},$

$(2)\; {N}_{A}*{N}_{B} = {N}_{B}*{N}_{A},$

$(3)\; \partial ({N}_{A}+{N}_{B}) = {\partial N}_{A}+\partial {N}_{B}, \partial > 0,$

$(4)\; \left({\partial }_{1}+{\partial }_{2}\right){N}_{A} = {{\partial }_{1}N}_{A}+{\partial }_{2}{N}_{A}, {\partial }_{1}, {\partial }_{2} > 0,$

$(5)\; {{(N}_{A}*{N}_{B})}^{\partial } = {N}_{A}^{\partial }*{N}_{B}^{\partial }, \mathrm{ }\partial \mathrm{ } > \mathrm{ }0,$

$(6)\; {N}_{A}^{\partial 1+\partial 2} = {N}_{A}^{\partial 1}\mathrm{*}{N}_{A}^{\partial 2}, {\partial }_{1}, {\partial }_{2} > 0.$

Proof: Easy to prove.

3.5.   Score and accuracy

The accuracy and score functions play an important role in decision-making. We introduce a novel accuracy and scoring function for this set, which will be used to compare two N-cubic Pythagorean fuzzy sets.

3.5.1.   Score function of N-cubic Pythagorean fuzzy sets

Definition: Let $N =$ < ${\grave{N}}_{P}\left(x\right)$, ${N}_{P}\left(x\right)$ > be the NCPFN then we define the score function of N as follows:

S(N) = 1/2 < S$\left({\grave{N}}_{P}\left(x\right)\right),$ S(${N}_{P}\left(x\right)$) > where S$\left({\grave{N}}_{P}\left(x\right)\right) =$½ {(-1${)}^{2\left(2\right)+1}$ [${\mathtt{μ} }_{3}^{-}(x{)}^{2\left(2\right)}$+ ${\mathtt{μ} }_{3}^{+}(x{)}^{2\left(2\right)}$

– ${\eta }_{3}^{-}(x{)}^{2\left(2\right)}$ – ${\eta }_{3}^{+}(x{)}^{2\left(2\right)}$] }, and S(${N}_{P}\left(x\right)$ = (-1${)}^{2\left(2\right)+1}$ { ${\mathtt{μ} }_{N}{\left(x\right)}^{2\left(2\right)}-{\eta }_{N}{\left(x\right)}^{2\left(2\right)}\}$

S(N)$= 1/2$ (-1${)}^{2\left(2\right)+1}\{1/2$ [${\mathtt{μ} }_{3}^{-}(x{)}^{2\left(2\right)}$ + ${\mathtt{μ} }_{3}^{+}(x{)}^{2\left(2\right)}$ – ${\eta }_{3}^{-}(x{)}^{2\left(2\right)}$ – ${\eta }_{3}^{+}(x{)}^{2\left(2\right)}$] + (${\mathtt{μ} }_{N}{\left(x\right)}^{2\left(2\right)}-{\eta }_{N}{\left(x\right)}^{2\left(2\right)}\left)\right\}$,

where S (${\grave{N}}_{P}\left(x\right)$) ∈ [-1, 1], S(${N}_{P}\left(x\right)$) ∈ [-1, 1] and S(N) ∈ [-1, 1].

Definition: Let ${N}_{A}, {N}_{B}$ ∈ NCPFN's, then ${N}_{A}$ ⊆ ${N}_{B}$, if and

and

${\{{\mathtt{μ} }_{A}\left(x\right)\le \mathtt{μ} }_{B}\left(x\right){, \eta }_{A}\left(x\right)\le {\eta }_{B}\left(x\right)OR{\mathtt{μ} }_{A}\left(x\right)\ge {\mathtt{μ} }_{B}\left(x\right), {\eta }_{A}\left(x\right)\ge {\eta }_{B}\left(x\right)$} ∀ x ∈ X.

Definition: Let ${N}_{A} =$ < ${\grave{N}}_{AP}\left(x\right)$, ${N}_{AP}\left(x\right)$ > and ${N}_{B} =$ < ${\grave{N}}_{BP}\left(x\right)$, ${N}_{BP}\left(x\right)$ > be two NCPFN. Then if

(1) S (${N}_{A}$) > S (${N}_{B}$) $\Rightarrow \; {N}_{A}$ > ${N}_{B}$,

(2) If we have S (${N}_{A}$) = S (${N}_{B}$). Then there is no difference disagreement, indicating that such a score function cannot achieve NCPFN's rating.

Due to the score function's inadequacy, the accuracy function was added in the following sense, which has two requirements.

(1) If a$\left({N}_{A}\right) =$a$\left({N}_{B}\right) \; \Rightarrow \; {N}_{A}$ = ${N}_{B}.$

(2) If a$\left({N}_{A}\right) >$a$\left({N}_{B}\right) \; \Rightarrow \; {N}_{A}$ > ${N}_{B}$.

To proceed with this scoring function based on the inability to compare sets due to certain situations, we develop the following accuracy function for the implications of the provided condition:

Definition: Let $N =$ < ${\grave{N}}_{P}\left(x\right)$, ${N}_{P}\left(x\right)$ > then we define the accuracy function as follows:

a ($N) = 1/2$ < $a\left({\grave{N}}_{P}\left(x\right)\right)$, $a({N}_{P}\left(x\right)$) > ,

where

a (${\grave{N}}_{P}\left(x\right)$) $=$1/2 {(-1${)}^{2\left(2\right)+1}$ [${\mathtt{μ} }_{3}^{-}(x{)}^{2\left(2\right)}$ + ${\mathtt{μ} }_{3}^{+}(x{)}^{2\left(2\right)}$ + ${\eta }_{3}^{-}(x{)}^{2\left(2\right)}$ + ${\eta }_{3}^{+}(x{)}^{2\left(2\right)}$] },

and

a (${N}_{P}\left(x\right)$) = (-1${)}^{2\left(2\right)+1}$ { ${\mathtt{μ} }_{N}{\left(x\right)}^{2\left(2\right)}+{\eta }_{N}{\left(x\right)}^{2\left(2\right)}\}$ then, a($N) = 1/2$ (-1${)}^{2\left(2\right)+1}\{1/2$ [${\mathtt{μ} }_{3}^{-}(x{)}^{2\left(2\right)}$ + ${\mathtt{μ} }_{3}^{+}(x{)}^{2\left(2\right)}$ + ${\eta }_{3}^{-}(x{)}^{2\left(2\right)}$ + ${\eta }_{3}^{+}(x{)}^{2\left(2\right)}$] + (${\mathtt{μ} }_{N}{\left(x\right)}^{2\left(2\right)}+{\eta }_{N}{\left(x\right)}^{2\left(2\right)}\left)\right\}$,

Where a (${\grave{N}}_{P}\left(x\right)$) ∈ [-1, 0], a (${N}_{P}\left(x\right)$) ∈ [-1, 0] and a ($N)$ ∈ [-1, 0].

Definition: Let S (${N}_{A}$) be the score of ${N}_{A}$, S (${N}_{B}$) be the score of N_B. Then

S (${N}_{A}$) < S (${N}_{B}$) if S $\left({\grave{N}}_{AP}\left(x\right)\right)$ ≤ S $\left({\grave{N}}_{BP}\left(x\right)\right)$, S (${N}_{AP}\left(x\right)$) ≤ S (${N}_{BP}\left(x\right)$), or S (${N}_{AP}\left(x\right)$) ≥ S (${N}_{BP}\left(x\right)$) ∀ x ∈ X.

And, let a (${N}_{A}$) be the accuracy of ${N}_{A}$, a (${N}_{B}$) be the accuracy of ${N}_{B}$then

a (${N}_{A}$) ≤ a (${N}_{B}$) if, a (${\grave{N}}_{AP}\left(x\right)$) ≤ a (${\grave{N}}_{BP}\left(x\right)$), a (${N}_{AP}\left(x\right)$) ≤ a (${N}_{AP}\left(x\right)$),

or a (${N}_{AP}\left(x\right)$) ≥ a (${N}_{AP}\left(x\right)$) ∀ x ∈ X.

Example: Suppose ${N}_{A}$ = < [-0.8, -0.6], [-0.5, -0.2], (-0.6, -0.2) > , ${N}_{B}$ = < [-0.8, -0.7], [-0.6, -0.4], (-0.8, -0.4) > , be two NCPFN's. Then

$S\left({N}_{A}\right)$ = 1/2 < S$\left({\grave{N}}_{AP}\left(x\right)\right),$ S (${N}_{AP}\left(x\right)$) > $S\left({N}_{A}\right)$ = -0.28.

Also,

$S\left({N}_{B}\right)$ = 1/2 < S$\left({\grave{N}}_{BP}\left(x\right)\right),$ S (${N}_{BP}\left(x\right)$) > = -0.22 $\Rightarrow \; S\left({N}_{B}\right) > \; S\left({N}_{A}\right) \; \Rightarrow \; {N}_{B} \; > {N}_{A}$.

Now if we have,

${N}_{A}$ = [-0.6, -0.4], [-0.6, -0.4], (-0.4, -0.4), ${N}_{B}$ = [-0.7, -0.6], [-0.7, -0.6], (-0.6, -0.6)$\Rightarrow \; S\left({N}_{A}\right)$ = 0, $S\left({N}_{B}\right)$ = 0 $\nRightarrow \; \left({N}_{A}\right)$ = $\left({N}_{B}\right)$.

Then, a (${N}_{A}$) = -0.08 also, a (${N}_{B}$) = -0.24 $\Rightarrow$ a (${N}_{A}$) > a (${N}_{B}$) $\Rightarrow$ (${N}_{A}$) > (${N}_{B}$).

Remark:

(1) If NIVPF membership is greater than non-membership, then score must be positive.

(2) If non-membership is greater than membership, then score must be negative.

(3) Similarly, if membership of NCPFN, s is greater than non-membership of NCPFN's, then score must be positive.

(4) Similarly, if non-membership of NCPFN, s is greater than membership, then score must be negative; similarly, if membership of NCPFN, s is greater than non-member.

4.   Triangular N-cubic Pythagorean fuzzy set

In this section we initiated the concept of triangular N-cubic Pythagorean fuzzy numbers to discuss AHP method.

Definition: The set T = < (a, b, c), $3\mathrm{p}\left(\mathrm{x}\right)$, Np(x) > is said to be triangular NCPFN, s defined as:

M(Ǹ_p(x)) = $\left\{\begin{array}{c}\frac{x-a}{b-a}\left(Ǹ\mathrm{p}\left(\mathrm{x}\right)\right), a\le x < b\\ \frac{x-b}{c-b}\left(Ǹ\mathrm{p}\left(\mathrm{x}\right)\right), b < x\le c\\ 0, x < a, x > c\end{array}\right.$

Also,

M(N_p(x)) = $\left\{\begin{array}{c}\frac{x-a}{b-a}\left(\mathrm{N}p\left(\mathrm{x}\right)\right), a\le x < b\\ \frac{x-b}{c-b}\left(\mathrm{N}p\left(\mathrm{x}\right)\right), b < x\le c\\ -1, x < a, x > c\end{array}\right.$

Where -1 ≤ a≤ b≤ c ≤ 0 and a represent lower value, b represents middle and c represent upper value of triangular NCPFN, s and M (${\grave{N}}$_p(x)) and M (N_p(x)) be the membership and non-membership of NCPFN. The graphical representation is given by Figure 4.

The region between lower value and upper value of the function demonstrates us that this is the membership region of triangular N-cubic Pythagorean fuzzy set, whereas grey shade form of triangle represents the non-membership value of the function. We formulate the three parametric functions in order to specify the triangular N-cubic Pythagorean fuzzy set.

Definition: Let ${T}_{1} = < \left({a}_{1}, {a}_{2}, {a}_{3}\right), {{μ} }_{P1}\left(x\right), {\mathtt{η}}_{P1}\left(x\right) >$, ${T}_{2} = < \left({b}_{1}, {b}_{2}, {b}_{3}\right), {{μ} }_{P2}\left(x\right), {\mathtt{η}}_{P2}\left(x\right) >$ be two triangular NCPFNs. Then,

(1) T₁ + T₂ = < ${(a}_{1}+{a}_{2}, {b}_{1}+{b}_{2}, {c}_{1}+{c}_{2})$, max[${\mathtt{μ} }_{p1}^{-}\left(x\right), {\mathtt{μ} }_{p2}^{-}\left(x\right)$], max[${\mathtt{μ} }_{p1}^{+}\left(x\right), {\mathtt{μ} }_{p2}^{+}\left(x\right)$],

min [${\eta }_{Np1}, {\eta }_{Np2}]$ for p', p'' ∈ µ_p1(x), q', q'' ∈ η_p1(x)

(2) T₁T₂ = < ${(a}_{1}{a}_{2}, {b}_{1}{b}_{2}, {c}_{1}{c}_{2})$, max [${\mathtt{μ} }_{p1}^{-}\left(x\right), {\mathtt{μ} }_{p2}^{-}\left(x\right)$], max [${\mathtt{μ} }_{p1}^{+}\left(x\right), {\mathtt{μ} }_{p2}^{+}\left(x\right)$],

min [${\eta }_{Np1}, {\eta }_{Np2}]$ for p, p ∈ µ_p1(x), q', q'' ∈ η_p1(x)

(3) ∂T = < (a∂, b∂, c∂),

(4) ${T}^{-1}$ = < ($\frac{1}{c}, \frac{1}{b}, \frac{1}{a}$), µ_p1(x), ηp₁(x) >

(5) T₁ - T₂ = < ${(a}_{1}-{a}_{2}, {b}_{1}-{b}_{2}, {c}_{1}-{c}_{2})$, max[${\mathtt{μ} }_{p1}^{-}\left(x\right), {\mathtt{μ} }_{p2}^{-}\left(x\right)$], max[${\mathtt{μ} }_{p1}^{+}\left(x\right), {\mathtt{μ} }_{p2}^{+}\left(x\right)$],

min [${\eta }_{Np1}, {\eta }_{Np2}]$ for p', p'' ∈ µ_p1(x), q', q'' ∈ η_p1(x) >

Definition: A function ῶ for N-interval valued Pythagorean fuzzy sets are given below,

$\tilde{\mathtt{ω}} = \left(\right[{\mathtt{μ} }_{1p}^{-}\left(x\right)+{\mathtt{μ} }_{1p}^{+}\left(x\right)\left]\right)\mathrm{ }(1\mathrm{ }-\mathrm{ }\mathrm{ß})\mathrm{ }+\mathrm{ }\mathrm{ß}(2-({\mathtt{μ} }_{2p}^{-}\left(x\right)+{\mathtt{μ} }_{2p}^{+}\left(x\right)$ If ß = 0(pessimistic value),

$\tilde{\mathtt{ω}} = ({\mathtt{μ} }_{1p}^{-}\left(x\right)+{\mathtt{μ} }_{1p}^{+}\left(x\right))/2, \mathrm{ }\mathrm{i}\mathrm{f}\mathrm{ }\mathrm{ß}\mathrm{ } = \mathrm{ }1$ (Optimistic value),

${\tilde{\mathtt{ω}} = \mathrm{ }(2\mathrm{ }-\mathtt{μ} }_{2p}^{-}\left(x\right)-{\mathtt{μ} }_{2p}^{+}\left)\right)/2$), and if ß = (1/2),

$\tilde{\mathtt{ω}} = (2 +{{\mathtt{μ} }_{1p}^{-}\left(x\right)+{\mathtt{μ} }_{1p}^{+}\left(x\right)+ \mathtt{μ} }_{2p}^{-}\left(x\right)-{\mathtt{μ} }_{2p}^{+}\left(x\right))/4)$, generally used.

Also, ß is a real number between 0 and 1 (always positive) that can never be negative used as a variable based on the contribution of each once.

Definition: Let η = < $a, b$ > be the N-Pythagorean fuzzy sets then,

$\tilde{\rm{v}}\left(\mathtt{η}\right) = 0.5\mathrm{ }(1-\pi)(1+\mathrm{a}),$ for $\pi$ = $(-1{)}^{2\left(2\right)+1}\sqrt{1-{{\mathtt{μ} }_{X}\left(x\right)}^{2\left(2\right)}-{\eta }_{X}{\left(x\right)}^{2\left(2\right)}}$

and,

η = 1- ṽ(η) where, $0 \le ṽ\left(\eta \right), \mathrm{ꜧ}\le 1$

to find weight, we define a function for this set: ${{\mathrm{Y}\mathrm{ } = \mathrm{ }0.5\mathrm{ }\left\{\right(2\mathrm{ }+\mathtt{μ} }_{1p}^{-}\left(x\right)+{\mathtt{μ} }_{1p}^{+}\left(x\right)-\mathtt{μ} }_{2p}^{-}\left(x\right)-{\mathtt{μ} }_{2p}^{+}\left(x\right))/4)\mathrm{ }+\mathrm{ }(1-\tilde{\rm{v}}(\mathtt{η}\left)\right)\},$

where, $0 \le ṽ\left(\eta \right), \mathrm{ꜧ}\le 1$ and ṽ (η) already defined.

5.   Applications

Investigation of downfall of IA (international airlines) using analytical hierarchy process under N-cubic Pythagorean fuzzy sets

The analytical hierarchy technique, first proposed by Thomas L. Saaty in 1970, was used to analyze difficult decisions and rank alternatives using mathematics and human reasoning in a hierarchy framework, however it failed to measure uncertainty. Following that, a fuzzy analytical hierarchy approach was established, although it is difficult to cover many challenges in decision-making. To put it another way, we're starting an NCPF analytical hierarchical process for complex decision-making. Our key goal is to determine what is causing international airlines to fall. To do so, we must first identify the essential key elements (both external and internal) that are related with airlines.

5.1.   Determining the factor affecting airlines

Following are the key factors that affect airline companies given in Table 1 with brief explanation.

5.2.   Investigation of N-cubic Pythagorean fuzzy analytic hierarchy process approach

These steps are followed to determine the perspective approach.

Assessment scale given by Saaty in 1970, by this we define linguistic triangular scale under NCPFN is given in Table 2.

Following are the steps of method (Chang's extent analysis of AHP under NCPFN's) are given by:

Step (a): the NCPF (${Q}_{K}$) extent value of kth criterion is given by, $\sum _{l}^{m}{\boldsymbol{M}}_{{h}_{k}}^{\boldsymbol{l}}{\left[\sum _{k}^{n}\sum _{l}^{m}{\boldsymbol{M}}_{{h}_{k}}^{\boldsymbol{l}}\right]}^{-1}$, where l, m and n represent lower, middle, and upper values of fuzzy relation.

Step(b): Degree of possibility can be analyzed by $v\left({Q}_{1}\le {Q}_{2}\right)$ = ${inf}_{r\le t}[\mathrm{max}(\left({{\grave{U}} }_{R!}\left(r\right)\right), {({\grave{U}} }_{R2}\left(t\right))\left)\right], {sup}_{r\le t}[\mathrm{m}\mathrm{i}\mathrm{n}(\left({\eta }_{S1}\left(r\right), \left({\eta }_{S2}\left(t\right)\right)\right].$ Where,

And for non-membership ${v}_{\eta }\left({S}_{1}\ge {S}_{2}\right) = {sup}_{r\le t}[\mathrm{m}\mathrm{i}\mathrm{n}(\left({\eta }_{S1}\left(r\right), \left({\eta }_{S2}\left(t\right)\right)\right] = {\eta }_{S1\cap S2}\left(d\right) =$

Note: Another matter we discuss here is that this structure is a combination of interval valued Pythagorean fuzzy sets and Pythagorean fuzzy sets, so how do we find the weight of these two sets in a single term? (For both the cases)

Step(c): R = Max {${R}_{5}\le {R}_{2}, {R}_{3}, {R}_{4}, {R}_{1}, {R}_{6}$}, S = min {${S}_{5}\ge {S}_{2}, {S}_{3}, {S}_{4}, {S}_{1}, {S}_{6}$}

Step (d): Analyzation of weights:

The weights of important elements can be determined for final ranking. The same scenario also determines the weights of sub criterion. Adding each sub-relative criterion's weight to the main criteria.

Step (e):

The primary criterion, sub criterion, and relative ranking are all ranked in this stage. Criteria are ranked, Ranking on the ground.

Before performing Chang's study, we create a comparison matrix of key factors based on expert judgments.

5.3.   Comparison matrix between main factors (see Table 3)

5.3.1.   Comparison matrix between sub-criteria (operational factor) (see Table 4)

5.3.2.   Comparison matrix between sub-criteria (economic and government factor) (see Table 5)

5.3.3.   Comparison matrix between sub-criteria (performance related factor) (see Table 6)

5.3.4.   Comparison matrix between sub-criteria (financial factor) (see Table 7)

5.3.5.   Comparison matrix between sub-criteria (market related factors) (see Table 8)

5.3.6.   Comparison matrix between sub-criteria (external factors) (see Table 9)

According to Chang's extent analysis (1992), we define some steps given below: By given Chang's method, the value of criterion ${h}_{k}$ can be evaluated by given scenario.

${M}_{{h}_{k}}^{l}$, (l = 1, 2, …, m), (k = 1, 2, …., n).

Step 1: The NCPF (${Q}_{K}$) extent value of kth criterion is given by $\sum _{l}^{m}{\boldsymbol{M}}_{{h}_{k}}^{\boldsymbol{l}}{\left[\sum _{k}^{n}\sum _{l}^{m}{\boldsymbol{M}}_{{h}_{k}}^{\boldsymbol{l}}\right]}^{-1}$,

Q (1) = (0.133, 0.23, 0.42), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)}, Q (2) = (0.08, 0.16, 0.30), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6), Q (3) = (0.06, 0.13, 0.27), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6), Q (4) = (0.12, 0.22, 0.39), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6), Q (5) = (0.08, 0.14, 0.25), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6), Q (6) = (0.05, 0.09, 0.16), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6).

Step 2: Analyzation of probabilistic degree:

Calculations for membership and non-membership

${v}_{{\grave{U}} }\left({R}_{1}\le {R}_{2}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{1}\ge {S}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{1}\le {R}_{3}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{1}\ge {S}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{1}\le {R}_{4}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{1}\ge {S}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{1}\le {R}_{5}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{1}\ge {S}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{1}\le {R}_{6}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{1}\ge {S}_{2}\right)$ = (-0.8, -0.6)

Max {${R}_{1}\le {R}_{2}, {R}_{3}, {R}_{4}, {R}_{5}, {R}_{6}$} = [-0.5, -0.3], [-0.4, -0.1], min {${S}_{1}\ge {S}_{2}, {S}_{3}, {S}_{4}, {S}_{5}, {S}_{6}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{2}\le {R}_{1}\right)$ = [-0.36, -0.14], [-0.25, -0.06], ${v}_{{\grave{U}} }\left({S}_{2}\ge {S}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{2}\le {R}_{3}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{2}\ge {S}_{3}\right)$ = (-0.20, -0.55)

${v}_{{\grave{U}} }\left({R}_{2}\le {R}_{4}\right)$ = [-0.33, -0.22], [-0.2, -0.07], ${v}_{{\grave{U}} }\left({S}_{2}\ge {S}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{2}\le {R}_{5}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{2}\ge {S}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{2}\le {R}_{6}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{2}\ge {S}_{6}\right)$ = (-0.19, -0.54)

Max {${R}_{2}\le {R}_{1}, {R}_{3}, {R}_{4}, {R}_{5}, {R}_{6}$} = [-0.33, -0.14], [-0.2, -0.06], min {${S}_{2}\ge {S}_{1}, {S}_{3}, {S}_{4}, {S}_{5}, {S}_{6}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{3}\le {R}_{1}\right)$ = [-0.27, -0.16], [-0.25, -0.1], ${v}_{{\grave{U}} }\left({S}_{3}\ge {S}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{3}\le {R}_{2}\right)$ = [-0.16, -0.08], [-0.13, -0.03], ${v}_{{\grave{U}} }\left({S}_{3}\ge {S}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{3}\le {R}_{4}\right)$ = [-0.25, -0.14], [-0.22, -0.04], ${v}_{{\grave{U}} }\left({S}_{3}\ge {S}_{4}\right)$ = (-0.18, -0.50)

${v}_{{\grave{U}} }\left({R}_{3}\le {R}_{5}\right)$ = [-0.4, -0.2], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{3}\ge {S}_{5}\right)$ = (-0.15, -0.44)

${v}_{{\grave{U}} }\left({R}_{3}\le {R}_{6}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{3}\ge {S}_{6}\right)$ = (-0.8, -0.6)

Max {${R}_{3}\le {R}_{1}, {R}_{2}, {R}_{4}, {R}_{5}, {R}_{6}$} = [-0.16, -0.08], [-0.13, -0.03], min {${S}_{3}\ge {S}_{1}, {S}_{2}, {S}_{4}, {S}_{5}, {S}_{6}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{4}\le {R}_{1}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{4}\ge {S}_{1}\right)$ = (-0.18, -0.49)

${v}_{{\grave{U}} }\left({R}_{4}\le {R}_{2}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{4}\ge {S}_{2}\right)$ = (-0.15, -0.43)

${v}_{{\grave{U}} }\left({R}_{4}\le {R}_{3}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{4}\ge {S}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{4}\le {R}_{5}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{4}\ge {S}_{5}\right)$ = (-0.21, -0.59)

${v}_{{\grave{U}} }\left({R}_{4}\le {R}_{6}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{4}\ge {S}_{6}\right)$ = (-0.8, -0.6)

Max {${R}_{4}\le {R}_{1}, {R}_{2}, {R}_{3}, {R}_{5}, {R}_{6}$} = [-0.5, -0.3], [-0.4, -0.1], min {${S}_{4}\ge {S}_{2}, {S}_{3}, {S}_{1}, {S}_{5}, {S}_{6}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{5}\le {R}_{1}\right)$ = [-0.22, -0.2], [-0.14, -0.05], ${v}_{{\grave{U}} }\left({S}_{5}\ge {S}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{5}\le {R}_{2}\right)$ = [-0.5, -0.2], [-0.3, -0.09], ${v}_{{\grave{U}} }\left({S}_{5}\ge {S}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{5}\le {R}_{3}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{5}\ge {S}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{5}\le {R}_{4}\right)$ = [-0.3, -0.16], [-0.25, -0.05], ${v}_{{\grave{U}} }\left({S}_{5}\ge {S}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{5}\le {R}_{6}\right)$ = [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({S}_{5}\ge {S}_{6}\right)$ = (-0.21, -0.59)

Max {${R}_{5}\le {R}_{2}, {R}_{3}, {R}_{4}, {R}_{1}, {R}_{6}$} = [-0.22, -0.16], [-0.14, -0.05], min {${S}_{5}\ge {S}_{2}, {S}_{3}, {S}_{4}, {S}_{1}, {S}_{6}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{6}\le {R}_{1}\right)$ = [-0.07, -0.04], [-0.06, -0.01], ${v}_{{\grave{U}} }\left({S}_{6}\ge {S}_{1}\right)$ = (-0.18, -0.52)

${v}_{{\grave{U}} }\left({R}_{6}\le {R}_{2}\right)$ = [-0.2, -0.1], [-0.2, -0.08], ${v}_{{\grave{U}} }\left({S}_{6}\ge {S}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{6}\le {R}_{3}\right)$ = [-0.3, -0.2], [-0.4, -0.07], ${v}_{{\grave{U}} }\left({S}_{6}\ge {S}_{3}\right)$ = (-0.20, -0.57)

${v}_{{\grave{U}} }\left({R}_{6}\le {R}_{4}\right)$ = [-0.1, -0.06], [-0.1, -0.04], ${v}_{{\grave{U}} }\left({S}_{6}\ge {S}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({R}_{6}\le {R}_{5}\right)$ = [-0.3, -0.24], [-0.3, -0.08], ${v}_{{\grave{U}} }\left({S}_{6}\ge {S}_{5}\right)$ = (-0.8, -0.6)

Max {${R}_{6}\le {R}_{2}, {R}_{3}, {R}_{4}, {R}_{5}, {R}_{1}$} = [-0.07, -0.04], [-0.06, -0.01], min {${S}_{6}\ge {S}_{2}, {S}_{3}, {S}_{4}, {S}_{5}, {S}_{1}$} = (-0.8, -0.6)

Sub-criteria (operational factor):

Step 1: The NCPF (${O}_{K}$) extent value of kth sub-criterion (operational factor) is given by:

O (1) = (0.06, 0.12, 0.23), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

O (2) = (0.078, 0.143, 0.26), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

O (3) = (0.073, 0.12, 0.23), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

O (4) = (0.076, 0.149, 0.278), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

O (5) = (0.052, 0.091, 0.16), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

O (6) = (0.08, 0.153, 0.274), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

O (7) = (0.077, 0.143, 0.25), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

O (8) = (0.034, 0.063, 0.132), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

Step 2:

${v}_{{\grave{U}} }\left({PO}_{1}\le {PO}_{2}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{1}\ge {OP}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{1}\le {PO}_{3}\right) =$ [-0.2, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({OP}_{1}\ge {OP}_{3}\right)$ = (-0.16, -0.47)

${v}_{{\grave{U}} }\left({PO}_{1}\le {PO}_{4}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{1}\ge {OP}_{4}\right) =$(-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{1}\le {PO}_{5}\right) =$ [-0.2, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({OP}_{1}\ge {OP}_{5}\right)$ = (-0.16, -0.48)

${v}_{{\grave{U}} }\left({PO}_{1}\le {PO}_{6}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{1}\ge {OP}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{1}\le {PO}_{7}\right) =$ [-0.2, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({OP}_{1}\ge {OP}_{7}\right)$ = (-0.16, -0.48)

${v}_{{\grave{U}} }\left({PO}_{1}\le {PO}_{8}\right) =$ [-0.2, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({OP}_{1}\ge {OP}_{8}\right)$ = (-0.16, -0.47)

Max {${PO}_{1}\le {PO}_{2}, {PO}_{3}, {PO}_{4}, {PO}_{5}, {PO}_{6}, {PO}_{7}, {PO}_{8}$} = [-0.2, -0.05], [-0.11, -0.007], min {${OP}_{1}\ge {OP}_{2}, {OP}_{3}, {OP}_{4}, {OP}_{5}, {OP}_{6}, {OP}_{7}, {OP}_{8}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{2}\le {PO}_{1}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{2}\ge {OP}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{2}\le {PO}_{3}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{2}\ge {OP}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{2}\le {PO}_{4}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{2}\ge {OP}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{2}\le {PO}_{5}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{2}\ge {OP}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{2}\le {PO}_{6}\right) =$ [-0.23, -0.06], [-0.12, -0.008], ${v}_{{\grave{U}} }\left({OP}_{2}\ge {OP}_{6}\right)$ = (-0.17, -0.51)

${v}_{{\grave{U}} }\left({PO}_{2}\le {PO}_{7}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{2}\ge {OP}_{7}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{2}\le {PO}_{8}\right) =$ [-0.22, -0.06], [-0.12, -0.008], ${v}_{{\grave{U}} }\left({OP}_{2}\ge {OP}_{8}\right)$ = (-0.17, -0.51)

Max {${PO}_{2}\le {PO}_{1}, {PO}_{3}, {PO}_{4}, {PO}_{5}, {PO}_{6}, {PO}_{7}, {PO}_{8}$} = [-0.22, -0.06], [-0.12, -0.008], min {${OP}_{2}\ge {OP}_{1}, {OP}_{3}, {OP}_{4}, {OP}_{5}, {OP}_{6}, {OP}_{7}, {OP}_{8}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{3}\le {PO}_{1}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{3}\ge {OP}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{3}\le {PO}_{2}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{3}\ge {OP}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{3}\le {PO}_{4}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{3}\ge {OP}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{3}\le {PO}_{5}\right) =$ [-0.20, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({OP}_{3}\ge {OP}_{5}\right)$ = (-0.16, -0.47)

${v}_{{\grave{U}} }\left({PO}_{3}\le {PO}_{6}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{3}\ge {OP}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{3}\le {PO}_{7}\right) =$ [-0.21, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({OP}_{3}\ge {OP}_{7}\right)$ = (-0.16, -0.48)

${v}_{{\grave{U}} }\left({PO}_{3}\le {PO}_{8}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{3}\ge {OP}_{8}\right)$ = (-0.8, -0.6)

Max {${PO}_{3}\le {PO}_{2}, {PO}_{1}, {PO}_{4}, {PO}_{5}, {PO}_{6}, {PO}_{7}, {PO}_{8}$} = [-0.20, -0.05], [-0.11, -0.007], min {${OP}_{3}\ge {OP}_{2}, {OP}_{1}, {OP}_{4}, {OP}_{5}, {OP}_{6}, {OP}_{7}, {OP}_{8}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{4}\le {PO}_{1}\right) =$ [-0.20, -0.05], [-0.11, -0.008], ${v}_{{\grave{U}} }\left({OP}_{4}\ge {OP}_{1}\right)$ = (-0.16, -0.48)

${v}_{{\grave{U}} }\left({PO}_{4}\le {PO}_{2}\right) =$ [-0.20, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({OP}_{4}\ge {OP}_{2}\right)$ = (-0.16, -0.47)

${v}_{{\grave{U}} }\left({PO}_{4}\le {PO}_{3}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{4}\ge {OP}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{4}\le {PO}_{5}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{4}\ge {OP}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{4}\le {PO}_{6}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{4}\ge {OP}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{4}\le {PO}_{7}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{4}\ge {OP}_{7}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{4}\le {PO}_{8}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{4}\ge {OP}_{8}\right)$ = (-0.8, -0.6)

Max {${PO}_{4}\le {PO}_{2}, {PO}_{3}, {PO}_{1}, {PO}_{5}, {PO}_{6}, {PO}_{7}, {PO}_{8}$} = [-0.20, -0.05], [-0.11, -0.007], min {${OP}_{4}\ge {OP}_{2}, {OP}_{3}, {OP}_{1}, {OP}_{5}, {OP}_{6}, {OP}_{7}, {OP}_{8}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{5}\le {PO}_{1}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{5}\ge {OP}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{5}\le {PO}_{2}\right) =$ [-0.22, -0.06], [-0.12, -0.008], ${v}_{{\grave{U}} }\left({OP}_{5}\ge {OP}_{2}\right)$ = (-0.17, -0.50)

${v}_{{\grave{U}} }\left({PO}_{5}\le {PO}_{3}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{5}\ge {OP}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{5}\le {PO}_{4}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{5}\ge {OP}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{5}\le {PO}_{6}\right) =$ [-0.21, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({OP}_{5}\ge {OP}_{6}\right)$ = (-0.17, -0.52)

${v}_{{\grave{U}} }\left({PO}_{5}\le {PO}_{7}\right) =$ [-0.18, -0.05], [-0.10, -0.007], ${v}_{{\grave{U}} }\left({OP}_{5}\ge {OP}_{7}\right)$ = (-0.18, -0.53)

${v}_{{\grave{U}} }\left({PO}_{5}\le {PO}_{8}\right) =$ [-0.18, -0.04], [-0.09, -0.005], ${v}_{{\grave{U}} }\left({OP}_{5}\ge {OP}_{8}\right)$ = (-0.17, -0.51)

Max {${PO}_{5}\le {PO}_{2}, {PO}_{3}, {PO}_{4}, {PO}_{1}, {PO}_{6}, {PO}_{7}, {PO}_{8}$} = [-0.18, -0.04], [-0.09, -0.005], min {${OP}_{5}\ge {OP}_{2}, {OP}_{3}, {OP}_{4}, {OP}_{1}, {OP}_{6}, {OP}_{7}, {OP}_{8}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{6}\le {PO}_{1}\right) =$ [-0.18, -0.05], [-0.11, -0.008], ${v}_{{\grave{U}} }\left({OP}_{6}\ge {OP}_{1}\right)$ = (-0.18, -0.54)

${v}_{{\grave{U}} }\left({PO}_{6}\le {PO}_{2}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{6}\ge {OP}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{6}\le {PO}_{3}\right) =$ [-0.18, -0.05], [-0.11, -0.008], ${v}_{{\grave{U}} }\left({OP}_{6}\ge {OP}_{3}\right)$ = (-0.18, -0.54)

${v}_{{\grave{U}} }\left({PO}_{6}\le {PO}_{4}\right) =$ [-0.18, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({OP}_{6}\ge {OP}_{4}\right)$ = (-0.18, -0.53)

${v}_{{\grave{U}} }\left({PO}_{6}\le {PO}_{5}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{6}\ge {OP}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{6}\le {PO}_{7}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{6}\ge {OP}_{7}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{6}\le {PO}_{8}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{6}\ge {OP}_{8}\right)$ = (-0.8, -0.6)

Max {${PO}_{6}\le {PO}_{2}, {PO}_{3}, {PO}_{4}, {PO}_{5}, {PO}_{1}, {PO}_{7}, {PO}_{8}$} = [-0.18, -0.05], [-0.11, -0.007], min {${OP}_{6}\ge {OP}_{2}, {OP}_{3}, {OP}_{4}, {OP}_{5}, {OP}_{1}, {OP}_{7}, {OP}_{8}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{7}\le {PO}_{1}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{7}\ge {OP}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{7}\le {PO}_{2}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{7}\ge {OP}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{7}\le {PO}_{3}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{7}\ge {OP}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{7}\le {PO}_{4}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{7}\ge {OP}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{7}\le {PO}_{5}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{7}\ge {OP}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{7}\le {PO}_{6}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{7}\ge {OP}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{7}\le {PO}_{8}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{7}\ge {OP}_{8}\right)$ = (-0.8, -0.6)

Max ${\{PO}_{7}\le {PO}_{2}, {PO}_{3}, {PO}_{4}, {PO}_{5}, {PO}_{6}, {PO}_{1}, {PO}_{8}\}$ = [-0.5, -0.3], [-0.4, -0.1], min {${OP}_{7}\ge {OP}_{2}, {OP}_{3}, {OP}_{4}, {OP}_{5}, {OP}_{6}, {OP}_{1}, {OP}_{8}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{8}\le {PO}_{1}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{8}\ge {OP}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{8}\le {PO}_{2}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{8}\ge {OP}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{8}\le {PO}_{3}\right) =$ [-0.23, -0.06], [-0.13, -0.008], ${v}_{{\grave{U}} }\left({OP}_{8}\ge {OP}_{3}\right)$ = (-0.18, -0.54)

${v}_{{\grave{U}} }\left({PO}_{8}\le {PO}_{4}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{8}\ge {OP}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{8}\le {PO}_{5}\right) =$ [-0.23, -0.06], [-0.13, -0.008], ${v}_{{\grave{U}} }\left({OP}_{8}\ge {OP}_{5}\right)$ = (-0.18, -0.54)

${v}_{{\grave{U}} }\left({PO}_{8}\le {PO}_{6}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{8}\ge {OP}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PO}_{8}\le {PO}_{7}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({OP}_{8}\ge {OP}_{7}\right)$ = (-0.8, -0.6)

Max {${PO}_{8}\le {PO}_{2}, {PO}_{3}, {PO}_{4}, {PO}_{5}, {PO}_{6}, {PO}_{7}, {PO}_{1}$} = [-0.23, -0.06], [-0.13, -0.008], min {${OP}_{8}\ge {OP}_{2}, {OP}_{3}, {OP}_{4}, {OP}_{5}, {OP}_{6}, {OP}_{7}, {OP}_{1}$} = (-0.8, -0.6)

Sub-criteria (economic/government factor)

Step1: The NCPF (${E}_{K}$) extent value of kth sub-criterion (economic/government factor) is given by,

E (1) = (0.21, 0.35, 0.34), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

E (2) = (0.18, 0.30, 0.29), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

E (3) = (0.48, 0.068, 0.061), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

E (4) = (0.15, 0.27, 0.26), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

Step 2:

${v}_{{\grave{U}} }\left({EC}_{1}\le {EC}_{2}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{1}\ge {GO}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({EC}_{1}\le {EC}_{3}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{1}\ge {GO}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({EC}_{1}\le {EC}_{4}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{1}\ge {GO}_{4}\right)$ = (-0.8, -0.6)

Max {${EC}_{1}\le {EC}_{2}, {EC}_{3}, {EC}_{4}$} = [-0.5, -0.3], [-0.4, -0.1], min {${GO}_{1}\ge {GO}_{2}, {GO}_{3}, {GO}_{4}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({EC}_{2}\le {EC}_{1}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{2}\ge {GO}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({EC}_{2}\le {EC}_{3}\right) =$ [-0.9, -0.1], [-0.4, -0.006], ${v}_{{\grave{U}} }\left({GO}_{2}\ge {GO}_{3}\right)$ = (-0.7, -0.5)

${v}_{{\grave{U}} }\left({EC}_{2}\le {EC}_{4}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{2}\ge {GO}_{4}\right)$ = (-0.8, -0.6)

Max {${EC}_{2}\le {EC}_{1}, {EC}_{3}, {EC}_{4}$} = [-0.5, -0.1], [-0.4, -0.006], min {${GO}_{2}\ge {GO}_{1}, {GO}_{3}, {GO}_{4}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({EC}_{3}\le {EC}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{3}\ge {GO}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({EC}_{3}\le {EC}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{3}\ge {GO}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({EC}_{3}\le {EC}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{3}\ge {GO}_{4}\right)$ = (-1, -0.6)

Max {${EC}_{3}\le {EC}_{2}, {EC}_{1}, {EC}_{4}$} = [-0.5, -0.3], [-0.4, -0.1], min {${GO}_{3}\ge {GO}_{2}, {GO}_{1}, {GO}_{4}$} = (-1, -0.6)

${v}_{{\grave{U}} }\left({EC}_{4}\le {EC}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{4}\ge {GO}_{1}\right)$ = (-1, -0.6)

${v}_{{\grave{U}} }\left({EC}_{4}\le {EC}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{4}\ge {GO}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({EC}_{4}\le {EC}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({GO}_{4}\ge {GO}_{3}\right)$ = (-1, -0.6)

Max {${EC}_{4}\le {EC}_{2}, {EC}_{3}, {EC}_{1}$} = [-0.5, -0.3], [-0.4, -0.1], min {${GO}_{4}\ge {GO}_{2}, {GO}_{3}, {GO}_{1}$} = (-1, -0.6)

Sub-criteria (performance related factor)

Step 1: The NCPF (${P}_{K}$) extent value of kth sub-criterion (performance related factor) is given by

P (1) = (0.089, 0.168, 0.30), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

P (2) = (0.080, 0.15, 0.293), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

P (3) = (0.093, 0.164, 0.296), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

P (4) = (0.06, 0.124, 0.23), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

P (5) = (0.098, 0.181, 0.32), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

P (6) = (0.04, 0.07, 0.13), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

P (7) = (0.07, 0.133, 0.24), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

Step 2:

${v}_{{\grave{U}} }\left({PE}_{1}\le {PE}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{1}\ge {RE}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{1}\le {PE}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{1}\ge {RE}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{1}\le {PE}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{1}\ge {RE}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{1}\le {PE}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{1}\ge {RE}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{1}\le {PE}_{6}\right) =$[-0.23, -0.06], [-0.13, -0.009], ${v}_{{\grave{U}} }\left({RE}_{1}\ge {RE}_{6}\right)$ = (-0.19, -0.55)

${v}_{{\grave{U}} }\left({PE}_{1}\le {PE}_{7}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{1}\ge {RE}_{7}\right)$ = (-0.8, -0.6)

Max $\{{PE}_{1}\le {PE}_{2}, {PE}_{3}, {PE}_{4}, {PE}_{5}, {PE}_{6}, {PE}_{7}\}$ = [-0.23, -0.06], [-0.13, -0.009], min ${\{RE}_{1}\ge {RE}_{2}, {RE}_{3}, {RE}_{4}, {RE}_{5}, {RE}_{6}, {RE}_{7}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{2}\le {PE}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{2}\ge {RE}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{2}\le {PE}_{3}\right) =$[-0.21, -0.06], [-0.12, -0.009], ${v}_{{\grave{U}} }\left({RE}_{2}\ge {RE}_{3}\right)$ = (-0.17, -0.48)

${v}_{{\grave{U}} }\left({PE}_{2}\le {PE}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{2}\ge {RE}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{2}\le {PE}_{5}\right) =$[-0.21, -0.05], [-0.11, -0.008], ${v}_{{\grave{U}} }\left({RE}_{2}\ge {RE}_{5}\right)$ = (-0.16, -0.48)

${v}_{{\grave{U}} }\left({PE}_{2}\le {PE}_{6}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{2}\ge {RE}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{2}\le {PE}_{7}\right) =$[-0.21, -0.06], [-0.12, -0.009], ${v}_{{\grave{U}} }\left({RE}_{2}\ge {RE}_{7}\right)$ = (-0.17, -0.49)

Max $\{{PE}_{2}\le {PE}_{1}, {PE}_{3}, {PE}_{4}, {PE}_{5}, {PE}_{6}, {PE}_{7}\}$ = [-0.21, -0.05], [-0.11, -0.009], min {${RE}_{2}\ge {RE}_{1}, {RE}_{3}, {RE}_{4}, {RE}_{5}, {RE}_{6}, {RE}_{7}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{3}\le {PE}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{3}\ge {RE}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{3}\le {PE}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{3}\ge {RE}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{3}\le {PE}_{4}\right) =$[-0.23, -0.06], [-0.13, -0.009], ${v}_{{\grave{U}} }\left({RE}_{3}\ge {RE}_{4}\right)$ = (-0.18, -0.53)

${v}_{{\grave{U}} }\left({PE}_{3}\le {PE}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{3}\ge {RE}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{3}\le {PE}_{6}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{3}\ge {RE}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{3}\le {PE}_{7}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{3}\ge {RE}_{7}\right)$ = (-0.8, -0.6)

Max ${\{PE}_{3}\le {PE}_{2}, {PE}_{1}, {PE}_{4}, {PE}_{5}, {PE}_{6}, {PE}_{7}$} = [-0.23, -0.06], [-0.13, -0.009], min {${RE}_{3}\ge {RE}_{2}, {RE}_{1}, {RE}_{4}, {RE}_{5}, {RE}_{6}, {RE}_{7}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{4}\le {PE}_{1}\right) =$[-0.22, -0.06], [-0.12, -0.009], ${v}_{{\grave{U}} }\left({RE}_{4}\ge {RE}_{1}\right)$ = (-0.18, -0.54)

${v}_{{\grave{U}} }\left({PE}_{4}\le {PE}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{4}\ge {RE}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{4}\le {PE}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{4}\ge {RE}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{4}\le {PE}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{4}\ge {RE}_{5}\right)$ = (-0.17, -0.51)

${v}_{{\grave{U}} }\left({PE}_{4}\le {PE}_{6}\right) =$[-0.20, -0.05], [-0.11, -0.009], ${v}_{{\grave{U}} }\left({RE}_{4}\ge {RE}_{6}\right)$ = (-0.17, -0.50)

${v}_{{\grave{U}} }\left({PE}_{4}\le {PE}_{7}\right) =$[-0.21, -0.05], [-0.11, -0.008], ${v}_{{\grave{U}} }\left({RE}_{4}\ge {RE}_{7}\right)$ = (-0.17, -0.51)

Max ${\{PE}_{4}\le {PE}_{2}, {PE}_{3}, {PE}_{1}, {PE}_{5}, {PE}_{6}, {PE}_{7}\}$ = [-0.21, -0.05], [-0.11, -0.008], min {${RE}_{4}\ge {RE}_{2}, {RE}_{3}, {RE}_{1}, {RE}_{5}, {RE}_{6}, {RE}_{7}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{5}\le {PE}_{1}\right) =$[-0.19, -0.05], [-0.11, -0.008], ${v}_{{\grave{U}} }\left({RE}_{5}\ge {RE}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{5}\le {PE}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{5}\ge {RE}_{2}\right)$ = (-0.18, -0.52)

${v}_{{\grave{U}} }\left({PE}_{5}\le {PE}_{3}\right) =$[-0.19, -0.05], [-0.11, -0.009], ${v}_{{\grave{U}} }\left({RE}_{5}\ge {RE}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{5}\le {PE}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{5}\ge {RE}_{4}\right)$ = (-0.16, -0.49)

${v}_{{\grave{U}} }\left({PE}_{5}\le {PE}_{6}\right) =$[-0.22, -0.05], [-0.12, -0.007], ${v}_{{\grave{U}} }\left({RE}_{5}\ge {RE}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{5}\le {PE}_{7}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{5}\ge {RE}_{7}\right)$ = (-0.8, -0.6)

Max $\{{PE}_{5}\le {PE}_{2}, {PE}_{3}, {PE}_{4}, {PE}_{1}, {PE}_{6}, {PE}_{7}\}$ = [-0.19, -0.05], [-0.11, -0.007], min {${RE}_{5}\ge {RE}_{2}, {RE}_{3}, {RE}_{4}, {RE}_{1}, {RE}_{6}, {RE}_{7}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{6}\le {PE}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{6}\ge {RE}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{6}\le {PE}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{6}\ge {RE}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{6}\le {PE}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{6}\ge {RE}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{6}\le {PE}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{6}\ge {RE}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{6}\le {PE}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{6}\ge {RE}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{6}\le {PE}_{7}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{6}\ge {RE}_{7}\right)$ = (-0.17, -0.51)

Max $\{{PE}_{6}\le {PE}_{2}, {PE}_{3}, {PE}_{4}, {PE}_{5}, {PE}_{1}, {PE}_{7}\}$ = [-0.5, -0.3], [-0.4, -0.1], min {${RE}_{6}\ge {RE}_{2}, {RE}_{3}, {RE}_{4}, {RE}_{5}, {RE}_{1}, {RE}_{7}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{7}\le {PE}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({RE}_{7}\ge {RE}_{1}\right)$ = (-0.17, -0.50)

${v}_{{\grave{U}} }\left({PE}_{7}\le {PE}_{2}\right) =$[-0.12, -0.04], [-0.07, -0.008], ${v}_{{\grave{U}} }\left({RE}_{7}\ge {RE}_{2}\right)$ = (-0.17, -0.51)

${v}_{{\grave{U}} }\left({PE}_{7}\le {PE}_{3}\right) =$[-0.13, -0.04], [-0.08, -0.007], ${v}_{{\grave{U}} }\left({RE}_{7}\ge {RE}_{3}\right)$ = (-0.16, -0.48)

${v}_{{\grave{U}} }\left({PE}_{7}\le {PE}_{4}\right) =$[-0.11, -0.03], [-0.06, -0.007], ${v}_{{\grave{U}} }\left({RE}_{7}\ge {RE}_{4}\right)$ = (-0.18, -0.52)

${v}_{{\grave{U}} }\left({PE}_{7}\le {PE}_{5}\right) =$[-0.17, -0.05], [-0.10, -0.007], ${v}_{{\grave{U}} }\left({RE}_{7}\ge {RE}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({PE}_{7}\le {PE}_{6}\right) =$[-0.10, -0.03], [-0.06, -0.008], ${v}_{{\grave{U}} }\left({RE}_{7}\ge {RE}_{6}\right)$ = (-0.16, -0.48)

Max $\{{PE}_{7}\le {PE}_{2}, {PE}_{3}, {PE}_{4}, {PE}_{5}, {PE}_{6}, {PE}_{1}\}$ = [-0.10, -0.03], [-0.06, -0.007], min $\{{RE}_{7}\ge {RE}_{2}, {RE}_{3}, {RE}_{4}, {RE}_{5}, {RE}_{6}, {RE}_{1}$} = (-0.8, -0.6)

Sub-criteria (financial factor)

Step 1: The NCPF (${F}_{K}$) extent value of kth sub-criterion (financial factor) is given by,

F (1) = (0.067, 0.122, 0.225), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

F (2) = (0.072, 0.128, 0.234), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

F (3) = (0.051, 0.090, 0.166), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

F (4) = (0.064, 0.112, 0.204), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

F (5) = (0.043, 0.076, 0.141), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

F (6) = (0.055, 0.095, 0.169), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

F (7) = (0.039, 0.065, 0.115), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

F (8) = (0.083, 0.153, 0.279), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

F (9) = (0.034, 0.065, 0.128), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

F (10) = (0.051, 0.090, 0.165), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

Step 2:

${v}_{{\grave{U}} }\left({IF}_{1}\le {IF}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{1}\ge {FI}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{1}\le {IF}_{3}\right) =$[-0.21, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({FI}_{1}\ge {FI}_{3}\right)$ = (-0.16, -0.49)

${v}_{{\grave{U}} }\left({IF}_{1}\le {IF}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{1}\ge {FI}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{1}\le {IF}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{1}\ge {FI}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{1}\le {IF}_{6}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{1}\ge {FI}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{1}\le {IF}_{7}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{1}\ge {FI}_{7}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{1}\le {IF}_{8}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{1}\ge {FI}_{8}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{1}\le {IF}_{9}\right) =$[-0.20, -0.05], [-0.11, -0.008], ${v}_{{\grave{U}} }\left({FI}_{1}\ge {FI}_{9}\right)$ = (-0.17, -0.50)

${v}_{{\grave{U}} }\left({IF}_{1}\le {IF}_{10}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{1}\ge {FI}_{10}\right)$ = (-0.8, -0.6)

Max {${IF}_{1}\le {IF}_{2}, {IF}_{3}, {IF}_{4}, {IF}_{5}, {IF}_{6}, {IF}_{7}, {IF}_{8}, {IF}_{9}, {IF}_{10}$} = [-0.20, -0.05], [-0.11, -0.007], min {${FI}_{1}\ge {FI}_{2}, {FI}_{3}, {FI}_{4}, {FI}_{5}, {FI}_{6}, {FI}_{7}, {FI}_{8}, {FI}_{9}, {FI}_{10}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{2}\le {IF}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{2}\ge {FI}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{2}\le {IF}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{2}\ge {FI}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{2}\le {IF}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{2}\ge {FI}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{2}\le {IF}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{2}\ge {FI}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{2}\le {IF}_{6}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{2}\ge {FI}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{2}\le {IF}_{7}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{2}\ge {FI}_{7}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{2}\le {IF}_{8}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{2}\ge {FI}_{8}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{2}\le {IF}_{9}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{2}\ge {FI}_{9}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{2}\le {IF}_{10}\right) =$[-0.21, -0.05], [-0.11, -0.008], ${v}_{{\grave{U}} }\left({FI}_{2}\ge {FI}_{10}\right)$ = (-0.17.-0.51)

Max {${IF}_{2}\le {IF}_{1}, {IF}_{3}, {IF}_{4}, {IF}_{5}, {IF}_{6}, {IF}_{7}, {IF}_{8}, {IF}_{9}, {IF}_{10}$} = [-0.21, -0.05], [-0.11, -0.008], min {${FI}_{2}\ge {FI}_{1}, {FI}_{3}, {FI}_{4}, {FI}_{5}, {FI}_{6}, {FI}_{7}, {FI}_{8}, {FI}_{9}, {FI}_{10}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{3}\le {IF}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{3}\ge {FI}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{3}\le {IF}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{3}\ge {FI}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{3}\le {IF}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{3}\ge {FI}_{4}\right)$ = (-0.16, -0.48)

${v}_{{\grave{U}} }\left({IF}_{3}\le {IF}_{5}\right) =$[-0.19, -0.05], [-0.10, -0.006], ${v}_{{\grave{U}} }\left({FI}_{3}\ge {FI}_{5}\right)$ = (-0.16, -0.49)

${v}_{{\grave{U}} }\left({IF}_{3}\le {IF}_{6}\right) =$[-0.18, -0.05], [-0.10, -0.006], ${v}_{{\grave{U}} }\left({FI}_{3}\ge {FI}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{3}\le {IF}_{7}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{3}\ge {FI}_{7}\right)$ = (-0.15, -0.47)

${v}_{{\grave{U}} }\left({IF}_{3}\le {IF}_{8}\right) =$[-0.19, -0.05], [-0.10, -0.006], ${v}_{{\grave{U}} }\left({FI}_{3}\ge {FI}_{8}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{3}\le {IF}_{9}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{3}\ge {FI}_{9}\right)$ = (-0.15, -0.46)

${v}_{{\grave{U}} }\left({IF}_{3}\le {IF}_{10}\right) =$[-0.20, -0.05], [-0.10, -0.005], ${v}_{{\grave{U}} }\left({FI}_{3}\ge {FI}_{10}\right)$ = (-0.8, -0.6)

Max {${IF}_{3}\le {IF}_{2}, {IF}_{1}, {IF}_{4}, {IF}_{5}, {IF}_{6}, {IF}_{7}, {IF}_{8}, {IF}_{9}, {IF}_{10}$} = [-0.18, -0.05], [-0.10, -0.005], min {${FI}_{3}\ge {FI}_{2}, {FI}_{1}, {FI}_{4}, {FI}_{5}, {FI}_{6}, {FI}_{7}, {FI}_{8}, {FI}_{9}, {FI}_{10}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{4}\le {IF}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{4}\ge {FI}_{1}\right)$ = (-0.17, -0.50)

${v}_{{\grave{U}} }\left({IF}_{4}\le {IF}_{2}\right) =$[-0.17, -0.05], [-0.09, -0.008], ${v}_{{\grave{U}} }\left({FI}_{4}\ge {FI}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{4}\le {IF}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{4}\ge {FI}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{4}\le {IF}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{4}\ge {FI}_{5}\right)$ = (-0.16, -0.49)

${v}_{{\grave{U}} }\left({IF}_{4}\le {IF}_{6}\right) =$[-0.21, -0.05], [-0.11, -0.006], ${v}_{{\grave{U}} }\left({FI}_{4}\ge {FI}_{6}\right)$ = (-0.16, -0.49)

${v}_{{\grave{U}} }\left({IF}_{4}\le {IF}_{7}\right) =$[-0.20, -0.05], [-0.11, -0.007], ${v}_{{\grave{U}} }\left({FI}_{4}\ge {FI}_{7}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{4}\le {IF}_{8}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{4}\ge {FI}_{8}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{4}\le {IF}_{9}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{4}\ge {FI}_{9}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{4}\le {IF}_{10}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{4}\ge {FI}_{10}\right)$ = (-0.8, -0.6)

Max {${IF}_{4}\le {IF}_{2}, {IF}_{3}, {IF}_{1}, {IF}_{5}, {IF}_{6}, {IF}_{7}, {IF}_{8}, {IF}_{9}, {IF}_{10}$} = [-0.17, -0.05], [-0.09, -0.006], min {${FI}_{4}\ge {FI}_{2}, {FI}_{3}, {FI}_{1}, {FI}_{5}, {FI}_{6}, {FI}_{7}, {FI}_{8}, {FI}_{9}, {FI}_{10}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{5}\le {IF}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{5}\ge {FI}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{5}\le {IF}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{5}\ge {FI}_{2}\right)$ = (-0.17, -0.51)

${v}_{{\grave{U}} }\left({IF}_{5}\le {IF}_{3}\right) =$[-0.19, -0.05], [-0.11, -0.008], ${v}_{{\grave{U}} }\left({FI}_{5}\ge {FI}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{5}\le {IF}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{5}\ge {FI}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{5}\le {IF}_{6}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{5}\ge {FI}_{6}\right)$ = (-0.16, -0.48)

${v}_{{\grave{U}} }\left({IF}_{5}\le {IF}_{7}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{5}\ge {FI}_{7}\right)$ = (-0.16, -0.48)

${v}_{{\grave{U}} }\left({IF}_{5}\le {IF}_{8}\right) =$[-0.16, -0.04], [-0.09, -0.006], ${v}_{{\grave{U}} }\left({FI}_{5}\ge {FI}_{8}\right)$ = (-0.15, -0.45)

${v}_{{\grave{U}} }\left({IF}_{5}\le {IF}_{9}\right) =$[-0.16, -0.04], [-0.09, -0.006], ${v}_{{\grave{U}} }\left({FI}_{5}\ge {FI}_{9}\right)$ = (-0.15, -0.47)

${v}_{{\grave{U}} }\left({IF}_{5}\le {IF}_{10}\right) =$[-0.19, -0.04], [-0.10, -0.005], ${v}_{{\grave{U}} }\left({FI}_{5}\ge {FI}_{10}\right)$ = (-0.8, -0.6)

Max {${IF}_{5}\le {IF}_{2}, {IF}_{3}, {IF}_{4}, {IF}_{1}, {IF}_{6}, {IF}_{7}, {IF}_{8}, {IF}_{9}, {IF}_{10}$} = [-0.16, -0.04], [-0.09, -0.005], min {${FI}_{5}\ge {FI}_{2}, {FI}_{3}, {FI}_{4}, {FI}_{1}, {FI}_{6}, {FI}_{7}, {FI}_{8}, {FI}_{9}, {FI}_{10}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{6}\le {IF}_{1}\right) =$[-0.17, -0.04], [-0.09, -0.005], ${v}_{{\grave{U}} }\left({FI}_{6}\ge {FI}_{1}\right)$ = (-0.15, -0.46)

${v}_{{\grave{U}} }\left({IF}_{6}\le {IF}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{6}\ge {FI}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{6}\le {IF}_{3}\right) =$[-0.18, -0.04], [-0.10, -0.005], ${v}_{{\grave{U}} }\left({FI}_{6}\ge {FI}_{3}\right)$ = (-0.17, -0.50)

${v}_{{\grave{U}} }\left({IF}_{6}\le {IF}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{6}\ge {FI}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{6}\le {IF}_{5}\right) =$[-0.14, -0.04], [-0.08, -0.007], ${v}_{{\grave{U}} }\left({FI}_{6}\ge {FI}_{5}\right)$ = (-0.4, -0.3)

${v}_{{\grave{U}} }\left({IF}_{6}\le {IF}_{7}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{6}\ge {FI}_{7}\right)$ = (-0.17, -0.51)

${v}_{{\grave{U}} }\left({IF}_{6}\le {IF}_{8}\right) =$[-0.19, -0.04], [-0.10, -0.005], ${v}_{{\grave{U}} }\left({FI}_{6}\ge {FI}_{8}\right)$ = (-0.17, -0.51)

${v}_{{\grave{U}} }\left({IF}_{6}\le {IF}_{9}\right) =$[-0.19, -0.05], [-0.11, -0.006], ${v}_{{\grave{U}} }\left({FI}_{6}\ge {FI}_{9}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{6}\le {IF}_{10}\right) =$[-0.19, -0.05], [-0.10, -0.006], ${v}_{{\grave{U}} }\left({FI}_{6}\ge {FI}_{10}\right)$ = (-0.16, -0.50)

Max {${IF}_{6}\le {IF}_{2}, {IF}_{3}, {IF}_{4}, {IF}_{5}, {IF}_{1}, {IF}_{7}, {IF}_{8}, {IF}_{9}, {IF}_{10}$} = [-0.14, -0.04], [-0.08, -0.005], min {${FI}_{6}\ge {FI}_{2}, {FI}_{3}, {FI}_{4}, {FI}_{5}, {FI}_{1}, {FI}_{7}, {FI}_{8}, {FI}_{9}, {FI}_{10}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{7}\le {IF}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{7}\ge {FI}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{7}\le {IF}_{2}\right) =$[-0.20, -0.05], [-0.11, -0.006], ${v}_{{\grave{U}} }\left({FI}_{7}\ge {FI}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{7}\le {IF}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{7}\ge {FI}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{7}\le {IF}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{7}\ge {FI}_{4}\right)$ = (-0.18, -0.53)

${v}_{{\grave{U}} }\left({IF}_{7}\le {IF}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{7}\ge {FI}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{7}\le {IF}_{6}\right) =$[-0.18, -0.05], [-0.10, -0.008], ${v}_{{\grave{U}} }\left({FI}_{7}\ge {FI}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{7}\le {IF}_{8}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{7}\ge {FI}_{8}\right)$ = (-0.17, -0.51)

${v}_{{\grave{U}} }\left({IF}_{7}\le {IF}_{9}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{7}\ge {FI}_{9}\right)$ = (-0.17, -0.51)

${v}_{{\grave{U}} }\left({IF}_{7}\le {IF}_{10}\right) =$[-0.14, -0.04], [-0.08, -0.006], ${v}_{{\grave{U}} }\left({FI}_{7}\ge {FI}_{10}\right)$ = (-0.15, -0.49)

Max {${IF}_{7}\le {IF}_{2}, {IF}_{3}, {IF}_{4}, {IF}_{5}, {IF}_{6}, {IF}_{1}, {IF}_{8}, {IF}_{9}, {IF}_{10}$} = [-0.14, -0.04], [-0.10, -0.006], min {${FI}_{7}\ge {FI}_{2}, {FI}_{3}, {FI}_{4}, {FI}_{5}, {FI}_{6}, {FI}_{1}, {FI}_{8}, {FI}_{9}, {FI}_{10}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{8}\le {IF}_{1}\right) =$[-0.13, -0.04], [-0.07, -0.006], ${v}_{{\grave{U}} }\left({FI}_{8}\ge {FI}_{1}\right)$ = (-0.16, -0.50)

${v}_{{\grave{U}} }\left({IF}_{8}\le {IF}_{2}\right) =$[-0.17, -0.04], [-0.09, -0.005], ${v}_{{\grave{U}} }\left({FI}_{8}\ge {FI}_{2}\right)$ = (-0.15, -0.48)

${v}_{{\grave{U}} }\left({IF}_{8}\le {IF}_{3}\right) =$[-0.15, -0.04], [-0.08, -0.005], ${v}_{{\grave{U}} }\left({FI}_{8}\ge {FI}_{3}\right)$ = (-0.16, -0.49)

${v}_{{\grave{U}} }\left({IF}_{8}\le {IF}_{4}\right) =$[-0.19, -0.04], [-0.10, -0.004], ${v}_{{\grave{U}} }\left({FI}_{8}\ge {FI}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{8}\le {IF}_{5}\right) =$[-0.17, -0.04], [-0.09, -0.005], ${v}_{{\grave{U}} }\left({FI}_{8}\ge {FI}_{5}\right)$ = (-0.18, -0.53)

${v}_{{\grave{U}} }\left({IF}_{8}\le {IF}_{6}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{8}\ge {FI}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{8}\le {IF}_{7}\right) =$[-0.11, -0.03], [-0.06, -0.007], ${v}_{{\grave{U}} }\left({FI}_{8}\ge {FI}_{7}\right)$ = (-0.6, -0.4)

${v}_{{\grave{U}} }\left({IF}_{8}\le {IF}_{9}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{8}\ge {FI}_{9}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{8}\le {IF}_{10}\right) =$[-0.17, -0.04], [-0.09, -0.005], ${v}_{{\grave{U}} }\left({FI}_{8}\ge {FI}_{10}\right)$ = (-0.8, -0.6)

Max {${IF}_{8}\le {IF}_{2}, {IF}_{3}, {IF}_{4}, {IF}_{5}, {IF}_{6}, {IF}_{7}, {IF}_{1}, {IF}_{9}, {IF}_{10}$} = [-0.11, -0.03], [-0.06, -0.004], min {${FI}_{8}\ge {FI}_{2}, {FI}_{3}, {FI}_{4}, {FI}_{5}, {FI}_{6}, {FI}_{7}, {FI}_{1}, {FI}_{9}, {FI}_{10}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{9}\le {IF}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{9}\ge {FI}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{9}\le {IF}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{9}\ge {FI}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{9}\le {IF}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{9}\ge {FI}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{9}\le {IF}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{9}\ge {FI}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{9}\le {IF}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{9}\ge {FI}_{5}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{9}\le {IF}_{6}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{9}\ge {FI}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{9}\le {IF}_{7}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{9}\ge {FI}_{7}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{9}\le {IF}_{8}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{9}\ge {FI}_{8}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{9}\le {IF}_{10}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{9}\ge {FI}_{10}\right)$ = (-0.15, -0.44)

Max {${IF}_{9}\le {IF}_{2}, {IF}_{3}, {IF}_{4}, {IF}_{5}, {IF}_{6}, {IF}_{7}, {IF}_{8}, {IF}_{1}, {IF}_{10}$} = [-0.5, -0.3], [-0.4, -0.1], min {${FI}_{9}\ge {FI}_{2}, {FI}_{3}, {FI}_{4}, {FI}_{5}, {FI}_{6}, {FI}_{7}, {FI}_{8}, {FI}_{1}, {FI}_{10}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{10}\le {IF}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{10}\ge {FI}_{1}\right)$ = (-0.15, -0.45)

${v}_{{\grave{U}} }\left({IF}_{10}\le {IF}_{2}\right) =$[-0.14, -0.04], [-0.08, -0.006], ${v}_{{\grave{U}} }\left({FI}_{10}\ge {FI}_{2}\right)$ = (-0.14, -0.42)

${v}_{{\grave{U}} }\left({IF}_{10}\le {IF}_{3}\right) =$[-0.13, -0.04], [-0.07, -0.06], ${v}_{{\grave{U}} }\left({FI}_{10}\ge {FI}_{3}\right)$ = (-0.14, -0.43)

${v}_{{\grave{U}} }\left({IF}_{10}\le {IF}_{4}\right) =$[-0.17, -0.04], [-0.09, -0.005], ${v}_{{\grave{U}} }\left({FI}_{10}\ge {FI}_{4}\right)$ = (-0.13, -0.41)

${v}_{{\grave{U}} }\left({IF}_{10}\le {IF}_{5}\right) =$[-0.15, -0.04], [-0.08, -0.005], ${v}_{{\grave{U}} }\left({FI}_{10}\ge {FI}_{5}\right)$ = (-0.14, -0.42)

${v}_{{\grave{U}} }\left({IF}_{10}\le {IF}_{6}\right) =$[-0.18, -0.04], [-0.09.-0.004], ${v}_{{\grave{U}} }\left({FI}_{10}\ge {FI}_{6}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{10}\le {IF}_{7}\right) =$[-0.16, -0.04], [-0.09, -0.005], ${v}_{{\grave{U}} }\left({FI}_{10}\ge {FI}_{7}\right)$ = (-0.16, -0.46)

${v}_{{\grave{U}} }\left({IF}_{10}\le {IF}_{8}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({FI}_{10}\ge {FI}_{8}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({IF}_{10}\le {IF}_{9}\right) =$[-0.12, -0.04], [-0.07, -0.007], ${v}_{{\grave{U}} }\left({FI}_{10}\ge {FI}_{9}\right)$ = (-0.6, -0.4)

Max {${IF}_{10}\le {IF}_{2}, {IF}_{3}, {IF}_{4}, {IF}_{5}, {IF}_{6}, {IF}_{7}, {IF}_{8}, {IF}_{9}, {IF}_{1}$} = [-0.12, -0.04], [-0.07, -0.004], min {${FI}_{10}\ge {FI}_{2}, {FI}_{3}, {FI}_{4}, {FI}_{5}, {FI}_{6}, {FI}_{7}, {FI}_{8}, {FI}_{9}, {FI}_{1}$} = (-0.8, -0.6)

Sub-criteria (market related factor)

Step 1: The NCPF (${M}_{K}$) extent value of kth sub-criterion (market related factor) is given by,

M (1) = (0.129, 0.229, 0.397), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

M (2) = (0.102, 0.175, 0.300), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

M (3) = (0.176, 0.316, 0.56), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

M (4) = (0.041, 0.078, 0.143), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

M (5) = (0.12, 0.20, 0.347), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

Step 2:

${v}_{{\grave{U}} }\left({RM}_{1}\le {RM}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{1}\ge {MR}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{1}\le {RM}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{1}\ge {MR}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{1}\le {RM}_{4}\right) =$[-0.23, -0.07], [-0.14, -0.01], ${v}_{{\grave{U}} }\left({MR}_{1}\ge {MR}_{4}\right)$ = (-0.24, -0.66)

${v}_{{\grave{U}} }\left({RM}_{1}\le {RM}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{1}\ge {MR}_{5}\right)$ = (-0.8, -0.6)

Max $\{{RM}_{1}\le {RM}_{2}, {RM}_{3}, {RM}_{4}, {RM}_{5}\}$ = [-0.23, -0.07], [-0.14, -0.01], min {${MR}_{1}\ge {MR}_{2}, {MR}_{3}, {MR}_{4}, {RM}_{5}$} = (-0.8, -0.66)

${v}_{{\grave{U}} }\left({RM}_{2}\le {RM}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{2}\ge {MR}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{2}\le {RM}_{3}\right) =$[-0.22, -0.06], [-0.12, -0.01], ${v}_{{\grave{U}} }\left({MR}_{2}\ge {MR}_{3}\right)$ = (-0.21, -0.61)

${v}_{{\grave{U}} }\left({RM}_{2}\le {RM}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{2}\ge {MR}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{2}\le {RM}_{5}\right) =$[-0.19, -0.06], [-0.11, -0.01], ${v}_{{\grave{U}} }\left({MR}_{2}\ge {MR}_{5}\right)$ = (-0.25, -0.67)

Max $\{{RM}_{2}\le {RM}_{1}, {RM}_{3}, {RM}_{4}, {RM}_{5}\}$ = [-0.19, -0.06], [-0.11, -0.01], min {${MR}_{2}\ge {MR}_{1}, {MR}_{3}, {MR}_{4}, {RM}_{5}$} = (-0.8, -0.67)

${v}_{{\grave{U}} }\left({RM}_{3}\le {RM}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{3}\ge {MR}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{3}\le {RM}_{2}\right) =$[-0.22, -0.06], [-0.12, -0.009], ${v}_{{\grave{U}} }\left({MR}_{3}\ge {MR}_{2}\right)$ = (-0.20, -0.59)

${v}_{{\grave{U}} }\left({RM}_{3}\le {RM}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{3}\ge {MR}_{4}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{3}\le {RM}_{5}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{3}\ge {MR}_{5}\right)$ = (-0.8, -0.6)

Max ${\{RM}_{3}\le {RM}_{2}, {RM}_{1}, {RM}_{4}, {RM}_{5}\}$ = [-0.22, -0.06], [-0.12, -0.009], min {${MR}_{3}\ge {MR}_{2}, {MR}_{1}, {MR}_{4}, {RM}_{5}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{4}\le {RM}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{4}\ge {MR}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{4}\le {RM}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{4}\ge {MR}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{4}\le {RM}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{4}\ge {MR}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{4}\le {RM}_{5}\right) =$[-0.07, -0.03], [-0.05, -0.01], ${v}_{{\grave{U}} }\left({MR}_{4}\ge {MR}_{5}\right)$ = (-0.20, -0.56)

Max ${\{RM}_{4}\le {RM}_{2}, {RM}_{3}, {RM}_{1}, {RM}_{5}\}$ = [-0.07, -0.03], [-0.05, -0.01], min {${MR}_{4}\ge {MR}_{2}, {MR}_{3}, {MR}_{1}, {RM}_{5}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{5}\le {RM}_{1}\right) =$[-0.11, -0.03], [-0.06, -0.007], ${v}_{{\grave{U}} }\left({MR}_{5}\ge {MR}_{1}\right)$ = (-0.18, -0.52)

${v}_{{\grave{U}} }\left({RM}_{5}\le {RM}_{2}\right) =$[-0.05, -0.03], [-0.06, -0.007], ${v}_{{\grave{U}} }\left({MR}_{5}\ge {MR}_{2}\right)$ = (-1, -0.52)

${v}_{{\grave{U}} }\left({RM}_{5}\le {RM}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({MR}_{5}\ge {MR}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({RM}_{5}\le {RM}_{4}\right) =$[-0.08, -0.03], [-0.05, -0.008], ${v}_{{\grave{U}} }\left({MR}_{5}\ge {MR}_{4}\right)$ = (-0.19, -0.54)

Max $\left\{{RM}_{5}\le {RM}_{2}, {RM}_{3}, {RM}_{4}, {RM}_{1}\right\}$ = [-0.05, -0.03], [-0.05, -0.007], min {${MR}_{5}\ge {MR}_{2}, {MR}_{3}, {MR}_{4}, {RM}_{1}$} = (-1, -0.6)

Sub-criteria (external factor)

Step1: The NCPF (${E}_{K}$) extent value of kth sub-criterion (external factor) is given by,

E (1) = (0.103, 0.17, 0.28), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

E (2) = (0.186, 0.312, 0.515), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

E(3) = (0.185, 0.309, 0.507), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

E(4) = (0.132, 0.207, 0.33), [-0.5, -0.3], [-0.4, -0.1], (-0.8, -0.6)

Step 2:

${v}_{{\grave{U}} }\left({XE}_{1}\le {XE}_{2}\right) =$ [-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({EX}_{1}\ge {EX}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({XE}_{1}\le {XE}_{3}\right) =$[-0.6, -0.16], [-0.10, -0.01], ${v}_{{\grave{U}} }\left({EX}_{1}\ge {EX}_{3}\right)$ = (-0.2, -0.7)

${v}_{{\grave{U}} }\left({XE}_{1}\le {XE}_{4}\right) =$[-0.6, -0.16], [-0.10, -0.01], ${v}_{{\grave{U}} }\left({XE}_{1}\ge {EX}_{4}\right)$ = (-0.2, -0.7)

Max {${XE}_{1}\le {XE}_{2}, {XE}_{3}, {XE}_{4}\}$ = [-0.5, -0.16], [-0.10, -0.01], min {${EX}_{1}\ge {EX}_{2}, {EX}_{3}, {EX}_{4}$} = (-0.8, -0.7)

${v}_{{\grave{U}} }\left({XE}_{2}\le {XE}_{1}\right) =$[-0.20, -0.05], [-0.11, -0.008], ${v}_{{\grave{U}} }\left({EX}_{2}\ge {EX}_{1}\right)$ = (-0.2, -0.6)

${v}_{{\grave{U}} }\left({XE}_{2}\le {XE}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({EX}_{2}\ge {EX}_{3}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({XE}_{2}\le {XE}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({EX}_{2}\ge {EX}_{4}\right)$ = (-0.8, -0.6)

Max {${XE}_{2}\le {XE}_{1}, {XE}_{3}, {XE}_{4}\}$ = [-0.20, -0.05], [-0.11, -0.008], min {${EX}_{2}\ge {EX}_{1}, {EX}_{3}, {EX}_{4}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({XE}_{3}\le {XE}_{1}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({EX}_{3}\ge {EX}_{1}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({XE}_{3}\le {XE}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({EX}_{3}\ge {EX}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({XE}_{3}\le {XE}_{4}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({EX}_{3}\ge {EX}_{4}\right)$ = (-0.8, -0.6)

Max {${XE}_{3}\le {XE}_{1}, {XE}_{2}, {XE}_{4}\}$ = [-0.5, -0.3], [-0.4, -0.1], min {${EX}_{3}\ge {EX}_{2}, {EX}_{1}, {EX}_{4}$} = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({XE}_{4}\le {XE}_{1}\right) =$[-0.2, -0.08], [-0.15, -0.01], ${v}_{{\grave{U}} }\left({EX}_{4}\ge {EX}_{1}\right)$ = (-0.2, -0.7)

${v}_{{\grave{U}} }\left({XE}_{4}\le {XE}_{2}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({EX}_{4}\ge {EX}_{2}\right)$ = (-0.8, -0.6)

${v}_{{\grave{U}} }\left({XE}_{4}\le {XE}_{3}\right) =$[-0.5, -0.3], [-0.4, -0.1], ${v}_{{\grave{U}} }\left({EX}_{4}\ge {EX}_{3}\right)$ = (-0.8, -0.6)

Max {${XE}_{4}\le {XE}_{2}, {XE}_{3}, {XE}_{1}\}$ = [-0.2, -0.008], [-0.15, -0.01], min {${EX}_{4}\ge {EX}_{2}, {EX}_{3}, {EX}_{1}$} = (-0.8, -0.7)

By the intersection of both membership functions and both non membership function of N-cubic Pythagorean fuzzy numbers is the abscissae of the common portion of upper and lower triangles are generated respectively.

Similarly, these steps (b, c) are followed by remaining other sub-criteria including (operational factors, economic/government factors, performance related factors, financial factors, market related factors, external factors).

Step 3: The weight of main factors that play role in downfall of PIA in hierarchy form is given by Table 10:

The weight of sub-criteria (operational factors) is as under (see Table 11):

The weight of sub-criteria (economic/government) factor is (see Table 12):

The weight of sub-criteria (performance related) factor as under (see Table 13):

The weight of sub-criteria (financial factor) is below (see Table 14):

The weight of sub-criteria (market related) is as under (see Table 15):

The weight of sub-criteria (external factors) as follows (see Table 16):

Step 4: Final ranking

The final ranking of main criteria's and sub-criteria's according to weights are (see Table 17):

The final ranking of the classification of several factors, it is observed that ranking of external factors is 1, which means external factors plays negative role in financial position to create the downfall of companies in airline sector. If we discuss further, then we may conclude that for other factors including Performance related factor contribute to negative sense after the external factors. Managers of the companies will be able to overcome the position of downfall after gaining information that which category is strong in their negative performance as well as weak.

6.   Comparison analysis

Following Table 18 shows that comparative analysis of fuzzy AHP and AHP under N-cubic Pythagorean fuzzy sets of the downfall of international airline.

7.   Conclusions

Experts must make decisions about how to improve the performance of airline companies for them to grow. There is a need to examine the inner and outside factors of airline's firms. For this purpose, and according to world research, the results will be more accurate. It is critical to observe both negative and positive factors when making decisions. To discuss the negative factors we initiated the study of N-cubic Pythagorean fuzzy set. This method reveals the behavior of variables in non-positive ways, which could be effective in overcoming the severity of this industry. This method will be used for ranking reasons in other real-world applications in future.

Acknowledgments

The authors express their appreciation for the Deanship of Scientific Research at King Khalid University for funding this work through the Public Research Project under Grant Number (R.G.P.2 / 48/43).

Conflict of interest

The authors declare that there is no conflict of interest regarding the publication of this article.