Group decision-making framework utilizing interval-valued intuitionistic fuzzy extended VIKOR methodology with applications in software quality assessment

2.1 IVIFS

Definition 2.1. (Atanassov and Gargov 1989) If $X$ is a non-empty set, then IVIFS $\tilde{A}$ is defined as follows:

where ${\mu }_{\tilde{A}}\left(x\right)$ and ${\nu }_{\tilde{A}}\left(x\right)$ represent the membership and non-membership of elements belonging to $X$, and both are interval numbers: ${\mu }_{\tilde{A}}\left(x\right)=\left({\mu }_{\tilde{A}}^l\left(x\right),{\mu }_{\tilde{A}}^u\left(x\right)\right)\subseteq \left(0,1\right),$ ${\nu }_{\tilde{A}}\left(x\right)=\left({\nu }_{\tilde{A}}^l\left(x\right),{\nu }_{\tilde{A}}^u\left(x\right)\right)\subseteq \left(0,1\right)$, where ${\mu }_{\tilde{A}}^u\left(x\right)=\sup {\mu }_{\tilde{A}}\left(x\right)$, ${\mu }_{\tilde{A}}^l\left(x\right)=\inf {\mu }_{\tilde{A}}\left(x\right)$, ${\nu }_{\tilde{A}}^u\left(x\right)=\sup {\nu }_{\tilde{A}}\left(x\right)$, ${\nu }_{\tilde{A}}^l\left(x\right)=\inf \nu \left(x\right)$, and ${\mu }_{\tilde{A}}^u\left(x\right)+{\nu }_{\tilde{A}}^u\left(x\right)\le 1$. The degree of hesitancy of elements in $X$ can be expressed as ${\pi }_{\tilde{A}}\left(x\right)=\left({\pi }_{\tilde{A}}^l\left(x\right),{\pi }_{\tilde{A}}^u\left(x\right)\right)$, and ${\pi }_{\tilde{A}}^l\left(x\right)=1-{\mu }_{\tilde{A}}^u\left(x\right)-{\nu }_{\tilde{A}}^u\left(x\right)$, ${\pi }_{\tilde{A}}^u\left(x\right)=1-{\mu }_{\tilde{A}}^l\left(x\right)-{\nu }_{\tilde{A}}^l\left(x\right)$. The following term is defined as an interval-valued intuitionistic fuzzy number (IVIFN) :

where $\left({\mu }^l,{\mu }^u\right)\subseteq \left(0,1\right),\left({\nu }^l,{\nu }^u\right)\subseteq \left(0,1\right)$, $0\le {\mu }^u+{\nu }^u\le 1$, ${\pi }^l=1-{\mu }^u-{\nu }^u,$ and ${\pi }^u=1-{\mu }^l-{\nu }^l$.

Definition 2.2. (Atanassov and Gargov 1989) Let $\tilde{\alpha }=\left(\left({\mu }^l,{\mu }^u\right),\left({\nu }^l,{\nu }^u\right)\right)$, ${\tilde{\alpha }}_1=\left(\left({\mu }_1^l,{\mu }_1^u\right),\left({\nu }_1^l,{\nu }_1^u\right)\right)$, and ${\tilde{\alpha }}_2=\left(\left({\mu }_2^l,{\mu }_2^u\right),\left({\nu }_2^l,{\nu }_2^u\right)\right)$ be any three IVIFNs, and define the following operation rules:

• (1)

  $\lambda \tilde{\alpha }=\left(\left(1-{\left(1-{\mu }^l\right)}^{\lambda },1-{\left(1-{\mu }^u\right)}^{\lambda }\right),\left({\left({\nu }^l\right)}^{\lambda },{\left({\nu }^u\right)}^{\lambda }\right)\right),\lambda >0;$

• (2)

  ${\tilde{\alpha }}_1\oplus {\tilde{\alpha }}_2=\left(\left({\mu }_1^l+{\mu }_2^l-{\mu }_1^l{\mu }_2^l,{\mu }_1^u+{\mu }_2^u-{\mu }_1^u{\mu }_2^u\right),\left({\nu }_1^l{\nu }_2^l,{\nu }_1^u{\nu }_2^u\right)\right);$

• (3)

  ${\tilde{\alpha }}_1\oplus {\tilde{\alpha }}_2=\left(\left({\mu }_1^l{\nu }_2^l,{\mu }_1^u{\mu }_2^u\right),\left({\nu }_1^l+{\nu }_2^l-{\nu }_1^l{\nu }_2^l,{\nu }_1^u+{\nu }_2^u-{\nu }_1^u{\nu }_2^u\right)\right).$

2.2 Projection measurement

Definition 2.3. (Xu and Hu 2010) Let $\tilde{\alpha }=\left({\tilde{\alpha }}_1,{\tilde{\alpha }}_2,\cdots ,{\tilde{\alpha }}_n\right)$ and $\tilde{\beta }=\left({\tilde{\beta }}_1,{\tilde{\beta }}_2,\cdots ,{\tilde{\beta }}_n\right)$ be two n-dimensional IVIFVs, where ${\tilde{\alpha }}_i=\left(\left({\mu }_{\tilde{\alpha }}^{il},{\mu }_{\tilde{\alpha }}^{iu}\right),\left({\nu }_{\tilde{\alpha }}^{il},{\nu }_{\tilde{\alpha }}^{iu}\right)\right)$ and ${\tilde{\beta }}_i=\left(\left({\mu }_{\tilde{\beta }}^{il},{\mu }_{\tilde{\beta }}^{iu}\right),\left({\nu }_{\tilde{\beta }}^{il},{\nu }_{\tilde{\beta }}^{iu}\right)\right)$. The projection of an IVIFV is defined as follows:

where $\tilde{\alpha }\tilde{\beta }=\sum _{i=1}^n\left({\mu }_{\tilde{\alpha }}^{il}{\mu }_{\tilde{\beta }}^{il}+{\mu }_{\tilde{\alpha }}^{iu}{\mu }_{\tilde{\beta }}^{iu}+{\nu }_{\tilde{\alpha }}^{il}{\nu }_{\tilde{\beta }}^{il}+{\nu }_{\tilde{\alpha }}^{iu}{\nu }_{\tilde{\beta }}^{iu}+{\pi }_{\tilde{\alpha }}^{il}{\pi }_{\tilde{\beta }}^{il}+{\pi }_{\tilde{\alpha }}^{iu}{\pi }_{\tilde{\beta }}^{iu}\right),$ ${\left(\tilde{\beta }\right)}^2=\sum _{i=1}^n\left({\left({\mu }_{\tilde{\beta }}^{il}\right)}^2+{\left({\mu }_{\tilde{\beta }}^{iu}\right)}^2+{\left({\nu }_{\tilde{\beta }}^{il}\right)}^2+{\left({\nu }_{\tilde{\beta }}^{iu}\right)}^2+{\left({\pi }_{\tilde{\beta }}^{il}\right)}^2+{\left({\pi }_{\tilde{\beta }}^{iu}\right)}^2\right),$ $\left(\tilde{\beta }\right)=\sqrt{ {\left(\tilde{\beta }\right)}^2}$, ${\pi }_{\tilde{\alpha }}^{il}=1-{\mu }_{\tilde{\alpha }}^{iu}-{\nu }_{\tilde{\alpha }}^{iu}$, ${\pi }_{\tilde{\alpha }}^{iu}=1-{\mu }_{\tilde{\alpha }}^{il}-{\nu }_{\tilde{\alpha }}^{il}$, ${\pi }_{\tilde{\beta }}^{il}=1-{\mu }_{\tilde{\beta }}^{iu}-{\nu }_{\tilde{\beta }}^{iu}$, ${\pi }_{\tilde{\beta }}^{iu}=1-{\mu }_{\tilde{\beta }}^{il}-{\nu }_{\tilde{\beta }}^{il}.$

As shown by Definition 2.3, the larger the value of $Pro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)$, the closer vector $\tilde{\alpha }$ is to $\tilde{\beta }$. The analysis indicates that Eq. (3) is not always reasonable as shown in Example 2.1 below.

Example 2.1. Let $\tilde{\alpha }=\left(\left(\left(0.9,0.92\right),\left(0,0.02\right)\right),\left(\left(0.9,0.91\right),\left(0.01,0.03\right)\right)\right)$ and $\tilde{\beta }=\left(\left(\left(0.3,0.33\right),\left(0,0.01\right)\right),\left(\left(0.31,0.35\right),\left(0,0.01\right)\right)\right)$, calculate $Pro{j}_{\tilde{\beta }}\left(\tilde{\beta }\right)=0.446$ and $Pro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)=1.4$ according to Eq. (3), then $Pro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)>Pro{j}_{\tilde{\beta }}\left(\tilde{\beta }\right)$, which leads to a contradiction. Eq. (3) becomes undefined when $\tilde{\beta }$ is a zero vector, and the projection value derived from Eq. (3) fails to meet the normalization requirement of $0\le Pro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)\le 1$. Additionally, the projection values can at times be excessively large or small, which hinders experts in their decision-making processes, this represents another limitation of the measure.

To address these limitations, some researchers (Yue et al., 2023) proposed the following normalized projection measure formula:

where ${\left(\tilde{\alpha }\right)}^2=\sum _{i=1}^n\left({\left({\mu }_{\tilde{\alpha }}^{il}\right)}^2+{\left({\mu }_{\tilde{\alpha }}^{iu}\right)}^2+{\left({\nu }_{\tilde{\alpha }}^{il}\right)}^2+{\left({\nu }_{\tilde{\alpha }}^{iu}\right)}^2+{\left({\pi }_{\tilde{\alpha }}^{il}\right)}^2+{\left({\pi }_{\tilde{\alpha }}^{iu}\right)}^2\right)$, others are the same as in Eq. (3) above.

Typically, the closer $NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)$ is to 1, the more similar vector $\tilde{\alpha }$ is to $\tilde{\beta }$; however, this study finds that Eq. (4) still has a hidden flaw: When $\tilde{\alpha }$ is determined, the smaller $\left(\tilde{\beta }\right)$ is, the closer the value of $NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)$ will be to 1. In particular, when $\left(\tilde{\beta }\right)=0$ constant $NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)=1$ will occur (see Example 2.2 below).

Example 2.2. Let $\tilde{\alpha }=\left(\left(\left(0.998,0.999\right),\left(0,0.001\right)\right),\left(\left(0.998,0.999\right),\left(0,0.001\right)\right)\right)$, $\tilde{\beta }=\left(\left(\left(0.8,0.9\right),\left(0,0.001\right)\right),\left(\left(0.8,0.9\right),\left(0,0.001\right)\right)\right)$ and $\tilde{\gamma }=\left(\left(\left(0.3,0.4\right),\left(0.3,0.\right)\right),\left(\left(0.2,0.4\right),0.2,0.4\right)\right)$, it can be seen the vector $\tilde{\beta }$ is closer to $\tilde{\alpha }$ than $\tilde{\gamma }$. However, calculate $NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)\approx 0.926$ and - $NPro{j}_{\tilde{\gamma }}\left(\tilde{\alpha }\right)\approx 0.962$ according to Eq. (4), then $NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)<NPro{j}_{\tilde{\gamma }}\left(\tilde{\alpha }\right)$, and therefore conclude that the vector $\tilde{\gamma }$ is closer to $\tilde{\alpha }$ than $\tilde{\beta }$, which is a mistake.

To address the shortcomings associated with classical and contemporary projection measures, this study develops a new normalized vector projection measure formulation as defined in Definition 2.4.

Definition 2.4. Let $\tilde{\alpha }$ and $\tilde{\beta }$ be two n-dimensional IVIFVs and define the normalized projection of the vector $\tilde{\alpha }$ on the $\tilde{\beta }$ as follows

The projection measure presented in equation (5) adheres to the properties outlined below.

Property 2.1. Let $\tilde{\alpha }=\left({\tilde{\alpha }}_1,{\tilde{\alpha }}_2,\cdots ,{\tilde{\alpha }}_n\right)$ and $\tilde{\beta }=\left({\tilde{\beta }}_1,{\tilde{\beta }}_2,\cdots ,{\tilde{\beta }}_n\right)$be two n-dimensional IVIFVs, where ${\tilde{\alpha }}_i=\left(\left({\mu }_{\tilde{\alpha }}^{il},{\mu }_{\tilde{\alpha }}^{iu}\right),\left({\nu }_{\tilde{\alpha }}^{il},{\nu }_{\tilde{\alpha }}^{iu}\right)\right)$, ${\tilde{\beta }}_i=\left(\left({\mu }_{\tilde{\beta }}^{il},{\mu }_{\tilde{\beta }}^{iu}\right),\left({\nu }_{\tilde{\beta }}^{il},{\nu }_{\tilde{\beta }}^{iu}\right)\right)$, $\left({\mu }_{\tilde{\alpha }}^{il},{\mu }_{\tilde{\alpha }}^{iu}\right)\subseteq \left(0,1\right),\left({\nu }_{\tilde{\alpha }}^{il},{\nu }_{\tilde{\alpha }}^{iu}\right)\subseteq \left(0,1\right)$, $\left({\mu }_{\tilde{\beta }}^{il},{\mu }_{\tilde{\beta }}^{iu}\right)\subseteq \left(0,1\right)$, $\left({\nu }_{\tilde{\beta }}^{il},{\nu }_{\tilde{\beta }}^{iu}\right)\subseteq \left(0,1\right)$ and $0\le {\mu }_{\tilde{\alpha }}^{iu}+{\nu }_{\tilde{\alpha }}^{iu}\le 1$, $0\le {\mu }_{\tilde{\beta }}^{iu}+{\nu }_{\tilde{\beta }}^{iu}\le 1$, then Eq. (5) satisfies $0<NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)\le 1$.

Proof. (a) It is known that the degree of hesitation ${\pi }_{\tilde{\alpha }}^{il}=1-{\mu }_{\tilde{\alpha }}^{iu}-{\nu }_{\tilde{\alpha }}^{iu}$, ${\pi }_{\tilde{\alpha }}^{iu}=1-{\mu }_{\tilde{\alpha }}^{il}-{\nu }_{\tilde{\alpha }}^{il}$, ${\pi }_{\tilde{\beta }}^{il}=1-{\mu }_{\tilde{\beta }}^{iu}-{\nu }_{\tilde{\beta }}^{iu}$, ${\pi }_{\tilde{\beta }}^{iu}=1-{\mu }_{\tilde{\beta }}^{il}-{\nu }_{\tilde{\beta }}^{il}$, obviously ${\pi }_{\tilde{\alpha }}^{il},{\pi }_{\tilde{\alpha }}^{iu},{\pi }_{\tilde{\beta }}^{il},{\pi }_{\tilde{\beta }}^{iu}\in \left(0,1\right)$, so $\tilde{\alpha }\tilde{\beta }=\sum _{i=1}^n\left({\mu }_{\tilde{\alpha }}^{il}{\mu }_{\tilde{\beta }}^{il}+{\mu }_{\tilde{\alpha }}^{iu}{\mu }_{\tilde{\beta }}^{iu}+{\nu }_{\tilde{\alpha }}^{il}{\nu }_{\tilde{\beta }}^{il}+{\nu }_{\tilde{\alpha }}^{iu}{\nu }_{\tilde{\beta }}^{iu}+{\pi }_{\tilde{\alpha }}^{il}{\pi }_{\tilde{\beta }}^{il}+{\pi }_{\tilde{\alpha }}^{iu}{\pi }_{\tilde{\beta }}^{iu}\right)>0$, and thus $1+\tilde{\alpha }\tilde{\beta }>0$. Also $1+\left(\tilde{\alpha }\right)\left(\tilde{\beta }\right)+\left(\tilde{\alpha }\tilde{\beta }-{\left(\tilde{\beta }\right)}^2\right)+\left({\left(\tilde{\alpha }\right)}^2-{\left(\tilde{\beta }\right)}^2\right)>0$, so $\left(1+\tilde{\alpha }\tilde{\beta }\right)/\left(1+\left(\tilde{\alpha }\right)\left(\tilde{\beta }\right)+\left(\tilde{\alpha }\tilde{\beta }-{\left(\tilde{\beta }\right)}^2\right)+\left({\left(\tilde{\alpha }\right)}^2-{\left(\tilde{\beta }\right)}^2\right)\right)>0$.

(b) According to the formula for the angle between two vectors it is known that $0\widetilde{<\alpha }\tilde{\beta }/\left(\tilde{\alpha }\right)\left(\tilde{\beta }\right)\le 1$, so $1+\tilde{\alpha }\tilde{\beta }\le 1+\left(\tilde{\alpha }\right)\left(\tilde{\beta }\right)+\left(\tilde{\alpha }\tilde{\beta }-{\left(\tilde{\beta }\right)}^2\right)+\left({\left(\tilde{\alpha }\right)}^2-{\left(\tilde{\beta }\right)}^2\right)$, then $\left(1+\tilde{\alpha }\tilde{\beta }\right)/\left(1+\left(\tilde{\alpha }\right)\left(\tilde{\beta }\right)+\left(\tilde{\alpha }\tilde{\beta }-{\left(\tilde{\beta }\right)}^2\right)+\left({\left(\tilde{\alpha }\right)}^2-{\left(\tilde{\beta }\right)}^2\right)\right)\le 1$.

Combining (a) and (b) leads to $0<\left(1+\tilde{\alpha }\tilde{\beta }\right)/\left(1+\left(\tilde{\alpha }\right)\left(\tilde{\beta }\right)+\left(\tilde{\alpha }\tilde{\beta }-{\left(\tilde{\beta }\right)}^2\right)+\left({\left(\tilde{\alpha }\right)}^2-{\left(\tilde{\beta }\right)}^2\right)\right)\le 1$, which is $0<NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)\le 1$.

The closer the value of $NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)$ in Eq. (5) is to 1, the closer $\tilde{\alpha }$ is to $\tilde{\beta }$. Eq. (5) can be used to measure the proximity of two real vectors and the proximity of two IVIFVs. Applying Eq. (5) to Example 2.1, we can obtain $NPro{j}_{\tilde{\beta }}\left(\tilde{\beta }\right)=1$, $NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)\approx 0.395$, and $NPro{j}_{\tilde{\beta }}\left(\tilde{\beta }\right)>NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)$. To calculate Example 2.2 using Eq. (5), we can get $NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)\approx 0.752$, $NPro{j}_{\tilde{\gamma }}\left(\tilde{\alpha }\right)\approx 0.375$, and $NPro{j}_{\tilde{\beta }}\left(\tilde{\alpha }\right)>NPro{j}_{\tilde{\gamma }}\left(\tilde{\alpha }\right)$; therefore, vector $\tilde{\beta }$ is closer to $\tilde{\alpha }$ than $\tilde{\gamma }$. Eq. (5) overcomes the limitations observed in Eqs. (3) and (4), and can therefore be used to accurately measure the proximity of two IVIFVs.

The following equation further extends the normalized projection measure formulation to two high-dimensional IVIFN matrices as defined in Definition 2.5.

Definition 2.5. Let $\tilde{A}={\left({\tilde{a}}_{kj}\right)}_{t\times n}$ and $\tilde{B}={\left({\tilde{b}}_{kj}\right)}_{t\times n}$ be two IVIF matrices, where ${\tilde{a}}_{kj}=\left(\left({\mu }_{kj}^l,{\mu }_{kj}^u\right),\left({\nu }_{kj}^l,{\nu }_{kj}^u\right)\right)$, ${\tilde{b}}_{kj}=\left(\left({\tau }_{kj}^l,{\tau }_{kj}^u\right),\left({\upsilon }_{kj}^l,{\upsilon }_{kj}^u\right)\right)$, then define the normalized projection of the IVIF matrix $\tilde{A}$ on the $\tilde{B}$ as:

where $\tilde{A}\tilde{B}=\sum _{k=1}^t\sum _{j=1}^n\left({\mu }_{kj}^l{\tau }_{kj}^l+{\mu }_{kj}^u{\tau }_{kj}^u+{\nu }_{kj}^l{\upsilon }_{kj}^l+{\nu }_{kj}^u{\upsilon }_{kj}^u+{\pi }_{kj}^l{\eta }_{kj}^l+{\pi }_{kj}^u{\eta }_{kj}^u\right)$, $\left(\tilde{A}\right)=\sqrt{ {\left(\tilde{A}\right)}^2}$, ${\left(\tilde{A}\right)}^2=\sum _{k=1}^t\sum _{j=1}^n\left({\left({\mu }_{kj}^l\right)}^2+{\left({\mu }_{kj}^u\right)}^2+{\left({\nu }_{kj}^l\right)}^2+{\left({\nu }_{kj}^u\right)}^2+{\left({\pi }_{kj}^l\right)}^2+{\left({\pi }_{kj}^u\right)}^2\right)$, $\left(\tilde{B}\right)=\sqrt{ {\left(\tilde{B}\right)}^2},$ ${\left(\tilde{B}\right)}^2=\sum _{k=1}^t\sum _{j=1}^n\left({\left({\tau }_{kj}^l\right)}^2+{\left({\tau }_{kj}^u\right)}^2+{\left({\upsilon }_{kj}^l\right)}^2+{\left({\upsilon }_{kj}^u\right)}^2+{\left({\eta }_{kj}^l\right)}^2+{\left({\eta }_{kj}^u\right)}^2\right)$, and ${\pi }_{kj}^l=1-{\mu }_{kj}^u-{\nu }_{kj}^u$, ${\pi }_{kj}^u=1-{\mu }_{kj}^l-{\nu }_{kj}^l$, ${\eta }_{kj}^l=1-{\tau }_{kj}^u-{\upsilon }_{kj}^u$, ${\eta }_{kj}^u=1-{\tau }_{kj}^l-{\upsilon }_{kj}^l$.

The closer the value of $GNPro{j}_{\tilde{B}}\left(\tilde{A}\right)$ in Eq. (6) is to 1, the closer the IVIF matrix $\tilde{A}$ is to $\tilde{B}$. As a result, equation (6) can be applied to assess the similarity between two IVIF matrices. It can also serve to evaluate the proximity of two real matrices.

3.1 Index sets

To streamline the decision-making process, the following index sets are utilized:

• (1)

  Record $M=\left(1,2,\cdots ,m\right)$, $N=\left(1,2,\cdots ,n\right)$, $T=\left(1,2,\cdots ,t\right)$.

• (2)

  The alternative set for the decision problem is $A=\left(A_i\left(i\in M\right)\right)$.

• (3)

  The attribute set of the decision problem is $u=\left(u_j\left(j\in N\right)\right)$.

• (4)

  The group of decision-making experts is noted as $D=\left(d_k\left(k\in T\right)\right)$.

• (5)

  The decision expert weight vector is denoted as $\Delta =\left({\delta }_k\left(k\in T\right)\right)$, where $0\le {\delta }_k\le 1$ and $\sum _{k=1}^t{\delta }_k=1$.

• (6)

  The evaluation attribute weights are represented by $\omega =\left({\omega }_j\left(j\in N\right)\right)$, where $0\le {\omega }_j\le 1$ and $\sum _{j=1}^n{\omega }_j=1$.

3.2 Determining expert weights

Evaluation results may vary due to differences in decision-makers’ knowledge structures, personal preferences, and perspectives on programs and attributes. Therefore, assigning appropriate weights to decision-makers is essential for rational decision-making.

Consider a decision-making situation where there are $m$ available alternatives, denoted as $A=\left(A_i\left(i\in M\right)\right)$. Experts from $t$ domains are invited to form an expert group for decision-making. The set of experts is denoted as $=\left(d_k\left(k\in T\right)\right)$. In reality, each domain includes more than one decision-making expert. In this study, we assume that the evaluation values of experts in the same domain have already been pooled and are ultimately represented by one expert. The expert group identifies the set of evaluation attributes for the decision problem, denoted as $u=\left(u_i\left(j\in N\right)\right)$. Multiple experts in each domain evaluate and score each alternative using IVIFNs. The scores from experts within the same domain are then aggregated, resulting in evaluation values that are also represented as IVIFNs. Each domain expert’s final evaluated values for all alternatives are presented in an IVIF matrix, denoted as $A_k,k\in T$, which is constructed as follows:

where ${\tilde{A}}_k={\left({\tilde{a}}_{ij}^k\right)}_{m\times n}$ ( $i\in M, j\in N$) denotes the evaluation information matrix of all alternatives by the $k$-th domain expert group, and ${\tilde{a}}_{ij}^k$ denotes the evaluation value of the $i$-th attribute of the $j$-th alternative by the $k$-th domain expert group, where ${\tilde{a}}_{ij}^k=\left(\left({\mu }_{ij}^{kl},{\mu }_{ij}^{ku}\right),\left({\nu }_{ij}^{kl},{\nu }_{ij}^{ku}\right)\right)$.

If the average proximity value between the evaluation matrix provided by the $k$-th decision-making expert and the evaluation matrices of other decision-making experts is greater, then the evaluation matrix provided by the $k$-th decision-making expert is deemed to be better, and the $k$-th decision-making expert should be assigned a higher weight. This is in combination with the proposed normalized projection measure. The average proximity between the $k$-th decision-making expert’s evaluation matrix and those of other decision-making experts can be calculated using the following formula:

where $GNPro{j}_{{\tilde{A}}_r}\left({\tilde{A}}_k\right)=\left(1+{\tilde{A}}_k{\tilde{A}}_r\right)\text{/}\left(1+\left({\tilde{A}}_k\right)\left({\tilde{A}}_r\right)+\left({\tilde{A}}_k{\tilde{A}}_r-{\left({\tilde{A}}_r\right)}^2\right)+\left({\left({\tilde{A}}_k\right)}^2-{\left({\tilde{A}}_r\right)}^2\right)\right)$.

Therefore, the weights for decision-making experts are calculated using the following formula:

where ${\delta }_k$ denotes the weight of the k-th expert, and ${\delta }_k$ satisfies conditions $0\le {\delta }_k\le 1$ and $\sum _{k = 1}^t{\delta }_k=1$.

3.3 GDM method for extending VIKOR based on projection measures

Next, we develop a new method for GDM in an IVIF environment using a developed projection measure formulation extended VIKOR technique.

Assume that the attribute-weight vector $\omega =\left({\omega }_j\left(j\in N\right)\right)$ is determined by a group of decision-making experts after consultation. Next, the attribute-weighted evaluation matrix of individual decision-making experts can be calculated as follows:

where ${\tilde{b}}_{ij}^k={\omega }_j{\tilde{a}}_{ij}^k=\left(\left({\phi }_{ij}^{kl},{\phi }_{ij}^{ku}\right),\left({\varphi }_{ij}^{kl},{\varphi }_{ij}^{ku}\right)\right)$, and ${\phi }_{ij}^{kl}=1-{\left(1-{\mu }_{ij}^{kl}\right)}^{{\omega }_j}$, ${\phi }_{ij}^{ku}=1-{\left(1-{\mu }_{ij}^{ku}\right)}^{{\omega }_j}$, ${\varphi }_{ij}^{kl}={\left({\nu }_{ij}^{kl}\right)}^{{\omega }_j}$, ${\varphi }_{ij}^{ku}={\left({\nu }_{ij}^{ku}\right)}^{{\omega }_j}$.

The expert weights ( ${\delta }_k$) determined in Eq. (9) are used to obtain the individual decision-making expert weighting and attribute-weighting matrix, denoted as individual weighted ${\tilde{C}}_k$, by assigning them to the attribute-weighted evaluation matrix ${\tilde{B}}_k$ in Eq. (10) as follows:

where ${\tilde{c}}_{ij}^k={\delta }_k{\tilde{b}}_{ij}^k=\left(\left({\tau }_{ij}^{kl},{\tau }_{ij}^{ku}\right),\left({\upsilon }_{ij}^{kl},{\upsilon }_{ij}^{ku}\right)\right)$, and ${\tau }_{ij}^{kl}=1-{\left(1-{\phi }_{ij}^{kl}\right)}^{{\delta }_k}$, ${\tau }_{ij}^{ku}=1-{\left(1-{\phi }_{ij}^{ku}\right)}^{{\delta }_k}$, ${\upsilon }_{ij}^{kl}={\left({\varphi }_{ij}^{kl}\right)}^{{\delta }_k}$, ${\upsilon }_{ij}^{ku}={\left({\varphi }_{ij}^{ku}\right)}^{{\delta }_k}$.

We transform the individual weighting matrix, ${\tilde{C}}_k,k\in T$, into the group weighting matrix, ${\tilde{C}}_i,i\in M$, according to each alternative as follows:

where ${\tilde{c}}_{kj}^i=\left(\left({\tau }_{kj}^{il},{\tau }_{kj}^{iu}\right),\left({\upsilon }_{kj}^{il},{\upsilon }_{kj}^{iu}\right)\right)$ is the same as the element ${\tilde{c}}_{ij}^k$ in ${\tilde{C}}_k$. According to the VIKOR method, we refer to ${\tilde{C}}_i$ as the group utility matrix.

Assuming that the decision attributes have all been transformed into benefit types, the ideal decision matrices based on the group utility matrix are determined as follows:

where ${\tilde{z}}_{kj}^{+}=\left(\left({\tau }_{kj}^{+l},{\tau }_{kj}^{+u}\right),\left({\upsilon }_{kj}^{+l},{\upsilon }_{kj}^{+u}\right)\right)$, and ${\tau }_{kj}^{+l}=\underset{i\in M}{\max }\left({\tau }_{kj}^{il}\right)$, ${\tau }_{kj}^{+u}=\underset{i\in M}{\max }\left({\tau }_{kj}^{iu}\right)$, ${\upsilon }_{kj}^{+l}=\underset{i\in M}{\min }\left({\upsilon }_{kj}^{il}\right)$, ${\upsilon }_{kj}^{+u}=\underset{i\in M}{\min }\left({\upsilon }_{kj}^{iu}\right)$, and ${\tilde{z}}_{kj}^{-}=\left(\left({\tau }_{kj}^{-l},{\tau }_{kj}^{-u}\right),\left({\upsilon }_{kj}^{-l},{\upsilon }_{kj}^{-u}\right)\right)$, and ${\tau }_{kj}^{-l}=\underset{i\in M}{\min }\left({\tau }_{kj}^{il}\right)$, ${\tau }_{kj}^{-u}=\underset{i\in M}{\min }\left({\tau }_{kj}^{iu}\right)$, ${\upsilon }_{kj}^{-l}=\underset{i\in M}{\max }\left({\upsilon }_{kj}^{il}\right)$, ${\upsilon }_{kj}^{-u}=\underset{i\in M}{\max }\left({\upsilon }_{kj}^{iu}\right)$.

Using Eq. (6), we calculate the proximity of the group utility matrix ${\tilde{C}}_i$ to obtain the positive ideal decision matrix ${\tilde{Z}}^{+}$ for each alternative as follows:

where $GNPro{j}_{{\tilde{Z}}^{+}}\left({\tilde{C}}_i\right)=\left(1+{\tilde{C}}_i{\tilde{Z}}^{+}\right)\text{/}\left(1+\left({\tilde{C}}_i\right)\left({\tilde{Z}}^{+}\right)+\left({\tilde{C}}_i{\tilde{Z}}^{+}-{\left({\tilde{Z}}^{+}\right)}^2\right)+\left({\left({\tilde{C}}_i\right)}^2-{\left({\tilde{Z}}^{+}\right)}^2\right)\right).$

The $G{U}_i^{+}$ is a utility measure based on the positive ideal decision matrix, where a value closer to 1 indicates a better alternative $A_i$.

Similarly, we calculate the proximity of the group utility matrix ${\tilde{C}}_i$ to the negative ideal decision matrix ${\tilde{Z}}^{-}$ for each alternative as follows:

where $GNPro{j}_{{\tilde{Z}}^{-}}\left({\tilde{C}}_i\right)=\left(1+{\tilde{C}}_i{\tilde{Z}}^{-}\right)\text{/}\left(1+\left({\tilde{C}}_i\right)\left({\tilde{Z}}^{-}\right)+\left({\tilde{C}}_i{\tilde{Z}}^{-}-{\left({\tilde{Z}}^{-}\right)}^2\right)+\left({\left({\tilde{C}}_i\right)}^2-{\left({\tilde{Z}}^{-}\right)}^2\right)\right).$

The $G{U}_i^{-}$ is defined as a utility measure based on the negative ideal decision matrix, where a value closer to 1 indicates a worse alternative $A_i$.

The relative closeness, determined by group utility, is defined as follows:

where a larger $U{C}_i$ indicates a better $A_i$.

The VIKOR method uses specific regret measures; however, when evaluating a GDM problem with IVIFNs, determining the specific group regret matrix is currently a challenge. This study proposes a solution by using the Euclidean distance measure of two IVIFNs. The group regret matrix corresponding to each alternative is provided in real number form, which is calculated as follows:

where $r_{kj}^i=D\left({\tilde{z}}_{kj}^{+},{\tilde{c}}_{kj}^i\right)=\left({\left({\tau }_{kj}^{il}-{\tau }_{kj}^{+l}\right)}^2+{\left({\tau }_{kj}^{iu}-{\tau }_{kj}^{+u}\right)}^2+{\left({\upsilon }_{kj}^{il}-{\upsilon }_{kj}^{+l}\right)}^2\right)$ ${\left(+{\left({\upsilon }_{kj}^{iu}-{\upsilon }_{kj}^{+u}\right)}^2+{\left({\pi }_{kj}^{il}-{\pi }_{kj}^{+l}\right)}^2+{\left({\pi }_{kj}^{iu}-{\pi }_{kj}^{+u}\right)}^2\right)}^{1/2}$JKSUS88_223 - Copy.eps].

The most valuable group regret matrices are defined as follows:

where $r_{kj}^{max}=\underset{i\in M}{\max }\left(r_{kj}^i\right)$, $r_{kj}^{\min }=\underset{i\in M}{\min }\left(r_{kj}^i\right)$.

We separately calculate the proximity of $R_i$ to $R_{max}$ for each alternative using Eq. (6) as follows:

where $GNPro{j}_{R_{max}}\left(R_i\right)=\left(1+R_i{R}_{max}\right)\text{/}\left(1+\left(R_i\right)\left(R_{max}\right)\right)+\left(R_i{R}_{max}-{\left(R_{max}\right)}^2\right)$ $\left(+\left({\left(R_i\right)}^2-{\left(R_{max}\right)}^2\right)\right)$jksus88_231 - Copy.eps] and the closer $G{R}_i^{max}$ is to 1, the worse $A_i$ is.

Similarly, we separately calculate the proximity of $R_i$ to $R_{min}$ for each alternative using Eq. (6) as follows:

where $GNPro{j}_{R_{min}}\left(R_i\right)=\left(1+R_i{R}_{min}\right)\text{/}\left(1+\left(R_i\right)\left(R_{min}\right)+\left(R_i{R}_{min}-{\left(R_{min}\right)}^2\right)\right)$, $\left(+\left({\left(R_i\right)}^2-{\left(R_{min}\right)}^2\right)\right)$jksus88_237 - Copy.eps] and the closer $G{R}_i^{min}$ is to 1, the better $A_i$ is.

The relative closeness based on group regret is defined as follows:

where a larger $R{C}_i$ indicates a better alternative $A_i$.

The regret matrix reflects the distance between each alternative and the positive ideal decision matrix. We argue that the VIKOR method’s overall evaluation capacity can be improved by incorporating information regarding the distance of each alternative from the negative ideal decision matrix. This information reflects the satisfaction level of the decision-making group with respect to the alternatives. In this study, the group satisfaction matrix for each alternative is presented in real form, derived from the Euclidean distance measure between the two IVIFNs, calculated as follows:

where $s_{kj}^i=D\left({\tilde{z}}_{kj}^{-},{\tilde{c}}_{kj}^i\right)=\left({\left({\tau }_{kj}^{il}-{\tau }_{kj}^{-l}\right)}^2+{\left({\tau }_{kj}^{iu}-{\tau }_{kj}^{-u}\right)}^2+{\left({\upsilon }_{kj}^{il}-{\upsilon }_{kj}^{-l}\right)}^2\right)$ ${\left(+{\left({\upsilon }_{kj}^{iu}-{\upsilon }_{kj}^{-u}\right)}^2+{\left({\pi }_{kj}^{il}-{\pi }_{kj}^{-l}\right)}^2+{\left({\pi }_{kj}^{iu}-{\pi }_{kj}^{-u}\right)}^2\right)}^{1/2}$JKSUS88_244 - Copy.eps].

The most valuable group regret matrices are defined as follows:

where $s_{kj}^{max}=\underset{i\in M}{\max }\left(s_{kj}^i\right)$, $s_{kj}^{\min }=\underset{i\in M}{\min }\left(s_{kj}^i\right)$.

We separately calculate the proximity of $S_i$ to $S_{max}$ for each alternative using Eq. (6) as follows:

where $GNPro{j}_{S_{max}}\left(S_i\right)=\left(1+S_i{S}_{max}\right)\text{/}\left(1+\left(S_i\right)\left(S_{max}\right)+\left(S_i{S}_{max}-{\left(S_{max}\right)}^2\right)\right)$ $\left(+\left({\left(S_i\right)}^2-{\left(S_{max}\right)}^2\right)\right)$jksus88_252 - Copy.eps], and the closer $G{S}_i^{max}$ is to 1, the better $A_i$ is.

Similarly, we separately calculate the proximity of $S_i$ to $S_{min}$ for each alternative using Eq. (6) as follows:

where $GNPro{j}_{S_{min}}\left(S_i\right)=\left(1+S_i{S}_{min}\right)\text{/}\left(1+\left(S_i\right)\left(S_{min}\right)+\left(S_i{S}_{min}-{\left(S_{min}\right)}^2\right)\right)$ $\left(+\left({\left(S_i\right)}^2-{\left(S_{min}\right)}^2\right)\right)$JKSUS88_258 - Copy.eps], and the closer $G{S}_i^{min}$ is to 1, the worse $A_i$ is.