# Probabilistic Hybrid Linguistic Approaches for Multiple Attribute Group Decision Making with Decision Hesitancy and the Prioritization of Attribute Relationships

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

#### 2.1. Probabilistic Hesitant Fuzzy Set (P-HFS)

**Definition**

**1.**

#### 2.2. Probabilistic Linguistic Term Set (PLTS)

**Definition**

**2.**

## 3. Probabilistic Hybrid Linguistic Term Set

#### 3.1. The Concept of Probabilistic Hybrid Linguistic Term Set (P-HLTS)

**Definition**

**3.**

**Definition**

**4.**

**Situation**

**1.**

**Situation**

**2.**

**Situation**

**3.**

**Definition**

**5.**

**[51]**, in the following section, we firstly detail the operational and comparison rules for P-UBLTS, then we propose novel distance and entropy measures for P-UBLTS in Section 3.3.

#### 3.2. Basic Operational Rules and Comparison Rules for P-UBLTS

**Definition**

**6.**

**Definition**

**7.**

**Definition**

**8.**

- (1)
- If$E\left({\tilde{L}}_{1}(p)\right)>E\left({\tilde{L}}_{2}(p)\right)$, then${\tilde{L}}_{1}(p)\succ {\tilde{L}}_{2}(p)$.
- (2)
- If$E\left({\tilde{L}}_{1}(p)\right)<E\left({\tilde{L}}_{2}(p)\right)$, then${\tilde{L}}_{1}(p)\prec {\tilde{L}}_{2}(p)$.
- (3)
- If$E\left({\tilde{L}}_{1}(p)\right)=E\left({\tilde{L}}_{2}(p)\right)$, then
- 1)
- If$\sigma \left({\tilde{L}}_{1}(p)\right)>\sigma \left({\tilde{L}}_{2}(p)\right)$, then${\tilde{L}}_{1}(p)\prec {\tilde{L}}_{2}(p)$.
- 2)
- If$\sigma \left({\tilde{L}}_{1}(p)\right)<\sigma \left({\tilde{L}}_{2}(p)\right)$, then${\tilde{L}}_{1}(p)\succ {\tilde{L}}_{2}(p)$.
- 3)
- If$\sigma \left({\tilde{L}}_{1}(p)\right)=\sigma \left({\tilde{L}}_{2}(p)\right)$, then${\tilde{L}}_{1}(p)\sim {\tilde{L}}_{2}(p)$.

#### 3.3. Proposed Distance Measure and Entropy Measure for P-UBLTS

**Definition**

**9.**

**Situation**

**1.**

**Situation**

**2.**

**Theorem**

**1.**

- (1)
- $0\le d({\tilde{L}}_{1}(p),{\tilde{L}}_{2}(p))\le 1$;
- (2)
- $d({\tilde{L}}_{1}(p),{\tilde{L}}_{2}(p))=0$if${\tilde{L}}_{1}(p)\sim {\tilde{L}}_{2}(p)$;
- (3)
- $d({\tilde{L}}_{1}(p),{\tilde{L}}_{2}(p))=d({\tilde{L}}_{2}(p),{\tilde{L}}_{1}(p))$.

**Definition**

**10.**

**Theorem**

**2.**

- (1)
- $0\le e(\tilde{L}(p))\le 1$;
- (2)
- $e(\tilde{L}(p))=0$, if and only if$\tilde{L}(p)=\left\{[{s}_{\tau},{s}_{\tau}](1)\right\}$or${\tilde{L}}_{2}(p)=\left\{[{s}_{-\tau},{s}_{-\tau}](1)\right\}$;
- (3)
- $e(\tilde{L}(p))=1$, if$\tilde{L}(p)=\left\{[{s}_{0},{s}_{0}](1)\right\}$;
- (4)
- $e(\tilde{L}(p))=e({\tilde{L}}^{c}(p))$, where${\tilde{L}}^{c}(p)=\left\{[{s}_{\tau -\beta},{s}_{\tau -\alpha}]({p}^{(l)})\right\}$;
- (5)
- $e({\tilde{L}}_{1}(p))=e({\tilde{L}}_{2}(p))$, if${\tilde{L}}_{1}(p)$is less fuzzy than${\tilde{L}}_{2}(p)$.

## 4. Prioritized Aggregation Operators for P-UBLTS

#### 4.1. PUBL-PWA Operator

**Definition**

**11.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

**Proof.**

- (1)
- Assume that $\left({\tilde{L}}_{1}^{\ast}(p),{\tilde{L}}_{2}^{\ast}(p),\dots ,{\tilde{L}}_{n}^{\ast}(p)\right)$ is any permutation of $\left({\tilde{L}}_{1}(p),{\tilde{L}}_{2}(p),\dots ,{\tilde{L}}_{n}(p)\right)$, then for each ${\tilde{L}}_{j}(p)$, there exists one and only one ${\tilde{L}}_{t}^{\ast}(p)={\tilde{L}}_{j}(p)$ and vice versa. Additionally, ${T}_{t}^{\ast}={T}_{j}$. Thus, based on Theorem 3, we have the following:$$\begin{array}{c}PUBL-PWA\left({\tilde{L}}_{1}(p),{\tilde{L}}_{2}(p),\dots ,{\tilde{L}}_{n}(p)\right)=\underset{j=1}{\overset{n}{\oplus}}\left(\frac{{T}_{j}{\tilde{L}}_{j}(p)}{{\displaystyle {\sum}_{i=1}^{n}{T}_{i}}}\right)\\ =\underset{j=1}{\overset{n}{\oplus}}\left(\frac{{T}_{t}^{\ast}{\tilde{L}}_{t}^{\ast}(p)}{{\displaystyle {\sum}_{i=1}^{n}{T}_{i}}}\right)=PUBL-PWA\left({\tilde{L}}_{1}^{\ast}(p),{\tilde{L}}_{2}^{\ast}(p),\dots ,{\tilde{L}}_{n}^{\ast}(p)\right).\end{array}$$
- (2)
- Suppose ${\tilde{L}}^{+}(p)=\left\{\langle \left[{s}_{\tau},{s}_{\tau}\right](1)\rangle \right\}$, ${\tilde{L}}^{-}(p)=\left\{\langle \left[{s}_{-\tau},{s}_{-\tau}\right](0)\rangle \right\}$, then we have the following:$${\tilde{L}}^{-}(p)\le PUBL-PWA\left({\tilde{L}}_{1}(p),{\tilde{L}}_{2}(p),\dots ,{\tilde{L}}_{n}(p)\right)\le {\tilde{L}}^{+}(p).$$

**Theorem**

**5.**

#### 4.2. PUBL-IPOWA Operator

**Definition**

**12.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

## 5. Approaches for MAGDM under Probabilistic Hybrid Linguistic Environments with Decision Hesitancy and Attribute Prioritization Relationships

**Approach**

**I.**

**MAGDM under P-HLTS environments with given prioritization relationships among the evaluated attributes**

**Step I-1.**Transform each individual probabilistic hybrid linguistic decision matrix ${R}^{q}={({r}_{ij}^{q})}_{n\times m}$ $(q=1,2,\dots ,t)$ to ${\tilde{R}}^{q}={({\tilde{r}}_{ij}^{q})}_{n\times m}$ in the form of probabilistic uncertain balanced linguistic term sets, then, reorganize ${\tilde{R}}^{q}={({\tilde{r}}_{ij}^{q})}_{n\times m}$ according to the prioritization relation ${C}_{1}\succ {C}_{2}\succ \dots {C}_{i}\succ \dots \succ {C}_{n}$.

**Step I-2.**Construct a synthesized group decision matrix $\tilde{R}={({\tilde{r}}_{ij})}_{n\times m}$ based on individual decision matrices ${\tilde{R}}^{q}={({\tilde{r}}_{ij}^{q})}_{n\times m}$ $(q=1,2,\dots ,t)$, where ${\tilde{r}}_{ij}=\left\{{\tilde{h}}_{{s}_{ij}}({p}_{ij})\right\}=\left\{{\tilde{s}}_{ij}^{{k}_{ij}}({p}_{ij}^{{k}_{ij}})\right\}$. All uncertain balanced linguistic terms ${\tilde{h}}_{{s}_{ij}}^{q}(q=1,2,\dots ,t)$ are integrated into the uncertain balanced linguistic term set ${\tilde{h}}_{{s}_{ij}}$.

**Step I-3.**Calculate the prioritized weights ${\omega}_{ij}(i=1,2,\dots ,n;j=1,2,\dots ,m)$ associated with the PUBL-PWA operator according to the following:

**Step I-4.**Obtain the aggregate results ${r}_{j}(j=1,2,\dots ,m)$ of each alternative by applying the PUBL-PWA operator:

**Step I -5.**Calculate $E\left({\tilde{r}}_{j}\right)$ and $\sigma ({\tilde{r}}_{j})$ according to Equations (14) and (17).

**Step I-6.**Based on the rules described in Definition 8, rank all the alternatives ${A}_{j}(j=1,2,\dots ,m)$ and select the most desirable one(s).

**Approach**

**II.**

**MAGDM under P-HLTS environments with unknown attribute prioritization relationships and unknown weights for DMs or DMUs**

**Step II-1.**Transform each individual probabilistic hybrid linguistic decision matrix ${R}^{q}={({r}_{ij}^{q})}_{n\times m}$ $(q=1,2,\dots ,t)$ to ${\tilde{R}}^{q}={({\tilde{r}}_{ij}^{q})}_{n\times m}$ in the form of probabilistic uncertain balanced linguistic term sets.

**Step II-2.**Aggregate all individual decision matrix ${\tilde{R}}^{q}={({\tilde{r}}_{ij}^{q})}_{n\times m}$ $(q=1,2,\dots ,t)$ into the group decision matrix $\tilde{R}={({\tilde{r}}_{ij}^{})}_{n\times m}$ by use of the PUBL-WA operator according to ${\tilde{r}}_{ij}^{}=PUBL-WA\left({\tilde{r}}_{ij}^{1},{\tilde{r}}_{ij}^{2},\dots ,{\tilde{r}}_{ij}^{t}\right)$:

**Step II-3.**Derive the order inducing vector ${\epsilon}_{i}$ according to descending order of divergence measures of the assessments under each attribute, where we denote the divergence measure by ${\overline{d}}_{i}$:

**Step II-4.**Calculate prioritized levels ${T}_{\sigma (i)j}^{}(i=1,2,\dots ,n;j=1,2,\dots ,m)$ in the group decision matrix $\overline{\tilde{R}}={({\overline{\tilde{r}}}_{\sigma (i)j}^{})}_{n\times m}$, such that the following is true:

**Step II-5.**Calculate the prioritized weights ${w}_{ij}^{}(i=1,2,\dots ,n;j=1,2,\dots ,m)$ associated with the PUBL-IPOWA operator, where the following is true:

**Step II-6.**Obtain the overall group aggregation results ${\tilde{r}}_{i}^{}(i=1,2,\dots ,n)$ of each alternative in the group matrix by applying the PUBL-IPOWA operator, where the following is true:

**Step II-7.**See Step I-5.

**Step II-8.**See Step I-6.

## 6. Illustrative Application Study

#### 6.1. Case Study on Governmental Website Usability Evaluation

**Case I:**Suppose that all three decision-making units have reached a consensus on the prioritization relationships among the evaluative attributes, that is ${C}_{3}\succ {C}_{2}\succ {C}_{5}\succ {C}_{6}\succ {C}_{4}\succ {C}_{1}$, we here firstly apply the first proposed approach to determine the most desirable alternative website(s). The steps for this are organized as follows:

**Step I-1.**Transform individual decision matrix ${R}^{1}$ in the form of P-CUBLTS, ${R}^{2}$ in the form of P-CBLTS, and ${R}^{3}$ in the form of P-UUBLTS into three decision matrices of ${\tilde{R}}^{1}$, ${\tilde{R}}^{2}$ and ${\tilde{R}}^{3}$ in the form of probabilistic uncertain balanced linguistic term sets, as shown in Table 4, Table 5 and Table 6.

**Step I-2.**Based on the three individual decision matrices of ${\tilde{R}}^{1}$, ${\tilde{R}}^{2}$ and ${\tilde{R}}^{3}$, we then construct a synthesized group decision matrix $\tilde{R}={({\tilde{r}}_{ij})}_{6\times 4}$, as shown in Table 7.

**Step I-3.**Calculate the prioritized weights ${\omega}_{ij}(i=1,2,\dots ,6;j=1,2,3,4)$ using Equation (27), where we get the following:

**Step I-4.**Obtain the overall group aggregation results of each alternative ${\tilde{r}}_{j}(j=1,2,3,4)$ by Equation (28). For brevity, the details of ${\tilde{r}}_{j}(j=1,2,3,4)$ are omitted here.

**Step I-5.**Calculate $E({\tilde{r}}_{j})$ according to Equations (14)–(16), where we have the following:

**Step I-6.**Rank the alternatives according to descending order of $E({\tilde{r}}_{j})$, where we then have the following:

**Case II**: Suppose that the decision-making units have reached a consensus that there is a prioritization relationship among the evaluative attributes, but they cannot explicitly determine the prioritization relationships. In this case, the administrative department advocates that the relative importance of the decision-making units should be differentiated objectively according to their given assessments. Then, we may apply the second approach to solve the above complicated decision-making problem.

**Step II-1**. See Step I-1.

**Step II-2.**Determine the weighting vector for the three decision-making units according to Equation (30): $\eta =\left\{0.3245,0.3302,0.3453\right\}$. Then, based on the PHUBL-WA operator and the individual matrices ${\tilde{R}}^{q}={({\tilde{r}}_{ij}^{q})}_{6\times 4}(q=1,2,3)$ in the form of probabilistic uncertain balanced linguistic term sets, we obtain the group decision matrix $\tilde{R}={({\tilde{r}}_{ij}^{})}_{6\times 4}$, as shown in Table 8.

**Step II-3.**By calculating the divergence measure ${\overline{d}}_{i}$ according to Equations (31) and (32), we derive an order-inducing vector $\epsilon $ in accordance with the descending order of ${\overline{d}}_{i}$, as listed in Table 9. Then, we can transform $\tilde{R}={({\tilde{r}}_{ij}^{})}_{6\times 4}$ to the reordered group decision matrix $\overline{\tilde{R}}={({\overline{\tilde{r}}}_{\sigma (i)j}^{})}_{6\times 4}$, according to the order-inducing vector $\epsilon $.

**Step II-4.**Calculate prioritized levels by Equations (33) and (34), where:

**Step II-5.**Suppose $f(x)=x$, then, according to Equation (35), we obtain the attributes’ weighting vectors as follows:

**Step II-6.**Obtain overall group aggregation results ${\tilde{r}}_{j}(j=1,2,3,4)$ by Equation (36). Please note that details about ${\tilde{r}}_{j}(j=1,2,3,4)$ are omitted here for brevity.

**Step II-7.**Calculate $E({\tilde{r}}_{j})$ according to Equations (14)–(16), and we have:

**Step II-8.**Rank the alternatives according to descending order of $E({\tilde{r}}_{j})$:

#### 6.2. Comparative Studies

#### 6.2.1. Comparative Experiments with Various Configurations of $\eta $ and $\epsilon $

#### 6.2.2. Comparative Experiments with Different Approaches

#### 6.3. Sensitivity Analysis

#### 6.4. Further Discussion: Vector Optimization Based Approach to Solving Website Usability Evaluation with Priority Attributes

**Approach**

**III.**

**Vector optimization-based**[78]

**decision-making steps in a general form for website usability evaluation**

**Step III-1.**Collect initial data of parameters $X=\{{X}_{1},{X}_{2},\dots ,{X}_{n}\}$ and six characteristics $C=\{{C}_{1},{C}_{2},{C}_{3},{C}_{4},{C}_{5},{C}_{6}\}$. If these were originally obtained in various forms of uncertain expressions, utilize appropriate transformative methods (such as the mapping function [30] for linguistic scales or the score function [51] for P-HLTS) to get the numerical values of these parameters and characteristics, as organized in Table 13. Determine constraints for both parameters and possible functional attributes. In the decision taken, it is desirable to obtain the values of all characteristics such that they are as high as possible (i.e., at a maximum).

**Step III-2.**Utilize regression analysis methods, such that the discrete data sets of ${C}_{1}(X)$, ${C}_{2}(X)$, ${C}_{3}(X)$, ${C}_{4}(X)$, ${C}_{5}(X)$, and ${C}_{6}(X)$ are respectively converted into six functions of ${f}_{1}(X)$, ${f}_{2}(X)$, ${f}_{3}(X)$, ${f}_{4}(X)$, ${f}_{5}(X)$, and ${f}_{6}(X)$. These six functions are then used as attributes in the vector problem of mathematical programming [78]: ${f}_{1}(X)\to \mathrm{max}$, ${f}_{2}(X)\to \mathrm{max}$, ${f}_{3}(X)\to \mathrm{max}$, ${f}_{4}(X)\to \mathrm{max}$, ${f}_{5}(X)\to \mathrm{max}$, and ${f}_{6}(X)\to \mathrm{max}$.

**Step III-3.**Solve the above vector problem of mathematical programming with equivalent attributes [79].

**Step III-4.**According to specific operational management arrangements, decision makers decide to choose priority attributes and determine the numerical value of the corresponding priority attributes.

**Step III-5.**With the given attribute priority, Mashunin and Mashunin’s [78] methods are used to obtain the optimal parameter vector ${X}^{oo}$ within the assigned error range.

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Definition**

**A1.**

## References

- Zimmerman, H.J. Fuzzy Set Theory and Its Applications; Springer: Dordrecht, The Netherlands, 2001. [Google Scholar]
- Liu, P.D.; Wu, X.Y. A competency evaluation method of human resources managers based on multi-granularity linguistic variables and vikor method. Technol. Econ. Dev. Econ.
**2012**, 18, 696–710. [Google Scholar] [CrossRef] [Green Version] - Liu, P. Multiple attribute group decision making method based on interval-valued intuitionistic fuzzy power heronian aggregation operators. Comput. Ind. Eng.
**2017**, 108, 199–212. [Google Scholar] [CrossRef] - Liu, P. A weighted aggregation operators multi-attribute group decision-making method based on interval-valued trapezoidal fuzzy numbers. Appl. Math. Model.
**2011**, 38, 1053–1060. [Google Scholar] [CrossRef] - Ju, Y.; Yang, S. Approaches for multi-attribute group decision making based on intuitionistic trapezoid fuzzy linguistic power aggregation operators. J. Intell. Fuzzy Syst.
**2014**, 27, 987–1000. [Google Scholar] [CrossRef] - Ju, Y.; Yang, S. A new method for multiple attribute group decision-making with intuitionistic trapezoid fuzzy linguistic information. Soft Comput.
**2015**, 19, 2211–2224. [Google Scholar] [CrossRef] - Ju, Y.; Yang, S.; Liu, X. A novel method for multiattribute decision making with dual hesitant fuzzy triangular linguistic information. J. Appl. Math.
**2014**, 2014, 12. [Google Scholar] [CrossRef] [Green Version] - Wu, J.; Chang, J.; Cao, Q.; Liang, C. A trust propagation and collaborative filtering based method for incomplete information in social network group decision making with type-2 linguistic trust. Comput. Ind. Eng.
**2019**, 127, 853–864. [Google Scholar] [CrossRef] - Qi, X.; Liang, C.; Zhang, J. Generalized cross-entropy based group decision making with unknown expert and attribute weights under interval-valued intuitionistic fuzzy environment. Comput. Ind. Eng.
**2015**, 79, 52–64. [Google Scholar] [CrossRef] - Qi, X.; Liang, C.; Zhang, J. Multiple attribute group decision making based on generalized power aggregation operators under interval-valued dual hesitant fuzzy linguistic environment. Int. J. Mach. Learn. Cybern.
**2016**, 7, 1147–1193. [Google Scholar] [CrossRef] - Pang, Q.; Wang, H.; Xu, Z. Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci.
**2016**, 369, 128–143. [Google Scholar] [CrossRef] - Zhao, S.; Wang, D.; Liang, C.; Leng, Y.; Xu, J. Some single-valued neutrosophic power heronian aggregation operators and their application to multiple-attribute group decision-making. Symmetry
**2019**, 11, 653. [Google Scholar] [CrossRef] [Green Version] - Zhang, J.; Qi, X.; Liang, C. Tackling complexity in green contractor selection for mega infrastructure projects: A hesitant fuzzy linguistic madm approach with considering group attitudinal character and attributes’ interdependency. Complexity
**2018**, 2018, 31. [Google Scholar] [CrossRef] - Wei, G.; Zhao, X. Some induced correlated aggregating operators with intuitionistic fuzzy information and their application to multiple attribute group decision making. Expert Syst. Appl.
**2012**, 39, 2026–2034. [Google Scholar] [CrossRef] - Wei, G. Interval valued hesitant fuzzy uncertain linguistic aggregation operators in multiple attribute decision making. Int. J. Mach. Learn. Cybern.
**2016**, 7, 1093–1114. [Google Scholar] [CrossRef] - Ju, Y.B. A new method for multiple criteria group decision making with incomplete weight information under linguistic environment. Appl. Math. Model.
**2014**, 38, 5256–5268. [Google Scholar] [CrossRef] - Ju, Y.B.; Wang, A.H.; You, T.H. Emergency alternative evaluation and selection based on anp, dematel, and tl-topsis. Nat. Hazards
**2015**, 75, S347–S379. [Google Scholar] [CrossRef] - Zadeh, L.A. Fuzzy sets. Inf. Control
**1965**, 8, 338–353. [Google Scholar] [CrossRef] [Green Version] - Atanassov, K.T. Intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1986**, 20, 87–96. [Google Scholar] [CrossRef] - Atanassov, K.T.; Gargov, G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1989**, 31, 343–349. [Google Scholar] [CrossRef] - Torra, V. Hesitant fuzzy sets. Int. J. Intell. Syst.
**2010**, 25, 529–539. [Google Scholar] [CrossRef] - Zhu, B.; Xu, Z.S.; Xia, M.M. Dual hesitant fuzzy sets. J. Appl. Math.
**2012**, 2012, 13. [Google Scholar] [CrossRef] - Ju, Y.B.; Liu, X.Y.; Yang, S.H. Interval-Valued dual hesitant fuzzy aggregation operators and their applications to multiple attribute decision making. J. Intell. Fuzzy Syst.
**2014**, 27, 1203–1218. [Google Scholar] [CrossRef] - Hao, Z.; Xu, Z.; Zhao, H.; Su, Z. Probabilistic dual hesitant fuzzy set and its application in risk evaluation. Knowl. Based Syst.
**2017**, 127, 16–28. [Google Scholar] [CrossRef] - Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning-i. Inf. Sci.
**1975**, 8, 199–249. [Google Scholar] [CrossRef] - Herrera-Viedma, E.; López-Herrera, A.G. A model of an information retrieval system with unbalanced fuzzy linguistic information. Int. J. Intell. Syst.
**2007**, 22, 1197–1214. [Google Scholar] [CrossRef] - Herrera, F.; Herrera-Viedma, E.; Martinez, L. A fuzzy linguistic methodology to deal with unbalanced linguistic term sets. IEEE Trans. Fuzzy Syst.
**2008**, 16, 354–370. [Google Scholar] [CrossRef] - Martínez, L.; Espinilla, M.; Liu, J.; Pérez, L.G.; Sánchez, P.J. An evaluation model with unbalanced linguistic information applied to olive oil sensory evaluation. J. Mult. Valued Log. Soft Comput.
**2009**, 15, 229–251. [Google Scholar] - Cai, M.; Gong, Z.; Yu, X. A method for unbalanced linguistic term sets and its application in group decision making. Int. J. Fuzzy Syst.
**2016**, 19, 1–12. [Google Scholar] [CrossRef] - Xu, Z.S. Uncertain linguistic aggregation operators based approach to multiple attribute group decision making under uncertain linguistic environment. Inf. Sci.
**2004**, 168, 171–184. [Google Scholar] [CrossRef] - Wei, G.; Zhao, X.; Lin, R.; Wang, H. Uncertain linguistic bonferroni mean operators and their application to multiple attribute decision making. Appl. Math. Model.
**2013**, 37, 5277–5285. [Google Scholar] [CrossRef] - Liu, P.; Yu, X. 2-Dimension uncertain linguistic power generalized weighted aggregation operator and its application in multiple attribute group decision making. Knowl. Based Syst.
**2014**, 57, 69–80. [Google Scholar] [CrossRef] - Liu, P.; He, L.; Yu, X. Generalized hybrid aggregation operators based on the 2-dimension uncertain linguistic information for multiple attribute group decision making. Group Decis. Negot.
**2016**, 25, 103–126. [Google Scholar] [CrossRef] - Liu, P. Some geometric aggregation operators based on interval intuitionistic uncertain linguistic variables and their application to group decision making. Appl. Math. Model.
**2013**, 37, 2430–2444. [Google Scholar] [CrossRef] - Meng, F.; Chen, X.; Zhang, Q. Some interval-valued intuitionistic uncertain linguistic choquet operators and their application to multi-attribute group decision making. Appl. Math. Model.
**2014**, 38, 2543–2557. [Google Scholar] [CrossRef] - Qi, X.-W.; Zhang, J.-L.; Liang, C.-Y. Multiple attributes group decision-making under interval-valued dual hesitant fuzzy unbalanced linguistic environment with prioritized attributes and unknown decision-makers’ weights. Information
**2018**, 9, 145. [Google Scholar] [CrossRef] [Green Version] - Rodríguez, R.M.; Martínez, L.; Herrera, F. Hesitant fuzzy linguistic term sets for decision making. IEEE Trans. Fuzzy Syst.
**2012**, 20, 109–119. [Google Scholar] [CrossRef] - Liao, H.; Jiang, L.; Xu, Z.; Xu, J.; Herrera, F. A linear programming method for multiple criteria decision making with probabilistic linguistic information. Inf. Sci.
**2017**, 415, 341–355. [Google Scholar] [CrossRef] - Xie, W.; Ren, Z.; Xu, Z.; Wang, H. The consensus of probabilistic uncertain linguistic preference relations and the application on the virtual reality industry. Knowl. Based Syst.
**2018**, 162, 14–28. [Google Scholar] [CrossRef] - Liu, H.; Le, J.; Xu, Z. Entropy measures of probabilistic linguistic term sets. Int. J. Comput. Intell. Syst.
**2018**, 11, 45–87. [Google Scholar] [CrossRef] [Green Version] - Wu, X.; Liao, H. An approach to quality function deployment based on probabilistic linguistic term sets and oreste method for multi-expert multi-criteria decision making. Inf. Fusion
**2018**, 43, 13–26. [Google Scholar] [CrossRef] - Xiang, C.; Jing, G.; Xu, Z. Venture capital group decision-making with interaction under probabilistic linguistic environment. Knowl. Based Syst.
**2018**, 140, 82–91. [Google Scholar] - Gao, J.; Xu, Z.; Ren, P.; Liao, H. An emergency decision making method based on the multiplicative consistency of probabilistic linguistic preference relations. Int. J. Mach. Learn. Cybern.
**2019**, 10, 1613–1629. [Google Scholar] [CrossRef] - Bai, C.Z.; Zhang, R.; Qian, L.X.; Wu, Y.N. Comparisons of probabilistic linguistic term sets for multi-criteria decision making. Knowl. Based Syst.
**2017**, 119, 284–291. [Google Scholar] [CrossRef] - Liu, P.; You, X. Probabilistic linguistic todim approach for multiple attribute decision-making. Granul. Comput.
**2017**, 2, 333–342. [Google Scholar] [CrossRef] - Zhang, X.; Xing, X. Probabilistic linguistic vikor method to evaluate green supply chain initiatives. Sustainability
**2017**, 9, 1231. [Google Scholar] [CrossRef] [Green Version] - Rodríguez, R.M.; Martínez, L.; Herrera, F. A group decision making model dealing with comparative linguistic expressions based on hesitant fuzzy linguistic term sets. Inf. Sci.
**2013**, 241, 28–42. [Google Scholar] [CrossRef] - Wang, H. Extended hesitant fuzzy linguistic term sets and their aggregation in group decision making. Int. J. Comput. Intell. Syst.
**2015**, 8, 14–33. [Google Scholar] [CrossRef] - Liu, P.; Fei, T. Some muirhead mean operators for probabilistic linguistic term sets and their applications to multiple attribute decision-making. Appl. Soft Comput.
**2018**, 68. [Google Scholar] [CrossRef] - Liu, P.; Li, Y. Multi-Attribute decision making method based on generalized maclaurin symmetric mean aggregation operators for probabilistic linguistic information. Comput. Ind. Eng.
**2019**, 131, 282–294. [Google Scholar] [CrossRef] - Lin, M.W.; Xu, Z.S.; Zhai, Y.L.; Yao, Z.Q. Multi-Attribute group decision-making under probabilistic uncertain linguistic environment. J. Oper. Res. Soc.
**2018**, 69, 157–170. [Google Scholar] [CrossRef] - Yager, R.R. Prioritized aggregation operators. Int. J. Approx. Reason.
**2008**, 48, 263–274. [Google Scholar] [CrossRef] [Green Version] - Yager, R.R. Prioritized owa aggregation. Fuzzy Optim. Decis. Mak.
**2009**, 8, 245–262. [Google Scholar] [CrossRef] - Yager, R.R. Prioritized aggregation operators and their applications. In Proceedings of the 6th IEEE International Conference Intelligent Systems (IS), Sofia, Bulgaria, 6–8 September 2012; pp. 2–5. [Google Scholar]
- Yu, X.H.; Xu, Z.S. Prioritized intuitionistic fuzzy aggregation operators. Inf. Fusion
**2013**, 14, 108–116. [Google Scholar] [CrossRef] - Yu, D.; Wu, Y.; Lu, T. Interval-Valued intuitionistic fuzzy prioritized operators and their application in group decision making. Knowl. Based Syst.
**2012**, 30, 57–66. [Google Scholar] [CrossRef] - Wei, G. Hesitant fuzzy prioritized operators and their application to multiple attribute decision making. Knowl. Based Syst.
**2012**, 31, 176–182. [Google Scholar] [CrossRef] - Jin, F.; Ni, Z.; Chen, H. Interval-Valued hesitant fuzzy einstein prioritized aggregation operators and their applications to multi-attribute group decision making. Soft Comput.
**2016**, 20, 1863–1878. [Google Scholar] [CrossRef] - Zhang, S.; Xu, Z.; He, Y. Operations and integrations of probabilistic hesitant fuzzy information in decision making. Inf. Fusion
**2017**, 38, 1–11. [Google Scholar] [CrossRef] - Marin, L.; Merigó, J.M.; Valls, A.; Moreno, A.; Isern, D. Induced unbalanced linguistic ordered weighted average. Sci. World J.
**2011**, 1, 1–8. [Google Scholar] - Marin, L.; Valls, A.; Isern, D.; Moreno, A.; Merigó, J.M. Induced unbalanced linguistic ordered weighted average and its application in multiperson decision making. Sci. World J.
**2014**, 2014, 19. [Google Scholar] [CrossRef] - Gou, X.; Xu, Z.; Liao, H. Multiple criteria decision making based on bonferroni means with hesitant fuzzy linguistic information. Soft Comput.
**2016**, 21, 1–15. [Google Scholar] [CrossRef] - Wei, G.; Lin, R.; Wang, H. Distance and similarity measures for hesitant interval-valued fuzzy sets. J. Intell. Fuzzy Syst.
**2014**, 27, 19–36. [Google Scholar] [CrossRef] - Zadeh, L.A. Probability measures of fuzzy events. J. Math. Anal. Appli.
**1968**, 23, 421–427. [Google Scholar] [CrossRef] [Green Version] - Yager, R.R.; Filev, D.P. Induced ordered weighted averaging operators. IEEE Trans. Syst. Man Cybern. Part B Cybern.
**1999**, 29, 141–150. [Google Scholar] [CrossRef] [PubMed] - Yager, R.R. Induced aggregation operators. Fuzzy Sets Syst.
**2003**, 137, 59–69. [Google Scholar] [CrossRef] - Chen, H.; Zhou, L. An approach to group decision making with interval fuzzy preference relations based on induced generalized continuous ordered weighted averaging operator. Expert Syst. Appl.
**2011**, 38, 13432–13440. [Google Scholar] [CrossRef] - Merigó, J.M.; Casanovas, M. Induced aggregation operators in the euclidean distance and its application in financial decision making. Expert Syst. Appl.
**2011**, 38, 7603–7608. [Google Scholar] [CrossRef] - Zhou, L.G.; Chen, H.Y. The induced linguistic continuous ordered weighted geometric operator and its application to group decision making. Comput. Ind. Eng.
**2013**, 66, 222–232. [Google Scholar] [CrossRef] - Verkijika, S.F.; de Wet, L. A usability assessment of e-government websites in Sub-Saharan Africa. Int. J. Inf. Manag.
**2018**, 39, 20–29. [Google Scholar] [CrossRef] - Baker, D.L. Advancing e-government performance in the united states through enhanced usability benchmarks. Gov. Inf. Q.
**2009**, 26, 82–88. [Google Scholar] [CrossRef] - Clemmensen, T.; Katre, D. Chapter 21—Adapting e-gov usability evaluation to cultural contexts. In Usability in Government Systems; Buie, E., Murray, D., Eds.; Morgan Kaufmann: Boston, MA, USA, 2012; pp. 331–344. [Google Scholar]
- Yager, R.R. On the inclusion of importances in owa aggregations. In The Ordered Weighted Averaging Operators; Yager, R.R., Kacprzyk, J., Eds.; Springer: Boston, MA, USA, 1997. [Google Scholar]
- Li, B.; Xu, Z. Prioritized aggregation operators based on the priority degrees in multicriteria decision-making. Int. J. Intell. Syst.
**2019**, 34, 1985–2018. [Google Scholar] [CrossRef] - Torra, V. The weighted owa operator. Int. J. Intell. Syst.
**1997**, 12, 153–166. [Google Scholar] [CrossRef] - Torra, V.; Narukawa, Y. Modeling Decisions: Information Fusion and Aggregation Operators; Springer: Berlin, Germany, 2007. [Google Scholar]
- Mashunin, K.Y.; Mashunin, Y.K. Simulating engineering systems under uncertainty and optimal decision making. J. Comput. Syst. Sci. Int.
**2013**, 52, 519–534. [Google Scholar] [CrossRef] - Mashunin, K.Y.; Mashunin, Y.K. Vector optimization with equivalent and priority criteria. J. Comput. Syst. Sci. Int.
**2017**, 56, 975–996. [Google Scholar] [CrossRef] - Mashunin, Y.K. Methods and Models of Vector Optimization; Nauka: Moscow, Russia, 1986. [Google Scholar]

**Table 1.**Decision matrix ${R}^{1}$ in the form of the probabilistic comparative unbalanced linguistic term set (P-CUBLTS).

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{1}$ | {<between QL and M, 0.2>, <at least VH, 0.7>} | {<between AL and QM, 0.6>} | {<between L and QM, 0.9>} | {<between QL and QM, 1>} |

${C}_{2}$ | {<between AN and QL, 0.4>, <between M and H, 0.6>} | {<between VL and L, 0.7>, < between AL and H, 0.2>} | {<between M and H, 0.8>} | {<between M and QM, 0.3>, < between H and VH, 0.5>} |

${C}_{3}$ | {<between AL and M, 0.2>, <at least VH, 0.8>} | {<between QM and VH, 0.7>} | {<between L and AL, 0.1>, <at least VH, 0.9>} | {<between M and QM, 0.6>} |

${C}_{4}$ | {< at least QM, 0.9>} | {<between QM and H, 0.3>, < at least VH, 0.3>} | {< at least H, 0.7>} | {<between VL and L, 0.7>, <between M and QM, 0.3>} |

${C}_{5}$ | {<between QL and M, 0.6>} | {< at least H, 0.8>} | {<between AL and VH, 0.9>} | {<between VL and QL, 0.4>, < between AL and M, 0.5>} |

${C}_{6}$ | {<between VL and QL, 0.6>, < at least H, 0.2>} | {<between H and VH, 1>} | {<between AN and VL, 0.5>, < between QM and H, 0.5>} | {< at least VH, 0.8>} |

**Table 2.**Decision matrix ${R}^{2}$ in the form of a probabilistic comparative balanced linguistic term set (P-CBLTS).

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{1}$ | {greater than ${s}_{5}$, 0.8} | {<between ${s}_{-2}$ and ${s}_{1}$, 0.1>, <greater than ${s}_{4}$, 0.8>} | {<between ${s}_{-4}$ and ${s}_{1}$, 0.7>, <between ${s}_{3}$ and ${s}_{5}$, 0.2>} | {<greater than ${s}_{3}$, 0.8>} |

${C}_{2}$ | {between ${s}_{2}$ and ${s}_{5}$, 0.8} | {at least ${s}_{3}$, 0.5} | {greater than ${s}_{3}$, 0.7} | {<between ${s}_{1}$ and ${s}_{2}$, 0.2>, <greater than ${s}_{3}$, 0.7>} |

${C}_{3}$ | {between ${s}_{-4}$ and ${s}_{-1}$, 0.9} | {between ${s}_{-2}$ and ${s}_{0}$, 0.9} | {at least ${s}_{6}$, 0.6} | {<between ${s}_{0}$ and ${s}_{3}$, 0.5>, <between ${s}_{4}$ and ${s}_{5}$, 0.4>} |

${C}_{4}$ | {<between ${s}_{-1}$ and ${s}_{1}$, 0.6>, <between ${s}_{2}$ and ${s}_{3}$, 0.4>} | {between ${s}_{-3}$ and ${s}_{2}$, 0.9} | {<between ${s}_{-6}$ and ${s}_{-4}$, 0.5>, <between ${s}_{1}$ and ${s}_{5}$, 0.5>} | {at least ${s}_{5}$, 0.9} |

${C}_{5}$ | {between ${s}_{4}$ and ${s}_{5}$, 0.4} | {between ${s}_{-3}$ and ${s}_{-1}$, 0.6} | {at least ${s}_{2}$, 0.9} | {between ${s}_{0}$ and ${s}_{4}$, 0.8} |

${C}_{6}$ | {<between ${s}_{0}$ and ${s}_{2}$, 0.8>, <great than ${s}_{3}$, 0.1>} | {<between ${s}_{-4}$ and ${s}_{-2}$, 0.4>, <great than ${s}_{4}$, 0.5>} | {between ${s}_{4}$ and ${s}_{7}$, 0.6} | {between ${s}_{-7}$ and ${s}_{-3}$, 0.9} |

**Table 3.**Decision matrix ${R}^{3}$ in the form of probabilistic uncertain unbalanced linguistic term set (P-UUBLTS).

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{1}$ | {<[H, AT], 0.9>} | {<[AL, H], 0.8>} | {<[AH, QH], 0.7>} | {<[AL, AH], 0.5>, <[H, QH], 0.5>} |

${C}_{2}$ | {<[M, H], 0.9>, <[VH, T], 0.1>} | {<[AL, H], 0.9>} | {<[L, M], 0.6>, <[H, QH], 0.3>} | {<[QH, AT], 0.6>} |

${C}_{3}$ | {<[M, QH], 0.9>} | {<[L, M], 0.5>, <[AT, T], 0.4>} | {<[VH, AT], 0.7>} | {<[H, QH], 0.2>, <[VH, T], 0.8>} |

${C}_{4}$ | {<[L, AH], 0.7>} | {<[L, M], 0.6>, <[VH, T], 0.3>} | {<[M, AH], 0.3>, <[AT, T], 0.6>} | {<[M, AT], 0.9>} |

${C}_{5}$ | {<[L, AH], 0.8>} | {<[AH, H], 0.3>, <[QH, VH], 0.5>} | {<[QH, AT], 0.6>} | {<[M, H], 0.7>, <[AT, T], 0.2>} |

${C}_{6}$ | {<[AL, M], 0.7>, <[VH, AT], 0.2>} | {<[QH, T], 0.9>} | {<[H, VH], 0.5>} | {<[AL, VH], 0.9>} |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{3}$ | {<[${s}_{-2}$,${s}_{0}$], 0.2>, <[${s}_{6}$,${s}_{8}$], 0.8>} | {<[${s}_{1}$,${s}_{6}$], 0.7>} | {<[${s}_{-4}$,${s}_{-2}$], 0.1>, <[${s}_{6}$,${s}_{8}$], 0.9>} | {<[${s}_{0}$,${s}_{1}$], 0.6>} |

${C}_{2}$ | {<[${s}_{-7}$,${s}_{-5}$], 0.4>, <[${s}_{0}$,${s}_{4}$], 0.6>} | {<[${s}_{-6}$,${s}_{-4}$], 0.7>, <[${s}_{-2}$,${s}_{4}$], 0.2>} | {<[${s}_{0}$,${s}_{4}$], 0.8>} | {<[${s}_{0}$,${s}_{1}$], 0.3>, <[${s}_{4}$,${s}_{6}$], 0.5>} |

${C}_{5}$ | {<[${s}_{-5}$,${s}_{0}$], 0.6>} | {<[${s}_{4}$,${s}_{8}$], 0.8>} | {<[${s}_{-2}$,${s}_{6}$], 0.9>} | {<[${s}_{-6}$,${s}_{-5}$], 0.4>, <[${s}_{-2}$,${s}_{0}$], 0.5>} |

${C}_{6}$ | {<[${s}_{-6}$,${s}_{-5}$], 0.6>, <[${s}_{4}$,${s}_{8}$], 0.2>} | {<[${s}_{4}$,${s}_{6}$], 1>} | {< [${s}_{-7}$,${s}_{-6}$], 0.5>, < [${s}_{1}$,${s}_{4}$], 0.5>} | {<[${s}_{6}$,${s}_{8}$], 0.8>} |

${C}_{4}$ | {<[${s}_{1}$,${s}_{8}$], 0.9>} | {<[${s}_{1}$,${s}_{4}$], 0.3>, <[${s}_{6}$,${s}_{8}$], 0.3>} | {<[${s}_{4}$,${s}_{8}$], 0.7>} | {<[${s}_{-6}$,${s}_{-4}$], 0.7>, <[${s}_{0}$,${s}_{1}$], 0.3>} |

${C}_{1}$ | {<[${s}_{-5}$,${s}_{0}$], 0.2>, <[${s}_{6}$,${s}_{8}$], 0.7>} | {<[${s}_{-2}$,${s}_{1}$], 0.6>} | {<[${s}_{-4}$,${s}_{1}$], 0.9>} | {<[${s}_{-5}$,${s}_{1}$], 1>} |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{3}$ | {[${s}_{-4}$,${s}_{-1}$], 0.9} | {[${s}_{-4}$,${s}_{0}$], 0.9} | {[${s}_{6}$,${s}_{8}$], 0.6} | {<[${s}_{0}$,${s}_{3}$], 0.5>, <[${s}_{4}$,${s}_{5}$], 0.4>} |

${C}_{2}$ | {[${s}_{2}$,${s}_{5}$], 0.8} | {[${s}_{3}$,${s}_{8}$], 0.5} | {[${s}_{4}$,${s}_{8}$], 0.7} | {<[${s}_{1}$,${s}_{2}$], 0.2>, <[${s}_{4}$,${s}_{6}$], 0.7>} |

${C}_{5}$ | {[${s}_{4}$,${s}_{5}$], 0.4} | {[${s}_{-3}$,${s}_{-1}$], 0.6} | {[${s}_{2}$,${s}_{8}$], 0.9} | {[${s}_{0}$,${s}_{4}$], 0.8} |

${C}_{6}$ | {<[${s}_{0}$,${s}_{2}$], 0.8>, <[${s}_{4}$,${s}_{8}$], 0.1>} | {<[${s}_{-4}$,${s}_{-2}$], 0.4>, <[${s}_{5}$,${s}_{8}$], 0.5>} | {[${s}_{4}$,${s}_{7}$], 0.6} | {[${s}_{-7}$,${s}_{-3}$], 0.9} |

${C}_{4}$ | {<[${s}_{-1}$,${s}_{1}$], 0.6>, <[${s}_{2}$,${s}_{3}$], 0.4>} | {[${s}_{-3}$,${s}_{2}$], 0.9} | {<[${s}_{-6}$,${s}_{-4}$], 0.5>, <[${s}_{1}$,${s}_{5}$], 0.5>} | {[${s}_{5}$,${s}_{8}$], 0.9} |

${C}_{1}$ | {[${s}_{6}$,${s}_{8}$], 0.8} | {<[${s}_{-2}$,${s}_{1}$], 0.1>, <[${s}_{5}$,${s}_{8}$], 0.8>} | {<[${s}_{-4}$,${s}_{1}$], 0.7>, <[${s}_{3}$,${s}_{5}$], 0.2>} | {<[${s}_{4}$,${s}_{8}$], 0.8>} |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{3}$ | {<[${s}_{0}$,${s}_{5}$], 0.9>} | {<[${s}_{-4}$,${s}_{0}$], 0.5>, <[${s}_{7}$,${s}_{8}$], 0.4>} | {<[${s}_{6}$,${s}_{7}$], 0.7>} | {<[${s}_{4}$,${s}_{5}$], 0.2>, <[${s}_{6}$,${s}_{8}$], 0.8>} |

${C}_{2}$ | {<[${s}_{0}$,${s}_{4}$], 0.9>, <[${s}_{6}$,${s}_{8}$], 0.1>} | {<[${s}_{-2}$,${s}_{4}$], 0.9>} | {<[${s}_{-4}$,${s}_{0}$], 0.6>, <[${s}_{4}$,${s}_{5}$], 0.3>} | {<[${s}_{5}$,${s}_{7}$], 0.6>} |

${C}_{5}$ | {<[${s}_{-4}$,${s}_{2}$], 0.8>} | {<[${s}_{2}$,${s}_{4}$], 0.3>, <[${s}_{5}$,${s}_{6}$], 0.5>} | {<[${s}_{5}$,${s}_{7}$], 0.6>} | {<[${s}_{0}$,${s}_{4}$], 0.7>, <[${s}_{7}$,${s}_{8}$], 0.2>} |

${C}_{6}$ | {<[${s}_{-2}$,${s}_{0}$], 0.7>, <[${s}_{6}$,${s}_{7}$], 0.2>} | {<[${s}_{5}$,${s}_{8}$], 0.9>} | {<[${s}_{4}$,${s}_{6}$], 0.5>} | {<[${s}_{-2}$,${s}_{6}$], 0.9>} |

${C}_{4}$ | {<[${s}_{-4}$,${s}_{2}$], 0.7>} | {<[${s}_{-4}$,${s}_{0}$], 0.6>, <[${s}_{6}$,${s}_{8}$], 0.3>} | {<[${s}_{0}$,${s}_{2}$], 0.3>, <[${s}_{7}$,${s}_{8}$], 0.6>} | {<[${s}_{0}$,${s}_{7}$], 0.9>} |

${C}_{1}$ | {<[${s}_{4}$,${s}_{7}$], 0.9>} | {<[${s}_{-2}$,${s}_{4}$], 0.8>} | {<[${s}_{2}$,${s}_{5}$], 0.7>} | {<[${s}_{-2}$,${s}_{2}$], 0.5>, <[${s}_{4}$,${s}_{5}$], 0.5>} |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{3}$ | {<[${s}_{-2}$,${s}_{0}$], 0.0714>, <[${s}_{6}$,${s}_{8}$], 0.2857>, <[${s}_{-4}$,${s}_{-1}$], 0.3214>, <[${s}_{0}$,${s}_{5}$], 0.3214>} | {< [${s}_{1}$,${s}_{6}$], 0.28>, <[${s}_{-4}$,${s}_{0}$], 0.56>, <[${s}_{7}$,${s}_{8}$], 0.16>} | {<[${s}_{-4}$,${s}_{-2}$], 0.0435>, <[${s}_{6}$,${s}_{8}$], 0.6522>, <[${s}_{6}$,${s}_{7}$], 0.3043>} | {<[${s}_{0}$,${s}_{1}$], 0.24>}, <[${s}_{0}$,${s}_{3}$], 0.2>, <[${s}_{4}$,${s}_{5}$], 0.24>}, <[${s}_{6}$,${s}_{8}$], 0.32>} |

${C}_{2}$ | {<[${s}_{-7}$,${s}_{-5}$], 0.1429>, <[${s}_{0}$,${s}_{4}$], 0.5357>, <[${s}_{2}$,${s}_{5}$], 0.2857>, <[${s}_{6}$,${s}_{8}$], 0.0357>} | {<[${s}_{-6}$,${s}_{-4}$], 0.3043>, < [${s}_{-2}$,${s}_{4}$], 0.4783>, <[${s}_{3}$,${s}_{8}$], 0.2174>} | {<[${s}_{0}$,${s}_{4}$], 0.3333>, <[${s}_{4}$,${s}_{8}$], 0.2917}>, <[${s}_{-4}$,${s}_{0}$], 0.25>, <[${s}_{4}$,${s}_{5}$], 0.125>} | {<[${s}_{0}$,${s}_{1}$], 0.1304>, <[${s}_{1}$,${s}_{2}$], 0.087>, <[${s}_{4}$,${s}_{6}$], 0.5217>, <[${s}_{5}$,${s}_{7}$], 0.2609>} |

${C}_{5}$ | {<[${s}_{-5}$,${s}_{0}$], 0.3333>, <[${s}_{4}$,${s}_{5}$], 0.2222>, <[${s}_{-4}$,${s}_{2}$], 0.4444>} | {<[${s}_{4}$,${s}_{8}$], 0.3636>, <[${s}_{-3}$,${s}_{-1}$], 0.2727> <[${s}_{2}$,${s}_{4}$], 0.1364>, <[${s}_{5}$,${s}_{6}$], 0.2273>} | {<[${s}_{-2}$,${s}_{6}$], 0.375>, <[${s}_{2}$,${s}_{8}$], 0.375>, <[${s}_{5}$,${s}_{7}$], 0.25>} | {<[${s}_{-6}$,${s}_{-5}$], 0.1538>, < [${s}_{-2}$,${s}_{0}$], 0.1923>, <[${s}_{0}$,${s}_{4}$], 0.5769>, <[${s}_{7}$,${s}_{8}$], 0.0769>} |

${C}_{6}$ | {<[${s}_{-6}$,${s}_{-5}$], 0.2308>, <[${s}_{4}$,${s}_{8}$], 0.1154>, <[${s}_{0}$,${s}_{2}$], 0.3077>, <[${s}_{-2}$,${s}_{0}$], 0.2692>, <[${s}_{6}$,${s}_{7}$], 0.0769>} | {<[${s}_{4}$,${s}_{6}$], 0.3571>, <[${s}_{-4}$,${s}_{-2}$], 0.1429>, <[${s}_{5}$,${s}_{8}$], 0.5>} | {<[${s}_{-7}$,${s}_{-6}$], 0.2381>, <[${s}_{1}$,${s}_{4}$], 0.2381>, <[${s}_{4}$,${s}_{7}$], 0.2857>, <[${s}_{4}$,${s}_{6}$], 0.2381>} | {<[${s}_{6}$,${s}_{8}$], 0.3077>, <[${s}_{-7}$,${s}_{-3}$], 0.3462>, <[${s}_{-2}$,${s}_{6}$], 0.3462>} |

${C}_{4}$ | {<[${s}_{1}$,${s}_{8}$], 0.3462>, <[${s}_{-1}$,${s}_{1}$], 0.2308>, <[${s}_{2}$,${s}_{3}$], 0.1538>, <[${s}_{-4}$,${s}_{2}$], 0.2692>} | {<[${s}_{1}$,${s}_{4}$], 0.125>, <[${s}_{6}$,${s}_{8}$], 0.25>, <[${s}_{-3}$,${s}_{2}$], 0.375>, <[${s}_{-4}$,${s}_{0}$], 0.25>} | {<[${s}_{4}$,${s}_{8}$], 0.2692>, <[${s}_{-6}$,${s}_{-4}$], 0.1923>, <[${s}_{1}$,${s}_{5}$], 0.1923>, <[${s}_{0}$,${s}_{2}$], 0.1154>, <[${s}_{7}$,${s}_{8}$], 0.2308>} | {<[${s}_{-6}$,${s}_{-4}$], 0.25>, <[${s}_{0}$,${s}_{1}$], 0.1071>, <[${s}_{5}$,${s}_{8}$], 0.3214>, <[${s}_{0}$,${s}_{7}$], 0.3214>} |

${C}_{1}$ | {<[${s}_{-5}$,${s}_{0}$], 0.0769>, <[${s}_{6}$,${s}_{8}$], 0.5769>, <[${s}_{4}$,${s}_{7}$], 0.3462>} | {<[${s}_{-2}$,${s}_{1}$], 0.3043>, <[${s}_{5}$,${s}_{8}$], 0.3478>, <[${s}_{-2}$,${s}_{4}$], 0.3478>} | {<[${s}_{-4}$,${s}_{1}$], 0.64>, <[${s}_{3}$,${s}_{5}$], 0.08>, <[${s}_{2}$,${s}_{5}$], 0.28>} | {<[${s}_{-5}$,${s}_{1}$], 0.3571>, <[${s}_{4}$,${s}_{8}$], 0.2857>, <[${s}_{-2}$,${s}_{2}$], 0.1786>, <[${s}_{4}$,${s}_{5}$], 0.1786>} |

${\mathit{A}}_{1}$ | ${\mathit{A}}_{2}$ | ${\mathit{A}}_{3}$ | ${\mathit{A}}_{4}$ | |
---|---|---|---|---|

${C}_{1}$ | {<[${s}_{3.336}$,${s}_{8}$], 0.144>, <[${s}_{5.459}$,${s}_{8}$], 0.504>} | {<[${s}_{-2}$,${s}_{2.23}$], 0.048>, <[${s}_{1.28}$,${s}_{8}$], 0.384>} | {<[${s}_{-1.446}$,${s}_{2.776}$], 0.441>, <[${s}_{0.926}$,${s}_{4.05}$], 0.126>} | {<[${s}_{-0.046}$,${s}_{8}$], 0.4>, <[${s}_{2.136}$,${s}_{8}$], 0.4>} |

${C}_{2}$ | {<[${s}_{-0.921}$,${s}_{2.668}$], 0.288>, <[${s}_{2.472}$,${s}_{8}$], 0.032>, <[${s}_{0.725}$,${s}_{4.362}$], 0.432>, <[${s}_{3.492}$,${s}_{8}$], 0.048>} | {<[${s}_{-0.872}$,${s}_{8}$], 0.315>, < [${s}_{0.046}$,${s}_{8}$], 0.09>} | {<[${s}_{0.68}$,${s}_{8}$], 0.336>, <[${s}_{2.99}$,${s}_{8}$], 0.168}>} | {<[${s}_{2.544}$,${s}_{4.6}$], 0.036>, <[${s}_{3.465}$,${s}_{5.636}$], 0.126>, <[${s}_{3.643}$,${s}_{5.737}$], 0.06>, <[${s}_{4.378}$,${s}_{6.426}$], 0.21>} |

${C}_{3}$ | {<[${s}_{-1.833}$,${s}_{2.072}$], 0.162>, <[${s}_{2.167}$,${s}_{8}$], 0.648>} | {< [${s}_{-2.074}$,${s}_{2.898}$], 0.315>, <[${s}_{3.729}$,${s}_{8}$], 0.252>} | {<[${s}_{4.423}$,${s}_{8}$], 0.042>, <[${s}_{6}$,${s}_{8}$], 0.378>} | {<[${s}_{1.703}$,..], 0.06>}, <[${s}_{3.043}$,${s}_{8}$], 0.24>, <[${s}_{2.991}$,${s}_{4.05}$], 0.048>}, <[${s}_{4.057}$,${s}_{8}$], 0.192>} |

${C}_{4}$ | {<[${s}_{-1.161}$,${s}_{8}$], 0.378>, <[${s}_{-0.013}$,${s}_{8}$], 0.252>} | {<[${s}_{-1.789}$,${s}_{2.19}$], 0.162>, <[${s}_{2.727}$,${s}_{8}$], 0.081>, <[${s}_{1.48}$,${s}_{8}$], 0.162>, <[${s}_{4.488}$,${s}_{8}$], 0.081>} | {<[${s}_{0.315}$,${s}_{8}$], 0.105>, <[${s}_{4.252}$,${s}_{8}$], 0.21>, <[${s}_{1.887}$,${s}_{8}$], 0.105>, <[${s}_{5.019}$,${s}_{8}$], 0.21>} | {<[${s}_{1.061}$,${s}_{8}$], 0.567>, <[${s}_{2.213}$,${s}_{8}$], 0.243>}. |

${C}_{5}$ | {<[${s}_{-0.5687}$,${s}_{2.76}$], 0.192>} | {<[${s}_{1.574}$,${s}_{8}$], 0.144>, <[${s}_{2.942}$,${s}_{8}$], 0.24>} | {<[${s}_{2.426}$,${s}_{8}$], 0.486>} | {<[${s}_{-1.593}$,${s}_{2.136}$], 0.224>, < [${s}_{3.321}$,${s}_{8}$], 0.064>, <[${s}_{-0.6}$,${s}_{2.991}$], 0.28>, <[${s}_{3.8}$,${s}_{8}$], 0.08>} |

${C}_{6}$ | {<[${s}_{-2.361}$,${s}_{-0.516}$], 0.336>, <[${s}_{2.056}$,${s}_{3.846}$], 0.096>, <[${s}_{-0.242}$,${s}_{8}$], 0.042>, <[${s}_{3.272}$,${s}_{8}$], 0.012>, <[${s}_{1.1}$,${s}_{8}$], 0.112>, <[${s}_{4.042}$,${s}_{8}$], 0.032>, <[${s}_{2.511}$,${s}_{8}$], 0.014>, <[${s}_{4.851}$,${s}_{8}$], 0.004>} | {<[${s}_{2.794}$,${s}_{8}$], 0.36>, <[${s}_{4.706}$,${s}_{8}$], 0.45>} | {<[${s}_{1.857}$,${s}_{5.009}$], 0.15>, <[${s}_{3.203}$,${s}_{6.008}$], 0.15>} | {<[${s}_{1.218}$,${s}_{8}$], 0.648>} |

Attributes | ${\overline{\mathit{d}}}_{\mathit{i}}$ | $\mathit{\epsilon}$ |
---|---|---|

${C}_{1}$ | 1.028 | 3 |

${C}_{2}$ | 0.64478 | 6 |

${C}_{3}$ | 1.115 | 2 |

${C}_{4}$ | 0.8592 | 5 |

${C}_{5}$ | 0.9954 | 4 |

${C}_{6}$ | 1.6372 | 1 |

**Table 10.**Comparative experiments on the proposed approaches with various configurations of $\eta $ and $\epsilon $.

Approach | Experiment | $\mathit{\eta}$ | $\mathit{\epsilon}$ | Ranking Results |
---|---|---|---|---|

Approach I | I-1 | Not considered | $w=\{1/6,1/6,1/6,1/6,1/6,1/6\}$ | ${A}_{1}\prec {A}_{4}\prec {A}_{2}\prec {A}_{3}$ |

I-2 | Not considered | $\epsilon =(3,2,5,6,4,1)$ given directly | ${A}_{2}\prec {A}_{1}\prec {A}_{4}\prec {A}_{3}$ | |

I-3 | Not considered | $\epsilon =(3,1,6,2,5,4)$ derived objectively | ${A}_{2}\prec {A}_{1}\prec {A}_{4}\prec {A}_{3}$ | |

Approach II | II-1 | $\{1/3,1/3,1/3\}$ | $w=\{1/6,1/6,1/6,1/6,1/6,1/6\}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{4}\prec {A}_{3}$ |

II-2 | $\{1/3,1/3,1/3\}$ | $\epsilon =(3,6,2,5,4,1)$ derived objectively | ${A}_{1}\prec {A}_{4}\prec {A}_{3}\prec {A}_{2}$ | |

II-3 | $\{0.3245,0.3302,0.3453\}$ | $\epsilon =(3,6,2,5,4,1)$ derived objectively | ${A}_{1}\prec {A}_{4}\prec {A}_{3}\prec {A}_{2}$ |

Approaches | $\mathit{\eta}$ | $\mathit{\epsilon}$ | Ranking Results |
---|---|---|---|

Extended TOPSIS [51] | Not considered | Not considered | ${A}_{3}\prec {A}_{1}\prec {A}_{4}\prec {A}_{2}$ |

Adapted Approach I (This paper) | Not considered | $\epsilon =(1,2,4,5,3)$ derived objectively | ${A}_{1}\prec {A}_{2}\prec {A}_{3}\prec {A}_{4}$ |

Prioritized TOPSIS (Constructed for comparison) | Not considered | $\epsilon =(1,2,4,5,3)$ derived objectively | ${A}_{1}\prec {A}_{3}\prec {A}_{2}\prec {A}_{4}$ |

**Table 12.**Ranking results by approaches I and II with various basic, unit interval, and monotonic (BUM) functions.

Approaches | Configurations | BUM Functions | Ranking Results |
---|---|---|---|

Approach I | $\eta $ not considered $\epsilon =(3,1,6,2,5,4)$ derived objectively | $f(x)={x}^{1/3}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{4}\prec {A}_{3}$ |

$f(x)={x}^{1/2}$ | ${A}_{1}\prec {A}_{2}\prec {A}_{4}\prec {A}_{3}$ | ||

$f(x)={x}^{1}$ | ${A}_{2}\prec {A}_{1}\prec {A}_{4}\prec {A}_{3}$ | ||

$f(x)={x}^{2}$ | ${A}_{2}\prec {A}_{1}\prec {A}_{4}\prec {A}_{3}$ | ||

$f(x)={x}^{3}$ | ${A}_{2}\prec {A}_{1}\prec {A}_{4}\prec {A}_{3}$ | ||

Approach II | $\eta =\{0.3245,0.3302,0.3453\}$ $\epsilon =(3,6,2,5,4,1)$ derived objectively | $f(x)={x}^{1/3}$ |