# A Programming-Based Algorithm for Probabilistic Uncertain Linguistic Intuitionistic Fuzzy Group Decision-Making

^{*}

## Abstract

**:**

## 1. Introduction

- (1)
- Most of the PRs fail to reflect the distribution of information given by DMs.
- (2)
- Most studies on PRs ignore the information that cannot be grasped by DMs or fail to take into account information loss caused by certain objective factors.
- (3)
- In the process of solving the priority weights, most of the GPMs only consider the principle of minimum consistency deviation and ignore the risk attitude of decision makers, which may result in the loss of original information and reduce the rationality of the ranking results.
- (4)
- Almost all methods, none can guarantee the consistency of PRs in the process of solving priority weights. They all need to test and improve the consistency of PRs, which greatly reduces the accuracy of the results.

- (1)
- We put forward PULIFS, which is of great significance for improving the application of LT in fuzzy theory and effectively promoting the application of qualitative information in GDM.
- (2)
- We extracted fuzzy and non-fuzzy uncertain information from PULIFPR, and then used it to define the distance measure of PULIFPRs, thus solving the problem of determining the individual weight in GDM.
- (3)
- We built a series of GPMs by taking into account the DMs’ qualitative preference, non-preference and fuzzy information, and then give a reasonable ranking results of PULIFPR.
- (4)
- We avoided the consistency test and correction of preference relation in GDM, thus simplifying the process of GDM and improving the accuracy of decision result.
- (5)
- The proposed method is applied to the industrial docking of virtual reality (VR) industry conference, which solves the problem of project selection before industrial docking.

- (1)
- Most of the decision-making methods directly use the information provided by the decision-maker to model and make judgments, but ignore the information that the decision-maker fails to grasp or the information loss caused by some objective factors. In this paper, uncertain information is divided into fuzzy uncertain information and non-fuzzy uncertain information for comprehensive discussion, which improves the utilization of information and ensures the rationality of decision-making results.
- (2)
- Most of the existing decision-making models fail to consider the risk attitude of DMs and fail to guarantee the consistency of preference information given by DMs. In this paper, two extreme attitudes of DMs under uncertain conditions are considered to establish programming models, which ensures the consistency of preference relations, simplifies GDM process and improves the accuracy of decision-making results.

## 2. Preliminaries

#### 2.1. PLTS and PULTS

**Definition**

**1**

**.**Let $S=\{{s}_{\alpha}|\alpha \in [0,2\tau ]\}$ be a continuous LTS, then a PLTS is defined as

#### 2.2. LIFS

**Definition**

**2**

**.**Let X be a finite universal set and $S=\{{s}_{\alpha}|\alpha \in [0,2\tau ]\}$ be a continuous linguistic term set. Then a LIFS L in X is given as

## 3. PULIFS and PULIFPR

#### 3.1. PULIFS

**Definition**

**3.**

#### 3.2. PULIFPR

**Definition**

**4.**

- (1)
- ${p}_{ij,}^{k}={p}_{ji}^{k},{p}_{ii}^{k}=1$;
- (2)
- ${s}_{{u}_{ij}^{k}}={s}_{{v}_{ji}^{k}},{s}_{{v}_{ij}^{k}}={s}_{{u}_{ji}^{k}}$;
- (3)
- $u{(p)}_{ii}=\{([{s}_{\tau},{s}_{\tau}],[{s}_{\tau},{s}_{\tau}]),1\}={s}_{\tau}$;
- (4)
- $\#u{(p)}_{ij}=\#u{(p)}_{ji}$;

#### 3.3. The Distance Measure of PULIFSs

**Definition**

**5.**

**Remark**

**1.**

**Definition**

**6.**

- (1)
- $0\le d({v}_{1},{v}_{2})\le 1$;
- (2)
- $d({v}_{1},{v}_{2})=0$ if and only if ${v}_{1}={v}_{2}$;
- (3)
- $d({v}_{1},{v}_{2})=d({v}_{2},{v}_{1})$.

**Example**

**1.**

**Remark**

**2.**

**Definition**

**7.**

#### 3.4. Deriving Individual Weights and Aggregating Individual PULIFPRs

**Definition**

**8.**

**Example**

**2.**

## 4. Consistency Analysis of PULIFPR and Acquisition of Its Priority Weight

#### 4.1. Consistency Analysis of PULIFPR

**Definition**

**9.**

**Definition**

**10.**

**Remark**

**3.**

**Definition**

**11.**

#### 4.2. Determine the Priority Weights of PULIFPR through the GPM

**Remark**

**4.**

- (1)
- At present, most of the programming models proposed in many literatures only consider the principle of minimum consistency deviation, such as literatures [27,31,32,33,34]. In this paper, the consistency of the newly proposed PULIFPR is considered comprehensively from the three aspects of fuzzy and non-fuzzy uncertain information and DM’s risk attitude. Therefore, the rationality of decision result is greatly improved.
- (2)
- Currently, most of the research on PR needs to test its consistency, and some literatures that needs to test the consistency of acceptable PR fails to provide a reasonable test method, such as the research on triangular FPR by Wang [35], and the research on interval-valued intuitionistic FPR by Wan et al. [36]. In this paper, the priority weight of consistent PULIFPR can be obtained directly through the proposed programming models without considering the consistency test, which greatly simplifies the DM process.

#### 4.3. A New Algorithm for Solving GDM with PULIFPR

## 5. Case Application and Comparative Analysis

#### 5.1. Application in VR Project Selection

**Step 1**: According to Equation (10), the distance measure between ${U}_{1}$ and ${U}_{2}$ is $d({U}_{1},{U}_{2})=\frac{1}{4\times 3}(|{f}_{12}^{1}-{f}_{12}^{2}|+|{g}_{12}^{1}-{g}_{12}^{2}|+|{f}_{13}^{1}-{f}_{13}^{2}|+|{g}_{13}^{1}-{g}_{13}^{2}|+|{f}_{14}^{1}-{f}_{14}^{2}|+|{g}_{14}^{1}-{g}_{14}^{2}|+|{f}_{23}^{1}-{f}_{23}^{2}|+|{g}_{23}^{1}-{g}_{23}^{2}|+|{f}_{24}^{1}-{f}_{24}^{2}|+|{g}_{24}^{1}-{g}_{24}^{2}|+|{f}_{34}^{1}-{f}_{34}^{2}|+|{g}_{34}^{1}-{g}_{34}^{2}|)=\frac{1}{12}(|0.2531-0.425|+|0.775-0.2719|+|0.3563-0.7031|+|0.225-0.7125|+|0.8625-0.2188|+|0.1781-0.5|+|0.1313-0.075|+|0.275-0.1188|+|0.0625-0.0938|+|0.125-0.0875|+|0.15-0.15|+|0.2313-0.25|)=0.2313$, Similarly, we can calculate $d({U}_{1},{U}_{3})=0.1930$ and $d({U}_{2},{U}_{3})=0.1398$ respectively, so the corresponding similarity degree is $s({U}_{1},{U}_{2})=0.7687,s({U}_{1},{U}_{3})=0.8070$ and $s({U}_{2},{U}_{3})=0.8602$ respectively.

**Step 2**: According to Equation (12), the confidence degree of the three teams can be calculated as $c{s}_{1}=s({U}_{1},{U}_{2})+s({U}_{1},{U}_{3})=1.5758,c{s}_{2}=1.6289$ and $c{s}_{3}=1.6672$, so the weight of each team can be further determined as ${w}^{1}=\frac{c{s}_{1}}{c{s}_{1}+c{s}_{2}+c{s}_{3}}=0.3234,{w}^{2}=0.3344$, and ${w}^{3}=0.3422$.

**Step 4**: The non-fuzzy uncertain information values ${f}_{ij}(i,j=1,2,3,4,i<j)$ and the fuzzy uncertain information values ${g}_{ij}(i,j=1,2,3,4,i<j)$ of PULIFPR $\tilde{U}$ are calculated respectively. By substituting them into Equation (22), the following model can be obtained

**Step 5**: By calculating the optimistic judgment values ${a}_{ij}={\sum}_{k=1}^{\#u(p)}{p}^{k}\times \frac{I({s}_{{\overline{u}}_{ij}^{k}})-I({s}_{{\underline{v}}_{ij}^{k}})+2\tau}{4\tau}$ and pessimistic judgment values ${b}_{ij}={\sum}_{k=1}^{\#u(p)}{p}^{k}\times \frac{I({s}_{{\underline{u}}_{ij}^{k}})-I({s}_{{\overline{v}}_{ij}^{k}})+2\tau}{4\tau},(i,j=1,2,3,4,i<j)$ and substituting them into Equations (24) and (25) respectively to solve the weight. But their feasible regions are all empty. Therefore, substitute the values of ${a}_{ij}$ and ${b}_{ij}$ into Equations (26) and (27) respectively, then the priority weights can be obtained as follows

**Step 6**: Combining the results obtained in steps 4 and 5, the comprehensive ranking weight of $\tilde{U}$ can be obtained as $\overline{{w}_{1}}=0.1505,\overline{{w}_{2}}=0.2404,\overline{{w}_{3}}=0.3195,\overline{{w}_{4}}=0.2896$. Therefore, the final ranking result is $\overline{{w}_{3}}>\overline{{w}_{4}}>\overline{{w}_{2}}>\overline{{w}_{1}}$, namely, ${x}_{3}$ is the best candidate partner of Microsoft.

#### 5.2. Comparison Analyses

- (1)
- Compared with models M-1 and M-3 in method [27], the model proposed in this paper can directly obtain the priority weight of preference relation without consistency test and correction.
- (2)
- Compared with Algorithms 1 and 2 in method [37], the algorithm proposed in this paper provides a method to determine the individual weight, and the consensus collective preference relation can be obtained directly without iterative calculation.
- (3)
- Compared with the model proposed by wan et al. [36], the model proposed in this paper considers the probability distribution of uncertain information, which is more suitable for large-scale GDM problems in complex environments and can ensure the consistency of collective preference relations.
- (4)
- Compared with the GPM proposed by liao et al. [39], the model proposed in this paper considers both the risk attitude of DMs and the information that they fail to grasp, which improves the rationality and accuracy of decision-making results.

- (1)
- Compared with the general preference relation, the PULIFPR proposed in this paper can express both individual preference and group preference, which is more suitable for the increasingly complex decision-making environment. Therefore, the decision-making method proposed in this paper has a broad application prospect. Such as the selection of investment projects, the formulation of enterprise marketing plans, the introduction of talents in institutions, etc.
- (2)
- The method proposed in this paper comprehensively considers the risk attitude and fuzzy uncertain information of DMs, which is more in line with the actual decision-making situation and is easily accepted and adopted by DMs.
- (3)
- The proposed model can guarantee the consistency of the collective preference relation without checking and revising, so it is more simple and accurate in practical application.

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- 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] - Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning—II. Inf. Sci.
**1975**, 8, 301–357. [Google Scholar] [CrossRef] - Zadeh, L.A. The concept of a linguistic variable and its application to approximate reasoning—III. Inf. Sci.
**1975**, 9, 43–80. [Google Scholar] [CrossRef] - Yager, R.R. A new methodology for ordinal multiobjective decisions based on fuzzy sets. Decis. Sci.
**1981**, 12, 589–600. [Google Scholar] [CrossRef] - Degadni, R.; Bortolan, G. The problem of linguistic approximation in clinical decision making. Int. J. Approx. Reason.
**1988**, 2, 143–162. [Google Scholar] [CrossRef] - Delgado, M.; Verdegay, J.L.; Vila, M.A. On aggregation operations of linguistic labels. Int. J. Intell. Syst.
**1993**, 8, 351–370. [Google Scholar] [CrossRef] - Herrera, F.; Martinez, L. A 2-tuple fuzzy linguistic representation model for computing with words. IEEE Trans. Fuzzy Syst.
**2000**, 8, 746–752. [Google Scholar] - Xu, Z.S. A method based on linguistic aggregation operators for group decision making with linguistic preference relations. Inf. Sci.
**2004**, 166, 19–30. [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] - Rodriguez, R.M.; Martinez, L.; Herrera, F. Hesitant Fuzzy Linguistic Term Sets for Decision Making. IEEE Trans. Fuzzy Syst.
**2012**, 20, 109–119. [Google Scholar] [CrossRef] - Meng, F.; Chen, X.; Zhang, Q. Multi-attribute decision analysis under a linguistic hesitant fuzzy environment. Inf. Sci.
**2014**, 267, 287–305. [Google Scholar] [CrossRef] - Zhang, H.M. Linguistic Intuitionistic Fuzzy Sets and Application in MAGDM. J. Appl. Math.
**2014**, 1–11. [Google Scholar] [CrossRef] - Ye, J. An extended TOPSIS method for multiple attribute group decision making based on single valued neutrosophic linguistic numbers. J. Intell. Fuzzy Syst.
**2015**, 28, 247–255. [Google Scholar] - Pang, Q.; Wang, H.; Xu, Z.S. Probabilistic linguistic term sets in multi-attribute group decision making. Inf. Sci.
**2016**, 369, 128–143. [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.
**2017**, 22, 1–15. [Google Scholar] [CrossRef] - Bai, C.; Zhang, R.; Shen, S.; Huang, C.; Fan, X. Interval valued probabilistic linguistic term sets in multi-criteria group decision making. Int. J. Intell. Syst.
**2018**, 33, 1301–1321. [Google Scholar] [CrossRef] - Zhang, R.C.; Li, Z.M.; Liao, H.C. Multiple-attribute decision-making method based on the correlation coefficient between dual hesitant fuzzy linguistic term sets. Knowl.-Based Syst.
**2018**, 159, 186–192. [Google Scholar] [CrossRef] - Pamucar, D.; Badi, I.; Sanja, K.; Obradovic, R. A Novel Approach for the Selection of Power-Generation Technology Using a Linguistic Neutrosophic CODAS Method: A Case Study in Libya. Energies
**2018**, 11, 2489. [Google Scholar] [CrossRef] - Liu, F.; Aiwu, G.; Lukovac, V.; Vukic, M. A multicriteria model for the selection of the transport service provider: A single valued neutrosophic DEMATEL multicriteria model. Decision Mak. Appl. Manag. Eng.
**2018**, 1, 121–130. [Google Scholar] [CrossRef] - Orlorski, S.A. Decision making with a fuzzy preference relation. Fuzzy Sets Syst.
**1978**, 3, 155–167. [Google Scholar] - Saaty, T.L. The Analytical Hierarchy Process; McGraw-Hill: New York, NY, USA, 1980. [Google Scholar]
- Herrera, F.; Herrera-Viedma, E.; Verdegay, J.L. A model of consensus in group decision making under linguistic assessments. Fuzzy Sets Syst.
**1996**, 78, 73–87. [Google Scholar] [CrossRef] - Xu, Z.S. On compatibility of interval fuzzy preference relations. Fuzzy Optim. Decis. Mak.
**2004**, 3, 217–225. [Google Scholar] [CrossRef] - Saaty, T.L.; Vargas, L.G. Uncertainty and rank order in the analytic hierarchy process. Eur. J. Oper. Res.
**1987**, 32, 107–117. [Google Scholar] [CrossRef] - Xu, Z.S. Intuitionistic preference relations and their apllication in group decision making. Inf. Sci.
**2007**, 177, 2363–2379. [Google Scholar] [CrossRef] - Xia, M.M.; Xu, Z.S.; Liao, H.C. Preference relations based on intuitionistic multiplicative information. IEEE Trans. Fuzzy Syst.
**2013**, 21, 113–133. [Google Scholar] - Meng, F.Y.; Tang, J.; Fujita, H. Linguistic intuitionistic fuzzy preference relations and their application to multi-criteria decision making. Inf. Fusion
**2019**, 46, 77–90. [Google Scholar] [CrossRef] - Zhai, Y.L.; Xu, Z.S.; Liao, H.C. Measures of Probabilistic Interval-Valued Intuitionistic Hesitant Fuzzy Sets and the Application in Reducing Excessive Medical Examinations. IEEE Trans. Fuzzy Syst.
**2018**, 26, 1651–1670. [Google Scholar] [CrossRef] - Tanino, T. Fuzzy preference orderings in group decision making. Fuzzy Sets Syst.
**1984**, 12, 117–131. [Google Scholar] [CrossRef] - Chiclana, F.; Herrera-Viedma, E.; Alonso, S.; Herrera, F. Cardinal consistency of reciprocal preference relations: A characterization of mulitiplicative transitivity. IEEE Trans. Fuzzy Syst.
**2009**, 17, 14–23. [Google Scholar] [CrossRef] - Meng, F.Y.; Tang, J.; Wang, P.; Chen, X.H. A programming-based algorithm for interval-valued intuitionistic fuzzy group decision making. Knowl.-Based Syst.
**2018**, 144, 122–143. [Google Scholar] [CrossRef] - Wan, S.P.; Wang, F.; Dong, J.Y. A group decision making method with interval valued fuzzy preference relations based on the geometric consistency. Inf. Fusion
**2018**, 40, 87–100. [Google Scholar] [CrossRef] - Wu, J.; Chiclana, F.; Liao, H.C. Isomorphic Multiplicative Transitivity for Intuitionistic and Interval-Valued Fuzzy Preference Relations and Its Application in Deriving Their Priority Vectors. IEEE Trans. Fuzzy Syst.
**2018**, 26, 193–202. [Google Scholar] [CrossRef] - Zhou, W.; Xu, Z.S. Probability Calculation and Element Optimization of Probabilistic Hesitant Fuzzy Preference Relations Based on Expected Consistency. IEEE Trans. Fuzzy Syst.
**2018**, 26, 1367–1378. [Google Scholar] [CrossRef] - Wang, Z.J. A Goal Programming Based Heuristic Approach to Deriving Fuzzy Weights in Analytic Form from Triangular Fuzzy Preference Relations. IEEE Trans. Fuzzy Syst.
**2018**. [Google Scholar] [CrossRef] - Wan, S.P.; Wang, F.; Dong, J.Y. Three-Phase Method for Group Decision Making With Interval-Valued Intuitionistic Fuzzy Preference Relations. IEEE Trans. Fuzzy Syst.
**2018**, 26, 998–1010. [Google Scholar] [CrossRef] - Xie, W.Y.; Ren, Z.L.; Xu, Z.S.; 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] - Zhang, Y.X.; Xu, Z.S.; Liao, H.C. A consensus process for group decision making with probabilistic linguistic preference relations. Inf. Sci.
**2017**, 414, 260–275. [Google Scholar] [CrossRef] - Liao, H.C.; Xu, Z.; Zeng, X.J.; Merigó, J.M. Framework of group decision making with intuitionistic fuzzy preference information. IEEE Trans. Fuzzy Syst.
**2015**, 23, 1211–1227. [Google Scholar] [CrossRef] - Zhao, M.; Ma, X.Y.; Wei, D.W. A method considering and adjusting individual consistency and group consensus for group decision making with incomplete linguistic preference relations. Appl. Soft Comput.
**2017**, 54, 322–346. [Google Scholar] [CrossRef] - Atanassov, K.; Gargov, G. Interval valued intuitionistic fuzzy sets. Fuzzy Sets Syst.
**1989**, 31, 343–349. [Google Scholar] [CrossRef] - Bucolo, M.; Fortuna, L.; La Rosa, M. Complex dynamics through fuzzy chains. IEEE Trans. Fuzzy Syst.
**2004**, 12, 289–295. [Google Scholar] [CrossRef]

**Figure 1.**The uncertainty space of probabilistic uncertain linguistic intuitionistic fuzzy preference relation (PULIFPR).

**Figure 3.**The variation trend of weights ${w}_{i}^{\prime}$ and $\overline{{w}_{i}}$ based on different parameter values $\lambda $.

Year | Event | |

Traditional | 1975 | Zadeh proposes the linguistic variable and introduced the fuzzy linguistic approach [1,2,3]. |

linguistic | 1981 | Yager presents an ordered structure model [4]. |

models | 1988 | Degani and Bortolan present the semantic model [5]. |

1993 | Delgado and Verdegay propose the symbolic model [6]. | |

2000 | Herrera and Martinez introduce the two-tuple linguistic model [7]. | |

2004 | Xu defines the virtual linguistic model [8]. | |

Complex | 2004 | Xu introduces the uncertain linguistic term (ULT) [9]. |

linguistic | 2012 | Rodriguez et al. present the concept of hesitant fuzzy linguistic term sets (HFLTS) [10]. |

expression | 2014 | Meng et al. propose the linguistic hesitant fuzzy sets (LHFS) [11]. |

2014 | Zhang gives the concept of linguistic intuitionistic fuzzy sets (LIFS) [12]. | |

2015 | Ye presents the single-valued neutrosophic linguistic sets (SVNLS) [13]. | |

2016 | Pang et al. present the probabilistic linguistic term sets (PLTS) [14]. | |

2107 | Lin et al. define the probabilistic uncertain linguistic term sets (PULTS) [15]. | |

2018 | Bai et al. present the interval-valued probabilistic linguistic term sets (IVPLTS) [16]. | |

2018 | Zhang et al. propose the dual hesitant fuzzy linguistic term sets (DHFLTS) [17]. |

$\mathit{\lambda}$ | $\overline{{\mathit{w}}_{1}}$ | $\overline{{\mathit{w}}_{2}}$ | $\overline{{\mathit{w}}_{3}}$ | $\overline{{\mathit{w}}_{4}}$ | Ranking Order |
---|---|---|---|---|---|

0.1 | 0.1350 | 0.2488 | 0.2945 | 0.3217 | ${x}_{4}>{x}_{3}>{x}_{2}>{x}_{1}$ |

0.2 | 0.1388 | 0.2467 | 0.3008 | 0.3137 | ${x}_{4}>{x}_{3}>{x}_{2}>{x}_{1}$ |

0.3 | 0.1427 | 0.2446 | 0.3070 | 0.3057 | ${x}_{3}>{x}_{4}>{x}_{2}>{x}_{1}$ |

0.4 | 0.1466 | 0.2425 | 0.3133 | 0.2976 | ${x}_{3}>{x}_{4}>{x}_{2}>{x}_{1}$ |

0.5 | 0.1505 | 0.2404 | 0.3195 | 0.2896 | ${x}_{3}>{x}_{4}>{x}_{2}>{x}_{1}$ |

0.6 | 0.1544 | 0.2382 | 0.3258 | 0.2816 | ${x}_{3}>{x}_{4}>{x}_{2}>{x}_{1}$ |

0.7 | 0.1583 | 0.2361 | 0.3321 | 0.2735 | ${x}_{3}>{x}_{4}>{x}_{2}>{x}_{1}$ |

0.8 | 0.1622 | 0.2340 | 0.3383 | 0.2655 | ${x}_{3}>{x}_{4}>{x}_{2}>{x}_{1}$ |

0.9 | 0.1660 | 0.2319 | 0.3446 | 0.2575 | ${x}_{3}>{x}_{4}>{x}_{2}>{x}_{1}$ |

Methods | Preference Relations | Considering the Non-Preference Information | Considering the Probability Distribution | Considering the Fuzzy Uncertainty (Ignorance Information) | Determining the Individual Weight | Avoiding Consistency Checks and Corrections | Considering the Risk Attitudes of DMs |
---|---|---|---|---|---|---|---|

The proposed method | PULIFPRs | Yes | Yes | Yes | Yes | Yes | Yes |

Meng et al.’s method [27] | LIFPRs | Yes | No | No | No | No | No |

Xie et al.’s method [37] | PULPRs | No | Yes | No | No | No | No |

Zhang et al.’s method [38] | PLPRs | No | Yes | No | No | No | No |

Wan et al.’s method [36] | IVIFPRs | Yes | No | Yes | Yes | No | Yes |

Liao et al.’s method [39] | IFPRs | Yes | No | No | No | No | No |

Wan et al.’s method [32] | IVFPRs | No | No | No | Yes | No | Yes |

Zhao et al.’s method [40] | LPRs | No | No | Yes | Yes | No | No |

Meng et al.’s method [31] | IVIFPRs | Yes | No | No | Yes | No | No |

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**MDPI and ACS Style**

Gong, K.; Chen, C.
A Programming-Based Algorithm for Probabilistic Uncertain Linguistic Intuitionistic Fuzzy Group Decision-Making. *Symmetry* **2019**, *11*, 234.
https://doi.org/10.3390/sym11020234

**AMA Style**

Gong K, Chen C.
A Programming-Based Algorithm for Probabilistic Uncertain Linguistic Intuitionistic Fuzzy Group Decision-Making. *Symmetry*. 2019; 11(2):234.
https://doi.org/10.3390/sym11020234

**Chicago/Turabian Style**

Gong, Kaixin, and Chunfang Chen.
2019. "A Programming-Based Algorithm for Probabilistic Uncertain Linguistic Intuitionistic Fuzzy Group Decision-Making" *Symmetry* 11, no. 2: 234.
https://doi.org/10.3390/sym11020234