An Algorithm for Interval-Valued Intuitionistic Fuzzy Preference Relations in Group Decision Making Based on Acceptability Measurement and Priority Weight Determination
Abstract
:1. Introduction
- (1)
- While checking the consistency degree of a given IVIFPR, existing research [17,18] has proposed the definitions of a consistent IVIFPR, which can be used to judge whether an IVIFPR is consistent or not. However, none of them can measure the consistency degree of an inconsistent IVIFPR. Other literature (References [10,12,19]) has also defined the consistency of an IVIFPR through extracting associated preference matrices from original IVIFPR, which cannot fully attach the initial preference information and needs to be constructed by the numerous computational efforts. Therefore, it is necessary to find a convenient solution to check the consistency degrees of IVIFPRs.
- (2)
- When a given IVIFPR is unacceptably consistent, only Liao et al. [18] and Wan et al. [12] proposed some iterative algorithms to improve its consistency degree. Nevertheless, it may be to repeat these algorithms several times to repair this unacceptably consistent IVIFPR until the acceptable consistency is achieved. This is time consuming and requires heavy workload. Furthermore, it is unknown how much of the repaired IVIFPR reserves preference information for the initial IVIFPR.
- (3)
- As for the determination of priority weights of alternatives for IVIFPRs, the existing studies [15,22,24] have directly adopted the formulas to rank the alternatives. However, such formulas are only applicable to consistent IVIFPRs and would be ineffective for inconsistent IVIFPRs. Other methods (References [9,11,25]) have established some optimizing or linear models to determine the priority weights, noting that these optimizing or linear models only minimize the deviation between the associated matrices form original IVIFPR and the converted consistent one to the most extent. However, for an extremely unacceptable consistent IVIFPR, the priority weights obtained by such models are unreasonable and cannot be accepted in decision making.
- (1)
- Liao et al. [18] defined the multiplicative consistency of IVIFPR to adjust or repair the inconsistent IVIFPRs. The consistent degree of an inconsistent IVIFPR cannot be measured by [18]. To measure the consistency degree of an IVIFPR, according to the multiplicative consistency definition in Reference [18], this paper proposes a new concept of multiplicative consistency index of an IVIFPR by considering various decision-making principles (i.e., the majority and minority principles). A notable feature is that multiplicative consistency index of an IVIFPR can be directly obtained from original IVIFPR without any transformation via Wan et al. [12]. Therefore, the definition of multiplicative consistency index of an IVIFPR is easy to operate.
- (2)
- An optimization model is directly established to repair and improve the consistency of IVIFPR. Compared with iterative algorithms proposed in References [12,18], the proposed model can not only rapidly obtain an acceptable consistent IVIFPR from the initial IVIFPR, but also enables the obtained IVIFPR to retain as much as possible the preference information hidden in the initial IVIFPR.
- (3)
- An error-analysis-based extension method is developed to derive IVIF priority weights from the acceptable consistent IVIFPR. Compared with some optimization models proposed in References [9,11,12], the extension method can simplify computation and save time. Moreover, in GDM problems, DMs’ weights are generated by using the proposed multiplicative consistency index of an IVIFPR, which is objective and reasonable to some extent.
2. Preliminaries
2.1. Some Associated Definitions on Preference Relations
- If, then.
- If, then
- If, then.
- If, then.
- If, then.
2.2. Error Propagation Formula
3. Acceptability Measurement of an IVIFPR
3.1. Multiplicative Consistency Index of an IVIFPR
3.2. An Optimization Model to Obtain the Acceptable Multiplicative Consistent IVIFPR
- Step 1.
- Set and .
- Step 2.
- Using Equations (14) and (15), obtain and . Then, by Equation (16), calculate the multiplicative consistency index Since , the IVIFPR is unacceptable multiplicative consistency index and go to next step.
- Step 3.
- Using Model (18), an optimization model is constructed as follows:
4. Priority Weights Determination of Alternatives for an IVIFPR
4.1. Extraction of Two Special IVFPRs from an IVIFPR
4.2. An Error-Analysis-Based Extension Method for Determining the IVIF Priority Weights
- Step 1.
- Using Equation (20), two special IVFPRs and can be extracted from the acceptable multiplicative consistent IVIFPR , which are shown in the following:
- Step 2.
- Using Equations (27) and (29), the priority vector with for the IVFPR can be determined as follows:
5. An Algorithm for GDM with IVIFPRs
5.1. Description of GDM Problems with IVIFPRs
5.2. The Method for Deriving DMs’ Weights
5.3. An Algorithm for GDM with IVIFPRs
- Step 1.
- Predefined consistency threshold and set the value of parameter .
- Step 2.
- Calculate the multiplicative consistency index for an individual IVIFPR using Equation (16).
- Step 3.
- Derive DMs’ weights using Equation (35).
- Step 4.
- Using formula , pick out the unacceptable multiplicative consistency of IVIFPR . If all IVIFPRs are acceptable multiplicative consistent, then go to Step 6; otherwise, go to Step 5.
- Step 5.
- Derive the acceptable multiplicative consistent IVIFPR from unacceptable multiplicative consistent IVIFPR by Equations (18) and (19).
- Step 6.
- Integrate the individual IVIFPR into a collective IVIFPR through Equation (33).
- Step 7.
- Using Equations (27) and (29), determine interval-valued priority weight .
- Step 8.
- Using Equations (30) and (31), determine interval-valued priority weight .
- Step 9.
- Generate the IVIF priority weights through Equation (32).
- Step 10.
- Using Equations (1) and (2), rank alternatives by calculating the score and the accurate values of IVIF priority weights
6. A Practical Example for GDM with IVIFPRs
6.1. A Practical Example of Enterprise Innovation Partner Selection
- Step 1.
- Predefine consistency threshold and set the value of parameter .
- Step 2.
- Calculate the multiplicative consistency index for individual IVIFPR using Equation (16), the results of the calculation are as follows:
- Step 3.
- Determine DMs’ weights using Equation (35).
- Step 4.
- Using formula , all the individual IVIFPRs are unacceptable multiplicative consistent.
- Step 5.
- Solving Model (18), the optimal solutions , , and for all and are derived from unacceptable multiplicative consistent IVIFPRs .Then, using Equation (19), the corresponding acceptable multiplicative consistent IVIFPRs are generated as
- Step 6.
- Integrating the acceptable multiplicative consistent IVIFPR by DMs’ weights , the collective IVIFPR is generated using Equation (33) as follows:
- Step 7.
- Using Equations (27) and (29), IVIFPR priority weight vector are generated as follows:
- Step 8.
- Using Equations (30) and (31), IVIFPR priority weight vector are generated as follows:
- Step 9.
- By Equation (32), the IVIFPR priority weights are derived as follows:
- Step 10.
- According to Definition 3, the scores function of IVIFV belongs to the interval [−1, 1], and its associating weight should be a non-negative number. Therefore, the score function defined in Definition 3 should be modified to facilitate this situation under the conditions of without changing any of the following basic properties
6.2. Comparative Analyses
6.2.1. Comparison with Wan’s Method
- (1)
- When measuring the consistency degree of an IVIFPR, the proposed algorithm is more straightforward than Wan’s method [12]. Wan’s method [12] needs resort to two particular IFPR matrices from original IVIFPR. In contrast, the proposed algorithm directly performs the measurement work only depending on original IVIFPR, which can avoid losing original information via Wan’s method [12].
- (2)
- In terms of repairing and improving the unacceptable multiplicative consistent IVIFPR, different from the iterative algorithms proposed in Wan’s method [12], this paper directly built an optimization model through considering various decision-making principles, which can quickly obtain the acceptable consistent IVIFPR and flexibly reflect the principles of decision making.
- (3)
- For determining priority weights of alternatives, Wan’s method [12] constructed a goal optimization model which needs numerous computations. On the contrary, the proposed algorithm determines IVIF priority weights by extending error-analysis-based priority method to IVIFPR environment, which is a time-saving and simple calculation.
6.2.2. Comparison with Other Existing Methods
- (1)
- The concerns of these methods are different. Method [6] studied the consistent and consensus in GDM with IFPR, whereas Method [18] and the proposed algorithm concentrate on the multiplicative consistency of IVIFPR. The discrepancy is that Method [18] focused on the multiplicative transitivity of IVIFPR and improving the consistency of an inconsistent IVIFPR, while the proposed algorithm is devoted to judging and measuring the multiplicative consistency degree of an IVIFPR and address the GDM with IVIFPRs.
- (2)
- Method [18] investigated an acceptable property of multiplicative consistency of IVIFPR and introduced some associated concepts of IVIFPR (i.e., the approximate, the perfect and the acceptable multiplicative consistent IVIFPR), but the multiplicative consistency degree of an IVIFPR cannot be obtained in Method [18]. Following the work of Method [18], based on two principles of decision making (the majority and minority principles), this paper proposes a new definition of consistency index of an IVIFPR, which can measure the multiplicative consistency degree of an IVIFPR.
- (3)
- Regarding to priority weight determination, Methods [6,18] cannot provide any tools or methods determine priority weights. However, this paper extends error-analysis-based approach in IVIFPR to determine priority weights, and it is worth noting that the proposed algorithm can reduce the complexity of the calculation.
7. Conclusions
- (1)
- A new definition of consistency index is defined to measure whether an IVIFPR is of acceptable multiplicative consistency. A common feature is that it can complete the measurement work only employing individual IVIFPRs themselves.
- (2)
- An optimization model is constructed to improve the consistency degree for those IVIFPRs that not attain the acceptable level. Moreover, the obtained IVIFPRs can retain the preference information given by the initial IVIFPRs, for the most part.
- (3)
- An error-analysis-based extension method is proposed to determine IVIF priority weights from the acceptable IVIFPR. It can help the decision maker to obtain the reasonable and identified decision-making results.
- (4)
- A step-by-step algorithm is developed for solving GDM with IVIFPRs, and a practical example is presented to demonstrate its effectiveness and practicality. The results in this paper are very important for the application of IVIFPR in GDM.
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
- Orlovsky, S.A. Decision-making with a fuzzy preference relation. Fuzzy Sets Syst. 1978, 1, 155–167. [Google Scholar] [CrossRef]
- Yan, H.B.; Ma, T. A group decision-making approach to uncertain quality function deployment based on fuzzy preference relation and fuzzy majority. Eur. J. Oper. Res. 2015, 241, 815–829. [Google Scholar] [CrossRef]
- Saaty, T.L. The Analytic Hierarchy Process: Planning, Priority Setting, Resource Allocation; McGraw-Hill: New York, NY, USA, 1980. [Google Scholar]
- Wang, Y.M.; Fan, Z.P.; Hua, Z.S. A chi-square method for obtaining a priority vector from multiplicative and fuzzy preference relations. Eur. J. Oper. Res. 2007, 182, 356–366. [Google Scholar] [CrossRef]
- Chu, J.; Liu, X.; Wang, Y.; Chin, K.S. A group decision making model considering both the additive consistency and group consensus of intuitionistic fuzzy preference relations. Comput. Ind. Eng. 2016, 101, 227–242. [Google Scholar] [CrossRef]
- Xu, G.L.; Wan, S.P.; Wang, F.; Dong, J.Y.; Zeng, Y.F. Mathematical programming methods for consistency and consensus in group decision making with intuitionistic fuzzy preference relations. Knowl.-Based Syst. 2016, 98, 30–43. [Google Scholar] [CrossRef]
- Wan, S.; Wang, F.; Dong, J. A group decision making method with interval valued fuzzy preference relation based on the geometric consistency. Inf. Fusion 2017, 40, 87–100. [Google Scholar] [CrossRef]
- Meng, F.Y.; An, Q.X.; Tan, C.Q.; Chen, X.H. An Approach for Group Decision Making With Interval Fuzzy Preference Relations Based on Additive Consistency and Consensus Analysis. IEEE Trans. Syst. Man Cybern. Syst. 2016, PP, 1–14. [Google Scholar] [CrossRef]
- Wan, S.P.; Wang, F.; Dong, J.Y. Additive consistent interval-valued Atanassov intuitionistic fuzzy preference relation and likelihood comparison algorithm based group decision making. Eur. J. Oper. Res. 2017, 263, 571–582. [Google Scholar] [CrossRef]
- Wan, S.P.; Wang, F.; Dong, J.Y. A 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]
- Chu, J.; Liu, X.; Wang, L.; Wang, Y. A Group Decision Making Approach Based on Newly Defined Additively Consistent Interval-Valued Intuitionistic Preference Relations. Int. J. Fuzzy Syst. 2017, 20, 1027–1046. [Google Scholar] [CrossRef]
- Wan, S.P.; Xu, G.L.; Dong, J.Y. A novel method for group decision making with interval-valued Atanassov intuitionistic fuzzy preference relations. Inf. Sci. 2016, 372, 53–71. [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. Decis. Mak. Appl. Manag. Eng 2018, 1, 121–130. [Google Scholar] [CrossRef]
- Petrović, I.; Kankaraš, M. DEMATEL-AHP multi-criteria decision making model for the selection and evaluation of criteria for selecting an aircraft for the protection of air traffic. Decis. Mak. Appl. Manag. Eng 2018, 1, 93–110. [Google Scholar] [CrossRef]
- Ze-Shui, X.U.; Chen, J. Approach to Group Decision Making Based on Interval-Valued Intuitionistic Judgment Matrices. Syst. Eng.-Theory Pract. 2007, 27, 126–133. [Google Scholar]
- Xu, Z.S.; Cai, X.Q. Incomplete interval-valued intuitionistic fuzzy preference relations. Int. J. Gener. Syst. 2009, 38, 871–886. [Google Scholar] [CrossRef]
- Xu, Z.; Cai, X. Group Decision Making with Incomplete Interval-Valued Intuitionistic Preference Relations. Group Decis. Negotiat. 2015, 24, 193–215. [Google Scholar] [CrossRef]
- Liao, H.C.; Xu, Z.S.; Xia, M.M. Multiplicative consistency of interval-valued intuitionistic fuzzy preference relation. J. Intell. Fuzzy Syst. 2014, 27, 2969–2985. [Google Scholar]
- 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]
- Mukhametzyanov, I.; Pamucar, D. A sensitivity analysis in MCDM problems: A statistical approach. Decis. Mak. Appl. Manag. Eng. 2018, 1, 1–20. [Google Scholar] [CrossRef]
- Wang, Z.J.; Lin, J. Acceptability measurement and priority weight elicitation of triangular fuzzy multiplicative preference relations based on geometric consistency and uncertainty indices. Inf. Sci. 2017, 402, 105–123. [Google Scholar] [CrossRef]
- Xu, Z.; Yager, R.R. Intuitionistic and interval-valued intutionistic fuzzy preference relations and their measures of similarity for the evaluation of agreement within a group. Fuzzy Optim. Decis. Mak. 2009, 8, 123–139. [Google Scholar] [CrossRef]
- Wu, J.; Chiclana, F. Non-dominance and attitudinal prioritisation methods for intuitionistic and interval-valued intuitionistic fuzzy preference relations. Expert Syst. Appl. 2012, 39, 13409–13416. [Google Scholar] [CrossRef]
- Yue, C. A geometric approach for ranking interval-valued intuitionistic fuzzy numbers with an application to group decision-making. Comput. Ind. Eng. 2016, 102, 233–245. [Google Scholar] [CrossRef]
- Zhou, H.; Ma, X.Y.; Zhou, L.G.; Chen, H.Y.; Ding, W.R. A Novel Approach to Group Decision-Making with Interval-Valued Intuitionistic Fuzzy Preference Relations via Shapley Value. Int. J. Fuzzy Syst. 2018, 20, 1172–1187. [Google Scholar] [CrossRef]
- Xu, Z.S. An error-analysis-based method for the priority of an intuitionistic preference relation in decision making. Knowl.-Based Syst. 2012, 33, 173–179. [Google Scholar] [CrossRef]
- Xu, Z. On Compatibility of Interval Fuzzy Preference Relations. Fuzzy Optim. Decis. Mak. 2004, 3, 217–225. [Google Scholar] [CrossRef]
- Atanassov, K.; Gargov, G. Interval Valued Intuitionistic Fuzzy-Sets. Fuzzy Set Syst. 1989, 31, 343–349. [Google Scholar] [CrossRef]
- Yager, R.R. Induced aggregation operators. Fuzzy Set Syst. 2003, 137, 59–69. [Google Scholar] [CrossRef]
- Pugh, E.M.; Winslow, G.H. The Analysis of Physical Measurements; Addison-Wesley: Reading, MA, USA, 1966. [Google Scholar]
- Xu, Y.; Herrera, F.; Wang, H. A distance-based framework to deal with ordinal and additive inconsistencies for fuzzy reciprocal preference relations. Inf. Sci. 2016, 328, 189–205. [Google Scholar] [CrossRef]
- Bustince, H. Conjuntos Intuicionistas e Intervalo Valorados Difusos: Propiedades y Construccion, Relaciones Intuicionistas Fuzzy. Ph.D. Thesis, Universidad Publica de Navarra, Pamplona, Spain, 1994. [Google Scholar]
- Shui, X.Z. Algorithm for priority of fuzzy complementary judgement matrix. J. Syst. Eng. 2001, 16, 311–314. [Google Scholar]
Ranking Order | |||||||||
---|---|---|---|---|---|---|---|---|---|
0 | 0.2449 | 0.9393 | 0.3092 | 0.9461 | 0.2752 | 0.9442 | 0.2122 | 0.9338 | |
0.1 | 0.2451 | 0.9393 | 0.3091 | 0.9461 | 0.2751 | 0.9443 | 0.2122 | 0.9339 | |
0.2 | 0.2451 | 0.9395 | 0.3090 | 0.9463 | 0.2750 | 0.9445 | 0.2123 | 0.9341 | |
0.3 | 0.2451 | 0.9395 | 0.3091 | 0.9464 | 0.2749 | 0.9445 | 0.2123 | 0.9341 | |
0.4 | 0.2450 | 0.9396 | 0.3093 | 0.9466 | 0.2749 | 0.9446 | 0.2121 | 0.9343 | |
0.5 | 0.2454 | 0.9394 | 0.3092 | 0.9466 | 0.2747 | 0.9444 | 0.2122 | 0.9344 | |
0.6 | 0.2454 | 0.9394 | 0.3094 | 0.9466 | 0.2745 | 0.9445 | 0.2121 | 0.9345 | |
0.7 | 0.2453 | 0.9389 | 0.3090 | 0.9461 | 0.2754 | 0.9440 | 0.2124 | 0.9342 | |
0.8 | 0.2452 | 0.9389 | 0.3089 | 0.9461 | 0.2755 | 0.9441 | 0.2125 | 0.9341 | |
0.9 | 0.2472 | 0.9370 | 0.3116 | 0.9456 | 0.2715 | 0.9421 | 0.2108 | 0.9348 | |
1 | 0.2460 | 0.9370 | 0.3101 | 0.9463 | 0.2722 | 0.9418 | 0.2110 | 0.9371 |
Parameter | Ranking Order | ||||||||
---|---|---|---|---|---|---|---|---|---|
0.2454 | 0.9394 | 0.3092 | 0.9466 | 0.2747 | 0.9444 | 0.2122 | 0.9344 | ||
0.2489 | 0.9348 | 0.3092 | 0.9423 | 0.2732 | 0.9400 | 0.2112 | 0.9310 | ||
0.2487 | 0.9350 | 0.3095 | 0.9426 | 0.2730 | 0.9401 | 0.2112 | 0.9312 |
0.4915 | 0.4898 | 0.4957 | |
0.6783 | 0.6836 | 0.6910 | |
0.3711 | 0.3644 | 0.3543 | |
0.3702 | 0.3665 | 0.3655 | |
R.O. |
© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
Share and Cite
Zhuang, H.; Tang, Y.; Li, M. An Algorithm for Interval-Valued Intuitionistic Fuzzy Preference Relations in Group Decision Making Based on Acceptability Measurement and Priority Weight Determination. Algorithms 2018, 11, 182. https://doi.org/10.3390/a11110182
Zhuang H, Tang Y, Li M. An Algorithm for Interval-Valued Intuitionistic Fuzzy Preference Relations in Group Decision Making Based on Acceptability Measurement and Priority Weight Determination. Algorithms. 2018; 11(11):182. https://doi.org/10.3390/a11110182
Chicago/Turabian StyleZhuang, Hua, Yanzhao Tang, and Meijuan Li. 2018. "An Algorithm for Interval-Valued Intuitionistic Fuzzy Preference Relations in Group Decision Making Based on Acceptability Measurement and Priority Weight Determination" Algorithms 11, no. 11: 182. https://doi.org/10.3390/a11110182