On consistency and priority weights for interval probabilistic linguistic preference relations
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On consistency and priority weights for interval probabilistic linguistic preference relations Xiangqian Feng1 · Xiaodong Pang2 · Lan Zhang2
© Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract When expressing preferences with different probability weights for different linguistic terms, only partial assessment information is usually to be provided. Then the probability information can be normalized to the interval probability, hence, using interval probabilistic linguistic term sets (IPLTs) is more appropriate. Considering this situation, interval probabilistic linguistic preference relation (IPLPR) is proposed. To measure the consistency of IPLPR, the consistency definition of IPLPR is put forward. For the consistent IPLPR, from which an expected consistent PLPR can be obtained, we can obtain interval weights as the final priorities by using the pairs of linear programming models. We also create the probabilistic linguistic geometric consistency index (PLGCI) of PLPRs to judge whether the IPLPR is satisfactorily consistent. For an unsatisfied consistency IPLPR, the adjusting algorithm is proposed. Probability information is firstly considered to be adjusted. If it is not possible to achieve satisfactory consistency through the adjustment of probability information, then the linguistic terms will be adjusted. In addition to examples of different situations, such as the consistency, satisfactory consistency and consistency improvement, the application example is also given to show the practicability of the proposed methods. Keywords Probabilistic linguistic term sets (PLTs) · Probabilistic linguistic preference relation (PLPR) · Probabilistic Linguistic Geometric Consistency Index (PLGCI) · Consistency measures · Interval weights
1 Introduction In practical decision making problems, the DMs or experts are more inclined to express their preferences for solutions by using the natural linguistic terms such as “poor”,
B
Xiangqian Feng [email protected]
1
School of Business, Nanjing Normal University, Nanjing 210023, China
2
School of Computer Science & Technology, Nanjing Normal University, Nanjing 210023, China
123
X. Feng et al.
“medium”, or “good”. In this case, the fuzzy linguistic approach was put forward in Zadeh (1975), which denotes quantitative aspects as linguistic values via the linguistic variables. There may exist uncertainties when DMs use linguistic information to express their preferences in practical applications. In order to represent uncertainty more accurately, a lot of extensions of the linguistic term sets (LTSs) have been devised such as the virtual LTSs (Xu and Wang 2017), the 2-tuple LTS (Herrera and Martinez 2000), hesitant fuzzy LTSs (HFLTSs) (Rodriguez et al. 2012) and probabilistic LTSs (PLTSs) (Liao et al. 2020; Pang et al. 2016). As an effective and flexible tool in expressing people’s subjective cognition, the LTS plays an irreplaceable role in practical applications. The LTS has been implemented to customer relationship management (Zhang et al. 20
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