1063-6706 (c) 2018 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information. This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TFUZZ.2018.2843330, IEEE Transactions on Fuzzy Systems 1 Abstract—The probabilistic linguistic term set is a powerful technique in representing linguistic evaluations of individuals or groups in the process of decision making. The aim of this paper is to propose a strongly robust method to solve multi-experts multi-criteria decision making problems with linguistic evaluations. To enrich the computation and to improve the measures of probabilistic linguistic term set, we firstly define an expectation function of it. In addition, we advance three kinds of probabilistic linguistic distance measures reflecting on the difference of linguistic terms and probabilities at the same time to make up for the defects of the existing distance measures, and then propose the similarity and correlation measures. Integrating the subjective opinions with the correlation coefficients between criteria, we put forward a combined weight determining method. The robustness of the ranking method, MULTIMOORA, is enhanced by the improved Borda rule. Based on these research findings, a probabilistic linguistic MULTIMOORA method is proposed. Finally, the developed method is applied to an empirical example concerning the selection of shared karaoke television brands. The effectiveness of the proposed method is verified by some comparative analyses. Index Terms—Multi-experts multi-criteria decision making, Probabilistic linguistic term set, MULTIMOORA, Distance measure, Combined weights, Borda rule. I. INTRODUCTION M ULTI-EXPERTS Multi-Criteria Decision Making (MEMCDM) is the process of selecting optimal scheme from a finite set of alternatives which are evaluated by multiple experts based on multiple criteria. A complete MEMCDM process includes three parts: (1) express the evaluations given by experts; (2) determine the weights of criteria; (3) aggregate the The work was supported by the National Natural Science Foundation of China (Nos. 71501135, 71771156, 71571123, 71771155). (Corresponding Author: Huchang Liao.) X. L. Wu, H. C. Liao and Z. S. Xu are with the Business School, Sichuan University, Chengdu 610064, China (e-mails: xingliwusly@foxmail.com; liaohuchang@163.com; xuzeshui@263.net). H. C. Liao, A. Hafezalkotob and F. Herrera are with the Department of Computer Science and Artificial Intelligence, University of Granada, E-18071 Granada, Spain (e-mails: hafez@correo.ugr.es; herrera@decsai.ugr.es). F. Herrera is also with Faculty of Computing and Information Technology, King Abdulaziz University, Jeddah, Saudi Arabia. alternatives’ evaluations and obtain their ranking. To describe the imprecise cognition, Zadeh [1] put forward the fuzzy set, which adopts the membership degree to depict the intensity of an element belonging to a set. Typically, in group decision making process, experts often hesitate among several evaluation values and they are reluctant to compromise, which makes the final judgments difficult to reach an agreement. Historically we can find different fuzzy set extensions to deal with the uncertainty [2]. In this case, Torra [3] proposed the Hesitant Fuzzy Set (HFS) which includes a set of possible membership degrees. A comprehensive overview on HFS can be found in Ref. [4]. The linguistic terms, however, are much closer to people's expression habits than the quantitative membership degrees. Thus, Rodríguez et al. [5] extended the HFS to linguistic context and introduced the Hesitant Fuzzy Linguistic Term Set (HFLTS) which allows experts to represent judgments in more than one linguistic terms. For example, the assessment on the project risk can be “between medium and high” and the evaluation of a teacher's teaching ability can be “at least strong”. In recent years, there has been a great deal of researches on HFLTSs [6-13]. However, the HFLTS assigns the same weight to each linguistic term and thus cannot reflect the actual opinions of experts in some cases. Thereafter, Pang et al. [14] extended the HFLTS to a general form, namely, the Probabilistic Linguistic Term Set (PLTS), by associating each linguistic term with a probability. The meaning of probability can be the weight, the possibility degree, or the trust degree. The PLTS is flexible to express both simple linguistic terms, such as “good”, “more than good” and complex linguistic expressions, such as “60% sure it is good and 40% sure it is very good”. A. Motivations Due to the flexibility and comprehensiveness of the PLTS, it has aroused growing concerns [14-19]. For calculation purposes, some basic operations of PLTSs were presented [14], but they are not very reasonable and the computation results may be beyond the given Linguistic Term Set (LTS). Despite some improved operations were developed [16], there are still some flaws: (1) it is difficult to give a reasonable explanation on the meaning of the operations, especially the multiplication and division; (2) the operations are extremely complex and Probabilistic Linguistic MULTIMOORA: A Multi-Criteria Decision Making Method Based on the Probabilistic Linguistic Expectation Function and the Improved Borda Rule Xingli Wu, Huchang Liao, Senior Member, IEEE, Zeshui Xu, Senior Member, IEEE, Arian Hafezalkotob, and Francisco Herrera, Senior member, IEEE