**Algorithm for Probabilistic Dual Hesitant Fuzzy Multi-Criteria Decision-Making Based on Aggregation Operators with New Distance Measures**

### **Harish Garg \* and Gagandeep Kaur**

School of Mathematics, Thapar Institute of Engineering & Technology (Deemed University) Patiala, Punjab 147004, India; gdeep01@ymail.com

**\*** Correspondence: harishg58iitr@gmail.com or harish.garg@thapar.edu; Tel.: +91-86990-31147

Received: 2 November 2018; Accepted: 21 November 2018; Published: 25 November 2018

**Abstract:** Probabilistic dual hesitant fuzzy set (PDHFS) is an enhanced version of a dual hesitant fuzzy set (DHFS) in which each membership and non-membership hesitant value is considered along with its occurrence probability. These assigned probabilities give more details about the level of agreeness or disagreeness. By emphasizing the advantages of the PDHFS and the aggregation operators, in this manuscript, we have proposed several weighted and ordered weighted averaging and geometric aggregation operators by using Einstein norm operations, where the preferences related to each object is taken in terms of probabilistic dual hesitant fuzzy elements. Several desirable properties and relations are also investigated in details. Also, we have proposed two distance measures and its based maximum deviation method to compute the weight vector of the different criteria. Finally, a multi-criteria group decision-making approach is constructed based on proposed operators and the presented algorithm is explained with the help of the numerical example. The reliability of the presented decision-making method is explored with the help of testing criteria and by comparing the results of the example with several prevailing studies.

**Keywords:** probabilistic dual hesitant fuzzy sets; distance measures; aggregation operators; consumer behavior; multi-criteria decision-making; maximum deviation method
