Journal of Intelligent & Fuzzy Systems 35 (2018) 5435–5448 DOI:10.3233/JIFS-171190 IOS Press 5435 New extension of TOPSIS method based on Pythagorean hesitant fuzzy sets with incomplete weight information Muhammad Sajjad Ali Khan a,∗ , Asad Ali a , Saleem Abdullah b , Fazli Amin a and Fawad Hussain c a Department of Mathematics, Hazara University, Mansehra, KPK, Pakistan b Department of Mathematics, Abdul Wali Khan University, Mardan, KPK, Pakistan c Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, KPK, Pakistan Abstract. Pythagorean Hesitant fuzzy set (PHFS) which permits the membership degree and non-membership degree of an element to a set represented by several possible values is deliberated as a powerful tool to express uncertain information in the process of multi-attribute decision making (MADM) problems. In this paper, we propose a novel approach based on TOPSIS method and the maximizing deviation method for solving MADM problems where the evaluation information provided by the decision makers (DMs) is expressed in form of Pythagorean hesitant fuzzy numbers and the information about attribute weights is incomplete. To determine the attribute weight we develop an optimization model based on maximizing deviation method. Finally we provide a practical decision-making problem to demonstrate the implementation process of the proposed method. Keywords: Pythagorean hesitant fuzzy set, maximizing deviation method, TOPSIS method 1. Introduction The researches which are keen on contributing to socially relevant research either through pure or applied streams are looking forward to solve real life problems. Multi-attribute decision making (MADM) which addresses the problem of making a finest choice that has the highest degree of satisfaction from a set of alternatives that are characterized in terms of their attributes, is a susual task in human activities and these kinds of MADM problems arise in many real-world situations. It is a major com- ponent of decision science, whose theory has been widely applied in the fields of economy [19, 27], ∗ Corresponding author. Muhammad Sajjad Ali Khan, Depart- ment of Mathematics, Hazara University, Mansehra, KPK, Pakistan. E-mail: sajjad maths@hu.edu.pk. management [12, 30], engineering [10, 11], etc. Many approaches have been proposed to handle the MADM problems, such as TOPSIS (Technique for Order Preference by Similarity to an Ideal Solution) [13], ELECTRE [28], and PROMETHEE [18]. In classi- cal MADM, the assessments of alternatives are surely known [6, 31]. In addition, for the uncertainty of the MADM problems, the DMs are difficult to pro- vide the precise assessments for alternatives. To solve this issue, fuzzy set theory [43] has been applied to MADM [4, 20], which provides a crucial means of describing the complex information. However, the fuzzy set theory is still confronted with some limita- tions when decision makers (DMs) intend to deal with some uncertain information induced from several sources of vagueness, the attributes involved in deci- sion making problems are not always expressed in real numbers, and some are better suited to be denoted 1064-1246/18/$35.00 © 2018 – IOS Press and the authors. All rights reserved