Journal of Fuzzy Set Valued Analysis 2016 No.2 (2016) 156-173 Available online at www.ispacs.com/jfsva Volume 2016, Issue 2, Year 2016 Article ID jfsva-00303, 18 Pages doi:10.5899/2016/jfsva-00303 Research Article A new and efficient method for elementary fuzzy arithmetic operations on pseudo-geometric fuzzy numbers F. Abbasi 1 ∗ , S. Abbasbandy 2 , J. J. Nieto 3 (1) Department of Mathematics, Ayatollah Amoli Branch, Islamic Azad University, Amol, Iran. (2) Department of Mathematics, Science and Research Branch, Islamic Azad University, Tehran, Iran. (3) Departamento de An ´ alisis Matem´ atico, Facultad de Matem´ aticas, Universidad de Santiago de Compostela, 15782, Santiago de Compostela, Spain Copyright 2016 c ⃝ F. Abbasi, S. Abbasbandy and J. J. Nieto. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract There are certain problems in the subtraction operator, division operator and obtaining the membership functions of operators and above all, dependence effect in the fuzzy arithmetic operations using the extension principle (in the domain of the membership function) or the interval arithmetics (in the domain of α - cuts). In this regard, this pa- per provide a new method regarding the effective practical computation of elementary fuzzy arithmetic operations on pseudo-geometric fuzzy numbers. Therefore we eliminated such deficiency with the new proposed method and demonstrated that the new operators are more efficient. Finally, several illustrative examples were given to show the accomplishment and ability of the proposed method. The future prospect of this paper is a new attitude to fuzzy mathematics. Keywords: Fuzzy arithmetic; pseudo-geometric fuzzy numbers; Extension principle (EP); Transmission average (TA). 1 Introduction In order to use fuzzy numbers and relations in any intelligent system one must be able to perform arithmetic op- erations, addition, subtraction, multiplication and division, employing these fuzzy quantities, the process of which is called fuzzy arithmetic. One of the most basic concepts of fuzzy set theory is the extension principle introduced by Zadeh, which was already implied in [10] in its rudimentary form and was finally presented in its fully-fledged form in [11, 12, 13]. This principle provides a general method for extending crisp mathematical concepts to fuzzy quantities, that is, it allows the domain of functional mapping definition to be extended from crisp elements to fuzzy sets as the arguments of the function. The usual arithmetic operations on real numbers can be extended to the ones defined on fuzzy numbers by means of Zadeh’s EP . In this context, direct implementation of this principle in fuzzy arithmetic is computationally expensive due to the requirement of solving a nonlinear programming problem [9]. To overcome this deficiency, many researchers consider fuzzy numbers as a collection of α -levels, in which case, fuzzy arithmetics are performed using conventional interval arithmetic according to the α -cut representation. Interval arithmetic can verify as addition (+) and multiplication (×) operations on closed intervals with commutative ∗ Corresponding author. Email address: k.9121946081@gmail.com; Tel:+98(912)1946081; Fax:+98(21)44865030. 156