A Review on Some Arithmetic Concepts of Z-Number and Its Application to Real-World Problems

Abstract

Zadeh introduced the concept of Z-numbers in 2011 to deal with imprecise information. In this regard, many research works have been published in an attempt to introduce some basic theoretical concepts of Z-numbers to model real-world problems. To understand the current challenges when dealing with Z-numbers and the feasibility of using Z-number in solving real-world problems, a comprehensive review of the existing work on Z-number is paramount. This paper consists of an overview of existing literature on Z-number and identifies some of the key areas that are required for further improvement.

Analytic Hierarchy Process Based on the Magnitude of Z-Numbers

“…The defuzzified reliability parts were then added to the restriction parts. It should be noted that the transformation of Z-numbers into regular fuzzy numbers has caused a significant loss of information (Abdullahi et al, 2020;Gardashova, 2019;Shen & Wang, 2018), especially since the original decision information in the form of Z-numbers was not preserved. On the other hand, the proposed AHP in the current study keeps the Znumbers in the form of paired fuzzy numbers, which are mainly in their original form, except for the quantification of their magnitudes to represent the criteria weights.…”

“…Azadeh et al (2013) extended the AHP based on Z-numbers to solve the selection of private self-financing technical institutions; however, in their proposed model, the Znumbers were converted into regular fuzzy numbers by integrating the second component into the first component using Kang et al's (2012) transformation method. The transformation of Z-numbers into regular fuzzy numbers causes a significant loss of information (Abdullahi et al, 2020;Gardashova, 2019;Shen & Wang, 2018). During the transformation, the reliability component of the Z-numbers was converted into a crisp value, which was added to the restriction component as a weight.…”

“…Instead of converting Z-numbers into regular fuzzy numbers, the direct calculation using the arithmetic operations of Z-numbers proposed by Aliev et al (2015) was used. Although this method could avoid the issue of information loss, the method may, however, lead to high computational complexity caused by the extensive use of linear programming to solve simple problems (Abdullahi et al, 2020).…”

“…Z-numbers are composed of restriction and reliability components; the reliability component describes how certain the preference on the restriction component is made. According to Abdullahi et al (2020), Z-numbers are the generalization of real, interval, and fuzzy numbers. Moreover, Z-numbers are very powerful in describing decision-making information due to their capability of modeling the real-world.…”

“…In this article, the strength of Z-numbers is adopted in developing the AHP based on the magnitude of Z-numbers. Since the magnitude of fuzzy numbers exhibits visual and natural meaning (Abbasbandy & Hajjari, 2009), the magnitude of Z-numbers is used to process the decision information instead of converting Z-numbers into regular fuzzy numbers that lead to information loss (Abdullahi et al, 2020;Gardashova, 2019;Shen & Wang, 2018).…”