Measures of Linear and Nonlinear Interval-Valued Hexagonal Fuzzy Number

Introduction

The theory of uncertainty plays an important role in modeling science and engineering problems. But there is a basic question, which is mostly raised is that how the uncertainty concept can be defined in our mathematical modeling. Researchers around the globe define many approaches to make it clear and describe various recommendations for this theory. The assimilation of the uncertain parameters is not unique, it varies from problem to problem, and modeling to decision makers, etc. The classification of these theories is also given as a flowchart in Figure 1. For instance, some basic difference of the uncertain concepts can be defined as,

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  The interval numbers (Karmakar, 2014) provide the information that belongs to a certain interval and it does not consider the concept of membership function.

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  Fuzzy numbers (Khan et al., 2014) elaborate the relevancy of the elements in a certain domain by defining a membership function.

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  Intuitionistic fuzzy numbers (Prakash et al., 2016) describe the relevancy and irrelevancy of the elements by taking into account membership and non-membership functions.

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  Neutrosophic numbers (Ye et al., 2018) illustrate the truthiness, falsity and indeterminacy of the elements with the help of three membership functions.

After the deliberation of fuzzy set theory by Zadeh (1965), it has undergone various improvements and has been broadened for different applications. Among many other fuzzy set studies, interval-valued fuzzy numbers are very important in various domains. Lee et al. (2016) employed interval-valued fuzzy numbers to grasp the vagueness of supplier selection problems of supply chain management and electronic markets more precisely. Mohagheghi et al. (2016) applied interval-valued fuzzy numbers in the composite risk, return index to evaluate the financial return and risk of the project portfolio selection. Mondal (2018) utilized interval-valued fuzzy numbers to solve differential equations that are studied in drug concentration and bank growth models. Thus, this study has been considered as an essential tool to construct realistic models for different aspects of applied sciences (Fei, 2018; Li et al., 2018, 2015, 2014, 2013; Wan et al., 2015, 2014, 2013; Li, 2016, 2016).