Utility-Numbers Theory

Fuzzy theory defined by Zadeh has been developed through both analytical and computational categories of diverse sciences in light of time. The goal of applying fuzzy theory is to incorporate uncertainties with conventional deterministic calculations to enhance the reliability of results. Since fuzzy theory has been developed and carried out by so many researchers, various types of generalizations, including Z-numbers, type-2 fuzzy sets, intuitionistic sets, and hesitant sets, have been revealed to estimate real-world variables more accurately. Toward these goals, utility-numbers or H-numbers theory is proposed with an ordered set of <inline-formula> <tex-math notation="LaTeX">$\left ({n, M\left ({n }\right),N(n) }\right)$ </tex-math></inline-formula>, to assign satisfaction degrees to corresponding uncertain numbers. The first component, n, is a constraint on the values, which a stochastic real-structured variable, X, may take. The following components, <inline-formula> <tex-math notation="LaTeX">$M\left ({n }\right),N(n)$ </tex-math></inline-formula>, represent the maximum and minimum utilities assigned to the first component, respectively. This paper first substantiates immediate H-numbers as <inline-formula> <tex-math notation="LaTeX">$\left ({n, M\left ({n }\right) }\right)$ </tex-math></inline-formula>, and the mathematical operations over them; then, these H-numbers are elaborated to <inline-formula> <tex-math notation="LaTeX">$\left ({n, M\left ({n }\right),N(n) }\right)$ </tex-math></inline-formula> in which both the minimum and maximum utility values are considered. Meanwhile, the main contribution of the proposed model leads to obtain satisfaction value for each fuzzy number component and helps practitioners to concentrate on total satisfaction that is aggregated by each alternative. The proposed model is validated by comparison tables and reasonable values over conventional fuzzy sets.