**1. Introduction**

Zadeh's fuzzy set theory [1] is an accurate mathematical tool that is able to model the uncertainty widely present in real-life problems. It has a wide range of applications in various scientific fields from medicine, engineering, and computer science to artificial intelligence. Dubois [2] emphasized that one of the roles of the fuzzy set theory is to facilitate a joint functionality of the numerical and qualitative approaches in decision-making.

Dzitac et al. [3] recently presented several important aspects of Zadeh's fuzzy logic theory that were proved to have useful applications. A discussion on the need of fuzzy logic and a nonstandard perspective on it was given in [4]. Wu and Xu [5] presented a wide range of applications of the fuzzy logic in decision making that proved fuzzy logic's ability in handling uncertain linguistic information. Shi [6] introduced several results from fuzzy group's theory that could represent a good foundation when the multivalued computer systems will be redeveloped in the future. N˘ad˘aban [7] presented a concise and unitary general view on the algebraic connections between classic, fuzzy, and quantum logics.

In this study, we restrict our attention to fuzzy mathematical programming. Zimmerman [8,9] emphasized the role of the fuzzy set theory in mathematical programming, introducing a solution approach to multiple objective optimization problems based on

**Citation:** Stanojevi´c, B.; Stanojevi´c, M.; N ˘ad ˘aban, S. Reinstatement of the Extension Principle in Approaching Mathematical Programming with Fuzzy Numbers. *Mathematics* **2021**, *9*, 1272. https://doi.org/10.3390/ math9111272

Academic Editor: Francisco Javier Cabrerizo-Lorite

Received: 7 May 2021 Accepted: 28 May 2021 Published: 1 June 2021

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aggregation of fuzzy goals and fuzzy constraints. Verdegay [10] emphasized that the fuzzy linear programming is one of the most studied topics in the theory of the fuzzy sets and systems. We focus especially on optimization problems that involve fuzzy numbers as coefficients and/or variables aiming to rehabilitate the position of Zadeh's extension principle [1] in approaching such problems.

When a solution approach to fuzzy optimization problems strictly follows the extension principle, the ranking of the involved fuzzy quantities is avoided. From our perspective, this fact is a real advantage since there are many ranking functions defined in the literature (Abbasbandy [11] mentioned more than thirty); each of them might generate a solution approach to certain classes of optimization problems, and any comparison of their effectiveness is almost impossible.

After a brief presentation of the basic notation and terminology related to fuzzy sets and mathematical programming given in Section 2, we include in Section 3 a discussion on the indispensability of the extension principle in solving mathematical programming problems with fuzzy numbers. In Section 4, we survey the main methodologies that address full fuzzy optimization problems and analyze the effects of neglecting the extension principle in some of their optimization steps. In Section 5, we sugges<sup>t</sup> research directions focusing on using the extension principle in all stages of the optimization process. Our concluding remarks are presented in Section 6.

#### **2. Notation and Terminology**
