*4.4. Summary*

A summary of the surveyed papers can be found in Table 3. In this table, the references are grouped with respect to the class of problems they address. The references marked with a superscript *r* are review papers, those written in bold introduce solution approaches that strictly comply to the extension principle, and those marked with a superscript ∗ need further investigations in order to conclude whether the presented methodologies strictly followed the extension principle or not. The rest of them elude the extension principle within the optimization step.

**Table 3.** A summary of the surveyed papers (the symbol *r* marks the review papers, the symbol ∗ marks the papers that need further investigations, and the references written in bold strictly comply to the extension principle).


We restricted our attention to about thirty papers from the literature that proposed solution approaches to full fuzzy mathematical programming problems, i.e., problems with both coefficients and variables assumed to be fuzzy quantities. This study is not meant to be an extensive survey of the above mentioned topic, but we hope that it succeeds to provide a hint on the extent to which the extension principle is currently ignored in the literature. We focused on basic methodologies for optimizing fuzzy number valued objective functions rather than on case studies dealing with fuzzy optimization problems involving fuzzy goals and/or fuzzy decisions.

## **5. Research Directions**

Our main reason for writing this review paper is our belief that the extension principle, which is one of the main principles in fuzzy sets theory, is neglected within mathematical programming with fuzzy numbers.

As can be seen from the literature review presented in previous section, there are various solving methods that comply with the extension principle when using fuzzy arithmetic but globally neglect it, since they rely on the ranking of fuzzy numbers when optimizing the objective function and interpreting the fuzzy constraints.

In the recent literature, a Monte Carlo simulation algorithm [29] was proposed to derive extension-principle-based fuzzy set solutions to linear programming problems with fuzzy numbers. Deriving empirically such fuzzy set solutions to a fuzzy problem has the same complexity as solving the crisp variant of that problem. Therefore, whenever a crisp problem is generalized to a problem with fuzzy numbers, the Monte Carlo simulation can be employed to disclose the shapes of the fuzzy set solutions. This method works for a wide range of problems; its application is not limited to the optimization problems. After disclosing the shapes of the fuzzy sets solutions, the next step is to develop algorithms able to derive numerically the membership functions of those fuzzy sets solutions.

Due to its simplicity, the class of linear programming problems, including the subclass of transportation problems, were already solved numerically using nonlinear models (see [28,29]). Two important research directions arise and deserve attention: (i) generalize the methodology to solve other classes of problems (for instance, problems from the field of data envelopment analysis), and (ii) find replacements for the nonlinear models in order to simplify the numerical approaches. Moreover, everything already concluded for mathematical optimization problems with fuzzy and/or intuitionistic fuzzy numbers is worth generalizing to another level of uncertainty, e.g., to Pythagorean [61] and Fermatean fuzzy numbers [62].

An argumen<sup>t</sup> that supports the relevance of the above mentioned research directions is illustrated in Tables 1 and 2. They disclose the nature of the differences among solutions that follow the extension principle and those that do not comply to it when solving transportation problems and general linear programming problems, respectively. Further on, due to the similarities of the existing solving approaches to full fuzzy LFP problems to those for LP problems, similar improvements are expected when solving full fuzzy LFP problems applying an extension principle-based approach. Such approaches have been lacking in the literature thus far.

Finally, inducing the same logic to a wider class of problems either relying or not on fuzzy LP but using fuzzy quantities within parameters, a similar increase in the accuracy and reliability of the derived solutions is expected when a methodology that strictly follows the extension principle is applied.
