*3.4. Defuzzification*

In this phase the fuzzy sets obtained of the inference engine are defuzzied into crisp outputs. There are several methods that can be used. Center of gravity is a method used in many studies. Let us assume that the result is the membership function *μ* : R → [0, 1] with the support the interval [*<sup>t</sup>*0, *tk*]. We consider an equidistant division Δ = {*<sup>t</sup>*0, *t*1, *t*2, ··· , *tk*−1, *tk*} ofthisinterval.Thecrispvalueisobtainedby

$$\hat{t}\_{\hat{j}} = \frac{\sum\_{i} t\_i \mu(t\_i)}{\sum\_{i} \mu(t\_i)}.$$

Other defuzzification methods are: max membership principle, weighted average method, mean max membership method, center of largest area etc.

#### **4. On Some Results Obtained by Ioan Dzitac**

*4.1. Fuzzy Method for Multiple Criteria Fractional Programming*

In paper [8], B. Stanojevi´c, I. Dzitac and S. Dzitac proposed a new method of solving a full fuzzy linear fractional programming (FFLFP) problems using *α*-cut representations for the triangular fuzzy coefficients involved in both the objective function and constraints. The decision variables were considered fuzzy valued, but their shapes were determined a posteriori.

The authors first discussed the issues arising when a ratio of fuzzy numbers had to be evaluated: (i) the error induced by approximating the exact membership function of the ratio with the membership function of the triangular fuzzy number that had the same support and the same value with maximal amplitude as the exact (non-triangular) fuzzy number representing the ratio; (ii) the misleading computation of the ratio of two functions of fuzzy numbers, in the case that the same fuzzy number appeared on both numerator and denominator of the ratio, and in the global *α*-cut evaluation distinct endpoints of the *α*-cut were used for that same fuzzy number on numerator and denominator respectively. The authors established the formula for computing the area between the graphic representations of the exact membership function of the ratio of fuzzy numbers and its above-mentioned approximation. The difference between the correct and a misleading evaluation of a rational function of one triangular fuzzy number valued variable, and the effect of the translation of the triangular fuzzy numbers on the rational function correctly evaluated were both illustrated in the paper.

Further on, the authors introduced a new methodology to solve the FFLFP problems by making use of two models to yield the left and the right endpoints of an arbitrary *α*-cut. Their approach properly handled the multiple occurrences of variables in the fractional objective function and in constraints as well. Discrete representations of the inverses of the exact membership functions of the products and ratios of fuzzy numbers were used to derive the fuzzy solutions expressed by (non-triangular) fuzzy numbers. The main advantage of the solution approach was in its ability to work with the exact membership functions of the ratios, and to avoid the usage of any special definition for fuzzy inequality evaluation.

By their approach, the authors disclosed a direction that can be useful in managing decision-making processes involving fractional objective functions under uncertainty. The illustrative example provided in the paper analyzed and solved a decision-making problem in production planning.

#### *4.2. Prudent Decision to Estimate the Risk in Insurance*

In paper [3], the authors investigated the issue of capital allocation with risk exposure starting from the regulations of the new solvency system "Solvency II" built by European Commission to regulate all insurance activities in European Union member countries.

As stated in the Solvency II Directive, 101 Article, the allocation of capital in insurance is done either using a standard formula or by developing an internal model taking into account the particularities of the insurance company.

Making prudent risk assessment decisions to which the insurance company is exposed ultimately means a capital allocation that better corresponds to the company's risk exposure. The risk measure advanced by the new Solvency II system under the standard formula is Value-at-Risk (VaR) or Tail Value-at-Risk (TVaR). The authors proposed for determination of VaR or TVaR, the use of the theory of extreme values. This decision proves more prudent in the sense of correlating the values of VaR and TVaR with the amplitude of risk exposure.

To argue quantitatively the usefulness of applying the theory of extreme events to set the level of prudence regarding the determination of the solvency capital requirement (SCR), the authors built a hypothetical portfolio composed of shares and bonds. Although the standard formula in the Solvency II system was built on the assumption of the normal distribution of the risk to which the company was exposed, this hypothesis was verified too few times, especially when it came to the occurrence of rare events. Under these conditions, the use of distributions better adapted to the company's risk exposure may prove to be a more prudent decision.

In the case of the built-up portfolio, compared with the use of normal law, the authors proved that more prudent decisions had been shown to be the use of specific distribution laws for determination of extreme quantiles such as: Chi2 (from the attraction domain of Fréchet—laws with heavy tails); Exponential, Gamma and Weibull (from the attraction domain of Gumbel—laws with exponential decreasing tails).

#### *4.3. Decision Support Model for Production Disturbance Estimation*

There are several models in the literature for the study of disturbance in manufacturing systems to optimize them. Unfortunately, these models do not give us a dynamic evaluation of multiple consequences of the disturbance. Many production bottlenecks have the effect of severe economic consequences, and they are caused by disturbances in technological lines. A good estimate of disturbance dynamics is needed to reduce losses.

In paper [4] the authors introduced a model which allows a dynamic evaluation of consequences of disturbance. The model used several indicators: time, energy and costs. Model testing was done using MATLAB.

#### *4.4. Manufacture System Control Using ANN Software*

In paper [5] the authors presented a technique for improving the autonomy of the flexible manufacturing systems. The paper presented a system capable of monitoring the tool flank and widening the manufacturing system whiteout input from the users.

The entire structure of the proposed system was presented as well as the different connections between the system itself and other components of the flexible manufacturing systems. The concept behind the system was the continuous monitoring of tools in the machine tool storage systems. From the hardware point of view the paper presented an experimental system designed to acquire the tool flank images.

The paper also presented the software component. The first step taken by the authors was the realization of an application able to extract significant data from the pictures obtained. All the steps taken were presented in the paper. Several parameters were identified and compared to determine their efficiency in determining the tool ware.

An alternative decisional process regarding the tool ware is based on artificial neural networks. The paper presented the realization and setting of such a network realized in MATLAB and performed an in-depth analysis of the network. Methods of Image Classification Using One-Hidden-Layer ANN on Image Data and Image Classification with Autoencoders on Image Data were presented.

The conclusion section of the paper makes an exhaustive presentation of the results for each method of image classification using ANN, a comparative network training parameters table is presented. Als the paper includes the statistical data for misclassification of images.

The paper offers significant theoretical and experimental data on the hardware component of the system, and it is also focusing on the software component. Several methods for image classification are presented and the results are compared to determine optimal solutions both from accuracy and computational time.

#### *4.5. Ant Colony Optimization for a Practical Economic Problem*

The source of inspiration for ant colony optimization (ACO) algorithm is the behavior of ants in search of food, which is a problem of optimization, more precisely of determining the optimal path between food and nest.

Several theoretical aspects of artificial ants were presented in papers [10,36].

In paper [7] the authors presented an algorithm based on Ant Colony Optimization (ACO) applied in a practical economic problem, more precisely, in finding an optimal solution in an electricity distribution network. A nonlinear function was used in the mathematical model of optimization. The mathematical model contains both equality constraints and inequality constraints. The results are validated on a network with 35 nodes.

#### **5. Conclusions and Future Trends**

We wrote this paper for a Special Issue dedicated to the centenary of birth of Lotfi A. Zadeh published by Mathematics. Therefore, this paper is an homage to Lofti A. Zadeh as a sign of respect for his grea<sup>t</sup> contributions to scientific knowledge. At the same time, this paper is dedicated to I. Dzitac, who passed away on 6 February 2021 as he stood out as one of the followers of Zadeh's work by publishing, between 2007 and 2020, tens of articles in the domain of soft computing.

I. Dzitac's most important contributions have just been presented, wishing and hoping that either his former partners or other researchers will continue his work.

We will finish with some challenges and future research directions, because the rapid development of the economy and of the modern industrial systems make the process of decision-making more complicated, leading to difficulties in dealing with vague information. In this environment, soft computing methods have been successfully applied. Nevertheless, as the human thinking process is very complex, raised attention and new research are necessary for the applications of soft computing methods in intelligent decision-making. We will raise a few problems, more general and larger, but that deserve the attention of researchers.


and it is necessary that in the future more attention to be granted and greater efforts to be made.


In the literature of genre can be found many models of consensus with linguistic information. Nevertheless, some questions remain open: the development of new models of consensus that include feedback mechanisms, the creation of new technologies to implement consensus models etc.

In conclusion, as the problems of decision-making become more and more difficult when vague linguistic information interferes, soft computing methods have gained more attention and a grea<sup>t</sup> interest in the latest years. In this paper, a comprehensive review upon some of I. Dzitac's results has been realized. The structure of a fuzzy system has been presented and its components have been analyzed. Finally, some challenges and future trends have been presented.

**Author Contributions:** Conceptualization, S.N.; formal analysis, S.D. and S.N.; methodology, S.D. and S.N.; supervision, S.D.; validation, S.D. and S.N.; writing-original draft, S.N.; writing-review and editing, S.D. and S.N. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Conflicts of Interest:** The authors declare no conflict of interest.
