**1. Introduction**

This paper is dedicated to Professor Ioan Dzitac (14 February 1953–6 February 2021). Therefore the paper begins with a short presentation of his life and work.

Ioan Dzitac was born in the village of Poienile de sub Munte, in the County of Maramures, Transylvania, Romania. He graduated from the Faculty of Mathematics of Babe¸s-Bolyai University of Cluj-Napoca in 1977 and continued as a high school math teacher in Bihor (Ale¸sd and Oradea), Romania. In 2002, Prof. Dzitac obtained his PhD Thesis at Babe¸s-Bolyai University of Cluj-Napoca and in the next few years he published several works in field of distributed and information systems.

In 2007, Dzitac had the grea<sup>t</sup> privilege of meeting the world-renowned scientist Lotfi A. Zadeh and since then, up to the end of his career, his scientific interest focused on different sub-fields:


He had the most important contributions in soft computing in a fuzzy environment. Some of them will be presented in this paper.

**Citation:** Dzitac, S.; N ˘ad ˘aban, S. Soft Computing for Decision-Making in Fuzzy Environments: A Tribute to Professor Ioan Dzitac. *Mathematics* **2021**, *9*, 1701. https://doi.org/ 10.3390/math9141701

Academic Editor: Daniel Gómez Gonzalez

Received: 20 May 2021 Accepted: 15 July 2021 Published: 20 July 2021

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**Copyright:** © 2021 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

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His collaboration with Lotfi A. Zadeh started in 2008. Zadeh was invited as speaker at the International Conference on Computers Communications and Control (ICCCC) (see Figure 1), an ISI indexed conference, founded and chaired by Dzitac.

**Figure 1.** Ioan Dzitac and Lotfi A. Zadeh at ICCCC 2008.

Just a week before passing away, I. Dzitac remembered: "I waited for him (Lofti A. Zadeh) at the airport in Budapest. At 87 years old, he was traveling unattended from San Francisco, where he lived and was the active director of the research institute BISC at the University of California, Berkeley (position that he held until his death in 2017, at 96 years old). He took a nap in the car. However, when he woke up, he started to tell me about his first visit in Romania and the encouragements offered by Grigore C. Moisil, in 1967, two years later, with grea<sup>t</sup> courage, he published "Fuzzy Sets". He really needed those pats on the back because many mathematicians, logicians and engineers met his theories with skepticism and sometimes even with mockery [14]".

L.A. Zadeh had a major influence on Ioan Dzitac's career, because, after their encounter, Ioan Dzitac had a very prosperous period from a scientific point of view, as he published many articles in well-known journals, either as a unique author or in cooperation with: F.G. Filip, M.J. Manolescu, S. Negulescu, A.E. Lascu, C. Butaci, S. Dzitac, G. Bologa, D. Benta, S. Nadaban, B. Barbat, I. Moisil, I. Felea, T. Vesselenyi, C. Secui, V. Lupse and abroad: B. Stanojevic (Serbia), H. Liu (China), S. Gao (China), R. Andonie (USA), A.M. Brasoveanu (Austria), Y. Shi (China), G. Kou (China), F. Cordova (Chile), H. Lee (Korea), etc.

One volume, very dear to Professor I. Dzitac, cannot be omitted. It is "From Natural Language to Soft Computing: New Paradigms in Artificial Intelligence", volume coedited by L.A. Zadeh (University of California), D. Tufis (Romanian Academy), F.G. Filip (Romanian Academy), I. Dzitac (Agora University), Romanian Academy Ed. House, 2008 [15].

The recognition of his results appeared very soon, his papers being quoted by authors from Romania, Chile, India, USA, Iran, Malaysia, Serbia, Canada, France, Russia, Turkey, Australia, Hungary, Lithuania, Morocco, Spain, Tunisia, Algeria, Czech Republic, in some prestigious journals.

To highlight the impact of his research it should be mentioned that Ioan Dzitac has in Web of Science h-index = 11 and 472 citations, of which 64 in the first 5 months of 2021. The paper [6] has 10 citations in the period January 2021–May 2021, while the paper [2] has 16 citations in the same period. The last citations from May 2021 are:


3. Hamzelou, N.; Ashtiani, M; Sadeghi, R. A propagation trust model in social networks based on the A\* algorithm and multi-criteria decision-making. *Computing* **2021**, *103*, 827–867, doi:10.1007/s00607-021-00918-w [18].

I. Dzitac was co-founder and Editor-in-Chief of an ISI Expanded quoted journal, *International Journal of Computers Communications & Control* (nominee by Elsevier for Journal Excellence Award - Scopus Awards Romania 2015) and member in Editorial Board of 12 scientific journals. Additionally, he is co-founder and General Chair of International Conference on Computers Communications and Control. He was member of the Program Committee of more than 80 international conferences.

He was Senior Member IEEE (since 2011). He was invited speaker and/or invited special sessions's organizer and chair in China (2013: Beijing, Suzhou, Chengdu, 2015: Dalian, 2016: Beijing), India (2014: Madurai, 2017: Delhi), Russia (2014: Moscow), Brazil (2015: Rio), Lithuania (2015: Druskininkai), South Korea (2016: Asan), USA (2018: Nebraska).

He was included among 100 Romanian computer scientists from all over the past 100 years in the volume "One Hundred Romanian Scientists in Theoretical Computer Science", Romanian Academy Publishing House, 2018.

I. Dzitac was full professor at Aurel Vlaicu University of Arad (since 2009), professor at Agora University of Oradea (since 2017) and Rector at Agora University of Oradea (2012– 2020). He was an Adjunct Professor at University of Chinese Academy of Sciences—Beijing, China (2013–2016) and since 2016 he was in Advisory Board Member at Graduate School of Management of Technology, Hoseo University, South Korea.

In 2019 he defended his Habilitation Thesis "Soft Computing for Decision-Making" at "Alexandru Ioan Cuza" University of Iasi, which conferred him the right to conduct doctorates. Thus, he became PhD supervisor at University of Craiova, Romania.

In all those years, I. Dzitac did not cease to show his gratitude towards the one who considered to be his mentor: Lofti A. Zadeh. Thus, in 2011, he edited a Special Issue of IJCCC at 90th Zadeh's birthday and another in 2015 at 50th Fuzzy Sets anniversary. In 2017, at Zadeh's death, I. Dzitac published a survey about his life and his famous contributions in scientific world [2]. In January 2021, just a month before his death, I. Dzitac edited another Special Issue of IJCCC dedicated to the centenary of the birth of Lotfi A. Zadeh (1921–2017).

As already mentioned, beginning with 2007, Ioan Dzitac's interest was in soft computing methods in fuzzy environments. Starting from here, the structure of the paper will continue as follows: Section 2 will present some general considerations regarding soft computing methods, highlighting the fundamental differences between soft computing and hard computing; considering that soft computing methods are numerous, in Section 3 we will resume to presenting some fundamental ideas of fuzzy logic, which is the most used Soft Computing method in a variety of decision-making problems; in Section 4 we will present a survey on I. Dzitac's contributions to this domain. Finally, in Section 5 we will have some conclusions but mostly some future trends will be discussed.

#### **2. Soft Computing Methods**

Hard computing (HC) is the conventional calculation and it needs an analytical model well defined and many times a log time for calculating. Many analytical models are valid only in ideal cases. The problems of the real world exist within a non-ideal frame. Thus, many complex systems that are found in engineering, biology, medicine, economy remain unsolved to HC.

Soft computing is a concept introduced for the first time by L.A. Zadeh [19]. According to Zadeh's definition, Soft computing (SC) methods are opposed to HC techniques, consisting of computational techniques in Informatics, machine learning and certain engineering subjects that study, shape and analyze a very complex reality for which the traditional methods prove ineffective. Soft computing can work with ambiguous data, and it is tolerant to vagueness, uncertainty, partial truth and approximation. The model for SC is human mind.

Lotfi A. Zadeh said about Natural Language (NL) Computation: "NL-Computation is of intrinsic importance because much of human knowledge is described in natural language. This is particularly true in such fields as economics, data mining, systems engineering, risk assessment and emergency management. It is safe to predict that as we move further into the age of machine intelligence and mechanized decision-making, NL-Computation will grow in visibility and importance. Computation with information described in natural language cannot be dealt with using machinery of natural language processing. The problem is semantic imprecision of natural languages. More specifically, a natural language is basically a system for describing perceptions. Perceptions are intrinsically imprecise, reflecting the bounded ability of sensory organs, and ultimately the brain, to resolve detail and store information. Semantic imprecision of natural languages is a concomitant of imprecision of perceptions. Our approach to NL-Computation centers on what is referred to as generalized-constraint-based computation, or GC-Computation for short. A fundamental thesis which underlies NL-Computation is that information may be interpreted as a generalized constraint." [20].

Soft Computing includes: (1) Fuzzy Logic; (2) Neural Computing: Perceptions, Artificial Neural Networks, Neuro-Fuzzy Systems; (3) Evolutionary Computation: Genetic Algorithms (GA), Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Artificial Life(AL); (4) Machine Learning: Intelligent Agents, Expert Systems, Data Mining; (5) Probabilistic Reasoning: Bayesian Networks, Markov Networks, Belief Networks.

To discuss all these methods would exceed the space for an article. Therefore, the next section will be limited to the presentation of only a few fundamental ideas of fuzzy logic, which is the most used Soft Computing method in a variety of decision-making problems.

The next table (see Table 1) is adapted from [9] and it presents the conclusions of the HC paradigm versus SC paradigm.


**Table 1.** Hard Computing vs. Soft Computing.

#### **3. Fuzzy Logic in Decision-Making**

In classical logic the sentences are bivalent, this meaning that all sentences that describe the state of an event are either true or false. With this bivalent logic computers were endowed, they can make massive computations, which are very difficult for people. On the other hand, computers cannot imitate the intuitive human mind and this because people can operate with vague linguistic information. It is difficult though for these to be modelled in classical logic. At the same time, in many real-world situations there are uncertainties and vague information.

To be able to deal with these uncertainties and ambiguities, L. A. Zadeh [21] introduced in 1965 the concept of fuzzy sets.

If *X* is an arbitrary set. A fuzzy set in *X* is a function *μ* : *X* → [0, 1]. The function *μ* is called the membership function and *μ*(*x*) represents the value of truth as *x* belongs to the fuzzy set.

In 1975, L.A. Zadeh [22] introduced the concept of linguistic variable. This is a variable whose values are words or sentences. For example, an important criterion in location problems is "accessibility". This is a linguistic variable. It can take values linguistic terms. For the linguistic variable "accessibility" the linguistic terms set is

> *T*("*accessibility*") = {*Very good*, *Good*, *Medium*, *Poor*, *Very poor*}.

The structure of a fuzzy logic system is presented in Figure 2. We notice that a fuzzy logic system is made of four components: fuzzyfication, fuzzy rules, inference engine, defuzzification.

**Figure 2.** The structure of a fuzzy logic system.
