Article Preview
TopIntroduction
The Fuzzy Sets (FS) theory was created in response of expressing mathematically real-world situations in which definitions have not clear boundaries. The “high mountains of a country”, “the young inhabitants of a city”, “the good players of a team”, etc. are some characteristic examples of FS. Fuzzy Logic (FL), based on FS theory, is an infinite-valued logic generalizing and completing the classical bi-valued logic. Despite the initial reserve of the classical mathematicians, fuzzy mathematics and logic have found nowadays many and important applications to almost all sectors of human activity (e.g. Klir & Folger, 1988; Voskoglou, 2017; Voskoglou, 2019a). Due to its nature of characterizing the ambiguous real-life situations with multiple values, FL offers, among others, rich resources for assessment purposes, which are more realistic than those of the classical logic (Voskoglou, 2012, 2019b).
FS and FL were proved to be important tools for the development of artificial intelligence (AI), a discipline that studies the theory and technology of constructing intelligent machines which mimic the human reasoning and behaviour. Within the general class of AI fuzzy systems together with probabilistic reasoning and artificial neural networks (ANN’s) are the three components of soft computing. AAN’s and fuzzy systems try to emulate the operation of the human mind. The AAN’s, the structure of which is analogous to that of the biological neural networks, concentrate on the “hardware” of the human mind having the ability to learn and to process rabidly the information. On the contrary, the fuzzy systems concentrate on the software emulating the human reasoning. A neuro-fuzzy system is a hybrid system that uses a learning algorithm derived from an AAN to determine its fuzzy parameters (FS and fuzzy rules). Characteristic examples of such kind of systems are the adaptive neuro-fuzzy inference systems (ANFIS). AI as a synthesis of ideas from mathematics, engineering, technology and science has rapidly developed during the last decades creating a situation that has the potential to generate enormous benefits to the human society. The spectrum of AI, apart from the soft computing, covers many other research areas and technologies, like knowledge engineering, data mining, reasoning methodologies, cognitive computing and modeling, machine learning, natural language processing and understanding, artificial planning and scheduling, vision and multimedia systems, intelligent tutoring and learning systems, etc.
Fuzzy numbers (FNs), which are a special form of FS on the set of real numbers, play an important role in fuzzy mathematics analogous to the role played by the ordinary numbers in the traditional mathematics. The simplest forms of FNs are the triangular FNs (TFNs) and the trapezoidal FNs (TpFNs).
In the present work we study applications of TFNs and of TpFNs to assessment processes and to linear programming (LP) under fuzzy conditions. The rest of the paper is formulated as follows: The Background section contains all the information about FS and FNs and about LP which is necessary for the understanding of the paper. The Main focus of the Article section is divided in two parts. In the first part an assessment method is developed using TFNs or TpFNs as tools, which enables the calculation of the mean performance of a group of uniform objects (individuals, computer systems, etc.) with respect to a common activity performed under fuzzy conditions. In the second part an analogous method is developed for solving LP problems with fuzzy data (Fuzzy LP). In the Solutions and Recommendations section examples are presented illustrating the applicability of both methods to real world situations related to hyper connectivity and computer systems. The assessment outcomes are validated with the parallel use of the GPA index, while the solution of the FLP problems is turned to the solution of ordinary LP problems by ranking the corresponding fuzzy coefficients. The article closes with a discussion for the future research directions on the topic and the final conclusions.