Moni—a system for monitoring nosocomial infections in intensive care units for adult and neonatal patients—employs knowledge bases that were written with extensive use of **fuzzy** **sets** and **fuzzy** **logic**, allowing the inherent un-sharpness of clinical terms and the inherent uncertainty of clinical conclusions to be a part of Moni’s output. Thus, linguistic as well as propositional uncertainty became a part of Moni, which can now report retrospectively on HAIs according to traditional crisp HAI surveillance definitions, as well as support clinical bedside work by more complex crisp and **fuzzy** alerts and reminders. This improved approach can bridge the gap between classical retrospective surveillance of HAIs and ongoing prospective clinical-decision-oriented HAI support.

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Keywords— **Fuzzy** **Sets**, Artificial Intelligence, **Fuzzy** **Logic**, Computational Intelligence, Soft Computing.
I. I NTRODUCTION
**Fuzzy** mathematics is the study of **fuzzy** structures, or structures that involve fuzziness i.e., such mathematical structures that at some points replace the two classical truth values 0 and 1 with a larger structure of degrees. Most of our traditional tools for formal modelling, reasoning, and computing are crisp, deterministic, and precise in character. Crisp means dichotomous, that is, yes-or-no type rather than more-or-less type [1]. In traditional dual **logic**, for instance, a statement can be true or false—and nothing in between. In set **theory**, an element can either belong to a set or not; in optimization a solution can be feasible or not. Precision assumes that the parameters of a model represent exactly the real system that has been modelled. This, generally, also implies that the model is unequivocal, that is, that it contains no ambiguities.

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Accepted Manuscript find that even the worst case of the 42-IT2-SOFLC and 42-GT2-SOFLC can still produce a lower error than the best case of the direct adaptive IT2-FLC in [16]. Nevertheless, it should be noted that the RMSE measured for direct adaptive IT2-FLC also counted the settling stage, which leads to a larger overall error. Hence, further research will provide a benchmark comparison to evaluate the performance of the two controllers more precisely. More recent research described in [17] uses genetic algorithms to design **fuzzy** PID controllers to regulate BIS. The parameters of the **fuzzy** PID controllers are optimized offline and a cost function is defined to evaluate the performance of the **fuzzy** PID controllers. However, this controlling structure is quite different from our study so it is difficult to directly compare the performance of this system with our approach. Future research could compare this control strategy to the SOFLCs if we change our monitoring of DoA to use BIS instead of BP as is the case in this study. In our simulations, we have used a fixed amount of initial bolus which is administered to patients to reach rapid anesthesia. The initial bolus is effective in muscle relaxation in the first 15 minutes and settles down towards the set point. The simulation results show that the bolus can let the system stabilize close to the desired set point which can then be regulated through the adaptive SOFLC. In a real clinical setting it is still however difficult for the anesthetists to decide the amount of the bolus to give. Different patients have different physiological response such as height and weight, and the dosage of initial bolus can therefore also vary in terms of its effectiveness on the patient. Anesthetists generally tend to guess the initial bolus and adjust the amount according to the patient’s physiological response. Future work will simulate different amount of initial bolus and see how the type-2 system can handle the related uncertainties and adjustment of the bolus amount to be initially administered.

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find that even the worst case of the 42-IT2-SOFLC and 42-GT2-SOFLC can still produce a lower error than the best case of the direct adaptive IT2-FLC in [16]. Nevertheless, it should be noted that the RMSE measured for direct adaptive IT2-FLC also counted the settling stage, which leads to a larger overall error. Hence, further research will provide a benchmark comparison to evaluate the performance of the two controllers more precisely. More recent research described in [17] uses genetic algorithms to design **fuzzy** PID controllers to regulate BIS. The parameters of the **fuzzy** PID controllers are optimized offline and a cost function is defined to evaluate the performance of the **fuzzy** PID controllers. However, this controlling structure is quite different from our study so it is difficult to directly compare the performance of this system with our approach. Future research could compare this control strategy to the SOFLCs if we change our monitoring of DoA to use BIS instead of BP as is the case in this study. In our simulations, we have used a fixed amount of initial bolus which is administered to patients to reach rapid anesthesia. The initial bolus is effective in muscle relaxation in the first 15 minutes and settles down towards the set point. The simulation results show that the bolus can let the system stabilize close to the desired set point which can then be regulated through the adaptive SOFLC. In a real clinical setting it is still however difficult for the anesthetists to decide the amount of the bolus to give. Different patients have different physiological response such as height and weight, and the dosage of initial bolus can therefore also vary in terms of its effectiveness on the patient. Anesthetists generally tend to guess the initial bolus and adjust the amount according to the patient’s physiological response. Future work will simulate different amount of initial bolus and see how the type-2 system can handle the related uncertainties and adjustment of the bolus amount to be initially administered.

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This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
Despite half a century of **fuzzy** **sets** and **fuzzy** **logic** progress, as **fuzzy** **sets** address complex and uncertain information through the lens of human knowledge and subjectivity, more progress is needed in the semantics of **fuzzy** **sets** and in exploring the multi-modal aspect of **fuzzy** **logic** due to the different cognitive, emotional and behavioral angles of assessing truth. We lay here the foun- dations of a postmodern **fuzzy** set and **fuzzy** **logic** **theory** addressing these issues by deconstructing **fuzzy** truth values and **fuzzy** set membership functions to re-capture the human knowledge and subjectivity structure in membership function evaluations. We formulate a fractal multi-modal **logic** of Kabbalah which integrates the cognitive, emotional and behavioral levels of humanistic systems into epistemic and modal, deontic and doxastic and dynamic multi-modal **logic**. This is done by creating a fractal multi-modal Kabbalah possible worlds semantic frame of Kripke model type. The Kabbalah possible worlds semantic frame integrates together both the multi-modal **logic** aspects and their Kripke possible worlds model. We will not focus here on modal operators and axiom **sets**. We constructively define a fractal multi-modal Kabbalistic L-**fuzzy** set as the central concept of the postmodern **fuzzy** set **theory** based on Kabbalah **logic** and semantics.

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A. **Fuzzy** Inferencing Systems
It is via the **Fuzzy** Inferencing System (FIS) 1 that **fuzzy** **sets** are put to use. An FIS is a decision making program which works by applying **fuzzy** **logic** operators to common- sense linguistic rules. In this paper we are concerned with the Mamdani FIS, in which a crisp numerical input passes through three stages: fuzzification, inferencing, and finally defuzzification. The output of inferencing is a **fuzzy** set known as the aggregated set. During the defuzzification stage the aggregated set is converted into a crisp number, which is the output of the FIS. Figure 1 provides a representation of a Mamdani-style type-2 FIS. A Mamdani-style complex FIS differs from the type-2 version only in that defuzzification is a one stage procedure. Defuzzification is beyond the scope of this paper; the focus will be on the inferencing stages of the FIS.

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Step 2: Definition of **Fuzzy** rules for the **fuzzy** **logic** system and inference method
**Fuzzy** **sets** and **fuzzy** operators are subjects and verves of **fuzzy** **logic**. These if-then rule statements were used to formulate the conditional statements that comprise **fuzzy** **logic**. CATs and EXAM were taken as parts of antecedent. All parts of antecedent were calculated simultaneously and resolved to a single number using the logical operator AND. The performance of the student was considered as the consequent affected by the antecedent. Next is sample of 10 rules taken from 25 if-then rules generated for the system :

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B. **FUZZY** INFERENCE SYSTEM Steps in **fuzzy** inference system:
1: STEP 1: FUZZIFICATION
The first step in a **fuzzy** inference system is the fuzzification of crisp inputs. It transforms the exact **logic** problem into a **fuzzy** **logic** problem. Unlike crisp **logic**, **fuzzy** **logic** deals with linguistic variables instead of numerical variables. The process of converting numerical variables of the problem into grades of membership for linguistic terms of **fuzzy** **sets** is called fuzzification. Thus it is a mapping from a certain input space to **fuzzy** **sets** in certain input universes of discourse.

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India
Abstract: F-Set **Theory** is a natural generalization of Goguen's L-**Fuzzy** Set **Theory** which itself is a generalization of Zadeh's, both **Fuzzy** and Interval Valued **Fuzzy** Set Theories. It naturally and neatly extends several of the crisp (Sub)Set-Map-Properties to: L-valued f-(sub) **sets**, f-maps between L-valued f-**sets** and M-valued f-**sets**, where the complete lattice L-may possibly different from the complete lattice M, M-valued f- image of an L-valued f-subset of the domain L-valued f-set and L-valued f-inverse image of an M-valued f-subset of the co-domain M-valued f- set. However, for several of the results in this **theory**, the complete homomorphisms are assumed to be one or a combination of: 0-preserving, 0- reflecting, 1-preserving and 1-reflecting. Further, some of the results use the infinite meet distributivity of the underlying complete lattice of the domain and/or range f-set.

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error) than FDM for out particular dataset, but there advantages of FDM are clear. It has less parameters and a more descriptive model if we use **fuzzy** concepts.
Also, consider the following: ”Usually, **fuzzy** controllers are implemented as software running on standard microprocessors. A few special-purpose microprocessors have been built that do **fuzzy** operations directly in hardware, but even these use digital binary signals at the lowest hardware level. There are some research prototypes of computer chips that use analog signals at the lowest level, but these chips simulate the operation of neurons rather than **fuzzy** **logic**.” So, if **fuzzy** com- puters become more widely available in the future it is conceivable that this kind of computations will become more efficient both for programming and running. 1

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PG Scholar, Dept. of EEE, Mar Athanasius College of Engineering, Kothamangalam, Kerala, India 1 Professor, Dept. of EEE, Mar Athanasius College of Engineering, Kothamangalam, Kerala, India 2,3
ABSTRACT: **Fuzzy** **logic** controller (FLC) is the most widely used applications of **fuzzy** set **theory**. A **Fuzzy** **Logic** Controlled (FLC) buck-boost converter for solar energy-battery systems is analyzed. Solar Photovoltaic system has been increasingly playing a significant role in new energy systems. PV system has some disadvantages including low conversion efficiency and inconsistent values of output voltage due to irregular solar power which is caused by weather changes and shading effects. To address these disadvantages, DC-DC converters have been used to control PV output voltage and output power. **Fuzzy** **logic** controllers are non linear controllers which do not require exact information of mathematical model. Hybridization of the two controllers can be done to use the positive sides of both controllers to obtain better performance. The comparison of result obtained from **fuzzy** **logic** controller, **fuzzy** - PI hybrid controller, **fuzzy** - PID hybrid controller for DC-DC converter shows the benefit of the hybrid algorithm in terms of transient response. General design of a **fuzzy** **logic** controller based on MATLAB/Simulink is performed. The control system has been developed, analyzed, and validated by simulation study.

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Now coming back to the developments in this side of this paper, Goguen further generalized the two types of **fuzzy** subsets of Zadeh, namely the **fuzzy** subset and the interval valued f-subset, to those that take the truth values in a complete lattice. However, even though Goguen unified both of them mathematically, one must observe here that, as mentioned earlier, when it comes to practical applications, the **fuzzy** subsets and the interval valued f-subsets are quite different because **fuzzy** **sets** require a specific real number between 0 and 1 to be associated with each of its elements while interval valued f-**sets** require a reasonable interval to be associated with each of its elements, offering a representation of even more uncertainty in belonging of certain elements to a set than the **fuzzy** **sets** themselves.

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Nanjing, China
Abstract—This paper investigates the use of grey **theory** to en- hance the concept of an R-**fuzzy** set, with regards to the precision of the encapsulating set of returned significance values. The use of lower and upper approximations from rough set **theory**, allow for an R-**fuzzy** approach to encapsulate uncertain **fuzzy** membership values; both collectively generic and individually specific. The authors have previously created a significance measure, which when combined with an R-**fuzzy** set provides one with a refined approach for expressing complex uncertainty. This pairing of an R-**fuzzy** set and the significance measure, replicates in part, the high detail of uncertainty representation from a type-2 **fuzzy** approach, with the relative ease and objectiveness of a type-1 **fuzzy** approach. As a result, this new research method allows for a practical means for domains where ideally a generalised type-2 **fuzzy** set is more favourable, but ultimately unfeasible due to the subjectiveness of type-2 **fuzzy** membership values.

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This approach to set **theory** was not applied to control systems until the 70's due to insufficient small-computer capability prior to that time. Professor Zadeh reasoned that people do not require precise, numerical information input, and yet they are capable of highly adaptive control. If feedback controllers could be programmed to accept noisy, imprecise input, they would be much more effective and perhaps easier to implement. Unfortunately, U.S. manufacturers have not been so quick to embrace this technology while the Europeans and Japanese have been aggressively building real products around it.

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SUMMARY
**POSSIBILITY** OF APPLYING **FUZZY** **LOGIC** IN DETERMINING PHASE ORDER FOR SIGNALIZED INTERSECTIONS
In this bachelor's thesis an adaptive control system for signalized intersections has been presented. The system uses **fuzzy** **logic** to fully optimize traffic flow, minimize wait times, queue length and controlling arrival and exit flow in order to prevent traffic congestion throughout the observed signalized intersection. Given that road traffic's use has been largely increased over years, it is necessary to optimize traffic flow in an already existing traffic infrastructure using technologies within intelligent transport systems. The designed system makes an exclusive use of changing the signal phase order to optimize functionality for all traffic demand scenarios covered in this work. **Fuzzy** **logic** enables the use of predefined rules which serve as a base for predicting possible cases of traffic flow, thus giving a better insight over further optimizing phase order in the intersection.

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© 2011 Elsevier Ltd. All rights reserved.
1. Introduction
To solve complicated problems in economics, engineering, environmental science and social science, methods in classical mathematics are not always successful because of various types of uncertainties presented in these problems. While probability **theory**, **fuzzy** set **theory** [1], rough set **theory** [2,3], and other mathematical tools are well known and often useful approaches to describing uncertainty, each of these theories has its inherent difficulties as pointed out in [4,5]. In 1999, Molodtsov [4] introduced the concept of soft **sets**, which can be seen as a new mathematical tool for dealing with uncertainties. This so-called soft set **theory** is free from the difficulties affecting existing methods.

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In this paper, a novel generalized interval-valued intuitionistic **fuzzy** **sets** (GIVIFS) is presented, which is the generalization of conventional intuitionistic **fuzzy** **sets** (IFS) and interval-valued intuitionistic **fuzzy** **sets** (IVIFS). By analyzing the degree of hesitancy, this paper introduces generalized interval-valued intuitionistic **fuzzy** **sets** with parameters (GIVIFSP). And then, it is proved that GIVIFS is a closed algebraic system as IFS and IVIFS.

Chapter 2
Fundamentals
This chapter is devoted to reviewing optimization concepts and the related compu- tational methods that will be used in the remaining chapters. In particular, we deal with three optimization concepts incorporating fuzziness and ambiguity of human judgments, uncertainty of events characterizing decision making problems, and mul- tiplicity of evaluation criteria. **Fuzzy** programming which is developed in order to take into account fuzziness and ambiguity of human judgments, and this method is ﬁrst presented together with the basic concepts in **fuzzy** set **theory**. Fundamentals of stochastic programming are also provided for decision problems under probabilistic uncertainty. Speciﬁcally, two-stage programming and chance constraint program- ming are covered. To meet expectations of diversiﬁed evaluations, multiobjective programming has been investigated. After presenting interactive methods for multi- objective linear programming problems, we give some techniques to deal with not only multiple objectives but also **fuzzy** goals for the objectives. In an organization with a hierarchical structure, we often ﬁnd decision making with two or more deci- sion makers (DMs) attempting to optimize their objective functions. To model such a decision making problem mathematically, a two-level programming problem is formulated, and cooperative and noncooperative solution methods are presented for two-level programming. Genetic algorithms are considered to be one of the most practical and proven meta heuristics for difﬁcult classes of optimization problems, and ﬁnally basic concepts of genetic algorithms are provided.

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Since in linguistics, the exact definitions of the aforementioned words do not exist, words are one of the original sources of uncertainty. This type of uncertainty could specially be seen in the problems where the available information is expressed in the form of words and phrases of a natural language. For example, we may have seen the following information in several cases: ‘The temperature is high’, ‘The pressure of the gas is medium’, etc. Hence, the use of the above words will lead us to define the fuzziness associated with the variables. Secondly, Membership functions provide a more mathematical view of fuzziness by defining a membership value for each value the **fuzzy** variable can take. Mostly used membership functions are trapezoid, triangular, belle shape, etc with a specific predefined mathematical formula for each. A Z-mouse offers a third choice.

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In biology, a mutation is defined as the permanent alteration of the nucleotide sequences of the genetic elements. In a colloquial manner, the progeny formed by the mating of any two individuals can be expressed as being different from expected. Mullin described the mutation process in algebraic sense in mathematics and he posed research problems and defined a mutation in a grupoid in [3]. In [4], he assumed that the set S is a population of individuals and the mating of any two individuals of the population is a binary operation on S. Thus, the progeny of the two individuals a and b both in S have written as a ∗ b which is also in S. According to [4], it could be divided S into equivalence classes according to some properties of elements of S (e.g. eye color, blood type). Sometimes, it may be ensued an offspring which is not in M from the mating of any two individuals that is in M. In that case, a mutation has taken place in the set M . So, in his related article [4], Mullin algebraically stated the definition of mutation such as M ∗ M ⊆ M c where M is a subset of the algebraic system (S, ∗) and M c is a complement of M . He said that the set M is a mutant set in (S, ∗). Afterwards, Mullin generalized and applied the results of mutant **sets** for group **theory** and ring **theory** in [5]. Iseki [6] introduced the definition of mutation in a semigroup as generalized sense. In [7], Iseki gave some results of the Cartesian product of mutant **sets**. In [8], Kim discussed some properties of mutant **sets** and gave some results in topological semigroups and algebraic semigroups.

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