The counting of fuzzy subsets of a finite set is of great interest in both practical and theoretical contexts in Mathematics. We have used some counting techniques such as the principle of Inclusion-Exclusion and the Mõbius Inversion to enumerate the fuzzy subsets of a finite set satisfying different conditions. These two techniques are interdependent with the M¨obius inversion generalizing the principle of Inclusion-Exclusion. The enumeration is carried out each time we redefine new conditions on the set. In this study one of our aims is the recognition and identification of fuzzy subsets with same features, characteristics or conditions. To facilitate such a study, we use some ideas such as the Hamming distance, mid-point between two fuzzy subsets and cardinality of fuzzy subsets. Finally we introduce the fuzzy scanner of elements of a finite set. This is used to identify elements and fuzzy subsets of a set. The scanning process of identification and recognition facilitates the choice of entities with specified properties. We develop a procedure of selection under the fuzzy environment. This allows us a framework to resolve conflicting issues in the market place.

In this thesis we use the natural equivalence of fuzzy subgroups studied by Murali and Makamba [25] to characterize fuzzy subgroups of some finite groups. We focus on the determination of the number of equivalence classes of fuzzy subgroups of some selected finite groups using this equivalence relation and its extension. Firstly we give a brief discussion on the theory of fuzzy sets and fuzzy subgroups. We prove a few properties of fuzzy sets and fuzzy subgroups. We then introduce the selected groups namely the symmetric group 3 S , dihedral group 4 D , the quaternion group Q8 , cyclic p-group pn G = Z/ , pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We also present their subgroups structures and construct lattice diagrams of subgroups in order to study their maximal chains. We compute the number of maximal chains and give a brief explanation on how the maximal chains are used in the determination of the number of equivalence classes of fuzzy subgroups. In determining the number of equivalence classes of fuzzy subgroups of a group, we first list down all the maximal chains of the group. Secondly we pick any maximal chain and compute the number of distinct fuzzy subgroups represented by that maximal chain, expressing each fuzzy subgroup in the form of a keychain. Thereafter we pick the next maximal chain and count the number of equivalence classes of fuzzy subgroups not counted in the first chain. We proceed inductively until all the maximal chains have been exhausted. The total number of fuzzy subgroups obtained in all the maximal chains represents the number of equivalence classes of fuzzy subgroups for the entire group, (see sections 3.2.1, 3.2.2, 3.2.6, 3.2.8, 3.2.9, 3.2.15, 3.16 and 3.17 for the case of selected finite groups). We study, establish and prove the formulae for the number of maximal chains for the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . To accomplish this, we use lattice diagrams of subgroups of these groups to identify the maximal chains. For instance, the group pn qm G = Z/ + Z/ would require the use of a 2- dimensional rectangular diagram (see section 3.2.18 and 5.3.5), while for the group pn qm r s G = Z/ + Z/ + Z/ we execute 3- dimensional lattice diagrams of subgroups (see section 5.4.2, 5.4.3, 5.4.4, 5.4.5 and 5.4.6). It is through these lattice diagrams that we identify routes through which to carry out the extensions. Since fuzzy subgroups represented by maximal chains are viewed as keychains, we give a brief discussion on the notion of keychains, pins and their extensions. We present propositions and proofs on why this counting technique is justifiable. We derive and prove formulae for the number of equivalence classes of the groups pn qm G = Z/ + Z/ , p q r G Z Z Z n m = / + / + / and pn qm r s G = Z/ + Z/ + Z/ where p,q and r are distinct primes and n,m, s Î N/ . We give a detailed explanation and illustrations on how this keychain extension principle works in Chapter Five. We conclude by giving specific illustrations on how we compute the number of equivalence classes of a fuzzy subgroup for the group p2 q2 r 2 G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of the group p q r G = Z/ + Z/ + Z/ 1 2 2 . This illustrates a general technique of computing the number of fuzzy subgroups of G = Z/ + Z/ + Z/ from the number of fuzzy subgroups of 1 -1 = / + / + / pn qm r s G Z Z Z . Our illustration also shows two ways of extending from a lattice diagram of 1 G to that of G .

In this research work, an approach for fire detection and estimation robots is presented. The approach is based on type-2 fuzzy logic system that utilizes measured temperature and light intensity to detect fires of various intensities at different distances. Type-2 fuzzy logic system (T2 FLS) is known for not needing exact mathematic model and for its capability to handle more complicated uncertain situations compared with Type-1 fuzzy logic system (T1 FLS). Due to lack of expertise for new facilities, a new approach for training experts’ expertise and setting up T2 FLS parameters from pure data is discussed in this thesis. Performance of both T1 FLS and T2 FLS regarding to same fire detection scenario are investigated and compared in this thesis. Simulation works have been done for fire detection robot of both free space scenario and new facility scenario to illustrate the operation and performance of proposed type-2 fuzzy logic system. Experiments are also performed using LEGO MINDSTROMS NXT robot to test the reliability and feasibility of the algorithm in physical environment with simple and complex situation.

This thesis focuses on a decision-making model for finding the locations for placement of utilities in roadway corridors. In recent years, there has been a rapid growth in the volume of traffic on roadways and in the number of utilities placed in Right of Ways. The increase in the demand for utilities is making it more difficult to place all the utilities within the Right of Way and also provide safe roads and highways with good carrying capacity. The public agencies approving the location for utilities are now using a first come first served method, which provide neither an efficient nor good economic solution. This model considers all the utilities within the corridor as a single system, including factors like installation costs, maintenance costs and also some future factors such as accident costs. A weighted coefficient optimization approach is used to find the solution in this model. These costs are modeled as fuzzy numbers or probabilistic random numbers depending on their characteristics. This algorithm will locate each utility at all its possible locations and find the total cost of all the utilities at all these locations, i.e. cost of the system. The least cost locations among all the possible locations are the good locations for utilities in the utility system. When utilities are placed in these locations the overall cost of the system will be lower compared to other locations. This model provides a flexible and interactive method for finding cost saving locations for the utilities in the highway corridor. Users will be able to change the parameters of the utility system according to their requirements and get reduced cost solutions.

Stress can be defined as the mental, physical, and emotional response of humans to stressors encountered in their personal or professional environment. Stressors are introduced in various activities, especially those found in dynamic task conditions when multiple task requirements must be performed. Stress and stressors have been described as activators and inhibitors of human performance. The ability to manage high levels of acute stress is an important determinant of successful performance in any occupation. In situations where performance is critical, personnel must be prepared to operate successfully under hostile or extreme stress conditions; therefore training programs and engineered systems must be tailored to assist humans in fulfilling these demands. To effectively design appropriate training programs for these conditions, it is necessary to quantitatively describe stress. A series of theoretical stress models have been developed in previous research studies; however, these do not provide quantification of stress levels nor the impact on human performance. By modeling acute stress under dynamic task conditions, quantitative values for stress and its impact on performance can be assessed. Thus, this research was designed to develop a predictive model for acute stress as a function of human performance and task demand. Initially, a four factor two level experimental design (2 (Noise) x 2 (Temperature) x 2 (Time Awareness) x 2 (Workload)) was performed to identify reliable physiological, cognitive and behavioral responses to stress. Next, multivariate analysis of variance (n=108) tests were performed, which showed statistically significant differences for physiological, cognitive and behavioral responses. Finally, fuzzy set theory techniques were used to develop a comprehensive stress index model. Thus, the resulting stress index model was constructed using input on physiological, cognitive and behavioral responses to stressors as well as characteristics inherent to the type of task performed and personal factors that interact as mediators (competitiveness, motivation, coping technique and proneness to boredom). Through using this stress index model to quantify and characterize the affects of acute stress on human performance, these research findings can inform proper training protocols and help to redesign tasks and working conditions that are prone to create levels of acute stress that adversely affect human performance.

This dissertation examines characteristics of a class of semi-infinite linear programming problems designated as C/C semi-infinite linear programming problems. Semi-infinite programming problems which belong to this class are problems of the form [See document] where S is a compact, convex subset of Euclidean m space and u_i : S→R, i=1,...,n are strictly concave functions while u _n+1 : S→R is convex. Certain properties of the C/C semi-infinite linear programming problems give rise to efficient solution techniques. The solution techniques are given as well as examples of their use. Of significant importance is the intimate relationship between the class of C/C semi-infinite linear programming problems and certain convex fuzzy set programming problems. The fuzzy set programming problem is defined as [See document] The convex fuzzy set programming problem is transformed to an equivalent semi-infinite linear programming problem. Characterizations of the membership functions are given which cause the equivalent semi-infinite linear programming problems to fall within the realm of C/C semi-infinite linear programming problems. Some extensions of the set inclusive programming problem are also given. / Ph. D.

Traditional methods for evaluating investments in state-of-the-art technology are sometimes found lacking in providing equitable recommendations for project selection. The major cause for this is the inability of these methods to handle adequately uncertainty and imprecision, and account for every aspect of the project, economic and non-economic, tangible and intangible. Fuzzy set theory provides an alternative to probability theory for handling uncertainty, while at the same time being able to handle imprecision. It also provides a means of closing the gap between the human thought process and the computer, by enabling the establishment of linguistic quantifiers to describe intangible attributes. Fuzzy set theory has been used successfully in other fields for aiding the decision-making process. The intention of this research has been the application of fuzzy set theory to aid investment decision making. The research has led to the development of a structured model, based on theoretical algorithms developed by Buckley and others. The model looks at a project from three different standpoints- economic, operational, and strategic. It provides recommendations by means of five different values for the project desirability, and results of two sensitivity analyses. The model is tested on a hypothetical case study. The end result is a model that can be used as a basis for promising future development of investment analysis models. / Master of Science / incomplete_metadata

In this work, fuzzy sets are used to express data or model uncertainty in structural systems where random numbers used to be utilized. / Master of Science

A structured and disciplined system engineering process is essential for the efficient and effective development of products and systems which are both responsive to customer needs and globally competitive. Rigor and discipline during the later life-cycle phases of design and development (preliminary and detailed) cannot compensate for an ill-conceived system concept and for premature commitments made during the conceptual design phase. This significance notwithstanding, the nascent stage of system design has been largely ignored by the research and development community. This research is unique. It focuses on conceptual system design and formalizes analysis and evaluation activities during this important life-cycle phase. The primary goal of developing a conceptual design analysis and evaluation methodology has been achieved, including complete integration with the system engineering process. Rather than being a constraint, this integration led to a better definition of conceptual design activity and the coordinated progression of synthesis, analysis, and evaluation. Concepts from fuzzy set theory and the calculus of fuzzy arithmetic were adapted to address and manipulate imprecision and subjectivity. A number of design decision aids were developed to reduce the gap between commitment and project specific knowledge, to facilitate design convergence, and to help realize a preferred system design concept. / Ph. D.

Available cost-variance investigation models are reviewed and evaluated in Chapter Three of this study. As shown in the chapter, some models suffer from ignoring the costs and benefits of the investigation. Other models, although meeting the cost-benefit test, fail to capture the essence of the real-world problem. For example, they fail to handle the imprecision (fuzziness) surrounding the investigation decision. They are also based on the unrealistic assumptions of (1) a two-state system, and (2) constant level of accuracy and precision. In addition, the models suffer from the lack of applicability. They require precise numerical inputs to the analysis that are difficult, if not impossible, to attain. This dissertation provides a new cost-variance investigation model that may overcome some of these problems. The new model utilizes the calculus of fuzzy set theory which was introduced by Zadeh in 1965 as a means for dealing with fuzziness. The theory is also intended to reduce the need for precise measures that are difficult to obtain. Consequently, the theory seems to be well suited for handling the investigation problem. (Chapter Two provides a summary of the theory and its applications in the decision making area.) The new model is presented in Chapter Four and extended in Chapter Five. The performance is assumed to be described by·a transformation function, S_t+1 = f(S_t,D_t), where S_t, D_t, and S_t+1 represent the sets of the input states, available decisions, and output states, respectively. The transformation function can be deterministic, stochastic, or fuzzy. Methods are suggested to obtain the optimal decision for the three cases of transformation functions. These methods are based on formulating a fuzzy optimal decision set D_O = {u_{D_O}(d_j)d_j}, where u_{D_O}(d_j) represents the compatibility (i.e., relative merit) of decision d_j with the optimal decision set. The optimal decision is the decision having the highest compatibility with the fuzzy optimal decision set. In addition to allowing for different transformation functions, the new model allows for varying degrees of out-of-controllness. The model also provides for the fuzziness (imprecision) surrounding (1) the states of performance, (2) the net benefits from the investigation, and (3) the probabilities. This is done by employing the basic concept in fuzzy set theory, namely, the membership function concept. The new model was examined (in Chapter Six) for feasibility. First, the model was computerized. Then, it was applied to an actual investigation problem encountered by a manufacturing company. As the application may indicate, the new model can be applied to real-world situations. / Ph. D.