Fuzzy sets and the modelling of physician decision processes,part I: The initial interview-information gathering session

Most mathematical models of physician decision processes offered to date, especially those relative to diagnosis and patient treatment, suffer from the inability to incorporate all useful data on the patient. Pertinent information so neglected or poorly modelled relate to variables that are intrinsically fuzzy but which describe the patient's health status. We present mathematical models based on fuzzy set theory for physician aided evaluation of a complete representation of information emanating from the initial interview including patient past history, present symptoms, signs observed upon physical examination, and results of clinical and diagnostic tests.     

Long-term inventory policy-making through fuzzy decision-making models

The inventory policy, meant as a replenishment rule, has a considerable impact on most firms. The paper considers the determination of optimal inventory policy of firms from a global viewpoint of top management. The inventory is represented as a fuzzy system with the fuzzy inventory level as the output, the fuzzy replenishment as the input and fuzzy demand. The control problem is formulated in terms of decision-making in a fuzzy environment with fuzzy constraints imposed on replenishments, a fuzzy goal for preferable inventory levels to be attained and the fuzzy decision as the intersection of fuzzy constraints and the fuzzy goal at subsequent stages. The planning horizon is infinite. The problem is to find an optimal time-invariant strategy relating the optimal replenishments to the current inventory levels, maximizing the membership function of fuzzy decision. The existence of such a strategy is proved and an algorithm for its determination is given. The optimal time-invariant strategy obtained is represented as a fuzzy conditional statement equated with a fuzzy relation which is the firm's optimal fuzzy replenishment rule.     

Fuzzy processes

In this paper, contributions to fuzzy probability and to differential equations with fuzzy parameters are made.After an introductory section, a review of fuzzy sets and fuzzy algebra is given in Section 2. The main new results of the investigation are contained in Section 3.In Section 3, Zadeh's definition of the probability of a ‘fuzzy event’ the average value of a fuzzy function are extended into the time domain. It is then shown that not only grades of membership, but also probabilistic processes with notions of fuzziness contained, can be defined which obey ordinary, matric, or integro-differential equations. Applications are also given in Section 3.     

Measures of fuzzy sets and measures of fuzziness

The measures presented in this paper are defined by using Weber's concept of decomposable measures m of crisp sets, having in particular the Archimedean decomposable operations in view (Section 2). Measures $\text{m}$ of fuzzy sets are introduced as integrals with respect to m. For the Archimedean cases, Weber's integral will be used as alternative to Sugeno's and Choquet's concepts (Section 3). What ‘fuzziness’ means will be described by functions of fuzziness F (another name: entropy N-functions) with respect to a negation. In addition to the types of functions of fuzziness which are induced by concave functions, we discuss also the ones which are induced by fuzzy connectives (Section 4). Now, using m for measuring the ‘importance of items’ and F for the ‘fuzziness’ of the possible values of a fuzzy set ?, m?(F ° ?) serves us as a measure of the fuzziness $\text{F}$? of ?. The concepts of De Luca and Termini, Capocelli and De Luca, Kaufmann, Knopfmacher, Loo, Gottwald, Dombi and, under the restriction to the Archimedean cases, also the concepts of Trillas and Riera and Yager turn out to be special cases (Section 5).     

Asymptotic structural characteristics of fuzzy measure and their applications

In fuzzy measure theory, as Sugeno's fuzzy measures lose additivity in general, the concept ‘almost’, which is well known in classical measure theory, splits into two different concepts, ‘almost’ and ‘pseudo-almost’. In order to replace the additivity, it is quite necessary to investigate some asymptotic behaviors of a fuzzy measure at sequences of sets which are called ‘waxing’ and ‘waning’, and to introduce some new concepts, such as ‘autocontinuity’, ‘converse-autocontinuity’ and ‘pseudo-autocontinuity’. These concepts describe some asymptotic structural characteristics of a fuzzy measure.In this paper, by means of the asymptotic structural characteristics of fuzzy measure, we also give four forms of generalization for both Egoroff's theorem, Riesz's theorem and Lebesgue's theorem respectively, and prove the almost everywhere (pseudo-almost everywhere) convergence theorem, the convergence in measure (pseudo-in measure) theorem of the sequence of fuzzy integrals. In the last two theorems, the employed conditions are not only sufficient, but also necessary.     

Bulletin

Anthony and Sherwood (1979, 1980) redefined the fuzzy subgroup, the idea introduced by Rosenfeld 1971. Foster 1979 introduced the concept of product fuzzy subgroups using Rosenfeld's definition. In this paper, the product fuzzy subgroups will be redefined according to Anthony and Sherwood's definition.     

Fuzzy closure spaces

Fuzzy topological spaces do not constitute a natural boundary for the validity of theorems, but many results can be extended to what are called fuzzy closure spaces (or fcs's, for short). The notions of a subspace, a sum, and a product are extended to fcs's. The hereditary, additivity, and productivity behaviour of compactness in fcs's is investigated and some weak forms of compactness and fuzzy continuous functions in fcs's are introduced. The interaction between fuzzy proximity spaces and fcs's is investigated; A necessary background is included for completeness.     

The outpatient appointment system: Design of a simulation study

This paper describes the model-construction of a simulation study. The purpose of this study was to produce a general method for determining a suitable appointment system for the clinics in the outpatient department of a hospital. The original model contained 11 variables. Investigation of the influence of each variable on patient's wainting-time and doctor's idle-time showed that a considerable reduction in the number of variables could be achieved. Only 5 variables were finally left in the stimulation.The use of the results of this study in a real-life clinic situation is discussed elsewhere.     

Simulating hospital emergency departments queuing systems: (GI/G/m(t)) : (IHFF/N/∞)

The queuing system to be considered in this paper is a real case study with the following characteristics: (1) general independent interarrival distribution, (2) general service-time distribution, (3) limited waiting room, (4) patient's priorities increase up to a certain number, (5) time-dependent number of servers (doctors), (6) infinite patient population, (7) each server meets the system only once within a certain period of time, while the total number of the available servers is known. The system to be considered is a hospital's Emergency Department and a general simulation algorithm is presented as well as the system's operating characteristics. This algorithm, implemented on a mini-computer or an inexpensive microcomputer solved a sophisticated operations research problem.     

Distinction between several subsets of fuzzy measures

In this paper, we place several fuzzy measure subsets in relation one with the other. The subsets under study are those corresponding to the definitions of probability measure. Sugeno's g_λ-measure, Shafer's belief function and Zadeh's possibility measure. We study the intersection of these subsets and we show the particular role of Dirac's measures in this comparison. We limit ourself to the case of mappins whose domain is the collection of all subsets of a finite set.Finally, the obtained partial results are summarized in only one figure which shoul clarify the specificity of each of the above definitions.     

The influence of some fuzzy implication operators on the accuracy of a fuzzy model-part II

The influence of fuzzy implication operators and the connective Also on the accuracy of a fuzzy model of a d.c. series motor is considered. Some typical fuzzy implication operators are applied to the construction of a fuzzy model of a d.c. series motor. A root-mean-square error is used as the criterion of the fuzzy model's adequacy to the real system. A number of mathematical operations necessary for the implementation of the fuzzy model are used as the criterion by which the fuzzy model's applicability if estimated from the point of view of computing techniques. The best types of fuzzy relations, representing fuzzy models of a real system, are chosen in order to secure the least root-mean-square error with minimal number of mathematical operations necessary for computer implementation.     

On the complexity of growth of the number of distinct fuzzy switching functions

Several attempts have been made to enumerate fuzzy switching (FSF's) and to develop upper and lower bounds for the number of FSF's of n variables in an effort to better understand the properties and the complexity of FSF's. Previous upper bounds are 2⁴ⁿ [9] and 2^{2–3n—2n—1} [7].It has also been shown that the exact numbers of FSF's of n variables for n = 0, 1, 2, 3, and 4 are 2, 6, 8, 84, 43 918 and 160 297 985 276 respectively.This paper will give a brief overview of previous approaches to the problem, study some of the properties of fuzzy switching functions and give improved upper and lower bounds for a general n.     

The fuzzy integral

In this paper we define the fuzzy integral of a positive, measurable function, with respect to a fuzzy measure. We show that the monotone convergence theorem and Fatou's lemma are still true in this new setting. We study some of the properties of this integral, and show that it coincides with another fuzzy integral defined in the literature. Our main result is a convergence theorem, that is in a way stronger than the Lebesgue-dominated convergence theorem. This holds when the fuzzy measure is also assumed to be subadditive.     

Towards fuzzy differential calculus part 1: Integration of fuzzy mappings

This paper deals with fuzzy-set-valued mappings of the real line, and more particularly focuses on mappings from the real line to the set of convex normal fuzzy sets of the real line. These mappings can also be viewed as fuzzy relations. Using Zadeh's extension principle, the integral of such fuzzy mappings over a crisp interval is defined. Provided a special analytical representation of the fuzzy mapping, the practical computation of such an integral is shown to be easy. Practically speaking, it yields the fuzzy surface of a fuzzily-bounded area.     

A note of fuzzy programming and the pareto optimal set

We indicate some errors made in Buckley's paper on fuzzy programming and the Pareto optimal set [1].     

General negations in fuzzy set theory

All possible negation operations which are optimal in some precise sense are fully described. They turn out to be Lowen's fuzzy complements, Yager's intuitionistic negation, and a dual to the latter.     

Remarks on the lattices of fuzzy subsets of a groupoid

In this paper we give a necessary and sufficient condition for a groupoid D such that the sup-min product is distributive over arbitrary intersection of fuzzy subsets of D, and correct some results from the paper [S. Ray, The lattice of all idempotent fuzzy subsets of a groupoid, Fuzzy Sets and Systems 96 (1998) 239–245]. Also, we prove that the set of all idempotent fuzzy sets forms a complete lattice, which is a complete join-sublattice of the lattice of all fuzzy subgroupoids. This result extends the corresponding result from the above mentioned paper.     

Series of fuzzy sets

Series of fuzzy sets with weakly closed, weakly compact or compact α-levels are considered. The basic space is Banach space. The subject of this paper is investigation of infinite addition of fuzzy sets and condition when the sum of α-levels is equal to the α-level of the sum. The results in this paper complement and complete some previous results proved by the same authors in Stojaković and Stojaković [Addition and series of fuzzy sets, Fuzzy Sets and Systems 83 (1996) 341–346] for the series with compact α-levels.     

Semantics and pragmatics of fuzzy sets and systems

A philosophical formalism of a new methodological aspect of humanistic systems design and evaluation is given. A requisite concept of context-dependency is highlighted, and some approaches to fuzzy sets and linguistics subsequently extended. It is consequently shown that a mathematical theory of pragmatic fuzzy subsets is not only conceptually possible but practically implementable in man-machine studies as well. Thus, the important context-dependent implication of the subjective nature of Zadeh's theory of fuzzy sets can be better exploited.As a pragmatic theory, the approach here is a seeming connection between pragmatism and ontology, concepts that are traditionally diametrically opposed to each other. The attitude adopted has been the equation of pragmatism and psychophysical measurements of ontological objects (noumena). Pragmatism is tacitly defined as a form of empiricism whereby linguistic constructs (i.e., linguistic-variable denotions) that represent any aspect(s) of a humanistic system are nothing more than an operational procedure used to achieve psychophysical measurements of the aspect(s). In this fashion, pragmatism would enable the contents of assertions, which are made through declarative propositions, about humanistic systems to be deciphered within relevant contexts. For pragmatism, direct sense experience provides both the meaning and the criterion of reality judgements. The context-dependent nature of the physical reality of ontological entities is, therefore, better understood pragmatically vis-a-vis the appropriate evaluative criteria and interpretative conventions.By emphasizing the distinction between a fuzzy system (naturally fuzzified) and a fuzzified mathematical structure (meta-mathematically fuzzified), the use of the standard fuzzy topologies is justified although it is conceptually possible to develop a yet more general topology or perhaps an alternative one especially in the case of a meta-mathematically fuzzified structure. However, for the express purpose of machine implementation, a naturally fuzzified system, such as a (complex) humanistic system, is more amenable to an initial test of the philosophy of pragmatic fuzzy subsets. Consequently, the fuzzy topology and spaces employed are intended to be devoid of extensive generalities, in this instance.