Fuzzy Arithmetic Research Articles

Latest theoretical approaches to uncertainty modelling in enterprise management: historical, conceptual and instrumental aspects

The article is devoted to fuzzy-set and interval approaches to modelling uncertainty within the issues of enterprise management. The nodal events of origin and development of fuzzy set theory, the main stages of formation and development of fuzzy-set approach as an applied methodology are considered. Particular attention is paid to the economic application of fuzzy set theory, including in relation to applied problems of enterprise management. The historical, conceptual and instrumental aspects of interval methodology, which is based on the apparatus of interval mathematics or analysis, as another significant approach to modelling uncertainty in economics and business, are highlighted. It is concluded that fuzzy set theory and interval analysis form a mathematical basis for powerful approaches to modelling uncertainty, which cannot be described using traditional probabilistic-statistical methods. Methods and models based on these theories are successfully used for a wide range of enterprise management tasks. It is proved that in the field of investment analysis and business planning one of the most important achievements of the theory of fuzzy sets should be considered the introduction of fuzzy numbers and the development of fuzzy arithmetic. Their use allows to carry out the entire volume of analytical work and financial and economic calculations for specific tasks in these areas at the level of fuzzy-multiple descriptions, starting from the assessment of initial parameters and ending with the finding and analysis of the resulting indicators. In general, the significance of both fuzzy set theory and interval analysis goes far beyond purely economic issues. The information presented in the publication convincingly proves that both the theory of fuzzy sets and interval analysis now determine the high degree of development of their theoretical and conceptual basis and instrumentalapplied component. At the same time, the issue of further formation and development of these methodologies remains relevant.

(2008-6074) Multidimensional interval type 2 epistemic fuzzy arithmetic

The article presents a new, multidimensional arithmetic of type 2 fuzzy numbers (M-IT2-F arithmetic) in which the result is a multidimensional fuzzy set. This arithmetic increases the accuracy of calculations and the scope of problems solved in relation to the currently used interval type 2 standard fuzzy arithmetic (IT2-SF arithmetic). The proposed M-IT2-F arithmetic has mathematical properties that IT2-SF arithmetic does not have. Thanks to these properties, it provides accurate calculation results that are not over- or under-estimated in terms of uncertainty. The paper contains comparisons of both types of arithmetic in application to two problems. Fuzzy arithmetic is not a finished work and is in a phase of continuous improvement and development. M-IT2-F arithmetic is a higher form of M-IT2 (non-fuzzy) arithmetic.

An alternate method for finding more for less solution to fuzzy transportation problem with mixed constraints

In this paper, we present a different process for the optimal solution to mixed constraints transportation problem with imprecise coefficients without changing to its crisp equivalent form which is very distinct from other existing methods. Here, all the variables are considered to be triangular fuzzy numbers. Utilizing a parametric form of triangular fuzzy numbers such as left fuzziness index, right fuzziness index, modal value and by using proposed algorithm, we obtained the initial basic fuzzy feasible solution to the problem without changing to its crisp equivalent form. To obtain the solution, we use fuzzy ranking dependent on left and right spreads at some S-level of fuzzy numbers and fuzzy arithmetic dependent on both location index function and fuzziness index function. We further discuss the more for less solution by obtaining shadow prices to the problem. The more for less situation occurs when we raise the supplies, and demand quantities may cause to reduce the optimal transportation cost. The more for less paradox is very useful to a manager in dynamic, for example expanding an item house loading level or plant age breaking point and publicizing attempts to build request at explicit markets. A numerical example is solved to show the adequacy of the proposed algorithm.

Cost Comparison of Selective Laser Sintering with Injection Molding in the Presence of Uncertainties

The application of additive manufacturing (AM) has increased exponentially in recent years. Industries are keen to explore this innovative technology but are apprehensive about the high processing cost of the process. Hence, it is crucial to carry out a cost analysis of the process. This paper presents an approach to compare the costs of an AM process (selective laser sintering (SLS)) and a traditional manufacturing process (injection molding (IM)) in the presence of uncertainties. Initially, the deterministic cost models comprising necessary cost variables for SLS and IM are described. The deterministic models are converted to fuzzy set-based models for tackling uncertainties. For this purpose, important uncertain variables are treated as fuzzy members and fuzzy arithmetic is employed. Only linear triangular fuzzy numbers are used in this work. Fuzzy cost estimates produce three values (low, most likely and high estimates) of cost corresponding to membership grades. A methodology to compare two fuzzy costs of the processes is proposed for a variable demand scenario. Concept of fuzzy reliability is suitably utilized and variability in demand is tackled from probability theory. Variable demand is assumed to follow uniform as well as normal probability distributions. The methodology is illustrated with the help of two examples.

Turtle biodiversity losses suggest coming sixth mass extinction

Biodiversity losses in the face of climate change are among the most important issues facing humanity today. Vertebrate extinction is at an all-time high, and losses of reptiles are no exception. The turtles are a unique vertebrate group of grave conservation concern. I compared recent declines in turtle biodiversity to the losses of Testudinata (Cryptodira + Pleurodira) and non-dinosaurian reptiles during the most recent mass extinction. Fuzzy arithmetic was used because of its suitability to deal with uncertainty in these kinds of data sets and the questions that arise with comparing geological data with those from recent times. This revealed that extinctions of turtles in modern times massively exceed losses observed at the Cretaceous–Paleogene border. Further, if turtle extinctions continued at their current level, massive losses of these charismatic animals could occur within our lifetime and loss of the entire group could occur in just a few centuries. We must ask ourselves, how much more can we lose before we also disappear.

The Fuzzy Arithmetic Operations of Transmission Average on Pseudo-Hexagonal Fuzzy Numbers and Its Application in Fuzzy System Reliability Analysis

In most natural science fields, triangular and trapezoidal fuzzy numbers are commonly used while in engineering and social science fields such as sociology and psychology while treating the uncertainties these numbers are not applicable and fuzzy numbers with more parameters and clear definitions of their arithmetic operations are needed. In order to fill this gap in the literature, we propose the new fuzzy arithmetic operations based on transmission average on pseudo-hexagonal fuzzy numbers, which was already implied in [1] in its rudimentary form and was finally presented in its fully fledged form in [2]. Several illustrative examples were given to show the accomplishment and ability of the proposed method. We present a new method for fuzzy system reliability analysis based on the fuzzy arithmetic operations of transmission average, where the reliabilities of the components of a system are represented by pseudo- hexagonal fuzzy numbers defined in the universe of discourse [0, 1]. The proposed method has the advantages of modeling and analyzing fuzzy system reliability in a more flexible and more intelligent manner. Finally, a marine power plant is considered in fuzzy environment. The reliability of components of the proposed model is considered as pseudo-hexagonal fuzzy numbers.

Fuzzy arithmetic in economic research

Many tasks in economic research are based on arithmetic calculations of indicators that reflect the state of economic development. The general incompleteness of publicly available data, designed to solve such problems, has led to the emergence of numerous decision support systems based on fuzzy arithmetic. The article presents a study on the approach aimed at integrating fuzzy information about economic indicators into economic models. The definition of arithmetic operations on fuzzy values is given, and the methods of obtaining the resulting fuzzy indicators with the help of some software tools are considered. Analytical and numerical methods of obtaining the resulting indicators in the form of fuzzy numbers are described and analyzed. A direct calculation algorithm for all arithmetic operations is proposed, utilized, and used for the evaluation of resulting indicators. Also, analytical and numerical methods for obtaining fuzzy results are considered in the article, and some of them are proposed for utilization. On the example of economic indicators used in the labor rationing, the results of an evaluation of some indicators for a technological operation in the form of fuzzy numbers were obtained by different methods and compared. The practical recommendations, given in the article, on the use of fuzzy arithmetic in decision support systems outline the directions of further research.

An Analytical Method for Assessing the Utility of Additive Manufacturing in an Organization

Although the applications of additive manufacturing are continuously increasing, but unless used judiciously, it cannot compete with traditional manufacturing processes. This article presents an analytical method for assessing the utility of additive manufacturing. For this, the concept of utilization factor is introduced and its effect on the production planning and associated costs is discussed. The uncertainty in the estimation of utilization factor and costs is addressed by the fuzzy set-based theory. For applying fuzzy arithmetic, linear triangular fuzzy numbers are used based on expert’s most likely, lower and upper estimates. Consideration of uncertainty with the help of fuzzy arithmetic provides a better picture for management to take decision. Further, a fuzzy set-based technique has been employed to compare an additive manufacturing process viz., selective laser sintering (SLS) with another process viz., injection moulding. The overall methodology is illustrated through case studies. It is demonstrated that even for SLS process, the unit cost of a part decreases with the quantities of production, which is reflected in the enhancement of the utilization factor. However, for very high volume of production, the SLS process cannot compete with injection moulding in terms of cost as well as production time.

Measuring the production performance indicators for food processing industry

Production performance measurement is the most important activity to achieve the higher productivity in food industries. The Overall Equipment Effectiveness (OEE) is a standard for measuring production productivity. However, the uncertainty of OEE components could mask the true production performance estimation in food processing lines. In this study, the production speed and stoppage duration measurements as major uncertainties to estimate the OEE and its components were investigated. In order to handle the uncertainties, two methods based on fuzzy arithmetic and interval arithmetic were conducted to supplement the classical-OEE. To survey the classical-OEE, the operational data field of edible oil purification process over a period of two years were examined. The comparison results of filed data revealed that the new OEE index has been increased around 4% by maintenance improvements. In such index, the total availability, performance efficiency and quality rate were improved by 3%, 2% and 5%, respectively. The research results showed that the spread values of Fuzzy-OEE are more explicit to show the worst and best values, while the interval arithmetic of OEE reflects the results by tighter limits. The proposed methods are more convenient to quantify uncertainty with limits to understand the OEE variations helping to make better decisions on uncertainties in food production lines.

Which Alternative for Solving Dual Fuzzy Nonlinear Equations Is More Precise?

To answer the question stated in the title, we present and compare two approaches: first, a standard approach for solving dual fuzzy nonlinear systems (DFN-systems) based on Newton’s method, which uses 2D FN representation and second, the new approach, based on multidimensional fuzzy arithmetic (MF-arithmetic). We use a numerical example to explain how the proposed MF-arithmetic solves the DFN-system. To analyze results from the standard and the new approaches, we introduce an imprecision measure. We discuss the reasons why imprecision varies between both methods. The imprecision of the standard approach results (roots) is significant, which means that many possible values are excluded.

Fuzzy regression model with input and output Z-numbers

The paper is devoted to the construction of a fuzzy regression model, the input and output data of which are Z-numbers. The coefficients of the regression model are crisp numbers. The definition of Z-number was proposed by prof. Lotfi Zadeh in 2011 for fuzzy information processing, the reliability of which is also fuzzy measured. In the paper the both components of Z-numbers are values of linguistic variables with the properties of completeness and orthogonality. Operations on Z-numbers are based not on classical fuzzy arithmetic, but on weighted fuzzy arithmetic, which greatly simplifies the operation, but at the same time, the original information features are preserved. The incoming information is formalized using weighted segments, based on which the distance between the initial and model data is determined. The optimization problem is solved with the condition of a minimum sum of squares of these distances, which allows to obtain unknown coefficients of the regression model.

Fuzzy extensions of PROMETHEE: Models of different complexity with different ranking methods and their comparison

Models of Fuzzy Multi-Criteria Decision Analysis (FMCDA) are based, as a rule, on different approaches to fuzzy extension of source MCDA methods. For this, simplified models are used to approximate the functions of fuzzy variables with propagation of parametric fuzzy numbers (FNs) through all calculations. In this paper, authors suggest a novel approach to fuzzy extension of MCDA methods, for PROMETHEE-I/II, through development of fuzzy PROMETHEE-I/II (FPOMETHEE-I/II) models of different complexity: in addition to simplified models, the standard fuzzy arithmetic (SFA), and transformation methods (TMs) are implemented for assessing functions of FNs corresponding to these models. For ranking of alternatives, two defuzzification based, and one pairwise comparison ranking methods are implemented within the developed models. Special attention is paid to analysis of the overestimation problem, which can occur when using SFA in the presence of dependent variables in corresponding expressions, and to “proper fuzzy extensions” of PROMETHEE-I/II (i.e., results of all functions of FNs within the model are in accordance with the extension principle) based on TMs and, for some models, on the SFA. One of the key goals of this contribution is comparison of the distinctions in ranking alternatives by different FPROMETHEE-II models. It is demonstrated by evaluating a large number of scenarios based on Monte Carlo simulation that the probability of distinction in ranking alternatives by “proper” and “approximated” FPROMETHEE-II models may be considered as significant for ranking multicriteria problems. Another goal of this paper is analysis of the correctness of FPROMETHEE-I/II models with respect to the basic MCDA axiom related to ranking of dominated and dominating alternatives. Authors demonstrate that the basic axiom can be violated, in the general case, by all developed FPROMETHEE-I/II models and suggest an approach to fix this problem.

A concrete reformulation of fuzzy arithmetic

Advancement in fuzzy arithmetic is essential to advancing the applicability of fuzzy set theory (FST) to approximate reasoning problems arose in engineering, medicine, managerial science and many other domains. In the recent FST literature, many of the noteworthy progresses in fuzzy arithmetic have been built upon increasingly sophisticated mathematical concepts and structures, leading to highly non-trivial results. In this article, we take a different approach, of offering a simple, concrete framework that could serve as an entry point for the classically-trained engineers (those who might not be well-acquainted with deep theoretical FST advances) to understand and use fuzzy arithmetic in their applications. Specifically, the contributions of this work are: (i) We extend a foundational model previously proposed in Ngan (2018) to arrive at a simple fuzzy arithmetic framework, in which each key element within the framework possesses simple, concrete meaning to these classically-trained engineers; (ii) We describe how to use these key elements as building blocks to create various concrete fuzzy arithmetic operators that are adaptive to various assumptions. This means that the classically-trained engineers can now easily customize and use fuzzy arithmetic robustly in their application domains; (iii) We demonstrate that the proposed framework can overcome the possibility of generating pathological results that were seen in other fuzzy arithmetic approaches in the FST literature; (iv) We demonstrate the utility of the proposed framework in solving multiple-criteria-decision-making (MCDM) problems concretely and in an easy-to-interpret manner (which are highly important considerations for the users of MCDM). Last but not least, we offer a new avenue for extending this fuzzy arithmetic framework to higher-order fuzzy settings, such as the type-2 fuzzy and other type-2-like fuzzy settings. Thus, our framework can potentially provide a unified, robust approach for developing fuzzy arithmetic in a wide range of FST settings.

Hybrid fuzzy system dynamics model for analyzing the impacts of interrelated risk and opportunity events on project contingency

Traditional risk analysis techniques are ineffective for capturing the dynamic causal interactions and subjective uncertainties involved in assessing risk and opportunity events since they treat risks independently and rely on the availability of sufficient historical data. In this paper, a hybrid fuzzy system dynamics (FSD) model is developed to analyze the impacts of interrelated and interacting risk and opportunity events on work package cost to determine work package and project contingencies using expert judgement and subjective assessment. A fuzzy decision-making trial and evaluation laboratory (DEMATEL) method is employed to structure and analyze the causal interactions among risk and opportunity events. This paper provides the following contributions: (1) a systematic risk assessment and prioritization procedure; (2) a structured method for defining the dynamic causal relationships among risk and opportunity events and quantifying their impact on work package and project contingencies using FSD; and (3) a method for representing linguistic variables and applying fuzzy arithmetic in FSD.

A critical review on theoretical drawbacks and mathematical incorrect assumptions in fuzzy OR methods: Review from 2010 to 2020

As a practical approach for handling uncertainty, various fuzzy sets combined with traditional OR (operations research) methods have found huge applications. However, the fuzzy arithmetic is still a key challenge in this field. Moreover, some researchers have ignored the critical fact that traditional arithmetic operations will produce meaningless or questionable results when applied to fuzzy numbers in general. Thus, many papers in the field of OR have applied fuzzy numbers in a wrong way. Despite the very extensive use of combination of fuzzy sets with OR methods, literature shows that it has been less concentrated on the study of fallacy that may be generated using these methods. Motivated by some theoretical drawbacks and incorrect assumptions in fuzzy sets in conjunction with traditional methods in OR literature, the papers about these pitfalls are reviewed. The papers are included in this review, if they disclose directly a certain pitfall or incorrect assumption in the integration of fuzzy sets with OR methods. Because many other researchers may employ the same assumptions to solve real-life problems, the aim of this study is to make general users aware that many of these fuzzy arithmetic operations are incorrectly used and can lead to untrue and misleading consequences. In this paper, we review 102 papers published in 29 influential journals from 2010 to 25th, January, 2020. ISI Web of Science is adopted as the database of our review and the majority of selected journals (around 70%) are ranked as Q1 determined by SJR.11SCImago Journal & Country Rank (https://www.scimagojr.com/). To help scholars obtain quick information, the collected articles are summarized in a tabular layout. We hope that researchers who apply fuzzy-based methods in the field of OR would understand the serious problems and deep pitfalls associated with these models.

Parameter Estimations of Fuzzy Forced Duffing Equation: Numerical Performances by the Extended Runge-Kutta Method

In this work, the forced Duffing equation with secondary resonance will be considered our subject by assuming that the initial values has uncertainty in terms of a fuzzy number. The resulted fuzzy models will be studied by three fuzzy differential approaches, namely, Hukuhara differential and its generalization and fuzzy differential inclusion. Applications of fuzzy arithmetics to the models lead to a set of alpha-cut deterministic systems with some additional equations. These systems are then solved by the extended Runge-Kutta method. The extended Runge-Kutta method is chosen as our numerical approach in order to enhance the order of accuracy of the solutions by including both function and its first derivative values in calculations. Among our fuzzy approaches, our simulations show that the fuzzy differential inclusion is the most appropriate approach to capture oscillation behaviors of the model. Using the aforementioned fuzzy approach, we then demonstrate how to estimate parameters to our generated fuzzy simulation data.

Fuzzy Logic in Student’s Achievement

The productiveness of teaching students in universities is important to evaluate for declaring the results of their work. Their learning mainly depends on such factors as attendance, productiveness of learning, the material-technical base of the university and etc. Generally, when one interesting to assess the dependencies of these factors, an inventory of all elements is usually carried out, and management is advised to make a certain decision. But the influence of factors on student learning is not carried out. This paper aims to present an assessment of this student learning ability using system fuzzy logic. In this investigation discussed a procedure that can be used to assess students' performance and learning ability. This process is completely based on the principle of obtaining fuzzy sets about factors affecting learning: class attendance, learning efficiency and infrastructure instruction environment, which are objects of fuzzy arithmetic and allows one to get crisp values of performance results. In the paper is applied the fuzzy set method, which is used for further analysis with additional factors affecting student learning.

A common-weights approach for efficiency evaluation in fuzzy data envelopment analysis with undesirable outputs: Application in banking industry

Data Envelopment Analysis (DEA) is a non-parametric approach based on linear programming for evaluating the performance of decision making units (DMUs) with multiple inputs and multiple outputs. The lack of the ability to generate the actual weights, not considering the impact of undesirable outputs in the evaluation process and the measuring of efficiencies of DMUs based upon precise observations are three main drawbacks of the conventional DEA models. This paper proposes a novel approach for finding the common set of weights (CSW) to compute efficiencies in DEA model with undesirable outputs when the data are represented by fuzzy numbers. The proposed approach is based on fuzzy arithmetic which formulates the fuzzy additive DEA model as a linear programing problem and gives fuzzy efficiencies of all DMUs based on resulting CSW. We demonstrate the applicability of the proposed model with a simple numerical example. Finally, in the context of performance management, an application of banking industry in Iran is presented for analyzing the influence of fuzzy data and depicting the impact of undesirable outputs over the efficiency results.

Implementation of Fuzzy Logic in Estimating Yield of a Vegetable Crop

Predicting the influence of climate variations on crop yield demands an appropriate model. In this study, Fuzzy logic based crop yield estimation is undertaken considering temperature, humidity and moisture of soil as input parameters. By subjecting these parameters to fuzzy arithmetic, crisp value of yield is obtained. Trapezoidal membership function is considered in the fuzzy modeling. The results are validated using available open source literature. It has been verified theoretically that air humidity 65%-75%, air temperature 18-290 C and soil moisture 60-80% would give high yield.

A new approach to fuzzy control using the distending function

Here, we develop a fuzzy controller using fuzzy arithmetics and a new type of membership function. The proposed new fuzzy control technique is simple, fast and computationally efficient, compared to the classical techniques (Mamdani, Takagi Sugeno) and it can also adapt to the process dynamics. The unique features are: 1) A new class of parametric membership function called the Distending Function (DF) is introduced; 2) A general parametric operator system is used. It utilizes most of the fuzzy operator systems for evaluating the knowledge base; 3) Inference is based on fuzzy arithmetic operations; 4) This leads to a computationally efficient single-step defuzzification. With these concepts, the paradigm of fuzzy control design changes radically. Using this technique with an optimization method, an adaptive fuzzy controller is designed. This adaptive controller adjusts to the changing dynamics of the non-linear processes by tuning our new type of membership function. The effectiveness of the proposed methodology is demonstrated on two industrial processes (a water tank system and continuously stirred tank reactor system).