Form Of Fuzzy Numbers Research Articles

Cyclic multi-hoist scheduling with fuzzy processing times in flexible manufacturing lines

This work addresses the problem of scheduling multiple hoists (robots), which arises on real automated electroplating lines. Several hoists operate on a shared track and should not collide with each other. The processing time for each operation is preset in interval form. The problem simultaneously takes into account two criteria: maximum processing quality and maximum line productivity. To do this, we estimate the dependence of the duration of operations on the level of their quality and, in accordance with these estimates, we represent the processing times in the form of fuzzy numbers with triangular membership functions. The novelty of this study is two-fold. First, we propose a new mathematical model based on the use of fuzzy numbers and fuzzy operations on them, which is designed to deal with complex galvanic lines served by multiple hoists. The second new feature is a new combinatorial method developed for scheduling multiple hoists. For this, we modify and refine the so-called “method of prohibited intervals”, which has been actively used in scheduling theory in recent decades to solve problems, but until now was known only for the case of a single hoist. Thus, the second novelty of this work is that it is actually the first attempt to use the powerful technique of fuzzy set theory to solve a practical problem of scheduling multiple hoists. Computational experiments on real and randomly generated instances show that the new algorithm can successfully solve real problems that arise in practice.

Assessing the investment potential of issuing companies in conditions of the stock market dynamics

Subject. Based on the developed procedure for fuzzy valuation of shares of issuing companies, the article considers topical issues related to forecasting their investment potential in conditions of stock market dynamics. Objectives. The purpose is to assess the investment potential of issuing companies for various scenarios of their development. Methods. The study rests on methods of analysis of economic phenomena and processes, systems approach to reviewing the development of issuing companies, estimation and prediction of their economic activities. It applies the fuzzy logic theory and the efficiency theory, the method of analogies, statistical processing of factual material, the comparative and expert analysis. Results. The developed methodology includes the formation of scenarios for companies’ development based on their financial and economic projections in the form of fuzzy numbers. I constructed membership functions of the initial fuzzy financial and economic indicators. The paper provides projected fuzzy estimates of the value of shares of the issuing company and its investment potential for the selected fuzzy financial and economic indicator and forecast scenario. The methodology was tested on the case of model forecast scenarios for ExxonMobil development. Conclusions. The presented methodology enables a forecast assessment of investment potential based on the database of quotations and financial and economic indicators of issuing companies. The predictive assessment is made within the formed fuzzy forecast scenario of the company's evolution in accordance with trends in the development of internal and external risk factors.

Visualization of uncertain data in the NEAT F-PROMETHEE method

One of the decision-making problems is the problem of description, which is supposed to explain the decision-making situation by describing the alternatives and their consequences. One of the basic multi-criteria decision analysis (MCDA) tools that are used when considering the problem of description are graphical visualizations. The article presents visualizations developed for the presentation of data in the MCDA method called NEAT F-PROMETHEE. Since the NEAT F-PROMETHEE method is used to consider decision-making problems characterized by uncertainty and imprecision, it operates on fuzzy numbers. Therefore, it was important that the developed visualizations also present data in the form of fuzzy numbers and consider uncertainty. The verification of the developed graphic representations consisted in comparing the ease of processing, understanding and interpreting information presented in a visual and tabular form. As a result of the conducted research, the advantages of the graphical representation over the tabular one were noticed. Data presented in the form of numerical values included in tables, if there are a lot of them, may cause information overload of the decision-maker. Fuzzy numbers included in the tables are often difficult to read, and a big challenge for the analyst is to establish mutual relationships and compare fuzzy numbers described numerically. It is much easier to process fuzzy numbers in a graphical form, because then you can immediately observe the relationships between the numbers.•Developing a visualization of uncertain data for the NEAT F-PROMETHEE method•Comparison of clarity of graphical and tabular representation.

Layer-specific evaluation-based Hierarchical Fuzzy Failure Mode and Effect Analysis

Reliability and safety have always been the main focus while developing critical automotive systems such as brakes. Failure Modes and Effect Analysis (FMEA) is one of the primary systematic reliability analysis tools. Due to the arising uncertainties and subjectivity, in models such as this one the fuzzy approach is highly popular, i.e., the fuzzy rule-based risk assessment methods can be used to model and depict the subjective opinions of estimators mathematically. In this paper, the authors propose a methodological approach to the implementation of fuzzy rule-based Hierarchical FMEA (H-FMEA), where the membership functions are different depending on the layer. Moreover, the information between each layer is transferred in the form of fuzzy numbers instead of crisp values, in order to further improve the reliability of the system.

Fuzzy Option Pricing Based on Fuzzy Number Binary Tree Model

In this article, a new method of option pricing in the form of fuzzy number is established based on fuzzy number binary tree model. Firstly,by using the operation rules of fuzzy number addition and number multiplication in cut set form, the fuzzy number binary tree model is equivalently transformed into two families of classical binary tree models which change with the level value, and some properties of the two binary tree model families are obtained, which are related to solve the problem of fuzzy option pricing. Then, based on the results obtained by us, the concepts of fuzzy set option price and fuzzy number option price are defined, and a sufficient condition is obtained for fuzzy set option price to be fuzzy number option price in the fuzzy number binary tree model. And then, for the convenience of practical application, the concepts of fold line fuzzy set and fuzzy number option price are introduced, a sufficient condition for fold line fuzzy set option price to be fold line fuzzy number option price is obtained, and a specific solution method of the fold line set and the fold line fuzzy number option pricing are set up. At last, two specific examples are given to show how to solve the fold line set and the fold line fuzzy number option price.

Maximum Likelihood Estimates and a survival function for fuzzy data for the Weibull-Pareto parameters

In this work, we discuss the estimation parameters and the reliability function for Weibull –Pareto distribution by maximum likelihood method. For that we used two iterative approximation techniques called “Newton-Raphson (NR)” and “Fisher’s Scoring (FS) method” when the available data is in the form of fuzzy numbers. The parameters’ and reliability function’s estimates are numerically contrasted using a Monte-Carlo simulation study in terms of their mean square error (MSE), integrated mean square error (IMSE), and mean absolute proportional error (MAPE) criteria. The FS technique outperforms the NR method in simulations, according to the empirical data.

Post-Disaster Temporary Shelters Distribution after a Large-Scale Disaster: An Integrated Model

This paper develops an integrated model for the distribution of post-disaster temporary shelters after a large-scale disaster. The proposed model clusters impacted areas using an Adaptive Neuro-Fuzzy Inference System (ANFIS) method and then prioritizes the points of clusters by affecting factors on the route reliability using a permanent matrix. The model’s objectives are to minimize the maximum service time, maximize the route reliability and minimize the unmet demand. In the case of ground relief, the possibility of a breakdown in the vehicle is considered. Due to the disaster’s uncertain nature, the demands of impacted areas are considered in the form of fuzzy numbers, and then the equivalent crisp counterpart of the non-deterministic is made by Jimenez’s method. Since the developed model is multi-objective, the Non-dominated Sorting Genetic Algorithm (NSGA-II) and Multi-Objective Firefly Algorithm (MOFA) are applied to find efficient solutions. The results confirm higher accuracy and lower computational time of the proposed MOFA. The findings of this study can contribute to the growing body of knowledge about disaster management strategies and have implications for critical decision-makers involved in post-disaster response projects. Furthermore, this study provides valuable information for national decision-makers in countries with limited experience with disasters and where the destructive consequences of disasters on the built environment are increasing.

A Fuzzy Version of the TOPSIS Method in Multi-Criteria Decision-Making Tasks

Choice and decision making are an integral part of the purposeful activities of people in all areas of public and private life. Tasks of multi-criteria decision making are characterised by the fact that alternative decisions are evaluated by a set of criteria and the concept of a decision and its outcome coincide. The defining concept in such problems is the concept of a set of Pareto optimal decisions (Pareto set). This set forms alternative decisions that are not comparable in terms of the set of evaluation criteria. The choice of the optimal decision in the Pareto set can be performed only on the basis of the subjective preferences of the decision maker. In recent decades, extensions of traditional methods of multi-criteria decision making to a fuzzy environment have been proposed. One of the well-known approaches to multi-criteria decision making is the TOPSIS method. In the paper, a fuzzy version of this method is considered in situations where the values of evaluation criteria are set in the form of fuzzy numbers.

On Solving Bi-level Multi-Objective Fully Quadratic Fractional Optimization Model with Fuzzy Demands

This paper has been designed to introduce the method for solving the Bi-level Multi-objective (BL-MO) Fully Quadratic Fractional Optimization Model through Fuzzy Goal Programming (FGP) approach by utilising non-linear programming. In Fully Quadratic Fractional Optimization Model, the objective functions are in fractional form, having quadratic functions in both numerator and denominator subject to quadratic constraints set. The motive behind this paper is to provide a solution to solve the BL-MO optimization model in which number of decision-makers (DM) exists at two levels in the hierarchy. First, the fractional functions with fuzzy demand, which are in the form of fuzzy numbers, are converted into crisp models by applying the concept of α-cuts. After that, membership functions are developed which are corresponding to each decision-maker’s objective and converted into simpler form to avoid complications due to calculations. Finally, the model is simplified by applying FGP approach, and a compromised solution to the initial model is obtained. An algorithm, flowchart and example are also given at the end to explain the study of the proposed model.

A Bi-level Neuro-Fuzzy System Soft Computing for Reservoir Operation

Reservoir operation studies purely based on the storage level, inflow, and release decisions during dry periods only fail to serve the optimal reservoir operation policy design because of the fact that the release decision during this period is highly dependent on wet season water conservation and flood risk management operations. Imperatively, the operation logic in the two seasons are quite different. If the two operations are not sufficiently coordinated, they may produce poor responses to the system dynamics. There are high levels of uncertainties on the model parameters, values and how they are logically operated by human or automated systems. Soft computing methods represent the system as an artificial neural network (ANN) in which the input- output relations take the form of fuzzy numbers, fuzzy arithmetic and fuzzy logic (FL). Neuro-Fuzzy System (NFS) soft computing combine the approaches of FL and ANN for single purpose reservoir operation. Thus, this study proposes a Bi-Level Neuro-Fuzzy System (BL-NFS) soft computing methodology for short and long term operation policies for a newly inaugurated irrigation project in Gidabo Watershed of Main Ethiopian Rift Valley Basin. Keywords: Bankruptcy rule, BL-NFS, Reservoir operation, Sensitivity analysis, Soft computing, Water conservation.

A type-1 OWA operator for Extended Comparative Linguistic Expressions with Symbolic Translation

Recently, a new representation model has been proposed to improve the interpretability and accuracy of Computing with Words (CW) processes, the so-called Extended Comparative Linguistic Expressions with Symbolic Translation (ELICIT). It extends the concept of comparative linguistic expressions based on hesitant fuzzy linguistic term sets into a continuous domain by means of the symbolic translation used in the linguistic 2-tuple model. The aggregation of this kind of information is vital when solving decision making problems, but as the field of ELICIT information is still quite new, not many aggregation operators have been defined for it. Therefore, taking into account the need to deal with the fuzzy representation of ELICIT information for its aggregation, we propose the ELICIT type-1 OWA (ELICIT-t1-OWA) operator based on the equivalent α−level type-1 OWA operator to simplify the CW processes by using ELICIT trapezoidal fuzzy representations. The ELICIT-t1-OWA operator will use interval weights induced by an interval-valued linguistic quantifier to aggregate ELICIT expressions and then its properties will be studied. Additionally, a way to deal with both the weights and arguments of the ELICIT-t1-OWA operator is provided in the form of fuzzy numbers. Finally, a decision making model dealing with ELICIT information that carries out the aggregation by using the proposed ELICIT-t1-OWA operator is developed and a numerical example to show its effectiveness is introduced.

On Estimation of Rayleigh Scale Parameter under Doubly Type-II Censoring from Imprecise Data

The scheme of doubly type-II censored sampling is an important method of obtaining data in lifetime studies. Statistical analysis of life time distributions under this censoring scheme is based on precise lifetime data. However, some collected lifetime data might be imprecise and are represented in the form of fuzzy numbers. This paper deals with the prob lem of estimating the scale parameter of Rayleigh distribution under doubly type-II censoring scheme when the lifetime observations are fuzzy and are assumed to be related to underlying crisp realization of a random sample. We propose a new method to determine the maximum likelihood estimate of the parameter of interest. The asymptotic variance of the ML estimate is then derived by using the missing information principle. Their performance is then assessed through Monte Carlo simulations. Finally, an illustrative example with real data concerning 25 ball bearings in a life test is presented.

Improving Air Transportation by Using the Fuzzy Origin–Destination Matrix

The work is devoted to the development of new methods and algorithms to support decision making when planning air travel using uncertainties in the form of fuzzy numbers. The proposed approach makes it possible to define rational methods of choice: how to change the transport graph to better meet the needs of the population. This is particularly relevant in the context of the reduced demand for air travel caused by the pandemic and the need to switch from large to smaller aircraft types. The problem is solved by restoring the fuzzy origin–destination matrix of current statistics on air traffic between airports. The problem is that we do not know what proportion of passengers moving between the specified points are forced to use large transport hubs as intermediate destinations. To determine the validity of the origin–destination matrix, we build a number of optimization models to determine fuzzy intervals and search for correspondence with the maximum value of the membership function. Algorithmic and software search for the fuzzy origin–destination matrix and fuzzy ranking of potentially promising routes are developed. The perspective of the given approach is shown by an example of a task concerning a choice of new routes between regional airports.

Project Delivery of Goods With Limited Resources and Minimum Time Using Fuzzy Logic Method

In solving the problem of fuzzy application for scheduling delivery of goods with a limited power source and minimal time, the problems that are still not considered in the RCPS problem modeling are the uncertainty characteristics of the parameters of the timing of activities in the delivery project. Even though this can be solved by using the PERT (Project Evaluation and Review Technoloque) method with a probabilistic approach, this technique still ignores the limitations of the supply of resources. Actually, a probabilistic approach can be used, if previously provided data about the experience in completing similar projects. But if the project is a new project or the techniques and methodologies used to complete a new project, such as new techniques and methodologies in software engineering, among others: object-oriented design and programming, computer-aided software design, user interface management systems, fourth generation languages, etc., then the probabilistic approach is not suitable. 
 In this situation, the decision maker must be able to estimate the cost and time duration, of all activities in the project based on existing experience, related to the level of knowledge they have, about new techniques and methodologies to be applied, and the level of human resource expertise. which are available. This method of estimating project costs and time, which is more precise, uses representations in the form of fuzzy numbers, namely fuzzy sets in the set of real numbers that are normal, convex, and closed intervals. The delivery time is modeled as a fuzzy number of LR types IKiri, IKanan, α, β, with three values ​​of α-cuts E = 0.3, L = 0.7, and I = 1.0. The fuzzy transformation model is based on three pairs of inferior and superior values ​​from α-cuts. The priority for scheduling delivery of goods is based on the smallest early start time EST value. Resources are solved by serial and parallel models. The smallest makespan value is used to determine the best solution. Goods delivery settlement uses fuzzy operations, namely arithmetic operations and relation operations. 
 Analysis of the output oftware based on testing with test scenarios (table ^ .50) for some input data shows that the parallel method is better than the serial method. This is indicated by the large difference in makespan value generated from the two methods. Based on input data from a software development project, the serial method gives makespan values ​​in the range between 675.0 and 867.0, while the parallel method gives makespan values ​​in the range between 116.0 and 259.0. The analysis of the output software in a fuzzy Gantt Chart representation shows that an activity can be scheduled with varying degrees of optimism. The degree of optimistic activity scheduling can be graded in linguistic terminology between two extreme pessimistic and optimistic values, namely very pessimistic, pessimistic, slightly pessimistic, slightly optimistic, optimistic, very optomistic.

Fuzzy Hypotheses Testing in the Framework of Type-2 Fuzzy Numbers

The estimation of various components in science always faces with errors and inaccuracies that are presented in the form of an uncertainty in identifying the considered components. This uncertainty can be represented in different shapes, one of which is the representation of estimated components in the form of fuzzy numbers. In a situation where a shortage of information or ambiguity in the studied system can be seen, fuzzy numbers can be used to express the ambiguity available in the components. In this study, we will introduce new approach for the formation of type-2 fuzzy numbers in which the mean and standard deviations of data are used to determine the uncertainty zone. Finally, with proposed type-2 fuzzy numbers, in proportion to any situation we have, we face with the claims related to the mean and assess with the subject of the statistical hypothesis test with fuzzy approach.

Predicting the optimal solution in fuzzy linear programming problem

In this paper we try to define a percentage form of LR fuzzy numbers which is a useful form of fuzzy numbers and its’ arithmetics. So, we show how the maximum variation range of optimal value of fuzzy objective function can be predicted by using this form of fuzzy numbers. Since fuzzy problems are generally solved through a complicated manner, the purpose of this study is releasing a kind of prediction for the final solution in the way that the manager can access to an outlook to optimal solution (Z∗) without solving the problem. Finally, optimal value of fuzzy objective function on fuzzy linear programming is predicted when maximum variation range of fuzzy variable have been predetermined.

A New Approach for Ranking Shadowed Fuzzy Numbers and its Application

In many decision situations, decision-makers face a kind of complex problems. In these decision-making problems, different types of fuzzy numbers are defined and, have multiple types of membership functions. So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are considered granule numbers which approximate different types and different forms of fuzzy numbers. In this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute decision making problem in which the evaluations of alternatives are expressed with different types of uncertain numbers. The comparative study for the results of different examples illustrates the reliability of the new method.

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.

A Fuzzy Best-Worst Multi-Criteria Group Decision-Making Method

Fuzzy best-worst method (BWM) has emerged as an efficient choice because of its comparison consistency to model the real-life and the consideration of fuzziness and uncertainties of decision-makers (DMs). However, how to extend the fuzzy BWM to group decision-making (GDM) environment has become an important topic because there are usually more than one decision-maker. For the GDM, decision makers may use different concepts to establish their individual assessment information due to the difference of cultural background and priori knowledge. To overcome this challenge, this article proposed a novel fuzzy best-worst multi-criteria group decision-making method to solve the GDM problem with multi-granular linguistic approach, which is an effective and promising technique to tackle this issue. In the proposed method, the selectable multi-granularity linguistic term sets (LTS) are firstly provided for experts to expressed their individual assessment information. Then, the improved fuzzy BWM is employed to calculate the weights of criteria with the form of fuzzy numbers. In current several studies using the BWM for group decision-making, only two unified best and worst criteria are given, which cannot reflect the evaluation of the best and worst criteria by different experts, resulting in the omission of information. Moreover, the difference between the best and worst criteria initially given and the experts’ ideas will cause the experts to be inaccurate in the comparison of each criterion. Therefore, in this article, in order not to omit too much information, each expert will determine the best and the worst criteria. Each expert’s assessment information which is based on his/her best and worst criteria is integrated into two vectors. What’s more, an improved input-based consistency measurement is proposed, which can provide the DMs with a clear guideline on the revision of the inconsistent judgement(s). Finally, two examples are given to illustrate the effectiveness and applicability of the proposed method.

A New Approach for Ranking Shadowed Fuzzy Numbers and its Application

In many decision situations, decision-makers face a kind of complex problems. In these decision-making problems, different types of fuzzy numbers are defined and, have multiple types of membership functions. So, we need a standard form to formulate uncertain numbers in the problem. Shadowed fuzzy numbers are considered granule numbers which approximate different types and different forms of fuzzy numbers. In this paper, a new ranking approach for shadowed fuzzy numbers is developed using value, ambiguity and fuzziness for shadowed fuzzy numbers. The new ranking method has been compared with other existing approaches through numerical examples. Also, the new method is applied to a hybrid multi-attribute decision making problem in which the evaluations of alternatives are expressed with different types of uncertain numbers. The comparative study for the results of different examples illustrates the reliability of the new method.