Exact Membership Function Research Articles

Design of single and double acceptance sampling plans based on interval type-2 fuzzy sets

Defectiveness of items is generally considered as a certain value in acceptance sampling plans (ASPs). It is clear that, it may not be certainly known in some real-case problems. Uncertainties of the inspection process such as measurement errors, inspectors’ hesitancies or vagueness of the process etc. should be taken into account to obtain more reliable results. The fuzzy set theory (FST) is one of the best methods to overcome these problems. There are some studies in the literature formulating the ASPs with the help of FST. Deciding the right membership functions of the fuzzy sets (FSs) has a vital importance on the quality of the uncertainty modeling. Additionally, the fuzzy set extensions have been offered to model more complicated uncertainties to achieve better modeling. As one of these extensions, type-2 fuzzy sets (T2FSs) gives an ability to model uncertainty in situations where it is not possible to determine exact membership function parameters. In this study, single and double ASPs based on interval T2FSs (IT2FSs) have been designed for binomial and Poisson distributions. Thus, it becomes possible to make more flexible, sensitive and descriptive sensitivity analyzes. The main characteristic functions of ASPs have been derived and the suggested formulations have been illustrated on a comparative application from manufacturing process. Results allowing for more comprehensive analysis as against to the traditional and T1FSs based plans have been obtained.

ON THE RATIO OF FUZZY NUMBERS – EXACT MEMBERSHIP FUNCTION COMPUTATION AND APPLICATIONS TO DECISION MAKING

In the present paper, we propose a new approach to solving the full fuzzy linear fractional programming problem. By this approach, we provide a tool for making good decisions in certain problems in which the goals may be modelled by linear fractional functions under linear constraints; and when only vague data are available. In order to evaluate the membership function of the fractional objective, we use the α-cut interval of a special class of fuzzy numbers, namely the fuzzy numbers obtained as sums of products of triangular fuzzy numbers with positive support. We derive the α-cut interval of the ratio of such fuzzy numbers, compute the exact membership function of the ratio, and introduce a way to evaluate the error that arises when the result is approximated by a triangular fuzzy number. We analyse the effect of this approximation on solving a full fuzzy linear fractional programming problem. We illustrate our approach by solving a special example – a decision-making problem in production planning.

Applications of type-2 fuzzy logic system: handling the uncertainty associated with candidate-well selection for hydraulic fracturing

The problem of selecting a target formation(s) in a reservoir among a vast number of zones/sub-layers within huge number of hydrocarbon producing wells for hydraulic fracturing (HF) by using interval type-2 fuzzy logic system (IT2-FLS) to maximize their net present value is studied in this paper. Classical fuzzy system which is called type-1 fuzzy logic system is not capable of accurately capturing the linguistic and numerical uncertainties in the terms used and the inconsistency of the expert's decision-making. IT2-FLS is very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set; hence it is very effective for dealing with uncertainties. In highlighting this need, the question has been answered why IT2-FLS should be used in this study. The procedure of applying this study in the area of HF candidate-well selection is illustrated through a case study in an oil reservoir.

Fuzzy efficiency ranking in fuzzy two-stage data envelopment analysis

Data envelopment analysis (DEA) is a useful tool for efficiency measurement of firms and organizations. Many production systems in the real world are composed of two processes connected in series. Measuring the system efficiency without taking the operation of each process into consideration will obtain misleading results. Two-stage DEA models show the performance of individual processes, thus is more informative than the conventional one-stage models for making decisions. When input and output data are fuzzy numbers, the derived efficiencies become fuzzy as well. This paper proposes a method to rank the fuzzy efficiencies when the exact membership functions of the overall efficiencies derived from fuzzy two-stage model are unknown. By incorporating the fuzzy two-stage model with the fuzzy number ranking method, a pair of nonlinear program is formulated to rank the fuzzy overall efficiency scores of DMUs. Solving the pair of nonlinear programs determines the efficiency rankings. An example of the ranking of the 24 non-life assurance companies in Taiwan is illustrated to explain how the proposed method is applied.

Model and algorithm for fuzzy joint replenishment and delivery scheduling without explicit membership function

We study a joint replenishment and delivery scheduling (JRD) problem in which a central warehouse serves n-retailers in the presence of vague operational conditions such as ordering cost and inventory holding cost. In the proposed fuzzy set-based approach, an exact membership function is not assumed and instead can be approximated using piecewise linear functions based on alpha level sets because of their easy handling and efficiency. Subsequently, the fuzzy total cost is defuzzified by the widely used signed distance method to ranking fuzzy numbers. However, due to the JRD's difficult mathematical properties, efficient and effective solution procedures for the problem have eluded researchers. To find an optimal solution, an effective and efficient differential evolution (DE) algorithm is designed. After determining the appropriate parameters of the DE by parameter tuning test, the effectiveness of the DE is verified by numerical examples. We compare the DE with the available best approach and results show that DE can solve this non-deterministic polynomial hard problem in a robust way with a high convergence rate and low average error.

Bearing condition prediction considering uncertainty: An interval type-2 fuzzy neural network approach

Rolling-element bearings are critical components of rotating machinery. It is important to accurately predict in real-time the health condition of bearings so that maintenance practices can be scheduled to avoid malfunctions or even catastrophic failures. In this paper, an Interval Type-2 Fuzzy Neural Network (IT2FNN) is proposed to perform multi-step-ahead condition prediction of faulty bearings. Since the IT2FNN defines an interval type-2 fuzzy logic system in the form of a multi-layer neural network, it can integrate the merits of each, such as fuzzy reasoning to handle uncertainties and neural networks to learn from data. The interval type-2 fuzzy linguistic process in the IT2FNN enables the system to handle prediction uncertainties, since the type-2 fuzzy sets are such sets whose membership grades are type-1 fuzzy sets that can be used in failure prediction due to the difficult determination of an exact membership function for a fuzzy set. Noisy data of faulty bearings are used to validate the proposed predictor, whose performance is compared with that of a prevalent type-1 condition predictor called Adaptive Neuro-Fuzzy Inference System (ANFIS). The results show that better prediction accuracy can be achieved via the IT2FNN.

Type-2 Fuzzy Sliding Mode Control of A Four-Bar Mechanism

This paper describes a new control scheme for the robust crank angular speed control of a four-bar mechanism driven by a DC motor. The proposed control method is based on type-2 fuzzy logic and sliding mode control (SMC) technique. Type-2 fuzzy control (FC) systems are characterized by type-2 membership functions which are useful in circumstances where it is difficult to determine an exact membership function. They are also very effective to handle uncertainties because type-2 fuzzy systems can model them and minimize their effects. SMC is a robust control method against parameter variations and external disturbances. Type-2 fuzzy logic and SMC can be combined to use the advantages of both methods and thus to improve the effectiveness of the controllers. One of the most important advantages of the use of SMC with type-2 FC is to reduce the number of fuzzy rules and thus to obtain a simpler and more practical control algorithm to use in real applications. To show the effectiveness of the proposed controller, simulation results of the Type-2 FC and Type-2 FSMC systems are presented in the paper.

The Extended Operations for Generalized Quadratic Fuzzy Sets

The extended algebraic operations are defined by applying the extension principle to normal algebraic operations. And these operations are calculated for some kinds of fuzzy numbers. In this paper, we get exact membership function as a results of calculation of these operations for generalized quadratic fuzzy sets.

Centroid defuzzification and the maximizing set and minimizing set ranking based on alpha level sets

Centroid defuzzification and the maximizing set and minimizing set methods are two commonly used approaches to ranking fuzzy numbers and often require membership functions to be known. In this paper, the two methods are reinvestigated when explicit membership functions are not known but alpha level sets are available. Two analytical formulas are derived under the assumption that the exact membership functions can be approximated by using piecewise linear functions based on alpha level sets. The derived analytical formulas are of significant importance and provide very useful decision supports for a wide variety of applications of the two methods in industrial engineering and other areas. Numerical examples are offered to test the derived formulas and illustrate their computational processes and applications in risk assessment of a software development.

A neuro-fuzzy approach to generating mold/die polishing sequences

A neuro-fuzzy approach to generating polishing sequences is presented. The measured initial surface roughness of the EDMed mold or die and the desired final polished surface roughness are the major inputs fed to the system. After receiving this input, the system consults its database for the polishing efficiency curves of the abrasive stone grain sizes available to it. With the outputs of this system being optimized for minimum polishing time, there is a selected sequence of grain sizes from among an available set, with each grain size used for each polishing step. There was a series of initial polishing experiments which were conducted for the different available grain sizes and workpiece roughness at different pressures and RPM's, with a change in roughness measured for polishing duration. From this, the database is constructed for designing the fuzzy logic rules. For constructing the exact membership function of the fuzzy interface, the neuro-fuzzy technique is combined with learning ability of neural network and the inferring ability of fuzzy logic system. Finally, while considering the stone-changing time, the actual experimental results from suggested polishing sequences are compared with the predicted value in order to establish the proposed approach.

A software tool: Type‐2 fuzzy logic toolbox

AbstractThe concept of type‐2 fuzzy set was initially proposed as an extension of classical (type‐1) fuzzy sets. Type‐2 fuzzy sets are very useful in circumstances where it is difficult to determine an exact membership function for a fuzzy set; hence they are very effective for dealing with uncertainties. However, type‐2 fuzzy sets are more difficult to use and understand than type‐1 fuzzy sets. Even in the face of these difficulties, type‐2 fuzzy logic has found applications in many fields. In this article, a new Type‐2 Fuzzy Logic Toolbox written in MATLAB programming language is introduced. The main aim is to help the user to understand and implement type‐2 fuzzy logic systems easily. Type‐2 Fuzzy Logic Controller Block is also prepared for use in SIMULINK. Since general type‐2 fuzzy logic systems are very complicated, they are not preferred in applications. Thus, only interval type‐2 fuzzy logic systems are considered in the proposed Type‐2 Fuzzy Logic Toolbox. Since MATLAB Fuzzy Logic Toolbox users are familiar with its windows, all the menus of the developed software are prepared in the same format of the MATLAB Fuzzy Logic Toolbox. © 2008 Wiley Periodicals, Inc. Comput Appl Eng Educ 16: 137–146, 2008; Published online in Wiley InterScience (www.interscience.wiley.com); DOI 10.1002/cae20138

A Time Optimized Process Planning System for Mold/Die Polishing

This paper presents a time-optimized process planning system for mold/die polishing using various sizes of abrasive stones on a flat mold or a die surface. The system is assigned a specific set of abrasive grain sizes. User inputs to this system are measured initial surface roughness of the EDMed mold or die and the desired final polished surface roughness. The outputs of this system, optimized for minimum polishing time, are a selected sequence of grain sizes from among the available set with each grain size used for one polishing step. With each polishing step is an associated polishing time, polishing pressure and workpiece RPM. A series of initial polishing experiments were conducted for the different available grain sizes and workpiece roughnesses at different pressures and RPM with the change in roughness measured for polishing duration. From this, an expert database is constructed for designing the fuzzy logic rules. Then a neuro-fuzzy technique combining the learning ability of neural network and the inferring ability of fuzzy logic system is used to construct the exact membership function of the fuzzy interface. Finally, actual experimental results from suggested polishing sequences are compared with the predicted value validating the proposed approach.

Rating design requirements in fuzzy quality function deployment via a mathematical programming approach

Quality function deployment (QFD) is an important tool for translating customer requirements into product features. Identification of critical design requirements is a key factor for product development to satisfy customer needs. When the relative weights of customer attributes and the relationship measures between customer attributes and design requirements are fuzzy numbers, the determination of the design requirements becomes more complicated in QFD planning process. The calculation of the technical importance of design requirements falls into the category of fuzzy weighted average. Since the derived membership functions of the technical importance of design requirements are not explicitly known, the paper devises a method that can prioritize the design requirements without knowing their exact membership functions. The ratings of the design requirements are determined by solving a pair of nonlinear programs for each design requirement. The impacts of correlation among design requirements and the budget factor are also considered for achieving higher customer satisfaction. An example of a flexible manufacturing system design is given to illustrate the application of the proposed method.

Fuzzy MCDM based on the concept of α‐cut

AbstractIn a fuzzy multiple criteria decision‐making (MCDM) problem, with a hierarchical structure of more than two levels and involving multiple decision‐makers (DMs), to find the exact membership functions of the final aggregation ratings of all feasible alternatives is almost impossible. Thus, ranking methods based on exact membership functions cannot be utilized to rank the feasible alternatives and complete the optimal selection. To resolve the above‐mentioned complexity and to incorporate assessments of all DMs' viewpoints, in this paper a fuzzy MCDM method with multiple DMs, based on the concepts of fuzzy set theory and α‐cut, is developed. This method incorporates a number of perspectives on how to approach the fuzzy MCDM problem with multiple DMs, as follows: (1) combining quantitative and qualitative criteria as well as negative and positive ones; (2) using the generalized means to develop the aggregation method of multiple DMs' opinions; (3) incorporating the risk attitude index β to convey the total risk attitude of all DMs by using the estimation data obtained at the data input stage; (4) employing the algebraic operations of fuzzy numbers based on the concept of α‐cut to calculate the final aggregation ratings and develop a matching ranking method for proposed fuzzy MCDM method with multiple DMs. Furthermore, we use this method to survey the site selection for free port zone (FPZ) in Taiwan as an empirical study to demonstrate the proposed fuzzy MCDM algorithm. The result of this empirical investigation shows that the port of Kaohsiung, the largest international port of Taiwan as well as the sixth container port in the world in 2004, is optimal for the Taiwan government in enacting the plan of FPZ. Copyright © 2005 John Wiley & Sons, Ltd.

On the compositional rule of inference under triangular norms

Abstract The aim of this paper is to provide a close upper bound of the membership function for the compositional rule of inference under Archimedean t-norm, where both the observation and the relation parts are given by Hellendoorn's φ-function (1980). In particular, if the left and right spreads of the observation part is the same as those of the relation part, then this upper bound is the exact membership function, which generalizes the earlier result by Fuller and Zimmermann (1982) in that assumption of twice differentiability is deleted.