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Du, R et al "Monitoring and Diagnosing Manufacturing Processes Using Fuzzy Set Theory" Computational Intelligence in Manufacturing Handbook Edited by Jun Wang et al Boca Raton: CRC Press LLC,2001 14 Monitoring and Diagnosing Manufacturing Processes Using Fuzzy Set Theory R Du* University of Miami Yangsheng Xu Chinese University of Hong Kong 14.1 Introduction 14.2 A Brief Description of Fuzzy Set Theory 14.3 Monitoring and Diagnosing Manufacturing Processes Using Fuzzy Sets 14.4 Application Examples 14.5 Conclusions Abstract Monitoring and diagnosis play an important role in modern manufacturing engineering They help to detect product defects and process/system malfunctions early, and hence, eliminate costly consequences They also help to diagnose the root causes of the problems in design and production and hence minimize production loss and at the same time improve product quality In the past decades, many monitoring and diagnosis methods have been developed, among which the fuzzy set theory has demonstrated its effectiveness This chapter describes how to use the fuzzy set theory for engineering monitoring and diagnosis It introduces various methods such as fuzzy linear equation method, fuzzy C-mean method, fuzzy decision tree method, and a newly developed method, fuzzy transition probability method By using good examples, it demonstrates step by step how the theory and the computation work Two practical examples are also included to show the effectiveness of the fuzzy set theory 14.1 Introduction According to Webster’s New World Dictionary of the American Language, “monitoring,” among several other meanings, means checking or regulating the performance of a machine, a process, or a system “Diagnosis” means deciding the nature and the cause(s) of a diseased condition of a machine, a process, or a system by examining the performance or the symptoms In other words, monitoring detects suspicious symptoms while diagnosis determines the cause of the symptoms There are several words and/or *This work was completed when Dr Du visited The Chinese University of Hong Kong ©2001 CRC Press LLC phrases that have similar or slightly different meanings, such as fault detection, fault prediction, inprocess verification, on-line inspection, identification, and estimation Monitoring and diagnosing play a very important role in modern manufacturing This is because manufacturing processes are becoming increasingly complicated and machines are much more automated Also, the processes and the machines are often correlated; and hence, even small malfunctions or defects may cause catastrophic consequences Therefore, a great deal of research has been carried out in the past 20 years Many papers and monographs have been published Instead of giving a partial review here, the reader is referred to two books One by Davies [1998] describes various monitoring and diagnosis technologies and instruments The reader should also be aware that there are many commercial monitoring and diagnosis systems available In general, monitoring and diagnosis methods can be divided into two categories: a model-based method and a feature-based method The former is applicable where a dynamic model (linear or nonlinear, time-invariant or time-variant) can be established, and is commonly used in electrical and aerospace engineering The book by Gertler [1988] describes the basics of model-based monitoring The latter uses the features extracted from sensor signals (such as cutting forces in machining processes and pressures in pressured vessels) and can be used in various engineering areas This chapter will focus on this type of method More specifically the objective of this chapter is to introduce the reader to the use of fuzzy set theory for engineering monitoring and diagnosis The presented method is applicable to almost all engineering processes and systems, simple or complicated There are of course many other methods available, such as pattern recognition, decision tree, artificial neural network, and expert systems However, from the discussions that follow, the readers can see that fuzzy set theory is simple and effective method that is worth exploring This chapter contains five sections Section 14.2 is a brief review of fuzzy set theory Section 14.3 describes how to use fuzzy set theory for monitoring and diagnosing manufacturing processes Section 14.4 presents several application examples Finally, Section 14.5 contains the conclusions 14.2 A Brief Description of Fuzzy Set Theory 14.2.1 The Basic Concept of Fuzzy Sets Since fuzzy set theory was developed by Zadeh [1965], there have been many excellent papers and monographs on this subject, for example [Baldwin et al., 1995; Klir and Folger, 1988] Hence, this chapter only gives a brief description of fuzzy set theory for readers who are familiar with the concept but are unfamiliar with the calculations The readers who would like to know more are referred to the abovementioned references It is known that a crisp (or deterministic) set represents an exclusive event Suppose A is a crisp set in a space X (i.e., A ⊂ X), then given any element in X, say x, there will be either x ∈ A or x ∉ A Mathematically, this crisp relationship can be represented by a membership function, µ(A), as shown in Figure 14.1, where x ∉ (b,c) Note that µ(A) = {0, 1} In comparison, for a fuzzy event, A′, its membership function, µ(A′), varies between and 1, that is µ(A) = [0, 1] In other words, there are cases in which the instance of the event x ∈ A′ can only be determined with some degree of certainty This degree of certainty is referred to as fuzzy degree and is denoted as µΑ’(x ∈ A ′ ) Furthermore, the fuzzy set is denoted as x/µA’(x), ∀ x ∈ A ′, and µA’(x) is called the fuzzy membership function or the possibility distribution It should be noted that the fuzzy degree has a clear meaning: µ(x) = means x is impossible while µ(x) = implies x is certainly true In addition, the fuzzy membership function may take various forms such as a discrete tablet, x: x1 x2 … xn µ(x): µ ( x 1) µ ( x 2) … µ ( x n) or a continuous step-wise function, ©2001 CRC Press LLC Equation (14.1) FIGURE 14.1 Illustration of crisp and fuzzy concept  x –a   b – a µ x = d – x   d –c  ( ) x ≤a a