Fuzzy Cognitive Maps in Reliability Modeling

Introduction

Success in applied modeling problem solving largely depends on the chosen mathematical tools. Reliability theory is like medicine; medicine deals with human health, while reliability theory deals with “the health of man-made systems”. The range of problems that may harm the reliability of system at the stages of design, production and operation makes reliability science an endless polygon, and presents opportunities for testing different mathematical theories and new methods for predicting, diagnosing and decision making.

Classical reliability theory [1, 2] arose in the middle of the 20th century when probability theory was the only way to formalize uncertainty. The models of classical reliability theory use the source data about failure probabilities and the laws of random value distribution ​​of time between failures and repairs. It is assumed that these inputs are known from reference books or can be obtained experimentally. In practice, though, obtaining the source data and addressing all the influencing factors is a difficult and often impossible task. These difficulties often force us to substitute approximate source data into the complex probabilistic models, and make it impossible to predict accurately the probability of failure. As a result, engineers do not believe in their own reliability calculations (Barnard, 2012).

The appearance in 1965 an article by L. Zadeh (Zadeh, 1965) laid the foundation for a new method of describing uncertainty in the form of fuzzy logic. This aroused natural interest among specialists in reliability, and as a result, led to the emergence of an independent direction, which can be called Fuzzy Reliability. The first specialized collection of articles in this area is (Onisawa & Kasprzyk, 1995), and the first monographs include (Cai, 1996; Rotshtein & Shtovba, 1997; Utkin & Shubinsky, 1998).

Our experience in using fuzzy mathematics in reliability modeling shows that the following fundamental concepts and principles have the greatest application:

• •

  Fuzzy numbers (Dubois & Prade, 1978) are a convenient means of presenting expert information about uncertain source data that are used in probabilistic reliability models (Onisawa et al., 1995; Rotshtein & Shtovba, 1997; Rotshtein & Shtovba, 1998).

• •

  The extension principle (Zadeh, 1973) allows converting probabilistic reliability models to the form necessary for working with uncertain source data. It is important to emphasize that calculations with fuzzy numbers are much simpler than similar calculations with probabilistic distribution functions (Onisawa et al., 1995; Rotshtein & Shtovba, 1997; Rotshtein & Shtovba, 1998).

• •

  Fuzzy if-then rules (Zadeh, 1973) make it possible to approximate non-linear dependences of the reliability level on influencing factors. Compared with regression models, fuzzy rules require less experimentation due to the use of expert knowledge in a natural language (Onisawa & Kasprzyk, 1995; Rotshtein & Shtovba, 1998; Rotshtein & Shtovba, 2009).

• •

  The membership function (Zadeh, 1965; Zadeh, 1973) is an indicator of the level of correctness of an element of the system depending on the value of some parameter. The structure of the system allows one to obtain the function of the correct operation of the system depending on the influencing parameters (Rotshtein, 2010; Rotshtein, 2011) and solve the problems of optimal redundancy without failure statistics (Katelnikov & Rotshtein, 2010; Rotshtein et al., 2012).

• •

  Fuzzy relation and compositional rule of inference (Zadeh, 1973) model cause-effect relationships in the form of a system of fuzzy logical equations. This allows you to solve the diagnostic problem, i.e. Failure Mode Effect Analysis - FMEA (International Electrotechnical Commission (IEC), 2006) as the restoration of causes from the observed effects (Rotshtein et al., 2006; Rotshtein & Rakytyanska, 2001; Rotshtein & Rakytyanska, 2008).

• •

  The transitive closure (Zadeh, 1973) of the fuzzy similarity relationship allows you to classify the elements of the system and determine the importance of their influence on the reliability of the system as a whole (Rotshtein & Dreyfuss, 2018; Rotshtein, 2017).

• •

  The principle of fuzzy criteria intersection by Bellmann – Zadeh (Bellman & Zadeh, 1970) allows to choose the values ​​of factors affecting the system’s reliability without complicated calculation of the failure probability (Rotshtein, 2018; Rotshtein, 2019).