Applying weakest t-norm based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers to evaluate reliability of PCBA fault

doi:10.1016/j.asoc.2014.06.036

Applied Soft Computing

Volume 23, October 2014, Pages 387-406

https://doi.org/10.1016/j.asoc.2014.06.036 Get rights and content

Highlights

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Weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers to evaluate fault interval and reliability interval.
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The proposed novel fuzzy arithmetic operations may obtain fitter decision values, which have smaller fuzziness accumulating and successfully analyze the system reliability.
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Also weakest t-norm arithmetic operations provide more exact fuzzy results and effectively reduce fuzzy spreads (fuzzy intervals).
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Fuzzy reliability of PCBA fault has been analyzed using the proposed approach.

Abstract

This research addresses system reliability analysis using weakest t-norm based approximate intuitionistic fuzzy arithmetic operations, where failure probabilities of all components are represented by different types of intuitionistic fuzzy numbers. Due to the incomplete, imprecise, vague and conflicting information about the component of system, the present study evaluates the reliability of system in terms of membership function and non-membership function by using weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers. In general, interval arithmetic (α-cut arithmetic) operations have been used to analyze the fuzzy system reliability. In complicated systems, interval arithmetic operations may occur the accumulating phenomenon of fuzziness. In order to overcome the accumulating phenomenon of fuzziness, this research adopts approximate intuitionistic fuzzy arithmetic operations under the weakest t-norm arithmetic operations (T_w) to analyze fuzzy system reliability. The approximate intuitionistic fuzzy arithmetic operations employ principle of interval arithmetic under the weakest t-norm arithmetic operations. The proposed novel fuzzy arithmetic operations may obtain fitter decision values, which have smaller fuzziness accumulating and successfully analyze the system reliability. Also weakest t-norm arithmetic operations provide more exact fuzzy results and effectively reduce fuzzy spreads (fuzzy intervals). Using proposed approach, fuzzy reliability of series system and parallel system are also constructed. For numerical verification of proposed approach, a malfunction of printed circuit board assembly (PCBA) is presented as a numerical example. The result of the proposed method is compared with the listing approaches of reliability analysis methods.

Introduction

Reliable engineering is one of the important engineering tasks in design and development of any technical system. The conventional reliability of a system is defined as the probability that the system performs its assigned function properly during a predefined period under the condition that the system behavior can be fully characterized in the context of probability measures. However, in the real world problems, the collected data or system parameters are often fuzzy/imprecise because of incomplete or non-obtainable information, and the probabilistic approach to the conventional reliability analysis is inadequate to account for such built-in uncertainties in data. To overcome this problem, fuzzy set theory [1] has been used in the evaluation of the reliability of a system. From a long period of time, efforts have been made in the design and development of reliable large-scale systems. In that period of time, considerable work has been done by researchers to build a systematic theory of reliability based on the probability theory. In a general sense, fuzzy reliability can be physically interpreted as the probability that no substantial performance deterioration occurs in a predetermined time interval [2].

In the work of Cai et al. [3], the fuzzy system reliability was established based on the binary state assumption and possibility assumption. However in the work of Cai et al. [4], the fuzzy system reliability was established based on the three-state assumption and possibility assumption. In the work of Cai et al. [5], the fuzzy system reliability was developed based on the basis of fuzzy state assumption and probability assumption. They presented a fuzzy set based approach to failure rate and reliability analysis, where profust failure rate is defined in the context of statistics. Cai et al. [6] also discussed the system reliability for coherent system based on the fuzzy state assumption and probability assumption.

Singer [7] used a fuzzy set approach for fault tree and reliability analysis in which the relative frequencies of the basic events are considered as fuzzy numbers. Cheng and Mon [8] analyzed fuzzy system reliability analysis by use of interval of confidence. Through theoretical analysis and computational results, they have shown that their proposed approach is more general and straightforward compared to Singer [7].

Huang et al. [9] proposed a fault-tree analysis based on posbist reliability theory. This method has the advantages of evaluating the probability of failure in a system when historical data are scarce or the failure probability is extremely small. Chen [10] presented a new method for fuzzy system reliability analysis using fuzzy number arithmetic operations in which reliability of each component is considered as fuzzy number and used simplified fuzzy arithmetic operations rather than complicated interval fuzzy arithmetic operations of fuzzy numbers [8] or the complicated extended algebraic fuzzy numbers [7]. However, the result of evaluating fuzzy system reliability in [7], [8], [10] are approximated values. It is also noted that in [11], Mon and Cheng addressed an approach for system reliability having components of different failure distribution. They also used α-cut and interval arithmetic operations. In all papers mentioned above, sup-min convolution has been basically used for fuzzy arithmetic operations to evaluate fuzzy system reliability. Hong and Do [12] used T_w (the weakest t-norm) instead of ‘min’ for fuzzy arithmetic operations based on the sup-t-norm convolution where the reliability of each system component is represented by L–R type fuzzy number.

Fuzzy set theory has been shown to be a useful tool to handle such situations by attributing a degree to which a certain object belongs to a set. In real life, a person may assume that an object belongs to a set to a certain degree, but it is possible that he is not so sure about it. In other words, there may be a hesitation or uncertainty about the membership degree of x in A. In fuzzy set theory, there is no means to incorporate that hesitation in the membership degrees. A possible solution was to use intuitionistic fuzzy sets (IFS), defined by Atanassov [13].

The concept of an intuitionistic fuzzy set (IFS) can be viewed as an alternative approach to define a fuzzy set in cases where available information is not sufficient for the definition of an imprecise concept by means of a conventional fuzzy set. In fuzzy set, the degree of acceptance is considered only but IFS is characterized by a membership function (acceptance) and a non-membership function (rejection) so that the sum of both values is less than one [14]. In fact, Biswas [15] pointed out that there were situations where IFS theory is more appropriate to deal. Bustince and Burillo [16] proposed that the concept of vague sets coincides with that of intuitionistic fuzzy sets. Therefore, it is expected that intuitionistic fuzzy sets could be used to simulate any activities and processes requiring human expertise and knowledge, which are inevitably imprecise or not totally reliable. IFS theory has been applied in different areas such as logic programming [17], [18], decision-making problems [19], [20], in medical diagnosis [21], and pattern recognitions [22]. Chen [23] presented a method for analyzing the fuzzy system reliability based on triangular vague sets. Shu et al. [24] proposed a method for the failure analysis problem of printed circuit board assembly (PCBA) to compute the intuitionistic fuzzy fault-tree interval, traditional reliability, and the intuitionistic fuzzy reliability interval. Chang et al. [25] proposed a vague fault-tree analysis procedure to determine the weapon system's reliability. Cheng et al. [26] proposed an intuitionistic fault-tree analysis procedure to determine the intuitionistic fuzzy reliability interval for liquefied natural gas terminal emergency shutdown system. Their approach integrated experts’ knowledge and experience in terms of providing the possibility of failure of bottom events, and used a triangular intuitionistic fuzzy set to perform the calculation. Kumar et al. [27] extended the concept of fuzzy set by idea of triangular intuitionistic fuzzy set and proposed a general procedure to construct the membership function and non-membership function of the reliability function using intuitionistic fuzzy failure rate. Here the failure rate of the system is represented by a triangular intuitionistic fuzzy number. Lin et al. [28] proposed approximated fuzzy arithmetic operations based on weakest t-norm to evaluate repairable reliability. Chang and Cheng [29] proposed a general approach to evaluating the PCBA for components with different membership functions and obtained fault interval and reliability interval of system using α-cut of vague set and interval arithmetic operations of vague sets.

In this paper, the concept of fuzzy number is extended by the idea of intuitionistic fuzzy number and weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations are proposed to calculate fault interval and reliability interval of a system, where failure probabilities of all components are represented by different types of intuitionistic fuzzy numbers. Since weakest t-norm (T_w) based arithmetic give a smaller fuzziness in reliability. So using proposed approach, it has been shown that length of reliability interval (uncertainty about the reliability) decreases i.e. smaller fuzzy spread may obtain and obtained results are more confident. For numerical verification, a malfunction of PCBA [29] is presented as a numerical example.

The rest part of this paper is organized as follows. In Section “Intuitionistic fuzzy set theory”, we give the review of basic concepts related to intuitionistic fuzzy sets and its operations. Weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers are introduced in Section “Weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers”. In Section “Fuzzy system reliability analysis using approximate intuitionistic fuzzy arithmetic operations”, novel approach has been developed for the fuzzy reliability analysis of series system and parallel system using weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations where the reliabilities of components of the system are represented by different types of intuitionistic fuzzy numbers. Section “Proposed approach” proposes a novel approach to calculate fault interval and reliability interval of PCBA. In Section “Numerical verification and comparison”, PCBA is used to illustrate the proposed approach and some comparisons are discussed. The final section makes conclusions.

Section snippets

Intuitionistic fuzzy set theory

Fuzzy set theory was first introduced by Zadeh [30]. The membership function of a classical fuzzy set assigns a number from the unit interval to each element of the universe of discourse to indicate the degree of belongingness to the set under consideration. Let X be the universe of discourse defined by $X = {x_{1}, x_{2}, ..., x_{n}}$ . The grade of membership in a fuzzy set defined on X indicates the evidence for $x_{i} \in X$ but does not indicate the evidence against $x_{i} \in X .$ Atanassov [13] introduced the concept of an

Weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers

In this section, weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers are introduced. Lin [28] developed the approximate fuzzy arithmetic operators with the weakest t-norm arithmetic. In this paper, approximate intuitionistic fuzzy arithmetic operations with weakest t-norm (T_w) are developed on different types of intuitionistic fuzzy numbers to calculate fault interval and reliability interval of system. The Tw-based

Fuzzy system reliability analysis using approximate intuitionistic fuzzy arithmetic operations

In this section, a new method has been developed for the fuzzy reliability analysis of series system and parallel system using weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations where the reliabilities of components of the system are represented by different types of intuitionistic fuzzy numbers.

Proposed approach

Numerical verification and comparison

In this section, example of a PCBA is presented to illustrate the proposed approach. Also, the results obtained using proposed approach are compared with the experimental results obtained using the traditional probability reliability, Huang et al.’s [9], Chang et al.’s [25] and Chang et al. (interval arithmetic) [29] methods. The fault tree for ‘PCBA fault’ is shown in Fig. 5 and case data from the mid-sized manufacturing factory located in Hsinchu Science Park in Taiwan [24]. This research

Conclusion

In this paper, weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations are developed to analyze fuzzy system reliability, where probabilities of failure of all components are represented by different types of intuitionistic fuzzy numbers. The proposed approximate intuitionistic fuzzy arithmetic operations can obtain less fuzzy accumulation and efficient computing results than interval arithmetic. We also find that our proposed arithmetic operations can effectively

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Applying weakest t-norm based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers to evaluate reliability of PCBA fault

Highlights

Abstract

Introduction

Section snippets

Intuitionistic fuzzy set theory

Weakest t-norm (Tw) based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers

Fuzzy system reliability analysis using approximate intuitionistic fuzzy arithmetic operations

Proposed approach

Numerical verification and comparison

Conclusion

Microelectronic Reliab.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Reliability Engineering System Safety

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

Fuzzy Sets Syst.

J. Comp. Syst. Sci.

Fuzzy Sets Syst.

Microelectronics Reliab.

Expert Syst. Appl.

Appl. Math. Model.

Appl. Soft Comput.

Inf. Contr.

Reliability and Safety Under Fuzziness

Weakest t-norm (T_w) based approximate intuitionistic fuzzy arithmetic operations on different types of intuitionistic fuzzy numbers