Reliability Analysis of the Engineering Systems Using Intuitionistic Fuzzy Set Theory

Abstract

The present paper investigates the reliability analysis of industrial systems by using vague lambda-tau methodology in which information related to system components is uncertain and imprecise in nature. The uncertainties in the data are handled with the help of intuitionistic fuzzy set (IFS) theory rather than fuzzy set theory. Various reliability parameters are addressed for strengthening the analysis in terms of degree of acceptance and rejection of IFS. Performance as well as sensitivity analysis of the system parameter has been investigated for accessing the impact of taking wrong combinations on its performance. Finally results are compared with the existing traditional crisp and fuzzy methodologies results. The technique has been demonstrated through a case study of bleaching unit of a paper mill.

1. Introduction

Engineering systems have become complicating day by day, and rapidly increasing the cost of the equipment challenges the plant personnel or job analyst that to maintain the system performance so that to produce the desirable profit under a predetermined time. However the failure is an inevitable phenomenon in an industrial system. Also the effects of product failures range from those that cause minor nuisances to catastrophic failures involving loss of life and property. Therefore it is difficult for the system analyst to maintain the performance of the system for a longer period of time. For this, it is common knowledge that large amount of data is required in order to estimate more accurately the failure, error or repair rates. However, it is usually impossible to obtain such a large quantity of data from any particular plant. From this point of view, fuzzy reliability is a novel concept in system engineering as fuzzy set can easily capture subjective, uncertain, and ambiguous information. Thus based on that system reliability has been evaluated by using various fuzzy arithmetic and interval of confidence operations [1–10].

After the successful applications of the fuzzy set theory since 1970, several researchers are engaged in their extensions. Out of existence of several extensions, intuitionistic fuzzy set theory (IFS) defined by Atanassov [11] is one of the most successful extensions and has been found to be well suited for dealing with problems concerning vagueness. In fuzzy set theory, it is assumed that the acceptance and rejection grades of membership are complementary in nature. But during deciding the degree of membership of an object there is always a degree of hesitation between the membership functions. This feature is highlighted in IFS theory by defining the membership and nonmembership values of an object by means of a mathematical relation. Thus there exists some degree of the indeterministic situation with respect to any object. Also an idea of fuzzy set theory has been extended to a vague set by Gau and Buehrer [12] while Bustince and Burillo [13] showed that there is an equivalence relation between vague sets and the IFSs. Apart from that a lot of work has been done to develop and enrich the IFS theory given in [14–19] and their corresponding references in terms of reliability evaluation of series-parallel system, as almost all the above researchers have analyzed the system reliability for performing the system behavior. But it is quite understood that the other reliability parameters such as failure rate, mean time between failure, and availability are also affecting the system performance directly. Thus in order to access all these reliability parameters, Garg [20] proposed an approach for analyzing the system behavior of various reliability parameters in terms of degree of membership and nonmembership values at different levels of confidence. Recently, IFS theory has been applied in different areas such as logic programming [21], decision making problems [22], in medical diagnosis [23], pattern recognitions [24], and reliability optimization [25].

In this paper, a structural framework has been presented for analyzing the behavior of the bleaching unit of a paper mill using intuitionistic/vague lambda-tau methodology (VLTM) [20]. For this, data related to various components of the system are extracted in the form of component failure rates and repair times from the various sources such as historical records, reliability databases, and system reliability expert opinion, and triangular intuitionistic fuzzy numbers (TIFNs) have been used in the analysis for handling the uncertainty between the data. Sensitivity on system MTBF as well as performance analysis of availability index has been conducted for different combinations of other reliability parameters. Finally obtained results are compared to crisp as well as fuzzy technique results. The rest of the paper is organized as follow. Section 2 discusses the basic concepts of the IFS theory that have been used during the analysis. In Section 3, the methodology for conducting the analysis has been presented through an illustrative example of the paper industry. Section 4 contains the sensitivity as well as performance analysis for finding the critical component of the system. Finally, some concrete conclusions have been presented in Section 5.

2. Basic Concepts of Intuitionistic Fuzzy Set Theory

The basic concepts and definition which are used for analyzing the behavior of the system are discussed in this section.

2.1. Intuitionistic Fuzzy Set [12]

A vague or IFS 𝐴 in the universe 𝑈 is characterized by a membership function 𝜇𝐴∶𝑈→[0,1] and a nonmembership function 𝜈𝐴∶𝑈→[0,1] with the condition 𝜇𝐴𝐴(𝑥)+𝜈(𝑥)≤1, for all 𝑥∈𝑈, where 𝜇𝐴(𝑥) is considered as the lower bound for the degree of membership of 𝑥 in 𝐴 (based on evidences) and 𝜈𝐴(𝑥) is the lower bound of the negation (derived from the evidence against of 𝑥) of the membership of 𝑥 in 𝐴. Therefore, the degree of membership of 𝑥 in the vague set 𝐴 is characterized by the interval 𝐴𝐴[𝜇(𝑥),1−𝜈(𝑥)]. A typical illustration of vague set 𝐴 is shown in Figure 1.

Figure 1 

A vague set.

2.2. 𝛼-Cut of the Vague Set

𝛼-cut of an IFS, denoted by 𝐴^(𝛼), contains all those elements of the universe at which its membership function is at least degree 𝛼 and is defined as 𝐴^(𝛼)=𝐴(𝑥)𝐴𝑥∈𝑈∶𝜇≥𝛼,1−𝜈(𝑥)≥𝛼, (1) where 𝛼 is the parameter in the range 0≤𝛼≤1. Every 𝛼-cut of a fuzzy number is a closed interval and is represented as 𝐴^(𝛼)=[𝐴_𝐿^(𝛼),𝐴_𝑈^(𝛼)], where 𝐴_𝐿^(𝛼)(𝛼)𝐴(𝑥),𝐴=inf𝑥∈𝑅𝜇⩾𝛼_𝑈(𝛼)(𝛼)𝐴=sup𝑥∈𝑅∣𝜇(𝑥)⩾𝛼. (2)

2.3. Triangular Intuitionistic Fuzzy Number (TIFN)

An IFS denoted by 𝐴=⟨[(𝑎,𝑏,𝑐);𝜇,𝜈]⟩, where 𝑎,𝑏,𝑐∈ℝ is said to be TIFN if its membership function is given by 𝜇𝐴(𝑥)=𝜇𝑥−𝑎𝜇𝑏−𝑎,𝑎≤𝑥≤𝑏𝜇,𝑥=𝑏𝑐−𝑥𝐴(𝑥)=()𝑐−𝑏,𝑏≤𝑥≤𝑐0,otherwise,1−𝜈1−𝜈𝑥−𝑎()𝑏−𝑎,𝑎≤𝑥≤𝑏1−𝜈,𝑥=𝑏1−𝜈𝑐−𝑥𝑐−𝑏,𝑏≤𝑥≤𝑐0,otherwise, (3) where the parameter 𝑏 gives the modal values of 𝐴, that is, 𝜇𝐴(𝑏)=1, and 𝑎,𝑐 are the lower and upper bounds of available area for the evaluation data. A triangular IFS defined by the triplet (𝑎,𝑏,𝑐) with 𝛼-cuts is defined below and shown graphically in Figure 2: 𝐴^(𝛼)=𝑎^(𝛼),𝑐^(𝛼),𝐴^(1−𝛼)=𝑎^(1−𝛼),𝑐^(1−𝛼), (4) where 𝑎^(𝛼)𝛼=𝑎+𝜇_𝑖(),𝑎𝑏−𝑎_(1−𝛼)𝛼=𝑎+1−𝜈_𝑖(),𝑐𝑏−𝑎_(𝛼)𝛼=𝑐−𝜇_𝑖(),𝑐𝑐−𝑏_(1−𝛼)𝛼=𝑐−1−𝜈_𝑖(𝑐−𝑏). (5)

Figure 2 

𝛼-cut of IFS 𝐴.

The four basic arithmetic operations that is, addition, subtraction, multiplication, and division, on two triangular vague sets 𝐴=⟨(𝑎₁,𝑏₁,𝑐₁);𝜇₁,𝜈₁⟩ and 𝐵=⟨(𝑎₂,𝑏₂,𝑐₂);𝜇₂,𝜈₂⟩ with 𝜇=min(𝜇₁,𝜇₂) and 𝜈=min(𝜈₁,𝜈₂) are given below:(i)addition: 𝐴+𝐵=[𝑎₁+𝑎₂,𝑏₁+𝑏₂,𝑐₁+𝑐₂], (ii)subtraction: 𝐴−𝐵=[𝑎₁−𝑐₂,𝑏₁−𝑏₂,𝑐₁−𝑎₂],(iii)multiplication: 𝐴⋅𝐵=[𝑎₁𝑎₂,𝑏₁𝑏₂,𝑐₁𝑐₂],(iv)division: 𝐴÷𝐵=[𝑎₁/𝑐₂,𝑏₁/𝑏₂,𝑐₁/𝑎₂], if 0∉𝐵.

3. System Description and Methodology

The present work is based on the evaluation of the bleaching unit of a paper mill, a complex repairable industrial system. The system consists of three components/subsystems, namely, tank, filter, and washer. The function of the tank is to store the washed pulp from the washing system, and chlorine is passed to it for brightening the pulp. On the other hand, filter and washer have two units arranged in parallel in which entrapped gases during the washing of the pulp are removed from filter component and chlorine is from the washer component after washing the fibers. The interactions among the various working unit of the system are shown through its reliability block diagram (RBD) in Figure 3. The described study is valid for the steady state period only, that is, during which the failure rate of the system can be considered as constant, that is, exponentially distributed. Exponential distribution is easy to understand and implement and has been demonstrated in the literature to provide good approximations to machine failure distributions [26]. The following assumptions were taken into account for modeling: (i)both failure rates (𝜆_𝑖) and repair times (𝜏_𝑖) of each component are constant and obey exponent distributions;(ii)separate maintenance facility is available for each component; (iii)after repairs, the repaired component is considered as good as new; (iv)repair or replacement is carried out in case of failures only;(v)subsystem either in the working or in failed state, that is, operating or failed state.

Figure 3 

Reliability block diagram of bleaching system.

The procedural step of the methodology is given below.

Step 1 (information extraction phase). The methodology starts with the information extraction phase in which information in the form of system components’ failure rate (𝜆_𝑖’s) and repair times (𝜏_𝑖’s) is extracted from various sources such as historical records, reliability databases, and system reliability expert opinion and is integrated with the help of plant personnel. This information is given in Table 1. The mission time for analysing the reliability parameter is taken to be 10 hours.

Table 1 

Failure rate and repair time of the system.

Step 2 (formulate the membership and nonmembership functions). The fact that the extracted database on which reliability analysis depends is either out of date or collected under different operating and environmental conditions leads to the problem of uncertainty in the current failure rates and repair times. So to account for the uncertainties in the analysis, the obtained data are converted into the triangular intuitionistic fuzzy numbers with some known spread say ±15% (also at ±25% and ±50%) suggested by decision maker/design maintenance expert/system reliability analyst, as depicted in Figure 4. Based on these triangular input data, the expressions of their systems’ failure rate (𝜆_𝑠) and repair time (𝜏_𝑠) corresponding to membership and nonmembership functions are obtained for AND- and OR-transitions at different 𝛼-cuts using interval arithmetic operations on TIFNs. These interval expressions are given below.
For Membership Functions Expressions for AND-Transitions. Consider the following: 𝜆^(𝛼𝜇)_𝑠=^𝑛_𝑖=1𝜆_𝑖2−𝜆_𝑖1𝛼_𝜇𝜇_𝑖+𝜆_𝑖1⋅^𝑛_𝑗=1^𝑛_{𝑖=1𝑖≠𝑗}𝜏_𝑖2−𝜏_𝑖1𝛼_𝜇𝜇_𝑖+𝜏_𝑖1,_𝑛𝑖=1−𝜆_𝑖3−𝜆_𝑖2𝛼_𝜇𝜇_𝑖+𝜆_𝑖3⋅^𝑛_𝑗=1^𝑛_{𝑖=1𝑖≠𝑗}−𝜏_𝑖3−𝜏_𝑖2𝛼_𝜇𝜇_𝑖+𝜏_𝑖3,𝜏^(𝛼𝜇)_𝑠=∏^𝑛_𝑖=1𝜏_𝑖2−𝜏_𝑖1𝛼_𝜇/𝜇_𝑖+𝜏_𝑖1∑^𝑛_𝑗=1∏^𝑛_{𝑖=1𝑖≠𝑗}−𝜏_𝑖3−𝜏_𝑖2𝛼_𝜇/𝜇_𝑖+𝜏_𝑖3,∏^𝑛_𝑖=1−𝜏_𝑖3−𝜏_𝑖2𝛼_𝜇/𝜇_𝑖+𝜏_𝑖3∑^𝑛_𝑗=1∏^𝑛_{𝑖=1𝑖≠𝑗}𝜏_𝑖2−𝜏_𝑖1𝛼_𝜇/𝜇_𝑖+𝜏_𝑖1. (6) Expressions for OR-Transitions. Consider the following: 𝜆^(𝛼𝜇)_𝑠=^𝑛_𝑖=1𝜆_𝑖2−𝜆_𝑖1𝛼_𝜇𝜇_𝑖+𝜆_𝑖1,_𝑛𝑖=1−𝜆_𝑖3−𝜆_𝑖2𝛼_𝜇𝜇_𝑖+𝜆_𝑖3,𝜏^(𝛼𝜇)_𝑠=^𝑛_𝑖=1𝜆_𝑖2−𝜆_𝑖1𝛼_𝜇𝜇_𝑖+𝜆_𝑖1⋅𝜏_𝑖2−𝜏_𝑖1𝛼_𝜇𝜇_𝑖+𝜏_𝑖1×^𝑛_𝑖=1−𝜆_𝑖3−𝜆_𝑖2𝛼_𝜇𝜇_𝑖+𝜆_𝑖3⁻¹,_𝑛𝑖=1−𝜆_𝑖3−𝜆_𝑖2𝛼_𝜇𝜇_𝑖+𝜆_𝑖3⋅−𝜏_𝑖3−𝜏_𝑖2𝛼_𝜇𝜇_𝑖+𝜏_𝑖3×^𝑛_𝑖=1𝜆_𝑖2−𝜆_𝑖1𝛼_𝜇𝜇_𝑖+𝜆_𝑖1⁻¹. (7)
For Nonmembership Functions Expressions for AND-Transitions. Consider the following: 𝜆^{(𝛼1−𝜈)}_𝑠=^𝑛_𝑖=1𝜆_𝑖2−𝜆_𝑖1𝛼_𝜈1−𝜈_𝑖+𝜆_𝑖1⋅^𝑛_𝑗=1^𝑛_{𝑖=1𝑖≠𝑗}𝜏_𝑖2−𝜏_𝑖1𝛼_𝜈1−𝜈_𝑖+𝜏_𝑖1,_𝑛𝑖=1−𝜆_𝑖3−𝜆_𝑖2𝛼_𝜈1−𝜈_𝑖+𝜆_𝑖3⋅^𝑛_𝑗=1^𝑛_{𝑖=1𝑖≠𝑗}−𝜏_𝑖3−𝜏_𝑖2𝛼_𝜈1−𝜈_𝑖+𝜏_𝑖3,𝜏^{(𝛼1−𝜈)}_𝑠=∏^𝑛_𝑖=1𝜏_𝑖2−𝜏_𝑖1𝛼_𝜈/1−𝜈_𝑖+𝜏_𝑖1∑^𝑛_𝑗=1∏^𝑛_{𝑖=1𝑖≠𝑗}−𝜏_𝑖3−𝜏_𝑖2𝛼_𝜈/1−𝜈_𝑖+𝜏_𝑖3,∏^𝑛_𝑖=1−𝜏_𝑖3−𝜏_𝑖2𝛼_𝜈/1−𝜈_𝑖+𝜏_𝑖3∑^𝑛_𝑗=1∏^𝑛_{𝑖=1𝑖≠𝑗}𝜏_𝑖2−𝜏_𝑖1𝛼_𝜈/1−𝜈_𝑖+𝜏_𝑖1. (8) Expressions for OR-Transitions. Consider the following: 𝜆^{(𝛼1−𝜈)}_𝑠=^𝑛_𝑖=1𝜆_𝑖2−𝜆_𝑖1𝛼_𝜈1−𝜈_𝑖+𝜆_𝑖1,_𝑛𝑖=1−𝜆_𝑖3−𝜆_𝑖2𝛼_𝜈1−𝜈_𝑖+𝜆_𝑖3,𝜏^{(𝛼1−𝜈)}_𝑠=^𝑛_𝑖=1𝜆_𝑖2−𝜆_𝑖1𝛼_𝜈1−𝜈_𝑖+𝜆_𝑖1⋅𝜏_𝑖2−𝜏_𝑖1𝛼_𝜈1−𝜈_𝑖+𝜏_𝑖1×^𝑛_𝑖=1−𝜆_𝑖3−𝜆_𝑖2𝛼_𝜈1−𝜈_𝑖+𝜆_𝑖3⁻¹,_𝑛𝑖=1−𝜆_𝑖3−𝜆_𝑖2𝛼_𝜈1−𝜈_𝑖+𝜆_𝑖3⋅−𝜏_𝑖3−𝜏_𝑖2𝛼_𝜈1−𝜈_𝑖+𝜏_𝑖3×^𝑛_𝑖=1𝜆_𝑖2−𝜆_𝑖1𝛼_𝜈1−𝜈_𝑖+𝜆_𝑖1⁻¹. (9)

(a) Membership functions of 𝜆𝑖

(b) Membership functions of 𝜏𝑖

Figure 4 

Input for the 𝑖th component of the system.

Step 3 (calculate various reliability parameters). Based on these expressions of the system 𝜆_𝑠 and 𝜏_𝑠, the various reliability parameters of the system, which depicts the behavior of the system, are analyzed at various membership grades along with left and right spreads. The expression of these reliability parameters is given in Table 2. The results obtained corresponding to these reliability indices are shown in Figure 5 that corresponds to ±15% spread along with the fuzzy lambda-tau (FLT) and traditional (crisp) methodologies results which indicate that results obtained from VLTM are in between them. That is, VLTM technique acts as a bridge between the Markov process (crisp values) and FLT technique. From these plots the following conclusions are drawn. (i)The values of all reliability indices computed by using traditional methods (crisp) are independent of the degree of confidence level (𝛼). It shows that while obtaining the results by these methods attention has not been paid to the uncertainties in the data. Thus this methodology is not practically sound, and hence their results will be suitable only for a system with precise data. (ii)The results computed by the FLT technique are presented in figure with FLT legend. In their technique there is a zero degree of hesitation between the membership functions. Moreover the domain of confidence level is taken to be one, that is, 𝛼=1. Therefore the results computed by FLT methodology are not that practical. (iii)The proposed approach provides improvement over the above shortcoming by considering 0.2 degree of hesitation between the degree of membership and nonmembership functions shown by solid and dotted lines respectively. In the proposed approach the domain of confidence level is clearly 𝛼≤0.8. In this, if 𝛾₁ is the degree of membership function for some reliability index and 𝛾₂ is the degree for the corresponding nonmembership function then there is 1−𝛾₁−𝛾₂ degree of hesitation between the degree of membership functions. For instance, corresponding to availability value 0.9975027, there is 0.3 degree of interminancy between the membership functions of the availability 0.9975027. Thus, the proposed technique is beneficial for the system analyst for analyzing the behavior of the system in more flexible ways in lesser time than the other existing techniques.

Table 2 

Some reliability parameters.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5 

Reliability indices plot at ±15% spread.

Step 4 (defuzzified values of the reliability parameters). In order to implement these results by the system analyst or plant personnel then it is necessary to convert these fuzzified results into the crisp or binary nature. For this, defuzzification methods, which convert fuzzified output into crisp value, are used, depending on the application. Here center of gravity method has been used for defuzzification because it gives the mean of the data [27]. Based on that, the crispness and defuzzified values of the reliability parameters at different spreads (±15%, ±25%, and ±50%) are computed and tabulated in Table 3. It follows from the tables that results computed by proposing an approach follows the same trend as the FLT results, and hence results are conservative in nature.

Table 3 

Defuzzified values of the reliability parameters.

4. Sensitivity and Performance Analyses

To analyze the effect of different reliability parameters on its performance, the following analysis have been done.

4.1. Sensitivity Analysis

To analyze the impact of change in values of reliability indices on to the system’s behavior, behavioral plots have been plotted for different combinations of reliability indices and are shown in Figure 6. Throughout the combinations, ranges of repair time and ENOF are fixed and have been varied, along the 𝑥- and 𝑦-axes, respectively, in the range computed by their membership functions (Figures 5(b) and 5(d)) at cut level 𝛼=0. The effects on MTBF by taking different combinations of the remaining parameters (reliability, failure rate, and availability) are computed and have been shown along the 𝑧-axis. For instance, in the first three combinations in these plots, reliability and availability parameters are set to be 0.9917 and 0.9965, respectively, while failure rate changes from 0.548×10⁻³ to 0.765×10⁻³ and further to 0.834×10⁻³. It is observed for this combination that the prediction ranges of MTBF shown in Figures 6(a)–6(c) are in the range 1781.8688, 1277.4099, and 1172.0125, respectively. The corresponding plots show that as the failure rate of the system increases then, for the prescribed ranges and values of the other indices, MTBF of the system decreases exponentially as shown in Table 4. This suggests that a slight change in system’s failure rate may change its MTBF largely and consequently behavior of the system. Similarly, for other combinations, behavior has been analyzed and corresponding results have been shown in Figures 6(a)–6(i). These plots will be beneficial for a plant maintenance engineer for preserving particular index and to analyze the impact of other indices on to the system’s behavior. Thus, based on the behavioral plots and corresponding table, the system manager can analyze the critical behavior of the system and plan for suitable maintenance.

Table 4 

Variation of the MTBF parameter by changing other reliability parameters.

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

Figure 6 

Effect of the variation of reliability parameters on system MTBF.

4.2. Performance Analysis

As the performance of the system is decreasing with the passage of time so it is necessary for the decision makers or system analyst that initial condition of the system should be changed for maintaining the performance of the system. For this, suitable maintenance strategy should be adopted for it. But the problem to the system analyst is that it is difficult to find the component measures of the system on which more attention should be given to enhance the performance of the system. For this, an analysis has been done on the availability index by varying the system component parameters, that is, failure rate and repair time. In their analysis, failure rate and repair time of each component of the system are varied simultaneously, and keeping the other parameter remain fixed, the corresponding effect on availability index has been shown graphically in Figure 7. This figure contains three subplots corresponding to three main components of the system while their corresponding ranges are summarized in Table 5. From this analysis, it has been seen that bleaching tank components produce a marginal effect on system performance while washer produces significantly. Thus it has been concluded from their analysis and corresponding table that, for increasing the performance of the system, it is necessary that special attention should be done on a washer component than filter and bleaching tank. Therefore importance measures for maintaining the system components and for saving money, manpower, and time are given as per preferential order, washer, filter, and bleaching tank.

Table 5 

Effect of simultaneous variations of system’s components’ failure and repair times on its availability index.

(a)

(b)

(c)

Figure 7 

Effect of varying components parameters on its performance.

5. Conclusion

The objective of this paper is to predict the behavior of an industrial system under imprecise and vague environment. To handle the uncertainty in the data, IFS theory has been used rather than fuzzy set theory as it offers additional information about the degree of hesitation. The conversion of the input data into triangular intuitionistic fuzzy numbers will greatly increase the relevance of the study. Six reliability parameters of the system have been computed in the terms of membership functions and their results with crisp and FLT results, and it was found that the proposed results act as a bridge between them. The crisp and defuzzified values of the indices at different levels of uncertainties will help the maintenance engineer for establishing a suitable maintenance action. Sensitivity as well as performance analysis has been done in analyzing the most critical component of the system; based on that system analyst or plant personnel may change their target goals and maintenance actions. From the results it has been concluded that, for saving money and time, necessary actions should be done in the components washer, filter, and bleaching tank as per preferential order. Apart from that system analyst may conduct cost benefit analysis from the computed results by considering the manufacturing and repairing cost for achieving the goals of higher profit at reasonable cost.