1 Introduction

Failure mode and effect analysis (FMEA) is an engineering technique used in wide range of industries to identify the potential failure modes, their causes and its consequences on the system and its surrounding (Stamatis 2003). It is considered as a powerful tool to improve the reliability and safety performance of wide variety of systems. FMEA is carried out by forming a team of experts from different disciplines associated with the system under consideration. The team generally contains maintenance experts, safety experts, instrumentation experts, design experts …etc. All the possible failure modes that can happen to the system and the effects of such failure modes are analyzed by the experts. Based on the analysis report, risk management decisions are taken to improve the reliability of the system to avoid future failures. Failure mode effect and criticality analysis (FMECA) is an extension of FMEA where the potential failure modes of components are prioritized based on the risk associated to the component.

To calculate risk, FMECA uses three risk parameters such as Occurrence, severity and non-detection. Occurrence value represents the probability of occurrence of failure modes; which illustrates the number of failures predicted in the entire life of a component. Severity value represents how severe the consequences if a particular failure occurs. Non-detection number represents the level of detection of failure mode; if a failure or future failure is identified or predicted at early stage, preventive measures can be taken to minimize its effects. By multiplying these three risk parameters, a numerical value called Risk priority number (RPN) is obtained for each potential failure modes. Finally, the failure modes are prioritized based on the RPN, where the largest number holds the highest rank. Due to its effectiveness, FMECA has been extensively used in wide variety industries including aerospace, manufacturing, chemical, medical, marine and offshore, food, nuclear, software…etc.

Even though it is broadly used, conventional FMECA technique has been widely criticized for several reasons. Many researchers have questioned the reliability of traditional FMEA and the use of conventional risk priority number (RPN) for failure mode prioritization. The reasons are as follows; (a) The weightage given to the three risk factors are same, but in real world scenario, the weightage may vary. (b) Exact determination of the value each risk factors for each failure mode are usually considered difficult for risk assessors; the risk factors which are elicited from expert’s knowledge is usually vague and uncertain; and conventional FMECA is incapable of handling uncertainty and ambiguity during the risk assessment procedure. (c) RPN considers only three risk factors (O, S&D), other risk factors such as maintenance time, repair cost, cost of replacement of equipment may be important as well. (d) There is no genuine proof to shows that the best way to compute the RPN score is by multiplying the three risk factors(O*S*D); while doing simple multiplication, the importance weightage given to all three risk factors are equal. But in real world scenario, the weightage may vary. (e) Different combinations of risk factors have high possibility to produce the same RPN score, however the hidden risk combinations might be totally different. (f) The experts prefer to express their opinions in qualitative terms rather than numbers used in conventional RPN method. (g) From the multiplication of three risk factors, only 120 numbers/combinations out of 1000 combinations can be formed; that means, holes are present (Fig. 1) between successive RPNs in multiplicative scales. (h) The RPN values are not continuous, rather discrete. (i) Even a small change in the value of one of the risk factors would lead to diverse effect on the RPN scores.

Fig. 1
figure 1

Traditional RPN values and its corresponding frequencies

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However, in recent years, many extensions of risk priority techniques are proposed in different kinds of areas in order to cope with the shortages of conventional FMECA. To minimize the drawbacks and improve the reliability of risk assessment, advanced methods of FMECA such as fussy set theory based FMECA (Fuzzy-FMECA) was introduced. Fuzzy methods are useful to model real-world scenarios where the fuzzy RPN method is able to handle vague and uncertain inputs. Experts prefer to give their opinions in qualitative manner, fuzzy FMECA is very effective for quantifying those qualitative opinions. Since the FRPN is continuous, the problems of multiplicative scales and duplication of the RPNs can be avoided.

In fuzzy FMECA, fuzzy input values obtained as a result of expert elicitation is used. The risk factors O, S &D are classified on a 10-point scale suggested by the standard IEC 60812. Based on the standard, experts choose their opinions for each failure mode. To minimize the time required for performing risk assessment through fuzzy FMECA method, software assistance such as MATLAB fuzzy tool box can be used. Using MATLAB fuzzy toolbox, based on the expert opinion, a fuzzy if–then rule base is constructed to link the inputs (O, S &D ranking) and output (FRPN). O, S& D are the inputs and the output is obtained as a modified form of RPN called fuzzy risk priority number (FRPN), a defuzzified crisp value, which is used to rank the failure modes in FRPN.

To improve the reliability even more by accounting the relative importance of OS&D and to take the weightage factor into consideration, grey theory can be used in combination with the fuzzy set theory. Grey theory consists of three steps, that’s are (a) defuzzification of linguistic variables, (b) calculation of grey relational coefficients and (c) prioritization of failure modes using grey relation. In grey theory, the fuzzy input values obtained as a result of expert elicitation is defuzzified into crisp values in the initial stage itself. Using an equation, Grey relational coefficients are obtained for each crisp values. Finally, the relative importance is taken into consideration and a fraction called the grey relation is obtained. The grey relation of each failure mode is ranked in order that the least value has the highest rank.

The purpose of this research is to reach a comparative case study between conventional FMECA, Fuzzy FMECA and Grey FMECA. Using a comparative case study of fuzzy FMECA and Grey FMECA, a much reliable decision-making tool is proposed. By comparing the results, the accuracy of decision making will be better than the Conventional FMECA method. Use of MATLAB fuzzy toolbox as software assistance can reduce the risk assessment time of and makes the process easier. This paper demonstrates the application of the proposed methodology by using a case study of atmospheric ammonia storage tank and its associated facility.

2 Literature review

FMEA was introduced in the 1960s by the US military for the reliability improvement in the aerospace industry as a powerful tool to assess the potential failure modes in a system (Bowles and Peláez 1995). Since then, it was widely used across the industries to improve the quality/reliability and safety of systems/products, particularly aerospace, automotive, nuclear, electronics, chemical, mechanical and medical industries (Chang et al. 2001; Sharma et al. 2005; Ying-ming Wang et al. 2009a, b). The FMECA method emphasizes on individual components and their failure modes, which is an efficient technique for ranking the criticality of each potential failure mode (Deng et al. 2015). In FMECA, failure modes having higher ranks are giving higher priority, and is useful in preventing the undesirable outcomes in complex systems (Sankar and Prabhu 2001). While performing FMECA, the failure modes are ranked based on a number called Risk Priority Number (RPN), which is the direct product of three risk parameters called Occurrence, Severity and Non-Detection (Scipioni et al. 2002). But the RPN obtained in the traditional FMECA methods does not account the possible weighing for the importance of the three risk parameters (Pillay and Wang 2003). Also, the difficulty of using numerical values to evaluate the occurrence of the potential failure modes encouraged the development of other techniques which supports linguistic assessment of failure modes (Xu et al. 2002).

The fuzzy logic theory was introduced by Zadeh (1965) in the 1960s and it is assumed to be a precise technique for handling the failure data and has a wide range of application while performing FMECA. (Chanamool and Naenna 2016) applied fussy set theory to improve the decision-making process in an emergency department of a hospital. As a result, several failure modes having an identical RPN value from different combinations of O, S and D could be differentiated. (Rachieru et al. 2015) evaluated risks of failure modes of CNC lathes using fuzzy logic toolbox in Matlab, and made comparisons with the results of traditional RPN method. (Renjith et al. 2018) used fuzzy logic toolbox in Matlab to prioritize the failure modes in an LNG storage facility. The analysis was performed by using fuzzy linguistic variables for O, S and D and a fuzzy if–then rule base to interconnect these variables to reach FRPN. This method could rectify some of the limitations of traditional RPN, such as identical RPN for different sets of failure modes. (Sankar and Prabhu 2001) suggested a Risk Priority Rank (RPR) that indicates all the 1000 possible combinations of severity-occurrence-detection, followed by an “if–then” rule base to prioritize the risks.

A modified FMECA was developed by (Bevilacqua et al. 2000) that included a weighted sum of six variables (cost of maintenance, safety, failure frequency, downtime length, operating conditions and machine importance for the process). A Multi-Attribute Failure Mode Analysis (MAFMA) approach was developed by (Braglia 2000) based on the Analytic Hierarchy Process (AHP) technique that takes into account the risk factors O, S, D and the expected cost due to failures as decisional criteria, the possible causes of failure as decisional alternatives and the selection of causes of failure as decisional goal. For better reprioritization of FMECA, (Seyed-Hosseini et al. 2006) suggested a new methodology called decision making trial and evaluation laboratory (DEMATEL). (Castiglia et al. 2015) proposed Fuzzy HEART (Human Error Reduction and Assessment Technique) methodology, a versatile tool to support safety studies in various industrial fields have been used to calculate Human error occurrence probabilities. (Abdelgawad and Fayek 2010) proposed a method of risk management in the construction industry using combined fuzzy FMEA and fuzzy Analytical Hierarchy Process (FAHP). (Gul et al. 2020) proposed an improved failure mode and effect analysis (FMEA) with fuzzy Bayesian Network (FBN) and Fuzzy Best Worth Method (FBWM) to assess failures in plastic production. (Kalathil et al 2020) proposed a Dempster Shafer theory based FMECA on LNG regasification plant. By far the most popular technique used to improve the traditional FMEA is by utilizing Fuzzy Inference Systems (FIS), also known as fuzzy controllers. The FIS approach to FMEA is prompted because it is logical to think of the O, S and D ratings in fuzzy linguistic terms such as ‘likely’, ‘very likely’, etc. (Renjith et al. 2018) proposed a fuzzy inference system FMECA on LNG regasification plant.

Grey theory is also widely used in improved RPN analysis, which is popular in decision making field and it has been widely applied to fields such as optimization, engineering, economy, history, geography, traffic management. Deng first proposed grey system theory in 1982 and made great improvements in decision making process (Ju-Long 1982). The grey relational analysis was utilized to determine the risk priorities of failure modes using a metric called grey relational coefficient. (Chang et al. 2001) applied grey theory to develop FMEA framework for determining a risk priority number by assigning relative weighing coefficient to each failure modes. (Wang et al. 2009a, b) applied grey theory to risk analyzing for steam turbine systemin power plant to eliminate deviation in traditional RPN analysis caused by excessive classified levels. (Pillay and Wang 2003) improved traditional FMEA and introduced fuzzy rule base approach, and grey theory was also applied to make analysis and the advantages of the improved method were illustrated through an example of fishing vessel. (Moon et al. 2013) demonstrated a quantitative method to facilitate service process design for identifying failure modes that are affected to service operations. (Liu et al. 2011) proposed a modified FMEA method based on fuzzy evidential reasoning approach and grey theory, which can prioritize failure modes under different types of uncertainties.(Wang et al. 2019) applied a combined method of fuzzy set theory and grey theory to prioritize the failure modes in the spindle system of CNC lathes. Grey theory and the results were used for inspection, decision making and maintenance of different equipment on tankers and suggested a method based on this for optimizing the maintenance resource by (Zhou and Thai 2016). (Sun et al. 2018) applied the Grey relational analysis to the Hesitant Fuzzy Sets (HFS) to deal with the pattern recognition problems. (Chen and Deng 2018) proposed a method of FMEA using Dempster-Shafer Evidence theory and Grey relational Projection Method (GRPM).

Using fuzzy TOPSIS method, a dynamic maintenance planning framework and its application in a food industry is proposed by (Selim et al. 2016).(Yazdi et al. 2017) and (Daneshvar et al. 2020) proposed Fuzzy FMEA methods to reduce the increasing accidents related to aircraft landing gear in Iranian airlines. (Yazdi 2019) applied and interactive analyzing of failure modes and its corrective actions in a gas refinery pipeline under construction. (Kengpol and Tuammee 2016) conducted an FMEA study in logistics industry to optimize green logistics route by considering weight from the user. (Dargahi et al. 2016) applied FMEA for environmental risk assessment in ore complex on wildlife habitats. (Liu et al. 2014) proposed a method of performing FMEA using a combination of D numbers and Grey relational projection method. (Lin et al. 2014) performed human reliability analysis for medical devices using fuzzy linguistic theory. (Yazdi et al. 2020) performed a Pythagorean fuzzy DEMATEL analysis in an offshore oil platform. To mitigate the welding defect risk, (Shoar et al. 2017) proposed a stochastic logic based methos for QRA using Monte Carlo simulation and fuzzy set theory. (Bian et al. 2018) proposed an FMEA using a combined method of D numbers and TOPIS and applied the method for risk assessment of the rotor blades of an aircraft turbine.

In this study, the failure modes of an atmospheric ammonia storage tank and its associated facility is identified and we try to demonstrates the application of the proposed methodology. Several researchers identified the various failure modes in an atmospheric ammonia storage facility using different tools. (Roy et al. 2015) conducted dynamic failure assessment of an ammonia storage unit using Event Tree Analysis (ETA). (Nemati and Heidary 2012) done a risk analysis of cryogenic ammonia storage tank using fault tree method.

3 Materials and methods

3.1 Traditional FMECA

To conduct FMECA study, the system under study is divided into subsystems/components. A team of experts are formed. The experts are from different areas related to the system/process under study. The expert team may consist of the process engineer, design engineer, maintenance engineer, electrical and instrumentation engineer, safety manager, a senior work supervisor or even an operator. FMECA consists of two parts that’s are (1) Identification of failure modes and their causes and consequences (known as FMEA) and (2) Prioritization of the failure modes using risk parameters (known as criticality analysis) by (Bowles and Peláez 1995). The conventional FMECA is performed by calculating a value called Risk Priority Number (RPN), which is calculated by the multiplication of three risk factors such as Occurrence, Severity and Detection (Eq. 1). Occurrence is the probability of occurrence(O) of the failure modes, Severity(S) is the degree of the consequence that can happen if a specific failure mode occurs, Non-detection(D) is the probability of not detecting the failure mode if it happened.

$${\text{RPN}} = {\text{O}} \times {\text{S}} \times {\text{D}}$$
(1)

The failure modes which have the highest RPN is the most critical. The failure modes are arranged in which the highest number holds the first position.

The following 5 steps are used for performing the conventional FMECA,

  1. 1.

    Selection of the process

This includes all the parts associated with the product. If necessary, the entire system will be divided into sub-assemblies or sub parts.

  1. 2.

    Forming a multi-disciplinary team for evaluation of the failure modes.

  2. 3.

    Collection of all the information available for the process under study

This includes collecting the past failure data, experienced worker’s comments, maintenance history

  1. 4.

    Conducting failure mode effect and criticality analysis

    1. a.

      Identification of each failure modes.

    2. b.

      Identification of the potential effect of each failure mode

    3. c.

      Taking expert elicitation on each failure mode

    4. d.

      Finding out RPN of each failure mode

    5. e.

      Ranking the failure modes based on the values of RPN

  2. 5.

    Taking corrective actions and look for any further improvement.

3.1.1 Limitations of conventional FMECA

As mentioned in the literature, even though traditional FMECA is used for wide range of applications, it has a lot of drawbacks. Some of the major limitations of the RPN evaluation by traditional FMECA are as follows (Liu et al. 2013; Sankar and Prabhu 2001; Seyed-Hosseini et al. 2006).

i. Different combinations of O, S and D may produce exactly the same value of RPN, but their hidden risk implications may be totally different. For example, two different events with the values of 2, 3, 2 and 4, 1, 3 for O, S and D, respectively, have the same RPN value of 12. However, the hidden risk implications of the two events may not necessarily be the same.

  1. i.

    The three factors are difficult to be precisely estimated. Much information in FMEA can be expressed in a linguistic way such as ‘Likely’, ‘Important’ or ‘Very High’ and ‘Soon’

  2. ii.

    The mathematical formula for calculating RPN is questionable.

  3. iii.

    The relative importance among O, S&D are not taken into consideration. The three risk factors are assumed to be equally important. This may not be the case when considering a practical application of FMEA.

  4. iv.

    The RPN considers only three factors in terms of safety and neglects other important factors such as economy and down time factors.

  5. v.

    The RPN is not a continuous function so that the meaning of the differences among the RPN values cause some interpretation problems.

  6. vi.

    The RPN is the product of O, S and D, and even a very small variation in any one of the parameters lead to a huge difference in RPN.

A lot of research work has been carried out to minimize the drawbacks of the traditional FMECA in the past decades and some significant efforts have been made by some of the researchers to improve the reliability of FMECA.

3.2 Application of fuzzy set theory in FMECA

The proposed method is derived from the extension of the method proposed by (Markowski and Mannan 2008). The fuzzy logic is often useful for performing FMECA using the expert’s information that are sometimes uncertain or vague. Compared to the traditional FMECA, the fuzzy inference technique results in a more realistic and practical reflection of a real scenario. The main components associated with fuzzy FMEA are fuzzification, fuzzy rule base, fuzzy inference system, and defuzzification. Figure 2 shows the flow chart of the fuzzy FMEA approach Tables 1, 2.

  1. 1.

    The fuzzifier maps crisp input into fuzzy sets, which means fuzzification of each risk factors (Occurrence, Severity, Non-detection).

  2. 2.

    The inference engine of the Fuzzy Logic System maps input fuzzy sets by means of a set of rules into fuzzy output sets. The fuzzy inference engine handles the way in which the rules are combined. These set of rules are defined by an expert team using if–then combination statements.

  3. 3.

    Defuzzification is the process of weighing and averaging the outputs from all the defined individual fuzzy rules into one single output value. The obtained output will be a crisp value.

Fig. 2
figure 2

Schematic diagram of a simple Fuzzy Logic System

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As like in the traditional FMECA, an expert team is formed and the expert team comprises of the process engineer, design engineer, maintenance engineer, electrical and instrumentation engineer, safety manager, a senior work supervisor, or even an operator. The expert team members use linguistic variables (such as likely, moderate, high, etc.) as inputs to describe the three risk parameters (O, S, and D) that characterizes the effect of each failure mode. The crisp input data are commonly measured on a 10- point scale suggested by (IEC 2006) Table 3,4,5. Using the membership functions, the crisp inputs are “fuzzified” to determine the degree of membership in each input class. Using a linguistic rule base and fuzzy logic operations, these resulting fuzzy inputs are evaluated in a fuzzy inference engine to yield a “fuzzy output”. The fuzzy inference engine has a predefined rule base made by the expert team. The rule base narrates the riskiness of the failure modes for each combination of input variables. These rules are expressed in linguistic terms rather than in numerical terms, and they are often expressed as “if–then” rules. The importance of fuzzy rule base arises from the fact that human expertise and knowledge can often be represented in the form of fuzzy rules. An example of rule base is given below:

“If occurrence is high (8), severity is moderate (5) and non-detection is high (7), then FRPN is high (9)”

To transfer the qualitative rules to quantitative results, Mamdani min/max inference algorithm is used (input method: min; aggregate method: max) Finally, the fuzzy output is “defuzzified” to obtain the Fuzzy Risk Priority Number (FRPN) using centroid method. The obtained crisp output (FRPN) is used for prioritization of failure modes.

3.3 Application of grey theory in FMECA

Grey theory is used to define relationships between discrete quantitative and qualitative series and solve a decision-making problem which is characterized by incomplete and partially known information. The grey theory applied to FMECA is used as another tool to validate the results obtained by the fuzzy FMECA method. So that the accuracy of the method will be increased. The application of Grey theory to Fuzzy FMECA contains three steps, that’s are:

  1. 1.

    Defuzzification of linguistic variables

  2. 2.

    Establishment of comparative series

  3. 3.

    Grey relational coefficient

3.3.1 Defuzzification of the linguistic variables

The linguistic terms of the risk factors are defuzzified to produce crisp number of fuzzy sets. (Chen and Klein 1997) have proposed an easy defuzzification method for obtaining the crisp number of a fuzzy set as shown by the Eq. (2) below.

$$K\left( x \right) = \frac{{\mathop \sum \nolimits_{i = 0}^{n} \left( {b_{i} - c} \right)}}{{\mathop \sum \nolimits_{i = 0}^{n} \left( {b_{i} - c} \right) - \mathop \sum \nolimits_{i = 0}^{n} \left( {a_{i} - d} \right)}}$$
(2)

where K(x) is the defuzzified crisp number, n is the maximum number of membership function levels. The values a0 and b0 are rating values at the extreme limits of the linguistic term when the membership function is 0. The values of ai and bi are the rating values of the membership term when the membership function is 1. The values of c and d are the minimum and maximum values of the linguistic variable scale, and the values of c and d remain same for the defuzzification of all the linguistic terms. (Pillay and Wang 2003) gave the pictorial representation and procedure.

Fig. 3
figure 3

Defuzzification of linguistic terms

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For example, consider the linguistic term moderate as shown in Fig. 3. The linguistic value moderate can be de-fuzzified to produce a crisp value.

$$K\left( x \right) = \frac{{\left[ {b_{0} - c} \right] + \left[ {b_{1} - c} \right]}}{{\left\{ {\left[ {b_{0} - c} \right] + \left[ {b_{1} - c} \right]} \right\} - \left\{ {\left[ {a_{0} - d} \right] + \left[ {a_{1} - d} \right]} \right\}}}$$
$$K\left( x \right) = \frac{{\left[ {7 - 0} \right] + \left[ {5.5 - 0} \right]}}{{\left\{ {\left[ {7 - 0} \right] + \left[ {5.5 - 0} \right]} \right\} - \left\{ {\left[ {4 - 10} \right] + \left[ {5.5 - 10} \right]} \right\}}}$$
$$K\left( x \right) = 0.521$$

3.3.2 Establishment of comparative series

In this step, an \(m\times n\) matrix is made, where the number of columns represents the number of risk parameters (O, S&D) taken under study and the number of rows represents the number of failure modes taken under study. The matrix is filled with crisp numbers made by the defuzzified values of linguistic variables for each risk parameter for each failure mode.

$$x = \left[ {\begin{array}{*{20}c} {x_{1} } \\ {x_{2} } \\ \vdots \\ {x_{n} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {x_{1} \left( 1 \right)} & {x_{1} \left( 2 \right)} & {x_{1} \left( 3 \right)} \\ {x_{2} \left( 1 \right)} & {x_{2} \left( 2 \right)} & {x_{2} \left( 3 \right)} \\ \vdots & \vdots & \vdots \\ {x_{n} \left( 1 \right)} & {x_{n} \left( 2 \right)} & {x_{n} \left( 3 \right)} \\ \end{array} } \right]$$

The value of xi(k) represents the de-fuzzified crisp values of the linguistic variables of the risk parameters of each failure modes identified. As an example, consider three failure modes FM1, FM2, FM3 and the linguistic variables of the risk parameter of the failure modes from Fig. 3 are shown in Table 1.

Table 1 Example of comparative series
Full size table

Based on the linguistic variable from Table1 the comparative series \(x=\left[\begin{array}{cccc}{x}_{1}\left(1\right),& {x}_{2}\left(2\right),& \dots ,& {x}_{n}(k)\end{array}\right]\) is made as shown below.

$$x = \left[ {\begin{array}{*{20}c} {FM1} \\ {FM2} \\ {FM3} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {Low} & {Moderate} & {Remote} \\ {Low} & {Very high} & {Moderate} \\ {Remote} & {High} & {Low} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {0.370} & {0.521} & {0.184} \\ {0.370} & {0.889} & {0.521} \\ {0.184} & {0.717} & {0.370} \\ \end{array} } \right]$$

3.3.3 Establishment of standard series

Standard series \({x}_{0}=\left[\begin{array}{cccc}{x}_{0}\left(1\right),& {x}_{0}\left(2\right),& \dots ,& {x}_{0}(k)\end{array}\right]\) is the optimal/ideal/desired level of all the risk factors. To ensure maximum safety, the standard/ideal matrix can be taken as the lowest level of all the risk factors or a zero matrix. Here in this example, we can take ‘Remote” as the ideal level of all the risk factors. That is \({x}_{0}=\left[\begin{array}{cccc}Remote,& Remote,& \dots ,& Remote\end{array}\right]\).

$$x_{0} = \left[ {\begin{array}{*{20}c} {FM1_{0} } \\ {FM2_{0} } \\ {FM3_{0} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {Remote} & {Remote} & {Remote} \\ {Remote} & {Remote} & {Remote} \\ {Remote} & {Remote} & {Remote} \\ \end{array} } \right]$$
$$x_{0} = \left[ {\begin{array}{*{20}c} {FM1_{0} } \\ {FM2_{0} } \\ {FM3_{0} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {0.184} & {0.184} & {0.184} \\ {0.184} & {0.184} & {0.184} \\ {0.184} & {0.184} & {0.184} \\ \end{array} } \right]$$

3.3.4 Difference matrix

To obtain the difference between the comparative series and standard series, a difference matrix \({D}_{0}={\left[\left|{x}_{0} -x\right|\right]}\) is formed.

$$D_{0} = \left[ {\begin{array}{*{20}c} {\Delta_{01} \left( 1 \right)} & {\Delta_{01} \left( 2 \right)} & \ldots & {\Delta_{01} \left( k \right)} \\ {\Delta_{02} \left( 1 \right)} & {\Delta_{02} \left( 1 \right)} & \ldots & {\Delta_{02} \left( k \right)} \\ \vdots & \vdots & \vdots & \vdots \\ {\Delta_{0n} \left( 1 \right)} & {\Delta_{0n} \left( 1 \right)} & \ldots & {\Delta_{0n} \left( k \right)} \\ \end{array} } \right]_{{}}$$

where \({\Delta }_{0j}\left(k\right)=\Vert {x}_{0}\left(k\right)-{x}_{i}(k)\Vert\), \({x}_{0}\left(k\right)\) is the standard series and \({x}_{i}\left(k\right)\) is the comparative series.

$$D_{0} = \left[ {\begin{array}{*{20}c} {\left\| {0.184 - 370} \right\|} & {\left\| {0.184 - 0.521} \right\|} & {\left\| {0.184 - 0.184} \right\|} \\ {\left\| {0.184 - 370} \right\|} & {\left\| {0.184 - 0.889} \right\|} & {\left\| {0.184 - 0.521} \right\|} \\ {\left\| {0.184 - 0.184} \right\|} & {\left\| {0.184 - 0.717} \right\|} & {\left\| {0.184 - 370} \right\|} \\ \end{array} } \right]$$
$$D_{0} = \left[ {\begin{array}{*{20}c} {0.186} & {0.337} & 0 \\ {0.186} & {0.705} & {0.337} \\ 0 & {0.533} & {0.186} \\ \end{array} } \right]$$

3.3.5 Calculation of grey relational coefficient

The grey relational coefficient \(\gamma \left({x}_{0}\left(k\right),{x}_{i}\left(k\right)\right)\) of each risk parameters for each failure modes will be found out using an equation proposed by (Cheng.et.al).

$$\gamma \left( {x_{0} \left( k \right),x_{i} \left( k \right)} \right) = \frac{{\min_{i} \min_{k} \left| {x_{0} \left( k \right) - \left. {x_{i} \left( k \right)} \right| + \xi \max_{i} \max_{k} \left| {x_{0} } \right.\left( k \right) - \left. {x_{i} \left( k \right)} \right|} \right.}}{{\left| {x_{0} } \right.\left( k \right) - \left. {x_{i} \left( k \right)} \right| + \xi \max_{i} \max_{k} \left| {x_{0} } \right.\left( k \right) - \left. {x_{i} \left( k \right)} \right|}}$$
(3)

where \({x}_{0}\left(k\right)\) is the value of risk factors in the standard series, \({x}_{i}\left(k\right)\) is the value of risk factors in the comparative series and \(\xi\) is an identifier, \(\xi \epsilon (\mathrm{0,1})\) only affecting the relative value of risk without affecting the priority, because the relative weighing coefficient is assigned without using any utility function. Considering the previous example as mentioned in Table 1 the grey relation of each risk parameters can be calculated, assuming \(\xi =0.5\). For failure mode 1, the grey relation for the linguistic variables of O, S&D can be calculated using Eq. (3) as,

\(\gamma _{Occurrence} = \frac{{0 + \left[ {0.5 \times 0.705} \right]}}{{0.186 + \left[ {0.5 \times 0.705} \right]}}\) = 0.655 \(\gamma _{Severity} = \frac{{0 + \left[ {0.5 \times 0.705} \right]}}{{0.337 + \left[ {0.5 \times 0.705} \right]}}\) = 0.511.

\(\gamma _{Non - detection} = \frac{{0 + \left[ {0.5 \times 0.705} \right]}}{{0 + \left[ {0.5 \times 0.705} \right]}}\) = 1.

The grey relational coefficient matrix will be,

$$\gamma \left( {x_{0} \left( k \right),x_{i} \left( k \right)} \right) = \left[ {\begin{array}{*{20}c} {0.655} & {0.408} & 1 \\ {0.655} & {0.705} & {0.511} \\ 1 & {0.398} & {0.350} \\ \end{array} } \right]$$

3.3.6 Calculation of grey relation

The degree of grey relation of each failure mode, \(\boldsymbol{\Gamma }({x}_{i} , {x}_{j})\) is calculated using Eq. (4), where the relative importance weights of the risk parameters are assigned.

$$\Gamma \left( {x_{0} , x_{i} } \right) = \beta_{k} \left\{ {\gamma \left( {x_{0} \left( k \right),x_{i} \left( k \right)} \right)} \right\}$$
(4)

where \({\beta }_{k}\) is the weighing coefficient of the linguistic variables of the risk factors. The weighing coefficients are obtained by considering the reliability of the available data and the objective of the analysis. The weighing factors has a large influence on the result of the risk analysis. It must be decided by consultation with all the experts associated with the system under study.

For example, by using the Eq. (4), the grey relation of the failure mode 1 in the example shown in Table 1 can be calculated as,

$${\varvec{\varGamma}}\left( {x_{i} , x_{j} } \right) = \beta_{O} \gamma \left( {x_{0} \left( 1 \right),x_{i} \left( 1 \right)} \right) + \beta_{S} \gamma \left( {x_{0} \left( 2 \right),x_{i} \left( 2 \right)} \right) + \beta_{D} \gamma \left( {x_{0} \left( 3 \right),x_{i} \left( 3 \right)} \right)$$

The weighing coefficients of the risk parameters O, S& D are represented as \({\beta }_{O}\),\({\beta }_{S}\), \({\beta }_{D}\) respectively. Assuming the value of weighing coefficients of \({\beta }_{O}\),\({\beta }_{S}\), \({\beta }_{D}\) as 0.4,0.4,0.2 respectively,

$${\varvec{\varGamma}}\left( {x_{i} , x_{j} } \right) = \left\{ {\left[ {0.4 \times 0.655 + 0.4 \times 0.408 + 0.2 \times 1} \right]} \right\} = 0.625$$

Similarly, the degree of grey relation of the remaining failure modes in the Table 1 can be obtained.

Based on the results, priority ranking of failure modes is made, where the least value holding failure mode will have highest priority. The summary of the results of the discussed example is shown in Table 2.

Table 2 Example of ranking of failure modes based on the degree of grey relation
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4 Case study: ammonia storage facility

4.1 Operations at ammonia storage facility

The process flow diagram of atmospheric ammonia storage facility is shown in Fig. 4 The storage tank is a double-walled double integrity tank with insulation on the outer side of the outer wall and it has a capacity of 10,000 tons. The storage tank maintains liquid ammonia at atmospheric pressure and − 33 °C temperature (boiling point of ammonia). The ammonia storage tank is associated with a refrigeration pack and related facilities to maintain the liquid ammonia at required pressure and temperature. The tank has inlet lines, outlet lines and refrigeration lines. Liquid ammonia (atmospheric pressure and − 33 °C temperature) is transferred from carrier ship to the storage tank through unloading arm. The transfer pipe between the unloading arm and the storage tank is designed to withstand a temperature of − 33 °C. To open/close the inlet flow, a motor operated valve (MOV-1) is provided at the inlet line. Even though the storage tank is insulated, a small amount of heat from the outside atmosphere will transfer into the liquid storage through the walls. As a result, a small amount of liquid ammonia inside the tank evaporates and ammonia gas will get accumulated at the top of the tank. This vapor accumulation creates excess pressure inside the tank and may result in tank rupture. To tackle this problem, a refrigeration unit is provided. Ammonia itself is a refrigeration liquid, so the vaporized ammonia is directly taking out from the storage tank and used for the refrigeration purpose. The refrigeration system condenses the ammonia vapor and pump it back to the storage tank. Liquid outlet line is used to pump ammonia to specially designed barge and truck using loading pumps (BLP, TLP).

Fig. 4
figure 4

Process flow diagram of the ammonia storage facility

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An automatic liquid level measuring instrument called level transfer continuously measure the liquid level inside the tank and transfer the information to the operator in the control room. During charging process (ammonia filling), whenever the storage tank exceeds the maximum storage capacity, level transfer-1(LT-1) operates the high-level alarm (LAH) and indicate the operator to close the inlet valve (MOV-1). If the LT-1 failed to indicate the operator or if any operator negligence happened, level inside the storage tank will rise and LT-4 send signal to close MOV-1 automatically to prevent overfilling accidents.

The design pressure inside the tank is well maintained between − 500 and 9800 Pa using refrigeration system and pressure relief valves. An automatic pressure measuring instrument called pressure transfer (PT), continuously measures the gauge and vacuum pressure level inside the tank and give signal to the operator. If the pressure inside the tank reached 7000 Pa, PT-1 send signal to start the refrigeration pack. Condensation starts and pressure inside the tank reduces. If the pressure inside the tank goes beyond 8000 Pa, in addition to the refrigeration pack start signal, PT-1 send a pilot signal to operate PRV-1 to release the excess pressure. If the pressure inside the tank goes beyond 9800 Pa, the pressure safety valve (PSV-1) opens and release the excess pressure to the atmosphere through flare. Similarly, to prevent vacuum formation due to the excess refrigeration, PT-2 and vacuum relief valve are used. If the pressure inside the tank goes below 1000 Pa, PT-2 sends signal to stop the refrigeration pack. If the pressure goes beyond − 500 Pa PSV-2 opens and sucks atmospheric air to prevent further vacuum formation.

In the refrigeration system, the vaporized ammonia from the storage tank is brought out and admitted into the saturator. The saturator separates liquid particles from the ammonia vapor and passes only ammonia gas to the compressor. After removal of the liquid particles, ammonia gas is admitted into the low-pressure reciprocating compressor (stage-1 compression). Before the second stage of compression, the compressed vapor is passed through a flash vessel to improve the efficiency of compression. After the second stage of compression, the compressed gas is passed through ammonia condenser where the condensation happens. The condenser is a shell and tube heat exchanger, which is supplied with cooling water to carry away the latent heat of vaporization. The shell and tubes of the condenser is specifically designed to have maximum heat exchange between ammonia and cooling water. The compressed ammonia passes through the condenser tubes and rejects heat to the cooling water through conduction and convection. The cooling water absorbs heat from the compressed ammonia and exchange this heat to the atmospheric air through a cooling tower. The condensed liquid ammonia is collected using ammonia receiver and is readmitted to the storage tank at intermediate pressure by passing through the flash vessel.

To maintain the liquid level inside the flash vessel, a level transfer (LT-3) and an automatic level indicated control valve (LIC-2) is used.

4.2 Application of the proposed methodology on ammonia storage facility.

This study used fuzzy logic to overcome the issues in prioritizing the critical failure modes having same Risk Priority Numbers. For expert elicitation, a team consists of safety analysts, design experts who have a detailed knowledge about ammonia storage facility is formed. The expert team divided the atmospheric ammonia storage facility into sub systems/components and potential failure modes are identified. The risk parameters (Occurrence, Severity and Non-detection) are classified in a 10- point scale (Table 3,4,5,) suggested by (IEC 2006). By analyzing the collected information like failure history, probability of failure, consequence of failures, likelihood of non-detection, the experts have described the risk factors of O, S & D using fuzzy linguistic expressions. A simple multiplication of the ranks of the classes of each failure modes is done to sort out the unnecessary failure modes. The RPN scale is listed in Table 6(Renjith et al. 2018). A total of 61 failure modes are identified, and out of the total failure modes, 30 failure modes with RPN > 36 have listed in this study as mentioned in the Tables 7, 8, 9.

Table 3 Occurrence ranking criteria of failure modes in FMECA. (Renjith et al. 2018)
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Table 4 Severity ranking in FMECA. (Renjith et al. 2018)
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Table 5 Detection ranking in FMECA. (Renjith et al. 2018)
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Table 6 RPN scale in FMECA. (Renjith et al. 2018)
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Table 7 Linguistic variables of risk parameters of each failure modes
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4.2.1 Ranking of the failure modes using Fuzzy-RPN

The schematic diagram of the fuzzy logic method of determining RPN using Matlab fuzzy toolbox is shown in Fig. 5.

Fig. 5
figure 5

Schematic of fuzzy logic process for FMECA developed in MATLAB FIS editor

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After assigning the linguistic terms to Occurrence, Severity and Non detection, 1000 if–then rules are generated with linguistic variables as input and Fuzzy RPN as output. The method follows three major steps, that’s are: (1) Fuzzification process of the crisp input into fuzzy membership functions of Occurrence, Severity and Non-Detection (Fig. 6,7,8). Occurrence values are identified from generic failure data book published by CCPS (CCPS 1989) and data book by International Atomic Energy Agency (IAEA 1988). Severity and non-detection values are obtained from the literature and consultation with experts. (2) Rule evaluation based on the expert knowledge (Giardina and Morale 2015) on multiple combinations of Occurrence, Severity and Non-detection is performed. For the evaluation of Fuzzy RPN, 1000 if–then rules were created. Since the number of risk factors is three and each risk factors are divided into 10 linguistic variables, the total number of combinations of fuzzy if–then rules are 103 = 1000. Using these if–then rules, all the possible combinations of O, S and D are entered in the rule base. The rule plot developed is shown in Fig. 9. The evaluation is based on Mamdani min/max method of inference system (input method: min; aggregate method: max). (3) Defuzzification process to generate crisp output as fuzzy risk priority number (FRPN) using centroid method. The FRPN values of each failure modes are calculated, and they are used to prioritize the failure modes by considering the highest value holding failure mode is the most critical one. The failure modes are prioritized based on the FRPN value and shown in Table 10.

Fig. 6
figure 6

Membership functions for input variable ‘Occurrence’ (on Log10 Scale)

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Fig. 7
figure 7

Membership functions for input variable ‘Severity’

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Fig. 8
figure 8

Membership functions for input variable ‘Non-Detection’

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Fig. 9
figure 9

Defuzzification at the rule plot in Matlab

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Matlab Fuzzy logic toolbox is used to do the entire process of fuzzification and defuzzification. To represent the input variables, a combination of trigonometric and trapezoidal membership functions are used (Fig. 6,7,8), and for the output (FRPN), Gaussion-membership function is used.

4.2.2 FMECA using grey theory

To make comparison with the result obtained using fuzzy RPN method, grey theory is applied. First, fuzzy numbers in Table 3,4,5, are defuzzified using Eq. (2) on the membership plots (Fig. 6,7,8) to get crisp numbers Table 8,9,10.

Table 8 Defuzzified crisp values of occurrence
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Table 9 Defuzzified crisp values of severity
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Table 10 Defuzzified crisp values of non-detection
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Defuzzified crisp value,

$$K\left( x \right) = \frac{{\mathop \sum \nolimits_{i = 0}^{n} \left( {b_{i} - c} \right)}}{{\mathop \sum \nolimits_{i = 0}^{n} \left( {b_{i} - c} \right) - \mathop \sum \nolimits_{i = 0}^{n} \left( {a_{i} - d} \right)}}$$

In the second step, the linguistic variables of each failure modes in Table 7 are used to make comparative series.

$${x}_{i}= \left[\begin{array}{c}{FM}_{1}\\ {FM}_{2}\\ {FM}_{3}\\ {FM}_{4}\\ {FM}_{5}\\ {FM}_{6}\\ {FM}_{7}\\ {FM}_{8}\\ {FM}_{9}\\ {FM}_{10}\\ {FM}_{11}\\ {FM}_{12}\\ {FM}_{13}\\ {FM}_{14}\\ {FM}_{15}\\ {FM}_{16}\\ {FM}_{17}\\ {FM}_{18}\\ {FM}_{19}\\ {FM}_{20}\\ {FM}_{21}\\ {FM}_{22}\\ {FM}_{23}\\ {FM}_{24}\\ {FM}_{25}\\ {FM}_{26}\\ {FM}_{27}\\ {FM}_{28}\\ {FM}_{29}\\ {FM}_{30}\end{array}\right] = \left[\begin{array}{ccc}Remote& Very extreme& Very low\\ Unlikely& Very extreme& Low\\ Unlikely& Extreme& Low\\ Very remote& Very extreme& Low\\ Very remote& Very extreme& Moderate\\ Very remote& Very extreme& Low\\ Low& Moderete& Remote\\ Low& Very extreme& Moderate\\ Very remote& Serious& Low\\ Low& Significant& Moderate\\ Low& Moderate& Very low\\ Very remote& Moderate& Low\\ Very low& Moderate& Moderately high\\ Unlikely& Significant & Low\\ Low& Extreme& High\\ Low& Low& Moderately high\\ Unlikely& Significant& Low\\ Low& Major& Very low\\ Very remote& Major& Moderate\\ Moderate& Extreme& Moderate\\ Very remote& Extreme& Very low\\ Remote& Extreme& Moderate\\ Very remote& Serious& Moderately high\\ Very remote& Extreme& High\\ Moderate& Extreme& Moderately high\\ Moderate& Major& High\\ Very remote& Very extreme& Moderately high\\ Moderate& Extreme& High\\ Unlikely& Very extreme& Moderate\\ Low& Low& High\end{array}\right] = \left[\begin{array}{ccc}0.412& 0.864& 0.630\\ 0.263& 0.864& 0.543\\ 0.263& 0.773& 0.543\\ 0.400& 0.864& 0.543\\ 0.400& 0.864& 0.456\\ 0.400& 0.864& 0.543\\ 0.563& 0.500& 0.717\\ 0.563& 0.864& 0.456\\ 0.400& 0.952& 0.543\\ 0.563& 0.591& 0.456\\ 0.563& 0.500& 0.630\\ 0.400& 0.500& 0.543\\ 0.525& 0.500& 0.370\\ 0.263& 0.591& 0.543\\ 0.563& 0.773& 0.283\\ 0.563& 0.409& 0.370\\ 0.263& 0.591& 0.543\\ 0.563& 0.682& 0.630\\ 0.400& 0.682& 0.456\\ 0.650& 0.773& 0.456\\ 0.400& 0.773& 0.630\\ 0.412& 0.773& 0.456\\ 0.400& 0.952& 0.370\\ 0.400& 0.773& 0.283\\ 0.650& 0.773& 0.370\\ 0.650& 0.682& 0.283\\ 0.400& 0.864& 0.370\\ 0.650& 0.773& 0.283\\ 0.263& 0.864& 0.456\\ 0.563& 0.409& 0.283\end{array}\right]$$

The standard series is taken to be the lowest level of the linguistic variables in all the three risk parameters or a zero matrix can be taken as the standard series. Whether we take zero or the lowest value matrix as standard series, there is no change in relative importance between the failure modes. Here in this case, we take a zero matrix as the standard series. Hence, the difference matrix \({D}_{0}={\left[\left|{x}_{0} -x\right|\right]}\) will be same as the comparative series matrix.

$${x}_{0}= \left[\begin{array}{ccc}0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\\ 0& 0& 0\end{array}\right]{D}_{0}= \left[\begin{array}{ccc}0.412& 0.864& 0.630\\ 0.263& 0.864& 0.543\\ 0.263& 0.773& 0.543\\ 0.400& 0.864& 0.543\\ 0.400& 0.864& 0.456\\ 0.400& 0.864& 0.543\\ 0.563& 0.500& 0.717\\ 0.563& 0.864& 0.456\\ 0.400& 0.952& 0.543\\ 0.563& 0.591& 0.456\\ 0.563& 0.500& 0.630\\ 0.400& 0.500& 0.543\\ 0.525& 0.500& 0.370\\ 0.263& 0.591& 0.543\\ 0.563& 0.773& 0.283\\ 0.563& 0.409& 0.370\\ 0.263& 0.591& 0.543\\ 0.563& 0.682& 0.630\\ 0.400& 0.682& 0.456\\ 0.650& 0.773& 0.456\\ 0.400& 0.773& 0.630\\ 0.412& 0.773& 0.456\\ 0.400& 0.952& 0.370\\ 0.400& 0.773& 0.283\\ 0.650& 0.773& 0.370\\ 0.650& 0.682& 0.283\\ 0.400& 0.864& 0.370\\ 0.650& 0.773& 0.283\\ 0.263& 0.864& 0.456\\ 0.563& 0.409& 0.283\end{array}\right]$$

The values of the difference matrix are used to find the grey relational coefficient \(\gamma \left({x}_{0}\left(k\right),{x}_{i}\left(k\right)\right)\). The grey relational coefficient of linguistic variables of each failure mode is calculated using Eq. (3) and made the grey relational coefficient matrix.

Grey relational coefficient,

$$\gamma \left( {x_{0} \left( k \right),x_{i} \left( k \right)} \right) = \frac{{min_{i} min_{k} \left| {x_{0} \left( k \right) - \left. {x_{i} \left( k \right)} \right| + \xi max_{i} max_{k} \left| {x_{0} } \right.\left( k \right) - \left. {x_{i} \left( k \right)} \right|} \right.}}{{\left| {x_{0} } \right.\left( k \right) - \left. {x_{i} \left( k \right)} \right| + \xi max_{i} max_{k} \left| {x_{0} } \right.\left( k \right) - \left. {x_{i} \left( k \right)} \right|}}$$

Grey relational coefficient matrix,

$$\gamma \left({x}_{0}\left(k\right),{x}_{i}\left(k\right)\right) = \left[\begin{array}{ccc}0.891& 0.669& 0.768\\ 1.000& 0.669& 0.813\\ 1.000& 0.704& 0.813\\ 0.899& 0.669& 0.813\\ 0.899& 0.669& 0.813\\ 0.899& 0.669& 0.813\\ 0.802& 0.837& 0.728\\ 0.802& 0.669& 0.863\\ 0.899& 0.638& 0.813\\ 0.802& 0.787& 0.863\\ 0.802& 0.837& 0.768\\ 0.899& 0.837& 0.813\\ 0.823& 0.837& 0.919\\ 1.000& 0.787& 0.813\\ 0.802& 0.704& 0.984\\ 0.802& 0.893& 0.919\\ 1.000& 0.787& 0.813\\ 0.802& 0.744& 0.768\\ 0.899& 0.744& 0.863\\ 0.758& 0.704& 0.768\\ 0.899& 0.704& 0.984\\ 0.891& 0.704& 0.863\\ 0.899& 0.638& 0.919\\ 0.899& 0.704& 0.984\\ 0.758& 0.704& 0.919\\ 0.758& 0.744& 0.984\\ 0.899& 0.669& 0.919\\ 0.758& 0.704& 0.984\\ 1.000& 0.669& 0.863\\ 0.802& 0.893& 0.984\end{array}\right]$$

The degree of grey relation of each failure mode, \(\Gamma ({x}_{i} , {x}_{j})\) is calculated using Eq. (4), where the relative importance weights of the risk parameters are assigned.

$$\Gamma \left( {x_{0} , x_{i} } \right) = \beta_{k} \left\{ {\gamma \left( {x_{0} \left( k \right),x_{i} \left( k \right)} \right)} \right\}$$

The weighing coefficients of the risk parameters \({\beta }_{O}\),\({\beta }_{S}\), \({\beta }_{D}\) are decided as 0.4, 0.4 and 0.2 respectively. The grey relation of each failure modes is calculated and prioritization were made Table 11.

Table 11 Ranking of the failure modes using Grey relation with weightage values of βo = 0.4, βs = 0.4, βd = 0.2.
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5 Results and discussions

Table 12. shows the comparative analysis of the prioritization of all the 30 failure modes using traditional RPN, Fuzzy-RPN and Grey relation. In traditional RPN method, the prioritization is difficult when identical values of RPN occurs. For example, the failure modes, 7, 9, 15, 22 have the same RPN of 120 and they are difficult to distinguish. But, in Fuzzy-RPN method and grey relation method, they are easy to distinguish. The failure mode 25 in Fuzzy-RPN method has a prioritization rank of 5. The same failure mode has prioritization rank of 3 when relative importance weight is considered in grey relation method. Using grey theory, it is clearly seen that the failure mode 25 is riskier than failure mode 1 and 18, because the ammonia compressor has both rotating and reciprocating parts and it needs close monitoring and maintenance than components involved in other two failure modes. It can be seen that the relative importance weights (weighing coefficients) of Occurrence and severity assigned by the experts are higher (\({\beta }_{O}\)=0.4,\({\beta }_{S}\)=0.4) compared to the weightage for Non-detection (\({\beta }_{D}\)=0.20). It created small changes to the importance ranks of few failure modes compared to the Fuzzy-RPN method.

In the comparison of the analytical results based on fuzzy set theory and grey theory shown in Table 12, prioritization ranking of 20 failure modes are almost same or at most a rank difference of 2, which shows the practicability and accuracy of the proposed method. The top ranked failure modes need much more attention to prevent any catastrophic accidents in ammonia storage facility. The designers, safety managers, maintenance managers can take decisions and plan their maintenance plans based on the results which help to improve the reliability and safety performance of the ammonia storage facility.

Table 12 Comparison of ranking
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The proposed method is of great benefit to engineering practice such that the proposed method is very suitable for complex systems. One can replicate the procedure while risk assessment in any kind of industry. Furthermore, the use of assistance software tools such as Matlab fuzzy tool box extensively helps to reduce the calculation time of performing fuzzy FMECA, which is a great advantage in the fast-moving world to save time required for risk assessment. Even though the software assistance can reduce the calculation time, dependence on fuzzy risk assessment is not suitable if the number of risk factors are very high. Setting up a rule base by building 104,105,106 combination statements are very difficult for the expert team. It needs a breakthrough to come up with a new method to account many more risk parameters in a simple and easy way. In future works, we are aiming to solve the above stated problem.

6 Conclusions

FMECA plays a vital role in safety management and risk analysis in complex systems. It helps managers to take decisions on maintenance and risk management strategies. In this paper, a risk assessment method based on fuzzy set theory and grey theory is proposed. The main intension of the proposed method was to overcome the drawbacks of the FMECA based on traditional RPN method. Through steps such as formation of team, collection of information, creation of fuzzy sets, expert rating, fuzzification, defuzzification, creation of rule base, calculation of grey relation and prioritization, the method is developed and is applied to an atmospheric ammonia storage tank. According to the results obtained, the following conclusions were drawn.

  1. i.

    By using fuzzy RPN, a more realistic and accurate rankings of the failure modes was done. The problem of handling vague and uncertain inputs was eliminated. The fuzzy inference system gives different and distinctive values for each failure modes, so that no any similar values obtained and one of the drawbacks of the conventional RPN approach was eliminated. By using Fuzzy inference system, duplication of the same RPN for different failure modes were eliminated. So that the ranking will be accurate. Even complex systems can be ranked based on risk.

  2. ii.

    By using the grey theory, the relative importance of Occurrence, Severity, Non-detection have been considered and evaluated in a linguistic manner, which makes the prioritization more realistic and objective. Grey theory in FMEA reflects the nature of relative ranking, because the ranking is based on the grey relational coefficient which is determined by the comparison between comparative and standard series.

  3. iii.

    Using a comparative case study of fuzzy FMECA and Grey FMECA, a much reliable decision-making tool is proposed. By comparing the results, the accuracy of decision making will be better than the Conventional FMECA method.

  4. iv.

    By the use of Matlab fuzzy tool box as software assistance, a great amount of time could be saved while doing calculations in fuzzy risk assessment.

This methodology was applied to an Ammonia storage tank for the prioritization of the failure modes and identified the critical events to overcome the drawbacks of the conventional RPN based approach to FMECA discussed in the literature. The results obtained by the Fuzzy FMECA and grey theory based FMECA show more accurate and reliable results than the conventional FMECA using RPN method. This method can be duplicated and extended to other complex and critical installations such as refineries, offshore drilling, nuclear, petrochemical and fertilizer industries.