1Chian Haur Jong, 1*Kai Meng Tay, 2Chee Peng Lim1Faculty of Engineering, Universiti Malaysia Sarawak, Malaysia

2Centre for Intelligent Systems Research, Deakin University, Australia.

1*kmtay@feng.unimas.my

A Single Input Rule Modules Connected Fuzzy FMEA Methodology for Edible Bird

Nest Processing

Presentation OutlineIntroductionProblem StatementsObjectivesPreliminaryThe Proposed Fuzzy FMEA procedure

Case study: EBN food processingConcluding Remarks

Failure Mode and Effects Analysis (FMEA) is a tool for quality assurance and reliability improvement. In FMEA, a failure mode occurs when a component, system, subsystem, or process fails to meet the designated intent. Traditionally, the Risk Priority Number (RPN) model is used to rank failure modes and it is defined by the equation below:

RPN=S x O x D An RPN model defined by 3 risk factors, i.e. Severity (S),

Occurrence (O) and Detect (D). S and O are the frequency and seriousness (effects) of a failure mode, and D is the effectiveness to detect a failure mode before it reaches the customer.

Introduction: FMEA methodology and its RPN model

Bowles and Paláez (1995) had suggested using an Fuzzy Inference System (FIS) to aggregate S, O, and D ratings (namely an FIS-based RPN model), instead of a simple product function

An FIS-based RPN was introduced, for the following reasons.

1. It allows expert knowledge and experience to be incorporated

2. It is robust against uncertainty and vagueness

3. It allows a nonlinear relationship between the RPN score and the three risk factors to be formed

4. The three risk factors can be captured qualitatively, instead of quantitatively

J.B. Bowles and C.E. Pelaez, Fuzzy Logic prioritization of failures in a system failure mode, effect and criticality analysis, Reliability Engineering and System Safety, Vol 50, No 2, pp. 203-213, (1995).

Introduction: The FIS-based RPN model

Introduction: The FIS-based RPN modelVarious FIS-based RPN models have since been developed

and applied to a variety of application domains, e.g.

1. Maritime Z. Yang, S. Bonsall and J. Wang, Fuzzy rule-based Bayesian reasoning approach for prioritization of failures in FMEA, IEEE Transactions on Reliability, Vol. 57, No.3, pp. 517-528, (2008).

2. Nuclear Engineering SystemsA.C.F. Guimarães and C.M.F. Lapa, Fuzzy inference to risk assessment on nuclear engineering systems, Applied Soft Computing, Vol 7, No1, pp17-28, (2007)

3. Semiconductor IndustryK.M. Tay, C.P. Lim, Fuzzy FMEA with a guided rules reduction system for prioritization of failures. International Journal of Quality And Reliability Management. Vol. 23, No. 8, pp. 1047 – 1066 (2006).

4. Engine system K Xu, L.C Tang, M Xie, S.L Ho, M.L Zhu, Fuzzy assessment of FMEA for engine systems, Reliability Engineering & System Safety, Vol 75, No 1, pp.17-29, (2002)

An FIS-based RPN model suffers from two major shortcomings viz.,

Shortcoming 1: combinatorial rule explosion

The first shortcoming suggests that an FIS-based RPN model requires a large number of fuzzy rules, and it is a tedious process to gather a complete fuzzy rule base in practice.

K.M. Tay and C.P. Lim, Fuzzy FMEA with a guided rules reduction system for prioritization of failures, International Journal of Quality & Reliability Management, Vol. 23, No 8, pp.1047 – 1066, (2006).Y.C. Jin, Fuzzy modeling of high-dimensional systems: complexity reduction and interpretability improvement, IEEE Transactions on Fuzzy Systems, Vol. 2 No. 8, pp. 212-21, (2000).

Problem Statements

Indeed, a search in the literature reveals that a lot of investigations for rule reduction in FIS-based RPN models have been reported for problem related to the combinatorial rule explosion.

Tay and Lim (2006) implement a guided rule reduction system to improve the FMEA methodology by identifying only the important fuzzy rules

An FIS-based RPN model with 125 fuzzy rules was reduced to 35 fuzzy rules with the use of the method in Pillay and Wang (2003)

K.M. Tay and C.P. Lim, Fuzzy FMEA with a guided rules reduction system for prioritization of failures, International Journal of Quality & Reliability Management, Vol. 23, No 8, pp.1047 – 1066, (2006).A. Pillay and J. Wang, Modified failure mode and effects analysis using approximate reasoning, Reliability Engineering and System Safety, Vol 79, No 1, pp. 69-85,(2003).

Problem Statements

A method which proposed by Guimara˜es and Lapa (2004) was successfully reduced FIS-based FMEA model with 125 fuzzy rules to 14 fuzzy rules. In Guimara˜es and Lapa (2006), the authors further reduced FIS-based FMEA model with 125 fuzzy rules to 6 fuzzy rules.

A.C.F. Guimarães and C.M.F. Lapa, Fuzzy FMEA applied to PWR chemical and volume control system, Progress in Nuclear Energy, Vol. 44, No. 3, pp. 191-213, (2004).A.C.F. Guimaraes and C.M.F. Lapa, Hazard and operability study using approximate

reasoning in light-water reactors passive systems, Nuclear Engineering and Design, Vol 236, No 12, pp. 1256-1263,(2006).

Problem Statements

Shortcoming 2: Monotonicity property fulfillmentFor an FIS-based RPN that fulfills the monotonicity

property, dRPN / dx ≥ 0 , where x ϵ [S, O, D]. It is essential to fulfill the monotonicity property.Fulfillment of the monotonicity property is difficult, and

yet important to ensure the validity of the RPN scores

K.M. Tay and C.P. Lim, On monotonic sufficient conditions of fuzzy inference systems and their applications, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol 19, No 5, pp. 731-757, (2011).

K.M. Tay and C.P. Lim, On the use of fuzzy inference techniques in assessment models: part I - theoretical properties, Fuzzy Optimization and Decision Making, Vol 7, No 3, pp.269-281, (2008).

K.M. Tay and C.P. Lim, On the use of fuzzy inference techniques in assessment models: part II: industrial applications, Fuzzy Optimization and Decision Making, Vol 7, No 3, pp. 283-302, (2008).

Problem Statements

A zero-order Single Input Rule Modules (SIRMs) connected Fuzzy Inference System (FIS) is used.

Theorems from Seki’s papers is simplified and used.

N. Yubazaki, J.Q. Yi and K. Hirota, SIRMs (Single Input Rule Modules) Connected fuzzy inference model, Journal of Advanced Computational Intelligence, Vol 1, No1, pp 22-29, (1997).

H. Seki, H. Ishii and M. Mizumoto, On the generalization of Single Input Rule Modules Connected Type fuzzy reasoning method, IEEE Transactions on Fuzzy Systems, Vol.16, No.5, pp.1180-1187, (2008).

Seki, H., Ishii, H., Mizumoto, M.: On the monotonicity of fuzzy-inference method related to T-S inference Method. IEEE Trans. Fuzzy Syst. 18, 629-634 (2010).

Seki, H.,Tay, K.M.: On the monotonicity of fuzzy inference models.J. Adv. Comput. Intell. Intell. Informat. 16, 592-602 (2012).

Objectives: Direction of the paper

A new fuzzy FMEA methodology with Single Input Rule Modules (SIRMs) connected Fuzzy Inference System (FIS)-FIS-based RPN model is proposed.

Monotonicity property of the SIRMs-connected FIS-based RPN model is preserved to ensure an valid output for risk evaluation.

A case study relating to edible bird nest (EBN) processing is demonstrated.

Objectives of the paper

Preliminary: SIRMs connected FIS modelThe use of an SIRMs connected FIS model in FMEA shall

reduce the number of fuzzy rules drastically. From the literature, an SIRMs connected FIS model require less fuzzy rule in FIS modeling.

EXAMPLES:

1. The zeros-order SIRMs connected FIS was proposed by Yubazaki et al (1997) for a plural input fuzzy control to reduce the number of fuzzy rules required in FIS modeling.

2. Seki et al (2008) proposed Functional-type SIRMs connected FIS, in which the consequences are generalized as functions.

N. Yubazaki, J.Q. Yi and K. Hirota, SIRMs (Single Input Rule Modules) Connected fuzzy inference model, Journal of Advanced Computational Intelligence, Vol 1, No1, pp 22-29, (1997).

H. Seki, H. Ishii and M. Mizumoto, On the generalization of Single Input Rule Modules Connected Type fuzzy reasoning method, IEEE Transactions on Fuzzy Systems, Vol.16, No.5, pp.1180-1187, (2008).

A zero-order SIRMs connected FIS model with 𝑛 inputs (i.e., 𝑦=𝑓ሺ𝑥ҧሻ), where 𝑥ҧ= ሺ𝑥1,𝑥2,…𝑥𝑛ሻ is considered. It consists of 𝑛 fuzzy rule modules, as in Fig. 1.

SIRM− 1: ቄ𝑅1𝑗1:𝑖𝑓𝒙𝟏𝑖𝑠𝐴1𝑗1𝑡ℎ𝑒𝑛𝑦1𝑗1 = c1𝑗1ቅ𝑗𝑚1

…….. …………………… SIRM− 𝑖: ቄ𝑅𝑖𝑗𝑖:𝑖𝑓𝒙𝒊𝑖𝑠𝐴𝑖𝑗𝑖𝑡ℎ𝑒𝑛𝑦𝑖𝑗𝑖 = 𝑐𝑖𝑗𝑖ቅ𝑗𝑚𝑖

…….. …………………… SIRM− 𝑛: ቄ𝑅𝑛𝑗𝑛:𝑖𝑓𝒙𝒏𝑖𝑠𝐴𝑛𝑗𝑛𝑡ℎ𝑒𝑛𝑦𝑛𝑗𝑛 = 𝑐𝑛𝑗𝑛ቅ𝑗𝑚𝑛

Fig. 1. Fuzzy rules for a zero-order SIRMs connected FIS model

SIRM− 𝑖 represents the i-th rule module, where 𝑥𝑖 is the sole variable in

the antecedent and 𝑖 = 1,2,…,𝑛. 𝑅𝑖𝑗𝑖 is the j-th rule in SIRM− 𝑖, where 𝑗𝑖 = 1,2,…,𝑚𝑖 . A fuzzy rule 𝑅𝑖𝑗𝑖 can be viewed as a mapping from 𝐴𝑖𝑗𝑖to c𝑖𝑗𝑖, i.e., 𝑅𝑖𝑗𝑖:𝐴𝑖𝑗𝑖 →c𝑖𝑗𝑖.c𝑖𝑗𝑖is a variable or a fuzzy singleton.

The output of SIRM− 𝑖, i.e., 𝑦𝑖ሺ𝑥𝑖ሻ,is obtained using Eq. (1).

𝑦𝑖ሺ𝑥𝑖ሻ= σ ቂ𝜇𝑖𝑗𝑖ሺ𝑥𝑖ሻ×𝑐𝑖𝑗𝑖ቃ𝑚𝑖𝑗𝑖=1σ ቂ𝜇𝑖𝑗𝑖ሺ𝑥𝑖ሻቃ𝑚𝑖𝑗𝑖=1 (1)

The membership function for 𝐴𝑖𝑗𝑖 is denoted as𝜇𝑖𝑗𝑖ሺ𝑥𝑖ሻ. The

zero-order SIRMs connected FIS model further aggregates the output for each rule module with a weighted addition, as in Eq. (2), where𝜔𝑖 is a numerical value that reflects the relative importance of the i-th rule module. 𝑦= σ ሾ𝜔𝑖𝑦𝑖ሺ𝑥𝑖ሻሿ𝑛𝑖=1 (2)

Definition 1.Consider two convex fuzzy sets, i.e., A and B, inthe 𝑹 domain, with universe of discourse ranged from 𝒙to𝒙, as shown in Fig. 2. The 𝜶-level sets of A and B are defined as follows.

ሾ𝐴ሿ𝛼 = ሾ𝑎𝛼,𝑏𝛼ሿ,ሾ𝐵ሿ𝛼 = ሾ𝑐𝛼,𝑑𝛼ሿ (3)

Fig. 2.Comparable fuzzy sets with a fuzzy ordering 𝐴≼ 𝐵

Preliminary: The α-level set and comparable fuzzy sets

Definition 2. A fuzzy ordering 𝐴≼ 𝐵exists and there are comparable, if the following condition is satisfied. 𝑎𝛼 ≤ cα,𝑏𝛼 ≤ 𝑑𝛼,α ∈[0,1]

Preliminary: The α-level set and comparable fuzzy sets

Seki, H., Ishii, H., Mizumoto, M.: On the monotonicity of fuzzy-inference method related to T-S inference Method.IEEE Trans. Fuzzy Syst. 18, 629-634 (2010).Seki, H.,Tay, K.M.: On the monotonicity of fuzzy inference models.J. Adv.

Comput. Intell. Intell. Informat. 16, 592-602 (2012).

Monotonicity Property of SIRMs connected FIS model

Let 𝑓(𝑥ҧ) denote an n-input function, where 𝑥ҧ=ሺ𝑥1,𝑥2,…𝑥𝑛ሻ∈X1 × X2 × …Xn. The i-th input in 𝑥ҧ is represented by 𝑥𝑖 , where 𝑥𝑖 ∈X𝑖 , 𝑖 = 1,2,..𝑛 . A sequence, 𝑠ҧ, denotes a subset of 𝑥ҧ, whereby 𝑥𝑖 is excluded from 𝑠ҧ, i.e.𝑠ҧ⊂ 𝑥ҧ;𝑥𝑖 ∉𝑠ҧ. The monotonicity property of 𝑓(𝑥ҧ) can be formally written as follows: Definition 3: An SIRMs-connected FIS model is said to fulfill the monotonicity property between its output, y, and its input, 𝑥𝑖, when y monotonically increases as 𝑥𝑖 increases, i.e., 𝑓(𝑠ҧ,𝑥𝑖) ≥ 𝑓(𝑠ҧ,𝑥′𝑖) where 𝑥𝑖 > 𝑥𝑖′ ∈𝑋𝑖.

Theorem 1. By simplifying the theorems in [9-10], a zero-order SIRMs connected FIS model fulfills the monotonicity property if the following conditions are satisfied.

o Condition 1: The fuzzy membership functions for the 𝑥𝑖 domain are compare-able, i.e., 𝐴𝑖𝑗𝑖 ≼ 𝐴𝑖𝑗𝑖+1, 𝑗𝑖 = 1,2,…,𝑚𝑖 − 1.

o Condition 2: 𝑐𝑖𝑗𝑖 ≤ 𝑐𝑖𝑗𝑖+1, 𝑗𝑖 = 1,2,…,𝑚𝑖 − 1.

Seki, H., Ishii, H., Mizumoto, M.: On the monotonicity of fuzzy-inference method related to T-S inference Method.IEEE Trans. Fuzzy Syst. 18, 629-634 (2010).Seki, H.,Tay, K.M.: On the monotonicity of fuzzy inference models.J. Adv.

Comput. Intell. Intell. Informat. 16, 592-602 (2012).

The Proposed Fuzzy FMEA procedure

Fig. 3. The proposed Fuzzy FMEA methodology

The proposed SIRMs-based FMEA methodology is summarized in Fig. 3. The details are as follows.

1. Define the scale tables for S, O, and D.

2. Construct the membership functions for each input factor (i.e., S, O and D). Condition 1 is used as the governing equation.

3. Gather expert knowledge to construct the fuzzy rule base. Condition 2 is imposed in the construction phase.

4. Construct the SIRMs connected FIS-based RPN model.

5. Study the intent, purpose, goal, objective for the product/process. Generally, it is identified by studying the interaction among the component/process flow diagram and is followed by a task analysis.

6. Identify the potential failures of a product/process, which include problems, concerns, and opportunities for improvement.

7. Identify the consequence of each failure to other components or the other processes, operation customers, and government regulations.

8. Identify the potential root causes of the potential failures.

9. Identify the method/procedure to detect/ prevent the potential failures.

10. Evaluate S, O, and D based on the predefined scale tables.

11. Calculate the fuzzy RPN (FRPN) scores using the SIRMs connected FIS-based RPN model.

12. Make any necessary corrections. Go back to step (5) if needed.

13. End

Modeling of the SIRMs connected FIS-based RPN Model

The SIRMs connected FIS-based RPN model has three inputs (i.e., S, O and D) and one output, i.e. the FRPN score. It consists of three fuzzy rule modules, as shown in Fig 4.

Fig. 4.Fuzzy rules for an SIRMs connected FIS-based RPN Model

Note that 𝑆𝐼𝑅𝑀− 1, 𝑆𝐼𝑅𝑀− 2, and 𝑆𝐼𝑅𝑀− 3 are the rule modules

for S, O, and D, respectively.

𝑆𝐼𝑅𝑀− 1: ቄ𝑅1𝑗1:𝑖𝑓𝐒𝑖𝑠𝐴1𝑗1𝑡ℎ𝑒𝑛𝐹𝑅𝑃𝑁1𝑗1 = 𝑐1𝑗1ቅ𝑗1=1𝑚1

𝑆𝐼𝑅𝑀− 2: ቄ𝑅2𝑗2:𝑖𝑓𝐎𝑖𝑠𝐴2𝑗2𝑡ℎ𝑒𝑛𝐹𝑅𝑃𝑁2𝑗2 = 𝑐2𝑗2ቅ𝑗2=1𝑚2

𝑆𝐼𝑅𝑀− 3: ቄ𝑅3𝑗3:𝑖𝑓𝐃𝑖𝑠𝐴3𝑗3𝑡ℎ𝑒𝑛𝐹𝑅𝑃𝑁3𝑗3 = 𝑐3𝑗3ቅ𝑗3=1𝑚3

The FRPN score is obtained using Eq. (4). 𝐹𝑅𝑃𝑁= 𝑓ሺ𝑆,𝑂,𝐷ሻ= 𝜔𝑆× 𝑅𝑃𝑁1 + 𝜔𝑂 × 𝑅𝑃𝑁2 + 𝜔𝐷× 𝑅𝑃𝑁3

=𝜔𝑆σ ቂ𝜇1𝑗1ሺ𝐒ሻ×𝐶1𝑗1ቃ𝑚1𝑗1=1σ ቂ𝜇1𝑗1(𝐒)ቃ𝑚1𝑗1=1 + 𝜔𝑂σ ቂ𝜇2𝑗2(𝐎)×𝐶2𝑗2ቃ𝑚2𝑗2=1σ ቂ𝜇2𝑗2(𝐎)ቃ𝑚2𝑗2=1 + 𝜔𝐷σ ቂ𝜇3𝑗3(𝐃)×𝐶3𝑗3ቃ𝑚3𝑗3=1σ ቂ𝜇3𝑗3(𝐃)ቃ𝑚3𝑗3=1 (4)

From the Eq. (4), the inference outputs of 𝑆𝐼𝑅𝑀− 1, 𝑆𝐼𝑅𝑀− 2,

and 𝑆𝐼𝑅𝑀− 3, i.e., 𝐹𝑅𝑃𝑁1, 𝐹𝑅𝑃𝑁2, and 𝐹𝑅𝑃𝑁3, respectively, are combined. Note that 𝜔S, 𝜔O , and 𝜔D are the degree of importance of each rule module towards the combined FRPN score.

In the SIRMs connected FIS module design, the important degree of each of the rule modules are set to equal i.e, ωS = ωO = ωD = 1/3

Define the scale tables First, the scale tables for S, O, and D is defined.

An example of scale table for O

The membership functions for S, O and D are designed in such fuzzy membership functions are compare-able (refer to Definition 1 and 2) to ensure the condition 1 is fulfilled (refer theorem 1). Example:

Fig. 5.The membership function for Occurrrence

Design of the membership function

A set of fuzzy rules, as summarized following is used, is used.

Gathering of the fuzzy rules

𝑆𝐼𝑅𝑀− 1

{𝑅11:𝑖𝑓 S 𝑖𝑠 𝑁𝑜𝑛𝑒 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁11 𝑖𝑠 𝐸𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝐿𝑜𝑤} ሼ𝑅12:𝑖𝑓 S 𝑖𝑠 𝐿𝑜𝑤 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁12 𝑖𝑠 𝐿𝑜𝑤ሽ ሼ𝑅13:𝑖𝑓 S 𝑖𝑠 𝑀𝑜𝑑𝑒𝑟𝑎𝑡𝑒 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁13 𝑖𝑠 𝐻𝑖𝑔ℎሽ ሼ𝑅14:𝑖𝑓 S 𝑖𝑠 𝐻𝑖𝑔ℎ 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁14 𝑖𝑠 𝑉𝑒𝑟𝑦 𝐻𝑖𝑔ℎ ሽ {𝑅15:𝑖𝑓 S 𝑖𝑠 𝑉𝑒𝑟𝑦 𝐻𝑖𝑔ℎ 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁15 𝑖𝑠 𝐸𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝐻𝑖𝑔ℎ}

𝑆𝐼𝑅𝑀− 2 {𝑅21:𝑖𝑓 O 𝑖𝑠 𝑅𝑒𝑚𝑜𝑡𝑒 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁21 𝑖𝑠 𝐸𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝐿𝑜𝑤} ሼ𝑅22:𝑖𝑓 O 𝑖𝑠 𝑉𝑒𝑟𝑦 𝐿𝑜𝑤 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁22 𝑖𝑠 𝑉𝑒𝑟𝑦 𝐿𝑜𝑤ሽ ሼ𝑅23:𝑖𝑓 O 𝑖𝑠 𝐿𝑜𝑤 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁23 𝑖𝑠 𝐿𝑜𝑤ሽ ሼ𝑅24:𝑖𝑓 O 𝑖𝑠 𝑀𝑜𝑑𝑒𝑟𝑎𝑡𝑒 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁24 𝑖𝑠 𝑀𝑒𝑑𝑖𝑢𝑚ሽ ൛𝑅25:𝑖𝑓 O 𝑖𝑠 𝐻𝑖𝑔ℎ 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁25 𝑖𝑠 𝑣𝑒𝑟𝑦 𝐻𝑖𝑔ℎൟ {𝑅26:𝑖𝑓 O 𝑖𝑠 𝑉𝑒𝑟𝑦 𝐻𝑖𝑔ℎ 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁26 𝑖𝑠 𝐸𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝐻𝑖𝑔ℎ}

𝑆𝐼𝑅𝑀− 3 {𝑅31:𝑖𝑓 D 𝑖𝑠 𝑉𝑒𝑟𝑦 𝐻𝑖𝑔ℎ 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁31 𝑖𝑠 𝐸𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝐿𝑜𝑤} {𝑅32:𝑖𝑓 D 𝑖𝑠 𝐻𝑖𝑔ℎ 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁32 𝑖𝑠 𝐿𝑜𝑤 } {𝑅33:𝑖𝑓 D 𝑖𝑠 𝑀𝑜𝑑𝑒𝑟𝑎𝑡𝑒 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁33 𝑖𝑠 𝐻𝑖𝑔ℎ } {𝑅34:𝑖𝑓 D 𝑖𝑠 𝐿𝑜𝑤 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁34 𝑖𝑠 𝑉𝑒𝑟𝑦 𝐻𝑖𝑔ℎ} {𝑅35:𝑖𝑓 D 𝑖𝑠 𝑉𝑒𝑟𝑦 𝐿𝑜𝑤 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁35 𝑖𝑠 𝑉𝑒𝑟𝑦 𝐻𝑖𝑔ℎ} {𝑅36:𝑖𝑓 D 𝑖𝑠 𝐸𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝐿𝑜𝑤 𝑡ℎ𝑒𝑛 𝐹𝑅𝑃𝑁36 𝑖𝑠 𝐸𝑥𝑡𝑟𝑒𝑚𝑒𝑙𝑦 𝐻𝑖𝑔ℎ}

Fig.6 Fuzzy rules for the SIRMs FIS-based RPN model

The fuzzy rules are designed in such a way that Condition 2 (refer theorem 1) is fulfilled

As an example, for SIRM-1, the fuzzy singletons are 1, 376, 750, 875, and 1000. Since 1 ≤ 376 ≤ 750 ≤ 875 ≤ 1000, Condition 2 is satisfied.

Data and information gathered from two swiftlets farms and two EBN production plants in Sarawak (as depicted in the Fig. 7), Malaysia are used to assess the efficacy of the proposed approach.

Fig 7: Geographical locations of two swiftlets farms and two EBN production plants in

Sarawak, Malaysia

Case study: EBN food processing

An analysis on fuzzy rule reduction

With the conventional FIS-based RPN model, the number of fuzzy rules required for an FIS-based RPN model is ς mi3i=1 .

With the proposed SIRMs connected FIS-based RPN model, the number of fuzzy rules required is σ mi3i=1 .

In this case study, there are 5, 6, and 5 membership functions, for S, O, and D, respectively. Hence, fuzzy rules required are 150 (5x6x5) and 11 (5+6+5) for the conventional FIS-based RPN and SIRMs connected FIS-based RPN models, respectively.

The efficiency of the fuzzy rule reduction is evaluated using Eq (5). In this study, the percentage of the number of fuzzy rules reduced is (150-11)/150×100=92.67%.

𝑃𝑒𝑟𝑐𝑒𝑛𝑡𝑎𝑔𝑒 𝑜𝑓 𝑓𝑢𝑧𝑧𝑦 𝑟𝑢𝑙𝑒𝑠 𝑟𝑒𝑑𝑢𝑐𝑒𝑑= ς 𝑚𝑖3𝑖=1 − σ 𝑚𝑖3𝑖=1ς 𝑚𝑖3𝑖=1 × 100% (5)

Table 2. Failure risk evaluation, ranking, and prioritization results using the RPN and SIRMs connected FIS-based RPN models for EBN processing

Failure Mode

Rating Scores RPN models Sev Occ Det RPN RPN

Rank FRPN FRPN

Rank 1 1 1 1 1 1 1 1 2 2 1 1 2 2 51 2 3 3 1 1 3 3 101 3 4 3 1 2 6 5 141 4 5 4 1 2 8 6 190 5 6 5 1 1 5 4 201 6 7 5 1 4 20 7 309 7 8 6 1 4 24 8 359 8 9 6 1 6 36 10 417 9

10 3 10 1 30 9 434 10 11 4 9 1 36 10 458 11 12 3 7 4 84 12 471 12 13 4 10 1 40 11 484 13

Risk evaluation results with the SIRMs FIS-based RPN model

From Table 3, Columns “Sev” (S), “Occ” (O), and “Det” (D) show the S, O, and D ratings that describe each failure mode.

The failure risk evaluation and prioritization outcomes from the conventional RPN model are explained in columns “RPN” and “RPN Rank”, respectively. For the SIRMs connected FIS-based RPN model, its risk evaluation and prioritization outcomes are summarized in columns “FRPN” and “FRPN Rank”, respectively.

From Table 3, the fulfillment of the monotonicity property can be observed. As an example, failure modes #1, #2, and #3 have the same S and O ratings, i.e., 1 and 1, respectively. The D scores are 1, 2, and 3, for each of the failure modes, respectively.

With the SIRMs connected FIS-based RPN model, the FRPN scores are 1, 51, and 101, respectively; hence satisfying the monotonicity property.

Surface plotTo easily observe the monotonity property of the

overall model (as shown in Table 3), a surface plot can be used.

Fig. 7. A surface plot of FRPN versus severity and occurrence (with detect = 10)

Fig. 7 depicts a surface plot for FRPN versus S and O, with D = 10. A monotonic surface can be observed easily.

SummaryA new fuzzy FMEA methodology with a zero-order SIRMs

connected FIS-based RPN model has been proposed. The theorems in [9-10] have been simplified and adopted as a

set of governing equations in the proposed fuzzy FMEA methodology.

The proposed approach constitutes a new fuzzy FMEA methodology with a reduced fuzzy rule base, which satisfies the monotonicity property.

A case study relating to EBN processing has been examined. The results have shown that the proposed approach is able to reduce the number of fuzzy rules effectively and yet, to satisfy the monotonicity property.

This paper also contributes to a new application of fuzzy FMEA to food processing (i.e., EBN processing).