LOS ESTANDARES PROBATORIOS

RPN for quantifying risk in the FMEA system has several flaws as previously explained, such as the challenge of aggregating imprecise multiple experts’ information. To holistically address these FMEA challenges, two novel methods are proposed for risk prioritisation for marine machinery systems:

(1) AVRPN: AVeraging technique for data aggregation and Risk Priority Number evaluation

(2) AVTOPSIS: AVeraging technique for data aggregation with TOPSIS. These are explained in the following sections.

3.3.1 AVRPN: AVeraging technique for data aggregation and Risk Priority Number evaluation

AVRPN is a combination of an averaging technique and the RPN. The averaging technique is applied in converting experts’ imprecise ratings into precise ratings while the RPN is used as a tool for the ranking of the failure modes.

3.3.1.1 Averaging technique for data aggregation:

The averaging technique is a data aggregation method principally designed for aggregating imprecise values of individual expert’s criteria ratings (O, S and D) such that the imprecisions are captured as an expectation interval. The mean value of the maximum and minimum bounds of the expectation interval is then used as the input to the chosen methodologies such as RPN, TOPSIS, VIKOR and CP for the ranking of the risk of each the failure modes. The steps are as follows:

(1) Formation of decision matrix. The values assigned by an expert to failure modes against risk criteria are used to form a decision matrix (m x n). Where m is the number of failure modes and n is the number of criteria.

Risk criteria rating information obtained from experts is used to form a matrix of m-failure modes with the rating value for each of n-decision criteria.

(2) Computation of the minimum and maximum risk criteria values

The risk criteria data for producing the decision matrix can take the following form (Chin et al., 2009a)

(a) A Precise rating is identified with single confidence of 100%. For example, if the rating is 5 this can be written as 5:100%.

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(b) A Complete distribution such as 5:80% and 7:20% means that a value of 5 at 80% confidence and 7 at 20% confidence is assigned to a failure mode against a risk criterion with the confidence summing to 100%.

(c) An incomplete or imprecise distribution such as 7:30% and 8:60% means a value of 7 at 30% confidence and 8 at 60% confidence with 10% confidence missing. The missing 10% confidence is usually called local ignorance and could be assigned to any rating between 1 and 10 (Shafer, 1976).

The incomplete or imprecise assessment can be represented as an expectation interval whose minimum and maximum risk criteria values are evaluated as follows (Chin et al., 2009b): 𝑥_𝑖𝑗𝑚𝑖𝑛 _{= 𝑥} 𝑖𝑗 1_{. 𝑝} 𝑖𝑗1 + 𝑥𝑖𝑗2. 𝑝𝑖𝑗2 + [1. (100% − 𝑝𝑖𝑗1 − 𝑝𝑖𝑗2)] (3.1) 𝑥_𝑖𝑗𝑚𝑎𝑥 _{= 𝑥} 𝑖𝑗1. 𝑝𝑖𝑗1 + 𝑥𝑖𝑗2. 𝑝𝑖𝑗2 + [10. (100% − 𝑝𝑖𝑗1 − 𝑝𝑖𝑗2)] (3.2) Where

𝑥_𝑖𝑗𝑚𝑖𝑛 _{is the minimum rating of failure mode 𝑖 with respect to risk criterion 𝑗}

𝑥_𝑖𝑗𝑚𝑎𝑥 is the maximum rating of failure mode 𝑖 with respect to risk criterion 𝑗 𝑥_𝑖𝑗1 _{and 𝑥}

𝑖𝑗2 are the distribution ratings of failure mode 𝑖 with respect to risk criterion 𝑗

assigned by an expert at percentage confidence 𝑝_𝑖𝑗1 _{and 𝑝}

𝑖𝑗2 respectively.

3. Computation of the mean rating of failure mode 𝑖 with respect to risk criteria 𝑗

After determination of the minimum and maximum rating values of failure mode 𝑖 with respect to risk criterion 𝑗 , the average may be calculated to obtain the mean rating of failure mode 𝑖 with respect to risk criterion 𝑗 as follows:

𝑥𝑖𝑗 =

𝑥_𝑖𝑗𝑚𝑖𝑛_{+ 𝑥}

𝑖𝑗𝑚𝑎𝑥

2 (3.3)

Where 𝑥_𝑖𝑗 is the mean rating of failure mode 𝑖 with respect to risk criterion 𝑗

The next step is to use the value of 𝑥_𝑖𝑗 as the input to the RPN calculation or any other risk of failure modes ranking tool such as TOPSIS, VIKOR and CP.

60 3.3.1.2 Failure mode ranking tool; RPN

The mean rating of failure modes 𝑖 with respect to risk criteria 𝑗 are used as inputs in the RPN model to evaluate the risk of each failure mode as follows:

𝑅𝑃𝑁_𝑖 = ∏ 𝑥_𝑖𝑗 𝑛 𝑗=1 ∶ 𝑖 = 1 , 2 … , 𝑚 , 𝑗 = 1, 2, … 𝑛 (3.4)

Where 𝑅𝑃𝑁_𝑖 is the risk priority number of the failure mode 𝑖.

An alternative approach is to feed Eq. (3.1) – (3.3) separately into the RPN model rather than feeding only Eq. (3.3) to obtain maximum, minimum and mean risks of each failure mode. However when dealing with a complete distribution risk criteria problem, Eq. (3.1) – (3.3) generate the same result as using Eq. (3.3) alone. In that case Eq. (3.1) and (3.2) are equal, since local ignorance will be zero. Where data is available from multiple experts, RPN values from the individuals are averaged to obtain the risk of each failure mode.

3.3.2 AVTOPSIS: AVeraging technique for data aggregation and TOPSIS method

AVTOPSIS is a combination of the averaging technique and TOPSIS. The averaging technique is used in aggregating imprecise rating of failure modes from experts while the TOPSIS is used in the ranking of risk of failure modes.

The averaging technique has been described in Section 3.3.1.1.

3.3.2.1 Failure mode ranking tool; TOPSIS

TOPSIS is a technique for order preference by similarity to the ideal solution and was first proposed by Hwang and Yoon in 1981 (Hwang and Yoon, 1981). The concept of TOPSIS is that the best alternative is usually the one which is closest to the ideal solution and farthest from the negative ideal solution (Yoon and Hwang, 1995). In this chapter the best alternative is the failure mode that poses the greatest risk to the system under investigation. Although TOPSIS has many advantages, the rating methodology uses precise values and in effect is incapable of dealing with some real life problems where data may be imprecise or incomplete. To address these challenges the averaging technique for data aggregation detailed in Section

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3.3.1.1 has been integrated with TOPSIS for prioritisation of risk in machinery systems. In this case, the mean values of O, S, and D are used as input data for the TOPSIS methodology. The TOPSIS methodology steps applied here are as shown in Çalişkan et al. (2013). Although the TOPSIS model is capable of incorporating more than three risk criteria, the number of risk criteria were was limited to three here for an unbiased comparison with the output of AVRPN. The steps involved in the TOPSIS methodology are as follows:

(1) Formation of decision matrix:

Since the problem is one of dealing with imprecise or incomplete risk criteria rating, a decision matrix is formed using values obtained at the aggregation stage. The decision matrix, X, may be represented as:

𝑋 = (𝑥𝑖𝑗)_𝑚.𝑛 (3.5)

(2) Normalization of the decision matrix.

Normalization of the decision matrix is carried out as follow: 𝑟𝑖𝑗 = 𝑥_𝑖𝑗 √∑𝑚𝑖=1𝑥𝑖𝑗2 , 𝑖 = 1, … 𝑚 ; 𝑗 = 1, … , 𝑛 (3.6) Where 𝑟_𝑖𝑗 are the normalised criteria ratings.

(3) Calculation of the weighted normalised decision matrix:

The weighted normalised decision matrix can be calculated by multiplying each row of the normalised decision matrix by the weight 𝑤_𝑗 of each criterion:

𝑣_𝑖𝑗 = 𝑤_𝑗𝑟_{𝑖𝑗 ,} 𝑖 = 1, … , 𝑚 ; 𝑗 = 1, … , 𝑛 (3.7) Where 𝑤𝑗 is the weight of the 𝑗𝑡ℎ criterion.

(4) Computation of the weights of decision criteria:

In the literature, many methods are reported for assigning the weight of risk criteria such as the entropy method, AHP, ANP etc. (Ölçer and Odabaşi, 2005, Chu et al., 2007b, Çalişkan et al., 2013, Liou and Chuang, 2010). For this particular solution to the risk prioritisation problem, the entropy method was adopted because of its dynamism and objectivity in

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weighting of risk criteria relative to the decision making process, as opposed to AHP and ANP and other prior weighting methods that assign weight subjectively and independent of the decision making process. The steps are as follows (Çalişkan et al., 2013):

Using the normalised decision matrix, the entropy value 𝑒𝑗 of 𝑗𝑡ℎ criterion is calculated as

follow: 𝑒_𝑗 = −𝑘 ∑ 𝑟_𝑖𝑗 𝑚 𝑖=1 ln 𝑟_𝑖𝑗 (3.8) Where 𝑘 = 1

ln 𝑚 is a constant which guarantees 0 ≤ 𝑒𝑗 ≤ 1and m is the number of failure

modes.

The objective weight for each risk criterion is then given by 𝑤_𝑗 = 1 − e𝑗

∑𝑛 1 − e_𝑗

𝑗=1

(3.9)

(5) Determination of the positive-ideal and negative-ideal solutions.

The reference values for risk prioritisation are the positive and negative ideal solutions. The positive ideal solution, 𝐴+_{, is the best value of each weighted criterion and the negative ideal}

solution, 𝐴−_{, is the worst value of each weighted criterion and are determined as follows:}

𝐴+ _{= {𝑣}

1+ , 𝑣2 ,…,+ 𝑣𝑛+} = {(max_𝑖 𝑣𝑖𝑗|𝑗𝜖𝐼) , (min_𝑖 𝑣𝑖𝑗|𝑗𝜖𝐼′)} (3.10)

𝐴− _{= {𝑣}

1− , 𝑣2 ,…,− 𝑣𝑛−} = {(min_𝑖 𝑣𝑖𝑗|𝑗𝜖𝐼) , (max_𝑖 𝑣𝑖𝑗|𝑗𝜖𝐼′)} (3.11)

Where 𝐼 is associated with the benefit criteria and 𝐼′ _{is associated with cost criteria}

(6) Determination of the distance from positive-ideal and negative-ideal solutions.

The distance of each failure mode from the positive-ideal solution, 𝐷_𝑖+, and from the negative- ideal solution, 𝐷_𝑖−, are evaluated, respectively as:

63 𝐷_𝑖+ = √∑( 𝑣𝑖𝑗 − 𝑣𝑗+) 2 𝑛 𝑗=1 𝑖 = 1,2 … . 𝑚; 𝑗 = 1, 2, … , 𝑛 (3.12) 𝐷_𝑖− _{= √∑( 𝑣} 𝑖𝑗 − 𝑣𝑗− ) 2 𝑛 𝑗=1 𝑖 = 1,2, … , 𝑚; 𝑗 = 1,2, … , 𝑛 (3.13)

(7) Computation of the relative closeness of failure mode 𝑖 to the positive ideal solution The relative closeness 𝑅𝐶𝑖 of each failure mode to the positive ideal solution is computed as:

𝑅𝐶𝑖 = 𝐷_𝑖− 𝐷_𝑖+ _{+ 𝐷} 𝑖− , 𝑖 = 1 , … , 𝑚 (3.14)

The 𝑅𝐶_𝑖 value is the risk index of the failure modes. The higher the value the greater the risk the failure mode poses to the system.

(8) Computation of mean risk of failure modes:

Finally, where data is available from multiple experts, 𝑅𝐶_𝑖 values from the individuals are averaged to obtain the mean risk of each failure mode.