A practical reliability allocation method considering modified criticality factors

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A practical reliability allocation method considering modified criticality factors Om Prakash Yadav, Xing Zhuang

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Reliability Engineering and System Safety

Received date: 1 October 2013 Revised date: 26 March 2014 Accepted date: 9 April 2014 Cite this article as: Om Prakash Yadav, Xing Zhuang, A practical reliability allocation method considering modified criticality factors, Reliability Engineering and System Safety, http://dx.doi.org/10.1016/j.ress.2014.04.003 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

 

A practical reliability allocation method considering modified  criticality factors     

  Om Prakash Yadav and Xing Zhuang      Department of Industrial and Manufacturing Engineering North Dakota State  University, Fargo, ND 58108   

        Abstract  Reliability allocation is a very critical step of product development process for setting achievable reliability goals. The majority of existing methods allocate system level failure rate target to subsystems or components instead of allocating failure rate reduction (improvement) targets. This approach of reliability allocation often allocates reliability goals lower than existing reliability levels. Further, the existing methods failed to effectively capture potential for improvement and impact of improvement efforts on failure effects while calculating reliability allocation weights. This paper investigates the limitations of existing approaches and proposes a modified criticality measure for allocating system level reliability improvement goal to subsystems. The modified criticality measure is developed considering non-linear phenomena for both severity rating and failure rate to capture potential for improvement. The failure rate reduction targets are assigned in proportion of improvement potential. A case example is presented to demonstrate the effectiveness of the proposed approach. Keywords Reliability allocation, criticality, severity rating, failure rate

1   

 

1. Introduction  Reliability allocation is an important and critical step of product development process for setting achievable reliability goals for individual subsystems or components, which are often developed by several suppliers or design teams at different locations. It is, therefore, important to communicate realistic reliability requirements that each design team must achieve and demonstrate during product development process. Further, to design a system for given reliability target, it is essential to set reliability goals considering system behavior and performance by understanding failure behavior, failure effects, and scope of improvement. Moreover, the purpose of reliability allocation methods should be not only to set reliability goals but also to prioritize improvement opportunities based on real potential for reliability improvement. Recognizing the criticality of these issues and concerns, the focus of this study is to devise an approach that provides realistic and achievable reliability targets based on the most crucial information for design improvement. Several approaches have been proposed to obtain reliability allocation weight for allocating the system level reliability goal to each subsystem or component. The existing approaches have attempted to combine several criteria in different combinations for obtaining reliability allocation weight and then allocate system level reliability goal to subsystems or components in proportion of their allocation weights. These methods include Aeronautical Radio Inc. (ARNIC) [1, 2, 3], Advisory Group on Reliability of Electronic Equipment (AGREE)[4], the integrated factor method [5, 6], Karmiol method [7], the maximum entropy ordered weighted averaging method [8, 9], and the feasibility of objectives method [2]. Few researchers have proposed equal apportionment reliability allocation method [10, 11], which appears to be the simplest among available methods. Additionally, several optimization techniques were formulated to deal with redundancy allocation, cost minimization, and enhancement of subsystem or

2   

  component reliability [12-17]. Kuo and Prasad [18] provide an overview of the methods that have been developed since 1977 for solving various reliability optimization problems. A few recent papers attempt to link reliability allocation weight to failure modes and effect analysis (FMEA) information using risk priority number (RPN) or criticality. A comprehensive reliability allocation method [19] considers criticality factor as one of the criteria in determining the allocation weight. The criticality number is obtained by multiplying severity rating and occurrence rating from FMEA that essentially captures the failure effect and frequency of failure. Yadav [20] proposes to capture product behavior through three dimensional analysis and functional dependency of a complex physical system. Itabashi and Yadav [21] use FMEA criticality analysis for reliability allocation. Though these methods derive reliability allocation weights from failure analysis information (RPN or criticality number) or warranty data (failure rate or probability of failure), the simple multiplication of these ratings (severity, occurrence, and detection) for calculating RPN or criticality is based on the assumption of linearity in rating scales. However, in reality there exist a non-linear relationship between any of these ratings and reliability improvement potential. Additionally, assigning system level reliability target to subsystems in proportion of allocation weights sometimes might assign lower reliability target to few subsystems than current level of reliability. Therefore, what is really needed is a more realistic approach to assign system level reliability improvement target (not reliability target) to subsystems in proportion of potential for reliability improvement. However, this requires calculation of allocation weights reflecting potential for reliability improvement and improvement impact on system performance. It has been observed in practice that potential for improvement is much higher when failure rate and/or severity rating is higher as compared to lower failure rate or severity rating situations. Recently, Kim et al. [22] propose a new reliability allocation approach based on the subsystem failure severity and its relative frequency. To reflect the non-linear phenomena of failure severity in reliability allocation, the transformed failure severity, a monotone increasing function of the severity ranking, is used to obtain allocation weight. 3   

  This paper proposes a realistic reliability allocation approach by taking into account the potential for reliability improvement, improvement impact on failure severity, and the degree of difficulty associated with reliability improvement efforts required to achieve given reliability goals. There is a common understanding within design community that the further improvement in reliability for a highly reliable system requires more intense efforts and hence costs more than benefits resulting through reliability improvement as compared to system with higher failure rate. We, therefore, capture this phenomena treating non-linear relationship between failure rate and improvement efforts required to improve subsystem reliability. Similarly, the improvement efforts will have more significant impact on reducing severity effect of failures if we reduce the occurrence of the unacceptable failure effect (severity rating 10 or 9) than reducing moderate failure effects (severity rating 5 or lower) [22]. This explains that there exists a non-linear relationship between improvement impact (reducing failure effects) and severity rating. Considering this non-linear phenomena, this paper proposes a modified criticality factor for calculating reliability allocation weights that essentially captures overall potential for reliability improvement. Further, to allocate system level reliability goal to subsystems, the paper proposes to consider system level reliability improvement target and allocate it to subsystems in proportion of their allocation weights in order to derive reliability goal for each subsystem. The effectiveness of the proposed approach is demonstrated by considering an example taken from Kim et el. [22]. The results of the proposed methodology are compared with Kim et al. [22], which demonstrates reliability allocation more logical and realistic.

2. Literature review    The reliability allocation process can be quite difficult when designing a new system where information on the operational and failure characteristics of the components is relatively unknown. Moreover, allocation of reliability target requires consideration of diverse aspects from component performance, cost factor, complexity, and several other criterions. In the early product development stage, the MIL-HDBK338B [2] is considered the main source to refer for reliability allocation purpose. It describes three 4   

  approaches namely equal apportionment, considering subsystem’s complexity, and feasibility-ofobjectives, respectively. In the absence of definitive information on the system, other than the fact that “n” subsystems are to be used in series, equal apportionment to each subsystem would seem reasonable. The equal apportionment technique assumes a series of “n” subsystems, each of which is to be assigned the same reliability goal. The equal apportionment model is given as:

 ∏ where

     

is the required system reliability and

   

 

(1)

is reliability target assigned to ith subsystem.

A prime weakness of the equal apportionment technique is that the subsystem goals are not assigned in accordance with the degree of difficulty associated with achievement of these goals. However, if additional information is available such as subsystem complexity, failure information, or any other criterion, more realistic approaches were suggested. The Advisory Group on Reliability of Electronic Equipment (AGREE) method [4] proposes to use complexity of a subsystem to derive its reliability allocation weight as follow: i i

∑

,

1,2, … ,

(2)

i

where ni is the number of components or parts in a subsystem i from a system consisting of k subsystems. The weight

i

is then used to allocate reliability target to subsystem i as given below:

 

(3)

Another typical approach mentioned in MIL-HDBK-338B is the feasibility-of-objectives (FOO), in which subsystems are appraised by four categories: intricacy (I), state-of-the-art (S), performance time (P), and environment (E). Each factor is rated on a scale from 1 to 10. The allocation weight is derived by considering the product of these four factors (I×S×P×E). Let Ai = Ii×Si×Pi×Ei represents the multiplication value of these four factors for subsystem i, the reliability allocation weight is calculated as:

5   

 

i i

∑

,

1,2, … ,

(4)

i

Bracha [23] used the similar factors to obtain the reliability allocation weight. Four factors considered in his approach are state of art

, subsystem complexity

, evaluated by number of components in

subsystem i, operating time

, and environmental conditions

. Using these four factors, the

allocation weight is calculated as follows: (5)

∑

After system is launched and put into operation for some time and historical data become available, it is important to take advantage of this information to enhance the accuracy of reliability allocation approach. Keeping this in mind, Karmiol [7] used historical information to derive reliability allocation weight. Considering four factors: state of the art (Ai1), subsystem intricacy (Ai2), operating time (Ai3), and criticality (Ai4), the system reliability allocation weight is defined as:

i1 i

∑

i2 i1

i3 i2

i4 i3

,

1,2, … ,

(6)

i4

It is important to note that in equations (1-6), equal importance is given to all four factors while deriving reliability allocation weights. Recently, several researchers [8, 9, and19] have considered weighted average method for reliability allocation purpose. Assuming

is the importance weight assigned to

factor j, the allocation weight is expressed as: ∑ ∑

Where n is the number of factors and

∑

(7)

is the rating of factor j for subsystem i.

Kuo et al. [24] proposed a similar way for reliability allocation known as Average Weighting Allocation Method. Considering experts rating method, the system is evaluated based on influential factors such as 6   

  complexity, state-of-the-art, system criticality, environment, safety, and maintenance. For each of these factors, the subsystem is rated on a scale of 1 to 10. Assuming q number of experts evaluating a complex system on the basis of p factors, and let

be the rating assigned by mth expert considering the jth factor

for ith subsystem, the average rating is given as: Yi j

∑

mij

/ ,

1,2, … ,

(8)

The reliability allocation weight for ith subsystem is expressed as: ∏ ∑

ij

∏

,

1,2, … , ,

(9)

ij

If sufficient warranty or historical failure information from previous design or experience is available, the failure rate of subsystem i λ can be easily estimated. Considering estimated failure rate information, Aeronautical radio Inc. (ARINC) method [1] can be used to obtain reliability allocation weights as given below: λ

∑

(10)

λ

Boyd [25] proposed a weighted sum method combining equal apportion method and ARINC method to determine reliability allocation method as given below:

1

where

λ

∑

λ

,           0

,

1,

1, 2, … ,

is a weight to combine the ARNIC and equal apportion method and

(11)

safety margin to allow for

design change. It is important to note that most of the above mentioned reliability allocation methods are primarily using qualitative rating methods for each factor to obtain weight

for subsystem i. However, the ARNIC

method seems to be the first reliability allocation approach that clearly considers historical (quantitative) information for obtaining allocation weights. Later, several papers have attempted to link reliability 7   

  allocation process to FMEA information and field failure data. The fundamental reasoning is that the field failure and failure analysis information captures system behavior and performance in real life and hence provides more realistic basis for reliability allocation. Thomas and Richard [26] present a warranty-based practical method to establish reliability targets for reliability improvement of subsystems or components. In this approach authors considered warranty costs as reliability allocation criteria. The rationale for this method is that since the relative quality of a product component is reflected through its warranty, then components that fail frequently or have high warranty expenses or both will have a correspondingly higher warranty burden. It is therefore argued that under reliability improvement initiative the goal should be to reduce failures of each component i in proportion to the relative fraction of warranty cost due to that component. Additionally, several other optimization techniques were formulated to deal with redundancy allocation, cost minimization, and improvement of component reliability. An extensive literature exists on methods to solve reliability allocation problems under a range of conditions on the associated structure, and conditions on variables. Recently, several papers proposed reliability allocation approaches considering failure analysis information generated during FMEA process and historical warranty data. These methods use either risk priority number (RPN) or criticality related information as a major criterion for reliability allocation [19, 21, 27, and 28]. Approaches proposed by [20, 21, and 27] consider the average RPN or criticality values of all failure modes Ci

in a subsystem to derive the criticality of the subsystem i as given below: ∑

ij

ij

,

1,2, … , ,

(12)

Considering the criticality of subsystem (C i , the reliability allocation weight is then calculated as: (13)

∑

where 1

∑

,           

1,2, … ,

8   

 

Wang et al. [19] also use criticality as one of the criteria for formulating a comprehensive reliability allocation method. Yadav [28] proposes a reliability allocation methodology considering failure criticality and functional dependency. The paper brings three dimensional understanding of the physical system to capture functional dependency as well as failure behavior and locates the single point (critical point) for failure. The reliability allocation weight for subsystem i is derived as:

D i

where  

i

 

∑

i D

C i

i C

,

1,2, … , ,

(14)

i

represents functional dependency and

i

measures criticality of subsystem i, whereas

denote the importance assigned to dependency and criticality factors respectively. In this

approach criticality is measured considering severity and occurrence ratings from FMEA documents for each single point failure as given below: ∑ where

∑

indicates the functions supported by subsystem i, and

(15) and

denote the severity and

occurrence rankings for failure mode j of function k supported by subsystem i. While calculating criticality or RPN, all of these methods used simple multiplication of these ratings assigned in the FMEA document on 10-point ordinal scale. Moreover, these ratings are based on the assumption of linearity in rating scale. Therefore, the simple multiplication of these ratings with linearity assumption does not allocate reliability in accordance with degree of difficulty associated in achieving target and also fails to capture potential for improvement. There exists a general understanding among design community that any subsystem with higher failure rate requires relatively less efforts for improvement as compared to lower failure rate. In other words, it is easy to improve reliability of a poor system than improving reliability of a highly reliable system.

9   

  Keeping in mind these observations and limitations of the existing methods, Kim et al. [22] propose a new approach to calculate allocation weights based on the subsystem failure severity and its relative frequency. This approach suggests exponential transformation of the original 10 point ordinal severity rating to reduce severe failure effect effectively. Therefore, authors propose the reliability allocation weight calculation considering the non-linear relationship between failure effect and transformed severity rating as given below:

i i

Si

i

,

∑ i

where

i

Si

1,2, … ,

(16)

i

is the number of failure modes in subsystems i having maximum severity rating,

transformed severity rating,

and

i

  is

is failure frequency of subsystem i caused by the

multiple failure modes. However, this method still assumes linearity between failure frequency and improvement efforts required for improvement. Further, only taking the non-linear relationship between failure effect and severity rating makes the severity rating dominate over failure frequency in the reliability allocation process. Moreover, careful investigation of their results shows that the method proposed by Kim et al. [22] sometimes assigns lower reliability goal to a subsystem already having lower failure frequency meaning the assigned reliability target is lower than current level of reliability. We believe this happens due to the dominance of severity rating over failure frequency in the model suggested and also occurrence rating or failure frequency used in the model still follows linearity assumption. It seems most of the existing methods have not been very successful in effectively capturing potential for improvement, improvement impact, and technological difficulty or complexity of a given subsystem and hence don’t allocate reliability in accordance with degree of difficulty in achieving target. It is therefore necessary to develop more effective and practical reliability allocation approach that considers failure effect and potential for

10   

  reliability improvement to allocate reliability in accordance with degree of difficulty for improvement efforts.

3. Proposed methodology for reliability allocation This section presents a new reliability allocation method based on the system behavior and performance information and potential for reliability improvement. The proposed methodology is developed based on the following considerations. 3.1 Modified criticality factor The criticality in failure analysis process is defined as the product of severity rating (S) of failure events and likelihood of occurrence or occurrence rating (O) and each failure mode is rated by its criticality. This could be a good measure for rating criticality of each failure mode and prioritizing action items for design changes and improvement, but it does not provide explicit means for reliability allocation. The reason is that traditional criticality measure does not give clear sense of potential for reliability improvement and the impact of reliability improvement efforts. The drawback with the traditional criticality measure is that it assumes linearity in existing failure rate and severity rating scales and therefore, considers that same level of improvement efforts are required irrespective of severity rating and occurrence rating values. Moreover, assuming that improvement impact on failure effects will be same by reducing severity ranking by one point anywhere on 10-point ordinal scale is not a rational consideration. We, therefore, propose a modified criticality index for reliability allocation purpose that takes into consideration the potential for reliability improvement (improvement efforts), impact of improvement on failure severity, and the degree of technological difficulty in achieving reliability goals. The following sections provide more detailed discussion on these observations. 3.1.1 Failure rate versus improvement effort

11   

  There is a common understanding among design community that subsystems with higher failure rate must be assigned higher reliability target for further improvement. The calculation of criticality factor considers occurrence rating (O), which essentially is based on possible failure rate. The direct use of occurrence rating or failure rate in calculating criticality factor has some drawbacks from reliability allocation point of view. The occurrence rating uses 10-point ordinal scale and hence does not distinguish effectively the potential for improvement. Some researchers, including most of the traditional reliability allocation methods, have used failure rate for reliability apportionment. However, making reliability allocation decision considering failure frequency directly ignores the potential for reliability improvement and the amount of efforts required to achieve given reliability target. It happens because the underlying assumption is that there exists a linear relationship between failure rate and improvement effort as shown in Figure 1. To give better understanding of our observation, consider two subsystems having different failure rates denoted as

and

, where

. Now suppose design team decides to improve

reliability of both the subsystems by reducing failure rate by same amount, say ∆λ1=∆λ2. Considering linear relationship, the corresponding improvement efforts required for both the systems are denoted as ∆E1 and ∆E2. However, the linearity assumption between failure rate and improvement effort suggests that efforts or costs required to improve the reliability of both the subsystems are same, i.e. ∆E2 = ∆E1 (see Figure 1). This conclusion essentially goes against well-established belief among design community that the further improvement in reliability for a highly reliable system requires more intense efforts and hence costs more.

12   

 

  Figure 1: Failure rate versus improvement efforts    On the contrary, in any given complex system, a subsystem with high failure rate (less reliable) is likely to have more potential for further improvement with relatively less effort. Based on this belief, we assume that the failure rate, λ, is a monotone decreasing function of the improvement effort, E, as shown in Figure 2. The monotone decreasing function confirms with the existing notion of the design community and clearly shows (see Figure 2) that further improvement of a highly reliable subsystem requires much more efforts than less reliable system for the same amount of improvement, i.e. ∆E1 < ∆E2. Based on this observation and understanding, we argue that higher allocation weight should be assigned to subsystems having higher failure rate by considering improvement effort as a criterion because it discriminates potential for improvement more effectively than failure rate.

13   

 

Figure 2: Non‐linear relationship between failure rate and improvement effort    Based on above observations, a mathematical model is proposed to capture the relationship between failure rate and improvement effort. Let E denote the improvement effort and λ denote the failure rate of a subsystem, the rate of change on the curve

is proportional to current failure rate and given as:

(17)

Where

represents constant, indicates decreasing rate of  and

which is always negative. The constant doesn’t influence the rate of change

is the rate of change on the curve

has got some unknown numerical value, which essentially given by the above equation. We therefore can treat the equation

(17) as given below: (18) Since

0, the above equation can be written as: dE

dE lnλ

E 14 

 

  λ

(19)

where C is a constant obtained after integration. The equation (19) gives failure rate in terms of improvement efforts. However, for practical purposes, it is easier to get failure rate information from different sources then to get improvement effort or cost of improvement and therefore, equation (19) can be written as: E

(20)

For the purpose of calculating improvement efforts, we ignore the constant C as it does not affect the overall outcome of the analysis. The failure rate λ of any given subsystem can be obtained from warranty data or test failure data. In the absence of quantitative data, the occurrence rating information from the FMEA document can be used to derive possible failure rate information using following equation [22]: 9.993

ij

where

0.77020

represents occurrence rating of ith subsystem corresponding to jth failure mode. Now assuming

that a system consists of n subsystems with m number of failure modes and let λ be the failure rate of ith subsystem corresponding to jth failure mode. The total failure rate of ith subsystem can be expressed as: λi

∑

ij

,          for j

1,2, … , m

(21)

Considering equations (20) and (21), the improvement effort of ith subsystem is derived as follows:

Ei

i

,             

 

1,2, … ,

(22)

The normalized improvement effort e i is given as:

15   

  ei

∑

(23)

3.1.2 Transformed Severity Rating The effectiveness of improvement efforts can be measured in terms of how much impact each design change has on severity of failure effects. Currently, severity of failure effects is measured on 10-point ordinal scale. Any change in severity rating after undertaking necessary improvement actions should reflect the impact of improvement efforts. However, the use of 10-point ordinal scale for measuring improvement impact and subsequently using for reliability allocation purpose is not a logical approach. The reason is that subtraction of ordinal scale assumes equal difference between any two given intervals. For example, the difference between “hazardous effect without warning” and “hazardous effect with warning” is same as the difference between “low effect” and “very low effect” that essentially shows that improvement impact is same, i.e. ∆S1=∆S2 (see Figure 3). However, in reality reducing severity rating from “hazardous effect without warning” to “hazardous effect with warning” has much more significant impact on failure effect than simply reducing from “low effect” to “very low effect”. This clearly indicates that the use of 10-point ordinal scale in severity rating is not an appropriate measure to be utilized directly for reliability allocation purpose. Logically, design engineers need to pay more attention to subsystems having higher severity rating because failures of those subsystems will have huge effect not only on cost and the environment, but also on public safety. Consequently, improvements on subsystems having higher severity rating are invested and impacts are expected. Since the purpose of design improvement is to reduce the occurrences of the unacceptable failure effect, the severity rating needs to be able to reflect the failure severity and its impact of reducing failures. However, the current ordinal scale based severity rating approach assumes linearity with failure effects and hence fails to distinguish the impact of improvement efforts. This proves that there exists non-linear relationship between improvement impact and severity rating.

16   

 

  Figure 3: Severity rating and failure effect  To capture the above observation and reflect improvement impact in terms of severity rating, Kim et al. [22] have proposed a transformed severity rating that clearly distinguishes impact of improvement efforts on reducing failure effect. The transformed severity rating (S ij ) is a monotone increasing function of the ordinal severity ranking 

( S ). The transformed severity rating clearly

, which is given as S ij = exp α

shows that impact of improvement efforts is much greater ∆

ij

∆

when reducing failure effects

from “hazardous without warning” to “hazardous with warning” than reducing failure effects from “low effects” to “very low effects” (see Figure 4). This phenomenon clearly explains that reducing failure occurrence of a subsystem having higher severity rating will have greater impact in terms of improving (reducing) failure effects and reliability improvement, and therefore, transformed severity rating must be considered for reliability allocation purpose.

17   

 

  Figure 4: Transformed severity ratings and failure effect [22]    Additionally, a subsystem may be subjected to several different failure modes, each having different level of severity rating. In that case we need to select the severity rating that truly represents severity of the subsystem under consideration and therefore, as proposed by Kim et al. [22], the most severe failure mode should be considered to present the system’s failure effect precisely. Accordingly, the severity of an ith subsystem having multiple failure modes is defined as:

S i = Max ( Si1 , Si 2 , … , Sij ), Where

Sij  

j=1, 2, …, n

is the transformed severity rating of failure mode j in subsystem i.

The normalized severity rating

is given as:

si

Si ∑

(24)

Si

18   

 

It is important to note here that both improvement efforts

and tranformed severity rating S i are of

different magnitude in terms of their numerical values. We, therefore, normalize both the measure to avoid dominance of one over the other. 3.1.3 Proposed criticality index The criticality index in FMEA process is basically derived by multiplying severity rating and occurrence rating and is given as c i

si

o i . Nevertheless, the traditional criticality index is not a very effective

criteria for reliability allocation as highlighted earlier. Therefore, to develop a relistic reliability allocation approach, one must be taking into account the potential of reliability improvement (or improvement efforts required), the improvement impact on severity of failure effects, and degree of difficulty (complexity) associated with achieving the given reliability goals. Based on the our observations, we propose the following modified criticality criteria for reliability allocation :                                

Considering both

 and

(25)

equal to one, the equation (25) can be written as:

i

i

,        

1,2, … , ,

(26)

i

where

represents number of failure modes having maximum severity rating and

denotes level of

technical difficulty (complexity) for improvement in achieving given reliability target for ith subsystem. Considering

and

equal to one, as one can see from equation (26), it basically follows the concept of

traditional criticality measure but provides more logical base for reliability allocation. The modified criticality index for each subsystem is then used to calculate reliability allocation weight by normalize the modified criticality index as given below: 19   

 

i i

∑

,          

1,2, … , ,

(27)

i

The modified criticality measure actually captures potential for reliability improvement

or

efforts/cost required for improvement and possible impact of each improvement effort on reducing the severity rating or failure effects

. It further allows to consider the degree of technological difficulty

associated with design improvement for setting reliability target for a subsystem, as compared to other subsystems, by assigning higher value to

, which redefines improvement potential as

. The

equations (25 and 26) clearly show that modified criticality rating will be higher if system has higher severity rating and higher potential for improvement (less efforts required for improvement and technologically feasible) We, therefore, strongly recommend to use modified criticality index for reliability allocation purposes. If other factors such as complexity, functionality, effectiveness, and quality are not very important for reliability allocation purpose, the modified criticality index can be used for reliability allocation. However, if there are few other impotant criteria that need to be considered for reliability allocation, those can be easily incorporated in it. The next section discusses the system level reliability allocation approach to subsystem level. 3.3 The reliability allocation The most of the existing reliability allocation methods allocate reliability target to subsystems in proportion to allocation weights as given below: λ where

λ

(28)

represents allocation weight for subsystem i and λ denotes system level reliability target.

Generally allocation weights are derived considering existing failure rates, severity ratings, cost, and several other factors and therefore, equation (28) provides a basis to allocate reliability target in 20   

  proportion to allocation weight. On the other hand, in the proposed approach the modified criticality factor considers potential for improvement, impact of improvement on failure severity, as well as technological difficulty and efforts required for achieving improvement targets. The higher criticality factor means higher potential for improvement, significant improvement in terms of reducing severity of failure modes, and lesser amount of efforts required to achieve the improvement targets. Therefore, in the proposed approach we consider the improvement target over existing reliability level ∆λ

λ

λ  and

assign the system level improvement target to subsystems in proportion to allocation weights as given below: ∆λ

∆λ

(29)

The reliability allocation target for each subsystem is then given as: λ

λ λ

λ

∆λ

(30) ∆λ

(31)

4. Case example This section discusses the effectiveness of the proposed reliability allocation approach by considering an example from Kim et al. [22]. Since the proposed method builds on the approaches suggested by these researchers, it is imperative to compare and discuss the merits of the proposed approach with their methodology.

4.1. Example: A heating, ventilation, and air conditioning (HVAC) system  Kim et al. [22] illustrate their reliability allocation method by taking HVAC system as an example. The current HVAC system failure rate is given as  λ

0.01815135. The authors considered 20%

improvement in failure rate and decided system level failure rate target  λ

0.01452108 and reliability 21 

 

  improvement target ∆λ

0.0036303. Table 1 shows the reliability allocation weights and failure rates

allocated to subsystems using the methodology suggested by Kim et al. [22]. As discussed earlier, this approach calculates allocation weight rating as  

exp 

by using equation (16), which considers transformed severity

and failure frequency of subsystem i caused by multiple failure modes. It is

important to note that in this case the system level failure rate target is assigned to each subsystem in proportion of allocation weight. To investigate and analyze the results of their approach we first consider two subsystems namely reservoir and pump 1. The severity rating of reservoir is higher than pump 1 and the failure rate of pump 1 is higher than reservoir, but calculated allocation weight of pump 1 is almost four times higher as compared to reservoir. This inconsistency in allocation weight calculation ends up assigning higher failure rate target to subsystem ‘reservoir’ than current failure rate meaning assigning lower reliability goal than current reliability level. The similar discrepancy can be observed when we analyze allocation weights for cooling tower versus pump 2 and pump 2 versus air handler. In these cases also higher failure rate targets are assigned to cooling tower and air handler as compared to their current level of failure rates. Although this approach captures nonlinearity in severity rating scale; it seems the equation used to calculate allocation weights diminishes the impact of transformed severity ratings due to higher numerical value and ends up assigning more importance to failure rates.

22   

  Table 1: Results of the reliability allocation weight Using Kim et al. [22] method.  i

Subsystem

FMs

Sij

Oij

λij

λi

Wi

λ* i

1

Reservoir

FM11

6

2

0.000213

0.000213

0.0354

0.00051367

2

Pump 1

FM21

4

3

0.000461

0.005107

0.1297

0.00188491

FM22

5

6

0.004646

FM31

6

3

0.000461

0.001383

0.1061

0.00154102

FM32

5

3

0.000461

FM33

4

3

0.000461

FM41

4

1

0.000099

0.000312

0.0157

0.00022758

FM42

8

2

0.000213

FM51

6

7

0.010036

0.010249

0.5149

0.00747327

FM52

8

2

0.000213

FM61

4

3

0.000461

0.000887

0.1983

0.00288063

FM62

3

2

0.000213

FM63

7

2

0.000213

3

4

5

6

Total

Cooling tower

Pump 2

Chiller

Air handler

0.0181514

0.0181514

0.00363027

On the other hand, the proposed approach ensures that both factors are normalized to avoid dominance of one factor over other. Further, the modified criticality equation provides more rationale and realistic way to calculate allocation weights reflecting potential for further improvement. Table 2 provides allocation weights and failure rates allocated to subsystems considering modified criticality factor given by equation (25). While calculating allocation weights, we considered degree of difficulty

as 3 for subsystem 1

(reservoir) and 5 for subsystem 4 (pump 2). For the reason that failure rates for both subsystems are very low (lowest as compared to other subsystem) and transformed severity ratings are very high resulting in higher allocation weights if we consider degree of difficulty level same for all subsystems. Essentially the existing lower failure rates of these subsystems indicate the difficulty in achieving improvement targets in proportion of allocation weights. Therefore, assigning

values lower than these values might end up

23   

  giving infeasible reliability improvement targets (negative failure rate target) that in essence validates our concern of difficulty in meeting improvement targets. To further underline our concern, it is very important to mention here that initially we calculated allocation weights considering the same level of degree of defficulty associated (technological defficulty 1) for all subsystems, which assigned relatively higher weights to subsystem 1 (reservoir) and subsystem 4 (pump 2). Although the current failure rates of these subsystems are very low as compared to other subsystems, the relatively higher severity ratings contributed to higher allocation weights leading to infeasible improvement targets. Here infeasible improvement target means the model allocates failure rate improvement target ∆λ failure rate targets λ

that is higher than the current failure rate and hence ends up assigning negative

to these subsystems. However, it does provide a very good insight that although

the transformed severity rating is very high, it is practically very difficulty to reduce failure rate beyond a certain point indicating degree of difficulty assocaited is higher as compared to other subsystems. It is, therefore, important to take into account the degree of difficulty assocaited while assigning reliability improvement targets.  

 

24   

  Table 2: Result of proposed allocation weight method  Ei

si

ei

ci

λi

Wi

∆λ*i

λ*i=λi-∆λ*i

121.5104

0.0845

0.0686

0.2114

0.1622278

0.000213

0.032010868

0.000116209

0.00009679

2

54.5982

0.0528

0.0308

0.1319

0.2335553

0.005107

0.046085237

0.000167303

0.00493970

3

121.5104

0.0658

0.0686

0.1646

0.4166505

0.001383

0.082213675

0.00029846

0.00108454

4

601.8450

0.0807

0.3397

0.2018

0.4207549

0.000312

0.083023553

0.0003014

0.00001060

5

601.8450

0.0458

0.3397

0.1145

2.9660443

0.010249

0.585261214

0.002124674

0.00812433

6

270.4264

0.0703

0.1526

0.1757

0.8686654

0.000887

0.171405453

0.000622253

0.00026475

0.0036303

0.01452108

i

Si

1

total

0.0181514

As shown in Table 2, the proposed approach assigns reliability improvement targets ∆λ

in proportion

of allocation weights and therefore, allocates large portion of system level improvement target ∆λ

to

two subsystems (chiller and air handler) because of higher potential for reliability improvement. The model assigns higher allocation weight to subsystem 5 (chiller) realizing lower improvement efforts required because of highest failure rate (λ

0.010246 among all subsystems and higher impact of

improvement on failure severity as severity rating is also very high for this subsystem. Similarly, the subsystem 6 (air handler) also has higher failure rate and severity rating among remaining subsystems indicating higher potential for reliability improvement. It is, therefore, reasonable to assign higher improvement targets to chiller and air handler. Consequently, the failure rate improvement targets ∆λ column of Table 2 clearly shows that each subsystem has been assigned failure rate improvement target in proportion of potential for improvement (or allocation weights), which confirms to the reasoning given by Thomas and Richard [26] for assigning failure rate reduction targets to subsystems or components. The last column λ  of the Table 2 shows failure rate target for each subsystem. 25   

  This clearly indicates that the proposed approach effectively captures the impact of severity, degree of difficulty (technological difficulty) associated, and the improvement efforts required achieving given reliability target of the subsystem while calculating allocation weights. Further, the variation in allocation weights calculated using proposed model and their comparison with corresponding weights calculated using approach suggested by Kim et al. [22] clearly reflect observations and understanding captured in the proposed approach. For example, the proposed approach is able to capture that subsystem reservoir has higher severity rating and therefore, impact of severity is higher as compared to pump 1 (see column

).

On the other hand, the failure rate of pump 1 is higher than reservoir and hence is likely to have higher potential for further improvement (see column 

in comparison to reservoir. However, when compared

with other subsystems, these two subsystems are relatively better placed from reliability point of view, meaning having lower failure rates and severity ratings. Therefore, considering the modified criticality factor and feasibility aspect, the reliability allocation weights assigned to reservoir and pump 1 are lower and having less variability as compared to Kim et al. [22] leading to lower improvement targets for these two subsystems. On the other hand, subsystem 5 (chiller) has very high (highest) failure rate and severity rating and therefore, assigned highest failure rate improvement target ∆λ

0.002184249 that

accounts for almost 60% of system level failure rate improvement target. To further compare the results, the failure rates allocated to three subsystems (reservoir, cooling tower, and air handler) using the approach suggested by Kim et al. [22] are higher than current failure rates λ

λ

, which essentially means the assigned reliability targets are lower than current reliability levels

of these three subsystems. On the other hand, the proposed approach allocates the reliability improvement targets ∆λ  to each subsystem in proportion of allocation weights that represent potential for reliability improvement for each subsystem. Therefore, it allocates the failure rate target to each subsystem that will always be lower or at least equal to current failure rate  λ

λ . Further, if any subsystem already has

very high reliability and does not show much potential for further improvement or design team decides not to undertake any improvement initiative for that subsystem, the proposed approach allows to assign 26   

  minimal weight (close to zero) that ensures the reliability improvement goal is close to current level of reliability. This way the proposed model provides more flexibility to design engineers to deal with infeasibility and technological difficulty in setting targets and achieving further reliability improvements.

5. Conclusion The proposed method provides a more realistic and effective reliability allocation approach considering potential for improvement. The consideration of non-linear phenomena in severity rating and failure rate to capture potential for reliability improvement has significantly improved the effectiveness of the proposed approach over other methods. The reliability improvement target allocation method ensures each subsystem gets improvement target assigned in proportion of potential for improvement as well as exposes technological difficulty in achieving improvement goals when negative failure rate target is assigned. It provides more flexibility to design teams in setting targets keeping mind infeasibility issues and strategic issues as well. The methodology provides effective link between failure analysis process and design and development process. It gives more rationale and logical means for developing strategic goals to improve product reliability but not necessary sole mean. The proposed approach is more useful for existing systems where objective is to identify improvement opportunities and set improvement goals for subsystems or components.

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