Case study A: a numerical example

118

𝐴𝐴(𝑡𝑡𝑣𝑣) =𝑖𝑖 − 0.5

𝑅𝑅 (5.100)

In case of right-censored data, the cumulative distribution F(ti) is given by:

𝐴𝐴(𝑡𝑡_𝑣𝑣) = 1 −(1 − 𝑘𝑘𝑣𝑣) + (1 − 𝑘𝑘𝑣𝑣−1)

2 (5.101)

𝑘𝑘𝑣𝑣= 1 − � � 𝑅𝑅 − 𝑖𝑖 𝑅𝑅 − 1 + 1�^{𝛿𝛿𝑖𝑖}

𝑣𝑣

(5.102)

An automated measurement system that continuously monitor the devices during the test should be implemented in order to acquire the exact Accelerated Time To Failure (ATTF) of each device. To convert the ATTF into the corresponding Time To Failure (TTF) in normal operating conditions the acceleration model described in section 3.6.1 could be implemented.

After that, the obtained data can be used to estimate the parameters of the failure distribution for the considered device. This is the final step of the procedure illustrated in Fig. 5.8. Then, as explained also in Fig. 5.6, if the reliability target come out from the RA process has been achieved it is possible to move to the following components. Otherwise, if the estimated parameters do not allow to fulfill the allocated reliability it is necessary to select another component available on the market and repeat the test. This is done until all the components included in the system have been analyzed.

Obviously, performing ALT leads to a significant increase of the design phase duration and increase the cost of the product. Thus, the use of non-constant failure rate could be sustained only in presence of extremely critical components in case of safety-related systems. Otherwise, the overall procedure could not maintain an adequate cost-benefits ratio.

119 Fig. 5.9. Reliability Block Diagram of Case Study A

The system reliability goal to achieve through the Reliability Allocation procedure is:

𝑅𝑅𝑆𝑆𝑆𝑆𝑆𝑆∗(𝑡𝑡𝑎𝑎)|_{𝑡𝑡𝑎𝑎=8760 ℎ}= 0.99 (5.103)

According to the proposed procedure, the first step required to allocate the component reliability to the system described by case study A is the system decomposition into hierarchical levels, as illustrated in Fig. 5.10.

Fig. 5.10. Case study A: System decomposition into two hierarchical levels.

Then, it is necessary to estimate the influence factors and the weight factors of the items composing the system. After a great number of tests and simulations, the MEOWA method has proven to be the best solution among all the methods presented in this work. The auxiliary vector W solves the problem arisen with the other methods to assign an appropriate weight to influence factors with very high/low values. This is possible precisely tuning the situation parameter

120

α to allocate the proper reliability to the items characterized by extremely high influence factors. Thus, this section applies the proposed method presented in section 5.8. using the influence factors and the weight factor of the 6-parameter MEOWA approach described in section 5.6. TABLE V.VI shows the influence factors of the 6-parameter MEOWA (namely Complexity C, Environmental factor E, State of the art A, Criticality K, maintainability M and safety R) and their assessed values in order to implement the proposed method.

TABLE V.VI

INFLUENCE FACTORS ACCORDING TO 6-PARAMETER MEOWA USED TO IMPLEMENT THE PROPOSED METHOD ON CASE STUDY A.

BRANCH ITEM INFLUENCE FACTORS 𝐂𝐂 𝐄𝐄 𝐀𝐀 𝐊𝐊 𝐌𝐌 𝐑𝐑

1

1.1 1 4 7 7 7 6

1.2 1 5 4 8 8 10

1.3 1 2 10 6 9 3

2

2.1 3 5 9 9 8 8

2.2 2 7 6 9 8 4

2.3 4 6 7 10 6 7

2.4 3 2 6 9 10 9

3 3.1 2 5 10 8 10 3

3.2 2 6 6 7 6 8

4 4.1 7 3 2 6 10 6

5

5.1 2 4 9 8 6 8

5.2 2 3 9 10 6 4

5.3 2 5 9 5 4 3

5.4 1 2 9 5 8 4

5.5 1 2 9 8 6 7

After the assessment of the influence factors for the 15 components included in the system (as in the RBD illustrated in Fig. 5.9.) it is necessary to evaluate the influence factors of the equivalent units (i.e. the red blocks in Fig. 5.10.) according to the rules described in section 5.8.2.

The results for the four equivalent units are illustrated in TABLE V.VII (N.B.

the fourth branch of the RBD is composed by one single component, thus it is not necessary to identify an equivalent unit of the branch.)

121 TABLE V.VII

ESTIMATION OF THE INFLUENCE FACTORS FOR THE EQUIVALENT SUBUNITS. BRANCH ITEM INFLUENCE FACTORS

𝐂𝐂 𝐄𝐄 𝐀𝐀 𝐊𝐊 𝐌𝐌 𝐑𝐑

1 EQ1 1 5 7 6 8 3

2 EQ2 4 7 7 9 8 4

3 EQ3 2 6 8 7 8 3

5 EQ5 2 5 9 5 6 3

Since the top level is composed by a parallel configuration the proposed procedure for redundant architecture as in section 5.8.3.2. must be used. Thus, the inversion of the influence factor as in equation (5.72) and equation (5.73) has been implemented and reported in TABLE V.VIII.

TABLE V.VIII

INVERTED INFLUENCE FACTORS ACCORDING TO THE PROPOSED PROCEDURE. BRANCH ITEM INFLUENCE FACTORS

𝐂𝐂� 𝐄𝐄� 𝐀𝐀� 𝐊𝐊� 𝐌𝐌� 𝐑𝐑�

1 EQ1 10 6 4 5 3 8

2 EQ2 7 4 4 2 3 7

3 EQ3 9 5 3 4 3 8

4 4.1 4 8 9 5 1 5

5 EQ5 9 6 2 6 5 8

The next step is the allocation of the reliability requirements to the top hierarchical level (i.e. the equivalent parallel composed by five branches as in the top side of Fig. 5.10.). Thus, according to the MEOWA method, the influence factor of the equivalent unit must be sorted in descending order, as in TABLE V.IX.

After that, the auxiliary influence factor con be evaluated multiplying the ordered influence factors 𝑏𝑏𝑣𝑣 as in TABLE V.IX. by the 6-parameter auxiliary array 𝑤𝑤𝑣𝑣 as in TABLE V.V. The value of the situation parameter α has been set equal to 0.8 to emphasize the items characterized by high influence factors close to 10. Thus, according to equation (5.77), the inverted situation parameter 𝛼𝛼�

for parallel architecture will be set to 0.7. The products 𝑤𝑤𝑣𝑣∙ 𝑏𝑏_𝑣𝑣, the overall factors 𝑍𝑍𝑃𝑃 as in equation (5.55) and the normalization factor 𝑀𝑀𝛼𝛼 as in equation (5.56) are reported in TABLE V.X.

122

TABLE V.IX

ORDERED INFLUENCE FACTORS FOR THE EQUIVALENT SUBUNITS REQUIRED TO EVALUATE THE WEIGHT FACTOR BY MEANS OF 6-PARAMETER MEOWA.

BRANCH ITEM INFLUENCE FACTORS 𝐛𝐛𝟏𝟏 𝐛𝐛𝟐𝟐 𝐛𝐛𝟑𝟑 𝐛𝐛𝟒𝟒 𝐛𝐛𝟓𝟓 𝐛𝐛𝟔𝟔

1 EQ1 10 8 6 5 4 3

2 EQ2 7 7 4 4 3 2

3 EQ3 9 8 5 4 3 3

4 4.1 9 8 5 5 4 1

5 EQ59 9 8 6 6 5 2

TABLE V.X

AUXILIARY INFLUENCE FACTORS AND OVERALL FACTOR. BRANCH ITEM 𝐰𝐰𝟏𝟏∙ 𝐛𝐛𝟏𝟏 𝐰𝐰𝟐𝟐∙ 𝐛𝐛𝟐𝟐 𝐰𝐰𝟑𝟑∙ 𝐛𝐛𝟑𝟑 𝐰𝐰𝟒𝟒∙ 𝐛𝐛𝟒𝟒 𝐰𝐰𝟓𝟓∙ 𝐛𝐛𝟓𝟓 𝐰𝐰𝟔𝟔∙ 𝐛𝐛𝟔𝟔 𝐙𝐙𝐤𝐤

1 EQ1 3.4749 1.9182 0.9927 0.5708 0.3151 0.16310 7.4348 2 EQ2 2.4325 1.6784 0.6618 0.4566 0.2363 0.10870 5.5743 3 EQ3 3.1274 1.9182 0.8272 0.4566 0.2363 0.1631 6.7289 4 4.1 3.1274 1.9182 0.8272 0.5708 0.3151 0.0544 6.8131 5 EQ5 3.1274 1.9182 0.9927 0.6850 0.3939 0.1087 7.2258

Mα = 33.7769

After that, the values in TABLE V.X. can be used to evaluate the weight factor according to equation (5.56) (MEOWA method) and then the allocated reliability of the five branches using equations (5.74)-(5.77) specifically proposed for parallel architectures. The results are included in TABLE V.XI.

TABLE V.XI

WEIGHT FACTORS AND ALLOCATED RELIABILITY TO THE FIVE PARALLEL BRANCHES.

BRANCH ITEM WEIGHT FACTOR 𝛚𝛚𝐢𝐢

ALLOCATED RELIABILITY

𝐑𝐑_𝐢𝐢^∗(𝐭𝐭𝐚𝐚)

1 EQ1 0.220114 0.637112

2 EQ2 0.165034 0.532337

3 EQ3 0.199215 0.600452

4 4.1 0.201709 0.605014

5 EQ5 0.213928 0.626627

123 This phase concludes the first iteration of the proposed method. Considering the branch number 4, the result included in TABLE V. XI. is directly the reliability to be allocated to component R4.1.

Then, it is necessary to repeat the procedure to the 2^nd hierarchical level in order to assign the reliability of the equivalent units among the components that make up each branch.

Thus, firstly the reliability of the branch number #1 𝑅𝑅𝐸𝐸𝐸𝐸1∗= 0.637112 become the target reliability of a second iteration in which the reliability values are assigned to component R1.1 R1.2 and R1.3 considering a series configuration.

Then, quite similar operation will lead to the assessment of the reliability requirements to each branch. For the sake of brevity, the complete assessment at 2^nd hierarchical level is not fully reported.

As an example, the different steps of the application to branch #3 are reported in TABLE V.XII where the reliability is calculated considering the reliability of the equivalent unit 𝑅𝑅𝐸𝐸𝐸𝐸3∗= 0.600452.

TABLE V.XII

PROPOSED ITERATIVE PROCEDURE:APPLICATION TO BRANCH #3 OF CASE STUDY A.

ITEM 𝐂𝐂 𝐄𝐄 𝐀𝐀 𝐊𝐊 𝐌𝐌 𝐑𝐑

3.1 2 5 10 8 10 3

3.2 2 6 6 7 6 8

ITEM 𝐛𝐛𝟏𝟏 𝐛𝐛𝟐𝟐 𝐛𝐛𝟑𝟑 𝐛𝐛𝟒𝟒 𝐛𝐛𝟓𝟓 𝐛𝐛𝟔𝟔

3.1 10 10 8 5 3 2

3.2 8 7 6 6 6 2

ITEM 𝐰𝐰𝟏𝟏∙ 𝐛𝐛𝟏𝟏 𝐰𝐰𝟐𝟐∙ 𝐛𝐛𝟐𝟐 𝐰𝐰𝟑𝟑∙ 𝐛𝐛𝟑𝟑 𝐰𝐰𝟒𝟒∙ 𝐛𝐛𝟒𝟒 𝐰𝐰𝟓𝟓∙ 𝐛𝐛𝟓𝟓 𝐰𝐰𝟔𝟔∙ 𝐛𝐛𝟔𝟔

3.1 4.7812 2.5475 1.0859 0.3616 0.1156 0.0411 3.2 3.8250 1.7833 0.8144 0.4339 0.2312 0.0411

ITEM 𝐙𝐙𝐤𝐤 𝐌𝐌𝛂𝛂 WEIGHT FACTOR 𝛚𝛚𝐢𝐢 RELIABILITY 𝐑𝐑𝐢𝐢∗(𝐭𝐭𝐚𝐚)

3.1 8.9329 16.0618 0.556160 0.753006

3.2 7.1289 16.0618 0.443840 0.797406

124

Finally, TABLE V. XIII shows the reliability allocation results using the developed iterative approach. As a result, lower reliability is allocated to components with high influence factors.

TABLE V.XIII

OUTPUT OF THE PROPOSED APPROACH FOR CASE STUDY A(WEIGHT FACTORS IN COMPLIANCE WITH 6-PARAMETER MEOWA).

BRANCH ALLOCATED RELIABILITY 1 0.878770 0.849513 0.853435

2 0.852271 0.861832 0.856324 0.846346 3 0.753006 0.797406

4 0.605013

5 0.906414 0.903343 0.921514 0.913622 0.908997

This case study highlights the huge benefits that are achievable using the conditional parameter, in particular when RA procedures are assessed during design phase with imprecise, incomplete or uncertain pieces of information.

The tool calculates also the failure rate to be apportioned to each item, assuming that all the blocks of the system are single elements and not subsystems in turn. Fig. 5 shows an example of the tool outcomes containing the simulation results for MEOWA technique.

Nel documento To my family, (pagine 154-160)