Coherent System Reliability Using a PCA Based Multi-Response Optimization Nasser Fard, PhD, Northeastern University Huyang Xu, Northeastern University Yuanchen Fang, Northeastern University Key Words: Covariate Analysis, Multi-Response Optimization, Weighted Principal Component Analysis SUMMARY & CONCLUSIONS A functional package of components in a system can be represented as a module [1]. For a modular coherent binary system (MCBS), a robust design of experiment is proposed in this paper. Optimization of system reliability in a design phase could be attained by simultaneous optimization of organizational patterns of modules and module reliabilities. On the other hand, among all the methods for multi-response optimization, the weighted principal component analysis (WPCA)-based approach has shown high efficiency, since it encompasses all the possible correlations between response variables without increasing the computational complexity. By applying WPCA, the original responses (cause-specific reliabilities) are transformed to a set of uncorrelated principal components (PCs) and a multi-response performance index (MPI) is obtained to maximize the system reliability. However, eigenvectors obtained from WPCA are not unique. To solve this problem, the proposed indexing method will determine a unique optimal solution for MCS reliability optimization. 1 INTRODUCTION In many practical situations, such as power supply systems [2], coal transportation subsystems [3], and production systems [4], the system has more than two levels of performance, varying from perfect operating to complete failures [5]. Consistently, when the perspective of research focuses on the modular design phase, the system is characterized by functional partitioning into a discrete scalable, reusable module and the variety of system reliability can be explained as the result of the cumulative effect on entire system performance caused by the different performance levels of each module. Therefore, the improvement of system reliability in a MCBS is the product of alternate design that comprises the use of alternate structures and components. Correspondingly, “Alternate structures” can be defined so that the system function can be performed by different implementations through a reorganization of the system structure. On the other hand, “alternative components” are designated when one or more components can be used to substitute the intended component in a module to perform an

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identical function. Additionally, considering the necessity of redundancy, some modules usually contain more than one component in order to approach the high reliability requirement of the corresponding module. The reliability of such systems depends on combinations of covariates affected by alternate system structure and alternative modular technical specifications, including alternative component and redundancy policy. In this paper, multi-response optimization with WPCA is proposed in order to find the optimal design scheme for a MCBS attained by the simultaneous optimization of organizational patterns of modules and module reliability. That is, if the reliability of each specific failure caused by system reliability paths is seen as a response variable, the optimization can be designed as a multi-response optimization problem, and the number of responses is equal to the number of cause-specific system failures. Accordingly, many approaches for multi-response optimization such as assigning weight to response variables, grey relational analysis and multiple regression models have been proposed in recent years. Among all the methods for multi-response optimization, WPCA-based approach has shown high efficiency, since it takes into account all the possible correlations between response variables without increasing the computational complexity. By applying WPCA, the original responses are transformed to a set of uncorrelated PCs. Weights are assigned to each PC, and the PCs are combined into one multi-response performance index (MPI). The choice of a significant factor-level combination is based on ANOVA of MPI. Through WPCA-based multi-response optimization, the optimal levels of covariates, which maximize the system reliability, will be obtained. However, each eigenvalue obtained from the application of PCA in the multi-response optimization has a set of eigenvectors, and different eigenvectors corresponding to each eigenvalue will lead to different results [6]. To solve this problem, a method for determining the optimal eigenvector combination is proposed. Once the optimal eigenvector combination is determined, these eigenvectors are used to calculate the multi-response performance indices, and then the optimal factor-level combination will be uniquely determined.

This paper is organized as follows. The mathematical descriptions for the MCBS are discussed in Section 2, then the corresponding multiple-response optimization considering index method is proposed. In Section 3, a numerical example is given, and Section 4 provides concluding remarks. 2 MODEL DESCRIPTIONS Suppose a system is consisted of ܯmodules, and the ݅th module is one parallel-redundant subsystem with ܥሺሻ components for ݅ אሼͳǡ ڮǡ ܯሽ. All of the components in each module are assumed to perform the same sub-function of the system. Let ܥ denote the ݆th component in the ݅th module, where ݅ ൌ ͳǡʹǡ ǥ ǡ ܯand ݆ ൌ ͳǡʹǡ ǥ ǡ ܥሺሻ . Figure 1 shows a simple example of a system with 5 modules.

Figure - 1 A five-modular system example 2.1 Alternative Approaches for Reliability Improvement In order to improve the reliability of such a system, one way is to adjust the system structure without loss of the system function. Suppose the system has S alternative structures with equivalent functions, which are labeled as 1, 2, …, S. For example, Figure 2 shows alternative configurations of the system in Figure 1.

corresponds to a price ୧୩ , where ൌ ͳǡʹǡ ǥ ǡ and ൌ ͳǡʹǡ ǥ ǡ ሺ୧ሻ . Additionally, the traditional method with the purpose of improving the reliability of a system is to increase the number of redundant components within each module, which usually deteriorate product properties such as cost, weight, and size etc. Therefore, the number of components in each module, ܥሺሻ , is a type of negative measurement for product design improvement, on the other hand, boosting system reliability performance. The corresponding tradeoff is usually decided in order to balance overall product design specifications. In this paper, based on the fact that the product cost is the most important factor for marketing, the cost-dependent types of components are a significant factor considered in the optimization process. In summary, there are three main factors that need to be determined in system reliability optimization, which are: alternative system structure, amount of redundancy, and alternative components. System structure - For alternative system structure, the factor is the structure reference number and its corresponding levels are denoted by 1, 2, …, ܵ, as stated above. Number of Components in each module - There are ܯ factors for a ܯ-modular system, which are the number of redundant components for module 1, the number of redundant components for module 2, …, and the number of redundant components for module ܯ. These factors are denoted by ሺపሻ be the maximum number of ܥሺଵሻ ǡ ܥሺଶሻ ǡ ǥ ǡ ܥሺெሻ . Let ܥ෪ redundant components that can be comprised in module݅ . ሺపሻ . The factor levels for module ݅ are 1, 2, …, ܥ෪ Component types in each module - For alternative components, the factors are component selection for module 1, component selection for module 2, …, and component selection for module ܯ. For simplicity, these factors are ሺሻ ݊ ۍଵ ې ݊ ێሺሻ ۑ denoted by CS1, CS2, …, and CSM. An array, ێଶ ۑ, is ڭ ێሺሻ ۑ ݊ ۏைሺሻ ے defined to store the number of components chosen from ሺሻ different options, where ݊ is the number of components of ሺሻ option ݇ in the module ݅, ݇ ൌ ͳǡʹǡ ǥ ǡ ܱሺሻ , ݊ ൌ Ͳǡͳǡ ǥ ǡ ܱሺሻ , ሺሻ ሺሻ ሺሻ and ݊ଵ ݊ଶ ڮ ݊ைሺሻ ൌ ܥሺሻ , and the factor levels ሺሻ

Figure 2 - Two alternative structures of the system in Figure 1 Another way to improve the system reliability is to choose different components within each module. Components of the same function, but made from different manufacturers, have different reliabilities and different prices. Suppose there are ሺ୧ሻ options for the components in the th module to choose from, and the th option for the th module

݊ ۍଵ ې ݊ ێሺሻ ۑ corresponding to CS݅ is represented by ێଶ ۑ. ڭ ێሺሻ ۑ ݊ ۏைሺሻ ے The factor levels and combinations for ܯ-modular system reliability optimization are summarized in Table 1. From table 1, it can be seen that to optimize the reliability of a system with ܯ modules, there are ݆ ெ ሺሻ ෪ ሺపሻ ݍൌ ܵ ςୀଵ ܥσୀଵ ൬ ሺሻ ൰൨ factorial combinations in total. ܱ

Table 1 - Factor levels of ܯ-modular system reliability optimization Factor Level

Factor Combination No.

Structure No.

ܥ

ሺଵሻ

ܥ

ሺଶሻ

ܥሺெሻ

ڮ

1

1

1

1

ڮ

1

2

1

1

1

ڮ

1

ڭ

ڭ

ڭ

ڭ

ෑ ܱሺሻ ͳ

1

2

1

ڭ

ڭ

ڭ

ڭ

1

ሺଵሻ ܥ෪

ሺଶሻ ܥ෪

ڮ

෫ ሺெሻ ܥ

݆ ሺపሻ ൬ ෑ ܥ෪ ൰ ͳ ܱሺሻ

2

1

1

ڮ

1

ڭ

ڭ

ڭ

ڭ

2

ሺଵሻ ܥ෪

ሺଶሻ ܥ෪

ڭ

ڭ

ڭ

݆ ሺపሻ ൬ ሺܵ െ ͳሻ ෑ ܥ෪ ൰ ͳ ܱ ሺሻ

ܵ

1

1

ڭ

ڭ

ڭ

ڭ

ܵ

ሺଵሻ ܥ෪

ሺଶሻ ܥ෪

ڭ

ெ

ڮ

1

ୀଵ

ڭ

ெ

ሺሻ

݆ ሺపሻ ൬ ෑ ܥ෪ ൰ ܱ ሺሻ ୀଵ

ୀଵ

ெ

ሺሻ

ୀଵ

ୀଵ

ሺሻ

ெ

ሺపሻ ൬ ෑ ܥ෪ ୀଵ

ୀଵ

݆ ൰ ܱ ሺሻ

ڭ ெ

ሺሻ

ୀଵ

ୀଵ

ெ

ሺሻ

ୀଵ

ୀଵ

݆ ሺపሻ ൬ ݍൌ ܵ ෑ ܥ෪ ൰ ܱሺሻ

2.2 Response Variables For a system with ܯmodules, there are ܯ ͳ response variables, including the cost ୡ , and the reliability of each module ܻ . The cost is associated with the number of components in each module and the choice of components. Let ܻǡ denote the cost of the th combination, ݈ ൌ ͳǡʹǡ ǥ ǡ ݍ, calculated by ሺሻ

ሺሻ

ை ݕǡ ൌ σெ ୀଵ σୀଵ ݊ ,

where is the cost of the component chosen form the th ሺሻ option for the th module, and ݊ is the number of components from option ݇ in the module ݅. Since every module is assumed to be a parallel-redundant subsystem with ܥሺሻ independent components, and the reliability of each component is known, the reliability of every module can be easily computed. Let ܴǡ ሺݐሻ denote the reliability of module for combination, computed by ሺሻ

ܴǡ ሺݐሻ ൌ ͳ െ ςୀଵሾͳ െ ܴ ሺݐሻሿ. The input of the system reliability multi-response optimization problem is given in Table 2.

ڭ ෫ ሺெሻ ܥ

ڮ

ڭ ڮ

1 ڭ ෫ ሺெሻ ܥ

ڮ

CS1

CS2

ͳ Ͳ ڭ Ͳ Ͳ ͳ ڭ Ͳ ڭ ʹ Ͳ ڭ Ͳ ڭ

ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ ͳ Ͳ ڭ Ͳ ڭ

Ͳ Ͳ ൦ ڭ൪ ሺଵሻ ܥ෪ ͳ Ͳ ڭ Ͳ ڭ

Ͳ Ͳ ൦ ڭ൪ ሺଶሻ ܥ෪ ͳ Ͳ ڭ Ͳ ڭ

Ͳ Ͳ ൦ ڭ൪ ሺଵሻ ܥ෪ ڭ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ሺଵሻ ܥ෪

Ͳ Ͳ ൦ ڭ൪ ሺଶሻ ܥ෪ ڭ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ሺଶሻ ܥ෪

ڮ

CSM ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ ͳ Ͳ ڭ Ͳ ڭ

ڮ

ڮ

ڮ

Ͳ Ͳ ൦ ڭ൪ ෫ ሺெሻ ܥ ͳ Ͳ ڭ Ͳ ڭ

ڮ

ڮ

Ͳ Ͳ ൦ ڭ൪ ෫ ሺெሻ ܥ ڭ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ෫ ሺெሻ ܥ

ڮ

ڮ

ڮ

Obviously, ܻ is a smaller-the-better response, and ܻଵ ǡ ǥ ǡ ܻெ are larger-the-better responses. 2.3 Weighted Principal Component Analysis Based MultiResponse Optimization Weighted principal component analysis (WPCA) based multi-response optimization approach [7] is applied here to determine the optimal factor-level combination which maximizes the reliabilities of modules and meanwhile minimizes the cost. The procedure is described as follows: Step 1: Compute the loss function of responseୡ , ଶ , ܮǡ ൌ ݇ݕǡ

(1)

and loss of responses ଵ ǡ ǥ ǡ , ܮǡ ൌ ݇ܧሾ

ଵ

మ ሺ௧ሻ ோǡ

ሿ,

(2)

where ݅ ൌ ͳǡʹǡ ǥ ǡ ܯ, ݈ ൌ ͳǡʹǡ ǥ ǡ ݍ, and ݇ is the loss coefficient, [6] [8] Step 2: Normalize loss function of response ୡ , ሺேሻ

ܮǡ ൌ

ǡ ି൛ǡభ ǡǡమ ǡǥǡǡ ൟ

,

௫൛ǡభ ǡǡమ ǡǥǡǡ ൟି൛ǡభ ǡǡమ ǡǥǡǡ ൟ

(3)

and normalize loss function of responses ܻଵ ǡ ǥ ǡ ܻெ ,

ǡ ି൛ǡభ ǡǡమ ǡǥǡǡ ൟ

ሺேሻ

ܮǡ ൌ

.

Step 3: Perform PCA on normalized data to identify the elements of the eigenvector corresponding to the th largest eigenvalue ߣሺሻ ܽଵ ǡ ܽଶ ǡ ǥ ǡ ܽǡெାଵ , and the th component for the th combination (୪୰ ): ሺேሻ

ሺேሻ

ݓ ൌ

(4)

௫൛ǡభ ǡǡమ ǡǥǡǡ ൟି൛ǡభ ǡǡమ ǡǥǡǡ ൟ

ሺேሻ

ݖ ൌ ܽଵ ܮǡ ܽଶ ܮଵǡ ڮ ܽǡெାଵ ܮெǡ .

(5)

Step 4: Transform normalized quality loss into a MultiResponse Performance Index (MPI) statistic. The explained variance of each component is considered as weight (ݓ ),

ఒሺሻ ఒሺభሻ ାఒሺమሻ ାڮାఒሺಾశభሻ

,

(6)

and all the components are combined into one MPI, ߗ ൌ ݓଵ ݖଵ ݓଶ ݖଶ ڮ ݓெାଵ ݖǡெାଵ ,

(7)

where ݈ ൌ ͳǡʹǡ ǥ ǡ ݍ. The computed MPIs are shown in Table 3 Step 5: The factor-level combination corresponding to the smallest MPI value ȳ is chosen as the optimal result, and its corresponding design scheme is used as the new plan to improve the system reliability.

Table 2 - Experimental data for system reliability multi-response optimization Factor Level

Factor Combination No.

Structure No.

ܥ

ሺଵሻ

ܥ

ሺଶሻ

ڮ

ܥ

ሺெሻ

1

1

1

1

ڮ

1

2

1

1

1

ڮ

1

ڭ

ڭ

ڭ

ܵ

ሺଵሻ ܥ෪

ሺଶሻ ܥ෪

ڭ ሺሻ

ெ

ሺపሻ ൬ ݍൌ ܵ ෑ ܥ෪ ୀଵ

ୀଵ

݆ ൰ ܱሺሻ

ڭ ڮ

෫ ሺெሻ ܥ

Response

CS1

CS2

ͳ Ͳ ڭ Ͳ Ͳ ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ሺଵሻ ܥ෪

ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ሺଶሻ ܥ෪

ڮ ڮ

ڮ

ڮ

CSM ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ෫ ሺெሻ ܥ

ܻ

ܻଵ

ڮ

ܻெ

ݕǡଵ

ܴଵǡଵ ሺݐሻ

ڮ

ܴெǡଵ ሺݐሻ

ݕǡଶ

ܴଵǡଶ ሺݐሻ

ڮ

ܴெǡଶ ሺݐሻ

ڭ

ڭ

ݕǡ

ܴଵǡ ሺݐሻ

ڭ ڮ

ܴெǡ ሺݐሻ

Table 3 - Compute MPI for system reliability multi-response optimization Factor Combination No.

Structure No.

ܥሺଵሻ

ܥሺଶሻ

1

1

1

1

2

1

1

1

ڭ

ڭ

ڭ

ܵ

ሺଵሻ ܥ෪

ሺଶሻ ܥ෪

ڭ ெ

ሺሻ

ୀଵ

ୀଵ

݆ ሺపሻ ൬ ݍൌ ܵ ෑ ܥ෪ ൰ ܱሺሻ

Factor Level CS1 ܥሺெሻ ͳ 1 ڮ Ͳ ڭ Ͳ Ͳ 1 ͳ ڮ ڭ Ͳ ڭ ڭ Ͳ Ͳ ෫ ሺெሻ ൦ ڭ൪ ڮ ܥ ሺଵሻ ܥ෪

ڮ

3 NUMERICAL EXAMPLE Figure 3 presents a reliability block diagram of an engineering system [9]. This is a five-modular system with nine different types of components. All the components are assumed to be independent. Among the five modules, the number of redundancies in module 2 can be increased to at

CS2 ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ሺଶሻ ܥ෪

ڮ ڮ

ڮ

ڮ

CSM ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ෫ ሺெሻ ܥ

Multi-Response Performance Index ȳ ȳଵ ൌ ݓଵ ݖଵଵ ݓଶ ݖଵଶ ڮ ݓெାଵ ݖଵǡெାଵ

ȳଶ ൌ ݓଵ ݖଶଵ ݓଶ ݖଶଶ ڮ ݓெାଵ ݖଶǡெାଵ ڭ ȳ ൌ ݓଵ ݖଵ ݓଶ ݖଶ ڮ ݓெାଵ ݖǡெାଵ

most three, and the number of redundancies in module 4 can be increased or decreased by at most one. Moreover, component ܥଶ and ܥଽ can be purchased from two suppliers with different prices and reliabilities. Suppose that the component reliability at the mission time of 500 hours has been estimated from life tests. The data are given in Table 4.

Figure 3 - Reliability block diagram of the five-modular system

Table 4 - Estimates of components reliabilities and prices Structure

Module 1

Component ܥଵ ܥଶ ܥଷ ܥସ ܥହ ܥ ܥ ଼ܥ

2 1

3 4 5

ܥଽ

Supplier 1 1 2 1 1 1 1 1 1 1 2

Price ($/component) 2.25 5.30 5.00 3.75 2.85 4.15 6.05 5.50 10.75 2.95 2.00

Reliability at 500 hours 0.9856 0.9687 0.9453 0.9355 0.9566 0.9651 0.9862 0.9421 0.9622 0.9935 0.9744

Table 5 - Experimental data for system reliability multi-response optimization Factor Combination No.

ܥ

ሺଶሻ

Factor Level CS2 ܥሺସሻ ሾͳሿ 1

Response ܻଶ ܻଷ

CS9

ܻ

ܻଵ

ܻସ

ܻହ

ሾͳሿ

50.25

0.9548

0.9958

0.9105

0.9421

0.9997

1

2

2

2

1

ሾͳሿ

ሾʹሿ

48.35

0.9548

0.9958

0.9105

0.9421

0.9984

3

2

1

ሾʹሿ

ሾͳሿ

49.95

0.9317

0.9958

0.9105

0.9421

0.9997

4

2

1

ሾʹሿ

ሾʹሿ

48.05

0.9317

0.9958

0.9105

0.9421

0.9984

5

2

2

ሾͳሿ

ሾͳሿ

55.75

0.9548

0.9958

0.9105

0.9966

0.9997

6

2

2

ሾͳሿ

ሾʹሿ

53.85

0.9548

0.9958

0.9105

0.9966

0.9984

7

2

2

ሾʹሿ

ሾͳሿ

55.45

0.9317

0.9958

0.9105

0.9966

0.9997

8

2

2

ሾʹሿ

ሾʹሿ

53.55

0.9317

0.9958

0.9105

0.9966

0.9984

9

2

3

ሾͳሿ

ሾͳሿ

61.25

0.9548

0.9958

0.9105

0.9998

0.9997

10

2

3

ሾͳሿ

ሾʹሿ

69.35

0.9548

0.9958

0.9105

0.9998

0.9984

11

2

3

ሾʹሿ

ሾͳሿ

60.95

0.9317

0.9958

0.9105

0.9998

0.9997

12

2

3

ሾʹሿ

ሾʹሿ

59.05

0.9317

0.9958

0.9105

0.9998

0.9984

13

3

1

ሾͳሿ

ሾͳሿ

54.00

0.9548

0.9997

0.9105

0.9421

0.9997

14

3

1

ሾͳሿ

ሾʹሿ

52.10

0.9548

0.9997

0.9105

0.9421

0.9984

15

3

1

ሾʹሿ

ሾͳሿ

53.70

0.9317

0.9997

0.9105

0.9421

0.9997

16

3

1

ሾʹሿ

ሾʹሿ

51.80

0.9317

0.9997

0.9105

0.9421

0.9984

17

3

2

ሾͳሿ

ሾͳሿ

59.50

0.9548

0.9997

0.9105

0.9966

0.9997

18

3

2

ሾͳሿ

ሾʹሿ

57.60

0.9548

0.9997

0.9105

0.9966

0.9984

19

3

2

ሾʹሿ

ሾͳሿ

59.20

0.9317

0.9997

0.9105

0.9966

0.9997

20

3

2

ሾʹሿ

ሾʹሿ

57.30

0.9317

0.9997

0.9105

0.9966

0.9984

21

3

3

ሾͳሿ

ሾͳሿ

65.00

0.9548

0.9997

0.9105

0.9998

0.9997

22

3

3

ሾͳሿ

ሾʹሿ

63.10

0.9548

0.9997

0.9105

0.9998

0.9984

23

3

3

ሾʹሿ

ሾͳሿ

64.70

0.9317

0.9997

0.9105

0.9998

0.9997

24

3

3

ሾʹሿ

ሾʹሿ

62.80

0.9317

0.9997

0.9105

0.9998

0.9984

The objective is to determine the sources of component 2 and 9, and the number of redundancies for module 2 and 4 such that the overall cost and system reliability is optimized. Here, we have four factors: ܥሺଶሻ , ܥሺସሻ , CS2, and CS9, which

denotes the number of redundancies of module 2 and module 4, and the supplier selection for component 2, and for component 9. Meanwhile, there are 6 response variables: ܻ , ܻଵ , ܻଶ , ܻଷ , ܻସ , and ܻହ , corresponding to the overall cost of the

system, and the reliability of each module. The experimental data are shown in table 5, where ܻ is smaller-the-better variable, and ܻଵ , ܻଶ , ܻଷ , ܻସ , and ܻହ are the larger-the-better variables.

Since there is no change within module 3, the response variable ܻଷ is deleted from the table. Following step 1 and 2 in section 2.3, from equation (1), (2), (3), and (4), the normalized loss function is shown in table 6:

Table 6 - Normalized loss functions for data in table 5 Factor Combination No.

ܥ

ሺଶሻ

Factor Level CS2 ܥሺସሻ ሾͳሿ 1

CS9

Normalized Response Quality Loss ܻଵ ܻଶ ܻସ

ܻ

ሾͳሿ

0.0865

0

1

1.0000

0

ܻହ

1

2

2

2

1

ሾͳሿ

ሾʹሿ

0.0116

0

1

1.0000

1

3

2

1

ሾʹሿ

ሾͳሿ

0.0745

1

1

1.0000

0

4

2

1

ሾʹሿ

ሾʹሿ

0.0000

1

1

1.0000

1

5

2

2

ሾͳሿ

ሾͳሿ

0.3196

0

1

0.0510

0

6

2

2

ሾͳሿ

ሾʹሿ

0.2363

0

1

0.0510

1

7

2

2

ሾʹሿ

ሾͳሿ

0.3063

1

1

0.0510

0

8

2

2

ሾʹሿ

ሾʹሿ

0.2235

1

1

0.0510

1

9

2

3

ሾͳሿ

ሾͳሿ

0.5770

0

1

0.0000

0

10

2

3

ሾͳሿ

ሾʹሿ

1.0000

0

1

0.0000

1

11

2

3

ሾʹሿ

ሾͳሿ

0.5623

1

1

0.0000

0

12

2

3

ሾʹሿ

ሾʹሿ

0.4711

1

1

0.0000

1

ሾͳሿ

0.2428

0

0

1.0000

0

13

3

1

ሾͳሿ

14

3

1

ሾͳሿ

ሾʹሿ

0.1622

0

0

1.0000

1

15

3

1

ሾʹሿ

ሾͳሿ

0.2299

1

0

1.0000

0

16

3

1

ሾʹሿ

ሾʹሿ

0.1497

1

0

1.0000

1

17

3

2

ሾͳሿ

ሾͳሿ

0.4925

0

0

0.0510

0

18

3

2

ሾͳሿ

ሾʹሿ

0.4035

0

0

0.0510

1

19

3

2

ሾʹሿ

ሾͳሿ

0.4782

1

0

0.0510

0

20

3

2

ሾʹሿ

ሾʹሿ

0.3897

1

0

0.0510

1

21

3

3

ሾͳሿ

ሾͳሿ

0.7663

0

0

0.0000

0

22

3

3

ሾͳሿ

ሾʹሿ

0.6690

0

0

0.0000

1

23

3

3

ሾʹሿ

ሾͳሿ

0.7507

1

0

0.0000

0

24

3

3

ሾʹሿ

ሾʹሿ

0.6539

1

0

0.0000

1

The normalized quality losses are transformed to multiresponse performance indices (MPI) through PCA (as demonstrated in equation (5), (6), and (7)). Table 7 shows the computed results. From Table 7, it is shown that the choice of 3 redundancies for both module 2 and 4, and the selection of component from supplier 1 for both component 2 and 9 would result in overall system optimization. 4 CONCLUSION MCBS modeling has practical values in many realistic applications with the purpose of setting an optimal design scheme in a system-based product design phase. Meanwhile, if covariates that affect the reliability of MCBS are considered

in modeling procedures, multi-response optimization is an appropriate method for system reliability optimization. In this paper, WPCA-based multi-response optimization is proposed based on the high efficiency of solving such an optimization problem, and an indexing procedure is presented to solve the information distortion caused by WPCA in order to improve the multi-response optimization method. The procedure for this approach is based on the comparison of assigned indices and determination of a set of eigenvectors which represents the new coordinate axes, such that the relative magnitudes of the projections of data in the new coordinate system have a minimal total difference between each pair of indices. Based on the proposed method only one combination of eigenvectors leads to a unique optimal factor level combination in a multi-

response optimization so that the optimal design scheme for

system reliability is obtained.

Table 7 - Multi-response performance indices for data in Table 5 Factor Combination No.

ܥ

ሺଶሻ

Factor Level CS2 ܥ

CS9

Multi-Response Performance Indices (MPI) ȳ

ሺସሻ

1

2

1

ሾͳሿ

ሾͳሿ

0.1385

2

2

1

ሾͳሿ

ሾʹሿ

0.4770

3

2

1

ሾʹሿ

ሾͳሿ

0.4676

4

2

1

ሾʹሿ

ሾʹሿ

0.8061

5

2

2

ሾͳሿ

ሾͳሿ

0.1287

6

2

2

ሾͳሿ

ሾʹሿ

0.4674

7

2

2

ሾʹሿ

ሾͳሿ

0.4578

8

2

2

ሾʹሿ

ሾʹሿ

0.7966

9

2

3

ሾͳሿ

ሾͳሿ

0.1196

10

2

3

ሾͳሿ

ሾʹሿ

0.4406

11

2

3

ሾʹሿ

ሾͳሿ

0.4488

12

2

3

ሾʹሿ

ሾʹሿ

0.7878

13

3

1

ሾͳሿ

ሾͳሿ

-0.0068

14

3

1

ሾͳሿ

ሾʹሿ

0.3319

15

3

1

ሾʹሿ

ሾͳሿ

0.3224

16

3

1

ሾʹሿ

ሾʹሿ

0.6611

17

3

2

ሾͳሿ

ሾͳሿ

-0.0172

18

3

2

ሾͳሿ

ሾʹሿ

0.3218

19

3

2

ሾʹሿ

ሾͳሿ

0.3120

20

3

2

ሾʹሿ

ሾʹሿ

0.6510

21

3

3

ሾͳሿ

ሾͳሿ

-0.0269

22

3

3

ሾͳሿ

ሾʹሿ

0.3124

23

3

3

ሾʹሿ

ሾͳሿ

0.3024

24

3

3

ሾʹሿ

ሾʹሿ

0.6417

REFERENCES 1. 2. 3.

4.

5.

Z. W. Birnbaum, J. Esary, "Modules of coherent binary systems," Journal of the Society for Industrial & Applied Mathematics, 1965, pp. 444-462. R. Billinton, R. N. Allan, Reliability evaluation of power systems, Springer, 1996. A. Lisnianski, I. Frenkel, Y. Ding, Multi-state system reliability analysis and optimization for engineers and industrial managers, Springer Science & Business Media, 2010. R. Abdelkader, Z. Abdelkader, R. Mustapha, M. Yamani, Optimal Allocation of Reliability in Series Parallel Production System, INTECH Open Access Publisher, 2013. B. Natvig, Multistate systems reliability theory with

6.

7.

8. 9.

applications, John Wiley & Sons, 2010. N. Fard, H. Xu, Y. Fang, "A unique solution for principal component analysis-based multi-response optimization problems," The International Journal of Advanced Manufacturing Technology, 2015, vol. 79. H.-C. Liao, "Multi-response optimization using weighted principal component," The International Journal of Advanced Manufacturing Technology, 2006, pp. 27(7-8): 720-725. K. Yang, Y. Guangbin, "Robust reliability design using environmental stress testing," Quality and Reliability Engineering International, 1998, pp. 14(6): 409-416. G. Yang, Life Cycle Reliability Engineering, New Jersey: John Wiley & Sons, Inc., 2007.

BIOGRAPHIES Nasser Fard, PhD Department of Mechanical and Industrial Engineering Northeastern University 334 Snell Engineering Center 360 Huntington Avenue Boston, MA 02115 USA e-mail: [email protected] Dr. Fard is an associate professor of Industrial Engineering at the Northeastern University. He served as an associate editor of the IEEE Transactions on Reliability and he has been on the editorial board of the International Journal of Reliability, Quality and Safety Engineering. His experience and most recent research and professional activities are in Survival Data Analysis, Reliability Predication and Estimation, and Big Data Analytics. He has been principal investigator for public and private research projects.

Huyang Xu Department of Mechanical and Industrial Engineering Northeastern University 360 Huntington Avenue Boston, MA 02115 USA e-mail: [email protected] Yuanchen Fang Department of Mechanical and Industrial Engineering Northeastern University 360 Huntington Avenue Boston, MA 02115 USA e-mail: [email protected] Huyang Xu and Yuanchen Fang are PhD candidates in the Industrial Engineering Program at Northeastern University. Their research interests are in the areas of Robust Design of Experiments, Big Data Analytics, and Reliability.

,(((

identical function. Additionally, considering the necessity of redundancy, some modules usually contain more than one component in order to approach the high reliability requirement of the corresponding module. The reliability of such systems depends on combinations of covariates affected by alternate system structure and alternative modular technical specifications, including alternative component and redundancy policy. In this paper, multi-response optimization with WPCA is proposed in order to find the optimal design scheme for a MCBS attained by the simultaneous optimization of organizational patterns of modules and module reliability. That is, if the reliability of each specific failure caused by system reliability paths is seen as a response variable, the optimization can be designed as a multi-response optimization problem, and the number of responses is equal to the number of cause-specific system failures. Accordingly, many approaches for multi-response optimization such as assigning weight to response variables, grey relational analysis and multiple regression models have been proposed in recent years. Among all the methods for multi-response optimization, WPCA-based approach has shown high efficiency, since it takes into account all the possible correlations between response variables without increasing the computational complexity. By applying WPCA, the original responses are transformed to a set of uncorrelated PCs. Weights are assigned to each PC, and the PCs are combined into one multi-response performance index (MPI). The choice of a significant factor-level combination is based on ANOVA of MPI. Through WPCA-based multi-response optimization, the optimal levels of covariates, which maximize the system reliability, will be obtained. However, each eigenvalue obtained from the application of PCA in the multi-response optimization has a set of eigenvectors, and different eigenvectors corresponding to each eigenvalue will lead to different results [6]. To solve this problem, a method for determining the optimal eigenvector combination is proposed. Once the optimal eigenvector combination is determined, these eigenvectors are used to calculate the multi-response performance indices, and then the optimal factor-level combination will be uniquely determined.

This paper is organized as follows. The mathematical descriptions for the MCBS are discussed in Section 2, then the corresponding multiple-response optimization considering index method is proposed. In Section 3, a numerical example is given, and Section 4 provides concluding remarks. 2 MODEL DESCRIPTIONS Suppose a system is consisted of ܯmodules, and the ݅th module is one parallel-redundant subsystem with ܥሺሻ components for ݅ אሼͳǡ ڮǡ ܯሽ. All of the components in each module are assumed to perform the same sub-function of the system. Let ܥ denote the ݆th component in the ݅th module, where ݅ ൌ ͳǡʹǡ ǥ ǡ ܯand ݆ ൌ ͳǡʹǡ ǥ ǡ ܥሺሻ . Figure 1 shows a simple example of a system with 5 modules.

Figure - 1 A five-modular system example 2.1 Alternative Approaches for Reliability Improvement In order to improve the reliability of such a system, one way is to adjust the system structure without loss of the system function. Suppose the system has S alternative structures with equivalent functions, which are labeled as 1, 2, …, S. For example, Figure 2 shows alternative configurations of the system in Figure 1.

corresponds to a price ୧୩ , where ൌ ͳǡʹǡ ǥ ǡ and ൌ ͳǡʹǡ ǥ ǡ ሺ୧ሻ . Additionally, the traditional method with the purpose of improving the reliability of a system is to increase the number of redundant components within each module, which usually deteriorate product properties such as cost, weight, and size etc. Therefore, the number of components in each module, ܥሺሻ , is a type of negative measurement for product design improvement, on the other hand, boosting system reliability performance. The corresponding tradeoff is usually decided in order to balance overall product design specifications. In this paper, based on the fact that the product cost is the most important factor for marketing, the cost-dependent types of components are a significant factor considered in the optimization process. In summary, there are three main factors that need to be determined in system reliability optimization, which are: alternative system structure, amount of redundancy, and alternative components. System structure - For alternative system structure, the factor is the structure reference number and its corresponding levels are denoted by 1, 2, …, ܵ, as stated above. Number of Components in each module - There are ܯ factors for a ܯ-modular system, which are the number of redundant components for module 1, the number of redundant components for module 2, …, and the number of redundant components for module ܯ. These factors are denoted by ሺపሻ be the maximum number of ܥሺଵሻ ǡ ܥሺଶሻ ǡ ǥ ǡ ܥሺெሻ . Let ܥ෪ redundant components that can be comprised in module݅ . ሺపሻ . The factor levels for module ݅ are 1, 2, …, ܥ෪ Component types in each module - For alternative components, the factors are component selection for module 1, component selection for module 2, …, and component selection for module ܯ. For simplicity, these factors are ሺሻ ݊ ۍଵ ې ݊ ێሺሻ ۑ denoted by CS1, CS2, …, and CSM. An array, ێଶ ۑ, is ڭ ێሺሻ ۑ ݊ ۏைሺሻ ے defined to store the number of components chosen from ሺሻ different options, where ݊ is the number of components of ሺሻ option ݇ in the module ݅, ݇ ൌ ͳǡʹǡ ǥ ǡ ܱሺሻ , ݊ ൌ Ͳǡͳǡ ǥ ǡ ܱሺሻ , ሺሻ ሺሻ ሺሻ and ݊ଵ ݊ଶ ڮ ݊ைሺሻ ൌ ܥሺሻ , and the factor levels ሺሻ

Figure 2 - Two alternative structures of the system in Figure 1 Another way to improve the system reliability is to choose different components within each module. Components of the same function, but made from different manufacturers, have different reliabilities and different prices. Suppose there are ሺ୧ሻ options for the components in the th module to choose from, and the th option for the th module

݊ ۍଵ ې ݊ ێሺሻ ۑ corresponding to CS݅ is represented by ێଶ ۑ. ڭ ێሺሻ ۑ ݊ ۏைሺሻ ے The factor levels and combinations for ܯ-modular system reliability optimization are summarized in Table 1. From table 1, it can be seen that to optimize the reliability of a system with ܯ modules, there are ݆ ெ ሺሻ ෪ ሺపሻ ݍൌ ܵ ςୀଵ ܥσୀଵ ൬ ሺሻ ൰൨ factorial combinations in total. ܱ

Table 1 - Factor levels of ܯ-modular system reliability optimization Factor Level

Factor Combination No.

Structure No.

ܥ

ሺଵሻ

ܥ

ሺଶሻ

ܥሺெሻ

ڮ

1

1

1

1

ڮ

1

2

1

1

1

ڮ

1

ڭ

ڭ

ڭ

ڭ

ෑ ܱሺሻ ͳ

1

2

1

ڭ

ڭ

ڭ

ڭ

1

ሺଵሻ ܥ෪

ሺଶሻ ܥ෪

ڮ

෫ ሺெሻ ܥ

݆ ሺపሻ ൬ ෑ ܥ෪ ൰ ͳ ܱሺሻ

2

1

1

ڮ

1

ڭ

ڭ

ڭ

ڭ

2

ሺଵሻ ܥ෪

ሺଶሻ ܥ෪

ڭ

ڭ

ڭ

݆ ሺపሻ ൬ ሺܵ െ ͳሻ ෑ ܥ෪ ൰ ͳ ܱ ሺሻ

ܵ

1

1

ڭ

ڭ

ڭ

ڭ

ܵ

ሺଵሻ ܥ෪

ሺଶሻ ܥ෪

ڭ

ெ

ڮ

1

ୀଵ

ڭ

ெ

ሺሻ

݆ ሺపሻ ൬ ෑ ܥ෪ ൰ ܱ ሺሻ ୀଵ

ୀଵ

ெ

ሺሻ

ୀଵ

ୀଵ

ሺሻ

ெ

ሺపሻ ൬ ෑ ܥ෪ ୀଵ

ୀଵ

݆ ൰ ܱ ሺሻ

ڭ ெ

ሺሻ

ୀଵ

ୀଵ

ெ

ሺሻ

ୀଵ

ୀଵ

݆ ሺపሻ ൬ ݍൌ ܵ ෑ ܥ෪ ൰ ܱሺሻ

2.2 Response Variables For a system with ܯmodules, there are ܯ ͳ response variables, including the cost ୡ , and the reliability of each module ܻ . The cost is associated with the number of components in each module and the choice of components. Let ܻǡ denote the cost of the th combination, ݈ ൌ ͳǡʹǡ ǥ ǡ ݍ, calculated by ሺሻ

ሺሻ

ை ݕǡ ൌ σெ ୀଵ σୀଵ ݊ ,

where is the cost of the component chosen form the th ሺሻ option for the th module, and ݊ is the number of components from option ݇ in the module ݅. Since every module is assumed to be a parallel-redundant subsystem with ܥሺሻ independent components, and the reliability of each component is known, the reliability of every module can be easily computed. Let ܴǡ ሺݐሻ denote the reliability of module for combination, computed by ሺሻ

ܴǡ ሺݐሻ ൌ ͳ െ ςୀଵሾͳ െ ܴ ሺݐሻሿ. The input of the system reliability multi-response optimization problem is given in Table 2.

ڭ ෫ ሺெሻ ܥ

ڮ

ڭ ڮ

1 ڭ ෫ ሺெሻ ܥ

ڮ

CS1

CS2

ͳ Ͳ ڭ Ͳ Ͳ ͳ ڭ Ͳ ڭ ʹ Ͳ ڭ Ͳ ڭ

ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ ͳ Ͳ ڭ Ͳ ڭ

Ͳ Ͳ ൦ ڭ൪ ሺଵሻ ܥ෪ ͳ Ͳ ڭ Ͳ ڭ

Ͳ Ͳ ൦ ڭ൪ ሺଶሻ ܥ෪ ͳ Ͳ ڭ Ͳ ڭ

Ͳ Ͳ ൦ ڭ൪ ሺଵሻ ܥ෪ ڭ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ሺଵሻ ܥ෪

Ͳ Ͳ ൦ ڭ൪ ሺଶሻ ܥ෪ ڭ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ሺଶሻ ܥ෪

ڮ

CSM ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ ͳ Ͳ ڭ Ͳ ڭ

ڮ

ڮ

ڮ

Ͳ Ͳ ൦ ڭ൪ ෫ ሺெሻ ܥ ͳ Ͳ ڭ Ͳ ڭ

ڮ

ڮ

Ͳ Ͳ ൦ ڭ൪ ෫ ሺெሻ ܥ ڭ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ෫ ሺெሻ ܥ

ڮ

ڮ

ڮ

Obviously, ܻ is a smaller-the-better response, and ܻଵ ǡ ǥ ǡ ܻெ are larger-the-better responses. 2.3 Weighted Principal Component Analysis Based MultiResponse Optimization Weighted principal component analysis (WPCA) based multi-response optimization approach [7] is applied here to determine the optimal factor-level combination which maximizes the reliabilities of modules and meanwhile minimizes the cost. The procedure is described as follows: Step 1: Compute the loss function of responseୡ , ଶ , ܮǡ ൌ ݇ݕǡ

(1)

and loss of responses ଵ ǡ ǥ ǡ , ܮǡ ൌ ݇ܧሾ

ଵ

మ ሺ௧ሻ ோǡ

ሿ,

(2)

where ݅ ൌ ͳǡʹǡ ǥ ǡ ܯ, ݈ ൌ ͳǡʹǡ ǥ ǡ ݍ, and ݇ is the loss coefficient, [6] [8] Step 2: Normalize loss function of response ୡ , ሺேሻ

ܮǡ ൌ

ǡ ି൛ǡభ ǡǡమ ǡǥǡǡ ൟ

,

௫൛ǡభ ǡǡమ ǡǥǡǡ ൟି൛ǡభ ǡǡమ ǡǥǡǡ ൟ

(3)

and normalize loss function of responses ܻଵ ǡ ǥ ǡ ܻெ ,

ǡ ି൛ǡభ ǡǡమ ǡǥǡǡ ൟ

ሺேሻ

ܮǡ ൌ

.

Step 3: Perform PCA on normalized data to identify the elements of the eigenvector corresponding to the th largest eigenvalue ߣሺሻ ܽଵ ǡ ܽଶ ǡ ǥ ǡ ܽǡெାଵ , and the th component for the th combination (୪୰ ): ሺேሻ

ሺேሻ

ݓ ൌ

(4)

௫൛ǡభ ǡǡమ ǡǥǡǡ ൟି൛ǡభ ǡǡమ ǡǥǡǡ ൟ

ሺேሻ

ݖ ൌ ܽଵ ܮǡ ܽଶ ܮଵǡ ڮ ܽǡெାଵ ܮெǡ .

(5)

Step 4: Transform normalized quality loss into a MultiResponse Performance Index (MPI) statistic. The explained variance of each component is considered as weight (ݓ ),

ఒሺሻ ఒሺభሻ ାఒሺమሻ ାڮାఒሺಾశభሻ

,

(6)

and all the components are combined into one MPI, ߗ ൌ ݓଵ ݖଵ ݓଶ ݖଶ ڮ ݓெାଵ ݖǡெାଵ ,

(7)

where ݈ ൌ ͳǡʹǡ ǥ ǡ ݍ. The computed MPIs are shown in Table 3 Step 5: The factor-level combination corresponding to the smallest MPI value ȳ is chosen as the optimal result, and its corresponding design scheme is used as the new plan to improve the system reliability.

Table 2 - Experimental data for system reliability multi-response optimization Factor Level

Factor Combination No.

Structure No.

ܥ

ሺଵሻ

ܥ

ሺଶሻ

ڮ

ܥ

ሺெሻ

1

1

1

1

ڮ

1

2

1

1

1

ڮ

1

ڭ

ڭ

ڭ

ܵ

ሺଵሻ ܥ෪

ሺଶሻ ܥ෪

ڭ ሺሻ

ெ

ሺపሻ ൬ ݍൌ ܵ ෑ ܥ෪ ୀଵ

ୀଵ

݆ ൰ ܱሺሻ

ڭ ڮ

෫ ሺெሻ ܥ

Response

CS1

CS2

ͳ Ͳ ڭ Ͳ Ͳ ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ሺଵሻ ܥ෪

ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ሺଶሻ ܥ෪

ڮ ڮ

ڮ

ڮ

CSM ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ෫ ሺெሻ ܥ

ܻ

ܻଵ

ڮ

ܻெ

ݕǡଵ

ܴଵǡଵ ሺݐሻ

ڮ

ܴெǡଵ ሺݐሻ

ݕǡଶ

ܴଵǡଶ ሺݐሻ

ڮ

ܴெǡଶ ሺݐሻ

ڭ

ڭ

ݕǡ

ܴଵǡ ሺݐሻ

ڭ ڮ

ܴெǡ ሺݐሻ

Table 3 - Compute MPI for system reliability multi-response optimization Factor Combination No.

Structure No.

ܥሺଵሻ

ܥሺଶሻ

1

1

1

1

2

1

1

1

ڭ

ڭ

ڭ

ܵ

ሺଵሻ ܥ෪

ሺଶሻ ܥ෪

ڭ ெ

ሺሻ

ୀଵ

ୀଵ

݆ ሺపሻ ൬ ݍൌ ܵ ෑ ܥ෪ ൰ ܱሺሻ

Factor Level CS1 ܥሺெሻ ͳ 1 ڮ Ͳ ڭ Ͳ Ͳ 1 ͳ ڮ ڭ Ͳ ڭ ڭ Ͳ Ͳ ෫ ሺெሻ ൦ ڭ൪ ڮ ܥ ሺଵሻ ܥ෪

ڮ

3 NUMERICAL EXAMPLE Figure 3 presents a reliability block diagram of an engineering system [9]. This is a five-modular system with nine different types of components. All the components are assumed to be independent. Among the five modules, the number of redundancies in module 2 can be increased to at

CS2 ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ሺଶሻ ܥ෪

ڮ ڮ

ڮ

ڮ

CSM ͳ Ͳ ڭ Ͳ ͳ Ͳ ڭ Ͳ ڭ Ͳ Ͳ ൦ ڭ൪ ෫ ሺெሻ ܥ

Multi-Response Performance Index ȳ ȳଵ ൌ ݓଵ ݖଵଵ ݓଶ ݖଵଶ ڮ ݓெାଵ ݖଵǡெାଵ

ȳଶ ൌ ݓଵ ݖଶଵ ݓଶ ݖଶଶ ڮ ݓெାଵ ݖଶǡெାଵ ڭ ȳ ൌ ݓଵ ݖଵ ݓଶ ݖଶ ڮ ݓெାଵ ݖǡெାଵ

most three, and the number of redundancies in module 4 can be increased or decreased by at most one. Moreover, component ܥଶ and ܥଽ can be purchased from two suppliers with different prices and reliabilities. Suppose that the component reliability at the mission time of 500 hours has been estimated from life tests. The data are given in Table 4.

Figure 3 - Reliability block diagram of the five-modular system

Table 4 - Estimates of components reliabilities and prices Structure

Module 1

Component ܥଵ ܥଶ ܥଷ ܥସ ܥହ ܥ ܥ ଼ܥ

2 1

3 4 5

ܥଽ

Supplier 1 1 2 1 1 1 1 1 1 1 2

Price ($/component) 2.25 5.30 5.00 3.75 2.85 4.15 6.05 5.50 10.75 2.95 2.00

Reliability at 500 hours 0.9856 0.9687 0.9453 0.9355 0.9566 0.9651 0.9862 0.9421 0.9622 0.9935 0.9744

Table 5 - Experimental data for system reliability multi-response optimization Factor Combination No.

ܥ

ሺଶሻ

Factor Level CS2 ܥሺସሻ ሾͳሿ 1

Response ܻଶ ܻଷ

CS9

ܻ

ܻଵ

ܻସ

ܻହ

ሾͳሿ

50.25

0.9548

0.9958

0.9105

0.9421

0.9997

1

2

2

2

1

ሾͳሿ

ሾʹሿ

48.35

0.9548

0.9958

0.9105

0.9421

0.9984

3

2

1

ሾʹሿ

ሾͳሿ

49.95

0.9317

0.9958

0.9105

0.9421

0.9997

4

2

1

ሾʹሿ

ሾʹሿ

48.05

0.9317

0.9958

0.9105

0.9421

0.9984

5

2

2

ሾͳሿ

ሾͳሿ

55.75

0.9548

0.9958

0.9105

0.9966

0.9997

6

2

2

ሾͳሿ

ሾʹሿ

53.85

0.9548

0.9958

0.9105

0.9966

0.9984

7

2

2

ሾʹሿ

ሾͳሿ

55.45

0.9317

0.9958

0.9105

0.9966

0.9997

8

2

2

ሾʹሿ

ሾʹሿ

53.55

0.9317

0.9958

0.9105

0.9966

0.9984

9

2

3

ሾͳሿ

ሾͳሿ

61.25

0.9548

0.9958

0.9105

0.9998

0.9997

10

2

3

ሾͳሿ

ሾʹሿ

69.35

0.9548

0.9958

0.9105

0.9998

0.9984

11

2

3

ሾʹሿ

ሾͳሿ

60.95

0.9317

0.9958

0.9105

0.9998

0.9997

12

2

3

ሾʹሿ

ሾʹሿ

59.05

0.9317

0.9958

0.9105

0.9998

0.9984

13

3

1

ሾͳሿ

ሾͳሿ

54.00

0.9548

0.9997

0.9105

0.9421

0.9997

14

3

1

ሾͳሿ

ሾʹሿ

52.10

0.9548

0.9997

0.9105

0.9421

0.9984

15

3

1

ሾʹሿ

ሾͳሿ

53.70

0.9317

0.9997

0.9105

0.9421

0.9997

16

3

1

ሾʹሿ

ሾʹሿ

51.80

0.9317

0.9997

0.9105

0.9421

0.9984

17

3

2

ሾͳሿ

ሾͳሿ

59.50

0.9548

0.9997

0.9105

0.9966

0.9997

18

3

2

ሾͳሿ

ሾʹሿ

57.60

0.9548

0.9997

0.9105

0.9966

0.9984

19

3

2

ሾʹሿ

ሾͳሿ

59.20

0.9317

0.9997

0.9105

0.9966

0.9997

20

3

2

ሾʹሿ

ሾʹሿ

57.30

0.9317

0.9997

0.9105

0.9966

0.9984

21

3

3

ሾͳሿ

ሾͳሿ

65.00

0.9548

0.9997

0.9105

0.9998

0.9997

22

3

3

ሾͳሿ

ሾʹሿ

63.10

0.9548

0.9997

0.9105

0.9998

0.9984

23

3

3

ሾʹሿ

ሾͳሿ

64.70

0.9317

0.9997

0.9105

0.9998

0.9997

24

3

3

ሾʹሿ

ሾʹሿ

62.80

0.9317

0.9997

0.9105

0.9998

0.9984

The objective is to determine the sources of component 2 and 9, and the number of redundancies for module 2 and 4 such that the overall cost and system reliability is optimized. Here, we have four factors: ܥሺଶሻ , ܥሺସሻ , CS2, and CS9, which

denotes the number of redundancies of module 2 and module 4, and the supplier selection for component 2, and for component 9. Meanwhile, there are 6 response variables: ܻ , ܻଵ , ܻଶ , ܻଷ , ܻସ , and ܻହ , corresponding to the overall cost of the

system, and the reliability of each module. The experimental data are shown in table 5, where ܻ is smaller-the-better variable, and ܻଵ , ܻଶ , ܻଷ , ܻସ , and ܻହ are the larger-the-better variables.

Since there is no change within module 3, the response variable ܻଷ is deleted from the table. Following step 1 and 2 in section 2.3, from equation (1), (2), (3), and (4), the normalized loss function is shown in table 6:

Table 6 - Normalized loss functions for data in table 5 Factor Combination No.

ܥ

ሺଶሻ

Factor Level CS2 ܥሺସሻ ሾͳሿ 1

CS9

Normalized Response Quality Loss ܻଵ ܻଶ ܻସ

ܻ

ሾͳሿ

0.0865

0

1

1.0000

0

ܻହ

1

2

2

2

1

ሾͳሿ

ሾʹሿ

0.0116

0

1

1.0000

1

3

2

1

ሾʹሿ

ሾͳሿ

0.0745

1

1

1.0000

0

4

2

1

ሾʹሿ

ሾʹሿ

0.0000

1

1

1.0000

1

5

2

2

ሾͳሿ

ሾͳሿ

0.3196

0

1

0.0510

0

6

2

2

ሾͳሿ

ሾʹሿ

0.2363

0

1

0.0510

1

7

2

2

ሾʹሿ

ሾͳሿ

0.3063

1

1

0.0510

0

8

2

2

ሾʹሿ

ሾʹሿ

0.2235

1

1

0.0510

1

9

2

3

ሾͳሿ

ሾͳሿ

0.5770

0

1

0.0000

0

10

2

3

ሾͳሿ

ሾʹሿ

1.0000

0

1

0.0000

1

11

2

3

ሾʹሿ

ሾͳሿ

0.5623

1

1

0.0000

0

12

2

3

ሾʹሿ

ሾʹሿ

0.4711

1

1

0.0000

1

ሾͳሿ

0.2428

0

0

1.0000

0

13

3

1

ሾͳሿ

14

3

1

ሾͳሿ

ሾʹሿ

0.1622

0

0

1.0000

1

15

3

1

ሾʹሿ

ሾͳሿ

0.2299

1

0

1.0000

0

16

3

1

ሾʹሿ

ሾʹሿ

0.1497

1

0

1.0000

1

17

3

2

ሾͳሿ

ሾͳሿ

0.4925

0

0

0.0510

0

18

3

2

ሾͳሿ

ሾʹሿ

0.4035

0

0

0.0510

1

19

3

2

ሾʹሿ

ሾͳሿ

0.4782

1

0

0.0510

0

20

3

2

ሾʹሿ

ሾʹሿ

0.3897

1

0

0.0510

1

21

3

3

ሾͳሿ

ሾͳሿ

0.7663

0

0

0.0000

0

22

3

3

ሾͳሿ

ሾʹሿ

0.6690

0

0

0.0000

1

23

3

3

ሾʹሿ

ሾͳሿ

0.7507

1

0

0.0000

0

24

3

3

ሾʹሿ

ሾʹሿ

0.6539

1

0

0.0000

1

The normalized quality losses are transformed to multiresponse performance indices (MPI) through PCA (as demonstrated in equation (5), (6), and (7)). Table 7 shows the computed results. From Table 7, it is shown that the choice of 3 redundancies for both module 2 and 4, and the selection of component from supplier 1 for both component 2 and 9 would result in overall system optimization. 4 CONCLUSION MCBS modeling has practical values in many realistic applications with the purpose of setting an optimal design scheme in a system-based product design phase. Meanwhile, if covariates that affect the reliability of MCBS are considered

in modeling procedures, multi-response optimization is an appropriate method for system reliability optimization. In this paper, WPCA-based multi-response optimization is proposed based on the high efficiency of solving such an optimization problem, and an indexing procedure is presented to solve the information distortion caused by WPCA in order to improve the multi-response optimization method. The procedure for this approach is based on the comparison of assigned indices and determination of a set of eigenvectors which represents the new coordinate axes, such that the relative magnitudes of the projections of data in the new coordinate system have a minimal total difference between each pair of indices. Based on the proposed method only one combination of eigenvectors leads to a unique optimal factor level combination in a multi-

response optimization so that the optimal design scheme for

system reliability is obtained.

Table 7 - Multi-response performance indices for data in Table 5 Factor Combination No.

ܥ

ሺଶሻ

Factor Level CS2 ܥ

CS9

Multi-Response Performance Indices (MPI) ȳ

ሺସሻ

1

2

1

ሾͳሿ

ሾͳሿ

0.1385

2

2

1

ሾͳሿ

ሾʹሿ

0.4770

3

2

1

ሾʹሿ

ሾͳሿ

0.4676

4

2

1

ሾʹሿ

ሾʹሿ

0.8061

5

2

2

ሾͳሿ

ሾͳሿ

0.1287

6

2

2

ሾͳሿ

ሾʹሿ

0.4674

7

2

2

ሾʹሿ

ሾͳሿ

0.4578

8

2

2

ሾʹሿ

ሾʹሿ

0.7966

9

2

3

ሾͳሿ

ሾͳሿ

0.1196

10

2

3

ሾͳሿ

ሾʹሿ

0.4406

11

2

3

ሾʹሿ

ሾͳሿ

0.4488

12

2

3

ሾʹሿ

ሾʹሿ

0.7878

13

3

1

ሾͳሿ

ሾͳሿ

-0.0068

14

3

1

ሾͳሿ

ሾʹሿ

0.3319

15

3

1

ሾʹሿ

ሾͳሿ

0.3224

16

3

1

ሾʹሿ

ሾʹሿ

0.6611

17

3

2

ሾͳሿ

ሾͳሿ

-0.0172

18

3

2

ሾͳሿ

ሾʹሿ

0.3218

19

3

2

ሾʹሿ

ሾͳሿ

0.3120

20

3

2

ሾʹሿ

ሾʹሿ

0.6510

21

3

3

ሾͳሿ

ሾͳሿ

-0.0269

22

3

3

ሾͳሿ

ሾʹሿ

0.3124

23

3

3

ሾʹሿ

ሾͳሿ

0.3024

24

3

3

ሾʹሿ

ሾʹሿ

0.6417

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4.

5.

Z. W. Birnbaum, J. Esary, "Modules of coherent binary systems," Journal of the Society for Industrial & Applied Mathematics, 1965, pp. 444-462. R. Billinton, R. N. Allan, Reliability evaluation of power systems, Springer, 1996. A. Lisnianski, I. Frenkel, Y. Ding, Multi-state system reliability analysis and optimization for engineers and industrial managers, Springer Science & Business Media, 2010. R. Abdelkader, Z. Abdelkader, R. Mustapha, M. Yamani, Optimal Allocation of Reliability in Series Parallel Production System, INTECH Open Access Publisher, 2013. B. Natvig, Multistate systems reliability theory with

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7.

8. 9.

applications, John Wiley & Sons, 2010. N. Fard, H. Xu, Y. Fang, "A unique solution for principal component analysis-based multi-response optimization problems," The International Journal of Advanced Manufacturing Technology, 2015, vol. 79. H.-C. Liao, "Multi-response optimization using weighted principal component," The International Journal of Advanced Manufacturing Technology, 2006, pp. 27(7-8): 720-725. K. Yang, Y. Guangbin, "Robust reliability design using environmental stress testing," Quality and Reliability Engineering International, 1998, pp. 14(6): 409-416. G. Yang, Life Cycle Reliability Engineering, New Jersey: John Wiley & Sons, Inc., 2007.

BIOGRAPHIES Nasser Fard, PhD Department of Mechanical and Industrial Engineering Northeastern University 334 Snell Engineering Center 360 Huntington Avenue Boston, MA 02115 USA e-mail: [email protected] Dr. Fard is an associate professor of Industrial Engineering at the Northeastern University. He served as an associate editor of the IEEE Transactions on Reliability and he has been on the editorial board of the International Journal of Reliability, Quality and Safety Engineering. His experience and most recent research and professional activities are in Survival Data Analysis, Reliability Predication and Estimation, and Big Data Analytics. He has been principal investigator for public and private research projects.

Huyang Xu Department of Mechanical and Industrial Engineering Northeastern University 360 Huntington Avenue Boston, MA 02115 USA e-mail: [email protected] Yuanchen Fang Department of Mechanical and Industrial Engineering Northeastern University 360 Huntington Avenue Boston, MA 02115 USA e-mail: [email protected] Huyang Xu and Yuanchen Fang are PhD candidates in the Industrial Engineering Program at Northeastern University. Their research interests are in the areas of Robust Design of Experiments, Big Data Analytics, and Reliability.