ME-OWA based DEMATEL reliability apportionment method Expert Systems with Applications 38 (2011) 9713–9723 Contents lists available at ScienceDirect Expert Systems with Applications j o u r n a l h o m e p a g e : w w w . e l s e v i e r . c o m / l o c a t e / e s w a ME-OWA based DEMATEL reliability apportionment method Cheng-Shih Liaw a,⇑, Yung-Chia Chang a, Kuei-Hu Chang b, Thing-Yuan Chang c a Department of Industrial Engineering and Management, National Chiao Tung University, Hsinchu 300, Taiwan b Department of Management Sciences, R.O.C. Military Academy, Kaohsiung 830, Taiwan c Department of Information Management, National Chin-Yi University of Technology, Taichung 411, Taiwan a r t i c l e i n f o Keywords: Reliability allocation Maximal entropy ordered weighted averaging (ME-OWA) Feasibility-of-objectives technique Decision making trial and evaluation laboratory (DEMATEL) 0957-4174/$ - see front matter � 2011 Elsevier Ltd. A doi:10.1016/j.eswa.2011.02.029 ⇑ Corresponding author. Tel.: +886 3 5731815; fax: E-mail address: liaw1158@yahoo.com.tw (C.-S. Lia a b s t r a c t The maximal entropy ordered weighted averaging (ME-OWA)-based decision making trial and evaluation laboratory (DEMATEL) method for reliability allocation has been examined. The assessment results show that most conventional reliability allocation methods have five fundamental problems. The first problem is the measurement scale; while the second problem is that the system allocation factors are not equally weighted to one another, the third problem is that most reliability allocations methods often neglect many important features, such as maintainability and risk issues. The fourth problem is that they do not consider indirect relations between subsystems or components, and the fifth problem is that they do not consider predicted failure rate in the apportionment process. This study evaluated reliability allo- cation using a fighter aircraft’s digital flight control computer (DFLCC). The proposed method offers sev- eral benefits compared with current military and commercial approaches. The computational results clearly demonstrate the advantages of the proposed approach for solving the five fundamental problems. � 2011 Elsevier Ltd. All rights reserved. 1. Introduction Reliability allocation is a top-down approach for apportioning accuracy goals in a system, which is essential when different de- sign teams, subcontractors, or manufacturers are involved, as it ef- fects the system safety and usability of the product. The purpose of the reliability allocation is to assign limited resources to the most important subsystems or components and ensure that the product can achieve its designed functions under specific operating condi- tions. The allocation technique significantly influences product life cycle cost and system operational effectiveness. Apportioning system reliability is a multiple-criteria decision- making task, which is usually allocated on the basis of the compo- nent performance and/or cost as criteria. Fuqua (1987) introduced the ‘‘Reliability engineering for electronic design’’, and MIL-HDBK- 338B (1988) defines four approaches to allocating reliability, which includes the equal apportionment technique, the ARINC apportion- ment technique, the feasibility-of-objectives (FOO) technique, and the minimization of effort algorithm. Kuo (1999) in his book ‘‘Reli- ability Assurance: Application for engineering and management’’, introduced four approaches to allocating reliability, which in- cluded the equal apportionment technique, the ARINC apportion- ment technique, the average weighting allocation method, and the pair comparison method. These conventional reliability methods have been widely and successfully applied in a great ll rights reserved. +886 3 5722392. w). many domains (Anderson, 1976; Fuqua, 1987; Kuo, 1999; Smedley, 1992). In addition to these methods, Bracha (1964) introduced an allocated reliability method using four factors: state of the art, sub- system complexity as estimated by the number of parts, environ- mental conditions, and relative operating time, whereas Karmiol (1965) evaluated the complexity, state of the art, operational pro- file, and criticality of the system to mission objectives to apportion subsystem reliability. Boyd (1992) proposed the Boyd method to combine the equal method with the ARINC method, while Falcone, Silvestri, and Bona (2002) used the integrated factors method (IFM) using four factors, criticality (C), complexity (K), functionality (F), and effectiveness (O), to calculate system reliability for an aero- space prototype project. However, the Karmiol method, the FOO technique, the average weighting allocation method, and the IFM method assume an equal interval between category labels; therefore, the operations of mul- tiplication and division are not meaningful on ordinal numbers, and addition and subtraction, while sometimes meaningful, must be done carefully because they assume equal intervals between category labels. Thus these methods all share a common weakness in their measurement scale. Take the IFM as an example: the four system factors used in allocation reliability—criticality (C), com- plexity (K), functionality (F), effectiveness (O)—obtain the subsys- tem reliability IGi = Ki ⁄ Fi ⁄ Oi/Ci (IGi: global index relative to the subsystem). In the use of the IFM method, the four system factors C, K, F, and O are rated from range 1 to 10. These four system factors are multiplied and divided to derive the IGi; i.e., IGi = Ki ⁄ Fi ⁄ Oi/Ci—the multiplied results inIG range from 1 to 1000. Higher http://dx.doi.org/10.1016/j.eswa.2011.02.029 mailto:liaw1158@yahoo.com.tw http://dx.doi.org/10.1016/j.eswa.2011.02.029 http://www.sciencedirect.com/science/journal/09574174 http://www.elsevier.com/locate/eswa 9714 C.-S. Liaw et al. / Expert Systems with Applications 38 (2011) 9713–9723 IGs are assumed to be more of an overall rating than those having a lower IG. From a calculation perspective, the IFM method is simple, easy to understand, straightforward to use, and well documented for ease of reference. However, the method has problems with its measurement scale. The first problem is that the four system fac- tors, K, F, O, and C, are evaluated according to discrete ordinal scales of measure, which represent serious flaws from a technical perspective; in particular, multiplication and/or division is not meaningful and in fact is misleading. The second problem is that the four system factors are not equally weighted, which makes the analysis and interpretation of the results problematic. For example, for two components with IG values of (8 � 2 � 2)/2 = 16 and (7 � 3 � 2)/2 = 21, respectively, the former should have had a higher reliability allocation overall rating than the latter, even though it has a lower IG value. In resolving these two problems, Chang, Chang, and Liaw (2009) provided an innovative reliability allocation using the maximal entropy ordered weighted averaging (ME-OWA) method. This approach uses Yager’s OWA (1988) and the ME-OWA (Chang, Cheng, & Chang, 2008; Fuller & Majlender, 2001) operators, which uses Lagrange multipliers on Yager’s OWA equation to derive a polynomial equation, and determines the optimal weighting vector under the maximal entropy operator. This method is a both simple and effective approach that can efficiently resolve the two short- comings of the conventional allocation methods. However, most current reliability allocation methods do not consider indirect rela- tions between subsystems or components; also, most reliability allocation methods do not consider predicted failure rate in the evaluation process. The Battelle Memorial Institute, through its Geneva Research Centre, first developed the decision making trial and evaluation laboratory (DEMATEL) method (Gabus & Fontela, 1973). It is a potent method that gathers group knowledge for cap- turing the causal relationships between criteria, which is an impor- tant analytical tool to prioritize the alternatives based on the type of relationships and severity of influences. The DEMATEL method to analyze indirect relations has been successfully used in many industrial fields, such as marketing strategies, R&D projects, e-learning evaluations, managers’ competencies, control systems, and airline safety problems (Chiu, Chen, Tzeng, & Shyu, 2006; Hori & Shimizu, 1999; Lin & Wu, 2008; Tzeng, Chiang, & Li, 2007). Chang et al. (2009) introduced the ordered weighted geometric averaging (OWGA) operator and DEMATEL method to evaluate the ordering of risks for failure problems. This is a more general risk priority number (RPN) methodology and provides a more general and reasonable risk assessment ranking. The proposed approach, using ME-OWA, was used throughout the DEMATEL calculation to determine subsystem allocation weighting factors, and the DEMATEL technique also can consider indirect relations. After DEMATEL calculation processes, the proposed method also consid- ers each subsystem’s predicted failure rate. The higher ME- OWA-based DEMATEL values should have a higher reliability allocation overall rating and apportion a higher reliability ratio into subsystems or components. Meanwhile, when using the situation parameter (a), considering the indirect relationship and predicted subsystem failure rate at same time, the proposed method can effi- ciently resolve the five shortcomings of the conventional reliability allocation methods. This study evaluates reliability allocation using a fighter aircraft digital flight control computer (DFLCC). The results from comparison with conventional reliability methods show that the proposed method is an effective methodology, with proven effectiveness yet flexible and yielding accurate results. The remainder of this paper is organized as follows: Section 2 introduces the ME-OWA operations and applications, Section 3 introduces the DEMATEL method, Section 4 introduces conven- tional reliability allocation methods, Section 5 proposes the ME-OWA-based DEMATEL method, and in Section 6, an example is drawn from an aircraft fighter’s DFLCC using the proposed approach for reliability allocation assessment. Section 7 is the conclusion. 2. ME-OWA operators and its operations 2.1. ME-OWA operators Yager (1988) first introduced the concept of OWA operators, which are important aggregation operators within the class of weighted aggregation methods. They have the ability to derive optimal weights of the attributes based on the rating of the weight- ing vectors after an aggregation process (see Definition 1). Definition 1. An OWA operator of dimension n is mapped F: Rn ? R, which has an associated n weighting vector W = [w1, w2, . . . , wn] T of the properties P iwi ¼ 1;8wi 2 ½0; 1�; i ¼ 1; . . . ; n, such that fða1; a2; . . . ; anÞ¼ Xn i¼1 wi bi ð1Þ where bi is the ith largest element in the vector (a1, a2, . . . , an), and b1 P b2 P . . . P bn. Yager (1988) also introduced two important characterizing measurements with respect to the weighting vector W of the OWA operator. One of these two measures is orness of the aggregation, which is defined in Definition 2. Definition 2. Assume F is an OWA aggregation operator with a weighting function W = [w1, w2, . . . , wn]. The degree of orness asso- ciated with this operator is defined as: ornessðWÞ¼ 1 n � 1 Xn i¼1 ðn � iÞwi ð2Þ where orness(W) = a is a situation parameter. It is clear that orness(W) 2 [0, 1] holds for any weighting vector. The second characterizing measurement introduced by Yager (1988) is a measure of dispersion of the aggregation, which is defined in Definition 3. Definition 3. Assume W is a weighting vector with elements w1, . . . , wn; then the measure of dispersion of W is defined as: dispersionðWÞ¼� Xn i¼1 wi ln wi: ð3Þ O’Hagan (1988) combined the principle of maximum entropy and OWA operators to propose a particular OWA weight that has maximum entropy with a given level of orness. This approach is based on the solution of the following mathematical programming problem: Maximize � Xn i¼1 wi ln wi ð4Þ Subject to : 1 n � 1 Xn i¼1 ðn � iÞwi ¼ a; 0 6 a 6 1; ð5Þ Xn i¼1 wi ¼ 1; 0 6 wi 6 1; i ¼ 1; . . . ; n: ð6Þ 2.2. Determination of ME-OWA weights Fuller and Majlender (2001) used the method of Lagrange mul- tipliers on Yager’s OWA equation to derive a polynomial equation, C.-S. Liaw et al. / Expert Systems with Applications 38 (2011) 9713–9723 9715 which can determine the optimal weighting vector under the max- imal entropy. By their method, the associated weighting vector is easily obtained by Eqs. (7)–(9): wi ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wn�j1 w j�1 n n�1 q ð7Þ and wn ¼ ððn � 1Þa � nÞw1 þ 1 ðn � 1Þa þ 1 � nw1 ð8Þ then w1½ðn � 1Þa þ 1 � nw1� n ¼ððn � 1ÞaÞn�1½ððn � 1Þa � nÞw1 þ 1� ð9Þ where w is the weight vector, n is the number of attributes, and a is the situation parameter. 3. DEMATEL methodology The Battelle Memorial Institute, through its Geneva Research Centre (Gabus & Fontela, 1973), first developed the DEMATEL method. It is a potent method that gathers group knowledge for capturing the causal relationship between criteria and can precisely ascertain the cause–effect relationship of criteria when measuring a problem. In recent years, the DEMATEL method has been successfully applied in different industries and in many fields (Chiu et al., 2006; Hori & Shimizu, 1999; Lin & Wu, 2008; Tzeng et al., 2007). The original DEMATEL method was aimed at the fragmented and antagonistic phenomena of world societies and the search for integrated solutions. It is especially practical and useful for visualizing the structure of complicated causal relationships with matrices or digraphs. The matrices or digraphs portray a contextual relation between the elements of the system in which a numeral represents the strength of influ- ence. Hence, the DEMATEL method can convert the relationship between the causes and effects of criteria into an intelligible structural model of the system, which are not only the direct influences taken into account but also the indirect influences among multiple factors. In this section, we briefly describe the DEMATEL method and procedure. 3.1. Outline of the DEMATEL method The essential of the DEMATEL method is reviewed below (Seyed-Hosseini, Safaei, & Asgharpour, 2006). Definition 4. The pair-wise comparison scale may be designated into four levels, where the scores of 0, 1, 2, and 3 represent ‘‘No influence’’, ‘‘Low influence’’, ‘‘High influence’’, and ‘‘Very high influence’’, respectively. Definition 5. The initial direct-relation matrix Z is an n � n matrix that is obtained by pair-wise comparisons in terms of influences and directions between criteria, in which Zij is denoted as the degree to which the criterion Di affects criterion Dj. Accordingly, all principal diagonal elements Zii of matrix Z are set to zero: ð10Þ Definition 6. Let s ¼ max 16j6n Xn j¼1 zij ! : ð11Þ Then, the normalized direct-relation matrix X can be obtained through the following formula: X ¼ Z s : ð12Þ Definition 7. The total relation matrix T can be acquired by using formula (13), in which the I is denoted as the identity matrix: T ¼ limit k!1 ðX þ X2 þ�� �þ XkÞ¼ XðI � XÞ�1: ð13Þ Definition 8. Let tij (i, j = 1, 2, . . . , n) be the elements of the total- relation matrix T; then, the sum of the rows and the sum of the col- umns, denoted as Ri and Cj, respectively, can be obtained through the following two formulas: Di ¼ Xn j¼1 tij ði ¼ 1; 2; . . . ; nÞ; ð14Þ Rj ¼ Xn i¼1 tij ðj ¼ 1; 2; . . . ; nÞ: ð15Þ Definition 9. A causal diagram can be acquired by mapping the ordered pairs of (R + C, R � C), where the horizontal axis (R + C) is named ‘‘Prominence’’ and the vertical axis (R � C) is named ‘‘Relation.’’ In the causal diagram, the horizontal axis ‘‘Prominence’’ shows how important the criterion is, whereas the vertical axis ‘‘Relation’’ may divide the criteria into the cause and effect groups. When the value (R � C) is positive, the criterion belongs to the cause group. If the value (R � C) is negative, the criterion belongs to the effect group. Hence, causal diagrams can visualize the complicated causal relationships between criteria into a visible structural model and provide valuable insight for problem-solving. Furthermore, with the help of a causal diagram, this study will allow proper decisions to be made by recognizing the difference between cause and effect criteria. 3.2. The procedure of the DEMATEL method The DEMATEL method can separate the relevant criteria of a system into the cause and effect groups to facilitate accurate decision-making. A DEMATEL procedure is explained as follows (Seyed-Hosseini et al., 2006): (1) A system designer or decision-maker evaluates the relation- ship between sets of paired alternatives. As a result of this evaluation, a matrix M is obtained as the initial data of the DEMATEL analysis. (2) The elements of the direct relative severity matrix (DRSM) are obtained by Eq. (11). It is the normalized version of matrix M. (3) The elements of the direct and indirect relative severity matrix (DIRSM) are obtained by Eq. (12). The DIRSM consists of all of the relations, including direct and indirect relations between alternatives. (4) Using the values of R + C and R � C, where C is the sum of the columns and R is the sum of the rows of the DIRSM, a level of influence and a level of relationship are defined. The value 9716 C.-S. Liaw et al. / Expert Systems with Applications 38 (2011) 9713–9723 R � C indicates the severity of influence for each alternative. Similarly, the value of R + C indicates the degree of relation between each alternative with one another. 4. Conventional reliability allocation methods This section presents a literature review of the applications of reliability allocation. Currently, many reliability allocation tech- niques are available, including the AGREE method (1957), ARINC apportionment technique (1976), Bracha method (1964), Karmiol method (1965), equalization allocation method, pair comparison allocation, the FOO technique, minimization of effort algorithm (1988), Boyd method (1992), average weighting allocation method (Kuo, 1999), Base method, and IFM method (Falcone et al., 2002), among others. The allocation technique significantly influences product life cycle cost and system operational effectiveness. Since the ARINC apportionment technique is an important method in military reliability allocation design, the basic definition and pro- cedure of the ARINC apportionment technique and the ME-OWA apportionment method are reviewed in this section. 4.1. ARINC apportionment technique This method assumes a series of subsystems with constant fail- ure rates. Other fundamental assumptions are: (1) series subsys- tems, (2) constant failure rates, (3) same mission duration time T for each subsystem, and (4) a pre-defined, known allowable system failure rate: k⁄. Suppose a system is composed of N subsystems. Let k�i be the failure rate allocated to subsystem i. The objective is to choose k�i such that (1988): Xn i¼1 k�i P k � : ð16Þ Determine the subsystem failure rates (ki) from past observa- tion or estimation; assign a weighting factor (wi) to each subsys- tem according to the failure rates determined by Eq. (17): wi ¼ kiPn i¼1ki : ð17Þ Allocate subsystem failure rate requirements as follows: k�i ¼ wik � : ð18Þ 4.2. ME-OWA apportionment method Chang et al. (2009) presented an innovative reliability allocation using the ME-OWA method for fighter aircraft RADAR programs. This method can determine the optimal weighting vector under maximal entropy, and the OWA operator has the ability to ascer- tain the optimal reliability allocation rating after an aggregation process. With the optimal weighting vector under maximal entro- py with respect to different a values, sensitivity analysis enables the identification of different a values to evaluate their impact on the reliability allocation rating using Eqs. (7)–(9) with n = 4. Re- sults from this analysis are presented in Table 1. Table 1 The optimal weighting vector under maximal entropy (n = 4). Weight wl w2 w3 w4 a = 0.5 0.250000 0.250000 0.250000 0.250000 a = 0.6 0.416657 0.233398 0.130859 0.073547 a = 0.7 0.493805 0.237305 0.113770 0.054918 a = 0.8 0.596466 0.251953 0.106445 0.045018 a = 0.9 0.764099 0.182129 0.043457 0.010365 a = 1.0 1.000000 0.000000 0.000000 0.000000 Suppose a system is composed of m subsystems; n is the num- ber of system factors. Let Rs be the system’s allocated rating, Ri be the allocated rating to the ith subsystem, and T be the mission duration; then the system failure rate ks is determined by Eq. (19) (1988): ks ¼� lnðRÞ=T: ð19Þ This method uses four subsystem allocation factors, which are computed as a function of a numerical rating, such as system intri- cacy (I), state of the art (S), performance time (P), environment (E), mission time (T), complexity (K), functionality (F), effectiveness (E), and operational profile (O). Each rating is based on a scale from 1 to 10 and is estimated using design engineering and expert judg- ments. In order to compare the different method capabilities, the same influential system reliability factors were selected: I, S, P, and E. Also, the same estimated rating derived from design engi- neering and expert judgment. Based on Table 1 and Eq. (1), calcu- late the aggregated values by OWA weights with respect to different values of (a = 0.5, 0.6, 0.7, 0.8, 0.9, 1); a = 1 is used to represent the situation when the decision-maker is maximally optimistic (a pure optimist), and a = 0.5 is used when the deci- sion-maker faces a moderate assessment. Use Eq. (20) to calculate complexity C0k;8k. C0k ¼ w0k W0 ; 8k: ð20Þ Use Eq. (21) to calculate the allocated subsystem failure rate kk, "k. kk ¼ C 0 kks; 8k: ð21Þ 5. Proposed ME-OWA-based DEMATEL apportionment method 5.1. Advantages of the ME-OWA-based DEMATEL apportionment method With the FOO technique, the average weighting allocation method and IFM method, such as the ISPE and IG values, are ordinal measurement scales; therefore, the operations of multiplication and division are not meaningful and misleading. As a result, some (I, S, P, E) scenarios produce an ISPE value that is lower than other combinations but potentially produce a higher reliability allocation overall rating. For example, the scenario with ISPE value 9 � 5 � 2 � 2 = 180 is lower than the scenario with ISPE value 7 � 5 � 3 � 2 = 210, even thought it should have a higher reliabil- ity allocation overall rating. Therefore, I, S, P, and E are not equally weighted with respect to one another in terms of overall rating. Meanwhile, most current reliability allocation methods do not con- sider indirect relations between subsystems or components; also, most reliability allocation methods do not consider predicted fail- ure rate in the evaluation process. In resolving the five conventional reliability allocation problems mentioned above, the proposed approach is to combine the ME-OWA, DEMATEL, and the ARINC methods. The ME-OWA is used to derive ISPE values and then uses the DEMATEL for capturing the causal relationship between criteria; this method also considers a subsystem’s predicted failure rate at the same time. After calculating each subsystem’s total indirect relationship, then apportion a reasonable reliability rating into subsystems or components. To maintain the R � C value in a positive state, this study pro- poses the R � c value. c represents the average severity of influence for each alternative; the c value can be obtained through the fol- lowing formula—then, the R � c value is derived: ci ¼ CiPn i¼1 Ci : ð22Þ C.-S. Liaw et al. / Expert Systems with Applications 38 (2011) 9713–9723 9717 5.2. Procedures of the ME-OWA-based DEMATEL apportionment method The detailed procedure of the proposed ME-OWA-based DEMA- TEL apportionment method is organized into 10 steps and is de- scribed as follows: Step 1: List the structure of systems and subsystems. Step 2: Define the system reliability and mission time. Step 3: Compute the system failure rate from system specifica- tions. Based on Eq. (19), derive the system failure rate ks. Step 4: Determine the scales for I, S, P, and E, respectively. Subsystem allocation factors are computed as a function of numerical ratings of I, S, P, and E. Step 5: Perform DEMATEL procedure. The procedure of the DEMATEL process is as follows (Seyed-Hosseini et al., 2006): (1) A system designer or decision-maker evaluates the relationship between subsystems of paired alterna- tives. The pair-wise comparison scale may be desig- nated into 10 levels, where the scores of 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 represent the influence levels ‘‘None’’, ‘‘Very minor’’, ‘‘Minor’’, ‘‘Low’’, ‘‘Moderate’’, ‘‘Significant’’, ‘‘Major’’, ‘‘Extreme’’, ‘‘Serious’’, and ‘‘Hazardous’’, respectively. (2) Obtain the elements of DRSM by Eq. (11). (3) Obtain the elements of DIRSM by Eq. (12). (4) Calculate the values of R + C, R � C, and R � c. Step 6: Compute reliability allocation values by assigning differ- ent values to the conditional parameter (a = 0.5, 0.6, 0.7, 0.8, 0.9, 1). Based on Table 1 and Eq. (1), calculate the aggregated val- ues by OWA weights with respect to different values of (a = 0.5, 0.6, 0.7, 0.8, 0.9, 1). Step 7: Compute the allocation rating ri for each subsystem and derive the overall rating wi for the kth subsystem. According to the results of DEMATEL’s calculation, using Eq. (22) to derive ci, then calculate R � c values and use Eq. (23) for allocation weighting factors; calculate the aggregated value by ME-OWA-based weights. The appor- tioning weighing ratio also consider the prediction failure rate in the apportionment process. Use Eq. (23) to calcu- late the allocate weighting factors wi, "i: wi ¼ Xn i¼1 ISPEi � Di � ki= Xn i¼1 Xm i¼1 ISPEi � Di � ki ð23Þ where n: number of ISPE, m: number of subsystems, ISPEi: ME-OWA values for subsystem i, Di: R � c values for subsystem i, and ki: pre- dicted failure rate for subsystem i. Step 8: Compute the complexity C0k for the kth subsystem, "k. Use Eq. (20) to calculate the complexity C0k;8k. Step 9: Compute the allocated subsystem failure rate. Use Eq. (21) to calculate the allocated subsystem failure rate kk, "k. Step 10: Analyze the results and select the optimal reliability allo- cation decision. 6. A case study of the ME-OWA-based DEMATEL apportionment method This paper presents a real-world illustrative example for imple- mentation of the ME-OWA-based DEMATEL apportionment method. A case study of a DFLCC that is installed on a fighter aircraft, drawn from an aircraft company in Taiwan, was used to demonstrate the proposed approach. The DFLCC is a triplex redundant digital flight control computer. The DFLCC receives discrete, analog, and digital input signals from various sensors and computers throughout the fighter aircraft. It processes these input signals through a series of redundant management and control law algorithms, which were developed and supplied by the customer. The DFLCC produces various discrete, analog, and digital output signals that are sent to provide the pilot with the digital flight control system status and to the primary servo actua- tors and leading edge flap, used to control the aircraft’s motions. The DFLCC consists of eight major shop replaceable units (SRUs). The SRUs are the CPU circuit card assembly (CPU card), Input/ Output Processor circuit card assembly (IOP card), ADIO circuit card assembly (ADIO card), 1553/DOUT circuit card assembly (1553/DOUT card), power supply assembly (P/S card), ACS circuit card assembly (ACS card), IBU circuit card assembly (IBU card), and motherboard. The DFLCC block diagram is show in Fig. 1. To accurately assess reliability requirements of subsystems or components, system design engineering needs to clearly define the diagram for each system and subsystem. The diagram for the DFLCC is shown in Fig. 2. 6.1. ARINC apportionment technique analysis To illustrate this method, consider a DFLCC that consists of eight major SRUs. The SRUs are the CPU card, IOP card, ADIO card, 1553/ DOUT card, P/S card, ACS card, IBU card, and motherboard. There are eight subsystems with predicted failure rates of k1 = 22.4 (CPU card), k2 = 23.3 (IOP card), k3 = 15.3 (ADIO card), k4 = 13.1 (1553/DOUT card), k5 = 44.8 (P/S card), k6 = 35.5 (ACS card), k7 = 11.4 (IBU card), and k8 = 16.6 (motherboard) failures per 106 h, respectively. The system has a mission time of 3 h, and a 0.999412 reliability is required. The apportioned system reliability goals are found by Eq. (19) as follows: ks ¼� lnðRÞ=T ¼� lnð0:999412Þ=3 ¼ 196:058 failures per 106 h: Based on the predicted subsystem failure rate for each subsystem, as follows: k1 ¼ 22:4; k2 ¼ 23:3; k3 ¼ 15:3; k4 ¼ 13:1; k5 ¼ 44:8 k6 ¼ 35:5; k7 ¼ 11:4; k8 ¼ 16:6: Using Eq. (17), the weighting factor (wi) for each subsystem is as follows: w1 ¼ 22:4 22:4 þ 23:3 þ 15:3 þ 13:1 þ 44:8 þ 35:5 þ 11:4 þ 16:6 ¼ 0:122923 w2 ¼ 23:3 22:4 þ 23:3 þ 15:3 þ 13:1 þ 44:8 þ 35:5 þ 11:4 þ 16:6 ¼ 0:127650 w3 ¼ 15:3 22:4 þ 23:3 þ 15:3 þ 13:1 þ 44:8 þ 35:5 þ 11:4 þ 16:6 ¼ 0:083677 w4 ¼ 13:1 22:4 þ 23:3 þ 15:3 þ 13:1 þ 44:8 þ 35:5 þ 11:4 þ 16:6 ¼ 0:071762 w5 ¼ 44:8 22:4 þ 23:3 þ 15:3 þ 13:1 þ 44:8 þ 35:5 þ 11:4 þ 16:6 ¼ 0:245880 w6 ¼ 35:5 22:4 þ 23:3 þ 15:3 þ 13:1 þ 44:8 þ 35:5 þ 11:4 þ 16:6 ¼ 0:194512 w7 ¼ 11:4 22:4 þ 23:3 þ 15:3 þ 13:1 þ 44:8 þ 35:5 þ 11:4 þ 16:6 ¼ 0:062637 w8 ¼ 16:6 22:4 þ 23:3 þ 15:3 þ 13:1 þ 44:8 þ 35:5 þ 11:4 þ 16:6 ¼ 0:090958: Channel A Channel B Channel C Start Mother board End ACS card IBU card ADIO card IOP card CPU card 1553/ DOUT card P/S card 1553/ DOUT card ACS card IBU card ADIO card IOP card CPU card P/S card ACS card IBU card ADIO card IOP card CPU card 1553/ DOUT card P/S card Fig. 1. The digital flight control computer (DFLCC) block diagram. Fig. 2. Diagram for the digital flight control computer (DFLCC). 9718 C.-S. Liaw et al. / Expert Systems with Applications 38 (2011) 9713–9723 C.-S. Liaw et al. / Expert Systems with Applications 38 (2011) 9713–9723 9719 Using Eq. (18), the corresponding allocated subsystem reliability requirements are: k�1 ¼ w1ks ¼ 24:10014 k�2 ¼ w2ks ¼ 25:02689 k�3 ¼ w3ks ¼ 16:40557 k�4 ¼ w4ks ¼ 14:06958 k�5 ¼ w5ks ¼ 48:20675 k�6 ¼ w6ks ¼ 38:13554 k�7 ¼ w7ks ¼ 12:28047 k�8 ¼ w8ks ¼ 17:83306: 6.2. ME-OWA method analysis The proposed approach uses the ME-OWA operator for weight calculations. A sensitivity analysis using different values of a is presented to evaluate their impact on the reliability allocation rat- ing. Based on Table 1, the optimal weighting under maximal entro- py (n = 4), and Eq. (1), the failure rate of the CPU card is calculated as follows. For a = 0.5 (used when the decision-maker faces a moderate assessment), if I, S, P, and E for the subsystem CPU card are 10, 9, 8, and 7, respectively, and the weighting vector contains w1 = 0.25, w2 = 0.25, w3 = 0.25, and w4 = 0.25, then w0k ¼ð10 � 0:25Þþð9 � 0:25Þþð8 � 0:25Þþð7 � 0:25Þ¼ 8:5 C0k ¼ 8:5=45:5 ¼ 0:187: As a result, the failure rate of the CPU card (for a = 0.5) = 196.058 � 0.187 = 36.62622 per 106 h. a = 1 is used to represent the situation when the decision- maker is maximally optimistic (a pure optimist). For a = 1, the OWA(a1, a2, a3) = Max(a1, a2, a3) if I, S, P, and E for the subsystem CPU card are 10, 9, 8, and 7, respectively. The weighting vector con- tains w1 = 1, w2 = 0, w3 = 0, and w4 = 0; consequently, w0k ¼ð10 � 1Þþð9 � 0Þþð8 � 0Þþð7 � 0Þ¼ 10 C0k ¼ 10=59 ¼ 0:169: As a result, the failure rate of the CPU card (for a = 1) = 196.058 � 0.169 = 33.23017 per 106 h. Following the calculation above, the aggregated values of OWA weights by different values of a (a = 0.5, 0.6, 0.7, 0.8, 0.9, 1) are cal- culated for the subsystem CPU card. The resulting failure rates are 33.23017, 33.79919, 34.42269, 34.31116, 34.95256, and 36.62622 per 106 h, respectively. The failure rates for the CPU card, IOP card, ADIO card, 1553/DOUT card, P/S card, ACS card, IBU card, and motherboard are also calculated, and the results are summarized Table 2 The reliability allocation results for DFLCC system (ME-OWA method). Method (1) (2) (3) (4) Intricacy (I) State-of-the-art (S) Performance time (P) Environment (E) CPU 10 9 8 7 IOP 9 9 5 6 ADIO 5 6 3 4 1553/DOUT 7 8 7 6 P/S 4 5 10 9 ACS 3 4 4 3 IBU 2 3 3 3 Motherboard 3 4 9 4 Total 43 48 49 42 in Table 2, columns (5) through (10). The reliability allocation re- sults for the DFLCC system (ME-OWA method) are also shown in Table 2. 6.3. Proposed ME-OWA-based DEMATEL apportionment method analysis The proposed approach is to combine the ME-OWA, DEMATEL, and ARINC methods. The ME-OWA uses the four system reliability factors I, S, P, and E after the estimated rating from design engi- neering and expert judgment and under situation parameters (a = 0.5, 0.6, 0.7, 0.8, 0.9, 1) to derive the ISPE values. The results are summarized in Table 2, columns (5) through (10). Then, the DEMATEL is used for capturing the causal relationships between criteria. This method also considers a subsystem’s predicted failure rate at the same time. Based on the results of system engineering and expert evaluation of the relationship between sets of paired subsystems or components, the initial direct-relation matrix Z is shown in Fig. 3. According to Eq. (12), the elements of the direct relative severity matrix (DRSM) are obtained and shown in Fig. 4. According to Eq. (13), the elements of the direct and indirect relative severity matrix (DIRSM) are obtained, which are shown in Fig. 5. According to Eq. (10)–(15) and using Eq. (22), the outcome of DEMATEL implementation for the DFLCC with respect to direct and indirect relationships is shown in Table 3. After calculating each subsystem’s total indirect relationship, then apportion reasonable reliability ratings into subsystems or components. The calculation results for this model are as follows. The subsystem failure rates (kk, "k) are estimated from past observations or estimations. The SRUs are the CPU card, IOP card, ADIO card, 1553/DOUT card, P/S card, ACS card, IBU card, and motherboard. The predicted failure rates of the eight subsystems are k1 = 22.4 (CPU card), k2 = 23.3 (IOP card), k3 = 15.3 (ADIO card), k4 = 13.1 (1553/DOUT card), k5 = 44.8 (P/S card), k6 = 35.5 (ACS card), k7 = 11.4 (IBU card), and k8 = 16.6 (motherboard) failures per 106 h, respectively. The ISPE values for these eight subsystems are summarized in Table 2, columns (1) through (4). Then, using Eq. (1), we compute the overall rating and complexity factors and then ascertain the allocation failure rate for the CPU board as follows. For a = 0.5 (used when the decision-maker faces a moderate assessment), if I, S, P, and E for the subsystem CPU card are 10, 9, 8, and 7, respectively, the CPU card’s predicted failure rate is k1 = 22.4. Based on Table 1 and Eq. (1) to derive ISPE = 8.5, accord- ing to Eq. (10)–(15), the outcome of the DEMATEL R � c values is 5.25; then ME-OWA (5) (6) (7) (8) (9) (10) a = 0.5 a = 0.6 a = 0.7 a = 0.8 a = 0.9 a = 1.0 36.62622 34.95256 34.31116 34.42269 33.79919 33.23017 31.24001 31.26718 30.90323 31.12875 30.76176 29.90715 19.39035 19.69331 19.55413 19.77479 19.86120 19.93810 30.16277 28.23552 27.62428 27.65346 27.01773 26.58414 30.16277 32.21480 32.23624 32.75853 33.23659 33.23017 15.08138 14.34666 14.06539 14.09318 13.75045 13.29207 11.84966 11.11608 12.70847 10.82105 10.41737 9.96905 21.54484 24.23189 24.65510 25.40554 27.21371 29.90715 196.05800 196.05800 196.05800 196.05800 196.05800 196.05800 Fig. 3. Corresponding initial direct-relation matrix Z. Fig. 4. Corresponding DRSM of the DFLCC. Fig. 5. Corresponding DIRSM of the DFLCC. Table 3 The R + C, R � C and R � c values of the DFLCC by the ME-OWA based DEMATEL method. No. R C c R + C R � C R � c 1 5.401 5.301 0.151 10.702 0.100 5.250 2 4.853 5.346 0.152 10.198 �0.493 4.700 3 4.382 4.768 0.136 9.150 �0.386 4.246 4 4.388 4.586 0.131 8.974 �0.199 4.257 5 5.156 4.914 0.140 10.070 0.242 5.016 6 3.502 3.839 0.109 7.341 �0.338 3.392 7 3.367 3.411 0.097 6.778 �0.044 3.270 8 4.018 2.902 0.083 6.920 1.117 3.936 9720 C.-S. Liaw et al. / Expert Systems with Applications 38 (2011) 9713–9723 w0k ¼ 8:5 � 5:25 � 22:4 ¼ 999:6 C0k ¼ 999:6=4899:96 ¼ 0:204: As a result, the failure rate of the CPU card (for a = 0.5) = 196.058 � 0.204 = 39.99635 per 106 h. Following the calculation above, the aggregated values of OWA weights (a = 0.5) for the subsystem CPU card, IOP card, ADIO card, 1553/DOUT card, P/S card, ACS card, IBU card, and motherboard are 39.99635, 31.76884, 11.6974, 15.61947, 62.93992, 16.86418, 4.10171, and 13.07012 failures per 106 h, respectively. The results are summarized in Table 4, column (7). a = 1 is used to represent the situation when the decision- maker is maximally optimistic (a pure optimist). For a = 1, the OWA(a1, a2, a3) = Max(a1, a2, a3). IfI, S, P, and E for the subsystem CPU card are 10, 9, 8, and 7, respectively, the CPU card’s predicted failure rate is k1 = 22.4. Based on Table 1 and Eq. (1), ISPE = 10 is de- rived. According to Eq. (10)–(15) and using Eq. (22), the outcome of the DEMATEL R � c values is 5.25; then w0k ¼ 10 � 5:25 � 22:4 ¼ 1176:01 C0k ¼ 1176:01=6426:23 ¼ 0:183: As a result, the failure rate of the CPU card (for a = 1) = 196.058 � 0.183 = 35.87878 per 106 h. Following the calculation above, the aggregated values of OWA weights (a = 1) for the subsystem CPU card, IOP card, ADIO card, 1553/DOUT card, P/S card, ACS card, IBU card, and motherboard are 35.87878, 30.0706, 11.89226, 13.61114, 68.55898, 14.69581, 3.41185, and 17.93859 failures per 106 h, respectively. The aggre- gated values of OWA weights by different values of (a = 0.5, 0.6, 0.7, 0.8, 0.9, 1) for the subsystem CPU card, IOP card, ADIO card, 1553/DOUT card, P/S card, ACS card, IBU card, and motherboard are also calculated, and the results are summarized in Table 4, columns (7) through (12). The reliability allocation re- sults for the DFLCC system (ME-OWA-based DEMATEL method) are shown in Table 4. 6.4. Method comparison As shown in Table 5, the failure rates using the ARINC appor- tionment technique for the CPU card, IOP card, ADIO card, 1553/ DOUT card, P/S card, ACS card, IBU card, and motherboard are cal- culated, and the results are summarized in Table 5, row (13). The ME-OWA method results are shown in Table 5, rows (7) through (12). The ME-OWA-based DEMATEL method results are shown in Table 5, Rows (1)–(6). In this case study, using the ARINC appor- tionment technique, the appointing failure rates were calculated for the subsystem P/S card (48.20675) > ACS card (38.13554) > IOP card (25.02689) > CPU card (24.10014) > Motherboard (17.83306) > ADIO card (16.40557) > 1553/DOUT card (14.06958) > IBU card (12.28047). Compared with the ME-OWA method (a = 0.9), the appointing failure rate of the subsystem CPU card (33.79919) > P/S card (33.23659) > IOP card (30.76176) > Mother- board (27.21371) > 1553/DOUT card (27.01773) > ADIO card (19.8612) > ACS card (13.75045) > IBU card (10.41737). Compared with the ME-OWA-based DEMATEL method (a = 0.9), the appoint- Table 4 The reliability allocation results for DFLCC system (ME-OWA based DEMATEL method). Method (1) (2) (3) (4) (5) (6) ME-OWA I S P E R � c (DEMATEL) Predicted failure rate (7) (8) (9) (10) (11) (12) a = 0.5 a = 0.6 a = 0.7 a = 0.8 a = 0.9 a = 1.0 CPU 10 9 8 7 5.250 22.400 39.99635 37.74110 37.22365 37.07745 36.36190 35.87878 IOP 9 9 5 6 4.700 23.300 31.76884 31.44028 31.22120 31.22401 30.81863 30.07060 ADIO 5 6 3 4 4.246 15.300 11.69740 11.74707 11.71919 11.76663 11.80378 11.89226 1553/DOUT 7 8 7 6 4.257 13.100 15.61947 14.45767 14.21155 14.12477 13.78338 13.61114 P/S 4 5 10 9 5.016 44.800 62.93992 66.46878 66.82744 67.42426 68.32559 68.55898 ACS 3 4 4 3 3.392 35.500 16.86418 15.86289 15.62541 15.54426 15.14792 14.69581 IBU 2 3 3 3 3.270 11.400 4.10171 3.80468 4.37027 3.69459 3.55246 3.41185 Motherboard 3 4 9 4 3.936 16.600 13.07012 14.53553 14.85930 15.20203 16.26432 17.93859 Total 43 48 49 42 34.067 182.400 196.05800 196.05800 196.05800 196.05800 196.05800 196.05800 Table 5 The allocation results for three methods. Method Conditional parameter (a) CPU IOP ADIO 1553/DOUT P/S ACS IBU Motherboard 1 ME-OWA based DEMATEL 0.5 39.99635 31.76884 11.69740 15.61947 62.93992 16.86418 4.10171 13.07012 2 0.6 37.74110 31.44028 11.74707 14.45767 66.48878 15.86289 3.80468 14.53553 3 0.7 37.22365 31.22120 11.71919 14.21155 66.82744 15.62541 4.37027 14.85930 4 0.8 37.07745 31.22401 11.76663 14.12477 67.42426 15.54426 3.69459 15.20203 5 0.9 36.36190 30.81863 11.80378 13.78338 68.32559 15.14792 3.55246 16.26432 6 1 35.87878 30.07060 11.89226 13.61114 68.55898 14.69581 3.41185 17.93859 7 ME-OWA 0.5 36.62622 31.24001 19.39035 30.16277 30.16277 15.08138 11.84966 21.54484 8 0.6 34.95256 31.26718 19.69331 28.23552 32.21481 14.34666 11.11608 24.23189 9 0.7 34.31116 30.90323 19.55413 27.62428 32.23624 14.06539 12.70847 24.65510 10 0.8 34.42269 31.12875 19.77479 27.65346 32.75853 14.09318 10.82105 25.40554 11 0.9 33.79919 30.76176 19.86120 27.01773 33.23659 13.75045 10.41737 27.21371 12 1 33.23017 29.90715 19.93810 26.58414 33.23017 13.29207 9.96905 29.90715 13 ARINC – 24.10014 25.02689 16.40557 14.06958 48.20675 38.13554 12.28047 17.83306 C.-S. Liaw et al. / Expert Systems with Applications 38 (2011) 9713–9723 9721 ing failure rate of the subsystem P/S card (68.32559) > CPU card (36.3619) > IOP card (30.81863) > Motherboard (16.26432) > ACS card (15.14792) > 1553/DOUT card (13.78338) > ADIO card (11.80378) > IBU card (3.55246). The allocation results for the three methods are shown in Table 5. From Table 5, because the ME-OWA-based DEMATEL method using ISPE values as a based, and followed by DEMATEL calcula- tion process, which considers direct and indirect relationships between each of the subsystems at the same time. The results indicate that the P/S card and CPU card have higher failure rates, while the ADIO card and IBU card have lower failure rates. The P/S card R � c value is 5.016 and has a higher predicted failure rate (k5 = 44.8) than the CPU card (R � c values is 5.25 and pre- dicted failure rate k1 = 22.4), whereas the IBU card R � c value is 3.27 and has a lower predicted failure rate (k7 = 11.4) than the ADIO card R � c values (4.246 and predicted failure rate k3 = 15.3). From a technical perspective, the subsystems have higher R � c criteria (higher direct and indirect relationships) and higher predicted failure rates; the proposed ME-OWA-based DEMATEL apportionment method is more reasonable in appoint- ing a more reliable ratio in subsystems. As shown in Fig. 6, the results of the ME-OWA-based DEMATEL apportionment method, which incorporates ME-OWA, DEMATEL, and ARINC methods, this method obtains a correct and discrimi- nating allocation ratio. The result is a more flexible and reasonable allocation rating than the conventional ARINC apportionment tech- nique and the ME-OWA method. The results of the proposed ME- OWA-based DEMATEL method are compared with the ARINC apportionment technique and the ME-OWA method, shown in Fig. 6 below. A comparison of the ARINC apportionment technique, the ME-OWA method, and the proposed ME-OWA-based DEMATEL apportionment method is summarized in Table 6. ‘‘O’’ indicates that the related factor is applicable, whereas ‘‘X’’ indicates that the related factor is not applicable. The proposed method has concluded a number of advantages with its potentialities: (1) Proposes a combined reliability allocation method using the ME-OWA-based DEMATEL method in apportioning system reliability, which combines the ME-OWA, DEMATEL, and the ARINC methods. It can overcome the three conventional shortcomings: measurement scale problem, the not equally weighted problem, and no consideration of indirect relation- ships between subsystems during the appointment pro- cesses. It also uses the ARINC’s concept to consider the predicted failure rate for appointing reliability into subsys- tems or components. (2) Considers situation parameter factors: the ME-OWA is based on estimated ratings from design engineering and expert judgment for appointing reliability. The four system reliabil- ity factors I, S, P, and E under the situation parameter (a = 0.5, 0.6, 0.7, 0.8, 0.9, 1) are used to derived the ISPE values. This assessment result shows that the ME-OWA-based DEM- ATEL method yielded results that not only were correct but also resulted in a discriminating allocation ratio that is flex- ible for real-world applications. (3) Considers indirect relationships between subsystems and components: the combined DEMATEL calculates processes and can consider indirect relationships between subsystems and components. This calculation holds that the higher indi- rect relationship subsystems are appointed a higher alloca- tion ratio, which can efficiently allocate limited resources in subsystems or components. Fig. 6. Comparison of the five methods of reliability allocation. Table 6 Comparison of the three methods. Method Consider factor Measurement scale Order weight Indirect relationship Predicted failure rate Proposed method O O O O ME-OWA O O X X ARINC X X X O Note: ‘‘O’’ represents that the factor is applicable, and ‘‘X’’ represents that the factor is not applicable. 9722 C.-S. Liaw et al. / Expert Systems with Applications 38 (2011) 9713–9723 (4) Provides an organized approach and a more flexible struc- ture in reliability allocation: the ME-OWA-based DEMATEL method is applicable to the different design phases. The sys- tem reliability factors are not limited to only I, S, P, and E, and the number of factors is not limited to 4 items (3 or more items are acceptable). Depending upon the selection of applicable variables, such as system intricacy, complexity, state of the art (technology), cost, maintenance, risk, failure rate, design maturity, operating environment, and repair times, the allocating ratio is more suitable for different alter- natives. There is no limitation for implementation of DEMATEL in very large and complex systems, and it can therefore provide an improved structured arrangement for reliability allocation. 7. Conclusion This paper has successfully demonstrated the application of an ME-OWA-based DEMATEL apportionment method for reliability allocation using a fighter aircraft DFLCC. It is an easy, proven, and effective approach, which uses the ME-OWA to derive the ISPE val- ues, followed by the DEMATEL calculation processes to ascertain indirect relations. The higher ISPE values, indirect relationship sub- systems, and higher predicted failure rate translates into a higher reliability allocation overall rating. The main advantages of the proposed approach are: (1) It pro- vides an accurate yet flexible reliability allocation method, which combines the ME-OWA, DEMATEL, and the ARINC methods. The proposed approach can efficiently resolve the measurement scale problem, equally-weighted problem, and considers indirect rela- tionships between subsystems during the appointment processes. (2) The ME-OWA-based DEMATEL method uses the conditional parameter (a) to derive the ISPE values. The assessment results show that the ME-OWA-based DEMATEL method yields results that not only are accurate but also yield discriminating allocation ratios that are flexible for real-world applications. The proposed ME-OWA-based DEMATEL method can better help managers or designers make correct decisions. (3) Using DEMATEL indirect rela- tionships between subsystems and components can be considered. The calculation holds that the higher indirect relationship subsys- tems are appointed a higher allocation ratio, which can result in more efficient allocation of limited resources in subsystems or components. (4) It provides an organized approach and a more flexible structure in reliability allocation. The ME-OWA-based DEMATEL method is applicable to different design phases. The sys- tem reliability factors are not limited to only I, S, P, and E (3 or more items are acceptable). Depending upon the selection of applicable variables, such as system intricacy, complexity, state of the art (technology), cost, maintenance, risk, failure rate, design maturity, operating environment, and repair times, the allocating ratio is more suitable for different alternatives. There is no limitation for implementation of DEMATEL in very large and complex systems, and it can thereby provide an improved structured arrangement for reliability allocation. 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ME-OWA based DEMATEL reliability apportionment method Introduction ME-OWA operators and its operations ME-OWA operators Determination of ME-OWA weights DEMATEL methodology Outline of the DEMATEL method The procedure of the DEMATEL method Conventional reliability allocation methods ARINC apportionment technique ME-OWA apportionment method Proposed ME-OWA-based DEMATEL apportionment method Advantages of the ME-OWA-based DEMATEL apportionment method Procedures of the ME-OWA-based DEMATEL apportionment method A case study of the ME-OWA-based DEMATEL apportionment method ARINC apportionment technique analysis ME-OWA method analysis Proposed ME-OWA-based DEMATEL apportionment method analysis Method comparison Conclusion References