A-IFM reliability allocation model based on ...

A-IFM reliability allocation model based on multicriteria approach 1. Introduction Reliability allocation permits to assign reliability parameters to the different system units, so that the whole system reaches the established reliability target. Reliability Analysis is based on the performance results of tests in labs and working products. These data can be used to measure and improve the reliability of the products being produced. Often, an initial cost reduction is achieved by using cheaper parts or cutting testing programs. Unfortunately, quick savings thanks to the use of cheaper components or few test samples, usually result in higher long-term costs in terms of warranty costs or loss of the customer confidence. Any Reliability allocation method requires the definition of several factors such as system and nits failures, system reliability target, unit technology (mean fault rate), relation between the unit fault and the system fault, operation cycles etc. Starting from the system target, it is necessary to define reliability characteristics of the components: mean fault rates, mean lives or reliability for an established time period. We have series, modified, redundant and multi-modal systems according to the different functional duplications. It’s also important to consider the Modal Design Adequacy (Aggarwal, 1993; Dai and Wang, 1992) that expresses the relation between the mission success probability and the modal efficiency (frame of components to guarantee the system functions). The component operative time refers to the operation cycles (low cycles, high reliability). Therefore we can affirm: • the component allocated reliability grows with the reduction of the technology, of the operative time, and with the increase of the factor of importance; • the units with the same importance, operative time and technology, have to have the same allocated reliability. The mathematics treatment of allocation is strongly simplified if the following hypotheses are made: • the units must be chosen by independent fault probabilities; • the unit state must be described by binary terms: on/offThe system reliability allocation to the units, involves the resolution of the following inequality: f(R1°,R2°,…,RN°) ≥ R (1) R°j: allocated reliability to unitj; R*: system reliability target; f ( ): functional relation between the unit reliability and the system one.

Reliability allocation is a top-down method to determine the components’ target in a system. The Advisory Group on Reliability of Electronic Equipment (AGREE), 1957) developed a reliability allocation method. This takes into account unit or subsystem complexity and criticality more than failure rates. In contrast to the AGREE method, Aeronautical Radio Inc. (Alven, 1964) published the ARINC technique, based on the failure rates of units or subsystems. In addition to these methods, Bracha (1964) introduced an allocation reliability method using four factors: state-of-theart, subsystem complexity as estimated by number of parts, environmental conditions, and relative operating time. Karmiol (1965) evaluated instead complexity, state-of the-art, operational profile, and criticality of the system, related to mission objectives, in order to allocate reliability to each subsystem. More recently, the engineering design guide, Reliability Design Handbook. (Anderson, 1976), featured the feasibility-of-objectives (FOO) technique, which was incorporated into the Mil-hdbk-338B handbook (United States Department of Defense, 1988), an established standard for military reliability design. The FOO technique specifically provides a detailed reliability allocation

procedure for mechanical–electrical systems. Smedley (1992) applied FOO to perform reliability analysis among low energy booster (LEB) ring magnet power systems in a superconducting supercollider. In addition, Kuo (1999) created an average weighting allocation method as a guide for commercial reliability allocation design, while Falcone et al since 2004 used the Integrated Factors Method (IFM) (Falcone et al., 2004; Di Bona et al., 2014 (a); Di Bona et al., 2014 (b)) for reliability and safety allocation for an aerospace prototype project and for industrial applications. More recent papers, in particular Kim et al. (2013) and Yadav et al. (2006, 2007, 2014) use the risk priority number (RPN) (Bowles, 2003; Itabashi-Campbell and Yadav, 2009) generated during qualitative criticality analysis FMEA (Department of the Army. TM 5-689-4, 2006). All conventional allocation methods assign the same weight to the considered factors, moreover many factors are subjectively evaluated. In order to solve these problems, Cheng et al. (2006) and Chang et al. (2008, 2009) consider a reliability allocation weight based on an ordered weighted averaging operator proposed by Yager (1988) (ME-OWA); moreover Liaw et al. (2011) assigned a different weight to the factors, basing on the DEMATEL decision making method together with ME-OWA. According to these methods, the weights assigned are the same for each unit or subsystem, while in reality the weight of each factor may vary depending on the unit in question and of its function in the system. These considerations suggest the implementation of an allocation method able to assign different weights to each factor and for each unit considered. In particular, the proposed method Analytic IFM method (A-IFM) uses the IFM method to allocate reliability, supported by an Analytic Hierarchy Process (AHP) a well-known technique introduced by Saaty (Saaty, 1980) that is used to determine how the reliability of a distributed system may be controlled by appropriately assigning weights to its components. The paper is organized as follows: Section 2 introduces conventional reliability allocation methods, Section 3 analyzes conventional reliability methods to aerospace prototype, Section 4 introduces Analytic Hirerarchy Process method, Section 5 proposes research architecture, and in Section 6, an example is drawn from an aerospace prototype using the proposed approach. Finally in section 7 results and conclusions are analyzed.

2. Reliability Allocation Methods In the present section, the principal procedures and techniques of reliability allocation are analysed. The different methodologies can be used together; it is often possible to use more techniques in the different phases of a complex system plan. Let R*(t) be the realibility target of a series system operating for a t>0. Let Ri*(t) be the reliability allocated to the ith unit for i=1,…,k. The system reliability is (Equation 2): k

R * (t ) = ∏ Ri* (t )

t>0 (2)

i =1

Let wi be a reliability allocation weight for unit i. The allocation procedure is an iterative one, it starts from the initial plan step, when few data about components are available. In this phase it is better to consider the sub-systems in series and to adopt one of the allocation methodologies for such systems. If no definitive information is available on the unit, an Equal Method is used where the system failure rate target λ* is assigned equally among all subsystem using (Equation 3 and 4): λ* = wi λ* (3) i*

1 i=1,…..,k (4) k When the failure rate of subsystem i can be predicted by λ*i* from data banks or previous experience

where λ* the is failure rate of unit I and w i * = i*

on similar subsystem, the ARINC Method can be used where: w i* =

λi

(5)

k

∑λ

i

i =1

The Boyd Method is a weighted sum of Equation (4) and Equation (5):    1 λ  w i* = K a + (1 − a ) k i   k λi  ∑   i =1 (6)

where “a” is a weight to combine ARINC method and Equal Method and K is a safety margin. Then, when more data are available (number of components and their interconnections), it is possible to use other methodologies. The AGREE method is based on the follow equation (Equation 7 and Equation 8): R *i * (t ) = R * (t ) wi (7) k

where

∑w i =1

i

= 1 and w i * =

ni

(8)

k

∑n

i

i =1

i=1,….,k where ni is the number of components in unit i In the second phase, when many parameters, about the number of components and their connections, are known , it is possible to use more Complex Methods, based on different factors: system criticality, technology, mission time etc. Let λ* the failure rate of unit i and λ* the failure rate of the system. The Bracha method (Equation 9) is based on Equation (7), where wi is obtained by considering four factors: - Ai1 : state of art-technology of unit i - Ai2 : complexity of unit i (evaluated by the number of components in unit i) - Ai3 : operating time factor of unit i (evaluated by the relative operating time of unit i during the total mission time) - Ai4 : environmental condition (evaluated by the externally applied effective stress to unit i) i*

wi =

Ai1 ( Ai 2 + Ai 3 + Ai 4 ) k

∑[ A

i1

i=1,…..,k (9)

( Ai 2 + Ai 3 + Ai 4 )]

i =1

The Feasibility of Objectives (FOO) method included in Mil-Hdbk-338Bis also based on Equation (3) after considering four factors: - Ai1 : state of art-technology of unit i - Ai2 : intricacy of unit i - Ai3 : operating time of unit i

- Ai4 : environmental condition An expert judgment evaluates each factor using a 10-point numerical scale and the final allocation is given by (Equation 10): (A A A A ) i=1,…..,k (10) w i = k i1 i 2 i 3 i 4 ∑ [ Ai1 Ai 2 Ai 3 Ai 4 ] i =1

Based on FOO method, Chang et al. and Liaw et al. consider a reliability allocation weight based on operator, proposed by Yager. When aggregating n factor ratings together to form the weight of subsystem i, two extreme cases are considered: - a high weight is assigned when all of the factors have high ratings; - a high weight is assigned when at least one of the factors has a high rating. The degree of a mathematical operator called “orness” (a situation parameter) is fixed between 0 and 1 depending on if all of the factor ratings must be high or if at least one factor must be high. Given the degree of orness, a weight of a factor, denoted by bj, is determined by an ordered position of a factor rating in a subsystem using the maximal entropy of the weight ME-OWA proposed by O'Hagan (1988) and Fuller and Majlender (2001). Then, the allocation weight is expressed by (Equation 11): n

∑b wi =

j

Aij

j =1 k

i=1,…..,k (11)

n

∑∑ b

j

Aij

i =1 j =1

where n is the number of factor sand Aij is the ordering of Aij within a subsystem i. The weight bj is not related with the importance of a factor but with its ordering position in a subsystem. When the subsystems have λ* = cons tan t , we further consider a case that the system reliability and all unit i*

reliabilities are closed to one. Using an approximation, 1 − e − λt = λ t , Equation (3) is reduced to (Equation 12): F i** (t ) = wi F * (t ) i=1,…..,k (12)

where F* is unreliability system and Fi*unreliability unit i. Karmiol Method uses Equation (12) considering four factors: - Ai1 : state of art-technology of unit i - Ai2 : intricacy of unit i - Ai3 : operating time of unit i - Ai4 : criticality of unit i (defined as the effect of the unit i failure on mission success) The allocation weight is (Equation 13): wi =

( Ai1 + Ai 2 + Ai 3 + Ai 4 ) k

∑[ A

i1

i=1,…..,k (13)

+ Ai 2 + Ai 3 + Ai 4 ]

i =1

where Aij are numerical rating on a 10-point scale. The Integrated Factors Method (IFM) considering six factors:

-

-

Ai1 : criticality of unit i (ratio between the number of sub-system functions that cause an undesirable event if not realised, and the number of total functions of the unit) Ai2 : intricacy of unit (ratio between the number of parts of the unit and the number of parts of the whole system) Ai3 : functionality of unit I (ratio between the number of total unit functions, and the number of total system function) Ai4 : Effectiveness index (ratio between the unit effectiveness time and the mission total time). A5:Technology index A5=0,5: traditional components; A5=1: innovative components). To discriminate against electronic systems, against mechanical ones, characterized by the same complexity, A6:Electronic Functionality index A6=1: completely electronic system; A6=0,1: completely mechanical system).

is develop where (Equation 14): −1

wi = GI % =

−1

( Ai1 Ai 2 Ai 3 Ai 4 Ai 5 Ai 6 ) k

∑[A

i1

−1

i=1,…..,k (14)

−1

Ai 2 Ai 3 Ai 4 Ai 5 Ai 6 ]

i =1

where Aij are numerical rating of four factors. Some recent papers consider the failure effect in reliability allocation using Risk Priority Number (RPN) generated during the FMEA analysis. Let unit I have Ni failure modes with severity ranking Sij, occurrence rating Oij and detection ranking Dij for j=1,….,N and i=1,…,k. The three factors are evaluated by an ordinal scale from 1 to 10. The RPN of failure mode j in unit I is given by (Equation 15): RPN= Sij x Oij x Dij (15) Then the reliability allocation weight is given by (Equation 16): wi =

ωi

where ω i = 1 −

k

∑ω i =1

i

Ci

and C i =

k

∑C

1 Ni

Ni

∑ (S

ij

xOij xDij ) (16)

j =1

i

i =1

Each analysed method shows own advantages and disadvantages. As already discussed, most of the methods use constant weights for the factors involved in the reliability allocation, regardless of the purpose of the system design. Furthermore, also methods which provide for the assignment of weights, do not consider that they can vary for each unit or subsystem, depending on the individual case. We can affirm that there isn’t an universal reliability allocation technique. Each method shows own advantages and disadvantages. Table 1 shows advantages and disadvantages of the literature methods that can be applicated to the aerospace prototype. Table 1: Principal advantages and disadvantages of the analysed methods for aerospace prototype The above analysis suggests some guide lines to develop the new technique; so we have analysed and applied those methods more suitable for the aerospace prototype. In particular we applied Integrated Factors Method, FOO, and Karmiol because :

-

-

-

aerospace prototype is an innovative system in pre-desing phase, but many parameters, about the number of components and their connections, are known (system criticality, technology, mission time etc); aerospace prototype units have λ=constant, so we further consider a case that the system reliability and all unit reliabilities are closed to one; aerospace prototype is a series configuration.

3. Reliability Allocation to Aerospace Prototype In the future, it will be possible to achieve the space by conventional vehicles, with characteristics coming from the traditional aviation, often called aerospace planes. The considered project analyses the vehicles reusability through flying demonstration, putting a system under the real operative conditions. A flying test bed (flying demonstrator) is the system under study: a propelled winged vehicle (Figure 1), with: - a flying segment of three stages (Stratospheric Ballon, Winged and Kick) ; - a ground segment of six stations.

Figure 1: Prototype Product Tree In order to obtain knowledge and skills about RLV (Reusable Launch Vehicles), a Sub-orbital Reentry Test will be programmed (Figure 2).

Figure 2: Sub-orbital Re-entry Test R.A.M.S. Analysis (Reliability Availability Maintainability Safety Analysis) involves the whole project development; in particular, the actual pre-design phase requires a first reliability evaluation. The first step has been to study reliability allocation techniques in literature, to point out those more suitable for prototype complex systems. The reliability target allocation to the system units has required a preliminary study, whose main steps are: Product Tree definition Reliability Block Diagram definition Preliminary Hazard Analysis Functional analysis Functional-FMECA Starting from Figure 3, the Product Tree (Figure 3) and the RDB (series configuration) have pointed out these units: -

Unit 1: Stratospheric Balloon Unit 2: Propeller Unit 3: Avionics Unit 4: Stage Separation System Unit 5: Instrumentation Unit 6: Self-destruction System Unit 7: Parachute System Unit 8: Splashdown System

Figure 3: Product Tree for Reliability Allocation The Preliminary Hazard Analysis has pointed out three top events: -

Top Event 1: Data loss (Significant Event) Top Event 2: Vehicle loss (Major Event) Top Event 3: Mission failure (Failure Event)

The unreliability target are: -

Top Event 1: F(t)=0,10 Top Event 2: F(t)=0,15 Top Event 3: F(t)=0,20

The target assignment to the units has been realised using IFM, FOO and Karmiol. We have valued the allocation indexes for each unit put in series, included in the Reliability Block Diagram, coming out from the aerospace prototype Product Tree (Table 2). It is important to note that in pre-design phase, it is impossible to value A2 IFM index as ratio between the number of parts of the unit and the number of parts of the whole system. So we decide to value A2 IFM index through a qualitative mode: referring to the technological and constructive structure; possible values are: 0,10 for simple system; 0,20-0,90 for not very complex system; 1,00 for complex one. To value the indexes, we have used both preliminary study results (Functional Analysis e FFMECA) and an Expert Judgement. Applied (14) and (12) we obtained the following IFM matrices with GI% and F(t)allocated: Table 2: IFM Matrix (Top Event 1-2-3) Applied (10) and (13) we have valued the FOO and Karmiol indexes for three Top Event. Then the allocated reliability has been evaluated by Equation 17: R(t)allocat = 1-F(t)allocate (17) Table 3 shows the IFM results, compared with the FOO and Karmiol ones.

Table 3: Comparison between IFM, FOO and Karmiol results The values of Table 3 are compared with the reliability data, obtained from databanks (NASA, 2007) or supplied by manufacturers. For the Top Event 3, the most critical one F*(t)=0.20, it is possible to notice that (Figure 4): • the allocated reliability values are comparable to the supplied reliability ones but not for unit 1 and unit 3; • the units performances are partly respected. Figure 4: Reliability data comparison (Failure Event)

In order to calculate the mean value of absolute percentile errors (έ), the results obtained by the considered allocation techniques (IFM, FOO and Karmiol) were compared with reliability data coming out from data banks or manufacturers (Table 4). The result shows that the IFM method obtains a more reasonable reliability allocation rating than FOO and Karmiol methods. It’s possible to affirm that the new methodology gives more consistent output-data with data banks (Equation 18): MADIFM < ( MADFOO; MADKarmiol-sum; MADBracha)

(18)

Also evaluating negative technological errors (Equation 19), the IFM provides more similar values to databanks (Table 4). (19)

έtechnological =[ R(t)databanks -R(t)allocat]

Table 4 Comparison between Reliabilty allocation methods and databanks (MAD, έtechnological) Negative technological errors (R(t)allocat> R(t)databanks) highlight the criticalities of the allocation technique. The sum ∑(-έ) for IFM is minimum (∑(-έ)= -8,85%). 4. Analytic Hierarchy Process The Analytic Hierarchy Process (AHP) breaks down a decision-making problem into several levels in such a way that they form a hierarchy with unidirectional hierarchical relationships between levels (De Felice and Petrillo, 2014). The AHP for decision making uses objective mathematics to process the inescapably subjective and personal preferences of an individual or a group in making a decision. With the AHP, one constructs hierarchies or feedback networks, then makes judgments or performs measurements on pairs of elements with respect to a controlling element to derive ratio scales that are then synthesized throughout the structure to select the best alternative (De Felice, 2012). The top level of the hierarchy is the main goal of the decision problem. The lower levels are the tangible and/or intangible criteria and sub-criteria that contribute to the goal. The bottom level is formed by the alternatives to evaluate in terms of the criteria. The modeling process can be divided into different phases for the ease of understanding which are described as follows: PHASE 1: Pairwise comparison and relative weight estimation. Pairwise comparisons of the elements in each level are conducted with respect to their relative importance towards their control criterion. Saaty suggested a scale of 1-9 when comparing two components (see Table 5). For example, number 9 represents extreme importance over another element. And number 8 represents it is between ‘‘very strong important” and ‘‘extreme importance” over another element.

Table 5: Semantics scale of Saaty For a general AHP application we can consider that A1, A2,…,Am denote the set of elements, while aij represents a quantified judgment on a pair of Ai, Aj. Through the 9-value scale for pairwise comparisons, this yields an [m x m] matrix A as follows:

A= aij=

A1 A2 … Am

A1 1 1/a12 … 1/a1m

A2 a12 1 … 1/a2m

… … … … …

Am a1m a2m … 1

where aij > 0 (i, j = 1, 2,..,,m), aii = 1 (i = 1, 2,…,m), and aij = 1/aji ( 1; 2;…,m). A is a positive reciprocal matrix. The result of the comparison is the so-called dominance coefficient aij that represents the relative importance of the component on row (i) over the component on column (j), i.e., aij=wi/wj. The pairwise comparisons can be represented in the form of a matrix. The score of 1 represents equal importance of two components and 9 represents extreme importance of the component i over the component j. In matrix A, the problem becomes one of assigning to the m elements A1, A2,…,Am a set of numerical weights w1, w2,…,wm that reflects the recorded judgments. If A is a consistency matrix, the relations between weights wi, wj and judgments aij are simply given by aij = wi/wj (for i,j = 1, 2, …, m) and

A=

A1 A2 … Am

w1/w1 w2/w1

w1/w2 w2/w2

… wm/w1

… wm/w2

w1/wm w2/wm … …

… wm/wm

If matrix w is a non-zero vector, there is a λmax of Aw = λmaxw, which is the largest eigenvalue of matrix A. If matrix A is perfectly consistent, then λmaxw = m. But given that aij denotes the subjective judgment of decision-makers, who give comparison and appraisal, with the actual value (wi/wj) having a certain degree of variation. Therefore, Ax = λmaxw cannot be set up. So the judgment matrix of the traditional AHP always needs to be revised for its consistency. PHASE 2: Priority vector. After all pairwise comparison is completed, the priority weight vector (w) is computed as the unique solution of Aw = λmaxw, where λmax is the largest eigenvalue of matrix A. PHASE 3: Consistency index estimation. Saaty (1990) proposed utilizing consistency index (CI) to verify the consistency of the comparison matrix. The consistency index (CI) of the derived weights could then be calculated by: CI = (λmax−n)/ n−1. In general, if CI is less than 0.10, satisfaction of judgments may be derived.

5. Research Architecture: the IFM based AHP allocation reliability method With IFM method it is possible to obtain a better allocation of reliability target as shown in Table 4 (MAD and error Technology smaller than the other values). However, IFM method, as FOO and Karmiol methods, presents two main weaknesses: - All Ai factors, like other methods, have a unit weight (equal importance); - All unitsi have equal importance in process allocation target. Definitively the above weaknesses cause deviations from databanks (MADIFM=10,09% e ∑(-έ)= 8,85%) which are often not acceptable especially in the design phase of complexes prototype systems complexes (for which values of databanks are not always available or unreliable). These deviations are mainly determined by the values placed for unit 1. έtechnological =[ R(t)databanks -R(t)allocat]= - 7,36% This is due because unit 1 requires performance that are not in the literature. In our opinion, these above aspects represent an evident limitation. Thus, in order to develop a methodology applicable to complex systems in the pre-design phase when the databanks are absent

or unreliable in this paper we propose a new allocation method, called Analytic IFM method (AIFM), in which the AHP method is integrated with the IFM method. The aim of A-IFM method is to assign to different factors and to different units a different weights according to their greater or lesser importance during the allocation process reliability. Through the pairwise comparisons between the units and factors, it will be assigned a higher weight and consequently a greater unreliability allocated to units with less responsible for the failure of the mission (upward trend GI%). In this way it will be possible to minimize MAD and έtechnological through the alignment of data allocated with databanks, according to target system's reliability. A-IFM method is structured in the following steps: -

Step 1: Definition of the system and uniti (Product Tree Design); Step 2: Construction of a Reliability Block Diagram; Step 3: Analysis of a Preliminary Hazard Analysis and definition of the Top Events; Step 4: Functional Analysis and FMECA Analysis; Step 5: Calculation of factors A1, A2,A3, A4, A5, A6 for the single uniti and definition of the IFM Matrix [ix6]. A1 A2 A3 A4 A5 A6 w11 w12 w13 w14 w15 w16 Unit 1 w21 w22 w23 w24 w25 w26 Unit 2 w31 w32 w33 w34 w35 w36 Unit 3 ……….. ……… ……… ………. ………. ……… ……... wi2 wi3 wi4 wi5 wi6 wi1 Unit i

-

Step 6: AHP method applied to units and to factors of allocation, as shown in the following AHP Matrix.

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 w1/w1 w1/w2 … … … …. … w1/w8 … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … … w8/w1 … … … … … … w8/w8

In the present step it will be defined a pairwise comparison among units according to Saaty’s Scale. The result is a ranking vector for units [ix1]. Unit 1 Unit 2 Unit 3 ……. Unit i

z11 z21 z31 … zi1

n

Where

∑z

i1

= 1 and unit with zi1max it will be the unit with a lower value of performance

i =1

(bigger is GI%, higher is reliability allocated). Then, according to Saaty’s Scale it will be defined a pairwise comparison among factors A1, A2,A3, A4, A5, A6 i. The result is a ranking vector for factors [1x6], as follows. A1 A2 A3 A4 A5 A6 h11 h12 h13 h14 h15 h16 6

Where

∑h

1i

= 1 and Ai with h1imax it will be will be decisive in the reliability allocation

i =1

process (greater impact on the calculation of GI%). Finally, in the present step there will be the definition of AHP weights matrix [ix6] obtained by multiplying the units vector ranking with the factors vector ranking. Where k ij = z ij × hij represent the importance of the i-th unit regard to Ai. Furthermore it is important to note that: 6


The sum of i-th row is equal to

∑K

1j

= z11 (weight of the ith unit distributed among the

j =1

factors Ai) n


The sum of j-th column is equal to

∑K

i1

= h11 (weight of factors Ai distributed among

i =1

units).

Consequently the sum is

6

n

j =1

i =1

∑ ∑K

A1

A2

ij

= 1 , as shown in the following AHP weighs matrix:

A3

A4

A5

A6

6

∑K

1j

j =1

Unit 1 Unit 2 Unit 3 ……. Unit i n

∑ K i1

k11 k12 k13 k14 k15 k16 k21 k22 k23 k24 k25 k26 k31 k32 k33 k34 k35 k36 …. … … … … … ki1 ki2 ki3 ki4 ki5 ki6 h11 h12 h13 h14 h15 h16

i =1

-

z11 z21 z31 … zi1 6

n

j =1

i =1

∑ ∑K

ij

=1

Step 7: Definition of A-IFM matrix, as follows, obtained by multiplying each other the values of the matrices IFM and AHP, where Pij = k ij × wij .

Unit 1 Unit 2 Unit 3 ……. Unit i

A1 A2 A3 A4 A5 A6 P11 P12 P13 P14 P15 P16 P21 P22 P23 P24 P25 P26 P31 P32 P33 P34 P35 P36 …. … … … … … Pi1 Pi2 Pi3 Pi4 Pi5 Pi6

GI% will be calculated by applying Equation (14). -

Step 8: Calculation of allocated reliability (17) of units.

-

Step 9: Analysis of results.

6. A-IFM application to aerospace prototype The proposed approach was applied to the aerospace system described in Section 3, relating to the “Sub-Orbital Re-entry” mission. Starting from the reliability analysis performed in Section 3 (Product Tree, RDB, PHA), the new method was initially tested on the Top Event 3 – F(t)=0,20, since the Top Event 3 will be probably more restrictive than the other two (all units have influence): Step 1: analyzed in Section 3 Step 2: analyzed in Section 3 Step 3: analyzed in Section 3 Step 4: analyzed in Section 3 Step 5: analyzed in Section 3 Step 6: AHP method applied to units and to factors of allocation: -

Through Saaty’s scale pairwise comparisons between units were obtained, as shown in the following AHP matrix for units. Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8

Unit 1 1,000 0,333 1,000 0,111 0,333 0,111 0,143 0,250

Unit 2 3,000 1,000 3,000 0,250 1,000 0,167 0,333 1,000

Unit 3 1,000 0,333 1,000 0,111 0,333 0,111 0,143 0,250

Unit 4 9,000 4,000 9,000 1,000 4,000 1,000 1,000 3,000

Unit 5 3,000 1,000 3,000 0,250 1,000 0,167 0,333 1,000

Unit 6 9,000 6,000 9,000 1,000 6,000 1,000 2,000 4,000

Unit 7 7,000 3,000 7,000 1,000 3,000 0,500 1,000 2,000

Unit 8 4,000 1,000 4,000 0,333 1,000 0,250 0,500 1,000

The final ranking for each unit is shown in the following vector: Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8

0,3149 0,1084 0,3040 0,0283 0,1084 0,0193 0,0424 0,0745

n

∑z i =1

i1

= 1,0000

Results show for the unit 1 the lower allocated reliabilty. According to Saaty’s Scale it will be defined a pairwise comparison among factors A1, A2,A3, A4, A5, A6 i. The result is a AHP matrix for factors [6x6]. A1 A2 A3 A4 A5 A6 A1 1,000 0,333 0,333 0,167 0,111 0,111

3,000 3,000 6,000 9,000 9,000

A2 A3 A4 A5 A6

1,000 1,000 2,000 5,000 5,000

1,000 1,000 2,000 5,000 5,000

0,500 0,500 1,000 2,000 2,000

0,200 0,200 0,500 1,000 1,000

0,200 0,200 0,500 1,000 1,000

The final ranking vector for factors is the following: A1

A2

A3

A4

A5

A6

6

∑h

1i

i =1

0,0388 0,1073 0,1055 0,2381 0,4846 0,0258 1,0000

The AHP weight matrix is obtained by multiplying the two final vectors (units and factors). 6

A1 0,0122 0,0042 0,0118 0,0011 0,0042 0,0007 0,0016 0,0029

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8 n

∑K i =1

-

i1

= h11

A2 0,0338 0,0116 0,0326 0,0030 0,0116 0,0021 0,0045 0,0080

A3 0,0332 0,0114 0,0321 0,0030 0,0114 0,0020 0,0045 0,0079

A4 0,0750 0,0258 0,0724 0,0067 0,0258 0,0046 0,0101 0,0177

A5 0,1526 0,0525 0,1473 0,0137 0,0525 0,0094 0,0205 0,0361

A6 0,0081 0,0028 0,0078 0,0007 0,0028 0,0005 0,0011 0,0019

0,0388 0,1073 0,1055 0,2381 0,4846 0,0258

∑K

1j

= z11

j =1

0,3149 0,1084 0,3040 0,0283 0,1084 0,0193 0,0424 0,0745 6

n

j =1

i =1

∑ ∑K

ij

=1

Step 7: Multiplying IFM matrix for Top Event 3 (Table 2) and AHP weight matrix the result is the A-IFM matrix (Table 6), from which applying Equation (14) and Equation (12) it is possible to obtain GI% and F(t)allocated for Top Event 3: Table 6: A-IFM matrix for aerospace prototype

-

Step 8: Calculation of allocated reliability (17) of units (Table 7). Table 7: IFM based AHP R(t)allocat

Results show that with A-IFM method the unit1 is worse than the other units of the aerospace system. -

Step 9: The results obtained were compared in terms of MAD and negative technological errors, with the results obtained in Section 3.

Table 8 shows comparison among literature methods and A-IFM method, according to Mean Absolute Deviation (MAD). Table 8: Comparison among literature methods and A-IFM (Mean Absolute Deviation)

Table 9 shows comparison among literature methods and A-IFM method according to έtechnological Table 9: Comparison among literature methods and A-IFM method (έtechnological) The final results obtained with A-IFM method can be summarize as following: - Reduction of R(t)allocat for unit1 (about 3,46%). This means a good alignment with respect to databank and a substantial savings in the choice of the unit less performing; - Reduction of MAD, about 2,49% (absolute) and about 23% (relative); - Reduction of ∑(-έtechnological), about 1,25% (absolute) and about 14% (relative). It is important to highlight that A-IFM method assigns to the units slightly higher values allocated (tenths percent or at most one percentage point) compared to literature data. This means to ensure a “safety” condition for complex systems prototype.

7. Conclusion The A-IFM method proposed in this paper has validated the application of the IFM based on a multicriteria method for reliability allocation for an aerospace prototype. It is an simple and effective approach which uses IFM method values adjusted by Analytic Hierarchy Process, depending on the importance of each factor and each unit of the system. The main advantages of the proposed approach are: - it provided an accurate yet flexible reliability allocation method, which combines AHP and IFM Method; - it is applicable into different design phases and can be also used in different industries and fields; - indirect relationships are considered between unit and factors Ai; - higher weight of unit corresponds to a higher unreliability allocation value (reduction of R(t)allocat for unit1 about 3,46%). The comparison with conventional reliability methods show that: - MADA-IFM is smaller than MAD of conventional methods and is equal to 7,61%; - ∑(-έtechnological) A-IFM is smaller than ∑(-έtechnological) of conventional methods and is equal to -7,60%. The results highlight how the proposed method is both accurate and realistic.

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Figure 1: Prototype Product Tree

Figure 2: Sub-orbital Re-entry Test)

Flying Segment

StratosphericBallo on

Winged Vehicle 2nd Stage

1st Stage PT

Avionics

Propeller

Instrument.

Stage Separation System

Self-destruction PT

Parachute System

Splashdown

Figure 3: Product Tree for Reliability Allocation

Figure 4: Reliability data comparison (Failure Event).

Table 1: Principal advantages and disadvantages of the analysed methods for aerospace prototype METHOD FOO

ADVANTAGES - Application simplicity

ARINC

- Application simplicity - Objectivity

BOYD

- Versatility

AGREE

- Good detail

KARMIOL - Very good detail - Applicable to innovative systems BRACHA - Exact analytical treatment

IFM

- Very good detail - Applicable to innovative systems - Applicable in every phase

DISADVANTAGES - Only applicable to systems in series - Only applicable in the initial phases - Only applicable to systems in series - Only applicable in the initial phases - Fault rate knowledge of similar systems - Only applicable to systems in series - Only applicable in the initial phases - Only applicable to systems in series - Applicable in advanced phase - Partial subjectivity of the analyst - Subjectivity of the analyst - Not easy determination of stress factors - Component criticality not considered - Only applicable to systems in series

Table 2: IFM Matrix (Top Event 1-2-3) Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8

A1 0,6 1 1 1 1 1 0 0

A2 1 0,4 0,7 0,3 0,5 0,3 0,5 0,3

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8

A1 0,4 1 1 1 0 1 1 1

A2 1 0,4 0,7 0,3 0,5 0,3 0,5 0,3

Unit 1 Unit 2 Unit 3 Unit 4 Unit 5 Unit 6 Unit 7 Unit 8

A1 0,6 1 1 1 1 1 1 1

A2 1 0,4 0,7 0,3 0,5 0,3 0,5 0,3

TOP EVENT 1 Data loss (Significant Event) A3 A4 A5 A6 GI 0,1724 0,67 0,5 0,3 0,3209 0,1379 0,67 0,5 0,5 0,0370 0,1724 1 0,5 1 0,0603 0,069 0,33 0,5 0,6 0,0057 0,1379 0,67 0,5 0,9 0,0257 0,069 0,33 0,5 0,6 0,0057 0,1379 0,33 0,5 0,5 not influent* 0,1034 0,33 0,5 0,6 not influent* GItot= 0,4552 TOP EVENT 2 Vehicle loss (Major Event) A3 A4 A5 A6 GI 0,1724 0,67 0,5 0,3 0,4813 0,1379 0,67 0,5 0,5 0,0370 0,1724 1 0,5 1 0,0603 0,069 0,33 0,5 0,6 0,0057 0,1379 0,67 0,5 0,9 not influent* 0,069 0,33 0,5 0,6 0,0057 0,1379 0,33 0,5 0,5 0,0228 0,1034 0,33 0,5 0,6 0,0085 GItot= 0,6212 TOP EVENT 3 Mission Failure (Failure Event) A3 A4 A5 A6 GI 0,1724 0,67 0,5 0,3 0,3209 0,1379 0,67 0,5 0,5 0,0370 0,1724 1 0,5 1 0,0603 0,069 0,33 0,5 0,6 0,0057 0,1379 0,67 0,5 0,9 0,0257 0,069 0,33 0,5 0,6 0,0057 0,1379 0,33 0,5 0,5 0,0228 0,1034 0,33 0,5 0,6 0,0085 GItot= 0,4865

GI% 70,49% 8,12% 13,26% 1,25% 5,64% 1,25% not influent* not influent*

GI% 77,47% 5,95% 9,71% 0,92% not influent* 0,92% 3,66% 1,37%

GI% 65,95% 7,60% 12,40% 1,17% 5,28% 1,17% 4,68% 1,75%

F(t)allocated 0,0705 0,0081 0,0133 0,0013 0,0056 0,0013 not influent not influent

F(t)allocated 0,0116 0,0009 0,0015 0,0001 not influent 0,0001 0,0005 0,0002

F(t)allocated 0,1319 0,0152 0,0248 0,0023 0,0106 0,0023 0,0094 0,0035

Table 3: Comparison between IFM, FOO and Karmiol results Top Event 1

Top Event 2

Top Event 3

IFM

FOO

KARMIOL

IFM

FOO

KARMIOL

IFM

FOO

KARMIOL

Unit 1

92,95%

95,29%

98,25%

88,38%

92,93%

97,12%

86,81%

90,57%

96,02%

Unit 2

99,19%

99,62%

98,90%

99,11%

99,43%

98,26%

98,48%

99,25%

97,50%

Unit 3

98,67%

97,64%

98,51%

98,54%

96,46%

97,63%

97,52%

95,29%

96,62%

Unit 4

99,88%

99,19%

99,22%

99,86%

98,79%

98,77%

99,77%

98,38%

98,24%

Unit 5

99,44%

99,76%

98,83%

-

99,65%

97,23%

98,94%

99,53%

97,36%

Unit 6

99,88%

99,92%

99,22%

99,86%

99,88%

98,77%

99,77%

99,84%

98,24%

Unit 7

-

98,65%

98,51%

99,45%

97,98%

98,56%

99,06%

97,31%

97,94%

Unit 8

-

99,92%

98,56%

99,79%

99,88%

98,67%

99,65%

99,84%

98,08%

90,23%

90,34%

90,42%

85,43%

85,76%

85,93%

81,05%

81,35%

81,65%

R(t) target

Table 4 Comparison between Reliabilty allocation methods and databanks (MAD, έtechnological) έtech (%)

FOO

έ (%)

έtech (%)

KARMIOL

Databanks

έ (%)

έtech (%)

Unit 1

86,81%

79,45%

7,36%

-7,36%

90,57%

79,45%

11,12%

-11,12%

96,02%

79,45%

16,57%

-16,57%

Unit 2

98,48%

98,41%

0,07%

-0,07%

99,25%

98,41%

0,84%

-0,84%

97,50%

98,41%

0,91%

0,91%

Unit 3

97,52%

96,13%

1,39%

-1,39%

95,29%

96,13%

0,84%

0,84%

96,62%

96,13%

0,49%

-0,49%

Unit 4

99,77%

99,75%

0,02%

-0,02%

98,38%

99,75%

1,37%

1,37%

98,24%

99,75%

1,51%

1,51%

Unit 5

98,94%

99,85%

0,91%

0,91%

99,53%

99,85%

0,32%

0,32%

97,36%

99,85%

2,49%

2,49%

Unit 6

99,77%

99,75%

0,02%

-0,02%

99,84%

99,75%

0,09%

-0,09%

98,24%

99,75%

1,51%

1,51%

Unit 7

99,06%

99,30%

0,24%

0,24%

97,31%

99,30%

1,99%

1,99%

97,94%

99,30%

1,36%

1,36%

Unit 8

99,65%

99,75%

0,10%

0,10% Tot - έtechIFM= -8,85%

99,84%

99,75%

0,09%

-0,09% Tot - έtechFOO= -12,13%

98,08%

99,75%

1,67%

1,67% Tot - έtechKarmiol= -17,06%

IFM

Databanks

MADIFM

έ (%)

10,09%

Databanks

MADFOO

16,66%

MADKarmiol

26,51%

Table 5: Semantics scale of Saaty Intensity of importance aij 1 3

Equal Importance Moderate importance

5

Strong importance

7

Very strong or demonstrated importance Extreme importance

9 2,4,6,8

Definition

For compromise between the above values

Explanation Two activities contribute equally to the objective Experience and judgment slightly favor one activity over another Experience and judgment strongly favor one activity over another An activity is favored very strongly over another; its dominance demonstrated in practice The evidence favoring one activity over another is of the highest possible order of affirmation Sometimes one needs to interpolate a compromise judgment numerically because there is no good word to describe it

Table 6: A-IFM matrix for aerospace prototype F(t)allocated GI% 83,24% 16,65%

Unit 1

A1 0,0073

A2 0,0338

A3 0,0057

A4 0,0502

A5 0,0763

A6 0,0024

GI 0,0415060481

Unit 2

0,0042

0,0046

0,0016

0,0173

0,0263

0,0014

0,0005660026

1,14%

0,23%

Unit 3

0,0118

0,0228

0,0055

0,0724

0,0737

0,0078

0,0072745976

14,59%

2,92%

Unit 4

0,0011

0,0009

0,0002

0,0022

0,0069

0,0004

0,0000059398

0,01%

0,01%

Unit 5

0,0042

0,0058

0,0016

0,0173

0,0263

0,0025

0,0003930574

0,79%

0,16%

Unit 6

0,0007

0,0006

0,0001

0,0015

0,0047

0,0003

0,0000027678

0,01%

0,01%

Unit 7

0,0016

0,0023

0,0006

0,0033

0,0103

0,0005

0,0000532455

0,11%

0,02%

Unit 8

0,0029

0,0024

0,0008

0,0059

0,0180

0,0012

0,0000617608

0,12%

0,02%

0,0498634197 100,00%

20,00%

GI Tot

Table 7: IFM based AHP R(t)allocat R(t)allocat Unit 1

83,35%

Unit 2

99,77%

Unit 3

97,08%

Unit 4

99,99%

Unit 5

99,84%

Unit 6

99,99%

Unit 7

99,98%

Unit 8

99,98%

R(t)SIST

80,56%

Table 8: Comparison among literature methods and A-IFM (Mean Absolute Deviation) IFM

Data

έ (%)

FOO

Data

έ (%)

KARMIOL

Data

έ (%)

IFM-AHP

Data

έ (%)

16,57%

83,35%

79,45%

3,90%

Unit 1

86,81%

79,45%

7,36%

90,57%

79,45%

11,12%

96,02%

Unit 2

98,48%

98,41%

0,07%

99,25%

98,41%

0,84%

97,50%

98,41%

0,91%

99,77%

98,41%

1,36%

Unit 3

97,52%

96,13%

1,39%

95,29%

96,13%

0,84%

96,62%

96,13%

0,49%

97,08%

96,13%

0,95%

99,75%

1,51%

99,99%

99,75%

0,24%

79,45%

Unit 4

99,77%

99,75%

0,02%

98,38%

99,75%

1,37%

98,24%

Unit 5

98,94%

99,85%

0,91%

99,53%

99,85%

0,32%

97,36%

99,85%

2,49%

99,84%

99,85%

0,01%

Unit 6

99,77%

99,75%

0,02%

99,84%

99,75%

0,09%

98,24%

99,75%

1,51%

99,99%

99,75%

0,24%

99,30%

1,36%

99,98%

99,30%

0,68%

99,75%

1,67%

99,98%

99,75%

0,23%

Unit 7

99,06%

99,30%

0,24%

97,31%

99,30%

1,99%

97,94%

Unit 8

99,65%

99,75%

0,10%

99,84%

99,75%

0,09%

98,08%

MAD

10,09%

MAD

16,66%

MAD

26,51%

MAD

7,61%

Table 9: Comparison among literature methods and A-IFM method (έtechnological) IFM

Data

έ (%)

FOO

Data

έ (%)

KARMIOL

Data

έ (%)

IFM-AHP

Data

έ (%)

79,45%

-3,90%

Unit 1

86,81%

79,45%

-7,36% 90,57%

79,45%

-11,12%

96,02%

79,45%

-16,57%

83,35%

Unit 2

98,48%

98,41%

-0,07% 99,25%

98,41%

-0,84%

97,50%

98,41%

0,91%

99,77%

98,41%

-1,36%

Unit 3

97,52%

96,13%

-1,39% 95,29%

96,13%

0,84%

96,62%

96,13%

-0,49%

97,08%

96,13%

-0,95%

99,75%

-0,24%

Unit 4

99,77%

99,75%

-0,02% 98,38%

99,75%

1,37%

98,24%

99,75%

1,51%

99,99%

Unit 5

98,94%

99,85%

0,91% 99,53%

99,85%

0,32%

97,36%

99,85%

2,49%

99,84%

99,85%

0,01%

Unit 6

99,77%

99,75%

-0,02% 99,84%

99,75%

-0,09%

98,24%

99,75%

1,51%

99,99%

99,75%

-0,24%

99,30%

-0,68%

99,75%

-0,23%

Unit 7

99,06%

99,30%

0,24% 97,31%

99,30%

1,99%

97,94%

99,30%

1,36%

99,98%

Unit 8

99,65%

99,75%

0,10% 99,84%

99,75%

-0,09%

98,08%

99,75%

1,67%

99,98%

Tot -έ (%)

-8,85%

Tot -έ (%)

-12,13%

Tot -έ (%)

-17,06%

Tot -έ (%)

-7,60%