The relationship between system functions, reliability and dependent failures

zyxwvutsrqpo zyx zyxw zyxwvutsrqpon zyxwvutsrqp zyxwvutsrqp The Relationship Between System Functions, Reliability and Dependent failures Jayant Trewn Industrial & Manufacturing Engineering Wayne State University Detroit, MI, 48202 Kai Yang Industrial & Manufacturing Engineering Wayne State University Detroit, MI, 48202 hierarchical model proposed in ref. [5]. Section 4 will link reliability into the model of system design and discuss the cost of failure of functions of the system. Section 5 will discuss dependent failure and its effect on system reliability. Section 6 is the conclusion of the paper. ABSTRACT This paper deals with the relationship between system design, reliability and dependent failure. A hierarchical multi-level system model is developed to characterize system design mapping from the functional domain to the physical domain. Based on this modeling framework, the reliability and cost of reliability are linked to system functions, design structures and dependent failures. The evaluation method for analyzing product system designs from a reliability perspective is proposed. 2. Engineering System Design Engineering design involves a continuous interplay between what we want to achieve (design objective) and how we want to achieve it (physical solution). Many attempts have been made to develop scientific principles to guide and evaluate the design process [3][5]. Two notable schools of thought are Num Suh’s axiomatic design principles and Hubka’s model for technical systems. 1. Introduction Onc of the fundamcntal inadcquacies o i traditional reliability enginccring is its lack of reliability deployment and evaluation strategy in the early concept design stage. Traditional reliability focuses on assessment of reliability of systemdproducts based on tcst data. Reliability assurance is often fcatured by the cycles of build-test-evaluate and fix. This approach is usually rime consuming and costly. Also, inherent in its delinition, testing is after-the-fact, much in opposition to Dermng’s philosophy of bringing quality improvements upstream. 2.1. Axiomatic Design Dr. Nam Suh proposes the use of axioms as the pursued scientific foundations of design [3]. In this approach, the design is defined as the creation of synthesized solutions in the form of products, processes or systems that satisfy perceived needs through the mapping between the functional requirements (FRs) in the functional domain and the design parameters (DPs) in the physical domain. This is illustrated in Figure 1. Classical reliability evaluation of systems is also based on the assumption that the components comprising the system work independently and their failures have no mutual interdependence [ 13. In reality, classifying failures as mutually independent events does not hold true in most system designs. A recent study in the automotive industry showed that, in some situations, single component failures cause only 15% of system failure 121. The remaining 85% of causes of system failure are due to sub-system interactions, poor sub-system interface, improper part installation and so on. These subsystem interdependencies also affect the severity of the damage and the cost to repair the damage. Fig 1: Axiomatic Design Process This paper is an attempt to develop a design evaluation tool that will guide design engineers to evaluate reliability of products in the conceptual design stage. The conceptual design, or product system design is featured by the mapping of the functional domain (product functions) into the physical domain (actual physical entities) [3][4][5]. In this paper, the multi level hierarchical (MLH) model proposed in ref. [5] is used to map the functional requirements onto design parameters, which is further mapped onto sub-system and components specifications. Next reliability concepts are introduced into this model to evaluate the soundness of system design from the reliability perspective. In modeling reliability, the failures of components can also be assumed to be mutually dependent. By integrating failure dependency and multi-level system models, the effects of dependent failure to system reliability can also be evaluated. The cost of failures and their effects to system performances will also be analyzed. I Functional Space ==> Physical Space FR - Desired output zy DP - System which delivers desired output Suh proposes the following 2 axioms as the universal principles which any ‘good’ system design should satisfy. Axiom 1: requirements Maintain the independence of finctional It means that in a good design, the independence of functional requirements is maintained., This paper will be subdivided into 6 sections. Section 2 will discuss general models for engineering system design. Section 3 proposes a mathematical model to describe the multi level Axiom 2: Minimize the information content of the design 4722 zyxwvutsrqpon EO3l zyxw zyxwvuts zyxwvutsrq zyxwvutsrq Figure 3. Mapping of Functional Requirements to component structure It means that among designs that satisfy axiom 1, the best design 'is the one that has the minimum information content. Here the information content is a measure of design complexity. So the second axiom indicates that the design simplicity should be pursued given the functional requirements can be met. I Funchonal Space ==> Physical Space ==> Component Space ~ Based on the axiomatic design theory, there are two major types of design vulnerabilities in most of engineering design solution entities: coupling (non-independence) and complexity. Intuitively, these conceptual design vulnerabilities will cause the system reliability to decrease. FR - Desired output DP System which delivers ~ 2.2 Multi Layer Hierarchical System I As an independent development in the theory of engineering design, Hubka defines a technical system as a multi layer hierarchical (MLH) system at various levels of abstraction [ 5 ] . The flow of customer requirements from the highest (most abstract) level to the lowest (least abstract) level is achieved in a conceptual design process. The layers of a MLH system are at various levels of abstraction and are shown in figure 2. Abstraction Level A Highest Functional Requirements Functional Structure B C Comparing with Suh's system design model, Hubka's MLH model resolves the abstraction in the design parameter level by extending the design structure to the component structure level. Specifically, in a design process, functional requirements are transformed into design parameters, which are further transformed into sub-systemskomponents or parts. The components and parts structure is the most concrete stage of the concept design process. This is shown in the 2-step mapping illustrated in Figure 3. Design Specifications - Organ Structure Lowest It is author's belief that this 2-step mapping provides more insights into the design process. There are many cases where several DP's can be encapsulated onto a single component. There are also cases where a single DP has to be delivered by using several components. Therefore, the structure of relationships among components is often different than that of Dps. However, the subsystedcomponents structures are often the final form of an engineering design. In reliability perspective, it is the subsystedcomponent structure that determines the reliability of the design. However, Hubka's two step mapping model is only a descriptive model. In comparison, Suh's axiomatic design model provided elaborate analytical model and numerical design evaluation criteria. SystedHardware model Component Structure Hubka states that, "in the process of abstracting, the number of specific systems covered by the model increases. Abstracting in this sense, is a relative deterministic and analytical process, it is a one to many mapping, and two different systems can deliver the same model. At each transition, a designer has a number of ways of fulfilling the requirements, and can choose among them according to various criteria. Different designers make different decisions in the various contexts of the sociotechnical systems (market, companies, etc.) which results in a number of solutions offered for solving the same problem" [ 5 ] . In this section, we will develop an analytical hierarchical multilevel model to describe the 2 step mapping model proposed by Hubka. This model will be used to develop a relationship between the component structure and the functional structure of designs. It is this relationship which forms the basis for an evaluation criteria to evaluate the systems ability to perform the intended functions. In effect, functional reliability is defined as the ability of a part or component to perform its intended functions, or on a system level, the effect of the reliability of a part on the ability of the system to perform its intended functions. As the system gets more complex, the dependence of the performance of the system on a complex structure of parts becomes more critical. Hence, it is a measure of this complexity that determines the ability of a concept design to reliably perform its functions. Hubka states "It is generally accepted that technical systems of lower degree of complexity find greater breadth of application and versatile use." [5] Terminology in terms of the MLH model can be stated as: - Function structures (as functional requirements) can be realized by various organ structures - Organ structures (as design specifications) can be realized from various component structures desired output 3. Analytical Multi Level Hierarchical Model Figure 2: A Multi Layer Hierarchical (MLH) system Layer CS -Component Structure It is the challenge of evaluating these different and competing system designs that satisfy the same functional requirements that is the motivation to develop a MLH evaluation model. Hubka's MLH model maps the design process from the conceptualization at the customers' level in the form of functional requirements, to the design formulization in the form of design specifications (parameters), to the component level in the form of component specification. This mapping takes the design process from an abstract functional structure to a concrete component structure. Figure 3 models this mapping process. 3.1 The MLH mapping methodology The multi level hierarchical mapping methodology is best understood by the aid of a design example. Consider a system design with 3 functions, 3 design parameters and 4 4723 zyxwvutsrqpon zyx 3.3 Physical to component space mapping components. The design process is split into two subprocesses. These are: the first process that relates the functional requirements to the design process and the second process that relates the design parameters to the component structure. Component structure construction is defined as the mapping process between the design parameters in the design space to the component structure in the component space. Figure 5 maps the relation between the design parameters and the component structure. 3.2 Function to physical space mapping Figure 5: Component Space Design is defined as the mapping process between the functional requirement in the functional space to the design parameters in the design space to the components in the component space. Figure 4 maps the relation between the functional requirements to the design parameters. Figure 4: Design Space DP. is the jth design parameter J zyxwv zyxwvutsr zyxwvutsrq CPk is the kth functional requirement Let the component space be characterized by a vector {CS) with k components. FRi is the ith functional requirement DP’s in the physical domain are characterized by vector {DP} with j components. zyx DP. is the jth design parameter J Let there be i components represented by a set of independent functional requirements. Let [B] be the design matrix that maps the components to the design parameters. [FR) is the functional requirement vector. DP’s in the functional domain are characterized by vector [ DP] with j components. (DP) = @ I ICPJ Let [A] be the design matrix of rank (ij) that maps the design parameters to the functional requirements. bjk is the element of the matrix [B] that maps the kth component to the jth design parameter . The element bjk is binary, 0 = no relationship and 1 = related. It defines the relationship between the jth design parameter and the kth component. Hence the relationship between design parameters and functional requirements is: {FRI = [AI {DPl 1 qj= (0>1>)1 (1) [: 11 1 [B]= 0 1 0 1 In the case of figure 4, the design matrix is as follows: Matrix ‘“’=I::I [B] maps the relationship between the design parameters to the component structure. For example: bll = 1, denotes that component 1 maps (is related to) to design parameter 1 . bI2 = 0, denotes that component 2 is not related to design parameter 2. zyxwvutsrqp Matrix [A] maps the relationship between the functional requirements to the design parameters. For example: 3.4 Analytical multi level hierarchical modeling a1 1 = 1, denotes that design parameter 1 maps (is related to) to functional requirement 1. (2) In the case of figure 5, the design matrix is as follows: is the element of the matrix [A] that maps the jth design parameter to the ith functional requirement. The element is binary, where 0 = no relationship and 1 = related. It definesihe relationship between the ith functional requirement and the jth design parameter. r1 0 11 { b&=(0,1,)1 The concept design process is the exercise of choosing the right set of components that conform to the design parameters that satisfy the functional requirements. AI2 = 0, denotes that design parameter 2 is not related to functional requirement 2 . 4724 zyx In reality, the customer relates to functional performance and the components that provide those functions. The design parameters are abstract to the customer/operator. 4. Reliability and Multi-Level Hierarchical Model Since [A] and [B] matrices are relationship vectors relating Functional, Design and Component spaces respectively through binary elements (O,l), we hypothesize that Reliability is usually defined as the probability that the product perform its stated function over a stated period of time. One of the most important tasks in reliability engineering is the relationship between component reliability and system reliability. For example, in a series system, system's reliability is equal to the product of their components' reliability. In reality, however, the failure of different components will cause different kind of failure modes for the system, hence the degree of damage and cost of rectifying the failure will also be different. For example, both the failure of the battery in an automobile and the failure of a piston rod will cause the car to stop functioning. Clearly the damage of failure caused by battery and the subsequent cost to repair is far less than that of piston rod. zyxwvutsrqp zyxwvutsrqpo zyxwvutsrqpon zyx zyxwvutsrqp zyxwvutsrq zyxwvutsrqponm [: ;:] I' The Resultant matrix [C] = [A] [B] is the relationship between the functional requirements and the components and it maps the relationship of each component of the system to the functional requirements that its existence satisfies. The operator ( 0 ) is a composite relational operator for binary matrices. The composite relation A defined as: o B AB = C = (cik) where ( Cik = (0,l)) In which The hierarchical multi-level model for engineering design outlined in section 3 provides a much more detailed description about the relationship between system functions and component structure. In this section, reliability model will be established by integrating failure probabilities of components to this hierarchical multilevel model. This model will enable us to analyze the exact impact of failure of components on the product and to assess the damage of the failure more precisely. the matrix [C] where [C] is (4) 4.1. Notations and Assumptions: For proof refer to Appendix 1. p k : Failure probability of component k, k=l, ...,n. P(FR, ) : Probability that FRi can be delivered Applying equation 4 to the example in section 3.2 and 3.3: C=AoB (5) C p k : kth component, k=l, ...,n. ccp, : Cost of replacing component k. c ,: Cost of losing Fri. 0 1 1. c=o 1 l o 0 1 0 1 1 0 1 0 ECCpk: Expected cost of failure due to the failure of kth Component. Re: System Reliability ECRe: Expected cost of product reliability It is assumed here, that all components will work independently. Matrix C translates to the relationship map as in figure 6. zyxwvuts 4.2. Functional Delivery Reliability and System Reliability Figure 6: Functional requirements to Component structure From the definition of reliability, it is clear that the system functions only if it delivers all its functions. Therefore: R, = fi P( FRi ) i=l For each FRi, it is clear that: P(FR, ) = fi (1 - pk)cik (7) k=l Where Cik is the entry of C matrix in the ith row and kth column. Clearly, Ck=l indicates that kth component will affect the ith functional requirement and vice versa. Therefore equation (7) is true since ERi can only be delivered if all its relevant components work successfully. For example, 47 25 zyxwvutsrqponm FRI (Figure 6) can only be delivered if CP1, CP3 and CP4 are In reality, however, this assumption may not be true. There are many cases where a single failure mode may cause several components to fail simultaneously. Also, the failure of one component may cause other component to failure. Bhattacharya, Misra and Balaguruswamy, and Humphreys and Jenkins have documented the dependent failure concept [l] [6] ~71. all in working states. Similarly: In last example, clearly, 4 p(FRl zyxwvutsrqpo zyxwvutsrqponml zyxwvu nFR2)=n(1-pk) 5.1. Notations. (9) p(kl j ) : Probability of failure of component k given the failure of component j, clearly, p(kl k ) = 1. p k : independent failure probability for component k. p k : : total probability of failure for component k. k=l Since all four components have to work properly in order to deliver FR,and FR2. For system reliability: In last example, reliability of product equal to the product of reliability of all four components. Figure 7: Example of Dependent Failure 4.3. Cost of failure When failure occurs, the product may lose some or all of its functions. If the failure is due to the failure of the components, these components may have to be repaired or replaced. In the multi-level hierarchical model discussed earlier, each component will affect the product function in different ways. Some of the components may only affect some minor functions, whereas, some of key component may affect all the functions of the product. Also, different components have different repairheplacement cost. I zyx zyxwvutsrqpon In this section, we will derive the expected cost of failure due to the failures of components. ECCpk= Expected cost of failure of component k Dependent failure relationship Functional relationship Clearly, Equation (11) states that the expected cost of failure due to component k is proportional to the component replacement cost and the cost of losing relevant product functions affected by the failure of component k. There are several ways to reduce the expected cost of failure. First, it can be reduced by reducing Pk, the failure probability of that component. Secondly, it can be reduced by improving the design such that the failure of the component will affect fewer functions of the product. By summing up the costs of failure of each component we can get the expected cost of reliability for the product. The function to component relationship matrix [C] is: [Cl = 5. Dependent Failures and Engineering Design In traditional reliability engineering, it is usually assumed that component failures are independent events. This assumption holds true for situations where failing components do not induce effects (stress, load, etc.) on other components. ‘:=I 0 0 1 0 0 0 0 1 Figure 7 shows that in this design case example, the relationship of components to the product functions are relatively independent. Without dependent failures, the failure of each component will only affect one or two product functions. However, with dependent failures, the failure of component 2 may also cause component 3 and 4 to fail, thus affecting more functions and possibly damaging more components. zyxwv 47 26 zyxwvutsr zyxwvutsrqp zyxw zy [ 2 ] P. Palady, Failure Modes and Effects Analysis, PT Publication Inc., 1995. The cost of failure due to component k will be: [3] N. Suh, The Principles ofDesign, Oxford University Press, N.Y., 1990. [4] G. Phal and W. Beitz, Engineering Design: A Systematic Approach, Springer-Verlag, 1988. j#k [ 5 ] V. Hubka and W.E. Eder, Theory of Technical Systems, Berlin, Springer-Verlag, 2ndEd., 1988. [6] K.B. Misra and E. Balaguruswamy, “Reliability analysis of k out of m: G systems with dependent failures,” Intl. Journal of Systems Science, Vol. 7 , 1976, pp. 853-861. For the example illustrated in Figure 7, if we assume that: p(112)=0.5 p(312)zl.O p(412)=0.8 [7] P. Humphreys and A.M. Jenkins, “Dependent failures Developments,” Reliability Engineering and System Safety, Vol. 34, No. 3, 1991, pp. 417-427. Then, the expected cost of failure of component 2 is: [SI R.E. Prather, Discrete Mathematical Structures for Computer Science, Houghton Mifflin Company, Boston, 1976. zyxwvutsrqpo zyxwv Appendix 1 Clearly, the failure of component 2 could cause the failure of other components, thus incurring extra costs of replacing those components and the cost of losing related product functions. Functional Relationship Operators In this paper, logical operations are based on Boolean algebra and the theory of Di-graphs [SI. Notations for logical operators are as follows: For component 1, the cost of failure is: o logical product logical sum V logical or Clearly, component 2 is a very critical component from a reliability perspective. A logicaland Operations performed on binary elements are based on Boolean operations as follows: The expected cost of system reliability is: zyxwv zyx zyxwvutsrqpo 6. Conclusion Define two relations: Multi Level Hierarchical systems approach extends traditional reliability analysis at a component level to the functional level. This approach aids in estimating failure costs for concept designs and hence competing concept designs can be evaluated for functional dependence. The more complex a design, (Le. designs with certain components being critical to providing multiple functional requirements) the more the costs due to non-performance of functions. In future research, the concept of component degradation will be applied to the multi level hierarchical approach and the analysis of dependent failure component systems. R from set U to V {r,J= (0,1)} SfromsetV t o w {sk,=(O.l)] Define a composite relation R o S from U to W U (R S)w a there exists v in V with U R v and v S w Using Boolean algebra: The composite relation R S has the matrix RS = T = (t,,) = (0,I)l In which 7. References n tij = v (rik A skj) [I] A. Bhattacharya, “Reliability Evaluation of Systems with Dependent Failures,” Intl. Journal of Systems Science, Vol. 27, No. 9, 1996, pp. 881-885. k=l Where (rjk= (0,l)) and 4727 {sk, = (0,1)}