Minimum Cost Test Plans for a Series System with ... - CiteSeerX

Minimum Cost Test Plans for a Series System with Imperfect Interfaces Mainak Mazumdar & Jayant Rajgopal Department of Industrial Engineering University of Pittsburgh Pittsburgh, PA 15261, USA

ABSTRACT In order to draw cost-effective inferences on the reliability of a system one approach is to design test plans that only test the components of the system.

However, such an approach is

inappropriate when the component failures are not independent or when interfaces between components cause system failures. In such instances one must plan on testing the system as well as its components and a natural question of interest is the relative extent to which the system and the components should be tested. This paper analyzes such a situation for a series system and determines the optimal plan for allocating the total testing effort between component and system tests. It also illustrates the use of mathematical programming techniques for obtaining test plans that guarantee conventional statistical properties.

Running Title: Optimum Test Plans Key Words: Reliability, Mathematical Programming, Series System, Test Plans AMS Subject Classification: 62 Statistics, 90 Economics, operations research, programming, games

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Introduction When the failures of components that constitute a given system occur independently, it should be possible to express the system reliability as a well-defined function of the component reliabilities based on the system structure. As a matter of fact, a substantial portion of the reliability literature is devoted to obtaining the mathematical expressions for system reliabilities in terms of component reliabilities, e.g., Barlow and Proschan (1981). Thus, it stands to reason that if the assumption of independence holds, statistical inferences on system reliability could be made based on component tests alone. Starting with Gal (1974), several researchers have examined the problem of designing component test plans for making inference on system reliability. This has been done for various system configurations and under various assumptions, e.g., Mazumdar (1977, 1980), Yan and Mazumdar (1986, 1987), Rajgopal and Mazumdar (1988, 1995, 1996) Easterling et al. (1990) and Altinel (1992, 1994). The objective in all of the preceding research has been to derive optimal plans where only the components are tested and system testing is avoided.

The

advantages of component testing over system testing have been well-documented, e.g., Rajgopal and Mazumdar (1995). One assumption that has been made in all this work is that the interfaces that link the components with each other (such as wires, welds, mechanical couplings, glue, etc) are completely reliable. However, in real life situations, system failures occur not only because of the component failures but also because of the failures of interface components. In this paper we consider a situation where for the purpose of making inferences on system reliability, it is not enough to make component tests in order to detect interface failures. We make the assumption that the detection of such failures is only achievable by means of system tests. We address the following 2

question: “if the costs of system and component tests are known, how should the total test effort be allocated between the system and components so as to make a valid statistical inference on system reliability?” We consider a series system and the problem of inference is formulated in terms of a test of hypothesis on system reliability. Consider a series system of n components. It is assumed that the time to failure of component j follows an exponential distribution with (unknown) mean 1/λj. It is also assumed that the interfaces that link the components have together a failure time that is an exponentially distributed random variable with (an unknown) mean equal to 1/λI. Component and interface failures are assumed to be independent of each other. The system failure rate thus is Σjλj + λI. The following test plan is in effect: each component i is tested (with replacement of failed components) for ti units of time at a (given) cost of ci units per unit time on test. The number of failures observed in each case is denoted by the symbol Xi. In addition, the entire system is assembled and tested (again, with replacement of failed systems) for tS units of time at a (given) cost of cS units per unit time on test; the total number of system failures is denoted by the symbol XS. It is clear that XS and Xi are independently distributed Poisson random variables. 2. Proposed Test Plan Without loss of generality, we start by assuming that the mission time for the system is equal to 1 unit so that the system reliability RS=exp(-λI - Σjλj). In designing the test plan, one must (a) decide on an appropriate test statistic and acceptance rule based on the observed number of failures and (b) determine values for ti and tS that minimize test costs while providing sufficient assurance on the value of RS. We formulate this problem as follows : Minimize c1t1+c2t2+…+cntn+cStS

(2.1) 3

subject to P[Plan accepts the system | RS ≥ R1) ≥ 1-α

(2.2)

P[Plan accepts the system | RS ≤ R0) ≤ β

(2.3)

where α and β are given small fractions, and R0 and R1 are given values such that the system is considered definitely acceptable if RS exceeds R1 and definitely unacceptable if RS is lower than R0 (< R1). The probability constraints given in (2.2) and (2.3) are the same as those used in the reliability test plans given in the document MIL-HDBK-781D (1987). Before attempting to derive an appropriate test statistic we make the following observation: in the absence of any prior information on the component reliabilities, each component in a series system must be tested for equal time periods, irrespective of the test costs. This is because a series system is only as good as its weakest link and we assume that we do not know which one it is. Thus, in this situation, each component should be tested for a minimum amount of time, and once it is so done, there is no reason to test any component for any additional time. We therefore simplify the exposition that follows by replacing the series system with a single component with failure rate λC, which is tested for tC units of time at a cost of cC=Σjcj units per unit time on test. The number of failures observed is denoted by XC. Also, let λS=λC+λI. Now, let λˆC and λˆI be maximum likelihood estimates of λC and λI, respectively. A logical rule might be that if λˆS = λˆC + λˆI >d* then the system is rejected (d* needs to be determined). To compute λˆC and

λˆI we note that the likelihood function is given by L( λC , λI ) = exp( −λC tC )

X X (λC tC ) C [( λ + λI )t S ] S exp[ −(λC + λI )t S ] C X S! XC!

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(2.4)

This yields the MLE’s as λˆC =(XC/tC) and λˆI =(XS/tS)-(XC/tC) so that λˆS = λˆC + λˆI =(XS/tS), which holds as long as λˆS - λˆC ≥0. Thus using the MLE with the proposed testing approach results in a strategy where only the system is tested. On the other hand we would like to consider component testing as well because of the cost savings and other associated benefits. To do so, we consider the situation where some a priori knowledge exists about component reliability relative to that of the interfaces. Specifically, while the values of λC and λI are still assumed to be unknown, we consider the following realistic (prior) constraint: λI≤δλC, where δ is some known positive constant. To arrive at the MLE for λˆS , we again maximize the logarithm of the likelihood function given by (2.4), but now subject to the preceding constraint. Setting up the appropriate Lagrangian function, it may be shown that the candidate Kuhn-Tucker points, i.e., points that satisfy the necessary conditions for a maximum (Bazaraa, Sherali and Shetty, 1993) are given by: 1. λI=λC=0 2. λC=(XC/tC); λI=(XS/tS)-(XC/tC); i.e., λˆS = (XS/tS) 3. λC=(XC+XS)/(tC+tS); λI=0; i.e., λˆS = (XC+XS)/(tC+tS) 4. λC=(XC+XS)/[tC+(1+δ)tS]; λI=[δ(XC+XS)]/[tC+(1+δ)tS]; i.e., λˆS = [(1+δ)(XC+XS)]/[tC+(1+δ)tS] Now define R=(XS/tS)/(XC/tC). While the first point does not impose any additional requirements, it may be shown that in order to satisfy the Kuhn-Tucker conditions, the second point also requires 1≤R≤(1+δ), the third point also requires R≥1, and the last point also requires R≥(1+δ). We thus have the following results:

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If Rm*. Case 2c: cS ≥ (1+δ)cC and [(1+δ)B(m0)-A(m0)]>0 and (-ln R0/-ln R1) > (1+δ) First note that the quantity [(1+δ)B(m)-A(m)] is monotone decreasing for m>m0. Let us define m1= min{m| m>m0, [(1+δ)B(m) - A(m)] (1+δ). As discussed above, the optimum strategy will depend on the value of cS, and is as follows: Case 1: cS m0=5, [(1+δ)B(m) - A(m)]