: Formulas and tables are provided making it possible for users of the truncated sequential probability ratio tests of MIL-STD 781C to make acceptance or rejection decisions when the test procedure is prematurely tuncated. (Author)

When a system fails it may not be obvious which components are at fault. Locating faulty components to be replaced may require a series of inspections each of which reveals the state (functioning/failed) of one of the components. The order in which components are inspected and replaced can greatly affect the cost to restore the system to an operating condition. This paper investigates inspection sequences for complex coherent systems.

We consider an extreme version of the replacement problem. A vital component of a system must be replaced before it fails, otherwise the system fails with no possibility of repairing the system. We assume that n spares are available and that the distribution of a component life is known; the objective is to schedule the replacements so that the expected life of the system is maximized. Our results range from an iterative formula for constructing the optimal schedule to more general mathematical properties of optimal schedules and expected times.

AbstractThis paper describes the background of the Office of Management Budget Circular A‐21, “Principles for Determining Costs Applicable to Grants, Contracts, and Other Agreements with Educational Institutions,” that describes the requirement for effort reporting. A sampling procedure is proposed as an alternative to 100% reporting.

The linear (circular) k-of-n:F system has n linearly (circularly) ordered components. Each component either functions or fails. The system fails i.f.f.k consecutive components fail. This paper provides, in the i.i.d. case, recursive formulas and bounds for computing system reliability. It considers properties of system life distributions and, in the non i.i.d. case, questions of optimal system des...

AbstractThis paper is concerned with the statistical test plans contained in Military Standard 781C, “Reliability Design Qualification and Production Acceptance Tests: Exponential Distribution” and the selection and use of these plans. Modifications to the fixed‐length test plans of MIL‐STD‐781C are presented which allow early‐accept decisions to be made without sacrificing statistical validity. The proposed plans differ from the probability ratio sequential tests in the Standard in that rejection is permitted only after a fixed number of failures have been observed.

We consider an N- server queue with arbitrary arrivals and service times which are random but with differing rates for different servers. Customers arriving when all servers are occupied do not enter the system. We show that the policy of always assigning an arrival to that free server whose service rate is largest (smallest) stochastically minimises (maximises) the number in the system. We then show that in a particular component-repair context with exponential repair times the policy of repairing failed components with the smallest failure rate stochastically maximises the number of working components.

AbstractThis paper reconsiders the classical model for selling an asset in which offers come in daily and a decision must then be made as to whether or not to sell. For each day the item remains unsold a continuation (or maintenance cost)cis incurred. The successive offers are assumed to be independent and identically distributed random variables having an unknown distributionF. The model is considered both in the case where once an offer is rejected it may not be recalled at a later time and in the case where such recall of previous offers is allowed.

A system e.g., a motor vehicle must operate for t units of time. A certain component e.g., a battery is essential for its operation and must be replaced each time it fails. There are n types of replacement components. A type i replacement costs Ci and has a random life with distribution depending on i. There is no salvage value associated with the particular component in use when the system terminates. The problem is to assign the initial component and subsequent replacements from among the n types so as to minimize the total expected cost of providing an operative component for the t units of time. This paper treats this problem when the life distributions are exponential for each type and when t is fixed or has a truncated exponential distribution. Related problems are also considered.

We have N stages to sequentially construct I successful components. At each stage, we allocate a certain amount of money for the construction of a component. If y is the amount allocated, then the component constructed will be a success with probability P y , where P is a continuous nondecreasing function satisfying P 0 = 0. After each component is constructed, we are informed as to whether or not it is successful. If, at the end of the N stages, we are i components short, then a final penalty cost C i is incurred. The problem is to determine at each stage how much money to allocate so as to minimize the total expected cost construction cost plus penalty cost incurred. The major result is that if C i + 1- C i ≤ C i + 2- C i + 1, and if y n i denotes the optimal value to allocate when i components are needed with n stages remaining, then y n i is nondecreasing in i and nonincreasing in n .

This paper considers the following model, described in terms of an investment problem. We have D units available for investment. During each of N time periods an opportunity to invest will occur with probability p. As soon as an opportunity presents itself, we must decide how much of our available resources to invest. If we invest y, then we obtain an expected profit P(y), where P is a nondecreasing continuous function. The amount y then becomes unavailable for future investment. The problem is to decide how much to invest at each opportunity so as to maximize total expected profit. When P(y) is a concave function, the structure of the optimal policy is obtained (§1). Bounds on the optimal value function and asymptotic results are presented in §2. A closed-form expression for the optimal value to invest is found in §3 for the special cases of P(y) = log y and P(y) = yα, for 0 < α < 1. §4 presents a continuous-time version of the model, i.e., we assume that opportunities occur in accordance with a Poisson process. Other applications of the model are also considered.

AbstractThe first problem considered in this paper is concerned with the assembly of independent components into parallel systems so as to maximize the expected number of systems that perform satisfactorily. Associated with each component is a probability of it performing successfully. It is shown that an optimal assembly is obtained if the reliability of each assembled system can be made equal. If such equality is not attainable, then bounds are given so that the maximum expected number of systems that perform satisfactorily will lie within these stated bounds; the bounds being a function of an arbitrarily chosen assembly. An improvement algorithm is also presented.A second problem treated is concerned with the optimal design of a system. Instead of assembling given units, there is an opportunity to “control” their quality, i.e., the manufacturer is able to fix the probability, p, of a unit performing successfully. However, his resources, are limited so that a constraint is imposed on these probabilities. For (1) series systems, (2) parallel systems, and (3) k out of n systems, results are obtained for finding the optimal p's which maximize the reliability of a single system, and which maximize the expected number of systems that perform satisfactorily out of a total assembly of J systems.

AbstractWe are concerned with the following reliability problem: A system has k different types of components. Associated with each component is a numerical value. Let {aj}, (j= 1,…,k), denote the set of numerical values of the k components. Let R(a1,…, ak) denote the probability that the system will perform satisfactorily (i. e., R(a1,…. ak) is the reliability of the system) and assume R(a1,…,ak) has the properties of a joint cumulative distribution function.Now suppose aj1 ≤… ≤ ajn are n components of type j (j= 1,…, k). Then n systems can be assembled from these components. Let N denote the number of systems that perform satisfactorily. N is a random variable whose distribution will depend on the way the n systems are assembled. Of all different ways in which the n systems can be assembled, the paper shows that EN is maximized if these n systems have reliability R(a1j,…, akj) (i = 1,…,n). The method used here is an extension of a well known result of Hardy, Littlewood, and Polya on sums of products. Furthermore, under certain conditions, the same assembly that maximizes EN minimizes the variance of N.Finally, for a similar problem in reliability, it is shown that for a series system a construction can be found that not only maximizes the expected number of functioning modules, but also possesses the stronger property of maximizing the probability that the number of functioning modules is at least r, for each 0 ≤ r ≤ n.

Suppose X1, X2, ···, Xn are independent identically distributed exponential random variables with parameter λ1. Let Y1, Y2, ···, Ym also be independent identically distributed exponential random variables with parameter λ2, and assume that X's and Y's are independent. The problem is to estimate R(t) = e−(λ1+λ2)t. A procedure for determining on exact (1 − α) level lower confidence bound for R(t) is presented. let U = min(Σn t = 1Xi, Σm i = 1Yi), and K = {largest i ≤ n: Σj i − 1Xi ≤ U} + {Largest i ≤ m: Σj i = 1 Yi ≤ U}. Then given K = k, it is shown that U has a gamma distribution with parameters k and λ1 + λ2. Hence, a lower confidence bound for R(t) can be obtained. The suggested procedure is then compared with others presented in the literature.

Naval Research Logistics QuarterlyVolume 16, Issue 1 p. 17-35 Article The status and impact of reliability methodology† Gerald J. Lieberman, Gerald J. Lieberman Stanford University This work was supported in part by the Army, Navy, and Air Force under contract Nonr-225(53)(NR-042-002) with the Office of Naval Research.Search for more papers by this author Gerald J. Lieberman, Gerald J. Lieberman Stanford University This work was supported in part by the Army, Navy, and Air Force under contract Nonr-225(53)(NR-042-002) with the Office of Naval Research.Search for more papers by this author First published: March 1969 https://doi.org/10.1002/nav.3800160102Citations: 12 † Presented at the Research and Technology Conference on Optimization/Statistics, Simulation, Information Processing, sponsored by the Office of Naval Research and the George Washington University on September 11–13, 1968. AboutPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onEmailFacebookTwitterLinkedInRedditWechat References 1 Barlow, R. E. and A. W. Marshall, “Bounds on Interval Probabilities for Restricted Families of Distributions,” 5th Berkeley Symposium (1965), pp. 1229–257. Google Scholar 2 Barlow, R. E., and F. Proschan, Mathematical Theory of Reliability (John Wiley and Sons, Inc., New York, N.Y., 1965). Google Scholar 3 Birnbaum, Z. W., J. D. Esary, and A. W. Marshall, “A Stochastic Characterization of Wear-Out For Components and Systems,” Ann. Math. Stat., 37, 816–825 (1966). 10.1214/aoms/1177699362 Web of Science®Google Scholar 4 Birnbaum, Z. W. and S. C. Saunders, “A New Family of the Distributions,” University of Washington Laboratory of Statistical Research, Technical Report No. 52 (1968). Google Scholar 5 Esary, J. D., F. Proschan, and D. Walkup, “A Multivariate Notion of Association, with Applications,” Ann. Math. Stat., 38, 1466–1474 (1967). 10.1214/aoms/1177698701 Web of Science®Google Scholar 6 Johns, M. V. and G. J. Lieberman, “An Exact Asymptotically Efficient Confidence Bound for Reliability in the Case of the Weibull Distribution,” Technometrics, 8, 135–175 (1966). 10.1080/00401706.1966.10490330 Web of Science®Google Scholar 7 Kramer, H. C., “One-Sided Confidence Interval for the Quality Indices of a Complex Item,” Technometrics, 5, 400–403 (1963). 10.1080/00401706.1963.10490110 Google Scholar 8 Marshall, A. W. and I. Olkin, “A Multivariate Exponential Distribution,” J. Am. Statist. Assoc. 62, 30–44 (1967). Web of Science®Google Scholar 9 Sarkar, T., Unpublished Manuscript on Reliability, Stanford University (1968). Google Scholar 10 Shooman, M. L., Probabilistic Reliability — An Engineering Approach (McGraw Hill Book Co., New York, N.Y., 1968). Web of Science®Google Scholar 11 U.S. Government, “Reliability Stress and Failure Rate Data,” MIL-HDBK-217A, Government Printing Office, Washington, D.C. (1965). Google Scholar 12 Woods, W. M., and J. R. Borsting, “A Method for Computing Lower Confidence Limits on System Reliability Using Component Failure Data with Unequal Sizes,” U.S. Naval Postgraduate School Technical Report NPS55 Wo/Bg 8061 A (1968). Google Scholar Citing Literature Volume16, Issue1March 1969Pages 17-35 ReferencesRelatedInformation

Two important classes of problems encountered by industry and government are those dealing with replacement policies and ordering policies. These problems are usually treated separately. A particular model which treats the replacement and stocking problems as one is considered in this paper. It is formulated as a denumerable state Markovian decision model. It is shown that the optimal decision policy has a particular structure. In attaining the optimal policy, this structure is exploited so that an optimal policy can be obtained in a finite number of steps.

When a linear relationship has been fitted by least squares, the methods for securing a prediction interval for the response at some fixed value of the independent variable are explained in many statistical text books. This paper describes the somewhat more complex problem of determining the joint prediction interval for the responses at each of K separate settings of the independent variables when all K predictions must be based upon the original fitted model.

In their paper (Derman, C., M. Klein. Inventory depletion management. Technical Report No. 2, Oct. 1957, Statistical Engineering Group, Columbia University.), Derman and Klein present sufficient conditions on the field life of an item function under which a LIFO policy is optimal. This paper presents an alternate set of sufficient conditions under which a LIFO policy is optimal.