Bayesian estimation and optimal designs in partially accelerated life testing

A method of life testing is proposed which combines both ordinary and accelerated life-testing procedures. It is assumed that an item can be tested either in a standard environment or under stress. The amount of stress is fixed in advance and is the same for all items to be tested However, the time x at which an item on lest is taken out of the standard environment and put under stress can be chosen by the experimenter subject to a given cost structure. When an item is put under stress its lifetime is changed by the factor α. Let the random variable T denote the lifetime of an item in the standard environment, and let γ denote its lifetime under the partially accelerated test procedure just described. Then Y = T if T ≦ x, and Y = x + α (T > x) if T > x. It is assumed that T has an exponential distribution with parameter θ. The estimation of θ and α and the optimal design of a partially accelerated life test are studied in the framework of Bayesian decision theory.     

Future supply uncertainty in EOQ models

This article deals with an inventory problem where the supply is available only during an interval of (random) length X. The unavailability of supply lasts for a random duration Y. Using concepts from renewal theory, we construct an objective function (average cost/time) in terms of the order-quantity decision variable Q. We develop the individual cost components as order, holding, and shortage costs after introducing two important random variables. Due to the complexity of the objective function when X and Y are general random variables, we discuss two special cases and provide numerical examples with sensitivity analysis on the cost and noncost parameters. The article concludes with a discussion of the comparison of the current model with random yield and random lead-time models. Suggestions for further research are also provided.     

The relationship between decision variables and penalty cost parameter in (Q,R) inventory models

This paper is concerned with the optimum decision variables found using order quantity, reorder point (Q, R) inventory models. It examines whether the optimum variables (Q* and R*) are necessarily monotonic functions of the backorder cost parameter (or equivalently of the performance objective). For a general class of models it is proved that R* must increase as the performance objective is raised, and an inequality condition is derived which governs how Q* will change. Probability distributions of lead time demand are cited or found for which Q* increases, Q* decreases, and Q* is independent of increases in performance objectives or backorder cost parameter.     

The estimation of P(x < Y) for distributions useful in life testing

In this paper the reliability function K = P(X < Y) has been estimated when X and Y follow gamma, exponential or bivariate exponential distributions. The paper is partly expository.     

On estimating the scale parameter of the exponential distribution with a known linear relation between the location and the scale parameter

In this article we study the estimation of the average excess life θ in a two-parameter exponential distribution with a known linear relationship between α (the minimum life) and θ of the form α = aθ, where a is known and positive. A comparison of the efficiencies of estimators which are linear combinations of the smallest sample value and the sample sum of deviations from the smallest sample value and the maximum likelihood estimators is made for various sample sizes and different values of a. It is shown that these estimators are dominated in the risk by the minimum-risk scale equivariant estimator based on sufficient statistics. A class of Bayes estimators for inverted gamma priors is constructed and shown to include a minimum-risk scale equivariant estimator in it. All the members of this class can be computed easily.     

Extrapolation of the mean lifetime of a large population from its preliminary survival history

A large population of independent identical units having finite mean lifetime T is observed. From the history A(y) of cumulative arrivals and the history B(y) of cumulative removals in the interval 0 ≦ y ≦ τ one must predict at time τ the desired T . Two lifetime predictors X(τ) and Y(τ) and related simple predictors obtained from A(y) and B(y) are shown to converge to T with a rate of convergence dependent on the structure of the failure rate function of the units. This dependence is studied theoretically and numerically.     

Exact analysis of (R,s, S) inventory control systems with lost sales and zero lead time

We study an (R, s, S) inventory control policy with stochastic demand, lost sales, zero lead‐time and a target service level to be satisfied. The system is modeled as a discrete time Markov chain for which we present a novel approach to derive exact closed‐form solutions for the limiting distribution of the on‐hand inventory level at the end of a review period, given the reorder level (s) and order‐up‐to level (S). We then establish a relationship between the limiting distributions for adjacent values of the reorder point that is used in an efficient recursive algorithm to determine the optimal parameter values of the (R, s, S) replenishment policy. The algorithm is easy to implement and entails less effort than solving the steady‐state equations for the corresponding Markov model. Point‐of‐use hospital inventory systems share the essential characteristics of the inventory system we model, and a case study using real data from such a system shows that with our approach, optimal policies with significant savings in inventory management effort are easily obtained for a large family of items.     

Testing fit with nuisance location and scale parameters

For each n, X₁(n),…, X_n(n) are independent and identically distributed random variables, each with cumulative distribution function F(x) which is known to be absolutely continuous but is otherwise unknown. The problem is to test the hypothesis that \documentclass{article}\pagestyle{empty}\begin{document}$ F(x) = G\left( {{\textstyle{{x - \theta _1 } \over {\theta _2 }}}} \right) $\end{document}, where the cumulative distribution function G_x is completely specified and satisfies certain regularity conditions, and the parameters θ₁, θ₂ are unknown and unspecified, except that the scale parameter θ₂, is positive. Y₁ (n) ≦ Y₂ (n) ≦ … ≦ Y_n (n)are the ordered values of X₁(n),…, X_n(n). A test based on a certain subset of {Y_i(n)} is proposed, is shown to have asymptotically a normal distribution when the hypothesis is true, and is shown to be consistent against all alternatives satisfying a mild regularity condition.     

Optimal control of a production‐inventory system with both backorders and lost sales

We consider the optimal control of a production inventory‐system with a single product and two customer classes where items are produced one unit at a time. Upon arrival, customer orders can be fulfilled from existing inventory, if there is any, backordered, or rejected. The two classes are differentiated by their backorder and lost sales costs. At each decision epoch, we must determine whether or not to produce an item and if so, whether to use this item to increase inventory or to reduce backlog. At each decision epoch, we must also determine whether or not to satisfy demand from a particular class (should one arise), backorder it, or reject it. In doing so, we must balance inventory holding costs against the costs of backordering and lost sales. We formulate the problem as a Markov decision process and use it to characterize the structure of the optimal policy. We show that the optimal policy can be described by three state‐dependent thresholds: a production base‐stock level and two order‐admission levels, one for each class. The production base‐stock level determines when production takes place and how to allocate items that are produced. This base‐stock level also determines when orders from the class with the lower shortage costs (Class 2) are backordered and not fulfilled from inventory. The order‐admission levels determine when orders should be rejected. We show that the threshold levels are monotonic (either nonincreasing or nondecreasing) in the backorder level of Class 2. We also characterize analytically the sensitivity of these thresholds to the various cost parameters. Using numerical results, we compare the performance of the optimal policy against several heuristics and show that those that do not allow for the possibility of both backordering and rejecting orders can perform poorly.© 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

Inequalities involving the lifetime of series and parallel systems

Let {Xⁱ} be independent HNBUE (Harmonic New Better Than Used in Expectation) random variables and let {Yⁱ} be independent exponential random variables such that E{Xⁱ}=E{Yⁱ} It is shown that \documentclass{article}\pagestyle{empty}\begin{document}$ E\left[{u\left({\mathop {\min \,X_i}\limits_{l \le i \le n}} \right)} \right] \ge E\left[{u\left({\mathop {\min \,Y_i}\limits_{l \le i \le n}} \right)} \right] $\end{document} for all increasing and concave u. This generalizes a result of Kubat. When comparing two series systems with components of equal cost, one with lifetimes {Xⁱ} and the other with lifetimes {Yⁱ}, it is shown that a risk-averse decision-maker will prefer the HNBUE system. Similar results are obtained for parallel systems.     

On estimating the reliability of a component subject to several different stresses (strengths)

This paper is concerned with estimating p = P(X₁ < Y …, X_n < Y) or q =P (X < Y₁, …, X < Y_n) where the X's and Y's are all independent random variables. Applications to estimation of the reliability p from stress-strength relationships are considered where a component is subject to several stresses X₁, X₂, …, X_N whereas its strength, Y, is a single random variable. Similarly, the reliability q is of interest where a component is made of several parts all with their individual strengths Y₁, Y₂ …, Y_N and a single stress X is applied to the component. When the X's and Y's are independent and normal, maximum likelihood estimates of p and q have been obtained. For the case N = 2 and in some special cases, minimum variance unbiased estimates have been given. When the Y's are all exponential and the X is normal with known variance, but unknown mean (or uniform between 0 and θ, θ being unknown) the minimum variance unbiased estimate of q is established in this paper.     

Estimating moments of the minimum order statistic from normal populations—a bayesian approach

Let Yⁱ ∽ N(θⁱ, σ), i = 1, …, p, be independently distributed, where θⁱ and σ are unknown. A Bayesian approach is used to estimate the first two moments of the minimum order statistic, W = min (Y¹, …, Y^p). In order to compute the Bayes estimates, one has to evaluate the predictive densities of the Yⁱ's conditional on past data. Although the required predictive densities are complicated in form, an efficient algorithm to calculate them has been developed and given in the article. An application of the Bayesian method in a continuous-review control model with multiple suppliers is discussed. © 1994 John Wiley & Sons, Inc.     

Estimating component characteristics from system failure‐time data

Suppose that failure times are available from a random sample of N systems of a given, fixed design with components which have i.i.d. lifetimes distributed according to a common distribution F. The inverse problem of estimating F from data on observed system lifetimes is considered. Using the known relationship between the system and component lifetime distributions via signature and domination theory, the nonparametric maximum likelihood estimator _N(t) of the component survival function (t) is identified and shown to be accessible numerically in any application of interest. The asymptotic distribution of _N(t) is also identified, facilitating the construction of approximate confidence intervals for (t) for N sufficiently large. Simulation results for samples of size N = 50 and N = 100 for a collection of five parametric lifetime models demonstrate the utility of the recommended estimator. Possible extensions beyond the i.i.d. framework are discussed in the concluding section. © 2010 Wiley Periodicals, Inc. Naval Research Logistics, 2010     

Stochastic control of queueing systems

Suppose that the state of a queueing system is described by a Markov process { Y_t, t ≥ 0}, and the profit from operating it up to a time t is given by the function f(Y_t). We operate the system up to a time T, where the random variable T is a stopping time for the process Yt. Optimal stochastic control is achieved by choosing the stopping time T that maximizes Ef(Y_T) over a given class of stopping times. In this paper a theory of stochastic control is developed for a single server queue with Poisson arrivals and general service times.     

On the application and extension of system signatures in engineering reliability

Following a review of the basic ideas in structural reliability, including signature‐based representation and preservation theorems for systems whose components have independent and identically distributed (i.i.d.) lifetimes, extensions that apply to the comparison of coherent systems of different sizes, and stochastic mixtures of them, are obtained. It is then shown that these results may be extended to vectors of exchangeable random lifetimes. In particular, for arbitrary systems of sizes m < n with exchangeable component lifetimes, it is shown that the distribution of an m‐component system's lifetime can be written as a mixture of the distributions of k‐out‐of‐n systems. When the system has n components, the vector of coefficients in this mixture representation is precisely the signature of the system defined in Samaniego, IEEE Trans Reliabil R–34 (1985) 69–72. These mixture representations are then used to obtain new stochastic ordering properties for coherent or mixed systems of different sizes. © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008     

Application of perturbation analysis to a class of periodic review (s,S) inventory systems

In this article we apply perturbation analysis (PA), combined with conditional Monte Carlo, to obtain derivative estimators of the expected cost per period with respect to s and S, for a class of periodic review (s, S) inventory systems with full backlogging, linear holding and shortage costs, and where the arrivals of demands follow a renewal process. We first develop the general form of four different estimators of the gradient for the finite-horizon case, and prove that they are unbiased. We next consider the problem of implementing our estimators, and develop efficient methodologies for the infinite-horizon case. For the case of exponentially distributed demand interarrival times, we implement our estimators using a single sample path. Generally distributed interarrival times are modeled as phase-type distributions, and the implementation of this more general case requires a number of additional off-line simulations. The resulting estimators are still efficient and practical, provided that the number of phases is not too large. We conclude by reporting the results of simulation experiments. The results provide further validity of our methodology and also indicate that our estimators have very low variance. © 1994 John Wiley & Sons, Inc.     

Bounds for strength-stress interference via mathematical programming

Problems of bounding Pr {X > Y}, when the distribution of X is subject to certain moment conditions and the distribution of Y is known to be of convexconcave type, are treated in the framework of mathematical programming. Juxtaposed are two programming methods; one is based on the notion of weak duality and the other on the geometry of a certain moment space.     

An integral equation for the second moment function of a geometric process and its numerical solution

In this article, an integral equation satisfied by the second moment function M₂(t) of a geometric process is obtained. The numerical method based on the trapezoidal integration rule proposed by Tang and Lam for the geometric function M(t) is adapted to solve this integral equation. To illustrate the numerical method, the first interarrival time is assumed to be one of four common lifetime distributions, namely, exponential, gamma, Weibull, and lognormal. In addition to this method, a power series expansion is derived using the integral equation for the second moment function M₂(t), when the first interarrival time has an exponential distribution.     

Coordinated pricing and inventory control problems with capacity constraints and fixed ordering cost

This article addresses a single‐item, finite‐horizon, periodic‐review coordinated decision model on pricing and inventory control with capacity constraints and fixed ordering cost. Demands in different periods are random and independent of each other, and their distributions depend on the price in the current period. Each period's stochastic demand function is the additive demand model. Pricing and ordering decisions are made at the beginning of each period, and all shortages are backlogged. The objective is to find an optimal policy that maximizes the total expected discounted profit. We show that the profit‐to‐go function is strongly CK‐concave, and the optimal policy has an (s,S,P) ‐like structure. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012     

Centralized inventory control in a two‐level distribution system with Poisson demand

This paper introduces a new replenishment policy for inventory control in a two‐level distribution system consisting of one central warehouse and an arbitrary number of nonidentical retailers. The new policy is designed to control the replenishment process at the central warehouse, using centralized information regarding the inventory positions and demand processes of all installations in the system. The retailers on the other hand are assumed to use continuous review (R, Q) policies. A technique for exact evaluation of the expected inventory holding and backorder costs for the system is presented. Numerical results indicate that there are cases when considerable savings can be made by using the new (α₀, Q₀) policy instead of a traditional echelon‐ or installation‐stock (R, Q) policy. © 2002 Wiley Periodicals, Inc. Naval Research Logistics 49: 798–822, 2002; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/nav.10040