A two-echelon multi-station inventory model for navy applications

The two inventory echelons under consideration are the depot, D, and k tender ships E₁, …, E_k. The tender ships supply the demand for certain parts of operational boats (the customers). The statistical model assumes that the total monthly demands at the k tenders are stationary independent Poisson random variables, with unknown means λ₁, …, λ_k. The stock levels on the tenders, at the heginning of each month, can be adjusted either by ordering more units from the depot, or by shipping bach to the depot an excess stock. There is no traffic of stock between tenders which is not via the depot. The lead time from the depot to the tenders is at most 1 month. The lead time for orders of the depot from the manufacturer is L months. The loss function due to erroneous decision js comprised of linear functions of the extra monthly stocks, and linear functions of shortages at the tenders and at the depot over the N months. A Bayes sequential decision process is set up for the optimal adjustment levels and orders of the two echelons. The Dynamic Programming recursive functions are given for a planning horizon of N months.     

A sequential bayes procedure for reliability demonstration

A common problem in life testing is to demonstrate that the mean time to failure, θ, exceeds some minimum acceptable value, say θ₁, with a given confidence coefficient γ. When this is true, it is said that “θ₁ has been demonstrated with a confidence γ”. In this paper a Sequential Bayes Procedure (SBP) for demonstrating (by means of. a probability statement) that θ exceeds θ₁ is presented. The SBP differs from the classical procedure in the sense that a prior distribution is assumed on the parameter θ, calling for a Bayesian approach. The procedure is based on the sequence of statistics.     

Technical note: Find a hidden “treasure”

This paper deals with a two searchers game and it investigates the problem of how the possibility of finding a hidden object simultaneously by players influences their behavior. Namely, we consider the following two‐sided allocation non‐zero‐sum game on an integer interval [1,n]. Two teams (Player 1 and 2) want to find an immobile object (say, a treasure) hidden at one of n points. Each point i ∈ [1,n] is characterized by a detection parameter λ_i (μ_i) for Player 1 (Player 2) such that p_i(1 ? exp(?λ_ix_i)) (p_i(1 ? exp(?μ_iy_i))) is the probability that Player 1 (Player 2) discovers the hidden object with amount of search effort x_i (y_i) applied at point i where p_i ∈ (0,1) is the probability that the object is hidden at point i. Player 1 (Player 2) undertakes the search by allocating the total amount of effort X(Y). The payoff for Player 1 (Player 2) is 1 if he detects the object but his opponent does not. If both players detect the object they can share it proportionally and even can pay some share to an umpire who takes care that the players do not cheat each other, namely Player 1 gets q₁ and Player 2 gets q₂ where q₁ + q₂ ≤ 1. The Nash equilibrium of this game is found and numerical examples are given. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007     

Optimum sequential search with discrete locations and random acceptance errors

Much work has been done in search theory; however, very little effort has occurred where an object's presence at a location can be accepted when no object is present there. The case analyzed is of this type. The number of locations is finite, a single object is stationary at one location, and only one location is observed each step of the search. The object's location has a known prior probability distribution. Also known are the conditional probability of acceptance given the object's absence (small) and the conditional probability of rejection given the object's presence (not too large); these Probabilities remain fixed for all searching and locations. The class of sequential search policies which terminate the search at the first acceptance is assumed. A single two-part optimization criterion is considered. The search sequence is found which (i) minimizes the probability of obtaining n rejections in the first n steps for all n, and (ii) maximizes the probability that the first acceptance occurs within the first n steps and occurs at the object's location for all n. The optimum sequential search policy specifies that the next location observed is one with the largest posterior probability of the object's presence (evaluated after each step from Bayes Rule) and that the object is at the first location where acceptance occurs. Placement at the first acceptance seems appropriate when the conditional probability of acceptance given the object's absence is sufficiently small. Search always terminates (with probability one). Optimum truncated sequential policies are also considered. Methods are given for evaluating some pertinent properties and for investigating the possibility that no object occurs at any location.     

Survival distributions in crossing fields containing clusters of mines with possible detection and uncertain activation or kill

The present paper presents an algorithm for the exact determination of survival distributions in crossing mine fields. The model under consideration considers clusters of mines, scattered at random in the field around specified aim points. The scatter distributions of the various clusters are assumed to be known The encounter process allows for a possible detection and destruction of the mines, for inactivation of the mines and for the possibility that an activated mine will not destroy the object. Recursive formulae for the determination of the survival probabilities of each object (tank) in a column of n crossing at the same path are given. The distribution of the number of survivors out of n objects in a column is also determined. Numerical examples are given.     

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.     

基于One-port模式的测试点选取问题

在故障诊断过程中 ,每个测试点检测故障所需的时间可能不同。对于每个测试点一次检测所有可检测故障点的问题已经获得解决。对于每个测试点一次只能检测一个故障点 ,分两种情况加以讨论。若要求检测时间之和最小 ,给出了最优算法 ;若要求最大检测时间最小 ,证明了其是NP完全问题 ,并给出近似算法。最后给出一个实例对算法加以说明     

J2摄动下卫星编队重构的多脉冲轨迹优化

针对考虑J_2摄动的椭圆参考轨道的编队重构问题,以消耗燃料最少为目标函数,基于高斯变分方程研究编队重构的多脉冲轨迹优化方法。推导考虑J_2摄动和轨道面内外耦合的轨道要素偏差线性动力学方程,采用遗传算法和序列二次规划结合的混合算法对总的速度增量进行优化。数值仿真表明该混合算法有效,可以高效地得到可行解。由于考虑了J_2摄动和椭圆参考轨道,该算法对航天任务中的轨迹优化具有一定的参考意义。     

超低轨道卫星摄动特性分析及轨道维持方法

针对超低轨道卫星长时间在轨飞行的轨道维持问题,分析了超低轨道平均偏心率矢量变化特性,提出了一种超低轨道维持的控制方法。分析了J2、J3摄动以及大气阻力摄动作用下超低轨道卫星偏心率矢量的变化特性;基于能量守恒原理设计了超低轨道高度维持的控制策略;通过仿真算例验证了控制策略的有效性。结果表明:在地球非球形引力摄动、大气阻力摄动和速度脉冲作用下超低轨道平均偏心率的变化是稳定的,所设计的轨道维持方法不仅能够实现超低轨道高度维持,确保平均偏心率矢量收敛至平衡位置,且用于轨道维持的燃料消耗合理,能够满足长时间的超低轨道飞行要求。     

On a sequential rule for estimating the location parameter of an exponential distribution

Let us assume that observations are obtained at random and sequentially from a population with density function In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions In this paper we consider a sequential rule for estimating μ when σ is unknown corresponding to the following class of cost functions Where δ(X_I,…,X_N) is a suitable estimator of μ based on the random sample (X₁,…, X_N), N is a stopping variable, and A and p are given constants. To study the performance of the rule it is compared with corresponding “optimum fixed sample procedures” with known σ by comparing expected sample sizes and expected costs. It is shown that the rule is “asymptotically efficient” when absolute loss (p=-1) is used whereas the one based on squared error (p = 2) is not. A table is provided to show that in small samples similar conclusions are also true.     

On the conditional residual lifetime of coherent systems under double regularly checking

In this paper, we consider a coherent system with n independent and identically distributed components under the condition that the system is monitored at time instances t₁ and t₂ (t₁ < t₂). First, various mixture representations for reliability function of the conditional residual lifetime of the coherent system are derived under different scenarios at times t₁ and t₂ (t₁ < t₂). Several stochastic comparisons between two systems are also made based on the proposed conditional random variables. Then, we consider the conditional residual lifetime of the functioning components of the system given that j components have failed at time t₁ and the system has failed at time t₂. Some stochastic comparisons on the proposed conditional residual lifetimes are investigated. Several illustrative graphs and examples are also provided.     

二维 DFT 和 DCT 的 Systolic 阵列

超级计算中一个活跃的研究领域是将某些有限和,如离散富里叶变换(DFT)、离散余弦变换(DCT),映射到多处理机阵列上。本文首先通过二维DFT的行列分解算法流程图,给出了计算二维DFT的二种Systolic阵列:一种是由N_1个处理器组成的线性阵列,所花时间步为O(N_1N_2)(设二维DFT为N_1×N_2长的),与行列分解算法在单处理机上顺序执行所花时间相比,加速比为O(N)(设N_1=N_2=N)。这一结果无论是在时间消耗,还是在PE数量上都是目前最优的。另一种是由N_1×N_2个处理器组成的矩形阵列,所需时间为O(N_1+N_2),与行列算法在单处理机上顺序运行所花时间相比,加速比为O(N~2)(这里仍假定N_1=N_2=N)。本文还给出了二维DCT的与二维DFT相似的Systoilc阵列结构。不难将上述阵列推广到多维的情况。     

Bayes sequential design of stock levels

Bayesian determination of optimal stock levels is studied for the case of Poisson distribution of the demand variable, and prior gamma distribution of the expected demand. Bayes sequential procedure is derived, assuming that stock level can be adjusted at the beginning of each period so that a shortage can be immediately replenished and an overstock can be corrected. The Bayes sequential procedure is more difficult to obtain if this assumption is removed. A dynamic programming method for obtaining the general Bayes sequential procedure is outlined. Finally, an empiric Bayes estimation procedure of the optimal Bayesian stock level is presented.     

Solution of minisum and minimax location–allocation problems with Euclidean distances

A new method for the solution of minimax and minisum location–allocation problems with Euclidean distances is suggested. The method is based on providing differentiable approximations to the objective functions. Thus, if we would like to locate m service facilities with respect to n given demand points, we have to minimize a nonlinear unconstrained function in the 2m variables x₁,y₁, ?,x_m,y_m. This has been done very efficiently using a quasi-Newton method. Since both the original problems and their approximations are neither convex nor concave, the solutions attained may be only local minima. Quite surprisingly, for small problems of locating two or three service points, the global minimum was reached even when the initial position was far from the final result. In both the minisum and minimax cases, large problems of locating 10 service facilities among 100 demand points have been solved. The minima reached in these problems are only local, which is seen by having different solutions for different initial guesses. For practical purposes, one can take different initial positions and choose the final result with best values of the objective function. The likelihood of the best results obtained for these large problems to be close to the global minimum is discussed. We also discuss the possibility of extending the method to cases in which the costs are not necessarily proportional to the Euclidean distances but may be more general functions of the demand and service points coordinates. The method also can be extended easily to similar three-dimensional problems.     

Assembly of systems having maximum reliability

The 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.     

基于Fiducial推断的Bayes验前分布的表示

先验分布的确定与表示是Bayes统计推断的出发点和关键点。提出了一种基于信仰推断 (FiducialInfer ence)观点确定Bayes先验分布的设想 ,有助于解决无验前信息或验前历史信息较少时先验分布的确定问题 ,文中给出了一个实例说明了该方法的应用     

Covariance of weibull order statistics

Let X₁ < X₂ <… < X_n denote an ordered sample of size n from a Weibull population with cdf F(x) = 1 - exp (?x^p), x > 0. Formulae for computing Cov (X_i, X_j) are well known, but they are difficult to use in practice. A simple approximation to Cov(X_i, X_j) is presented here, and its accuracy is discussed.     

New results on a Ruckle problem in discrete games of ambush

This article deals with a two‐person zero‐sum game in which player I chooses in integer interval [1, N] two integer intervals consisting of p and q points where p + q < N, and player II chooses an integer point in [1, N]. The payoff to player I equals 1 if the point chosen by player II is at least in one of the intervals chosen by player II and 0 otherwise. This paper complements the results obtained by Ruckle, Baston and Bostock, Lee, Garnaev, and Zoroa, Zoroa and Fernández‐Sáez. © 2001 John Wiley & Sons, Inc. Naval Research Logistics 48: 98–106, 2001     

Estimating value in a uniform auction

Consider an auction in which increasing bids are made in sequence on an object whose value θ is known to each bidder. Suppose n bids are received, and the distribution of each bid is conditionally uniform. More specifically, suppose the first bid X₁ is uniformly distributed on [0, θ], and the i^th bid is uniformly distributed on [X_i?1, θ] for i = 2, …?, n. A scenario in which this auction model is appropriate is described. We assume that the value θ is un known to the statistician and must be esimated from the sample X₁, X₂, …?, X_n. The best linear unbiased estimate of θ is derived. The invariance of the estimation problem under scale transformations in noted, and the best invariant estimation problem under scale transformations is noted, and the best invariant estimate of θ under loss L(θ, a) = [(a/θ) ? 1]² is derived. It is shown that this best invariant estimate has uniformly smaller mean-squared error than the best linear unbiased estimate, and the ratio of the mean-squared errors is estimated from simulation experiments. A Bayesian formulation of the estimation problem is also considered, and a class of Bayes estimates is explicitly derived.     

Sojourn time analysis of a queueing system with two‐phase service and server vacations

We consider a two‐phase service queueing system with batch Poisson arrivals and server vacations denoted by M^X/G₁‐G₂/1. The first phase service is an exhaustive or a gated bulk service, and the second phase is given individually to the members of a batch. By a reduction to an M^X/G/1 vacation system and applying the level‐crossing method to a workload process with two types of vacations, we obtain the Laplace–Stieltjes transform of the sojourn time distribution in the M^X/G₁‐G₂/1 with single or multiple vacations. The decomposition expression is derived for the Laplace–Stieltjes transform of the sojourn time distribution, and the first two moments of the sojourn time are provided. © 2006 Wiley Periodicals, Inc. Naval Research Logistics, 2007