Estimation of ordered parameters from k stochastically increasing distributions

There are given k (? 2) univariate cumulative distribution functions (c.d.f.'s) G(x; θ_i) indexed by a real-valued parameter θ_i, i=1,…, k. Assume that G(x; θ_i) is stochastically increasing in θ_i. In this paper interval estimation on the i^th smallest of the θ's and related topics are studied. Applications are considered for location parameter, normal variance, binomial parameter, and Poisson parameter.     

A note on minimizing the expected makespan in flowshops subject to breakdowns

Consider a two machine flow shop and n jobs. The processing time of job j on machine i is equal to the random variable X_ij One of the two machines is subject to breakdown and repair. The objective is to find the schedule that minimizes the expected makespan. Two results are shown. First, ifP(X_2j ≧ X_1j) = 1 for all j and the random variables X₁₁, X₁₂,…, X_1n are likelihood ratio ordered, then the SEPT sequence minimizes the expected makespan when machine 2 is subject to an arbitrary breakdown process; if P(X_1j≧X_2j) = 1 and X₂₁, X₂₂,….,X_2n are likelihood ratio ordered, then the LEPT sequence minimizes the expected makespan when machine 1 is subject to an arbitrary breakdown process. A generalization is presented for flow shops with m machines. Second, consider the case where X_1j and X_2j are i.i.d. exponentially distributed with rate λ_j. The SEPT sequence minimizes the expected makespan when machine 2 is subject to an arbitrary breakdown process and the LEPT sequence is optimal when machine 1 is subject to an arbitrary breakdown process. © 1995 John Wiley & Sons, Inc.     

An algorithm for separable piecewise convex programming problems

We present a branch and bound algorithm to solve mathematical programming problems of the form: Find x =|(x₁,…x_n) to minimize Σ?_i0(x₁) subject to x?G, l≦x≦L and Σ?_i0(x₁)≦0, j=1,…,m. With l=(l₁,…,l_n) and L=(L₁,…,L_n), each ?_ij is assumed to be lower aemicontinuous and piecewise convex on the finite interval [l_i.L_i]. G is assumed to be a closed convex set. The algorithm solves a finite sequence of convex programming problems; these correspond to successive partitions of the set C={x|l ≦ x ≦L} on the bahis of the piecewise convexity of the problem functions ?_ij. Computational considerations are discussed, and an illustrative example is presented.     

Confidence intervals for ranked means

Suppose that observations from populations π₁, …, π_k (k ≥ 1) are normally distributed with unknown means μ₁., μ_k, respectively, and a common known variance σ². Let μ_[1] μ … ≤ μ_[k] denote the ranked means. We take n independent observations from each population, denote the sample mean of the n observation from π₁ by X _i (i = 1, …, k), and define the ranked sample means X _[1] ≤ … ≤ X _[k]. The problem of confidence interval estimation of μ₍₁₎, …,μ_[k] is stated and related to previous work (Section 1). The following results are obtained (Section 2). For i = 1, …, k and any γ(0 < γ < 1) an upper confidence interval for μ_[i] with minimal probability of coverage γ is (? ∞, X _[i]+ h) with h = (σ/n^1/2) Φ^?1(γ^1/k-i+1), where Φ(·) is the standard normal cdf. A lower confidence interval for μ_[i] with minimal probability of coverage γ is (X _i[i] – g, + ∞) with g = (σ/n^1/2) Φ^?1(γ^1/i). For the upper confidence interval on μ_[i] the maximal probability of coverage is 1– [1 – γ^1/k-i+1]ⁱ, while for the lower confidence interval on μ_[i] the maximal probability of coverage is 1–[1– γ^1/i] ^k-i+¹. Thus the maximal overprotection can always be calculated. The overprotection is tabled for k = 2, 3. These results extend to certain translation parameter families. It is proven that, under a bounded completeness condition, a monotone upper confidence interval h(X ₁, …, X _k) for μ_[i] with probability of coverage γ(0 < γ < 1) for all μ = (μ_[1], …,μ_[k]), does not exist.     

Openshop scheduling under linear resources constraints

Consider n jobs (J₁, …, J_n), m working stations (M₁, …, M_m) and λ linear resources (R₁, …, R_λ). Job J_i consists of m operations (O_i1, …, O_im). Operation O_ij requires P_k(i, j) units of resource R_k to be realized in an M_j. The availability of resource R_k and the ability of the working station M_h to consume resource R_k, vary over time. An operation involving more than one resource consumes them in constant proportions equal to those in which they are required. The order in which operations are realized is immaterial. We seek an allocation of the resources such that the schedule length is minimized. In this paper, polynomial algorithms are developed for several problems, while NP-hardness is demonstrated for several others. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 51–66, 1998     

Scheduling with earliest start and due date constraints on multiple machines

We consider the scheduling of n jobs on m identical machines when the jobs become available for processing at ready times a_i, a_i, ? 0, require d_i time units for processing and must be completed by times b_i for i = 1, 2, … n. The objective chosen is that of minimizing the total elapsed time to complete all jobs subject to the ready time and due date constraints, preemption is not allowed. We present a multi-stage solution algorithm for this problem that is based on an implicit enumeration procedure and also uses the labelling type algorithm which solves the problem when preemption is allowed.     

Checking policy for deteriorating multistate systems

In this article a multistate system under some checking policy is considered. The system has n + 1 states: 0,1, …,n, and deteriorates gradually. State 0 is a normal (full capacity) state and states 1, …,n are considered unsatisfactory. Transition from state 0 to state 1 is considered a system failure. This failure can be detected only through checking, which entails a fixed cost c. The holding time in the undiscovered state i (i = 1, …,n) results in cost d_i per unit of time. For such a system, the algorithm of determining optimum checking times is given.     

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.     

Readiness and the optimal redeployment of resources

This paper considers the problem of the optimal redeployment of a resource among different geographical locations. Initially, it is assumed that at each location i, i = 1,…, n, the level of availability of the resource is given by a₁ ≧ 0. At time t > 0, requirements R_f(t) ≧ 0 are imposed on each location which, in general, will differ from the a₁. The resource can be transported from any one location to any other in magnitudes which will depend on t and the distance between these locations. It is assumed that ΣR_j > Σa_t The objective function consideis, in addition to transportation costs incurred by reallocation, the degree to which the resource availabilities after redeployment differ from the requirements. We shall associate the unavailabilities at the locations with the unreadiness of the system and discuss the optimal redeployment in terms of the minimization of the following functional forms: \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{j = 1}^n {kj(Rj - yj) + } $\end{document} transportation costs, Max \documentclass{article}\pagestyle{empty}\begin{document}$ \mathop {Max}\limits_j \,[kj(Rj - yj)] + $\end{document} transportation costs, and \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_{j = 1}^n {kj(Rj - yj)^2 + } $\end{document} transportation costs. The variables y_j represent the final amount of the resource available at location j. No benefits are assumed to accrue at any location if y_j > R_j. A numerical three location example is given and solved for the linear objective.     

Discrete repairable reliability systems

Consider a reliability system consisting of n components. The failures and the repair completions of the components can occur only at positive integer-valued times k ϵ N₊₊ ϵ (1, 2, …). At any time k ϵ N₊₊ each component can be in one of two states: up (i.e., working) or down (i.e., failed and in repair). The system state is also either up or down and it depends on the states of the components through a coherent structure function τ. In this article we formulate mathematically the above model and we derive some of its properties. In particular, we identify conditions under which the first failure times of two such systems can be stochastically ordered. A variety of special cases is used in order to illustrate the applications of the derived properties of the model. Some instances in which the times of first failure have the NBU (new better than used) property are pointed out. © 1993 John Wiley & Sons, Inc.     

IFR results for repairable systems

Consider a k-out-of-n system with independent repairable components. Assume that the repair and failure distributions are exponential with parameters {μ₁, ?,μ_n} and {λ₁, ?,λ_n}, respectively. In this article we show that if λ_i – μ_i = Δ for all i then the life distribution of the system is increasing failure rate (IFR).     

A minmax search for the critical level of a system: The asymmetric case

We consider a system composed of k components, each of which is subject to failure if temperature is above a critical level. The failure of one component causes the failure of the system as a whole (a serially connected system). If z_i is the critical temperature of the ith component then z^* = min{z_i: i = 1,2,…, k} is the critical level of the system. The components may be tested individually at different temperature levels, if the temperature is below the critical level the cost is $1, otherwise the test is destructive and the cost is m > 1 dollars. The purpose of this article is to construct, under a budgetary constraint, an efficient (in a minmax sense) testing procedure which will locate the critical level of the system with maximal accuracy.     

A diffusion model for the control of a multipurpose reservoir system

This paper develops a methodology for optimizing operation of a multipurpose reservoir with a finite capacity V. The input of water into the reservoir is a Wiener process with positive drift. There are n purposes for which water is demanded. Water may be released from the reservoir at any rate, and the release rate can be increased or decreased instantaneously with zero cost. In addition to the reservoir, a supplementary source of water can supply an unlimited amount of water demanded during any period of time. There is a cost of C_i dollars per unit of demand supplied by the supplementary source to the i^th purpose (i = 1, 2, …, n). At any time, the demand rate R_i associated with the i^th purpose (i = 1, 2, …, n) must be supplied. A controller must continually decide the amount of water to be supplied by the reservoir for each purpose, while the remaining demand will be supplied through the supplementary source with the appropriate costs. We consider the problem of specifying an output policy which minimizes the long run average cost per unit time.     

Decomposition algorithms for minimal cut problems

Consider a network G(N. A) with n nodes, where node 1 designates its source node and node n designates its sink node. The cuts (Z_i, =), i= 1…, n - 1 are called one-node cuts if 1 ? Z_i,. n q Z_i, Z₁-_? {1}, Z_i ? Z_i+1 and Z_i and Z_i+l differ by only one node. It is shown that these one-node cuts decompose G into 1 m n/2 subnetworks with known minimal cuts. Under certain circumstances, the proposed one-node decomposition can produce a minimal cut for G in 0(n² ) machine operations. It is also shown that, under certain conditions, one-node cuts produce no decomposition. An alternative procedure is also introduced to overcome this situation. It is shown that this alternative procedure has the computational complexity of 0(n³).     

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.     

Graph distance‐dependent labeling related to code assignment in computer networks

For nonnegative integers d₁, d₂, and L(d₁, d₂)‐labeling of a graph G, is a function f : V(G) → {0, 1, 2, …} such that |f(u) − f(v)| ≥ d_i whenever the distance between u and v is i in G, for i = 1, 2. The L(d₁, d₂)‐number of G, λ(G) is the smallest k such that there exists an L(d₁, d₂)‐labeling with the largest label k. These labelings have an application to a computer code assignment problem. The task is to assign integer “control codes” to a network of computer stations with distance restrictions, which allow d₁ ≤ d₂. In this article, we will study the labelings with (d₁, d₂) ∈ {(0, 1), (1, 1), (1, 2)}. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2005     

A search game with a protector

A classic problem in Search Theory is one in which a searcher allocates resources to the points of the integer interval [1, n] in an attempt to find an object which has been hidden in them using a known probability function. In this paper we consider a modification of this problem in which there is a protector who can also allocate resources to the points; allocating these resources makes it more difficult for the searcher to find an object. We model the situation as a two‐person non‐zero‐sum game so that we can take into account the fact that using resources can be costly. It is shown that this game has a unique Nash equilibrium when the searcher's probability of finding an object located at point i is of the form (1 − exp (−λ_ix_i)) exp (−μ_iy_i) when the searcher and protector allocate resources x_i and y_i respectively to point i. An algorithm to find this Nash equilibrium is given. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47:85–96, 2000     

The asymptotic joint distribution of the state occupancy times in an alternating renewal process

An alternating renewal process starts at time zero and visits states 1,2,…,r, 1,2, …,r 1,2, …,r, … in sucession. The time spent in state i during any cycle has cumulative distribution function F_i, and the sojourn times in each state are mutually independent, positive and nondegenerate random variables. In the fixed time interval [0,T], let U_i(T) denote the total amount of time spent in state i. In this note, a central limit theorem is proved for the random vector (U_i(T), 1 ≤ i ≤ r) (properly normed and centered) as T → ∞.     

Flowshop/no-idle or no-wait scheduling to minimize the sum of completion times

This paper deals with flowshop/sum of completion times scheduling problems, working under a “no-idle” or a “no-wait” constraint, the former prescribes for the machines to work continuously without idle intervals and the latter for the jobs to be processed continuously without waiting times between consecutive machines. Under either of the constraints the problem is unary NP-Complete for two machines. We prove some properties of the optimal schedule for n/2/F, no-idle/σC_i. For n/m/P, no-idle/σC_i, and n/m/P, no-wait/σC_i, with an increasing or decreasing series of dominating machines, we prove theorems that are the basis for polynomial bounded algorithms. All theorems are demonstrated numerically.     

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.