Computational results for a stopping rule problem on averages

Suppose x₁, x₂, … are independently distributed random variables with Pr (x_i = 1) = Pr(x_i = ?1) = 1/2, and let s_n =

     

The queueing system MX/G/1 and its ramifications

This paper deals with the bulk arrival queueing system M^X/G/1 and its ramifications. In the system M^X/G/1, customers arrive in groups of size X (a random variable) by a Poisson process, the service times distribution is general, and there is a single server. Although some results for this queueing system have appeared in various books, no unified account of these, as is being presented here, appears to have been reported so far. The chief objectives of the paper are (i) to unify by an elegant procedure the relationships between the p.g.f.'s

     

Sandwich approximation of univariate convex functions with an application to separable convex programming

In this article an algorithm for computing upper and lower ? approximations of a (implicitly or explicitly) given convex function h defined on an interval of length T is developed. The approximations can be obtained under weak assumptions on h (in particular, no differentiability), and the error decreases quadratically with the number of iterations. To reach an absolute accuracy of ? the number of iterations is bounded by

     

On the minimization of a certain convex function arising in applied decision theory

The author, in an expository paper [4], has presented an algorithm for choosing a non-negative vector

     

Convolutions and generalization of logconcavity: Implications and applications

Additive convolution of unimodal and α‐unimodal random variables are known as an old classic problem which has attracted the attention of many authors in theory and applied fields. Another type of convolution, called multiplicative convolution, is rather younger. In this article, we first focus on this newer concept and obtain several useful results in which the most important ones is that if is logconcave then so are and for some suitable increasing functions ?. This result contains and as two more important special cases. Furthermore, one table including more applied distributions comparing logconcavity of f(x) and and two comprehensive implications charts are provided. Then, these fundamental results are applied to aging properties, existence of moments and several kinds of ordered random variables. Multiplicative strong unimodality in the discrete case is also introduced and its properties are investigated. In the second part of the article, some refinements are made for additive convolutions. A remaining open problem is completed and a conjecture concerning convolution of discrete α‐unimodal distributions is settled. Then, we shall show that an existing result regarding convolution of symmetric discrete unimodal distributions is not correct and an easy alternative proof is presented. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 109–123, 2016     

Scheduling deteriorating jobs to minimize makespan

We consider a single-machine problem of scheduling n independent jobs to minimize makespan, in which the processing time of job J_j grows by w_j with each time unit its start is delayed beyond a given common critical date d. This processing time is p_j if J_j starts by d. We show that this problem is NP-hard, give a pseudopolynomial algorithm that runs in time and O(nd) space, and develop a branch-and-bound algorithm that solves instances with up to 100 jobs in a reasonable amount of time. We also introduce the case of bounded deterioration, where the processing time of a job grows no further if the job starts after a common maximum deterioration date D > d. For this case, we give two pseudopolynomial time algorithms: one runs in O(n²d(D − d) time and O(nd(D − d)) space, the other runs in p_j)²) time and p_j) space. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 511–523, 1998     

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.     

A finiteness proof for modified dantzig cuts in integer programming

Let be a basic solution to the linear programming problem subject to: where R is the index set associated with the nonbasic variables. If all of the variables are constrained to be nonnegative integers and x_u is not an integer in the basic solution, the linear constraint is implied. We prove that including these “cuts” in a specified way yields a finite dual simplex algorithm for the pure integer programming problem. The relation of these modified Dantzig cuts to Gomory cuts is discussed.     

Skewness correction X̄ and R charts for skewed distributions

This paper proposes a skewness correction (SC) method for constructing the and R control charts for skewed process distributions. Their asymmetric control limits (about the central line) are based on the degree of skewness estimated from the subgroups, and no parameter assumptions are made on the form of process distribution. These charts are simply adjustments of the conventional Shewhart control charts. Moreover, the chart is almost the same as the Shewhart chart if the process distribution is known to be symmetrical. The new charts are compared with the Shewhart charts and weighted variance (WV) control charts. When the process distribution is in some neighborhood of Weibull, lognormal, Burr or binomial family, simulation shows that the SC control charts have Type I risk (i.e., probability of a false alarm) closer to 0.27% of the normal case. Even in the case where the process distribution is exponential with known mean, not only the control limits and Type I risk, but also the Type II risk of the SC charts are closer to those of the exact and R charts than those of the WV and Shewhart charts. © 2003 Wiley Periodicals, Inc. Naval Research Logistics 50: 555–573, 2003     

Asymptotic inference about a density function at an end of its range

For each n, X₁(n),…X_n(n) are independent and identically distributed random variables, with common probability density function Where c, θ, α, and r(y) are all unknown. It is shown that we can make asymptotic inferences about c, θ, and α, when r(y) satisfies mild conditions.     

Bounds for circular error probabilities

Bounds for P(X + X ⩽ k²σ) are given where X₁ and X₂ are independent normal variables having zero means and variances σ, σ, respectively. This is generalized when X₁ and X₂ are dependent variables with known covariance matrix.     

The knapsack problem: A survey

A unifying survey of the literature related to the knapsack problem; that is, maximize \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_i {v_i x_{i,} } $\end{document}, subject to \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_j {w_i x_i W} $\end{document} and x_i ? 0, integer; where v_i, w_i and W are known integers, and w_i (i = 1, 2, …, N) and W are positive. Various uses, including those in group theory and in other integer programming algorithms, as well as applications from the literature, are discussed. Dynamic programming, branch and bound, search enumeration, heuristic methods, and other solution techniques are presented. Computational experience, and extensions of the knapsack problem, such as to the multi-dimensional case, are also considered.     

Inventory policies for a make‐to‐order system with a perishable component and fixed ordering cost

We consider a make‐to‐order production system where two major components, one nonperishable (referred to as part 1) and one perishable (part 2), are needed to fulfill a customer order. In each period, replenishment decisions for both parts need to be made jointly before demand is realized and a fixed ordering cost is incurred for the nonperishable part. We show that a simple (s_n,S,S) policy is optimal. Under this policy, S along with the number of backorders at the beginning of a period if any and the availability of the nonperishable part (part 1) determines the optimal order quantity of the perishable part (part 2), while (s_n,S) guide when and how much of part 1 to order at each state. Numerical study demonstrates that the benefits of using the joint replenishment policy can be substantial, especially when the unit costs are high and/or the profit margin is low. © 2009 Wiley Periodicals, Inc. Naval Research Logistics, 2009     

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.     

On some stochastic inequalities involving minimum of random variables

Let X_i be independent IFR random variables and let Y_i be independent exponential random variables such that E[X_i]=E[Y_i] for all i=1, 2, ? n. Then it is well known that E[min (X_i)] ≥E[min (X_i)]. Nevertheless, for 1≤i≤n exponentially distributed X_i's and for a decreasing convex function ?(.). it is shown that .     

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.     

Some results of the queueing system E/M/c

This paper analyses the E/M/c queueing system and shows how to calculate the expected number in the system, both at a random epoch and immediately preceding an arrival. These expectations are expressed in terms of certain initial probabilities which are determined by linear equations. The advantages and disadvantages of this method are also discussed.     

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     

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.     

Scheduling multiple products on parallel machines with setup costs

We consider a class of production scheduling models with m identical machines in parallel and k different product types. It takes a time p_i to produce one unit of product type i on any one of the machines. There is a demand stream for product type i consisting of n_i units with each unit having a given due date. Before a machine starts with the production of a batch of products of type i a setup cost c is incurred. We consider several different objective functions. Each one of the objective functions has three components, namely a total setup cost, a total earliness cost, and a total tardiness cost. In our class of problems we find a relatively large number of problems that can be solved either in polynomial time or in pseudo‐polynomial time. The polynomiality or pseudo‐polynomiality is achieved under certain special conditions that may be of practical interest; for example, a regularity pattern in the string of due dates combined with earliness and tardiness costs that are similar for different types of products. The class of models we consider includes as special cases discrete counterparts of a number of inventory models that have been considered in the literature before, e.g., Wagner and Whitin (Manage Sci 5 (1958), 89–96) and Zangwill (Oper Res 14 (1966), 486–507; Manage Sci 15 (1969), 506–527). © 2008 Wiley Periodicals, Inc. Naval Research Logistics, 2008