Optimal sequential replenishment of ships during combat

A carrier battle group is operating in an area where it is subject to attack by enemy aircraft. It is anticipated that air raids will occur in large waves. The uncertain time between raids is available for the replenishment of supplies. We consider the problem of how best to schedule ammunition replenishment during this period. The theory of Gittins indices provides the technical background to the development of a range of models which yield a hierarchy of index-based heuristics for replenishment. One such heuristic is assessed computationally in a more realistic scenario than is explicitly allowed for by the models.     

Inventory rotation policies for slow moving items

In this article we present a stochastic model for determining inventory rotation policies for a retail firm which must stock many hundreds of distinctive items having uncertain heterogeneous sales patterns. The model develops explicit decision rules for determining (1) the length of time that an item should remain in inventory before the decision is made on whether or not to rotate the item out of inventory and (2) the minimum sales level necessary for retaining the item in inventory. Two inventory rotation policies are developed, the first of which maximizes cumulative expected sales over a finite planning horizon and the second of which maximizes cumulative expected profit. We also consider the statistical behavior of items having uncertain, discrete, and heterogeneous sales patterns using a two-period prediction methodology where period 1 is used to accumulate information on individual sales rates and this knowledge is then used, in a Bayesian context, to make sales predictions for period 2. This methodology assumes that over an arbitrary time interval sales for each item are Poisson with unknown but stationary mean sales rates and the mean sales rates are distributed gamma across all items. We also report the application of the model to a retail firm which stocks many hundreds of distinctive unframed poster art titles. The application provides some useful insights into the behavior of the model as well as some interesting aspects pertaining to the implementation of the results in a “real-world” situation.     

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.     

Dynamic evacuation routes for personnel on a naval ship

We propose a dynamic escape route system for emergency evacuation of a naval ship. The system employs signals that adapt to the causative contingency and the crew's physical distribution about the ship. A mixed‐integer nonlinear programming model, with underlying network structure, optimizes the evacuation process. The network's nodes represent compartments, closures (e.g., doors and hatches) and intersections, while arcs represent various types of passageways. The objective function integrates two potentially conflicting factors: average evacuation time and the watertight and airtight integrity of the ship after evacuation. A heuristic solves the model approximately using a sequence of mixed‐integer linear approximating problems. Using data for a Spanish frigate, with standard static routes specified by the ship's designers, computational tests show that the dynamic system can reduce average evacuation times, nearly 23%, and can improve a combined measure of ship integrity by up to 50%. In addition, plausible design changes to the frigate yield further, substantial improvements. Published 2008 Wiley Periodicals, Inc. Naval Research Logistics 2008     

Robustness of the normal approximation of lead-time demand in a distribution setting

Previous studies criticize the general use of the normal approximation of lead-time demand on the grounds that it can lead to serious errors in safety stock. We reexamine this issue for the distribution of fast-moving finished goods. We first determine the optimal reorder points and quantities by using the classical normal-approximation method and a theoretically correct procedure. We then evaluate the misspecification error of the normal approximation solution with respect to safety stock, logistics-system costs, total costs (logistics costs, including acquisition costs), and fill rates. The results provide evidence that the normal approximation is robust with respect to both cost and service for seven major industry groups. © 1997 John Wiley & Sons, Inc. Naval Research Logistics 44: 165–186, 1997     

Nonpreemptive scheduling of independent tasks with prespecified processor allocations

In this article we study the problem of scheduling independent tasks, each of which requires the simultaneous availability of a set of prespecified processors, with the objective of minimizing the maximum completion time. We propose a graph-theoretical approach and identify a class of polynomial instances, corresponding to comparability graphs. We show that the scheduling problem is polynomially equivalent to the problem of extending a graph to a comparability graph whose maximum weighted clique has minimum weight. Using this formulation we show that in some cases it is possible to decompose the problem according to the canonical decomposition of the graph. Finally, a general solution procedure is given that includes a branch-and-bound algorithm for the solution of subproblems which can be neither decomposed nor solved in polynomial time. Some examples and computational results are presented. © 1994 John Wiley & Sons, Inc.     

A generalized model for the economic design of x̄control charts for production systems with increasing failure rate and early replacement

This article generalizes the model for the economic design of x̄-control charts of Duncan [4], starting from the more recent papers of Lorenzen and Vance [8] and Banerjee and Rahim [3]. The classical model of Duncan [4] and its several extensions including the unified model of Lorenzen and Vance [8] assumed exponentially distributed in-control periods and provided uniform sampling schemes. Banerjee and Rahim [3], however, assumed a Weibull-distributed in-control period having an increasing failure rate and used variable sampling intervals. The present article is an extension of the work of Banerjee and Rahim [3], where a general distribution of in-control periods having an increasing failure rate is assumed and the possibility of age-dependent repair before failure is considered. Several different truncated and nontruncated probability models are chosen. It is proposed that economic benefits can be achieved by adopting a nonuniform inspection scheme and by truncating a production cycle when it attains a certain age. Numerical examples are presented to support this proposition. Finally, the effect of model specification in the choice of failure mechanism is investigated. © 1993 John Wiley & Sons, Inc.     

An inventory model of incentives for on-time delivery in just-in-time purchasing contracts

This article formulates an analytic model of just-in-time purchasing contracts and compares the minimum cost solution with the cost attainable through vertical integration. The models use standard inventory theory cost parameters and decision variables. The results quantify the increase in cost of buying an item rather than making it. Optimal incentives are characterized when JIT purchasing contracts are used. When JIT purchasing is implemented, buffer inventories are typically reduced. This inventory reduction makes on-time delivery critical to the buyer; yet timeliness is controlled by the supplier. As an incentive to provide on-time delivery, the buyer offers the supplier a bonus for on-time delivery. The supplier chooses a flow time allowance based upon the bonus offered. First- and second-order conditions are characterized in general, and examples are provided for exponentially and uniformly distributed flow times. The delivery timeliness obtainable in a vertically integrated firm is determined and compared with timeliness obtainable between separate firms. This comparison indicates that buyers who choose to purchase materials from a separate firm are more likely to experience late deliveries. The relationship between the value of the bonus and the proportion of on-time deliveries is also considered. The bonus required to achieve the same probability of on-time delivery as under vertical integration is also determined. © 1993 John Wiley & Sons, Inc.     

On the bivariate negative binomial distribution of mitchell and paulson

The bivariate negative binomial distribution of Mitchell and Paulson [17] for the case b = c = 0 is shown to be equivalent to the accident proneness model of Edwards and Gurland [4] and Subrahmaniam [19,20]. The diagonal series expansion of its joint probability function is then derived. Two other formulations of this distribution are also considered: (i) as a mixture model, which showed how it arises as the discrete analogue to the Wicksell-Kibble bivariate gamma distribution, and (ii) as a consequence of the linear birth-and-death process with immigration.