An analysis of single item inventory systems with returns

Inventory systems with returns are systems in which there are units returned in a repairable state, as well as demands for units in a serviceable state, where the return and demand processes are independent. We begin by examining the control of a single item at a single location in which the stationary return rate is less than the stationary demand rate. This necessitates an occasional procurement of units from an outside source. We present a cost model of this system, which we assume is managed under a continuous review procurement policy, and develop a solution method for finding the policy parameter values. The key to the analysis is the use of a normally distributed random variable to approximate the steady-state distribution of net inventory. Next, we study a single item, two echelon system in which a warehouse (the upper echelon) supports N(N ? 1) retailers (the lower echelon). In this case, customers return units in a repairable state as well as demand units in a serviceable state at the retailer level only. We assume the constant system return rate is less than the constant system demand rate so that a procurement is required at certain times from an outside supplier. We develop a cost model of this two echelon system assuming that each location follows a continuous review procurement policy. We also present an algorithm for finding the policy parameter values at each location that is based on the method used to solve the single location problem.     

Optimal ordering policies when anticipating parameter changes in EOQ systems

The classical Economic Order Quantity Model requires the parameters of the model to be constant. Some EOQ models allow a single parameter to change with time. We consider EOQ systems in which one or more of the cost or demand parameters will change at some time in the future. The system we examine has two distinct advantages over previous models. One obvious advantage is that a change in any of the costs is likely to affect the demand rate and we allow for this. The second advantage is that often, the times that prices will rise are fairly well known by announcement or previous experience. We present the optimal ordering policy for these inventory systems with anticipated changes and a simple method for computing the optimal policy. For cases where the changes are in the distant future we present a myopic policy that yields costs which are near-optimal. In cases where the changes will occur in the relatively near future the optimal policy is significantly better than the myopic policy.     

Three ways of obtaining the average cost expression in a problem related to joint replenishment inventory control

This paper does not present a new result, rather it is meant to illustrate the choice of modelling procedures available to an analyst in a typical inventory control problem. The same “average cost per unit time” expression is developed by three quite different procedures. This variety of approaches, as well as the recounting of the author's chronological efforts to solve the problem, should be of interest to the reader. The specific inventory problem studied is one where the controller of an item is faced with random opportunities for replenishment at a reduced setup cost; the problem is an integral component of the broader problem of inventory control of a group of items whose replenishments are coordinated to reduce the costs of production, procurement, and/or transportation.     

An explicit general solution in linear fractional programming

A complete analysis and explicit solution is presented for the problem of linear fractional programming with interval programming constraints whose matrix is of full row rank. The analysis proceeds by simple transformation to canonical form, exploitation of the Farkas-Minkowki lemma and the duality relationships which emerge from the Charnes-Cooper linear programming equivalent for general linear fractional programming. The formulations as well as the proofs and the transformations provided by our general linear fractional programming theory are here employed to provide a substantial simplification for this class of cases. The augmentation developing the explicit solution is presented, for clarity, in an algorithmic format.     

On the theory of semi-infinite programming and a generalization of the kuhn-tucker saddle point theorem for arbitrary convex functions

We first present a survey on the theory of semi-infinite programming as a generalization of linear programming and convex duality theory. By the pairing of a finite dimensional vector space over an arbitrarily ordered field with a generalized finite sequence space, the major theorems of linear programming are generalized. When applied to Euclidean spaces, semi-infinite programming theory yields a dual theorem associating as dual problems minimization of an arbitrary convex function over an arbitrary convex set in n-space with maximization of a linear function in non-negative variables of a generalized finite sequence space subject to a finite system of linear equations. We then present a new generalization of the Kuhn-Tucker saddle-point equivalence theorem for arbitrary convex functions in n-space where differentiability is no longer assumed.     

A cost-benefit analysis of military aircraft replacement policies

This paper describes a method of solving aircraft service life problems. The particular application concerns aircraft in the Naval Advanced Jet Training Command. The method of solution is comparative present value analysis of alternative replacement policies. The likely risks of estimation errors are reflected in the comparisons of present values. Differences are noted in the benefits associated with each policy, but external to Naval Aviation. Since the values of these benefits can be determined only at a higher level of decision-making, the result of the study is not a conclusive selection among policies, but a schedule of present values on the basis of which, together with values of the external benefits, a decision can be reached. This paper discusses replacement policies for aircraft used in the Naval Advanced Jet Pilot Training mission. Taking engineering technology and the training syllabus as given, four feasible plans for introducing replacement aircraft into service are evaluated in terms of the present values of differential costs associated with the plans and in terms of the likely errors in cost estimates used in calculation of the present values. The trade-off between present value of costs and planning flexibility is emphasized in choosing a recommended time pattern of aircraft replacement. The specific aircraft mixes considered are the TF–9J/TAF–9J and the TA–4F/A–4B. The first is the currently employed mix; the second is the proposed replacement. The problem is to select an optimal time-pattern of replacement of F–9's by A–4's, given technological differences favoring the A–4 and increasing costs of maintaining squadrons of F–9's. Replacements by aircraft types other than the A–4 are considered impractical. Four feasible plans for introducing A–4's through a 5-year period are evaluated in terms of current best estimates of the related costs of the plans and in terms of the flexibility of modifying each plan given future better information concerning the relevant costs. The method of analysis is comparative present value of expected costs.     

A characterization of the wald distribution

Suppose X is a random variable having an absolutely continuous distribution function F(x). We assume that F(x) has the Wald distribution. A relation between the probability density function of X⁻¹ with that of X is used to characterize the Wald distribution.     

The capital supply curve in capacity expansion models: Some economic and algorithmic aspects

Capacity expansion models typically minimize the discounted cost of acquisition and operation over a given planning horizon. In this article we generalize this idea to one in which a capital supply curve replaces the usual discount rate. A capital supply curve is a means to model financial outlook, investment limits, and risk. We show that when such a curve is included in a capacity expansion model, it will, under certain conditions, provide a less capital intensive solution than one which incorporates a discount rate. In this article, we also provide an algorithm that solves capacity expansion models that incorporate a capital supply curve. The attractive feature of this algorithm is that it provides a means to utilize the “discount rate” models efficiently. Throughout, we give applications in power generation planning and computational experience for this application is also presented.     

Design and estimation for small sample sensitivity testing

In sensitivity testing for the Department of Defense, the high cost of experimental units necessitates the use of small sample sizes and accentuates the importance of design. This article compares five data collection-estimation procedures. Four of these are modifications of the Robbins-Monro method, and the other is the Langlie. The simulation study is designed as a factorial experiment with response function, sample size, initial design point, gate width, and noise as factors. The estimated V₅₀ and its MSE are the responses compared to assess the small sample behavior of each method. Although there is no single clear-cut winner, the Delayed Robbins-Monro (DRM) with maximum likelihood estimation and the Estimated Quantal Response Curve (Wu [21]) are shown to perform well over a broad variety of conditions.