Return policies for an inventory system with positive and negative demands

We consider a single item inventory system with positive and negative stock fluctuations. Items can be purchased from a central stock, n items can be returned for a cost R + rn, and a linear inventory carrying cost is charged. It is shown that for minimizing the asymptotic cost rate when returns are a significant fraction of stock usage, a two-critical-number policy (a,b) is optimal, where b is the trigger level for returns and b – a is the return quantity. The values for a and b are found, as well as the operating characteristics of the system. We also consider the optimal return decision to make at time zero and show that it is partially determined by a and b.     

Finding the Kth shortest path in a time‐schedule network

We consider the problem of finding the Kth shortest path for a time‐schedule network, where each node in the network has a list of prespecified departure times, and departure from the node can take place only at one of these departure times. We develop a polynomial time algorithm independent of K for finding the Kth shortest path. The proposed algorithm constructs a map structure at each node in the network, using which we can directly find the Kth shortest path without having to enumerate the first K − 1 paths. Since the same map structure is used for different K values, it is not necessary to reconstruct the table for additional paths. Consequently, the algorithm is suitable for directly finding multiple shortest paths in the same network. Furthermore, the algorithm is modified slightly for enumerating the first K shortest paths and is shown to have the lowest possible time complexity under a condition that holds for most practical networks. © 2004 Wiley Periodicals, Inc. Naval Research Logistics, 2005.     

Multiple-attribute decision making with partial information: The comparative hypervolume criterion

A new approach is presented for analyzing multiple-attribute decision problems in which the set of actions is finite and the utility function is additive. The problem can be resolved if the decision makers (or group of decision makers) specifies a set of nonnegative weights for the various attributes or criteria, but we here assume that the decision maker(s) cannot provide a numerical value for each such weight. Ordinal information about these weights is therefore obtained from the decision maker(s), and this information is translated into a set of linear constraints which restrict the values of the weights. These constraints are then used to construct a polytope W of feasible weight vectors, and the subsets H_i (polytopes) of W over which each action a_i has the greatest utility are determined. With the Comparative Hypervolume Criterion we calculate for each action the ratio of the hypervolume of H_i to the hypervolume of W and suggest the choice of an action with the largest such ratio. Justification of this choice criterion is given, and a computational method for accurately approximating the hypervolume ratios is described. A simple example is provided to evaluate the efficiency of a computer code developed to implement the method.     

An approximate dynamic programing approach to the development of heuristics for the scheduling of impatient jobs in a clearing system

A single server is faced with a collection of jobs of varying duration and urgency. Each job has a random lifetime during which it is available for nonpreemptive service. Should a job's lifetime expire before its service begins then it is lost from the system unserved. The goal is to schedule the jobs for service to maximize the expected number served to completion. Two heuristics have been proposed in the literature. One (labeled π^S) operates a static priority among the job classes and works well in a “no premature job loss” limit, whereas the second (π^M) is a myopic heuristic which works well when lifetimes are short. Both can exhibit poor performance for problems at some distance from the regimes for which they were designed. We develop a robustly good heuristic by an approximative approach to the application of a policy improvement step to the asymptotically optimal heuristic π^S, in which we use a fluid model to obtain an approximation for the value function of π^S. The performance of the proposed heuristic is investigated in an extensive numerical study. © 2010 Wiley Periodicals, Inc. Naval Research Logistics 2010     

Parallel machine scheduling: A probabilistic analysis

The minimum makespan of the general parallel machine scheduling problem with m machines and n jobs is studied. As for a number of other important combinatorial problems, the theory of empirical processes proves to be a very elegant and powerful tool for the probabilistic analysis of the solution value. It is used in this paper to derive a scheduling constant θ such that, for random processing times, the minimum makespan almost surely grows as θn when n goes to infinity. Moreover, a thorough probabilistic analysis is performed on the difference between the minimum makespan and θn. Explicit expressions for the scheduling constant are given for an arbitrary number of unrelated machines with identically distributed processing times (with an increasing failure rate), and for an arbitrary number of uniform machines and generally distributed processing times. © 1996 John Wiley & Sons, Inc.     

Optimal policies for M/M/m queue with two different kinds of (N,T)‐policies

In this paper, two different kinds of (N, T)‐policies for an M/M/m queueing system are studied. The system operates only intermittently and is shut down when no customers are present any more. A fixed setup cost of K > 0 is incurred each time the system is reopened. Also, a holding cost of h > 0 per unit time is incurred for each customer present. The two (N, T)‐policies studied for this queueing system with cost structures are as follows: (1) The system is reactivated as soon as N customers are present or the waiting time of the leading customer reaches a predefined time T, and (2) the system is reactivated as soon as N customers are present or the time units after the end of the last busy period reaches a predefined time T. The equations satisfied by the optimal policy (N*, T*) for minimizing the long‐run average cost per unit time in both cases are obtained. Particularly, we obtain the explicit optimal joint policy (N*, T*) and optimal objective value for the case of a single server, the explicit optimal policy N* and optimal objective value for the case of multiple servers when only predefined customers number N is measured, and the explicit optimal policy T* and optimal objective value for the case of multiple servers when only predefined time units T is measured, respectively. These results partly extend (1) the classic N or T policy to a more practical (N, T)‐policy and (2) the conclusions obtained for single server system to a system consisting of m (m ≥ 1) servers. © 2000 John Wiley & Sons, Inc. Naval Research Logistics 47: 240–258, 2000     

Variance of the number of units produced on a transfer line with buffer inventories during a period of length T

The transfer-line models in the literature are planning models rather than operational models. That is, they are very useful for planning or designing the transfer line, but are less useful for controlling the daily operation of the line. The performance measure used in these models is the efficiency of the line A. The expected number of units produced during a period of length T cycles is AT. In this article a procedure is presented for calculating the variance of the number of units produced by the transfer line during a period of length T cycles. These two performance measures can be used to construct an interval estimate for, say, the number of units produced during a shift. This interval estimate is an operational guide for the production manager.     

Optimal replacement policies with two or more loaded sliding standbys

This article defines optimal replacement policies for identical components performing different functions in a given system, when more than one spare part is available. The problem is first formulated for two components and any number of spare parts and the optimal replacement time y(x) at time x is found to have a certain form. Sufficient conditions are then provided for y(x) to be a constant y* for x > y*, and y(x) = x for x > y* (single-critical-number policy). Under the assumption that the optimal policies are of the single-critical-number type, the results are extended to the n-component case, and a theorem is provided that reduces the required number of critical numbers. Finally, the theory is applied to the case of the exponential and uniform failure laws, in which single-critical-number policies are optimal, and to another failure law in which they are not.     

Multiple resource leveling in construction systems through variation of activity intensities

The resource leveling problem for a construction system producing a stream of output units is considered. The system is modeled using a critical-path-analysis activity network, from which an extended network is developed for an integrated planning effort of all output units. Activity intensity variables are defined which measure activity demand rates for resources and consequent activity durations for the production of each output unit. A heuristic approach consisting of an iterative nonlinear programming procedure is presented which computes activity durations (intensities) for the minimization of resource capacity costs subject to meeting construction due dates. The application to a major ship overhaul project is described, in which the procedure was used to level workloads of the various labor–trade shops.     

Quasi-separable utility functions

This paper is concerned with a method for the assessment of utility functions of multi-numeraire consequences. It is proven that given von Neumann and Morgenstern's axioms of “rational behavior” and two additional assumptions, the utility function for (x, y) consequences can be written as U(x, y) = U_x(x) + U_y(y) + KU_x(x) U_y(y). K is a constant that must be evaluated empirically. This form shall be designated as a quasi-separable utility function. It is more general than the separable utility function and is shown to be nearly as easy to use. Implications and ramifications of such a utility function and its requisite assumptions are discussed. A technique for practical application of this work is presented.     

Optimal policies for a multi-product inventory system with negotiable lead times

The objective of this paper is to determine the optimum inventory policy for a multi-product periodic review dynamic inventory system. At the beginning of each period two decisions are made for each product. How much to “normal order” with a lead time of λ_n periods and how much to “emergency order” with a lead time of λ_e periods, where λe = λ_n - 1. It is assumed that the emergency ordering costs are higher than the normal ordering costs. The demands for each product in successive periods are assumed to form a sequence of independent identically distributed random variables with known densities. Demands for individual products within a period are assumed to be non-negative, but they need not be independent. Whenever demand exceeds inventory their difference is backlogged rather than lost. The ordering decisions are based on certain costs and two revenue functions. Namely, the procurement costs which are assumed to be linear for both methods of ordering, convex holding and penalty costs, concave salvage gain functions, and linear credit functions. There is a restriction on the total amount that can be emergency ordered for all products. The optimal ordering policy is determined for the one and N-period models.     

An assumption-free convergence analysis for a perturbation of the scaling algorithm for linear programs,with application to the L1 estimation problem

This article is concerned with the scaling variant of Karmarkar's algorithm for linear programming problems. Several researchers have presented convergence analyses for this algorithm under various nondegeneracy types of assumptions, or under assumptions regarding the nature of the sequence of iterates generated by the algorithm. By employing a slight perturbation of the algorithm, which is computationally imperceptible, we are able to prove without using any special assumptions that the algorithm converges finitely to an ε-optimal solution for any chosen ε > 0, from which it can be (polynomically) rounded to an optimum, for ε > 0 small enough. The logarithmic barrier function is used as a construct for this analysis. A rounding scheme which produces an optimal extreme point solution is also suggested. Besides the non-negatively constrained case, we also present a convergence analysis for the case of bounded variables. An application in statistics to the L₁ estimation problem and related computational results are presented.     

Stochastic control of queueing systems

Suppose that the state of a queueing system is described by a Markov process { Y_t, t ≥ 0}, and the profit from operating it up to a time t is given by the function f(Y_t). We operate the system up to a time T, where the random variable T is a stopping time for the process Yt. Optimal stochastic control is achieved by choosing the stopping time T that maximizes Ef(Y_T) over a given class of stopping times. In this paper a theory of stochastic control is developed for a single server queue with Poisson arrivals and general service times.     

A note on the costly surveillance of a stochastic system

We consider the costly surveillance of a stochastic system with a finite state space and a finite number of actions in each state. There is a positive cost of observing the system and the system earns at a rate depending on the state of the system and the action taken. A policy for controlling such a system specifies the action to be taken and the time to the next observation, both possibly random and depending on the past history of the system. A form of the long range average income is the criterion for comparing different policies. If R ^Δ denotes the class of policies for which the times between successive observations of the system are random variables with cumulative distribution functions on [0, Δ], Δ < ∞, we show that there exists a nonrandomized stationary policy that is optimal in R ^Δ. Furthermore, for sufficiently large Δ, this optimal policy is independent of Δ.     

A class of Bayes-optimal two-stage screens

Items are characterized by a set of attributes (T) and a collection of covariates (X) associated with those attributes. We wish to screen for acceptable items (T ∈ C_T), but T is expensive to measure. We envisage a two-stage screen in which observation of X_ is used as a filter at the first stage to sentence most items. The second stage involves the observation of T for those items for which the first stage is indecisive. We adopt a Bayes decision-theoretic approach to the development of optimal two-stage screens within a general framework for costs and stochastic structure. We also consider the important question of how much screens need to be modified in the light of resource limitations that bound the proportion of items that can be passed to the second stage. © 1996 John Wiley & Sons, Inc.     

Estimating the lead-time demand distribution for an autocorrelated demand by the pearson system and a normal approximation

The exact first four moments of lead-time demand L are derived for an AR(1) and a MA(1) demand structures where the arbitrary lead-time distribution is assumed to be independent of the demand structure. These moments then form a basis for the Pearson curve-fitting procedure for estimating the distribution of L. A normal approximation to L, a version of the central limit theorem, is obtained under some general conditions. Reorder points (ROPs) of an inventory system are then estimated based on the Pearson system and a normal approximation. Their performances are evaluated. Numerical investigation shows that the Pearson system performs extremely well. The normal approximation, however, is good only for some limited cases, and is sensitive to the choice of the lead-time distribution. A possible improvement is noted.     

A note on testing for exponentiality

In this note some extensions are made to previous work by a number of authors on the development of tests for exponentiality. The most recent example is due to Fercho and Ringer in which they compare the small sample powers of a few well-known test statistics for the hypothesis of a constant failure rate. It is the primary intent of this current work to extend Gnedenko's F test to situations with hypercensoring and to provide guidance for its use, particularly when a log-normal distribution is the alternative.     

Optimal control of entry to an M/Ek/1 queue serving several classes of customers

The individual and social optimum control policies for entry to an M/M//1 queue serving several classes of customers have been shown to be control-limit policies. The technique of policy iteration provides the social optimum policy for such a queue in a straightforward manner. In this article, the problem of finding the optimal control policy for the M/E_k/1 system is solved, thereby expanding the potential applicability of the solutions developed. The Markovian nature of the queueing system is preserved by considering the service as having k sequential phases, each with independent, identically distributed, exponential service times, through which a customer must pass to be serviced. The optimal policy derived by policy iteration for such a system is likely to be difficult to use because it requires knowledge of the number of phases rather than customers in the system when an arrival occurs. To circumvent this difficulty, a heuristic is used to find a good usable (implementable) solution. In addition, a mixed-integer program is developed which yields the optimal implementable solution when solved.     

The “signature” of a coherent system and its application to comparisons among systems

Various methods and criteria for comparing coherent systems are discussed. Theoretical results are derived for comparing systems of a given order when components are assumed to have independent and identically distributed lifetimes. All comparisons rely on the representation of a system's lifetime distribution as a function of the system's “signature,” that is, as a function of the vector p= (p₁, … , p_n), where p_i is the probability that the system fails upon the occurrence of the ith component failure. Sufficient conditions are provided for the lifetime of one system to be larger than that of another system in three different senses: stochastic ordering, hazard rate ordering, and likelihood ratio ordering. Further, a new preservation theorem for hazard rate ordering is established. In the final section, the notion of system signature is used to examine a recently published conjecture regarding componentwise and systemwise redundancy. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 507–523, 1999     

An optimum t-test for the scale parameter of an extreme-value distribution

A Student's t-test proposed by Ogawa is considered for the hypothesis Ho: σ=σo against the alternative hypothesis H₁: σ ≠ σo, where σ is the scale parameter of the Extremevalue distribution of smallest values with known location parameter μ. The test is based on a few sample quantiles chosen from a large sample so as to give asymptotically maximum power to the test when the number of sample quantiles is fixed. A table which facilitates the computation of the test statistic is given. Several schemes for determining the ranks of the sample quantiles by the optimal spacings are compared and the effect of the bias of the estimate of σ on the test is investigated through a Monte Carlo study.