Expected time delay in multi‐item inventory systems with correlated demands

This paper considers multi‐item inventory systems where a customer order may require several different items (i.e., demands are correlated across items) and customer satisfaction is measured by the time delays seen by the customers. Most inventory models on time delay in the literature assume each demand only requires one item (i.e., demands are not correlated across items or are independent). In this paper, we derive an exact expression for the expected total time delay. We show that when items are actually correlated, assuming items are independent leads to an overestimate of the total time delay. However, (1) it is extremely difficult in practice to obtain the demand information for all demand types (especially in a system with tens of thousands of part numbers), and (2) the problem becomes too complicated to be of practical interest when the correlation is considered. We then explore the possibility of including the demand information partially and develop bounds for the time delays. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 671–688, 1999     

Measures of component importance for markov chain imbeddable reliability structures

In this article we study the reliability importance of the components for the wide class of Markov chain imbeddable systems (MIS). Methods for the evaluation of Birnbaum importance are developed for a general MIS, and some generating function techniques are demonstrated for the special case of homogeneous MISs. As an application, the reliability ordering for the components of a k‐out‐of‐n and consecutive‐k‐out‐of‐n structure is examined in some detail. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 613–639, 1999     

Northwest corner and banded matrix approximations to a Markov chain

In this paper, we consider approximations to discrete time Markov chains with countably infinite state spaces. We provide a simple, direct proof for the convergence of certain probabilistic quantities when one uses a northwest corner or a banded matrix approximation to the original probability transition matrix. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 187–197, 1999     

Planning accelerated life tests for censored two‐parameter exponential distributions

We consider optimal test plans involving life distributions with failure‐free life, i.e., where there is an unknown threshold parameter below which no failure will occur. These distributions do not satisfy the regularity conditions and thus the usual approach of using the Fisher information matrix to obtain an optimal accelerated life testing (ALT) plan cannot be applied. In this paper, we assume that lifetime follows a two‐parameter exponential distribution and the stress‐life relationship is given by the inverse power law model. Near‐optimal test plans for constant‐stress ALT under both failure‐censoring and time‐censoring are obtained. We first obtain unbiased estimates for the parameters and give the approximate variance of these estimates for both failure‐censored and time‐censored data. Using these results, the variance for the approximate unbiased estimate of a percentile at a design stress is computed and then minimized to produce the near‐optimal plan. Finally, a numerical example is presented together with simulation results to study the accuracy of the approximate variance given by the proposed plan and show that it outperforms the equal‐allocation plan. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 169–186, 1999     

The impact of inventory information distortion due to customer order cancellations

In this paper we study the impact of cancellations of customer orders on an inventory system. We develop a periodic review (s, S) inventory model with Poisson demands, deterministic demand leadtimes and supply leadtimes. When no set up cost is present for replenishment, the behavior of the system cost can be studied analytically. For the case with a fixed set up cost, we derive the operating characteristics of the model via an embedded Markov chain analysis. Based on this, we formulate the total cost function and suggest a two‐phase approach to optimization. Our model can be used to compute cancellation fees and to evaluate the impacts of various conditions of cancellation. We find that cancellations, as major sources of inventory information distortion, increase total system costs, and the magnitude of the cost impact depends on the probability of cancellation and the expected cancellation time. Other relevant lessons and insights are also discussed. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 213–231, 1999     

Incorporating learning curve costs in acquisition strategy optimization

Each year, the U.S. Army procures billions of dollars worth of weapons and equipment. The process of deciding what to buy, when to buy, and in what quantities is extremely complex, requiring extensive analysis. Two techniques used in this analysis are mathematical programming and cost estimation. Although they are related through constraints on available procurement funds, the use of nonlinear cost learning curves, which better represent system costs as a function of quantity produced, have not been incorporated into the mathematical programming formulations that compute the quantities of items to be procured. As a result, the solutions obtained could be either suboptimal, or even infeasible with respect to budgetary limitations. In this paper we present a piecewise linear approximation of the learning curve costs for a more accurate portrayal of budgetary constraints used in a mixed integer linear programming for acquisition strategy optimization. In addition, implementation issues are discussed, and performance results are given. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 255–271, 1999     

A general model for accelerated life testing with time‐dependent covariates

This paper introduces a general or “distribution‐free” model to analyze the lifetime of components under accelerated life testing. Unlike the accelerated failure time (AFT) models, the proposed model shares the advantage of being “distribution‐free” with the proportional hazard (PH) model and overcomes the deficiency of the PH model not allowing survival curves corresponding to different values of a covariate to cross. In this research, we extend and modify the extended hazard regression (EHR) model using the partial likelihood function to analyze failure data with time‐dependent covariates. The new model can be easily adopted to create an accelerated life testing model with different types of stress loading. For example, stress loading in accelerated life testing can be a step function, cyclic, or linear function with time. These types of stress loadings reduce the testing time and increase the number of failures of components under test. The proposed EHR model with time‐dependent covariates which incorporates multiple stress loadings requires further verification. Therefore, we conduct an accelerated life test in the laboratory by subjecting components to time‐dependent stresses, and we compare the reliability estimation based on the developed model with that obtained from experimental results. The combination of the theoretical development of the accelerated life testing model verified by laboratory experiments offers a unique perspective to reliability model building and verification. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 303–321, 1999     

Probabilistic analysis of an asymptotically optimal solution for the completion time variance problem

Scheduling a set of n jobs on a single machine so as to minimize the completion time variance is a well‐known NP‐hard problem. In this paper, we propose a sequence, which can be constructed in O(n log n) time, as a solution for the problem. Our primary concern is to establish the asymptotical optimality of the sequence within the framework of probabilistic analysis. Our main result is that, when the processing times are randomly and independently drawn from the same uniform distribution, the sequence is asymptotically optimal in the sense that its relative error converges to zero in probability as n increases. Other theoretical results are also derived, including: (i) When the processing times follow a symmetric structure, the problem has 2^{⌊(n−1)/2⌋} optimal sequences, which include our proposed sequence and other heuristic sequences suggested in the literature; and (ii) when these 2^{⌊(n−1)/2⌋} sequences are used as approximate solutions for a general problem, our proposed sequence yields the best approximation (in an average sense) while another sequence, which is commonly believed to be a good approximation in the literature, is interestingly the worst. © 1999 John Wiley & Sons, Inc. Naval Research Logistics 46: 373–398, 1999     

Efficient multinomial selection in simulation

Consider a simulation experiment consisting of v independent vector replications across k systems, where in any given replication one system is selected as the best performer (i.e., it wins). Each system has an unknown constant probability of winning in any replication and the numbers of wins for the individual systems follow a multinomial distribution. The classical multinomial selection procedure of Bechhofer, Elmaghraby, and Morse (Procedure BEM) prescribes a minimum number of replications, denoted as v*, so that the probability of correctly selecting the true best system (PCS) meets or exceeds a prespecified probability. Assuming that larger is better, Procedure BEM selects as best the system having the largest value of the performance measure in more replications than any other system. We use these same v* replications across k systems to form (v*)^k pseudoreplications that contain one observation from each system, and develop Procedure AVC (All Vector Comparisons) to achieve a higher PCS than with Procedure BEM. For specific small-sample cases and via a large-sample approximation we show that the PCS with Procedure AVC exceeds the PCS with Procedure BEM. We also show that with Procedure AVC we achieve a given PCS with a smaller v than the v* required with Procedure BEM. © 1998 John Wiley & Sons, Inc. Naval Research Logistics 45: 459–482, 1998