Stochastics and Statistics
Performance analysis for assemble-to-order systems with general renewal arrivals and random batch demands

https://doi.org/10.1016/j.ejor.2007.01.009Get rights and content

Abstract

We study multi-product and multi-item assemble-to-order systems under general assumptions on demand patterns and replenish leadtime distributions. We only assume that the demand process of each product being a renewal process, and the replenish leadtimes follow general distributions. Based upon techniques from renewal theory, we developed procedures for approximating key performance measures of these inventory systems, such as average inventory and immediate order fill rate. We also obtain qualitative results that reveal the impacts of changes in demand patterns and leadtime variability upon the performance of the systems.

Introduction

Assemble-to-order is an important mathematical model of multi-item inventory systems in both practice and theory. The success of its applications can be found in various industries, such as computer manufacture (most notably Dell Computer), automobile and telecommunications. At the theoretical front, an assemble-to-order system is a natural generalization of much studied single-item inventory systems. Although a simple two-stage system, an assemble-to-order system has the essentially fundamental elements of a multi-item inventory system, namely, the computational complexity caused by multidimensional features and correlations between different items. The study of assemble-to-order systems, hence, has drawn considerable attentions in the operations management research community, see, e.g. Song and Zipkin [7] for a detailed survey and references for related works.

The system under consideration in this paper is a generalization of that in Lu et al. [5]. One important assumption in [5] is that the arrival process for each product type is an independent Poisson process. The memoryless property of Poisson processes, then, played an crucial role in obtaining the explicit probability generating function of the state process of the system, which is the queue length process of a parallel MX/G/∞ queues. However, in practical data, it is not realistic to expect to observe memorylessness and homogeneity that are required for this assumption. In this paper, we relax this assumption to allow the arrival processes to be general renewal processes. In this case, the state process of the system becomes the system size process of a parallel GX/G/∞ queue with correlated arrivals, hence, the resulting probability generating functions can not be obtained in closed form. We are able to derive the integral equations that the generating functions satisfy from which we derive the recursions for the moments of the state process. Similar results for single GX/G/∞ queue and its variations were derived in [4] and its sequels. Thus, we generalized their results into the case of correlated parallel queues, furthermore, we relax their assumptions on the random batch size. In many applications, we can observe the demand arrivals to have periodic behavior or dependence upon exterior factors, hence can be modeled as a Markov modulated processes. The methods developed in this paper can then be extended to this case as well, the resulting mathematics characterization is a system of integral equations, and similar techniques can be employed to obtain computational methods for calculating the statistics, such as moments, of the system process.

Immediate order fill rate and long run average inventory are two key performance measures for an assemble-to-order system, since they characterized its performance from customers and system perspectives respectively. Immediate order fill rate is the probability of an order can be fulfilled immediately, noted that partial fulfillment is not allowed, i.e. only when all the required components are available will an order be shipped. We will obtain approximations to both quantities through the analysis of the outstanding orders, i.e. the components that have been ordered but not yet replenished. Since base-stock policy has been adapted for component inventory, the component outstanding orders are the same as the system size of a parallel queueing system with correlated renewal batch arrivals and infinite servers. We obtained probabilistic characterization of the system size process in the form of its probability generating function, then derive recursive expressions for its factorial moments. The long run average inventory of each component can be expressed as a function of the first two moments. The immediate order fill rate is equivalent to the distribution of the so called system size before arrivals, i.e. the system size observed by the arrivals. We can obtain the first two moments of system size before arrivals in closed form by conditioning upon the time of the arrivals, the approximation of the fill rate then can be achieved through normal approximations. Numerical experiments indicate that the approximations are usually fairly satisfactory. The second component of this analysis is the dependence of the system performance upon the demand patterns and replenishment variability. We developed stochastic ordering results to demonstrate the general directions of changes of the system performance when the demand or the replenishment behavior change.

The rest of the paper is ordered as the following, in Section 2, we will give details of the mathematics model, and some necessary notations; in Section 3 we present our main results including the derivation of the probabilistic characterization of the state process and the computation of the performance measures; in Section 4, we will discuss qualitative results of the change of demand and leadtime through stochastic order techniques; numerical results are collected in Section 5; finally, the paper is concluded in Section 6.

Section snippets

Model description and notations

In the following, let us give the detailed model descriptions of our assemble-to-order system. We assume that there are m different components in the system, indexed by 1,2,,m. Components are the “items” in a multi-item system, so we will use the terms item and component interchangeably throughout the paper. Different types of product orders are denoted by the subsets of {1,2,,m}. For any K{1,2,,m}, a type-K order requires assembling of components only in K. The total number possible types

Characterization of the outstanding orders process

To derive the probabilistic characterization of random vector X(t), we will make use of techniques in the theory of integral equations and renewal theory. To elaborate our approach, let us consider a special case, where there is only one class of arrival. Without loss of generality, assume that the order requires every i{1,2,,m}. Therefore, for the ease of exposition, we omit the superscript for the job type identification. We assume that the batch size distribution has finite moment

Dependence structure of the outstanding orders distribution

In this section, we will establish some fundamental structural properties of the random variable of system size, in both transient and stationary cases, as a multivariate random variable. This is mainly achieved through a series of stochastic ordering results. These results include that of stochastic order, moment generating order, orthant order and positive dependence.

We focus on the impact of the service time and the interarrival time. Let us consider an alternative system that differs from

Numerical examples

We examine the effectiveness of our approximations through the following numerical example based on an inventory system of personal computers. An variation of the example was originally presented in [2]. In this example, there are six different types of computers, and 14 different components. Table 1 provides the list of all the components and mean and standard deviation of their leadtimes, which follows normal distribution, and component requirements for each type of products. We denote

Conclusions

Two very important performance measures for assemble-to-order systems, namely average inventory and immediate order fill rate, have been derived under very general assumptions. Naturally, it intrigues us to carry out optimization study for better designs and operations of assemble-to-order systems. However, these performance measures, even in the form of normal approximations, are still very complicated functions of the original statistics. This, as well as the multi-dimensional nature of the

Acknowledgements

The author would like to thank Roger Lederman of Brown University for bringing the problem of analyzing assemble-to-order systems with renewal arrival to his attention; he would also like to thank two referees for their comments and suggestions.

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