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This paper addresses the optimization of the continuous-flow model of a single-stage single-product manufacturing system with constant demand and transportation delay from the machine to the inventory. The machine is subject to either time-dependent or operation-dependent failures. The production is controlled by a hedging point policy. The goal is to determine the optimal hedging point, which minimizes the long-run average inventory holding and backlogging cost. Sample path analysis shows that the cost function is convex and sample gradient estimators are derived. A bi-section search algorithm based on simulation and sample gradients is proposed to determine the optimal hedging point.

Continuous-flow models have been widely used for optimal control and design of manufacturing systems. From the point of view of optimization, continuous parameter optimization is often simpler than discrete parameter optimization. Kimemia and Gershwin (1983) were the first to use continuous-flow model to address the production control of a failure-prone manufacturing system. They showed that the optimal production policy has a special structure called hedging point policy in which a nonnegative production surplus should be maintained at times of excess capacity in order to hedge against future capacity shortages caused by machine failures. This pioneer work has triggered many interests and a very rich literature is now available for flow control of failure-prone manufacturing systems. In this paper, we focus on simulated-based techniques. In a pioneer work, motivated by buffer allocation optimization problem in a production line, Ho et al. (1979) developed an efficient technique called perturbation analysis (PA). It enables one to compute the sensitivity of a performance measure with respect to some system parameters by a single simulation run. Ho and Cao (1991) developed an infinitesimal perturbation analysis (IPA) technique for the efficient computation of n-dimensional gradient vector of performance measure, J(θ), of a discrete event dynamic system (DEDS) with respect to its parameter vector θ (such as buffer size, inflow rate, service rate, etc.) using only one statistical experiment of the system. This is opposed to the traditional method of estimating sensitivity information such as dJ(θ)/dθdJ(θ)/dθ. IPA calculates directly the sample derivative dL(θ, ξ)/dθ using information on the nominal trajectory (θ, ξ) alone, where L denotes the sample performance measure and ξ, represents a vector of random variables. The basic idea is the following: if the perturbations introduced into the trajectory (θ, ξ) are sufficiently small, then the event sequence of the perturbed trajectory (θ+Δθ, ξ) remains unchanged from the nominal, i.e., the two trajectories are deterministically similar in the order of their event sequence. In this case, the derivative dL(θ, ξ)/dθ can be calculated easily (see also Glasserman, 1990). Yan (1995) investigated a manufacturing system with a failure-prone machine. He illustrated how to devise gradient estimators based on the observations in a single simulation run and then designed an iterative algorithm, a constant step-size stochastic approximation (SA) procedure for finding the optimal number of circulating kanbans that minimizes the long-run average cost. Caramanis and Liberopoulos (1992) applied IPA to calculate the gradient estimates for the single unreliable machine-multiple product system. Haurie et al. (1994) proposed an IPA-based SA algorithm for the parameter optimization problem of continuous-flow model of a failure-prone manufacturing system with multiple part types. Sufficient conditions for convergence of SA are given for single-machine system and they observed that it is difficult to extend the results to two-machine problem. For failure-prone tandem manufacturing systems, Yan et al. (1994) applied PA to obtain consistent gradient estimates. They estimated the optimal threshold values by using SA algorithm and proved its convergence to the optimal threshold values. Kushner and Vázequez-Abad (1996) generalized these results by weakening the conditions. Yan et al. (1999) studied the two-machine case of failure-prone tandem systems and derived an optimal buffer-control policy to minimize a long-run average cost function. Xie (2002a) considered continuous-flow transfer lines composed of two machines subject to time-dependent failures and separated by a buffer of finite capacity. He established a set of evolution equations that determines the continuous state variables, i.e., cumulative productions and the buffer level, at epochs of discrete events. Based on these evolution equations, he proved the concavity of the throughput rates of the machines and derived gradient estimators and proposed a single sample path optimization algorithm. Xie (2002b) extended this approach to the performance evaluation and optimization of failure-prone discrete-event system by using a fluid-stochastic-event graph model, which is a decision-free Petri net. Fu and Xie (2002) estimated the derivatives of the throughput rate with respect to buffer capacity for continuous-flow models of a transfer line comprising two machines separated by a buffer of finite capacity and subject to operation-dependant failures. They showed that IPA leads to biased gradient estimators and proposed smoothed PA estimators. Stochastic fluid models (SFMs) have recently been considered as an alternative paradigm to queuing networks for modeling and simulation of telecommunication networks (see Cassandras et al., 2002; Panayiotou and Cassandras, 2004; Wardi et al., 2002; Wardi and Melamed, 2001; Wardi and Riley, 2002; Yu and Cassandras, 2004). SFM networks offer two advantages over their queuing-networks: (i) they can be faster to simulate, (ii) they give unbiased IPA gradient estimators for a large number of networks configurations, queuing disciplines and performance functions. Cassandras et al. (2003) used SFMs for control and optimization (rather than performance analysis) of communication network nodes processing two classes of traffic with one being uncontrolled and the other subject to threshold-based buffer control. All these works to date have been limited to a single node SFM. Only one exception is that of Sun et al. (2003), which considered a SFM consisting of several single-class nodes in tandem and performed PA for the node queue contents and associated event times with respect to a threshold parameter at the first node. Continuous-flow models are natural models for process industry. They are also widely used for discrete manufacturing systems in the case of high production volume or in the investigation of the impact and strategies to cope with infrequent but important random events such as machine failures and demand changes. In such situations, it is cumbersome to track individual parts part by part either in performance evaluation or real-time flow control as it needed in a pure discrete flow model. The number of possible states is huge and is usually beyond reasonable limits, the number of events to consider in a simulation study is very large as a result of large number of “minor” events such as start and end of processing of each individual part. For such discrete manufacturing systems, continuous-flow models offer an interesting way to reduce the complexity inherent to discrete flow modeling by approximating the discrete material flows with continuous material flows and hence allowing us to focus on important events such as machine failures and demand fluctuations. Although continuous-flow models offer an interesting way to reduce the complexity inherent to discrete manufacturing, the advantages and disadvantages of existing continuous-flow models can be summarized as follows: (i) all continuous-flow models capture the production capacity constraints of manufacturing resources by taking into account the throughput rate of these resources. As a result, these models offer rather satisfactory throughput estimation of the manufacturing system under consideration (see David et al. (1990) for analytical results and Suri and Fu (1994) for experimental results); (ii) continuous-flow models are simple and easy to analyze. For example, optimal control policies are established for continuous-flow models in Bielecki and Kumar (1988) and sample path gradient estimators with respect to buffer capacities are derived in Caramanis and Liberopoulos (1992) and Xie, 2002a and Xie, 2002b for various systems. These results cannot be established for corresponding discrete flow models; (iii) most existing continuous-flow models assume instantaneous material flows and neglect the impact of possible delays such as production lead times and transportation delays which could have significant on the performances of manufacturing systems in both process industry and discrete manufacturing; (iv) most existing continuous-flow models approximate the production capacity of each machine by its average throughput rate and hence neglects the impact of process time variability. Explicit modeling of the processing time variability requires more sophisticated throughput rate models of manufacturing resources in continuous-flow models. This paper is the first step of our effort to investigate how to integrate delays in continuous-flow models while preserving the simplicity and analyticity. Many manufacturing processes have significant delays in the material flow, such delays occur in oven processes (e.g. semiconductor diffusion), drying processes and testing. These delays usually have great impact on performance measures such as customer response time and work-in-process. Unfortunately, most existing continuous-flow models do not take into account these delays. There are only two exceptions. Van Ryzin et al. (1991) explicitly considered the impact delays for optimal flow control of job shops in order to minimize the discount and infinite-horizon average cost. A heuristic control policy for a flow shop with delay is derived using theoretical arguments and approximations. They considered a system composed of a set of flexible unreliable machines and neglected setup changes. Mourani et al. (2005) extended the model of van Ryzin et al. (1991) and proposed a continuous Petri net model with delays for performance modeling and optimization of transfer lines. The new model is more realistic than classical continuous-flow models yet keeps the simplicity and analyticity of continuous-flow models. In this paper, we consider a continuous-flow failure-prone manufacturing system composed of one machine, producing a single product and with transportation delay. The machine is subject to failures with generally distributed times to failure (TBF) and times to repair (TTR). Material flow produced by the machine arrives at a finished good inventory after a fixed transportation delay. The demand arrives at a constant speed, is filled from the inventory if it is not empty and is backlogged otherwise. The production of the machine is controlled by a hedging point policy characterized by a single control parameter called hedging point h. The goal of this paper is to determine the optimal hedging point h*h* in order to minimize the average inventory holding and backlogging cost. No analytical expression is available for performance evaluation of such systems with either continuous-flow models or discrete ones. Instead, we use a simulation based optimization approach. Through thorough sample path analysis and by using IPA technique, the convexity of the cost function with respect to the hedging point is established and sample gradient estimators of the cost function with respect to the hedging point are derived. The sample path gradient estimators that can be evaluated along the simulation are then used to design a bi-section search algorithm for determining optimal hedging point. It is worth noticing that gradient estimation with respect to the hedging point is not possible for discrete flow model as the hedging point would be an integer. The rest of the paper is organized as follows. Section 2 introduces the continuous-flow model with delays, the hedging point control policy and cost function. Section 3 presents a thorough analysis of sample paths and derives sample path gradients of the cost function with respect to the hedging point of the control policy for both time-dependent and operation-dependent failures (ODF). The cost function is proved to be convex in the hedging point. Sample path gradients of the cost function are used in Section 4 to design a bi-section search algorithm for determination of optimal hedging point. Numerical results and comparison between time-dependent and operation-dependant failures are given in Section 5. We conclude in Section 6.

In this paper, we have considered the optimization of the continuous-flow model of a single-stage failure-prone manufacturing system with transportation delay and a constant demand. Machine is subject to either TDF or ODF. A hedging point policy is used to control the production. Analytical expression of the cost function is not available and a simulation based optimization approach is taken in this paper. Using IPA technique, sample gradient estimators of the cost function with respect to the hedging point are derived. The sample path gradient estimators that can be evaluated along the simulation are then used to design sample path optimization algorithm for determining optimal hedging point. An interesting future research direction is the extension of these results to more general manufacturing systems such as transfer lines and assembly/disassembly manufacturing systems with random demand. Existing continuous-flow models have the advantage of being simple and easy to analyze but delays that have significant impact on the performances of manufacturing systems cannot be taken into account in existing continuous-flow models. The continuous-flow models with delay considered in this paper are an attractive way for including delays in continuous-flow models while preserving the simplicity and analyticity of the continuous-flow models. More sophisticated sample path analysis and sample path optimization algorithms are needed for general continuous-flow models with delays. Another interesting research direction is the optimization of other control policies such as Base Stock policy and CONWIP policy using continuous-flow models with delays for general manufacturing and supply chain systems. The combination of continuous-flow models with delay and simulation based optimization seems an interesting approach for optimal setting of parameters of various control policies that are used to operate a manufacturing and supply chain system. Continuous-flow models with delays seem to be of the right modeling granularity for control policy optimization.