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Article Excerpt
1. Introduction
Stochastic production control, characterized by uncertain demand, product defects and machine failure, is typically considered the prerogative of closed-loop or on-line approaches (see, for example, the pioneering work of Kimemia (1982), Kimemia and Gershwin (1983). Ghosh et al. (1993) and Akella and Kumar (1986)). The goal is to find an optimal production rate and inventory policy, which will offset for the uncertainty along a production horizon. Since the inventory is stochastic, the optimal production rate, which minimizes the expected inventory holding and backlog costs, is usually a function of the inventory realized and thereby accounts for each possible scenario (inventory realization) separately. The approach is thus based on the assumption that the inventory levels and machine states are observable.
Unfortunately, in certain manufacturing systems, information about either machine states or inventory levels may at best be imprecise, if not unobtainable. For example, in chip fabricating facilities, yield or machine breakdowns are due to complex causes which are difficult to identify. Since the part inspection, at a much later stage of the production, will eventually unveil the culprits, the system, like many modern ones, could continue producing at the same rate even when there has been a malfunction. Moreover, the production of such a manufacturing system, like many other classical. Computer Numerically Controlled (CNC)-based systems, may be periodically halted (i.e., bounce between a full load and no load at all), but not adjusted instantaneously to follow intermediate production rates in response to inventory updates. As a result, only a gradual build-up or adjustment of the production capacity will improve the efficiency of such a system.
This reality warrants the exploration of an open-loop or off-line control methodology, which provides better system management when the above-mentioned information is either absent or impossible to utilize. This implies that only one control function is chosen from the very beginning of the production horizon, which optimizes the expected cost over all possible scenarios.
In this paper we address both approaches with the goal of analyzing the consequences of applying an open-loop control instead of its closed-loop counterpart under uncertain demands and yields.
The incorporation of random demand and yield into manufacturing system models has been of interest since the work of Karlin (1958). Since then, many authors have considered such problems in various forms. Yano and Lee (1995) provided a comprehensive review of the existing literature. Based on the system modeling characteristics, they arranged random yield lot sizing problems into the following categories: discrete-time models, which include single-stage models (both single and multi-period); multiple stages in tandem: assembly systems; and continuous-time models with constant demand rates or random demand rates.
Subsequently, more generic models have been studied including such extensions as uncertain supply, backlogged demand, imperfect production and late-stage inspection. Liu and Yang (1996) considered multiple defect types (re-workable and non-reworkable defects) and determined the optimal lot size. Bollapragada and Morton (1999) used myopic heuristics for the random yield problem and obtained promising results. Yao and Zheng (1999) studied a two-slage problem in which, in order to coordinate the inspection procedures at the (two stages, the optimal policy is characterized by a sequence of thresholds at stage I and by a priority structure at stage 2. For an assembly system under random demand and production yield, Gurnani (2000) circumvented the difficulty of solving the original problem by modifying the exact cost function with an approximate one and determined a bound on the difference. Grosfeld-Nir el al. (2000) included inspection costs as a key part of the problem in a general multiple production run model.
In this paper we utilize Wiener increment-based modeling of random production yields and demands (see, for example. Tapiero (1988), Ghosh et al. (1993) and Haurie (1995) and for similar stochastic production models, while earlier examples of stochastic demand models based on Brownian motion theory and specifically the Wiener diffusion process can be found in Freidenfelds (1980, 1981)). Specifically, we extend the Wiener increment-based model presented in Kogan and Lou (2005) and treat it with an open-loop approach to incorporate: (i) both uncertain yield and demand; (ii) production cost in addition to the inventory cost; (iii) an infinite production horizon and thereby a steady-state analysis; and (iv) both closed-loop and open-loop approaches.
We derive analytical solutions along with the probability distributions of the inventories for both approaches. Accordingly, a steady state is determined, which implies that the open-loop production does not require any adjustment. This is unlike feedback control, which is stochastic and therefore requires adjustments at a steady state as well (only the expectation of the feedback control remains unchanged at a steady state). We show that the consequences of the inability to instantaneously adjust production and thereby adopt a feedback policy are quite significant in terms of expected costs and inventory policy and depend on the type of uncertainty (random shocks) with which the system is faced. The gap. however, narrows when the production system is either underloaded (incurs high production and/or low backlog costs) or overloaded (incurs low operational costs). Moreover, the relative difference in the expected costs may reduce to a few percentage points when there is a high cost of idle production facilities, as is typically the case in the pharmaceutical industry and bio-manufacturing. In such a case, the open-loop production rate as well as its expected inventory level tends to the expected closed-loop production rate and its inventory level respectively.
2. Problem formulation
Consider a machine producing a single product type at time t at a rate, u(t), in response to stochastic demand. The machine is characterized by a random production...
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