سرمایه گذاری بهینه در کاهش نصب در سیستم های تولید با موجودی WIP

A number of models have been proposed to predict optimal setup times, or optimal investment in setup reduction, in manufacturing cells. These have been based on the economic order quantity (EOQ) or economic production quantity (EPQ) model formulation, and have a common limitation in that they neglect work-in-process (WIP) inventories, which can be substantial in manufacturing systems. In this paper a new model is developed that predicts optimal production batch sizes and investments in setup reduction. This model is based on queuing theory, which permits it to estimate WIP levels as a function of the decisions variables, batch size and setup time. Optimal values for batch size and setup time are found analytically, even though the total cost model was shown to be strictly non-convex.

Setup time reduction in manufacturing operations is widely recognised to provide significant benefits in areas such as cost, agility and quality. Many techniques are available to improve setup times, such as revising setup procedures, modifying tooling for standardised locating and clamping, or introducing robotic change-over equipment, to name a few. Each technique will provide a certain level of benefit, and have an associated cost. Given that the goal of the firm is to select the setup reduction techniques that will minimize their overall costs, this paper develops an analytic model that can be used to predict optimal investments in setup reduction, while taking into account WIP inventory costs. Although arguments for setup time reduction have been made for some years now (e.g., Schonberger, 1982 and Hall, 1983), the issue of justifying investments and optimally allocating finite resources has to a large extent been glossed over. For instance, one of the primary manuals on setup reduction techniques (Shingo, 1985) advocates reducing setup times to less than 10 minutes, without offering reasons for this particular target, nor mentioning what target is appropriate in the admittedly significant number of cases where 10 minute setups cannot be achieved. The philosophy that organizations should strive for zero setup times is laudable, but may not be a pragmatic objective in and of itself. For example, a manufacturer of N products may achieve zero setup times by providing N parallel production systems, each dedicated to one product. Clearly, N does not need to be very large before the costs of achieving zero setup times will far outstrip any potential economic advantages. Some analytic models that deal with the problem of selecting appropriate levels of setup reduction investment are found in the literature. Probably the first is that due to Porteus (1985). This model extended the standard economic order quantity (EOQ) model to include the ordering cost (assumed to represent the setup cost in a manufacturing operation) as a decision variable in the model. By assuming a suitable functional relationship for setup time vs. cost, the optimal setup time and batch size that minimised total costs were determined. This model has been followed by a number of others incorporating various extensions. For example, Spence and Porteus (1987) found optimal setup reduction with a multi-product capacitated EOQ model, followed by similar models by Hwang et al., 1996 and Moon, 1994 and Banerjee et al. (1996). Dynamic lot sizing models with setup reduction have also been developed by Zangwill, 1987, Hong et al., 1996 and Mekler, 1993 and Diaby (1995). A variant of the EOQ model, the economic production quantity (EPQ) model, was adapted by Billington (1987) to include setup reduction as a decision variable, with variations of the EPQ model by Cheng, 1989, Kim et al., 1992, Kim and Arinze, 1992 and Trevino et al., 1993 and Sarker and Coates (1997). A fundamental limitation to these EOQ-based models, however, is that while they take finished goods cycle-stock inventories into account, they do not consider the costs of work-in-process (WIP) inventories. This is a significant omission as, for instance, Boucher (1984) has pointed out “observation of large-scale discrete parts production facilities will show that the largest component of inventory cost is work-in-process inventory”. Models which only capture finished goods inventory costs may not adequately represent manufacturing systems, and predictions made with such models may lead to sub-optimal or erroneous decisions. For instance, when using an EPQ-based model, Kim (1990, p. 85) found that the benefits of setup reduction decreased as the utilization of the manufacturing system increased. This finding runs counter to intuition, and can be explained by a property of the EPQ model: as system utilization increases, the EPQ model predicts that finished goods inventory will decrease, leading to less benefit from reducing setup time. WIP inventories, on the other hand, are known to increase as utilization increases (e.g., see Tijms, 1994). By not including WIP inventories, these existing models neglect one of the more significant cost justifications for setup reduction investment. Karmarkar (1987) developed a model of a manufacturing operation that captures WIP costs. This model is based upon the M/M/1 queue, and treated production batch sizes as an independent variable. Analogous to the EOQ model, Karmarkar showed that smaller batches led to higher ordering (or setup) costs while larger batches led to higher WIP holding costs (determined from batch waiting times in the M/M/1 queue). An optimal batch size could be determined which minimised the sum of setup and inventory costs. Hong (1995) used a deterministic model of the value of a product (based on a model by Groover, 1987) as it progressed through a manufacturing system to capture WIP costs. Both models assume that added value of a part increases linearly with its time in the system which is the sum of setup, operation and non-operation (i.e., waiting) times at each stage. Hong's model allowed WIP costs to be estimated, and minimization of total costs led to an expression for optimal lot size. Reduction of setup and non-operation times was shown to be important to reducing overall manufacturing lead times, but as the costs of bringing about reductions in setup and non-operation times were not considered, specific recommendations for reduction targets could not be made. While both Groover (1987) and Hong (1995) provide useful insights, their deterministic approach does not capture the non-linear impact of changes in setup times on WIP levels (and hence WIP costs) as completely as a queueing model (e.g. Karmarkar, 1987) does. Furthermore, they assume that an increase in non-operation (queueing) times represents an increase in added value, whereas we propose that a part's value does not increase while it is waiting in a queue prior to setup and operation time. Our paper incorporates both investment in setup reduction and queuing behaviour to predict WIP holding costs. This model is new in that it incorporates both an M/G/1 queuing model to predict WIP levels, and also determines the optimal investment in setup reduction. Section 2 of this paper describes the development of the analytic model of the physical production cell (i.e., predicting WIP levels as a function of batch size and setup times). In Section 3, an economic model is incorporated with the physical system model, such that total operating costs can be estimated as a function of investment in setup reduction. This model is then optimised in Section 4 and results are discussed in Section 5 with conclusions drawn in Section 6.

This paper describes the development of a new analytic model for the investment in setup time reduction in a manufacturing system. A major contribution of this model is that it captures the costs of work-in-progress (WIP) inventory, a major limitation of previous models in this area. Although the total relevant cost function for the manufacturing cell with WIP inventories was proved strictly non-convex, analytic expressions were developed for the batch size and setup time that minimise costs. As a consequence of the non-convex total cost function, two distinct optimal solutions were found. Either no investment should be made and an optimal batch size used, or enough investment should be made in setup reduction to reduce the optimal batch size to a single unit. This result, derived for the specific case of the linear investment function, was shown to represent the behaviour of the system under various non-convex and mildly convex investment functions. Only use of a strongly convex investment function will lead to potential optimal solutions between the extremes of no- or full-investment. Obtaining analytic results under the condition of a strongly convex investment function, however, remains an open area of research. Through the course of this work, additional research opportunities were found. For instance, although many stochastic models have been developed of manufacturing systems (e.g., see Buzacott and Shanthikumar, 1993), little has been published on the stochastic characteristics of typical setup and production times. This information would be very useful in determining which model forms are most appropriate in constructing cost models such as the one presented in this paper. Two areas of potential extension for this model are to consider multi-product/multi-stage production systems and to consider queuing-based lot sizing models for manufacturing. The latter extension has been developed to some extent in this paper. To be specific, of the two optimal cases derived here, the first was based on no investment in setup reduction, only batch size optimisation Eq. (16). Thus, optimal batch sizes can be found analytically for systems with WIP inventories, although more investigation needs to be performed.