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...shortage actually results lost sales or is satisfied by a subcontractor. The model considers only rotation policies, ones in which a cycle is complete when one batch of each product type has been set up and produced. We assume that the firm is a price taker and profit maximizer, and that demand and production occur at constant, known rates. The costs of carrying inventory in the model may make lost sales during the cycle economically attractive.
Until recently, most research on inventory problems has assumed lost sales are an anathema to the firm: shortages are either not permitted or are backordered (see, for example, the optimization and heuristic algorithms for ELS models described by Maxwell (1964), Doll and Whybark (1973). Delporte and Thomas (1977), Elmaghraby (1978). Hsu (1983), Spence and Porteus (1987), Jones and Inman (1989), Roundy (1989), Carreno (1990), Zipkin (1991), Dobson (1992) and Gallego (1996)). Although backordering allows analytic tractability in many models, it may unnecessarily restrict the firm's options. As Fisher (1997) has observed, there may be practical, economic reasons for a firm to decide not to satisfy all demand. He identified companies that should incur lost sales: those that sell stable, predictable products and typically avoid proactive price adjustments. As one example. Fisher cites a firm with a product whose contribution margin is around 10% and its average stockout rate is about 1%. The corresponding lost contribution to profit is only approximately 0.1% of sales, "a negligible cost that doesn't warrant the significant investments required to improve responsiveness". In another example, Lariviere and Porteus (1999) describe a mail order firm that regularly declines sales from certain customers. In a third case, shortages are lost to the firm, but not to customers who receive the goods from a subcontractor (Downs et al. 2001). In return, the subcontractor charges the firm a premium cost per unit.
In addition to lost sales, our model accounts for setup times, not setup costs. We believe this is a reasonable approach in many applications. While an incremental setup cost might make sense if those involved in setup are paid for each setup they perform, this situation is atypical. Employees who set up a facility are normally on the payroll that is, within reasonable bounds, invariant to the number of setups. There may be an incremental setup cost if material is consumed during setup, but even under these circumstances the material costs would likely be negligible. (See, for example, Meier et al. (1982, p. 161)). We are not the first (Magee, 1958) to raise the issue: Is setup cost a valid surrogate for the effects that setup times have on manufacturing capacity? Eilon (1964) also expressed reservations about the use of setup cost surrogates. Goldratt and Cox (1984) and Karmarkar and Rummel (1990) echoed the concerns of Magee (1958) and Eilon (1964). This awareness, which began with informal observations, prompted others to incorporate setup times directly in lot size analysis.
Bollapragada and Rao (1999) formulated an ELS problem that incorporates lost sales and setup times on nonidentical machines. Their objective was to minimize the average total costs of production, setup, inventory and lost sales by allocating item production to machines. Although they found this problem very difficult, they were able to obtain good heuristic solutions to small problems (20 products on four machines) for the case of no setup times. They argued that "ignoring setup times may not be very restrctive ...", a perspective quite different from ours. However, they did suggest a method for extending their heuristic to the setup time case.
We cover applications of the setup time, lost sales, rotation model in the following section. Then we establish the main structural result of the model, that the facility should never be idle, and despite assumptions of deterministic demands, production rates and setup times, lost sales may occur in a profit maximizing solution. We use these properties to construct an algorithm for the model and test it on a variety of problems ranging in size from 100 to 1000 products. The data for these problems are drawn randomly from a population whose parameters are consistent with published industrial data. The solution time in our experiments appears to grow according to [n.sup.2], where n is the number of products, although this has not been proven.
2. Applications
Many firms that offer an assortment of products made on a single machine or assembly line fit the setup time, lost sales, rotation profile. Applications include plastics extrusion, metal stamping, textile manufacturing, printing, bottling, and packing (Bollapragada and Rao, 1999). Kool King (Miller and Jaikumar, 1979) and Midcentral Foods (Meier et al., 1982) provide other case examples in the operations management teaching literature. Kool King (disguised company name) rotates through six models of home appliances on a labor-intensive assembly line with production rates and setup times that differ by model. Midcentral Foods (also disguised) uses a commercial oven to make various baked goods. The baking time per unit and changeover time (for cleaning and setup) vary by product. Both companies accept the economic realities of losing sales to their competition. Lost contribution and goodwill incur opportunity costs, but not enough to justify additional equipment or inventory carrying cost.
We have encountered three other applications. Titus Tool, Northwest Kayaks and Chateau St. Michelle, that fit the setup time, lost sales, rotation profile. In all three, demand for their products is fairly constant throughout the year, and competitors offer substitutes. In addition, the setup of each company's facility involves time, but little direct setup cost for material, equipment or labor.
Management at Titus Tool in Kent. WA, has realized the benefits of incurring some lost sales during its rotation production cycle of fasteners it supplies to furniture makers (Schmitt, 1995). Their bottleneck operation involves a machine that inserts a cam into one side of the fastener. allowing it to snap into the other side so that it might be used for securing legs onto tables or chairs. Before resuming production, the machine requires about 30 minutes to set up the next fastener model. When Titus runs short of fasteners, its customers are able to find enough elsewhere, and shortages by Titus have little effect on prices, supply and demand in the marketplace.
At Northwest Kayaks in Seattle, WA, the pacing operation involves a specially designed rotisserie that cures kayaks. The rotisserie machine requires several hours for cleaning and changeover. In another application, the Chateau St. Michelle winery in Bothell, WA, uses a bottling machine to rotate through its product line, requiring about 30 minutes for cleaning and changeover. In both companies, the cost of any residual materials removed during cleaning is negligible, the investment in equipment is a sunk cost, and the employees are on a fixed payroll. Additionally. both companies incur lost sales when their products are out of stock at retailers.
3. A setup time, lost sales, rotation ELS model
We maximize marginal net profit per unit time, defined as contribution less holding cost, subject to the requirement that there is sufficient time on the machine for setup and production. For each product i, let [p.sub.i] be the unit contribution to profit, [h.sub.i] the unit inventory holding cost, [d.sub.i] the demand rate (at the market price), [u.sub.i] the production rate, and [s.sub.i] the time required to set up the facility for each run of product i. We do not require that all demand be satisfied, but assume the production rate [u.sub.i] exceeds the demand rate [d.sub.i] for each product i (see Appendix I for relaxation of this assumption), and that at least one setup time is positive. We consider only rotation policies where a product is set up and produced exactly once in a cycle. For a comparison to the cyclic case where a product may be produced two or more times in a cycle, see Appendix 2. We do not examine the larger issue of which products should be included in or excluded from the firm's product...
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