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Weekly Updates1. Introduction
Inventory management is a key business function for companies operating with inventories that may quickly become obsolete or spoil, or have a future that is uncertain beyond a single period. This paper derives from a real-life problem faced by suppliers of garment accessories for producers of fashion industry goods. The demand for these accessory items is unknown prior to the selling season. Once the season starts, it is too late to produce since fashion good manufacturers cannot halt their production to wait for deliveries. There are two ways of dealing with this problem.
One way involves maintaining production overcapacity in order to prevent both surpluses and shortages during the selling season. This method has a drawback. The selling season of fashion goods is short, whereas the demand can be high, causing significant production overcapacity in garment accessories, which will not be utilized most of the time. On the other hand, urgent subcontracting during the selling season (a temporary increase of capacity) with just-in-time supplies leads to very high costs due to either production (if a local manufacturer contracts out), or overseas shipping (e.g., airfare).
As a result of these constraints, the common way of handling demand uncertainty is to accumulate high levels of inventory by advance ordering at a relatively low price, rather than to maintain an overcapacity. In such a case, the manufacturer's capacity is typically used to smooth demand fluctuation and compensate inaccuracy in estimating demands. The advantage is that no capital is required for excessive capacity. However, capital is tied up as excessive inventories are held at points of overestimated demand during the selling season, while sales and clients can be lost when there are not enough products. Moreover, fashionable garment accessory goods left over at the end of the season are virtually worthless.
To prevent loss of sales and clients, accessory manufacturers tend to keep large stocks of end items, tying up sizeable amounts of their capital. By accurately forecasting demand, manufacturers can reduce uncertainty. Therefore, much time and effort is normally devoted to professional exhibitions of leading fashion designers in order to foresee future trends and likely accessory needs for the upcoming season. That by itself presents another problem since the world is divided into five leading fashion centers: US West Coast (sports casual); US East Coast (business casual); London (the most innovative nowadays); Paris and Milan (traditional leaders). In addition to analyzing trends of the major fashion designers, uncertainty can be tackled by approaching major producers of garment accessories in Europe and China in order to find out what is being produced for which markets. That issue will be further elaborated in Section 4 of this paper.
The described conditions involve two well-known problems: production smoothing and newsboy. In regard to given demand for goods, there are various dynamic models and solution methods for optimal choice of production rates over a selling season in terms of both production and inventory holding costs. An extensive literature review can be found, for example, in Maimon et al. (1998). If, however, the demand is unknown in advance and production smoothing during the selling season has negligible efficiency, then an optimal choice of inventory orders is referred to as a newsboy model.
The classical, single-period newsboy problem is to find a product order quantity that either maximizes the expected profit or minimizes the expected costs of overestimating and underestimating probabilistic demand. The newsboy problem has attracted considerable attention since the pioneering papers of Arrow et al. (1951), and Morse and Kimball (1951). An extensive literature review on various extensions of the classical newsboy problem and related inventory models can be found, for example, in Silver et al. (1998) and Khouja (1999).
Among the numerous extensions to this problem suggested so far, one can find different models with respect to objectives (see, for example, Chung (1990) and Eeckhoudt et al. (1995)); news-vendor pricing policies and discount structures (Lau and Lau, 1988; Khouja, 1995); random yield of defective units (Henig and Gerchak, 1990) or of production capacity (Ciaralo et al., 1994); multi-products (Chang and Lin, 1991; Lau and Lau, 1996), and a number of subperiods to prepare for the selling season (Hausman and Peterson, 1972; Bitran et al., 1986; Matsuo, 1990). The idea behind the last type of extension is that there may be many periods to produce the items, which will be sold in a single season. Such dynamic models stress the importance of timing in producing or purchasing the items and production smoothing before the selling season. These models commonly utilize special product (or product family) and demand parameters to optimize operations under limited production capacity over each subperiod (Hausman and Peterson, 1972; Bitran et al., 1986). The former work formulated the single and multi-product cases as dynamic programming problems, whereas the latter work deals with several families of style goods and resulted in a stochastic, mixed-integer programming problem. Both studies suggest heuristic methods to provide an approximate solution. Matsuo (1990) observed that a limitation of these works is that they include discrete production subperiods to assign production and suggested a continuous-time heuristic approach for improving the objective function value when approximating the optimal solution to the problem.
In this paper, we consider an integrated continuous-time model. The model allows for both advance ordering of products to be delivered by the beginning of a selling season and continuous-time production smoothing during the selling season. As in production smoothing problems, the production rate is controllable over the selling season. As in the classical single-period newsboy problem, the demand is assumed to be unknown and its exact cumulative value is determined only by the end of the selling season.
There are two main streams in research related to continuous-time inventory control. One is to develop various approximation and heuristic solutions under time-dependent demands (see Maimon et al. (1998)). The other is to derive a closed-form solution under constant deterministic demand and no advance orders. For example, analytical solutions were suggested by Khmelnitsky and Caramanis (1998) for inventory control with setups under constant demand rates; by Kogan and Lou (2002) for optimal inventory control under deterministic seasonal demands for fashion goods of constant rates; and by Kogan et al. (2004) who analyzed a single-period inventory review under deterministic constant demand rate for fashion goods and random yield. Following the analytical stream, we assume that the demand shape is known. However, the demand shape may depend on time and its amplitude is random. Consequently, shortage or surplus (inventory holding) costs are incurred continuously at time intervals when the production falls short or exceeds the demand. For similar relationships characterized by continuous-time inventory costs and incomplete inventory information see, for example, Pai and Hsu (2003) and Kogan et al. (2004).
In practice, although inventory levels and losses associated with inventories may not be known precisely for a period of time, inventory costs (bookkeeping costs, material tracking costs, transportation costs, space costs, material transformation costs, labor costs, depreciation, etc.) are incurred continuously. At the end of the period, it is possible to determine exactly how much was spent, when and why. This explains the use of the expected inventory cost as the objective function in inventory control models.
The objective of this study is to determine advance order quantity before the selling season and adjust the production rates during the season in order to minimize total expected costs. In addition to shortage or surplus (inventory holding) costs, total costs include advance order costs by the beginning of the selling season and production costs during the selling season.
The stochastic problem, its deterministic equivalent and a dual formulation are presented in Section 2. In Section 3 analytical properties of the optimal solutions are derived with the aid of the maximum principle. As a result, the continuous-time problem is reduced to a number of discrete problems of determining optimal production conditions and a finite number of production switching points. The advantage of this approach is that a closed-form optimal solution is derived. An example is presented in Section 4. Section 5 summarizes the results.
2. Problem formulation
Consider a manufacturing system producing a product type during a selling season and containing a buffer to collect finished products. The products are produced to satisfy a demand rate, a(t)d, for the product type along the selling season, T. The production process is described by the following balance equation:
X(t) = u(t) - a(t)d, (1)
where X(t) is the surplus level in the buffer by time t and u(t) is the production rate at time t. The dynamic process (1) is determined by two decision variables, production rate, u(t), which is bounded by the maximum capacity of the system U:
0[less than or equal to]u(t)[less than or equal to]U, (2)
and the initial inventory level X(0) accumulated in the buffer. The initial level X(0) is due to advance orders contracted out and delivered at unit cost s by the beginning of the selling season:
X(0)[greater than or equal to]0. (3)
The demand a (t)d is a time-dependent parameter representing at time t the amount of the product type required per time unit, where a(t), a(t) > 0 for 0 [less than or equal to] t < T, is a known demand shape and d is a random demand amplitude. For a selling season T, there will be a single realization of d, D, which is known only by time T. Therefore, a decision has to be made under these uncertain conditions before production starts based on probability density [phi](D) and cumulative distribution [PHI](D) functions. We assume [phi](D) is differentiable.
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