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Capturing the risk-pooling effect through demand reshape.

Article Excerpt
1. Introduction

Variability of products' demand frequently attracts managers' attention because of its costly implications. By carrying high inventory levels, firms may satisfy customers' orders and materialize high revenue in periods when demand is high, but often end up with excess inventory in periods when demand is low. On the other hand, carrying low inventory levels to avoid the costs of excess inventory may result in a poor service level and loss of potential profit. To facilitate this burden associated with demand variability, firms attempt to employ various approaches to take advantage of the risk-pooling effect.

A company that sells a single product through several outlets to satisfy demand in several markets (locations) may consider instead establishing one central facility to satisfy demand from the various markets. Eppen (1979) demonstrated the cost benefits resulted from using a centralized inventory system to satisfy normally distributed demands generated in several locations. Chen and Lin (1989) extended Eppen's (1979) results by incorporating concave holding and penalty cost functions. Mehrez and Stulman (1984) replaced the penalty cost for stockouts with a binding constraint on the maximum probability of stockouts at each location and showed the superiority of centralized systems over decentralized systems. Stulman (1987) used a first-come, first-served inventory allocation to determine the minimal starting inventory level subject to service-level constraints, and demonstrated the reduction in inventory as a result of centralization. Chen and Lin (1990) provided a counterexample in which a centralized configu ration requires a higher inventory level. Eynan (1999) considered a profit-maximizing firm whose markets may be characterized with different selling prices as well as different shortage penalty costs, and showed that: (1) centralization is recommended, and (2) managers should not fear "supply cannibalization," where customers from low-paying markets arrive early causing the firm to be unable to satisfy later demand from high-paying markets.

When a company offers multiple products it can capture the advantages of the risk-pooling effect by using common components for different products and employ an assemble-to-order strategy. Baker et al. (1986) analyzed inventory changes resulting from the use of commonality. Gerchak et al. (1988) argued that the objective function should be minimization of inventory value rather than the number of units. Eynan and Rosenblatt (1996) suggested that a common component may be more expensive than each of the components it replaces because of its wider functionality However, even a more expensive component may still result in a cost reduction. Eynan (1996) studied the effect of correlation of demand on the benefits of commonality and showed that smaller correlations correspond with larger savings, and consequently more expensive common components can be afforded. Rutten and Bertrand (1993) and Eynan and Rosenblatt (1997) explored the employment of variable recipes (flexible design) in which products can be made foll owing several bills of material (recipes) by allowing concurrent usage of the original (cheaper) and the common (expensive) components.

In this paper, we consider a new approach that may be used by multiproduct companies to take advantage of the risk-pooling effect. The suggested approach is called "demand reshape" as it takes a given aggregate (total) demand and changes its distribution (allocation) among the various products. This reshaping is obtained by making an effort to persuade some of the customers to purchase another (usually a substitute) item instead of the original item they had in mind. The cost of such effort can vary widely, and can be as minimal as displaying posters in the store to make customers more aware of the available substitute item. Some Taco Bell drive-through facilities employ another example: When placing their order, customers are asked: "Would you like to try our new Gordita?" The question makes some customers switch from another item and apparently changes demands for the two items. This practice increases the demand mean and variability of one product while reducing them for the other product. This paper shows that demand reshape reduces the sum of demand variabilities and consequently increases total profit.

The topic of item substitution, where customers may switch to an available item upon stockout of another, has been previously explored. We mention only a few examples: McGillvary and Silver (1978) used substitution to reduce the total cost of holding and shortage. Parlar and Goyal (1984) considered profit maximization and assumed shortage cost and salvage value to be zero. Pasternack and Drezner (1991) extended these works suggesting that substitution results in a reduced (per-unit) revenue. Parlar (1985) studied a special case of perishable items where old items can substitute for fresh items and vice versa. It should be noted that the substitution (switch) in these works and this branch of research takes place only when inventory of one item is exhausted, or, in other words, when customers are "forced" to switch. In this work, however, switches take place due to customers' change of preference when inventory of their originally intended item may be available.

In the next section, assumptions and definitions are presented. In [section]3, the demand reshape model is established and analyzed for a fixed switching rate followed by a numerical example. Sections 5 and 6 generalize the model to include probabilistic switching. Section 7 extends the model to the multipleperiod case. Conclusions and suggestions for future research appear in the summary.

2. Assumptions and Definitions

The following definitions will be used throughout the paper.

[P.sub.i] = selling price of product i (i = 1,2).

[C.sub.i] = purchasing cost of product i.

[V.sub.i] = salvage value of product i.

To make the environment meaningful, we assume [p.sub.i] > [c.sub.i] > [v.sub.i].

[q.sub.i] = penalty for each unit short of product i.

[[micro].sub.i], [[sigma].sub.i] = mean, standard deviation of the original periodic demand for product i.

p = correlation coefficient between products' demands.

f([x.sub.0], [y.sub.0]) = joint density function of the original demand for the two products.

[alpha] = switching parameter (0 [less than or equal to] [alpha] [less then or equal to] 1)-proportion of customers who originally intended to purchase Product 1, however, because of the company's effort, have switched to Product 2.

Hence, any realization of an original demand ([x.sub.0], [Y.sub.0]) for Products 1 and 2, respectively, will be transformed into x = (1- [alpha])[x.sub.0], y = [alpha] [x.sub.0], + [y.sub.0]. It should be noted that occasionally reshaping efforts will lead to steering customers from an available product to an unavailable one. We "let" such scenarios take place assuming that reshape efforts may not be suspended instantaneously, or even if paused, their impacts remain unaffected. (Clearly, if reshape can be paused or discontinued, the benefits of the suggested approach will further increase.)

The effort to reshape demand (switch customers) results in a linear transformation of variables: Consequently, the parameters should be modified as follows (the "

" sign designates the respective parameters after reshaping):

[[micro].sub.1] = (1 - [alpha])[[micro].sub.1], (1)

[[micro].sub.2]...