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A particularly challenging problem for retail managers is to decide on stocking policies and pricing policies for new items which face high uncertainty in demand and which have limited selling seasons. Example items include a new clothing fashion, or a new toy during holiday periods.
The problem is challenging for several reasons. First, the retail buyer has little or no solid information to use in forecasting the level of demand at various prices. Second, this is a multi-period decision problem where pricing and stocking decisions made in earlier weeks affect options and decisions in future weeks. Third, the primary selling season is relatively short, say to 3 to 6 months. As a result, the retailers may be restricted in their ability to reorder during the short selling season. Finally, any inventory remaining at the end of the selling season is greatly reduced in value.
The challenge of making good pricing and stocking decisions under these or related circumstances is well known to retailers (Cash and Frankel, 1986, chp. 6) and researchers. As a result, a number of mathematical models have been proposed to assist retail buyers in making these decisions.
In one stream of studies on retail pricing policies, researchers such as Lazear (1986), Balvers and Cosimano (1990) and Keifer (1989), have demonstrated the advantages of incorporating Bayesian updating or "learning" in the model. Lazear has shown that expected profits can be increased by up to 30% if the retailer modifies both the prior distribution of reservation prices and the future pricing policies after observing the sales rate in the early weeks of the season. Models of this type can be used to determine optimal prices assuming that inventory levels are given.
In a second stream of studies, researchers such as Bradford and Sugrue (1990), Murray and Silver (1966) and Hausman and Peterson (1972) have determined optimal inventory levels and have included Bayesian updating of information. These studies assume that prices are given for the selling season.
In a third category of studies, researchers such as Lodish (1980) and Rajan, Rakesh and Steinberg (1992) have addressed the question of how to determine both optimal pricing and optimal stocking policies. Studies of this type have assumed that the seller knows the parameters of the demand distributions with certainty and that no learning or revision of the demand distributions takes place during the selling season.
The purpose of this study is to combine the results from these three separate streams of research. We seek to develop a new model for use by retailers that accomplishes three goals. First, it permits the retailer to incorporate learning and update the forecasts of demand as the season progresses. Second, it determines optimal pricing policies over the selling season. Third, it determines the optimal stocking policies. To the best of our knowledge, this formulation of the retailer decision problem has not been addressed in prior studies. A new model is developed.
The new model can be solved via dynamic programming with learning. Several example retailer decision problems are solved to demonstrate the general model and solution procedures. One particular example problem demonstrates that expected retailer profits can be substantially higher when the model includes learning. The increase in expected profits with learning is a function of the decision maker's degree of uncertainty about the distribution of demand.
II. DETERMINING OPTIMAL PRICES AND INVENTORY LEVELS WHEN PARAMETERS OF THE DEMAND DISTRIBUTION ARE KNOWN WITH CERTAINTY
As noted, other models have been developed to determine both the optimal pricing and inventory policy (Lodish, 1980; Rajan et al., 1992). In these studies it is assumed that the parameters of the demand distribution are known with certainty and are not updated during the selling season. A specific example of this assumption is shown in Table 1 of Lodish (1980). This table indicates the distributions of demand for specific price levels and specific weeks. These distributions remain fixed over the selling season and are not updated as new evidence on demand is observed. This assumption of fixed demand distributions may be well satisfied for an established product or service, such as selling T.V. spots. However, it is probably not well satisfied for many new items.
Rajan et al. (1992) developed a model to determine the optimal pricing and ordering decisions when faced with a known demand function, where there is a physical deterioration of inventory. The model explains the reduction in prices when the retail goods are subject to decay.
III. A MODEL TO DETERMINE OPTIMAL PRICING AND INVENTORY POLICIES WHEN PARAMETERS OF THE DEMAND DISTRIBUTION ARE NOT KNOWN WITH CERTAINTY
This section describes a general model for determining the optimal level of initial inventory and the optimal initial price for relatively new items with a short selling season (3 to 6 months) and high initial uncertainty about the probability distribution of demand. The model also determines the optimal prices and reordering decisions in subsequent periods, contingent on the observed level of sales in prior periods. We first present the general model and then consider and solve specific example problems. The following assumptions are made in the general model.
1. The entire selling season has been divided into a series of T equal periods or stages (t = 1, ..., T). Actual sales (units sold) during period t are denoted by [s.sub.t]. The inventory at the beginning of period t is denoted by [I.sub.t] and any orders placed at the beginning of period t are denoted by [o.sub.t]. The number of units demanded in period t is denoted by [n.sub.t]. Sales in period t, [s.sub.t], will equal demand, [n.sub.t], when demand is less than the inventory, [I.sub.t], at the beginning of period t. However in periods when demand ([n.sub.t]) exceeds the inventory, [s.sub.t], will equal [I.sub.t].
2. At the beginning of each period t, the retailer sets a price [p.sub.t] for that period. If desired, the retailer can state as a matter of store policy that the price will never be increased in succeeding periods. This is an option.
3. The retailer can place orders prior to the selling season to have an inventory of [I.sub.1] units on hand at the beginning of period 1. Depending upon the item and the supplier, the retailer may or may not have the option to order at the beginning of each period. If orders during the season, are permitted, it is assumed that there is a delivery lag of L periods. That is, an order, [o.sub.t], placed at the beginning of period t, is available for sale at the start of period t + L.
4. Demand ([n.sub.t]) at a given price level is stochastic, even if the retailer knows which of several possible demand distribution functions is the "true" one and knows the "true" parameter values. This contribution to stochasticity of demand is analogous to the random variation in demand for an established item when the true underlying demand distribution function and its parameters are known (Lodish, 1980, p. 204).
5. A second contribution to the stochastic nature of demand is due to the fact that for an "uncertain seasonal" item, such as a new fashion, the retailer is also uncertain as to which demand distribution is the "true" function (i.e., which of several sets of parameter values is the "true" set). For example, a retailer might believe that a specific new item could either be a "hot," "medium" or a "slow" selling item with certain probabilities. In this model, the retailer is asked to specify prior probabilities, [[Pi].sub.j,t], to each possible demand distribution function, [f.sub.j,t] ([n.sub.t] [where] [p.sub.t]), (j = 1, 2,... J), where J is the total number of possible demand distributions being considered by the retailer, [p.sub.t] is the price and [n.sub.t] is the number demanded in period [[Pi].sub.j,t] denotes the probability that the true underlying distribution of demand at time t is the jth distribution. Note that since sales levels are in integer values, the distribution functions are discrete and exist only for non-negative …