ارزش مدل سازی همراه با اثر مرجع در مشکلات موجودی تصادفی و قیمت گذاری

We analyze a periodic review inventory system in which the random demand is contingent on the current price and the reference price. The reference price captures the price history and acts as a benchmark against which the current price is compared. The randomness is due to additive and multiplicative random terms. The objective is to maximize the discounted expected profit over the selling horizon by dynamically deciding on the optimal pricing and replenishment policy for each period. We study three key issues using numerical computation and simulation. First, we study the effects of reference price mechanism on the total expected profit. It is shown that high dependence on a good history increases the profit. Second, we investigate the value of dynamic programming and show that the firm that ignores the dynamic structure suffers from the revenue. Third, we analyze the value of estimating the correct demand model with reference effects. We observe that this value is significant when the inventory related costs are low.

The reference price, or anchor price, is a benchmark which is developed by customers in repeated transactions. It is an internal standard against which prices are compared (Kalyanaram & Winer, 1995). The customers perceive the current price as high and the difference between the current price and the reference price as a loss, if the current price is greater than the reference price. Otherwise they perceive the current price as low and the difference as a gain. The reference price brings a challenge to companies. Although low prices may increase the current profit, they decrease the price expectations of the customers, therefore reduce the future profits. Hence there is a trade-off between current and future benefits in the presence of a reference price. In this study, we analyze joint pricing and inventory decisions when the random demand is subject to reference effects using simulation and computational analysis. Our aim is to find the effects of reference price mechanism on the optimal decisions and system profit. Moreover we investigate the value of modeling the problem with its underlying dynamic and reference effect structure. Adaptation level theory, which states that expectation-based reference price is the adaptation level against which current prices are judged (Monroe, 1973), constitutes the theoretical basis of reference price (Helson, 1964). The reference price is dynamic in the sense that it evolves through time with the announcements of new price levels. Although there are different models used for the evolution mechanism (see e.g. Nasiry & Popescu, 2011), the exponential smoothing model which depends on the adaptive expectation model is the most commonly used form (Nerlove, 1958). Fibich, Gavious, and Lowengart (2003) study dynamic pricing under deterministic linear demand with reference effects in a continuous time framework. They explicitly calculate the steady state prices and show that a firm should adopt a skimming or a penetration strategy depending on the initial reference price. Güler and Akan (2013) extend the study of Fibich et al. (2003) by incorporating an inventory decision and inventory related costs. They show that the cumulative structure of the holding cost yields an increase in the optimal prices. Popescu and Wu (2007) study the same problem for more general demand models when the time is discrete. They show that the skimming or penetration strategy can be generalized to non-linear models as well and explicitly calculate the steady state price levels. These papers study deterministic demand models. Urban (2008) analyzes a single period model with random demand. He studies joint inventory-and-pricing model with both symmetric and asymmetric reference price effect, and provide numerical analysis which indicates that accounting for reference prices has a substantial impact on the firm’s profitability. Chen et al., 2011 and Taudes and Rudloff, 2012 study periodic review linear demand models with stochastic demand. Taudes and Rudloff (2012) study a single period and a two-period model. In the single period case, they show that the optimal inventory level and the optimal price increase in the reference price. They prove the optimality of an state-dependent order-up-to (SDO) policy for the two-period case. In an SDO policy there is an optimal order-up-to level and price pair which depends on the state, i.e., the reference price. Chen et al. (2011) analyze finite and infinite horizon models and show the optimality of an SDO policy. They prove that the reference price converges to a steady state and provide characterizations of the steady state solution. In particular, they show that the optimal order-up-to level increases with the reference price. Güler et al., 2013a and Güler et al., 2013b study concave demand models with stochastic demand. Güler et al. (2013a) show that the optimality of the SDO policy can be generalized to some concave demand models. Güler et al. (2013b) analyze models in which the randomness is due to an additive random term. They characterize the optimal pricing and inventory policy by showing that the problem can be decomposed into two subproblems and provide the solutions for the optimal parameters. These studies analyze analytical solutions for stochastic multi-period period problems with reference effects. Gimpl-Heersink, Rudloff, Fleischmann, and Taudes (2008) make a simulation of the multi-period problem with the linear demand where the randomness is due to an additive random term. They show that a base-stock list-price policy is optimal in the simulation. In such a policy, there is an optimal order-up-to level and a price pair at every period which are used if the on-hand inventory level is lower than the optimal order-up-to level, otherwise the firm does not order and goes to a discount. They also show that joint decision making for the inventory and the price brings a substantial increase in the profit for the demand models with reference effects. For a detailed review on reference price, we refer the reader to Mazumdar et al., 2005 and Arslan and Kachani, 2010. There is also a vast amount of literature on joint inventory and pricing without reference effects. Here we only give some of this literature and refer the reader to Elmaghraby and Keskinocak, 2003 and Chen and Simchi-Levi, 2012 for a detailed review of joint pricing and replenishment/inventory decisions. Federgruen and Heching (1999) study the periodic review multi-period problem where the unsatisfied orders are backordered. Chen and Simchi-Levi, 2004a and Chen and Simchi-Levi, 2004b introduce a setup cost to the setting of Federgruen and Heching (1999). In these studies above, the optimal policy turns out to be a variant of an (s, S, p) policy. This policy states that if on hand inventory is below s, then the firm places an order to bring its inventory level to S such that s ⩽ S and announces the price p. Otherwise it orders nothing and announces a state dependent price. Although there are some analytical results for the joint inventory and pricing problem with reference effects under random demand, these results are limited due to the stochastic structure of the problem. The number of variables, together with stochasticity, increases the complexity of the problem. There are few studies which resort to numerical investigations to increase understanding of important and complex structure of the problem. Gimpl-Heersink (2008) provide computational studies for the linear model with an additive random term and show the effect of reference price on the optimal price and optimal inventory level. Gimpl-Heersink et al. (2008) show numerically the effects of different distributions for the additive random term. Both studies deal with the linear models. Güler et al. (2013a) provide some numerical illustrations for non-linear (concave) models which shows there is an evidence that the analytical results for the deterministic pricing problems hold for the stochastic problem as well. These two studies provide some insights for the problem; however there are quite a number of questions regarding the reference effects on pricing and inventory decisions. In this paper we provide a computational study and simulation to explore three research questions. First, we study the effects of reference price mechanism, i.e., the evolution mechanism and the initial reference price, on the total expected profit. Second, we investigate the value of dynamic programming. Third, we analyze the value of estimating the correct demand model with reference effects. For the last two goals, we compare two firms using Monte Carlo simulation. We use the demand models in Güler et al. (2013a) and use their optimality results in our computations. The rest of this paper is organized as follows. We describe the demand model, formulate the problem with dynamic programming and set the values of the parameters in Section 2. The main analysis is given in Section 3. Finally, Section 4 concludes the paper and points to interesting topics for future research.

We analyze a periodic review joint pricing and inventory problem of a single item with stochastic demand subject to reference effects. The demand is dependent on the current price and the price history which is captured by the reference price. The randomness is due an additive and a multiplicative random term. In general, we show the following. First we analyze the effect of the reference price mechanism and show that ‘high’ dependence on a ‘good’ history increases the profit. Next we study the value of dynamic programming by comparing an optimally acting firm with a myopic firm. We show that dynamic programming yields a substantial increase in the profit. It turns out that dynamic programming has greater importance when (i) time horizon is short, (ii) profit margin is low, (iii) inventory related costs are low and (iv) the variance is high. It is observed that the myopic firm suffers more from the revenue, rather than the inventory related costs. The myopic firm undershoots both the optimal order-up-to level and price. Although effect of α seems not to have a certain structure, the difference in profits decreases with initial reference price levels. Finally, we study the value of (correctly) modeling the problem with the reference effects. We show that if a firm cannot recognize the existence of the reference effect, then it loses relatively high profits when (i) initial reference price is low, (ii) reference price depends heavily on the past and (iii) the planning horizon is short. The decrease in profits is shown to be statistically significant especially when inventory related costs (or optimal inventory level) are low. It turns out that estimating the initial reference price correctly is more crucial than estimating the correct evolution parameter. We study the problem of joint inventory and pricing under reference effect on a particular demand model given in the literature. We provide some insights using numerical computation and simulation. The analytical results for this problem is the major future direction. The second future direction is to study (numerical or analytical) the case where the unsatisfied orders are lost, instead of being backordered.