Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 269695, 8 pages

http://dx.doi.org/10.1155/2015/269695

## Dynamic Inventory and Pricing Policy in a Periodic-Review Inventory System with Finite Ordering Capacity and Price Adjustment Cost

^{1}School of Business, Shanghai Dianji University, Shanghai 201306, China^{2}Department of Management Science, Southwestern University of Finance and Economics, Chengdu 611130, China^{3}Business School, Beijing Institute of Fashion Technology, Beijing 100029, China^{4}Department of Logistics Management, Southwestern University of Finance and Economics, Chengdu 611130, China

Received 23 April 2015; Accepted 21 September 2015

Academic Editor: Ruihua Liu

Copyright © 2015 Baimei Yang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider a dynamic inventory control and pricing optimization problem in a periodic-review inventory system with price adjustment cost. Each order occurs with a fixed ordering cost; the ordering quantity is capacitated. We consider a sequential decision problem, where the firm first chooses the ordering quantity and then the sale price to maximize the expected total discounted profit over the sale horizon. We show that the optimal inventory control is partially characterized by a policy in four regions, and the optimal pricing policy is dependent on the inventory level after the replenishment decision. We present some numerical examples to explore the effects of various parameters on the optimal pricing and replenishment policy.

#### 1. Introduction

Traditional literature on the multistage inventory system mainly focuses on replenishment decision with or without setup cost. The well-known result is that the order-up-to policy is optimal for the systems without setup cost and the policy is optimal for the systems with setup cost. Increasing researchers are devoted to the study of joint price and inventory control in the multistage inventory system. Our paper belongs to this stream, but our paper considers a sequential decision problem in a periodic-review inventory system with fixed ordering cost and price adjustment cost. The ordering quantity is capacitated; this may be limited by the storage capacity or the supply capability. The firm first decides its inventory level and then chooses a sale price to maximize its long-run profit. Our result shows that the optimal inventory and pricing decision still preservers a threshold-type structure.

Our paper is related to literature on the optimal control of a single product system with finite capacity and setup cost. Several studies have been devoted to this area. Shaoxiang and Lambrecht [1] obtain the generally known result; that is, the optimal policy can only be partially characterized in the form of - bands. In particular, when the inventory level is below the first band , then produce/order the capacity, and when the inventory level is over the second band , produce/order nothing. If the inventory level is between the two bands, the ordering policy is complicated and depends on the instance. Gallego and Scheller-Wolf [2] extend their work. They derive the structure of the policy between the bands. The optimal policy is characterized by two numbers and which divide the state space into four possible regions. However, none of them have studied the pricing problem in the inventory control problem. Zhang et al. [3] consider a single-item, finite-horizon, periodic-review coordinated decision model on pricing and inventory control with capacity constraints and fixed ordering cost. They show that the profit-to-go function is strongly -concave, and the optimal policy has an -like structure. However, the price adjustment cost has not been addressed. Chao et al. [4] recently consider the joint pricing and inventory decisions. They study a periodic-review inventory system with setup cost and finite ordering capacity in each period. They show that the optimal inventory control is characterized by an policy in four regions of the starting inventory level. However, in their paper, the selling price can be adjusted without any cost.

In reality, changing price is costly and incurs a price adjustment cost. In the economics literature, there are two major types of price adjustment costs: the managerial costs and the physical costs. Rotemberg [5], Levy et al. [6], Slade and Groupe de Recherche en Economie Quantitative d’Aix-Marseille [7], Aguirregabiria [8], Bergen et al. [9], and Zbaracki et al. [10] have stated that both types of costs are significant in retailing and other industries. According to these empirical studies, Chen et al. [11] consider a periodic-review inventory model with price adjustment cost. The price adjustment cost consists of both fixed and variable components. They develop the general model and characterize the optimal policies for two special scenarios, a model with inventory carryover and no fixed price-change costs and a model with fixed price-change costs and no inventory carryover. Although there is price adjustment cost, they do not consider the finite ordering capacity.

Under the assumption of random additive demand model, our paper tries to investigate the structure of the optimal inventory control and pricing policy in each period. We show that the optimal inventory policy is partially characterized by an policy on four regions; in two of these regions the optimal policy is completely specified while, in the other two, it is partially specified. More specifically, the optimal ordering quantity in the first region is the full capacity, while in the last region it is optimal to order nothing; in the two middle regions, the optimal decision is either to order to the maximum capacity, to order to at least a prespecified level , or to order nothing. The optimal pricing policy in each period is dependent on the inventory level after the replenishment decision, , which is in general not a monotone function. The key concept utilized is strong -concavity, which is an extension of -concavity, and was first introduced by Gallego and Scheller-Wolf [2].

The rest of this paper is organized as follows. In Section 2, we induce the model description. The structural properties of the optimal inventory and pricing policy are characterized in Section 3. We present some numerical examples to show the effects of various parameters on the optimal control policy in Section 4. Finally, we conclude with some future research direction in Section 5.

#### 2. The Model

Consider a periodic-review inventory system with finite ordering capacity and price adjustment cost. There are periods, with the first period being 1 and last period being . In each period, the sequence of events is given as follows: (1) inventory level is reviewed and replenishment order is placed; (2) replenishment order arrives; (3) a selling price is set; (4) random demand is realized; and (5) all costs are computed.

In period , the selling price is , which is taken in interval , and the demand is . We assume that the demand is sensitive to the selling price . Moreover, we consider an additive demand function. The demand function is , , where is a random variable with mean zero and is the average demand. Furthermore, is a decreasing linear function of . When the selling price increases from to , the average demand decreases from to ; that is, and . Each demand arrives requiring only one unit of product and is satisfied from inventory if any. If the demand cannot be satisfied from the on-hand inventory immediately, then it is backlogged and incurs a backorder cost. The structure of demand function indicates that determining the selling price is equivalent to setting the average demand .

Each replenishment incurs a fixed ordering cost and the variable unit ordering cost . There is a finite ordering capacity for each period, which means the ordering quantity in each period cannot exceed , where . If is sufficiently large, it generalizes to the incapacitated case. Let be the inventory level at the beginning of period before placing an order and let be the inventory level after the order delivered. At the end of each period, the demand is realized and a revenue is received. The expected revenue is given by , which is assumed to be a concave function. Meanwhile, an inventory holding and shortage cost occurs denoted by . If , represents the holding cost; if , represents the shortage cost. For ease of presentation, we let Therefore, given that the inventory level after replenishment is and the expected demand for period is , the expected holding and shortage cost is .

We assume that there is a fixed guide price for deciding the selling price . Price changing from the guide price to the actual price is costly. The cost of a price adjustment from guide price to the actual selling price in period is denoted by . Zbaracki et al. [10] and Chen et al. [11] pointed out that as the price adjustment cost becomes larger, it would cost more on decision and internal communication. Here, we assume that the variable cost is convex and increases with . The forms of could be either piecewise linear functions or quadratic functions. The ordering quantity in period is ; therefore, we have due to the capacitated ordering quantity . Therefore, the expected total cost incurs in period including setup cost, ordering cost, holding and shortage cost and price adjustment price is given bywhere is the indicating function taking value 1 if statement is true and zero otherwise.

We aim to obtain the optimal pricing and inventory decisions in each period to maximize the expected total discounted profit over the periods. Let denote the maximum expected total discounted profit from period to the end of the planning horizon with the starting inventory level (before ordering decision) . The optimality equation iswhere is the one-period discount factor, . The terminal condition is . Note that the price can be indicated in the form of demand by the inverse demand function; that is, , and the price adjustment cost can be written in the form of instead of , that is, , such that optimizing over the selling price is equivalent to optimizing over the average demand . Therefore, the optimality equation is rewritten as follows:

For notation convenience, we define another functionThen the optimality equation is further simplified to

#### 3. The Optimal Policy

In order to characterize the structural properties of the optimal replenishment and pricing policy, we first introduce the definition of strongly -concave and properties of -concave functions as well, which is defined in Chao et al. [4]. This definition and the properties are very important in studying inventory models with finite capacity and setup cost.

*Definition 1. *A function is strongly -concave if, for all , , and , we have

The structure of strong -concave function is shown in Figure 1. If is strong -concave, it implies that the slope of the line made of points and is smaller than the slope of the line made of points and .