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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

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

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 [

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 [

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

The rest of this paper is organized as follows. In Section

Consider a periodic-review inventory system with finite ordering capacity and price adjustment cost. There are

In period

Each replenishment incurs a fixed ordering cost

We assume that there is a fixed guide price

We aim to obtain the optimal pricing and inventory decisions in each period to maximize the expected total discounted profit over the

For notation convenience, we define another function

In order to characterize the structural properties of the optimal replenishment and pricing policy, we first introduce the definition of strongly

A function

The structure of strong

Chao et al. [

If

If

If

If

In the following, we aim to show that

Considering that

Let

Due to the concavity of

If

The proof of Lemma

Lemma

Due to the property of strong

The strong

Suppose that

If

order capacity

order at least up to

either order nothing or order at least up to

order nothing if

If

order capacity

either order nothing or order

either order nothing or order at least up to

order nothing if

The optimal pricing decision is characterized by

Suppose

Define

The optimal pricing decision is determined by the maximizer in Lemma

The structure of the optimal inventory policy is presented in Figure

The structure of the optimal replenishment policy.

Our results are similar to Gallego and Scheller-Wolf [

In order to explore the effects of the setup cost, the ordering capacity, the guide price, and the adjustment cost function on the optimal control policy, we conduct several numerical experiments for a simple inventory problem with

We study the effect of the setup cost

Optimal replenished inventory level

Optimal selling price

Figure

The optimal selling price

When

The effects of ordering capacity

Optimal replenished inventory level

Optimal selling price

The effects of guide price

Optimal replenished inventory level

Optimal selling price

In Figure

Higher guide price does not indicate higher optimal selling price.

The effects of price adjustment cost on the optimal ordering and pricing policy are shown in Figures

Optimal replenished inventory level

Optimal selling price

In this paper, we consider a dynamic inventory control and pricing optimization problem in a periodic-review inventory system with fixed ordering cost and price adjustment cost. At the same time, the ordering quantity is limited. Here, we assume that the price adjustment cost functions are piecewise linear. We show that the optimal inventory control, similar to Chao et al. [

There are still many interesting issues worth studying in the future research. Our paper studied increasing convex price adjustment cost; exploring price adjustment cost function with more complicated form may be one of potential research directions. In our paper, the decision sequence is first inventory decision and then price decision, but in reality the firm may first set price to serve the target market and then build up the inventory. In this case, what is the optimal pricing and replenishment policy? Does the optimal control policy still possess the similar structure?

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work was supported in part by the National Natural Science Foundation of China under Grants nos. 71201128 and 71201127, Project for Training and Supporting Young Teachers in Shanghai Universities (no. ZZSDJ13007), and Leading Academic Discipline Project of Shanghai Dianji University (no. 10XKJ01).