^{1}

^{1}

^{1}

In this paper, we consider the case that the consumers may cancel their reservations during the booking period so that consumers are divided into two parts: cancellation request and booking request. Simultaneously, we assume that consumers act strategically in booking products. Upon receiving the request, the firm has to decide whether to accept the cancellation request and what fees to charge the booking request in the presence of strategic consumers. We propose optimal price strategy and show the effectiveness of the decision models. We also obtain the optimal control of cancellation demands policy which is achieved by the number of remaining inventory. Numerical experiments are presented to explore the effect of various parameters on the performance of optimal strategy.

In revenue management, the fundamental problem is the decision to find the optimal price or inventory level in order to maximize total revenue. However, even though a good decision is made, a perfect match between predicted inventory and realized inventory still cannot be guaranteed. If realized inventory exceeds predicted inventory, some inventory will lose its value after the sale period. On the other hand, some consumers will become lost sales. Generally, the gap between predicted inventory and realized inventory comes from cancellation demand. In such circumstances, cancellation cause damage to firm’s sale that extra capacity becomes available and could be used to accommodate other potential consumers, which makes the system more complicated to operate.

How could the firms to overcome this dilemma? Many organizations use the overbooking policy. Although this approach reduces the likelihood of being left with much unsold inventory, it may also lead to a difficult situation when the number of reservations exceeds the available inventory at the time of delivery. In such cases, the firm not only will lose the consumer but also damage the firm’s reputation. We present a possible business strategy in this paper. That is control of cancellation demand―whereby firm can decide whether or not accept a cancellation request in each period. The control of cancellation demands, combined with inventory vary dynamically, makes the determination of how best to gain revenue become a difficult problem in the firm’s decision systems: How could the firm to charge price for the consumer who wants to buy an item? If a cancellation request arrives, whether or not accept her request? The models contained in this paper will help to answer these questions.

Dynamic pricing models have been well studied in the literature. For reviews of pricing models for revenue management, please refer to Bitran and Caldentey (2003) [

There are a number of the microeconomic literatures in existence, which focus on cancellation. Karaesmen and Van Ryzin (2004) [

The remainder of the paper is organized as follows. In the next two sections, we formulate our model and its associated dynamic programming value function. Section 4 analyzes the structure of the optimal policy and unfolds the basic property of the value function. A numerical example is presented in Section 5. We conclude with a brief discussion, as well as directions for future research in Section 6.

We assume that the supplier has

At any time

1) Accept or reject a cancellation request when a cancellation request arrives.

2) Charge fees of the booking request in the presence of strategic consumers.

The supplier maximizes the expected revenue by simultaneously determining whether or not accept the cancellation request and the optimal prices for booking request in the presence of strategic consumers at any given time and remaining inventory.

The following notation is used throughout the paper:

Let us formally make some assumption for our model.

1) In each time-period

2) The refund per cancellation,

3) The cumulative probability distribution of the reservation price of a booking request,

Given price

The boundary conditions of Equation (1) are:

1)

2)

In Equation (1),

We can solve the maximize of

where

Then, in any period

Let

For each

where

The boundary conditions of Equation (3) are:

1)

2)

Denote:

A natural question arises as to whether or not our model is better than the traditional revenue management model without control of cancellation demands. First, we know that the traditional revenue management model with cancellation demands is defined as follow, which mean that the firm always accepts the cancellation request in each period when cancellation demands arrives:

If the firm always rejects the cancellation request in each period when cancellation demands arrives, the expected revenue can be expressed as

That is, for all

Theorem 1. The control of cancellation demands model in Equation (3) performs no worse than the non- control of cancellation demands model in Equation (6) and Equation (7), i.e., for any

Proof. We can prove this theorem by induction.

1) Clearly,

where the first inequality follows by

2) Similar to case 1), we can prove that for any

where the first inequality follows by

This completes the proof of this theorem.

To derive the optimal policy, we begin with a marginal analysis of inventory. We first study the optimal value function

Lemma 1. (1) For any fixed

(2) For any fixed

This lemma shows that more stork and/or time leads to higher expected revenues.

In the sequel, we use the following notations for the quantities marginal in

request at state

Lemma 2. For any fixed

Proof. 1) Let

But the expression on the right-hand side is positive, which is a contradiction.

2) Let

This completes the proof.

Several revenue management models, such as Gallego and Van Ryzin (1994) [

Theorem 2.

Proof. Let

For any fixed

because

So we have

The proof is by inverse induction on time. The basis for induction is the terminal conditions. When

We can see that

Then:

Obviously, the expression on the right-hand side of

This completes the proof.

Theorem 3. The optimal price satisfies the equation:

Furthermore, if

Proof. Taking the first derivative of

The second derivative of

From Equation (9), we have:

Because

We have

So if

This completes the proof.

Although Theorem 3 gives the sufficient condition for the optimal price, obtaining it in closed form is quite difficult. However, we can make a number of qualitative statements about the optimal prices. We summarize these in the following theorem.

Corollary 1.

Proof. By theorem 2, we know that

Then, we have

According to Corollary 1, at a given point in time, the optimal price drops as the inventory increases. This property is useful if one wants to compute the optimal price policy numerically because they significantly reduce the set of prices over which one needs to optimize.

Under additional assumptions on

Corollary 2.

Proof. Similar to Corollary 1, from the theorem 2 and Lemma 1, we can obtain the conclusion.

From Equation (3) we know, a booking request for the first_{ }type of consumers is accepted if and only if_{ }type of consumers is accepted if and only if the benefit of accepting it is more than the benefit of rejecting it. To derive the optimal control policy, we have the following theorem.

Theorem 4. Let:

1)

2)

if

Then, the firm accepts a cancellation request if and only if the number of selling units is

Proof. If

and

It implies that for

If

Thus, when

This completes the proof.

The following corollary immediately follows from the optimal capacity control policy described in the above theorem.

Corollary 3. The optimal threshold level for the cancellation request,

Proof. We prove it by contradiction. Suppose that

Because

This contradicts the definition of

Corollary 3 demonstrates that the higher the refund is, the lower

Now we present a computation procedure for the optimal policy. The procedure leads to a closed-form solution which satisfies Equation (3). Moreover, as part of the control rules, all thresholds and price are jointly determined. We describe the recursive solution process as follows.

First, we note

We present a numerical experiment in this section to illustrate how the proposed approach works in a constructed example.

We tested sets of examples with

First, we will illustrate the refund value

Next, we consider the ratio of expected total revenue. The ratio is shown in

We can see that if the probability of cancellation request is increasing, the value of

In this paper we research the control policy of cancellation demands in revenue management. By building multi- period dynamic programming model, we obtained the price relation which meet to strategic consumers. In addition, we proved that the firms optimal revenue function and structure properties of pricing strategy. Finally, we use numerical experiments to explore the effect of various parameters on the performance of optimal strategy and show the effectiveness of the decision models. The results show that:

1) When the probability of cancellation request is large, there will be fewer cancellation inventory retained for those consumers in order to gain more profit trough dynamic pricing.

2) The control of cancellation demands model is no worse than the non-control of cancellation demands model with different type of I.

Our research can provide theoretical basis for the optimal pricing decision to firms in control of cancellation demands. There are a number of important research avenues that can be pursued. How to decide the refund per cancellation? In this paper, we only take strategic consumer into booking products, we can explore the normal purchase in the presence of strategic consumers in the future.

This work was supported by the National Natural Science Foundation of China (Grant No: 71402012), and the Natural Science Foundation of Education in Chongqing(KJ130402).

Hao Li,Meirong Tan,Liuxin Zou, (2016) Dynamic Pricing for Perishable Products with Control of Cancellation Demands and Strategic Consumer. Open Journal of Social Sciences,04,1-9. doi: 10.4236/jss.2016.47001