Optimal Pricing and Ordering Policy for Deteriorating Items with Stock-and-Price Dependent Demand and Presale Rebate

This paper considers an EOQ inventory model with presale policy for deteriorating items, in which the demand rate depends on both on-hand inventory and selling price. Under the assumption that all the presale orders are fully backlogged with waiting-time dependent rebate, this study develops several propositions and derives optimal pricing and ordering policy by designing an effective algorithm. Two numerical examples are also given to illustrate the effectiveness of the algorithm. Finally, the sensitivity analysis of the main parameters is provided.


Introduction
A large number of products have deteriorating property which manifests the decay or devaluation.This kind of products has much shorter life cycle, for example, some seasonal fresh products that cannot be stored for a long time even in modern storage conditions, electronic products with a fast updating speed, and fashion clothing.These products should be sold in a much shorter spot-sale period to reduce deteriorating cost and holding cost.Presale policy is an effective marketing strategy to reduce the spot-sale period, which is that customers are encouraged to order in advance to get a rebate related to the preorder time.All the orders will be satisfied when the products launch into the market.In addition, it is crucial to find out what factors mostly impact the demand of product, so we can decide proper ordering quantity.
It is obvious that the price of product highly influences the demand rate; thus the pricing for the product plays an important role in marketing strategy.Meanwhile, the stock level of product can also affect the demand rate.Many researchers have focused on such topic.Hou and Lin [1] considered an EOQ inventory model for deteriorating items with price-and-stock dependent selling rate, in which the proposed model allowed shortage and complete backorder while the shortage cost is constant.You and Hsieh [2] developed an inventory model for a seasonal item over a finite planning time by determining the optimal ordering quantity and price setting/changing strategy.Chang et al. [3] studied the optimal selling price and order quantity under EOQ model for deteriorating items, and the demand depends on the selling price and stock on display.Dye and Hsieh [4] developed an inventory model for deteriorating items with demand rate depending on selling price and stock, and shortages are allowed with the backlogging rate to be a decreasing function of the waiting time, while Giri and Bardhan [5] studied an integrated single-manufacturer and single-retailer supply chain model for deteriorating items with stock-and-price dependent demand.Relevant references can be found in Teng and Chang [6], Panda et al. [7], and so forth.Furthermore, more inventory models for deteriorating items with a variety of demand rate are summarized in Table 1.The difference between the relevant references and our study can be found in the table.
In our study, presale rebate depending on waiting time is given to encourage customers to order in advance.Although presale rebate cost is similar in form to the traditional shortage cost, it is different in meaning; moreover, the presale rebate cost in this paper is more complex and realistic than the traditional shortage cost which is generally considered to be constant.In addition, we assume the demand is price-andstock dependent according to the actual situation.Eventually, we derive optimal pricing and ordering policy by designing an effective algorithm and provide the sensitivity analysis of the parameters to assess their effects on the optimal policy.
The rest of the paper is organized as follows: we introduce some basic notations and assumptions in Section 2. Section 3 establishes an inventory model under presale policy and provides an effective procedure to find the optimal pricing and presale strategy.In Section 4, we use several numerical examples to illustrate the procedure of the optimal strategies and analyze the sensitivity of parameters involved.Finally, we provide a summary of the paper.

Notation and Assumptions
The fundamental notation and assumptions used in this paper are given as below.

Notation
: the purchase price of unit item; ∏( 1 ,  2 , ): the average total profit per unit time.

Assumptions
(i) The replenishment rate is infinite; that is, replenishment is instantaneous.
(ii) Assume the customers are loyal, and all the presale orders will not be canceled until the products are launched into the market.
(iii) The demand rate, (, ): we assumed it to be where () is a function of ,   () < 0, and   () ≥ 0, where  denotes the stock-dependent consumption rate.
(v) The marginal gross profit with respect to price is decreasing; that is, ( −   )() is a strictly concave function of .

Inventory Model with Presale Policy
In this section, we consider an inventory model with presale policy for deteriorating items, and the behavior of the system is depicted in Figure 1.From  = 0 to  1 , the total presale quantity accumulates on account of demand which is fully backlogged and achieves its maximum at time  =   zero.According to the above notations and assumptions, such inventory system can be described by following differential equations: ( With boundary condition The solution of ( 2) can be obtained as ( The maximum inventory level per cycle is The presale quantity per cycle is The ordering quantity over the inventory cycle is Based on (3), the total cost per cycle consists of the following elements: (i) Ordering cost  0 .
(ii) Purchase cost (iii) Sales revenue (iv) Deteriorating cost (v) Holding cost (vi) Presale rebate cost Therefore, the total profit per unit time of the model can be obtained as follows: where Δ =  ℎ + ( + )  +    − .
Taking the first-order partial derivative of ∏( 1 ,  2 , ) with respect to  1 ,  2 , and , respectively, we have For any given selling price , the optimal solution ( * 1 ,  * 2 ) is determined by the following equations: By simplifying ( 14), we have From the analysis of ( 15) and ( 16), we have the following propositions.
Proof.(i) From ( 15) and ( 16), we have If Δ < 0, it is obvious that there does not exist any nonzero feasible solution to satisfy (17), which means that the optimal solution of the maximum profit ∏( 1 ,  2 | ) does not exist.
(ii) Let Taking the first-order derivative of ( 18) with respect to , it yields which means that () is a nondecreasing function in [0, +∞).
Similarly, we know that the right hand-side of ( 17) is a strictly increasing function of  1 and goes to infinite as  1 → ∞.
Proof.Taking the second-order partial derivative of ∏( 1 ,  2 | ) with respect to  1 and  2 , respectively, we have Then, the Hessian matrix satisfies ] 2 > 0. (26) Therefore, we conclude that the stationary point ( * 1 ,  * 2 ) is a global optimal solution for the considered problem.
In the following, we will show that the optimal selling price also exists and is unique for the problem.Proposition 3.For any given ( 1 ,  2 ), there exists a unique optimal selling price such that ∏( |  1 ,  2 ) is maximized.
From the above analysis, we know that the solution of () + ( −   )  () = 0 is the lower bound for the optimal selling price  such that  ∏( |  1 ,  2 )/ = 0. Combining Propositions 2 and 3, we establish an algorithm to obtain the optimal policy of the considered model as follows.

Numerical Examples and Sensitivity Analysis
To demonstrate our theoretical results, we study several numerical examples to explain the algorithm proposed in the above section.) is strictly concave in , as shown in Figure 3.
In order to illustrate the effect of the parameters (, , ) on the optimal policy, that is, optimal pricing, the optimal ordering quantity, and the optimal total profit for the inventory models, the sensitivity analysis is performed on the base of Example 2 by changing the value of only one parameter at a time and keeping the rest of the parameters at their initial values.The results are shown in Table 2.
From Table 2, we can observe the following: (   means that if the presale rebate increases, the presale period will get shorter to reduce the rebate cost.Moreover, the optimal order quantity and profit will decrease, while the selling price and spot-sale period will increase. (2) As  increases,  * 2 ,  * ,  * , and ∏( * 1 ,  * 2 ,  * ) will decrease, and  * 1 will increase.It implies that the increase in deterioration cost can lengthen the presale period and then shorten the spot-sale period.As a result, the selling price will decrease for sales promotion; moreover, the optimal order quantity and profit will decrease.
(3) As  increases,  * 1 will decrease, and  * 2 ,  * ,  * , and ∏( * 1 ,  * 2 ,  * ) will increase.It means that higher demand rate in spot-sale period will lead to shortening the presale period, while the selling price, the optimal order quantity, and profit will increase.

Conclusion
In this paper, we study an EOQ inventory model with presale policy for deteriorating items in which the demand rate depends on both on-hand inventory and the price of items.By analyzing the inventory model, an optimal pricing and inventory policy is proposed.We also use several numerical examples to illustrate the solution procedure.Moreover, the sensitivity analysis of the parameters is provided to assess their effects on the optimal policy of the studied problem.From the results of numerical experiments, we find that  and  have a negative effect on the profit of inventory system, while  has a positive effect on the profit of inventory system.In addition, this paper provides an interesting topic for further study of inventory models.It also can be extended in other ways, that is, considering the nonconstant or noninstantaneous deterioration rate and others.

Figure 1 :
Figure 1: Graphical representation of inventory level over the cycle.

( 4 )
In general, the fluctuation of  has more effect on  * 1 , ∏( * 1 ,  * 2 ,  * ) than that of  and , and the fluctuation of , , and  each has more effect on  * 2 ,  * than that of  * .

Table 2 :
The effect of parameters on the optimal policy for presale model.