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

Authors: Lianxia Zhao; Jianxin You
Journal: Scientific Programming
 (2016)

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## Abstract

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.


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## Body

## 1. 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.Table 1
Our paper versus literatures for key assumptions of inventory models with deteriorating items.
Author/authors Demand rate Deterioration rate Shortage or not Shortage cost Decisionvariable∗ Skouri et al. [8] Time-dependent Weibull distribution Yes Constant t,  T Sicilia et al. [9] Time-dependent Constant Yes Constant t,  T Cheng et al. [10] Time-dependent Constant Yes Constant t Zhao [11, 12] Time-dependent Weibull distribution Yes Constant t Begum et al. [13] Price-dependent Weibull distribution No No p,  T Dye [14] Price-dependent Time-dependent Yes Constant p, t, and  T Yang et al. [15] Price-dependent No Yes Constant t,  T B. Sarkar and S. Sarkar [16] Stock-dependent Time-dependent Yes Constant t,  T Padmanabhan and Vrat [17] Stock-dependent Constant Yes Constant t,  T Avinadav et al. [18] Price-and-time Constant No No p,  Q, and  T Maihami and Nakhai Kamalabadi [19] Price-and-time Noninstantaneous Yes Constant p,  T Hou and Lin [1] Price-and-stock Constant Yes Constant t,  T Giri and Bardhan [5] Price-and-stock No No No t,  n  This paper Price-and-stock Constant Presale Time-dependent p,  t, and  T p   ∗ = selling price,  t = replenishment/out of stock time,  Q = order quantity,  T = inventory cycle, and  n = the number of replenishment.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-and-stock 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 Section2.  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.


## 2. Notation and Assumptions

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


### 2.1. Notation

c p: the purchase price of unit item; A 0: the fixed cost per order; S: the maximum inventory level; Q: the ordering quantity; p: the selling price of unit item θ: the constant deteriorating rate, 0≤θ≪1; t 1: presale period; t 2: spot-sale period; I ( t ): the level of inventory at time t,  0≤t≤T, where T=t1+t2; c d: the cost of each deteriorated item; c h: the inventory holding cost per unit per time; D ( t , p ): the demand rate; c s ( x ): the presale rebate cost; ∏ ( t 1 , t 2 , p ): the average total profit per unit time.


### 2.2. 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,D(t,p): we assumed it to be (1)Dt,p=gp,0≤t≤t1gp+αIt,t1≤t≤T, whereg(p) is a function of p,  g′(p)<0, and g′′(p)≥0, where α denotes the stock-dependent consumption rate. (iv)
c s ( x ) is the presale rebate determined by the waiting time x; that is, cs(x)=c0(eλx-1), where c0 is positive constant and λ∈(0,1). (v)
The marginal gross profit with respect to price is decreasing; that is,(p-cp)g(p) is a strictly concave function of p.


## 2.1. Notation

c p: the purchase price of unit item; A 0: the fixed cost per order; S: the maximum inventory level; Q: the ordering quantity; p: the selling price of unit item θ: the constant deteriorating rate, 0≤θ≪1; t 1: presale period; t 2: spot-sale period; I ( t ): the level of inventory at time t,  0≤t≤T, where T=t1+t2; c d: the cost of each deteriorated item; c h: the inventory holding cost per unit per time; D ( t , p ): the demand rate; c s ( x ): the presale rebate cost; ∏ ( t 1 , t 2 , p ): the average total profit per unit time.


## 2.2. 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,D(t,p): we assumed it to be (1)Dt,p=gp,0≤t≤t1gp+αIt,t1≤t≤T, whereg(p) is a function of p,  g′(p)<0, and g′′(p)≥0, where α denotes the stock-dependent consumption rate. (iv)
c s ( x ) is the presale rebate determined by the waiting time x; that is, cs(x)=c0(eλx-1), where c0 is positive constant and λ∈(0,1). (v)
The marginal gross profit with respect to price is decreasing; that is,(p-cp)g(p) is a strictly concave function of p.


## 3. 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 Figure1. From t=0 to t1, the total presale quantity accumulates on account of demand which is fully backlogged and achieves its maximum at time t=t1. During the time interval [t1,t1+t2), due to customers’ demand and deterioration, on-hand inventory level gradually decreases to zero. According to the above notations and assumptions, such inventory system can be described by following differential equations:(2)dItdt=-gp0≤t≤t1;dItdt=-θIt-gp-αItt1≤t≤t1+t2.With boundary condition I(0)=0,  I(t1+t2)=0.Figure 1
Graphical representation of inventory level over the cycle.The solution of (2) can be obtained as(3)It=-gpt0≤t≤t1,It=gpα+θeα+θt1+t2-t-1t1≤t≤t1+t2.The maximum inventory level per cycle is(4)It1+=gpα+θeα+θt2-1.The presale quantity per cycle is(5)It1-=-gpt1.The ordering quantity over the inventory cycle is(6)Q=It1+-It1-=gpα+θeα+θt2-1+gpt1.Based on (3), the total cost per cycle consists of the following elements:(i)
Ordering costA0.(ii)
Purchase cost(7)cpgpα+θeα+θt2-1+gpt1.(iii)
Sales revenue(8)pgpt1+∫t1t1+t2gp+αItdt.(iv)
Deteriorating cost(9)cdgpα+θeα+θt2-1-∫t1t1+t2gp+αItdt.(v)
Holding cost(10)chgpα+θ∫t1t1+t2eα+θt1+t2-t-1dt.(vi)
Presale rebate cost(11)c0gp∫0t1eλt1-1-1tdt.Therefore, the total profit per unit time of the model can be obtained as follows:(12)∏t1,t2,p=1t1+t2sales revenue-holding cost-ordering cost-presale rebate cost-deteriorated items cost-purchase cost=-gpt1+t2Δα+θ2eα+θt2-1+c0λ2eλt1-12λ2t12-λt1-1-p-cpt1-1α+θpθ+ch+cdθt2-A0t1+t2,where Δ=ch+(α+θ)cp+cdθ-pα.Taking the first-order partial derivative of∏(t1,t2,p) with respect to t1,  t2, and p, respectively, we have(13)∂∏t1,t2,p∂t1=-∏t1,t2,pt1+t2+gpt1+t2p-cp-c0λeλt1-λt1-1,∂∏t1,t2,p∂t2=-∏t1,t2,pt1+t2+gpt1+t2pθ+ch+cdθα+θ-Δα+θeα+θt2,∂∏t1,t2,p∂p=1t1+t2gpα-g′pΔα+θ2eα+θt2-α+θt2-1-c0g′pλ2eλt1-12λ2t12-λt1-1+gp+g′pp-cp.For any given selling price p, the optimal solution (t1∗,t2∗) is determined by the following equations:(14)∂∏t1,t2,p∂t1=0,∂∏t1,t2,p∂t2=0.By simplifying (14), we have(15)∏t1,t2,p-gpp-cp-c0λeλt1-λt1-1=0,(16)∏t1,t2,p-gppθ+ch+cdθα+θ-Δα+θeα+θt2=0.From the analysis of (15) and (16), we have the following propositions.Proposition 1.
For any given pricep, (i) if Δ≤0, then there does not exist any optimal solution (t1,t2) so as to maximize the profit ∏(t1,t2,p); (ii) if Δ>0, then the optimal solution (t1∗,t2∗) which solves (15) and (16) simultaneously not only exists but also is unique.Proof.
(i) From (15) and (16), we have (17)Δα+θeα+θt2-1=c0λeλt1-λt1-1.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 ∏t1,t2∣p does not exist.
IfΔ=0, by solving (17), we have t1=0. Substituting it into (16), we have ∏(0,t2,p)=pθ+ch+cdθ/α+θg(p). Thus, from (12), we obtain A0=0, which means a contradictory. Therefore, if Δ<0, there does not exist any optimal solution (t1,t2) to minimize the profit ∏t1,t2∣p.
(ii) Let(18)fx=Δα+θeα+θx-1-c0λeλt1-λt1-1.Taking the first-order derivative of (18) with respect to x, it yields(19)f′x=Δeα+θx≥0which means that f(x) is a nondecreasing function in [0,+∞).
Similarly, we know that the right hand-side of (17) is a strictly increasing function of t1 and goes to infinite as t1→∞.
On the other hand, from (17), we have Δeα+θt2(dt2/dt1)=c0(eλt1-1)>0, which means dt2/dt1>0, and(20)t2=1α+θln⁡α+θc0λΔeλt1-λt1-1+1.Therefore, for any given t~1, there exists a unique t~2 such that t~2=(1/α+θ)ln(α+θc0/λΔ)eλt~1-λt~1-1+1.
To prove the uniqueness of the solution, substitute (12) into (15), and let(21)Ft1=-gpΔα+θ2eα+θt2-1+p-cpt2-t2α+θpθ+ch+cdθ-A0-c0gpλ2eλt1-12λ2t12-λt1-1-λeλt1-λt1-1t1+t2.Taking the first-order derivative of F(t1) with respect to t1, we have(22)dFt1dt1=-Δgpα+θeα+θt2-1dt2dt1+c0gpλeλt1-λt1-1dt2dt1+λeλt1-1t1+t2=c0gpeλt1-1t1+t2≥0which means that F(t1) is an increasing function in [0,+∞). Using (17), it deduces t2=0 as t1=0; then we have F(0)=-A0<0 and limt1→+∞⁡F(t1)=+∞. With the Intermediate Value Theorem, we can find a unique root t1∗∈[0,+∞) such that F(t1∗)=0.
From the above analysis, it is concluded that ifΔ>0, the optimal solution (t1∗,t2∗) which solves (15) and (16) simultaneously not only exists but also is unique.Proposition 2.
For any givenp, the solution (t1∗,t2∗) of (15) and (16) simultaneously is the global maximum of the profit per unit time.Proof.
Taking the second-order partial derivative of∏t1,t2∣p with respect to t1 and t2, respectively, we have(23)∂2∏t1,t2∣p∂t12t1,t2=t1∗,t2∗=-c0gpt1∗+t2∗eλt1∗-1<0,(24)∂2∏t1,t2∣p∂t22t1,t2=t1∗,t2∗=-Δgpt1∗+t2∗eα+θt2∗<0,(25)∂2∏t1,t2∣p∂t1∂t2t1,t2=t1∗,t2∗=0.Then, the Hessian matrix satisfies(26)H=∂2∏t1,t2∣p∂t12t1,t2=t1∗,t2∗×∂2∏t1,t2∣p∂t22t1,t2=t1∗,t2∗-∂2∏t1,t2∣p∂t1∂t2t1,t2=t1∗,t2∗2>0.Therefore, we conclude that the stationary point (t1∗,t2∗) 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(t1,t2), there exists a unique optimal selling price such that ∏p∣t1,t2 is maximized.Proof.
For any givent1 and t2, the first-order derivation of ∏p∣t1,t2 with respect to p is(27)d∏p∣t1,t2dp=1t1+t2gpα-g′pΔα+θ2eα+θt2-α+θt2-1-c0g′pλ2eλt1-12λ2t12-λt1-1+gp+g′pp-cp.Because g′(p)<0, we have g(p)α-g′(p)Δ>0. Thus, d∏p∣t1,t2/dp=0  provides a solution only if g(p)+p-cpg′(p)<0.
Let(28)Gp=1t1+t2gpα-g′pΔα+θ2eα+θt2-α+θt2-1-c0g′pλ2eλt1-12λ2t12-λt1-1+gp+g′pp-cp.
Since the gross profitg(p)(p-cp) is a strictly concave function of p, which means 2g′(p)+g′′(p)(p-cp)<0, we have(29)G′p=1t1+t2g′pα-g′′pΔα+θ2eα+θt2-α+θt2-1-c0g′′pλ2eλt1-12λ2t12-λt1-1+2g′p+g′′pp-cp<0.Therefore, there exists a unique optimal selling price p∗ which maximizes ∏p∣t1∗,t2∗ and completes the proof.From the above analysis, we know that the solution ofg(p)+p-cpg′(p)=0 is the lower bound for the optimal selling price p such that d∏p∣t1,t2/dp=0. Combining Propositions 2 and 3, we establish an algorithm to obtain the optimal policy of the considered model as follows.Algorithm 4.
Step 1. Start with i=0 and let pi be a solution of gp+p-cpg′(p)=0.
Step 2. Put pi into (15) and (16) to obtain the corresponding values of (t1∗,t2∗); then substitute them into (27) and determine the optimal pi+1.
Step 3. If |pi+1-pi|≤ε, where ε is any small positive number, then set p∗=pi+1 and (t1∗,t2∗,p∗) is the optimal solution to minimize ∏(t1,t2,p); otherwise, set i=i+1 and go back to Step 2.


## 4. Numerical Examples and Sensitivity Analysis

To demonstrate our theoretical results, we study several numerical examples to explain the algorithm proposed in the above section.Example 1.
Consider an inventory system with the following data:A0=25$/order,  α=0.03$,  cp=10.0$/unit,  cd=1.2$/unit,  ch=1.0$/unit,  c0=0.5$/unit,  θ=0.2,  λ=0.3, and g(p)=50e-0.04p, where p∈[10,+∞).
For the inventory model with presale policy, by using Algorithm4, we have t1∗=2.090,  t2∗=0.552,  p∗=37.709$,  Q∗=32.104$, and ∏(t1∗,t2∗,p∗)=209.707$.Example 2.
Consider an inventory system with the following data:A0=25$/order,  α=0.05$,  cp=10.0$/unit,  cd=1.2$/unit,  ch=1.0$/unit,  c0=0.5$/unit,  θ=0.02,  λ=0.6, and g(p)=100-5p, where p∈[10,20].
For the inventory model with presale policy, by using Algorithm4, we have t1∗=1.739,  t2∗=0.669,  p∗=15.130$,  Q∗=59.014$, and ∏(t1∗,t2∗,p∗)=108.783$. The three-dimensional total profit per unit time graph for any given p is shown in Figure 2 and the numerical results indicate that ∏p∣t1∗,t2∗ is strictly concave in p, as shown in Figure 3.Figure 2
Total profit per unit time∏t1,t2∣p∗.Figure 3
Total profit per unit time∏p∣t1∗,t2∗.In order to illustrate the effect of the parameters (λ,θ,α) 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.Table 2
The effect of parameters on the optimal policy for presale model.
Para. Value of para. t1∗ t2∗ p ∗ Q ∗ ∏ ( t 1 ∗ , t 2 ∗ , p ∗ ) λ 0.2 2.792 0.480 15.086 80.609 113.323 0.4 2.087 0.594 15.111 65.851 110.583 0.6 1.739 0.669 15.130 59.014 108.783 0.8 1.517 0.725 15.246 54.882 107.431  θ 0.01 1.725 0.744 15.131 60.532 109.017 0.02 1.739 0.669 15.130 59.014 108.783 0.03 1.749 0.608 15.128 57.236 108.542 0.04 1.758 0.557 15.127 56.752 108.337  α 0.04 1.749 0.608 15.128 57.685 108.542 0.05 1.739 0.669 15.130 59.014 108.783 0.07 1.728 0.731 15.136 60.396 109.128 0.10 1.709 0.851 15.152 63.142 109.486From Table2, we can observe the following:(1)
Asλ increases, t1∗,  Q∗, and ∏(t1∗,t2∗,p∗) will decrease, while t2∗ and p∗ will increase simultaneously, which 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)
Asθ increases, t2∗,  p∗,  Q∗, and ∏(t1∗,t2∗,p∗) will decrease, and t1∗ 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)
Asα increases, t1∗ will decrease, and t2∗,  p∗,  Q∗, and ∏(t1∗,t2∗,p∗) 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.(4)
In general, the fluctuation ofλ has more effect on t1∗,  ∏(t1∗,t2∗,p∗) than that of θ and α, and the fluctuation of λ,  θ, and α each has more effect on t2∗,  Q∗ than that of p∗.


## 5. 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 thatλ and θ have a negative effect on the profit of inventory system, while α 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.



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