optimal ordering and pricing policy for an inventory system with order cancellations

OR Spectrum (2007) 29:661–679DOI 10.1007/s00291-006-0067-y

R E G U L A R A RT I C L E

Optimal ordering and pricing policy for an inventorysystem with order cancellations

Peng-Sheng You · Mon-Ting Wu

Published online: 14 September 2006© Springer-Verlag 2006

Abstract This paper investigates the problem of ordering and pricing over afinite time planning horizon for an inventory system with advance sales andspot sales. It is assumed that the planning horizon is divided into several salescycles, each of which is divided into an advance sales period and a spot salesperiod. During the advance sales period, all customers are required to makereservations for their orders and will receive them at the arrival time of thereplenishment orders. In the case of the spot sales periods, all customers receivetheir orders at the time of the purchase. In actual practice, since customers withreservations may cancel their orders before receiving them, this paper considersthis phenomenon and develops a continuous time inventory model to deal withthe proposed problem. This paper maximizes the total profit over a finite timeplanning horizon by determining the optimal advance sales price, spot salesprice, order size, and replenishment frequency. Analysis of results shows that asimple algorithm can be developed to arrive at an optimal decision.

Keywords Inventory · Price · Order · Backorder · Cancellation

1 Introduction

Advance sales systems are prevalent in the service and retailing industries.Examples in the real world include airlines selling discount tickets to ad-vance purchase customers, hotels selling discount rooms to advance booking

P.-S. You (B) · M.-T. WuGraduate Institute of Transportation and Logistics, National Chia-Yi University,300 Shiue-Fu Road, Chia-Yi 600, Taiwane-mail: [email protected]

662 P.-S. You, M.-T. Wu

customers, and publishers offering discount prices to customers who purchasebooks in advance (e.g., Weatherford and Pfeifer 1994).

The reasons for firms to offer advance purchase discounts are partially de-rived from the effects of exploiting economies of scale and revenues. For exam-ple, it is a business practice that consumers with relatively lower valuations haveincentives to make advance purchase because consumers with higher valuationsmay increase the price they are willing to pay in the spot market. This scenarioleads to a situation whereby firms use advance purchase discount policies toattract customers with lower valuations in order to improve their profits. Xieand Shugan (2001) suggested that offering discount prices during the advancesales period can improve the overall sales revenues as compared to the modelusing only a spot sales price. Tang et al. (2004) showed that customers whopurchased products such as moon cakes in advance can benefit by paying lessand gaining the seller’s credibility.

Many works have discussed the applications or advantages of the advancesales systems. Moe and Fader (2002) used the pattern of advance sales to forecastfuture product sales and market potentials. Shugan and Xie (2000) discussed themarketing implications of separating purchase from consumption with empha-sis on advance sales. They suggested that the advance sales can increase theprofit of service providers. McCardle et al. (2004) pointed out that some newphysical products such as music CDs, video games, or books can be appropriatefor an advance booking discount program. For instance, in order to forecastthe appropriate quantities for pre-launching a new book, a retailer can hold anadvance sale to predict the future sales. Under an advance sales system, custom-ers can purchase books in advance at advance sales prices during the advanceselling period. Bookstores can deliver books to customers who purchase themin advance once the replenishment orders arrive. Then, they can leave the addi-tional volumes for customers who purchase them at spot sales prices during thespot sales period. After all the books are sold out, another cycle begins.

Engaging in advance sales has great potential to yield higher profits for manyindustries. However, when adopting such a system, organizations may face someproblems. One of these problems involves the cancellation of orders. It is notunusual for customers to withdraw their advance purchase orders during theadvance sales period. Ignoring this phenomenon may lead to the situation ofover-estimating the demand and reduced profits due to higher inventory pur-chasing and holding costs. To avoid such a situation, it is worthwhile to considerthe phenomenon of order cancellations when making inventory decisions.

Another problem is that of pricing. It can be noted that many practicesreveal that the advance purchase customers are expected to purchase a productat a price lower than the spot sales price. Usually, a strategy using a discountprice during the advance sales period seems to be more attractive to advancepurchase customers than a strategy without any discount. In other words, suchprice differentiation may play a strategic role in a firm whose earnings may beaffected to a certain extent. Since a poor price setting may result in tremendouslosses, how to set the spot sales price and the advance sales price has becomeimportant decision-making problems.

Optimal ordering and pricing policy 663

This paper tries to deal with the aforementioned problems from the deter-ministic aspect. The purpose is to maximize the profit over a finite planninghorizon by determining the optimal advance sales price, spot sales price, ordersize, and replenishment frequency.

The literature which is particularly relevant to this study include studies oninventory models with backorders, price sensitive demand, order cancellations,and non-linear demand functions. Hariga (1995) dealt with an inventory systemwherein shortages are fully backordered and all the replenishment cycle are thesame. His paper aimed to minimize the total inventory cost by determining theoptimal number of replenishments over a finite time horizon. Luo (1998) sug-gested an integrated inventory model for perishable goods with considerationsof the impact of the backorders, pricing and advertising decisions.

Hsu and Lowe (2001) investigated variants of the economic lot sizing problemwhen the backorder cost functions depend on the periods in which inventory isproduced and the periods in which backorders are filled. Chu et al. (2001) ana-lyzed a two-segment partial backorder inventory model with a time-dependentbackorder cost. A solution procedure was developed for finding the optimalorder quantity.

Meanwhile, Argon et al. (2001) investigated an inventory system when thedemands are deterministic but sensitive to backorder realization. Boyaci andGallego (2001) investigated an inventory model with a constraint on expectedbackorders. They presented an iterative-substitution procedure to compute theoptimal policy. Lau and Lau (2002) addressed a continuous-review order-quan-tity-reorder-point inventory system with backorders. They presented a simplerexpression to approximate the average inventory level. Dye and Ouyang (2005)considered a deterministic EOQ model for perishable items when the backor-der demand is time-dependent. They showed that the optimal replenishmentpolicy uniquely exists. Resh et al. (1976) investigated an inventory model witha linear time dependent demand. They developed an algorithm for finding theoptimal replenishment amount and times.

Barbosa and Friedman (1978) investigated an inventory model with a power-form demand rate. Chang and Dye analyzed an inventory model when thebackorder rate declines with the length of the waiting time. Numerical exam-ples with exponential and linear demand functions were used to explain theanalytic result. Teng et al. (2003) extended Chang and Dye (1999) work to acase with a non-constant purchase cost. A solution procedure was proposed tofind the optimal replenishment schedule.

Barbosa and Friedman (1978) investigated a deterministic inventory lot sizemodel wherein shortages are not allowed and the demand rate is of the powerform b(t) = krt where k > 0 and r > 0 are known constants. Yang et al.(2002) extended Barbosa and Friedman’s (1978) work by allowing shortages.They presented a forward recursive method to find the optimal schedule ofreplenishments.

Research overviews on pricing and inventory policies can be found in theliterature reviews of McGill and Ryzin (1999), Petruzzi and Dada (1999) andWeatherford and Bodily (1992). Earlier research on the coordination of

664 P.-S. You, M.-T. Wu

ordering and pricing dates back to the work of Whitin (1955) who dealt with thecombined problem of price and inventory control under a linear price depen-dent demand. Kunreuther and Richard (1971) developed the optimal pricingand ordering decisions for an inventory system under the assumption that all de-mand must be satisfied. Kunreuther and Schrage (1973) proposed an algorithmfor determining the pricing and ordering decision for an item over a fixed timehorizon. In addition, Baker and Urban (1988) considered a pricing inventorymodel without a backorder, in which the demand is dependent on the inventorylevel and is assumed to be the form of d(i) = αiβ , where i is the inventory level,and α and β are respectively a scale and a shape parameters. A non-linear pro-gramming algorithm was applied to obtain the optimal order level and the orderpoint. Gallego and Ryzin (1994) proposed a dynamic pricing model for sellinga fixed number of items by a deadline. They found the optimal pricing policy ina closed form under the exponential demand. Abad (1996) developed a pricingand lot sizing problem with partial backorder for perishable commodities. Asolution procedure was developed to solve the problem optimally.

Bolander et al. (1999) developed an inventory model for finding lot sizequantity decisions and related transfer pricing policies in a decentralized man-ufacture. Burnetas and Simth (2000) developed an adaptive policy for a pricingand ordering problem of a perishable inventory under incomplete informationon the distribution of demand.

Additionally, Abad (2001) took the reselling case into consideration whereinthe backorder is dependent on waiting time and the selling prices are treated asexogenous parameters. Abad (2003) investigated a pricing and lot-sizing modelfor a perishable product under a finite production rate, an exponential decayand a partial backordering. A solution procedure was developed for obtainingthe optimal decisions.

Chen (2003) considered a pricing inventory model with backorders. Theconcept of symmetric k-convex functions was introduced to derive the charac-terization of the optimal ordering and pricing policy. Chun (2003) developedan inventory model for determining the optimal list price and order quantityfor a seasonal/perishable product which is sold for a limited period of time andwhere the list price is posted at the start of a sales period. You (2003) dealt withthe combined problems of dynamic pricing, lot sizing and overbooking. Tengand Chang (2005) provided the necessary conditions to determine the optimalsolution for an EPQ model in which demand is dependent on price and stock.Other related pricing literature are found in the field of revenue managementproblems (e.g., Weatherford and Bodily 1992; You 1999), although, most ofthese problems focus on advance sales system only.

Although the aforementioned studies do not consider the situation ofadvance sales, spot sales and order cancellations at the same time, this problemoccurs in actual practice.

Considering this situation, this paper addresses the simultaneous determina-tion of ordering policy, advance sales price and spot sales price. The decisionmaker is assumed to be able to shift the demand rate by means of pricingpolicy.


The purpose of this paper is to find the optimal decision for maximizing thetotal profit over a finite time planning horizon. The decision rules include (1)the order size, which is placed at the start of every cycle; (2) the advance salesprice, which is set during advance sales periods; and (3) the spot sales price,which is set during the spot sales periods.

2 Assumptions and formulation

In this section, we develop the mathematical model for this problem. Consideran inventory system where a retailer purchases and sells an item over a finiteplanning horizon L. Assume that the demand for the item depends on the salesprice and follows a known demand function d(p), where d(p) is decreasing in thesales price and d(p)p is concave in the sales price. The concavity of d(p)p stemsfrom the standard economic assumption that the marginal revenue decreasesas output increases (Gallego and Ryzin 1994).

Suppose the firm divides the total planning horizon into n (a decision vari-able) equal time sales cycles with equal time intervals. The decision variable nalso represents the replenishment frequency. Let T denote the length of a cycle.Then, T is dependent on n and is given by L/n. Postulate that the firm employsthe advance and spot sales systems to sell their purchased items. A cycle isdivided into an advance sales period and a spot sales period. The sequence ofthe sales periods in a cycle is that of an advance sales period followed by a spotsales period. The length of an advance sales period in any cycle is assumed tobe a constant value k, while the length of a spot sales period is T − k, whichdepends on the number of sales periods, n with n < Nmax where Nmax is apredetermined maximum number of cycles.

During the advance sales period, all customers are required to make reserva-tions and are offered a unit advance sales price p1 for their purchases. Assumethat an order quantity Q arrives at the end of each advance sales period. Inaddition, we assume that all the demands during the advance sales periods areimmediately satisfied upon the arrival time of replenishments. During the spotsales period, customers purchase the item at unit spot sales price p2 and imme-diately receive their purchased items. Moreover, we assume that the fraction ofthe number of order cancellation is a constant rate β and that those customerswho cancel their order are partially refunded. The refund amount is assumed tobe gp1, where g represents the ratio of refund to advance sales price. Assumethat a fixed setup cost, s, is incurred each time when a replenishment is madeand that an inventory holding cost, h, is incurred when one unit of inventory iscarried for one unit of time. Finally, we assume that the unit purchasing cost isc. The components of revenues and costs considered in this paper include salesrevenues, refund, setup cost, inventory holding cost and purchasing cost. Thebenefits from cancelled orders are included in the sales revenues. It is noted thatthe sales revenues are influenced by the sales price and sales volume. If the firmwanted to increase the sales volume, the firm would have to reduce the salesprice. However, economies of scale in pricing may motivate the firm to increase

666 P.-S. You, M.-T. Wu

the sales price since increasing it usually increases the sales profit per unit sold.In addition, if the firm wanted to reduce the holding cost per unit sold, thefirm would have to reduce the cycle inventory. However, economies of scale inreplenishment motivate the firm to increase the cycle inventory since increasingit usually decreases the setup cost per unit sold. Taken together, economies ofscale in pricing and replenishment motivate the firm to make the trade-offsthat maximize the sum of the above revenues and costs by determining theoptimal advance sales price, spot sales price, order size, and replenishment fre-quency. Now, we will develop our model. We summarize the notation as follows:

Notation:

L = total planning horizon,

n = the number of cycles (a decision variable),

T = the length of a cycle time with T = L/n,

p1 = the advance sales price (a decision variable),

p2 = the spot sales price (a decision variable),

g = the ratio of refund to advance sales price,

d(p) = unit time demand rate when a sale price is set at p,

β = the fraction of the number of order cancellations,

k = the length of advance sales period,

c = unit purchasing cost,

s = ordering cost per order, and

h = unit inventory carrying cost per unit time.

For a cycle, let I1(t) and I2(t) be the inventory level at time t during an advancesales period and a spot sales period, respectively. Let Imax and Ib be the max-imum inventory level and the maximum advance sales amount, respectively.Figure 1 depicts the inventory level at any time. At time t = 0 of a cycle, theinventory level is zero. Since, during the advance sales period, there is demandat a rate, d(p1), and a cancellation fraction at a constant rate, β, the inventorylevel will change at a rate −d(p1) − βI1(t). At time k, the cumulated advancesales demand reaches −Ib. Immediately after the time k, a replenishment ordersize Q = Imax + Ib arrives, the advance sales demand during this period issatisfied and the inventory level reaches I2(k) = Imax. After time t = k, theinventory level decreases at a rate d(p2) due to demand. Thus, the inventorylevel at this stage changes at a rate −d(p2). At time T, the inventory levelreaches zero again and another cycle starts. This process repeats until the endof the planning horizon.


Fig. 1 Behavior of inventory level with time

Now, we will develop the mathematical model. According to the previousanalysis, the inventory system I1(t) and I2(t) can be expressed as the followingdifferential equations

dI1(t)dt

= −d(p1) − βI1(t), 0 ≤ t < k, (1)

dI2(t)dt

= −d(p2), k ≤ t ≤ T. (2)

Solving the above equations by using the boundary conditions of I1(0) = 0and I2(k) = Imax yields

I1(t) = − (1 − e−βt)d(p1)

β, 0 ≤ t < k, (3)

I2(t) = d(p2)(k − t) + Imax, k ≤ t ≤ T. (4)

Using Eqs. (3) and (4), we develop the profit function. The components ofprofit function are composed of sales revenues, refund, holding cost, purchas-ing cost and setup cost. Let Ns and Ns respectively denote the number of salesbetween time interval [0, k] and [k, T] when the length of a cycle is set at thevalue of T. Then, we have

Ns = d(p1)k, (5)

Ns = d(p2)(T − k). (6)

Let Nc denote the number of order cancellations in a cycle. Then, since thecumulated reservation amount at time k is Ib and the sales amount during [0, k]is Ns, it follows that

Nc = Ns − Ib = d(p1)k − Ib. (7)

Let H denote the inventory carrying cost in a cycle. Then

H =T∫

k

hI2(t)dt = −0.5hd(p2)(T2 − k2) + h(Imax + d(p2)k)(T − k). (8)

668 P.-S. You, M.-T. Wu

From above, we obtain that the total revenues from the advance sales isnNsp1, the total revenues from spot sales is nNsp2, the total refund is nNcgp1,the total holding cost is nH, the total purchasing cost is n(Imax + Ib)c, andthe total setup cost is ns when the replenishment frequency is set at n. LetF(n, p1, p2) denote the total profit when replenishment frequency is set at n.Then

F(n, p1, p2) = n(Nsp1 + Nsp2 − Ncgp1 − H − (Imax + Ib)c − s). (9)

This paper aims to determine n, p1, p2 and Q = Imax + Ib to maximizeF(n, p1, p2) subject to the constraints of p1 ≥ 0, p2 ≥ 0, d(p1) ≥ 0 and d(p2) ≥ 0.

3 Analysis

In this section, we will develop a solution procedure to find the optimal decisionfor the proposed problem. First, we will derive the relationship between Q withn, p1 and p2. Using the boundary conditions of I1(k) = −Ib and I2(T) = 0, wehave

Ib = d(p1)(1 − e−βk)/β, (10)

Imax = d(p2)(T − k) (11)

from which we obtain

Q = Ib + Imax = d(p1)(1 − e−βk)/β + d(p2)(T − k). (12)

Let u1(β) = βk − 1 + e−βk. Then, since u1(β) is increasing in β and u1(0) = 0.Thus, by (7) and (10) we have

Nc = d(p1)/βu1(β) > 0. (13)

Substituting (10) and (11) into (9) yields

F(n, p1, p2) = nd(p1)p1w1/β + nd(p2)(T − k)p2 − nc(1 − e−βk)d(p1)/β

− 0.5nd(p2)(T − k)(h(T − k) + 2c) − ns (14)

where

w1 = g(1 − e−βk) + βk(1 − g). (15)

Lemma 3.1 Suppose d(p) is convex in p and d(p)p is concave in p. Then, theprofit function F(n, p1, p2) is concave in p1.


Proof Taking the first and second derivatives with respective to p1 gives

∂F(n, p1, p2)

∂p1=n

ddp1

d(p1)w1p1/β+nd(p1)w1/β− nc(1−e−βk)d

dp1d(p1)/β,

(16)

∂2F(n, p1, p2)

∂p21

=nw1

(d2

dp21

d(p1)p1 + 2d

dp1d(p1)

)/β− nc(1 − e−βk)

d2

dp21

d(p1)/β.

(17)

Note the assumption that d(p)p is concave in p implies that d2

dp21d(p1)p1 +

2 ddp1

d(p1) ≤ 0. Thus, we have completed the proof. ��

Lemma 3.2 Suppose d(p) is convex in p and d(p)p is concave in p. Then, theprofit function F(n, p1, p2) is concave in p2.

Proof Taking the first and second derivatives with respective to p2 gives

∂F(n, p1, p2)

∂p2= n(T−k)

(d

dp2d(p2)p2 + d(p2) − d

dp2d(p2)(0.5h(T − k) + c)

),

(18)

∂2F(n, p1, p2)

∂p22

= n(T − k)

(d2

dp22

d(p2)p2 + 2d

dp2d(p2)

− d2

dp22

d(p2)(0.5h(T − k) + c)

). (19)

Since d2

dp22d(p2)p2 + 2 d

dp2d(p2) ≤ 0, it follows that ∂2F

∂p22

< 0. ��

Definition For fixed n, let p1(n) and p2(n) be respectively the solutions to the firstorder conditions of ∂F(n,p1,p2)

∂p1= 0 and ∂F(n,p1,p2)

∂p2= 0.

Lemma 3.3 Suppose d(p) is convex in p and d(p)p is concave in p. Then, thedeterminant of the Hessian matrix is negative definite.

Proof Since ∂2

∂p1∂p2F(n, p1, p2) = 0, we have ∂2F(n,p1,p2)

∂p21

∂2F(n,p1,p2)

∂p22

−(∂2F(n,p1,p2)

∂p1∂p2)2 > 0. Combining this and Lemmas 3.1 and 3.2, the results fol-

lows. ��According to Lemmas 3.1–3.3, p1(n) and p2(n) are the optimal sales prices if

d(p) is convex in p, d(p)p is concave in p, and the constraints of p1 ≥ 0, p2 ≥ 0,d(p1) ≥ 0 and d(p2) ≥ 0 are satisfied.

670 P.-S. You, M.-T. Wu

3.1 Case in which demand is a linear function of price

In this subsection, demand is assumed to be price-dependent and follow thelinear function of d(p) = a − bp, where a and b are constants. For a givenparameter a∗ = b(c+ (L−k)h), if parameter a is not larger than a∗, for demandto be positive the sales price must be set at a level less than c + (L − k)h. In thiscase, even if an optimal pricing policy is employed, a firm cannot get positiveprofit since the sales price cannot cover the sum of purchasing cost, holding andsetup costs, and so the firm should avoid this business. Below, we only considerthe case a > a∗. Substituting d(p) = a − bp into Eqs. (16) and (18), we obtain

∂

∂p1F(n, p1, p2) = −nbw1p1/β + n(a − bp1)w1/β + nbc(1 − e−βk)/β, (20)

∂

∂p2F(n, p1, p2) = −n(T − k)(a − 2bp2 + 0.5b(h(T − k) + 2c)). (21)

Solving the above equation system gives

p1(n) = 0.5a/b + 0.5c(1 − e−βk)/w1, (22)

p2(n) = 0.5a/b + 0.5c + 0.25h(T − k). (23)

Now, we will show that for fixed n, p1(n) and p2(n) are the optimal sales prices.To show this, we need the following lemmas.

Lemma 3.4 0 < p1(n) ≤ a/b.

Proof It is clear that p1(n) > 0. The difference between p1(n) and a/b is given byp1(n)−a/b = −0.5a/b+0.5c(1−e−βk)/w1. Note a ≥ b(c+(L−k)h) ≥ bc. Thus,p1(n)−a/b ≤ 0 if (1−e−βk) < w1. By (15) and the fact that u1(β) > 0 as shownbefore, we obtain (1−e−βk)−w1 = −(1−g)(e−βk−1+βk) = −(1−g)u1(β) < 0and thus we have completed the proof. ��Lemma 3.5 0 < p2(n) ≤ a/b.

Proof The left-hand side is clear. Let u2(a) = a/b − p2(n). Then, we havedu2(a)

da = 0.5/b > 0 and u2(a) = 0.25h(T − k) > 0 for a = b(c + (L − k)h) fromwhich we have u2(a) > 0 for a ≥ b(c + (L − k)h). Thus, we obtain p2(n) ≤ a/b.

��Theorem 3.1 For fixed n, p1(n) and p2(n) are the optimal sales prices.

Proof Since d2d(p)

dp2 ≥ 0 and d2d(p)pdp2 ≤ 0, we have from Lemmas 3.1 to 3.3 that

the determinant of the Hessian matrix is negative definite. Note the statementsof 0 < p1(n) ≤ a/b from Lemma 3.4 and 0 < p2(n) ≤ a/b from Lemma 3.5imply that d(p1(n)) > 0 and d(p2(n)) > 0. Thus, we have completed the proof.

��


Lemma 3.6 p1(n) and p2(n) are respectively independent of and decreasing withreplenishment frequency n.

Proof The former is clear. The later is since p2(n+1)−p2(n) = −0.25hL/(n(n+1)) < 0. Thus, we have completed the proof. ��Theorem 3.2 F(n, p1(n), p2(n)) is concave in n.

Proof Since p1(n) is independent of n, the statement holds if G(a) = F(n +2, p1, p2(n)) − 2F(n + 1, p1, p2(n)) + F(n, p1, p2(n)) ≤ 0. Substituting p2 = p2(n)

and T = L/n into F(n, p1, p2) gives

F(n, p1, p2(n)) = n(w2 + w3 − w4 − w5 − w6 − w7) + w8 + w9/n + w10/n2

(24)

where

w2 = w1(a − bp1)p1/β, (25)

w3 = cb(1 − e−βk)p1/β, (26)

w4 = 1/16k3bh2, (27)

w5 = 0.25k2(a − bc)h, (28)

w6 = 0.25(a − bc)2k/b, (29)

w7 = (ac + βs − ace−βk)/β, (30)

w8 = 1/16L(2a − 2bc + hbk)(2a − 2bc + 3hbk)/b, (31)

w9 = 1/16L2h(4bc − 3hbk − 4a), (32)

w10 = 1/16bL3h2 (33)

from which we have

G(a) = −hL2(3hbk − 4cb + 4a)

8n(n + 1)(n + 2)+ L3h2b(2 + 6n + 3n2)

8n2(n + 1)2(n + 2)2 . (34)

Note a > b(c + (L − k)h). Thus, we have G(a) ≤ G(b(c + (L − k)h)) = V(k)

for all a > c + (L − k)h where

V(k) = bh2L2(−4Ln3 − 9Ln2 − 2nL + kn3 + 3kn2 + 2kn + 2L)

8n2(n + 2)2(n + 1)2 . (35)

Since V(k) is increasing in k and k ≤ T = L/n, we have G(a) ≤ V(L/n) =−h2bL3(4n3 + 8n2 − n − 4)/(8n2(n + 1)2(n + 2)2) < 0. Thus we have completedthe proof. ��Solution procedure:

1. let n = 0 and F∗ = 0.

672 P.-S. You, M.-T. Wu

2. n = n + 1.3. compute p1(n), p2(n) by (22) and (23), respectively.4. substitute d(p) = a − bp, p1(n) and p2(n) into (14) to compute F(n, p1(n),

p2(n)).5. if F∗ < F(n, p1(n), p2(n)), then

let n∗ = n, p∗1 = p1(n), p∗

2 = p2(n) and F∗ = F(n, p1(n), p2(n)).return to step 2.

elsesubstitute p∗

1 and p∗2 into (12) to compute Q(n∗) and let Q∗ = Q(n∗).

go to step 6.end if

6. print out the optimal solutions n∗, p∗1, p∗

2 and Q∗.

Example 1 We provide the following example to illustrate the problem. Sup-pose a firm purchases and sells an item over L = 500 units of time. The unit costfor this item is c = $10. The decision-maker divides the planning horizon intoseveral cycles. The sequence in a cycle is that of an advance sale period followedby a spot sale period. The sales length of an advance sales period is pre-specifiedand assumed to be k = 40 and the length of a spot sales period is dependent onthe number of cycles, n. Demand for the item is assumed to follow the linearfunction of d(p) = 10 − 0.25p. During the lead time period, we assume that thefirm keeps no inventory on hand and asks customers to make reservations fortheir purchases. It is assumed that all demand during the advance sales periodis immediately provided upon the arrival time of replenishments.

Suppose that the item is sold at an advance sales price p1 during advancesales period and a spot sales price p2 is offered during spot sales period. Fur-thermore, we assume that customers with reservations may cancel their ordersand the fraction of cancellations to the number of advance sales amount isassumed to be β = 0.02. In addition, the fraction of refund to the advance salesprice is assumed to be g = 0.8. Finally, we assume that the inventory carryingcost per unit per unit time is h = 0.01, a setup cost s = 100 is charged for eachreplenishment and the maximum number of cycles is Nmax = 10.

The firm aims to maximize total profit by simultaneously determining (1) thereplenishment frequency, (2) the advance sales price, (3) the spot sales price,and (4) the order size per replenishment.

Using the solution procedure, we can sequentially compute the advance salesprice, spot sales price and order size for distinct replenishment frequency. Thecomputational results are shown in Table 1. This table can be illustrated asfollows: Suppose the replenishment frequency is set at one. Then, from Table 1,we see that the profit, the advance sales price, the spot sales price and theorder size are respectively set at $23,743.39, $24.58, $26.15 and 1698.86 whenthe replenishment frequency is one.

This table also reveals that the total profit of $25,533.43 is maximum whenthe replenishment frequency is set at n = 3, and the optimal advance sales price,the spot sales price and the order size for each cycle are set at $24.58, $25.32


Table 1 Computational results for Example 1

n T(n) k(n) p1(n) p2(n) d(p1) d(p2) Q(n) F(n) Ib(n) Imax

1 500.00 40.00 24.58 26.15 3.85 3.46 1698.86 23743.39 106.11 1592.752 250.00 40.00 24.58 25.52 3.85 3.62 866.05 25367.79 106.11 759.943 166.67 40.00 24.58 25.32 3.85 3.67 571.08 25533.43 106.11 464.974 125.00 40.00 24.58 25.21 3.85 3.70 420.34 25322.17 106.11 314.235 100.00 40.00 24.58 25.15 3.85 3.71 328.86 24958.20 106.11 222.756 83.33 40.00 24.58 25.11 3.85 3.72 267.44 24517.33 106.11 161.337 71.43 40.00 24.58 25.08 3.85 3.73 223.35 24032.31 106.11 117.248 62.50 40.00 24.58 25.06 3.85 3.74 190.17 23519.62 106.11 84.069 55.56 40.00 24.58 25.04 3.85 3.74 164.29 22988.44 106.11 58.1810 50.00 40.00 24.58 25.02 3.85 3.74 143.55 22444.29 106.11 37.44

and 571.08, respectively. In addition, under the optimal policy, the maximumnumber of advance sales is 106.11 and the maximum inventory level is 464.97.

3.2 Case in which demand is an exponential function of price

In this subsection, demand is assumed to be price-dependent and follows theexponential function of d(p) = ae−bp, where a > 0 and b > 0 are constants.In this case, d(p) is convex in p. However, d(p)p is not necessary concave in p.Substituting d(p) = ae−bp into Eqs. (16) and (18), we have

∂

∂p1F(n, p1, p2) = nae−bp1(−bw1p1 + w1 + bc(1 − e−βk))/β, (36)

∂

∂p2F(n, p1, p2) = −nae−bp2(−bp2 + 1 + 0.5bh(T − k) + cb)(T − k) (37)

from which we obtain

p1(n) = 1/b + c(1 − e−βk)/w1, (38)

p2(n) = 1/b + c + 0.5h(T − k). (39)

Below, we will show that the stationary points of p1(n) and p2(n) are optimalsales prices.

Lemma 3.7 p1(n) > 0 and p2(n) > 0.

Proof The statement is immediately from Eqs. (38) and (39). ��

674 P.-S. You, M.-T. Wu

Theorem 3.3 For fixed n, p1(n) and p2(n) are optimal sales prices.

Proof First, we have d(p1(n)) > 0 and d(p2(n)) > 0. Second, by substitutingp1(n), p2(n) and d(p) = ae−bp into Eqs. (17) and (19), we have

∂2

∂p21

F(n, p1, p2)∣∣p1=p1(n),p2=p2(n) = −nw1abe−(w1+bc(1−e−βk))/w1/β < 0, (40)

∂2

∂p22

F(n, p1, p2)∣∣p1=p1(n),p2=p2(n) = −nab(T − k)e−0.5bh(T−k)−bc−1 < 0 (41)

from which we have

∂2F(n, p1, p2)

∂p21

∣∣∣∣p1=p1(n),p2=p2(n)

∂2F(n, p1, p2)

∂p22

∣∣∣∣p1=p1(n),p2=p2(n)

−(

∂2F(n, p1, p2)

∂p1∂p2

∣∣∣∣p1=p1(n),p2=p2(n)

)2

> 0. (42)

Thus, the determinant of the Hessian matrix is negative definite. Combiningthis and Lemma 3.7, we have completed the proof. ��Lemma 3.8 p1(n) and p2(n) are respectively independent of and decreasing withthe replenishment frequency.

Proof The former is immediately from (38). The latter is due to p2(n + 1) −p2(n) = −0.5hL/(n(n + 1)) < 0. ��

The optimal decisions can be found by the similar solution procedure as solu-tion procedure I, except that the n-loop should be proceeded until n = Nmax.

Example 2 In this example, all parameters are assumed to be the same as exam-ple 1 except that d(p) is assumed to be the form of d(p) = 25e−0.1p. Using asimilar solution procedure as in Sect. 3.1, we can sequentially compute theadvance sales price, the spot sales price, and the order size for distinct replen-ishment frequency. The computational results are described in Table 2.

From Table 2, we see that the total profit of $15,079.48 is maximum whenthe replenishment frequency is set at n = 3, and the optimal advance salesprice, spot sales price and order size for each cycle are set at $19.17, $20.63 and503.48, respectively. In addition, under optimal policy, the cumulated backorderat the immediate arrival time of a replenishment is 101.22 and the maximuminventory level is 402.26.

4 Case in which advance sales period is decision variable

In the previous section, we have determined the advance and spot sales pricesunder the condition that the length of the advance sales period is predeter-mined. In this section, we will discuss the case in which advance sales period,



n T(n) k(n) p1(n) p2(n) d(p1) d(p2) Q(n) F(n) Ib(n) Imax

1 500.00 40.00 19.17 22.30 3.68 2.69 1337.80 13369.65 101.22 1236.582 250.00 40.00 19.17 21.05 3.68 3.05 740.91 14801.56 101.22 639.693 166.67 40.00 19.17 20.63 3.68 3.18 503.48 15079.48 101.22 402.264 125.00 40.00 19.17 20.42 3.68 3.24 376.84 15040.37 101.22 275.625 100.00 40.00 19.17 20.30 3.68 3.28 298.23 14869.57 101.22 197.006 83.33 40.00 19.17 20.22 3.68 3.31 244.69 14631.54 101.22 143.477 71.43 40.00 19.17 20.16 3.68 3.33 205.90 14354.56 101.22 104.688 62.50 40.00 19.17 20.11 3.68 3.35 176.50 14053.01 101.22 75.279 55.56 40.00 19.17 20.08 3.68 3.36 153.44 13734.97 101.22 52.2210 50.00 40.00 19.17 20.05 3.68 3.37 134.89 13405.32 101.22 33.67

k, is a control variable. Here, the total profit function F(n, p1, p2) in the pre-vious section is replaced with F(n, p1, p2, k). Let p1(n), p2(n) and k(n) be thestationary points to the equation system of ∂F(n,p1,p2,k)

∂p1= 0, ∂F(n,p1,p2,k)

∂p2= 0 and

∂F(n, p1, p2, k)/∂k = n(gβe−βk + 1 − g)d(p1) − nce−βkd(p1)

+ n(h(T − k) − p2 + c)d(p2) = 0. (43)

Then, the stationary points are optimal solutions if p1(n) > 0, p2(n) > 0,d(p1(n)) > 0, d(p2(n)) > 0 and the Hessian matrix

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

∂2F(n, p1, p2, k)

∂p21

∂2F(n, p1, p2, k)

∂p1∂p2

∂2F(n, p1, p2, k)

∂p1∂k

∂2F(n, p1, p2, k)

∂p1∂p2

∂2F(n, p1, p2, k)

∂p22

∂2F(n, p1, p2, k)

∂p2∂k

∂2F(n, p1, p2, k)

∂p1∂k∂2F(n, p1, p2, k)

∂p2∂k∂2F(n, p1, p2, k)

∂k2

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

(44)

evaluated at the stationary points is negative definite.To solve the equation system of ∂F(n,p1,p2,k)

∂p1= 0, ∂F(n,p1,p2,k)

∂p2= 0 and

∂F(n,p1,p2,k)∂k = 0, we can firstly substitute p1 = p1(n) and p2 = p2(n) into (42) for

solving k(n) and then substitute k = k(n) into p1(n) and p2(n) to compute p1(n)

and p2(n). It is difficult to solve the nonlinear equation∂F(n,p1,p2,k)

∂k

∣∣p1=p1(n),p2=p2(n)

= 0 analytically. However, it can be solved numeri-cally by using iterative approach such as Newton’s method or bivariate search.

Example 3 In this example, we will investigate the case in which demand fol-lows the type of d(p) = a − bp. Substituting d(p) = a − bp, p1 = p1(n) in (22)

676 P.-S. You, M.-T. Wu


n T(n) k(n) p1(n) p2(n) d(p1) d(p2) Q(n) F(n) Ib(n) Imax H1th H2th H3th

1 500.0 29.38 24.70 26.18 3.83 3.46 1711.4 23767.0 85.0 1626.4 −11.8 2782.5 −1242.82 250.0 13.08 24.87 25.59 3.78 3.60 896.9 25743.9 43.5 853.4 −11.8 2801.5 −3423.03 166.7 8.43 24.91 25.40 3.77 3.65 607.0 26359.5 29.3 577.7 −11.8 2811.3 −5637.54 125.0 6.23 24.94 25.30 3.77 3.68 458.6 26621.6 22.0 436.6 −11.9 2816.7 −7860.75 100.0 4.93 24.95 25.24 3.76 3.69 368.5 26740.3 17.7 350.8 −11.9 2820.0 −10087.56 83.3 4.09 24.96 25.20 3.76 3.70 308.0 26786.6 14.8 293.2 −11.9 2822.3 −12316.17 71.4 3.49 24.96 25.17 3.76 3.71 264.6 26791.4 12.7 251.9 −11.9 2824.0 −14545.78 62.5 3.04 24.97 25.15 3.76 3.71 231.8 26770.1 11.1 220.8 −11.9 2825.3 −16775.99 55.6 2.70 24.97 25.13 3.76 3.72 206.3 26731.5 9.9 196.5 −11.9 2826.3 −19006.610 50.0 2.42 24.98 25.12 3.76 3.72 185.9 26680.6 8.9 177.0 −11.9 2827.1 −21237.4

Hjth represents the jth minor determinants of the Hessian matrix

and p2 = p2(n) in (23) into (42) gives

∂F(n, p1, p2, k)

∂k= n

2

(a − bc(1 − e−βk)

w1

) (1 − g + ge−βk

2

(ab

+ c(1 − e−βk)

w1

)− ce−βk

)

−3nh2bk2

16− h(−3hbL + 4an − 4cnb)k

8

− (2an − 2cnb − hbL)(2an − 2cnb − 3hbL)

16nb. (45)

Now, we will use the numerical analysis to find the optimal solutions. Thedata in example 1 is used in this example. We assumed that Nmax = 10. Thevalues of p1(n), p2(n) and k(n) are computed and listed in Table 3. The deter-minants of the Hessian matrix are computed by Maple Software. The principalminor determinants of the Hessian matrix are also shown in Table 3. Table 3shows that p1(n) > 0, p2(n) > 0, d(p1(n)) > 0 and d(p2(n)) > 0 for all n,and the Hessian matrix is negative definite. Thus, for any n, the optimal solu-tions are p1(n), p2(n) and k(n). From these values, we can compute the optimalorder quantities and profits. Table 3 reveals that the total profit of $26,791.4 ismaximum when the replenishment frequency is set at n = 7, and the optimaladvance sales period, advance sales price, spot sales price and order size are setat 3.49, $24.96, $25.17 and 264.6, respectively. In addition, under optimal policy,the cumulated backorder at the immediate arrival time of a replenishment is12.7 and the maximum inventory level is 251.9.

Example 4 In this example, we will investigate the case in whichdemand follows the type of d(p) = ae−bp. Solving the equation∂F(n,p1,p2,k)

∂k

∣∣d(p)=ae−bp,p1=p1(n),p2=p2(n)

= 0 by the similar approach as in Exam-

ple 3, we can obtain k(n) and then p1(n) and p2(n). The data in Example 2 areused in this example. The computational results for all n are shown in Table 4.



n T(n) k(n) p1(n) p2(n) d(p1) d(p2) Q(n) F(n) Ib(n) Imax H1th H2th H3th

1 500.00 52.79 18.90 22.24 3.78 2.71 1333.07 13383.2 123.2 1209.9 −13.8 1674.6 −255.12 250.00 21.81 19.55 21.14 3.54 3.02 751.34 14885.4 62.5 688.8 −13.1 1803.7 −1023.53 166.67 13.87 19.72 20.76 3.48 3.13 521.10 15369.6 42.2 478.9 −13.0 1870.2 −1867.44 125.00 10.19 19.79 20.57 3.45 3.19 398.62 15571.5 31.8 366.8 −13.0 1907.2 −2734.55 100.00 8.05 19.84 20.46 3.44 3.23 322.69 15655.7 25.6 297.1 −13.0 1930.5 −3611.76 83.33 6.65 19.87 20.38 3.43 3.26 271.04 15679.9 21.4 249.7 −13.0 1946.5 −4494.27 71.43 5.67 19.89 20.33 3.42 3.27 233.63 15669.3 18.4 215.3 −13.0 1958.1 −5379.88 62.50 4.94 19.90 20.29 3.42 3.29 205.30 15636.6 16.1 189.2 −13.0 1967.0 −6267.39 55.56 4.38 19.91 20.26 3.41 3.30 183.09 15589.2 14.3 168.8 −13.0 1974.0 −7156.210 50.00 3.93 19.92 20.23 3.41 3.31 165.21 15531.5 12.9 152.3 −13.0 1979.6 −8046.1

From Table 4, we find that p1(n) > 0, p2(n) > 0, d(p1(n)) > 0 and d(p2(n)) > 0for all n, and the Hessian matrix is negative definite. Thus, for any n, the opti-mal solutions are p1(n), p2(n) and k(n). In addition, Table 4 shows that thetotal profit of $15,679.9 is maximum when the replenishment frequency is set atn = 6, and the optimal advance sales period, advance sales price, spot sales priceand order size are set at 6.65, $19.87, $20.38 and 271.04, respectively. In addition,under optimal policy, the cumulated backorder at the immediate arrival timeof a replenishment is 21.4 and the maximum inventory level is 249.7.

5 Conclusion

An inventory model wherein the demands are backordered during the advancesales period while providing immediate deliveries during spot sales period wasdeveloped. Various inventory models have been addressed in the past decade,but few have considered a situation wherein the backorders can be cancelled.

This paper developed a continuous time inventory model by considering thephenomenon of order cancellations. The purpose of the paper is to maximizetotal profit over a finite time planning horizon by determining the optimaladvance sales price, spot sales price, order size, and replenishment frequency.Analysis of results showed that the advance and spot sales prices and the ordersize for each distinct service frequency can be derived by closed form formulasfor linear and exponential demand cases. As a result, we developed a solutionprocedure to find the optimal decisions.

The case with infinite horizon is worth studying. However, the model for thiscase is a complicated nonlinear system since neither the analytical approachnor the numerical solution method used in this paper can be applied to solvethis problem. Thus, we leave it as a future work.

In addition, the cancellation fraction β is assumed to be constant over time.Since the cancellation phenomenon may be dependent on time, a study withnon-stationary β cannot be ignored and is worthy of further research.

678 P.-S. You, M.-T. Wu

Acknowledgements The authors would like to thank two anonymous referees for their helpfulcomments and suggestions that greatly improved the presentation of this paper. This research ispartially supported by National Science Council, Taiwan, under grant NSC 93-2213-E-415-003.

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