int. j. production economics - dost sci-net phil

Int. J. Production Economics 128 (2010) 209–222

Contents lists available at ScienceDirect

Int. J. Production Economics

0925-52

doi:10.1

� Corr

E-m

mverma

journal homepage: www.elsevier.com/locate/ijpe

Comparison of two periodic review models for stochasticand price-sensitive demand environment

Saibal Ray a, Yuyue Song b,�, Manish Verma b

a Desautels Faculty of Management, McGill University, QC, Canada H3A 1G5b Faculty of Business Administration, Memorial University of Newfoundland, NL, Canada A1B 3X5

a r t i c l e i n f o

Article history:

Received 30 October 2009

Accepted 14 July 2010Available online 1 August 2010

Keywords:

Periodic review

Review cycle length

Backordering cost structure

Stochastic demand

Pricing

73/$ - see front matter Crown Copyright & 2

016/j.ijpe.2010.07.019

esponding author.

ail addresses: [email protected] (S. Ray),

@mun.ca (M. Verma).

a b s t r a c t

In this paper, we study two periodic review inventory models which primarily differ in terms of how

backordering cost is charged: time-independent backordering (TIB) model where the backordering cost

is charged per unit backordered and is independent of the length of time for which backorders persist;

and the time-dependent backordering (TDB) model where the backordering cost is charged based on

the number of backorders as well as the length of time for which they are on the books (i.e., it is charged

per unit per unit time). Our objective is to investigate the impact of these two different backordering

cost structures on the optimal decisions of a firm in a stochastic and price-sensitive demand

environment. In order to do so, we first develop a general framework, where both such costs exist, in a

profit maximizing context. Subsequently, we analyze two special cases of this general framework with

either one of the costs—that is, TIB and TDB models—and derive some analytical results regarding the

values and behavior of the optimal decisions for both of them. We then concentrate on comparing the

two models through extensive numerical experiments. Our investigation demonstrates that the TIB

model generally results in longer review periods and lower retail prices. As far as the base stock level is

concerned, we show that it can be higher in either setting; however, the safety stock is considerably lower

for the TIB model. Lastly, indeed if a firm’s backordering cost is indeed time-dependent, then use of the

TIB model for making decisions results in significant profit penalty under most market/operating

conditions (specifically for innovative products), except when demand uncertainty and/or the

backordering cost are quite low (i.e., for mature, commodity products).

Crown Copyright & 2010 Published by Elsevier B.V. All rights reserved.

1. Introduction

The continuing rise of globalization and rapid technologicaladvancements has made effective supply chain management(SCM) a core competency in modern business environment. Thetwo most critical and strategic SCM decisions are setting pricesand managing inventory (Stern and El-Ansary, 1992). There aretwo techniques discussed in the literature regarding inventorymanagement for random demand—continuous review and peri-odic review (see Silver et al., 1998). In the former case, theinventory level is reviewed continuously and a new order isplaced whenever this level reaches the reorder point. Normally,all orders are of the same size, but the time interval betweenorders is random. On the other hand, for periodic review systems,the inventory level is reviewed only at regular intervals of time,and at that point orders are placed to bring the stock position up

010 Published by Elsevier B.V. All

[email protected] (Y.R. Song),

to the base-stock level1; so, order sizes for different periods vary,but the time between orders (i.e., review length) remainsconstant. In reality, suppliers tend to prefer regularity in termsof order intervals. So, periodic review systems are quite popularamong managers (Silver et al., 1998). In this paper we focus on aperiodic review system for inventory management. Moreover, inthe tradition of recent integrated operations-marketing literature(Yano and Gilbert, 2003), we also take into account theinteractions between pricing and inventory decisions whiledeciding on the optimal strategy for a firm.

There is a significant literature related to different aspects ofperiodic review systems like random yield (Wang and Gerchak,1996), emergency replenishments (Bylka, 2005) and variablepurchasing costs (Gavirneni, 2004). The inventory modelingframework in recent dynamic pricing and inventory controlliterature is also a periodic review one (e.g., refer to Polatogluand Sahin, 2000; Chen and Simchi-Levi, 2004; Huh and Janakira-man, 2008; Song et al., 2009, and references therein).

rights reserved.

1 Assuming the fixed cost for ordering to be zero.

www.elsevier.com/locate/ijpe

dx.doi.org/10.1016/j.ijpe.2010.07.019

mailto:[email protected]



dx.doi.org/10.1016/j.ijpe.2010.07.019

S. Ray et al. / Int. J. Production Economics 128 (2010) 209–222210

However, the primary decision variables in all these papers areinventory control and/or retail price. The review cycle length isassumed to be exogenously given. Nevertheless, one of the mostfundamental operational issues in periodic review paradigm is:What should be the right review frequency for a particular system? Inthis paper we consider a periodic review inventory system for asingle item where the demand process is stationary, stochastic,and price sensitive. In our setting, the firm needs to simulta-neously determine the profit maximizing review cycle length,base stock level and retail pricing strategies.

The key feature of our model framework is the backorderingcost structure. In a large proportion of real-life supply contracts,backordering penalties are assessed based on the length of thedelay. The usual contract specifications related to delivery delaysare as follows:

yIf the supplier defaults on his delivery commitments, heshall pay a contractual penalty amounting to—% of the value ofthe relevant order per day (or week) of the delay to theorderer.

Such penalty clauses appear in a variety of industrial sectors likeautomotive components, large machineries, oil and gas, alumi-nium, electronic equipment and retail as well as in governmentcontracts.2 As has been indicated in academic literature, time-dependent penalties are also the norm for steel mills (Gupta andWang, 2007) and for maintenance/repair items (Silver et al.,1998). In fact, recent theoretical inventory management assumesthat customers care more about how long they have to waitcompared to whether or not they have to wait, i.e., based onbackordering cost per unit per unit time (Zipkin, 2000). Never-theless, there are situations in real-life where backordering costdepends only on the number of backorders and not on theirlengths (refer to Silver et al., 1998 for examples).3 Motivated byabove, we analyze and compare two backordering cost scenariosin this paper—one where backordering cost is calculated basedonly on the number of orders backordered in each period andignores the time duration for which backorders persist (time-independent backordering (TIB) model), and the other wherebackordering cost is calculated based on how long each backorderremains on the book in each period (time-dependent back-ordering (TDB) model). One of the primary questions that weinvestigate in this context is that, if indeed the backordering costis time-dependent but the TIB model is used for decision-making,how does it affect managerial decisions and what is the extent ofprofit penalty that the firm will incur.

Although not very popular, there has been some research ininventory management literature investigating the concurrentdetermination of the optimal review cycle length and the optimalbase stock level.4 Chiu (1995) deals with this issue in the context of(partially) perishable items and propose heuristics to determinethe best review length and order quantity. Eynan and Kropp (1998)develop a near-optimal approach to determine the review cyclelength which minimizes the average ordering and holding costs, fora given base stock level. Subsequently, Eynan and Kropp (2007)generalize the above model by incorporating backordering costs forany excess demand. They propose a simple EOQ-like approach tosimultaneously decide on the optimal values of the review cycle

2 Refer to http://www.hansentransmissions.com/data/hansen_GPC.pdf and

http://www.alcoa.com/howmet/en/info_page/pdf/suplperf.pdf for some sample

contracts. More examples are available from the authors on request.3 For example, when product shortages are met by expedited shipping of the

short units and so the major shortage cost is the FedEx waybill which is mostly

independent of time.4 Chiang (2008) studies a model framework where review cycle length is a

random variable, but not a decision for the firm.

length and the base stock level that minimize the average totalinventory-related costs (ordering+holding+backordering). Silverand Robb (2008), on the other hand, use a model like Eynan andKropp (2007) to demonstrate and explain some counterintuitiveresults regarding the behavior of the optimal review cycle lengthwith respect to changes in various model parameters. Note that thebackordering cost in all the above papers is charged per unitbackordered ignoring the time duration for which a backorderpersists (i.e., like TIB model). Flynn (2000) is different from the onesabove since multiple deliveries between two subsequent reviews(rather than one) are allowed. Flynn establishes the optimal reviewinterval and ordering policy for different scenarios and show thatunlinking delivery and review intervals can lead to cost savings.However, the backordering costs in Flynn (2000) is still charged perunit, not taking the time duration into account. Note that Chiang(2008) also analyzes a problem setting similar to Flynn (2000) inTIB context without review length being a decision variable. Theonly periodic review model in this stream of literature which, likeour TDB model, takes time duration into consideration in thebackordering cost calculation is by Rao (2003). Rao establishes thejoint convexity of the average cost function in terms of the base-stock level and the review period, and also performs sensitivityanalysis of the optimal decision variable values.

The distinguishing elements of this paper compared to theprevious literature are two-fold. First, we formulate a generalmodel framework where backordering cost depends both on theamount of backorders and their lengths in a profit maximizingframework by assuming demand to be stochastic and price-sensitive

(all the papers in the previous paragraph use random, but price-independent, demand functions, with the aim of minimizing costs).Subsequently, we analyze the two special cases of the generalframework—TIB and TDB models, and simultaneously optimize forthe period length, base stock level and the retail price. We thenshow that the objective function for both models can be reduced toone-variable optimization problems, and analytically establish anumber of sensitivity results about the optimal decisions. Moreimportantly, the focus of this paper is quite different. We areprimarily interested in understanding the effects of backordering cost

structures on the values and behavior of the optimal decisions as well

as on profits by comparing TIB and TDB models, which has not beenaddressed in the literature before.

Our analysis reveals that, in general, decision making based onTIB model can be significantly misleading if a firm’s backorderingcost is indeed time-dependent. Managers in that case would review

the inventory levels less frequently, charge lower prices to the

customers and hold significantly less safety stock, compared to thetrue optimal. We pinpoint the reason behind such results, and alsoillustrate that the behavior of the TIB model can be somewhatunusual because of its underlying assumptions. From a managerialviewpoint, we have identified the specific business scenarios underwhich the use of TIB model in the place of (true) TDB model resultsin most profit penalty, and cases when it might not be so harmful.

As regards the remainder of the paper, in Section 2, we presentthe general model framework as well as the specific frameworksfor TIB and TDB models. Then, in Sections 3 and 4, we providesome analytical results for TIB and TDB models, respectively. InSection 5, an extensive numerical study is carried out to comparethe two models, and generate managerial insights. Section 6presents our concluding remarks and discusses some potentialfuture research issues.

2. Model framework

Demand function: Our model framework is based on a retailerselling a single item to end customers in a risk-neutral setting.

http://www.hansentransmissions.com/data/hansen_GPC.pdf

http://www.alcoa.com/howmet/en/info_page/pdf/suplperf.pdf

S. Ray et al. / Int. J. Production Economics 128 (2010) 209–222 211

The retail demand function per unit time xð1Þ comprises twoelements. One part is D(p), a deterministic function of the retailprice p. Note that D(p) is non-negative and strictly decreasing on(0, Pu), where Pu is the maximal retail price such that D(p) ispositive on (0, Pu). The second part is e, a non-price-sensitive,positive random variable having support on ð0,1Þ, with mean mand standard deviation s. We specifically assume thatxð1Þ ¼DðpÞe. Such multiplicative demand form is quite commonin the literature and is supported by empirical study (refer toPetruzzi and Dada, 1999; Granot and Yin, 2005). Without loss ofgenerality, we assume m¼ 1 in this paper. Hence the mean andstandard deviation of the per unit time demand random variablexð1Þ are D(p) and DðpÞs, respectively. In addition, suppose thatD(p) exhibits the following properties:

Assumption 1. D(p) is continuously differentiable and strictlydecreasing on (0, Pu). Furthermore, DðpÞ=DuðpÞ ¼ k1þk2p wherek1ðr0Þ and k2ð4�1Þ are two constants.

Note that the above assumption is not at all restrictive. Itmeans that the curvature of D(p), defined as E¼DðpÞD00ðpÞ=DuðpÞ2,is equal to 1�k28p, i.e., the expected demand function has aconstant curvature and it is not too high ðEo2Þ.5 Most of thedemand functions used for studying joint pricing-inventorydecisions, e.g., iso-elastic, exponential and linear, do exhibitconstant curvatures6 (refer to Petruzzi and Dada, 1999; Wanget al., 2004; Granot and Yin, 2005; Song et al., 2008).

Inventory management: The inventory status is reviewed atsome equally distributed discrete time points over the planninghorizon and a fixed cost of Kð40Þ is charged for each review. Theconstant time duration between any two adjacent review pointsdenoted by t is a decision variable. Since there is no fixed costassociated with ordering, a base-stock policy is used for inventorycontrol. That is, after each review, a replenishment order is placed,if necessary, to raise the inventory position up to the base stocklevel S, and the order is received after a lead time L, a positiveconstant. The per unit procurement cost is cð40Þ. The inventoryholding cost is charged for any on-hand inventory at the rate of h

per unit time per unit item. On the other hand, all excess demandsare backordered, and a backordering cost is charged. The primarydifference between the two models is in the way this back-ordering cost is charged; we would discuss this later on. Note thatin the following analysis, for analytical convenience, we use thetime period between arrivals of two successive orders, rather thanbetween placements of two successive orders, for calculatinginventory costs (since L is constant, the two formulations areequivalent).

The order-up-to base stock level S consists of two components:the average demand over the protection interval t+L, and thesafety stock to reduce the risk of stock out. Suppose that therandom variable representing demand during t+L is denoted byxðtþLÞ. Clearly, this random variable has a mean of (t+L)D(p) anda standard deviation of s

ffiffiffiffiffiffiffiffiffitþLp

DðpÞ. So, S can be expressed as

S¼ ½ðtþLÞþzsffiffiffiffiffiffiffiffiffitþL

p�DðpÞ, ð1Þ

where z represents the safety stock factor. In this paper, weapproximate xðtþLÞ by a normal distribution N� ððtþLÞ

5 Economically speaking, E is the elasticity of the slope of inverse demand

p(D). Loosely, one might think of the assumption that E is a constant to be a

smoothness condition on the demand function. Furthermore, Eo2 implies that

demand should not be too convex. This assumption ensures that the firm’s

marginal revenue is downward-sloping, and hence guarantees an interior solution.

Refer to Bresnahan and Reiss (1985) and Ritz (2005) for more details.6 Examples include concave functions like DðpÞ ¼ ða�kpÞg ða40,k40,

0ogo1Þ, as well as convex ones like D(p)¼ap�kða40,k41Þ, DðpÞ ¼ ða�kpÞg

(a40,k40,gZ1 or a40,ko0,go�1), D(p)¼ae�kpða40,k40Þ.

DðpÞ,sffiffiffiffiffiffiffiffiffitþLp

DðpÞÞ. For tractability reason, a common approach inthe inventory literature is to use such a second-order approxima-tion for the demand during the protection interval, i.e., it isassumed to be normally distributed with the precise mean andvariance values given by (t+L)D(p) and s


DðpÞ, respectively(Hadley and Whitin, 1963; Silver et al., 1998; Zipkin, 2000,Chapter 7; Axsaster, 2000, Chapters 3 and 5). Tyworth and O’Neill(1997) have shown this approximation to be quite robust (alsorefer to Lau and Lau, 2003) for a detailed discussion about theaccuracy of the approximation). Our research adds to thecomplexity, compared to the inventory models, in the form ofpricing. Moreover, since our goal is to develop managerialinsights, rather than accurate decision support systems, the aboveapproach is reasonable for this paper. We denote the density andcumulative functions of the standard normal distribution by f(z)and F(z), respectively. Clearly, for a given t, L and D(p), there is anone-to-one relation between S and z. So, the relevant decisionvariables for us are the length of the review cycle, t, the safetystock factor, z, and the retail price of the item, p. The objective is tomaximize the average profit (i.e., per unit time) for this singleitem system by choosing the optimal values of these threedecisions.

Based on the above notations, we are ready to formulate theaverage total profit function pðt,z,pÞ in terms of (t, z, p). We startby developing a general model framework where the back-ordering cost depends both on the number of backorders at theend of the period as well as on the length for which backorderspersist. Specifically:

pðt,z,pÞ ¼ ðp�cÞDðpÞ�1

tfKþhIðt,z,pÞþb1B1ðt,z,pÞþb2B2ðt,z,pÞg,

where Iðt,z,pÞ is the average on-hand inventory level over onereview cycle of length t, h is the holding cost per unit per unittime, B1ðt,z,pÞ is the average number of backorders at the end of areview cycle, b1 is the backordering cost per unit charged on theaverage number of backorders at the end of a review cycle,B2ðt,z,pÞ is the average backordering level over one review cycle oflength t and b2 is the backordering cost per unit per unit timecharged on the backorders during a cycle (B2ðt,z,pÞ can be thoughtof as weighted backordering level where the weight on eachbackorder is the length of time for which that particularbackorder is on the books).7

As discussed before, in this paper, we focus on analyzing twospecial cases of the above general framework—time-independentbackordering (TIB) model in which case b2¼0 and time-dependent backordering (TDB) model for which b1¼0. Our aimis to investigate the effects of different backordering coststructures on the values and behavior of the optimal decisionsas well as profits by comparing TIB and TDB models. The TIBmodel was first proposed by Hadley and Whitin (1963), and isdriven by the following two assumptions:

�

as a

pap

refe

det

Backordering cost is charged per unit backordered, indepen-dent of the time duration for which the demand is back-ordered.
� Backorders are incurred only in very small quantities. This
implies that when a new replenishment order arrives, it isalmost always sufficient to meet any outstanding backorders.8

7 Although most periodic review models with the length of the review period

decision variable assume b2¼0, there are continuous review inventory model

ers where backordering cost includes both b1 and b2 (e.g., Jose et al., 2006 and

rences therein).8 This means I ðt,z,pÞ is also approximate in the TIB model; see below for more

ails.


Note that the TIB model takes the time duration intoconsideration while calculating the holding cost (i.e., it is chargedper unit per unit time). On the other hand, in the TDB model, bothbackordering and holding costs are charged per unit per unit time.In the next two sections we present the average profit expressionsfor the two models.

2.1. Time-independent backordering (TIB) model

In this model: (i) the backordering cost depends only on theaverage number of backorders at the end of a review period, i.e.,the average number of backorders at the end of a review cycle isdetermined and each of them is charged a cost of b and (ii) theholding cost is approximated based on the assumption thatthe number of backorders is small.9 Based on the expressions forthe average inventory holding and backordering costs in Hadleyand Whitin (1963), the profit per unit time for this model can beexpressed as

Gðt,z,pÞ ¼ pDðpÞ�K

tþ

tDðpÞ

2hþhzs

ffiffiffiffiffiffiffiffiffitþL

pDðpÞ

�

þb

t

Z þ1S½x�S�f ðxÞdxþcDðpÞ

�,

where f ðxÞ is the density function of xðtþLÞ. In the above equationthe first term is for the revenue, the second term represents thereview cost, the third and fourth terms are for cycle and safetystock holding costs, respectively, the fifth one represents thebackordering cost, while the last one denotes the procurementcost. Based on normal approximation demand during (t+L), i.e.,xðtþLÞ �NððtþLÞDðpÞ,s


DðpÞÞ, the above expression can berewritten as

Gðt,z,pÞ ¼ ðp�cÞDðpÞ�K

tþ

tDðpÞ

2hþhzs


pDðpÞ

�

þbsffiffiffiffiffiffiffiffiffitþLp

t½f ðzÞ�zð1�FðzÞÞ�DðpÞ

�, ð2Þ

where f(z) and F(z) are the density and cumulative distributionfunctions of the standard normal distribution. As indicated before,the firm wants to maximize G by optimally selecting the reviewperiod t, the safety stock factor z and the retail price p.

2.2. Time-dependent backordering (TDB) model

In this model: (i) each backorder is charged b for each unit oftime it remains unfulfilled; and (ii) the number of backorders canbe arbitrarily large so that a new order might not be able to satisfyall backorders. When the firm places a procurement order at timel, it raises the inventory position to S and this order arrives at thetime point l+L. So, the net inventory level at lþLþt is S�xðLþtÞfor any tA ½0,tÞ. The total holding and backordering cost rate ofnet inventory at the time point lþLþt can then be expressed as:hR S�1½S�x�f ðxÞdxþb

R þ1S ½x�S�f ðxÞdx where f ðxÞ is the density

function of the total demand xðtþLÞ over a time duration oftþLð0rtrtÞ. For simplicity, let

d¼ z

ffiffiffiffiffiffiffiffiffitþLpffiffiffiffiffiffiffiffiffiffitþLp þ

t�tffiffiffiffiffiffiffiffiffiffitþLp

1

s: ð3Þ

9 The ‘‘small’’ number of backorders assumption implies that the holding cost

in each review period can be calculated as the holding cost rate times the average

stock level.

Recall that xðtþLÞ �NððtþLÞDðpÞ,sffiffiffiffiffiffiffiffiffitþLp

DðpÞÞ. This implies thatthe total holding cost rate of net inventory at lþtþL can berewritten as

h

Z S

�1

½S�x�f ðxÞdx¼ hDðpÞ

Z d

�1

½ðt�tÞþsðzffiffiffiffiffiffiffiffiffitþL

p�y

ffiffiffiffiffiffiffiffiffiffitþL

pÞ�f ðyÞdy

¼ hDðpÞf½ðt�tÞþzsffiffiffiffiffiffiffiffiffitþL

p�FðdÞþs


pf ðdÞg

¼ hDðpÞsffiffiffiffiffiffiffiffiffiffitþL

p½dFðdÞþ f ðdÞ� ¼ hsDðpÞIðt,z,tÞ,

where d is as defined in (3) and Iðt,z,tÞ ¼ffiffiffiffiffiffiffiffiffiffitþLp

½dFðdÞþ f ðdÞ�.Similarly, the total backordering cost rate of the net inventory atlþtþL can be rewritten as b

R þ1S ½x�S�f ðxÞdx¼ bDðpÞ

R þ1d ½ðt�tÞþ

sðyffiffiffiffiffiffiffiffiffiffitþLp

�zffiffiffiffiffiffiffiffiffitþLp

Þf ðyÞdy¼ bsDðpÞBðt,z,tÞ where Bðt,z,tÞ ¼ffiffiffiffiffiffiffiffiffiffitþLp

½f ðdÞþdðFðdÞ�1Þ�.Combining the average revenue, the average review cost, and

the average inventory holding and backordering costs, the profitper unit time for the firm can be expressed as follows:

Eðt,z,pÞ ¼ ðp�cÞDðpÞ�1

tKþhsDðpÞ

Z t

0Iðt,z,tÞdt

�

þbsDðpÞ

Z t

0Bðt,z,tÞdt

�: ð4Þ

The firm’s objective again is to maximize E by optimally selectingthe review period t, the safety stock factor z and the retail price p.

3. Analysis of the TIB model

In this section, we analyze the behavior of the profit functionfor the TIB model—Gðt,z,pÞ given in (2), in order to determine therelevant optimal decision variable values. First, we show that themaximization of Gðt,z,pÞ can be reduced to a one-variablemaximization problem in terms of t. Then, we investigate thebehavior of this one-variable function.

For any given review cycle length t and retail price p, partialderivative of G with respect to z yields

@Gðt,z,pÞ

@z¼ sDðpÞ


p b

t½1�FðzÞ��h

� �: ð5Þ

The above expression means that the optimal safety stock factor isonly related to t (independent of p). For any given (t,p), the uniquez(t) maximizing Gðt,z,pÞ is as specified in the following lemma.10

Lemma 1. If tZb=2h, then z(t)¼0; otherwise, z(t) is the unique

solution of 1�FðzÞ ¼ ðh=bÞt. As a consequence of this, z(t) is

decreasing11 for any tAð0,b=2hÞ.

The above lemma clearly shows the effect of ignoring the timeduration while determining the backordering cost in the approx-imate model, alluded to in Section 1. When t is low, the firm triesto reduce backordering costs by holding safety stock; however, ast increases, z(t), and hence the safety stock, decreases. Since thelength for which backorders persist does not matter, when t

becomes larger the firms’s optimal strategy is to focus more oncontrolling the holding cost through reduction of safety stock. Infact, when t is sufficiently large ðZb=2hÞ, the firm does not hold any

safety stock (z(t)¼0). Obviously, the threshold t value (i.e., b=2h) ishigher if the per unit backordering cost is higher, and is lower ifthe cost of holding is higher.

The optimization of the original three-variable functionGðt,z,pÞ can now be reduced to an optimization of a two-variable

10 All proofs are provided in the Appendix.11 Throughout the paper we use increasing/decreasing in the weak sense

unless otherwise specified.


function Gðt,pÞ by substituting z(t):

Gðt,pÞ ¼ ðp�cÞDðpÞ�K

tþ

DðpÞ

2htþbs


tf ðzðtÞÞDðpÞ

� �: ð6Þ

For notational convenience, let BðtÞ ¼ ht=2þbsðffiffiffiffiffiffiffiffiffitþLp

=tÞf ðzðtÞÞ forany tA ð0,þ1Þ. The above equation can then be rewritten as

Gðt,pÞ ¼ ðp�cÞDðpÞ�BðtÞDðpÞ�K

t: ð7Þ

We can analyze the above profit function further to determinethe optimal retail price p, in terms of t.

Lemma 2. For any given review cycle length t, Gðt,pÞ is unimodal in

terms of p. So, the solution of @Gðt,pÞ=@p¼ 0 provides the value of the

optimal price p(t). Specifically, for any tAð0,þ1Þ, we have

pðtÞ ¼c�k1

1þk2þ

1

1þk2BðtÞ: ð8Þ

Substituting the above p(t) in Gðt,pÞ enables us to represent thefirm’s profit function solely in terms of t, and we denote it by GðtÞ.In order to determine the optimal length of the review cycle,which is the maximizer of GðtÞ over ð0,þ1Þ, we first characterizethe behavior of GðtÞ in terms of t on ½b=2h,þ1Þ (note that B(t) hasa unique minimizer for tA ½b=2h,þ1ÞÞ.

Theorem 1. Let tmin be the unique minimizer of B(t) on ½b=2h,þ1Þ,t ¼maxfb=2h,tming, and t0 be the minimal value on ½t ,þ1Þ such that

GðtÞ ¼ ½B00=Bu2þ2=tBu�½k1þk2cþk2B�þ140. Then, there exists at

most one local maximizer of GðtÞ on ½b=2h,þ1Þ and it is on ðt ,t0Þ.Furthermore, GðtÞ is quasi-concave on ðt ,t0Þ.

Next, we focus on the behavior of GðtÞ in terms of t on ð0,b=2hÞ.Differentiation of GðtÞ results in

dGðtÞdt¼�BuðtÞDðpðtÞÞþ

K

t2

and

d2GðtÞdt2

¼�B00D�BuðtÞ2

1þk2Du�

2K

t3:

From dGðtÞ=dt¼ 0, we get BuðtÞDðpðtÞÞ ¼ K=t240. Hence, in order tocharacterize the shape of GðtÞ, it is crucial to understand thebehavior of B(t). So, in what follows, we study the behavior of B(t)over ð0,b=2hÞ. Before doing that, we first present a result aboutrðzÞ ¼ f ðzÞ=ð1�FðzÞÞ in terms of z on ð0,þ1Þ [recall that f(z) and F(z)are the density and cumulative distribution functions of N(0, 1)].

Lemma 3. Let rðzÞ ¼ f ðzÞ=ð1�FðzÞÞ and g(z)¼z[r(z)�z]. Then, g(z)and r(z) exhibit the following properties:

1.
0ogðzÞo1 for any zAð0,þ1Þ. Moreover, g(z) is unimodal in
terms of z on ð0,þ1Þ.
2. r(z) is strictly increasing and convex in terms of z on ð0,þ1Þ.
Now we are ready to study the behavior of B(t) over ð0,b=2hÞ.As BðtÞ ¼ ht=2þbsð


=tÞf ðzðtÞÞ and z(t) is the solution of1�FðzÞ ¼ ðh=bÞt for any tAð0,b=2hÞ, we have

BuðtÞ ¼h

2�bs tþ2L

2t2ffiffiffiffiffiffiffiffiffitþLp f ðzÞ�

h

b


tz

� �ð9Þ

and

B00ðtÞ ¼ bs 3t2þ12tLþ8L2

4t3ðtþLÞ3=2f ðzÞ�

tþ2L

t2ffiffiffiffiffiffiffiffiffitþLp

h

bz�

h

b

� �2 ffiffiffiffiffiffiffiffiffitþLp

t

1

f ðzÞ

( ): ð10Þ

Lemma 4. BuðtÞ is quasi-concave on ð0,b=2hÞ. Hence, there exists at

most one maximizer of BuðtÞ on ð0,b=2hÞ.

Note that if BuðtÞ ¼ 0 has no solution in ð0,b=2hÞ, the profitfunction would be increasing in t for tAð0,b=2hÞ. We can thenfocus on ðb=2h,1Þ for the optimal t. On the other hand, if BuðtÞ ¼ 0has solutions in ð0,b=2hÞ, let t1 be the smallest value and t2ðZt1Þ

be the largest value in ½0,b=2h� such that BuðtÞ40 on (t1, t2). Wecan then focus our search only on tA ðt1,t2Þ to determine themaximizer of GðtÞ for tAð0,b=2hÞ as shown below.

Theorem 2. Any maximizer of GðtÞ in terms of t on ð0,b=2hÞ must be

in (t1, t2).

So, overall we can conclude that a simple one-dimensionalsearch over tA ðt ,t0Þ and/or tAðt1,t2Þ is enough for managers toselect the optimal value of t. Substituting this t in thecorresponding p(t) and z(t) gives us the optimal values of all thethree decision variables for the approximate model. We wouldlike to point out that the characterization of the profit function ofthe TIB model as shown in Theorems 1 and 2 is one of our novelcontributions to the literature.

Based on the above characterization, we can go further andanalyze how the optimal decision variable values behave withrespect to change in certain model parameters (i.e., sensitivityanalysis). Note that we focus on the sensitivity analysis for twooptimal decision variables—price (p*) and review length (t*).Sensitivity analysis for the optimal base stock level (S*) isconsiderably more involved since it depends on z*, p* and t*.Consequently, that sensitivity analysis is done numerically inSection 5. For expositional convenience, we summarize the firstorder conditions for optimal price and review length below:

p� ¼c�k1

1þk2þ

1

1þk2

h

2t�þbs

ffiffiffiffiffiffiffiffiffiffiffit�þLp

t�f ðz�Þ

� �,

K

ðt�Þ2¼Dðp�Þ

h

2�bs t�þ2L

2ðt�Þ2ffiffiffiffiffiffiffiffiffiffiffit�þLp f ðz�Þ�

h

b

ffiffiffiffiffiffiffiffiffiffiffit�þLp

t�z�

" #( ):

Proposition 1. The following results about the change of the optimal

decision variable values when certain model parameters are changed

are true.

(1)
@t�=@b40 if and only if @p�=@b40. (2) @t�=@K40 if and only if @p�=@K40. (3) @t�=@s40 if and only if @p�=@s40. (4) @t�=@L40 if and only if @p�=@L40.
Two remarks are in order here regarding the above result. First,it shows that the behavior of the optimal price and the optimalreview length are quite similar (i.e., either they both decrease orthey both increase). The managerial intuition behind this result isas follows. If the review length increases, this increases the holdingcost for the firm, but does not significantly change the back-ordering cost (recall that Proposition 1 is for TIB model). In order tocounterbalance this cost increase, managers then need to increasethe retail price (this will reduce the uncertainty level of thedemand, hence the consequent reduction in safety stock also helpsin reducing holding cost). On the other hand, if the review lengthdecreases, managers can reduce the price, and increase the firm’sdemand (as well as revenue), without significantly increasing theholding cost. As we show in Section 5, this relation betweenoptimal review length and retail price is not necessarily true for theTDB model (e.g., in that case behavior of p* and t* with respect to b

might be totally opposite). Second, the above result implies thatmanagers should be able to decide about how to change some oftheir decision variables as the business environment changes quiteeasily. For example, it is intuitive that the optimal review length (t*)will always increase in the fixed review cost K. The above result


implies that managers should also increase the optimal retail pricep* as K increases. Similarly, since it is expected that p* will increasein b (backordering cost per unit), the behavior of t* with respect to b

then becomes evident for the managers.

4. Analysis of the TDB model

In this section we analyze the profit function for the TDBmodel—E(t,z,p) given by (4)—to determine the optimal decisionvariable values. First, for any given review cycle length t and retailprice p, differentiating E(t,z,p) with respect to z we get

@Eðt,z,pÞ

@z¼ sDðpÞ


t

Z t

0½b�ðhþbÞFðdÞ�dt:

Note thatR t

0½b�ðhþbÞFðdÞ�dt is strictly decreasing in terms of z.

If z tends to infinity, then d tends to infinity and FðdÞ tends to 1.This implies that @Eðt,z,pÞ=@zo0 when z is quite large. If z tends

to zero, thenR t

0½b�ðhþbÞFðdÞ�dt tends toR t

0½b�ðhþbÞFððt�tÞ=ðs

ffiffiffiffiffiffiffiffiffiffitþLp

ÞÞ�dt. Therefore, we get the following result:

Lemma 5. There exists a unique positive constant t0 such that for

any tA ½t0,þ1Þ, we have z(t)¼0; Otherwise, the unique z(t) satisfiesR t0½b�ðhþbÞFðdÞ�dt¼ 0.

Recall that d¼ zffiffiffiffiffiffiffiffiffitþLp

=ffiffiffiffiffiffiffiffiffiffitþLp

þððt�tÞ=ffiffiffiffiffiffiffiffiffiffitþLp

Þ1=s by (3) and itis strictly increasing in terms of z. Hence, E(t, z, p) can beexpressed in terms of (t, p) as follows (by substituting z(t)):

Eðt,pÞ ¼ ðp�cÞDðpÞ�1

tKþðhþbÞsDðpÞ

Z t

0


pf ðdðt,tÞÞdt

� �:

For any given t, taking partial derivative of E(t, p) with respect to p

we get

@Eðt,pÞ

@p¼Du pþ

D

Du�c�rðtÞ

� �,

where rðtÞ ¼ ðhþbÞs=tR t

0


f ðdðt,tÞÞdt. Hence, there exists aunique p(t) such that E(t, p) is maximized and it satisfiespþD=Du¼ cþrðtÞ, i.e.,

pðtÞ ¼c�k1

1þk2þ

rðtÞ1þk2

: ð11Þ

Therefore, the maximization of E(t, p) can be reduced to a one-variable optimization problem in terms of t and this issummarized in the following theorem.

Theorem 3. For any given t, there exist a unique z(t) and a unique

p(t) such that E(t, z, p) is maximized. Hence, the maximization of

E(t, p, z) can be reduced to a one-variable optimization problem in

terms of t.

Therefore, the TDB model involves maximization of thefollowing function.

EðtÞ ¼ ½pðtÞ�c�DðpðtÞÞ�K

t�DðpðtÞÞrðtÞ:

Differentiating E(t) with respect to t, we get

dEðtÞ

dt¼

K

t2�ruðtÞDðpðtÞÞ

and

d2EðtÞ

dt2¼�

2K

t3þr00ðtÞDþ 1

1þk2ruðtÞ2Du

� �:

It is quite difficult to characterize E(t)¼E(t,p(t)) in terms of t. However,once again a one-dimensional search is enough to determine theoptimal t, and substituting this t in z(t) and p(t), we can determine thevalues of all the three decision variables for the TDB model.

Similar to what we did for the TIB model in Section 3, in thefollowing, we would like to investigate the impact of some modelparameters on the behavior of the optimal retail price (p*) and theoptimal review length (t*). In this context, note that rðtÞ ¼ðhþbÞs=t

R t0


f ðdðt,tÞÞdt and dðt,tÞ ¼ zðtÞffiffiffiffiffiffiffiffiffitþLp

=ffiffiffiffiffiffiffiffiffiffitþLp

þððt�tÞ=ffiffiffiffiffiffiffiffiffiffitþLp

Þð1=sÞ. For convenience, we summarize the first orderconditions for p* and t* below:

p� ¼ �k1

1þk2þ

cþrðt�Þ1þk2

and

K

ðt�Þ2¼ ruðt�ÞDðp�Þ:

From the expressions of dEðtÞ=dt and d2EðtÞ=dt2, it is clear that we

always have ruðt�Þ40 and 2K=ðt�Þ3þr00ðt�ÞDðp�Þþ1= ð1þk2Þ�

ruðt�Þ2Duðp�ÞZ0. These two facts are useful in the followingsensitivity analysis result.

Proposition 2. Let r0 ¼ k1þk2cþrðt�Þ, a constant. Then the

following results about the change of the optimal decision variable

values when certain model parameters are changed are true.

(1)
If r0o0, then both @t�=@ho0 and @t�=@bo0. (2) If r040, then both @t�=@h40 and @t�=@b40. (3) Both @t�=@K40 and @p�=@K40. (4) If @t�=@s40, then @p�=@s40.
The important issue to note is that, for the TDB model, thebehavior of the optimal review length with respect to the unitholding and backordering costs are quite similar. Specifically, ifthe holding and backordering costs are quite high (which makesr0 high), then the optimal review length increases in h and b; ifthese costs are relatively low (which makes r0 relatively low),then the optimal review length decreases in h and b. So, managersneed to be careful about changing the review length as unitbackordering or holding costs change. The intuition of the aboveresult is as follows. When h and b are relatively low, any increasein either of them should be associated with a decrease in t* sincethis allows the firm to reduce the length of time for whichbackorders persist, and hence the backordering and holding costs(recall that Proposition 2 is for TDB model). As we will show in thenext section, this similarity of behavior for t* with changes in h orb is not necessarily true for the TIB model.

With any increase of the fixed review cost K, the optimal reviewcycle length t* should be decreased in order to reduce the averagereview cost. As a consequence of a longer t*, the average holding costis higher. In order to counterbalance this, the optimal retail price p*

should be increased (hence, the demand uncertainty level is lowerand the safety stock level is also lower). Similarly, if t* is higher withhigher s, then the optimal retail price p* should be increased in orderto balance the increased holding and backordering costs.

5. Comparison of the two models

Our above discussion indicates that TIB and TDB models arequite different in terms of how backordering cost is charged. Wehave also analytically characterized the behavior of the profitfunctions for both models showing that a one-dimensional searchprocedure would enable a manager to determine the optimaldecision variable values and profits for each of them. Our (partial)analytical results regarding the impact of different modelparameters on the optimal decision variables also show thedistinct behavior of the two models. However, for both models

TDBTIB

TDBTIB

TDBTIB

TDBTIB

TDBTIB

TDBTIB

0

5

10

15

20

25

σ L g

0

10

20

30

40

5

10

15

20

25

30

12

15

18

21

24

10

15

20

25

30

10

15

20

25

30

35

K b h

1 10 1.11.5 2 2.5 3 3.5 4 4.5 5 5.5 20 30 40 50 60 70 80 90 100 1.3 1.5 1.7 1.9

2 0.08 0.0066 10 14 18 0.1 0.12 0.14 0.16 0.18 0.008 0.01 0.012 0.014

Fig. 1. Impact on optimal review lengths for D(p)¼ap�g.


the first-order-condition which determines optimal t is ratherinvolved. So, any further comparison of the optimal decisionvariables and profit values is analytically intractable. Keeping thisin mind, we resort to extensive numerical experiments in order todemonstrate the difference in the optimal decisions/profits for thetwo models, and generate relevant managerial insights.

5.1. Numerical setting for comparison of the two models

For our numerical experiments, we consider two deterministic(price-sensitive) demand forms: iso-elastic for which D(p)¼ap�g

ða40,g41Þ, and linear for which D(p)¼A�gp ðA40,g40Þ.12 Thesetwo are widely used to model price sensitive demands in theoperations management literature (refer to Petruzzi and Dada, 1999;Wang et al., 2004; Granot and Yin, 2005). We focus on understandingthe effects of the following parameters on the optimal decision valuesand the corresponding profits while comparing between the models:demand uncertainty ðsÞ, leadtime (L), price-elasticity (g), review cost (K),backordering cost (b),13 and holding cost (h).

The base data set for the numerical study is as follows: a¼200and g¼1.5 (for iso-elastic demand), A¼30 and g¼2.0 (for lineardemand), K¼10, c¼1.0, h¼0.01, b¼0.2, L¼100, s¼ 3:0 and m¼ 1.In order to capture the effects of various parameters, we vary them(one at a time) around the base level. The three optimal values forthe two models—review cycle length, retail price and base stocklevel—as well as the profits are then compared. We first discuss thecomparison of the decisions and then the corresponding profits.

5.2. Optimal review cycle length

The effects of all the relevant parameters on the optimalreview length are shown in Figs. 1 and 2. Notice that, in general,the optimal review cycle length for the TIB model is much higher than

that for the TDB model. The underlying reason for this is as follows.Since there is no penalty in the TIB model for having backorders inthe books for a long time (recall that the TIB model ignores the

12 Note that, although g represents price-sensitivity in both demand forms,

their values might be quite different.13 Recall that the unit for the backordering cost is $/unit for the TIB model and

$/unit/unit time in the TDB model.

time duration of any backorder), this model increases the reviewlength. In fact, such an action enables the firm to reduce thereview and backordering costs (refer to G in (2)). Moreover, longreview length does not significantly increase the safety stockholding costs. Note again from the expression of G in (2) that therate of increase of safety stock holding costs is proportional onlyto the square root of the length of the review cycle t, while thebackordering and review costs decrease as t increases. The reasonthat the review cycle length is not even longer is that the cyclestock holding costs would then become very large (that portion ofthe costs increases linearly in t). However, in the TDB model toolong review length is not good for the firm since it will result inlengthy backorders, and hence large backordering costs.

Interestingly, even the behavior of the optimal review length

might be different in the two models. Based on Figs. 1 and 2, it isclear that the behavior of the optimal cycle length as b changesare totally opposite in the two models. These phenomena are alsorelated to the difference in how holding and backordering costsare charged in the two models. For example, as the value of b

becomes higher, they increase the potential cost of backorders.Since in the TDB model the backordering cost depends on howlong backorders persist, the TDB model then tries to reduce thelength of the review period (so that the backorders are in thebooks for relatively less time). On the other hand, length ofbackorders has no effect on the backordering cost for the TIBmodel. So, in that case we note that the model tries to increase theperiod length. As is clear from the expression of G in (2), such anaction allows it to reduce review and backordering costs. More-over, as we showed (partially) in our sensitivity analysis results inthe last two sections, Figs. 1 and 2 reveal that, while the optimalreview length behaves in the same manner when h or b changesfor the TDB model, the effects of the two parameters on t* aretotally opposite in the TIB model.

When s is low, both models are similar to a deterministic one;hence, both their optimal review lengths are also quite close tothat of the deterministic model.

Another interesting issue to note is that sometimes the TIBmodel can give rise to rather unreasonable results. As K tends tozero, it is natural that the optimal t should also tend towards zero.It is the case in the TDB model. However, for the TIB model, theoptimal review length has a significantly large positive value evenwhen the value of K is almost 0. Again, this is because of the

TDBTIB

TDBTIB

TDBTIB

TDBTIB

TDBTIB

TDBTIB

σ L g

14

18

22

26

30

34

051015202530

0

5

10

15

20

25

30

15

18

21

24

14

18

22

26

10

15

20

25

30

35

K b h

1.2 10 1.21.6 2 2.4 2.8 20 30 40 50 60 70 80 90 100 1.6 2 2.4 2.8

2 0.08 0.0086 10 14 18 0.1 0.12 0.14 0.16 0.18 0.006 0.01 0.012 0.014

Fig. 2. Impact on optimal review lengths for D(p)¼A�gp.


backordering cost structure in the TIB model. Note from theexpression of G in (2) that as t tends to zero, the shortage tends toinfinity. So, the TIB model will never make the review periodlength very small (and so, this model cannot approximate acontinuous review model in the limiting case).

5.3. Optimal retail prices

The effects of the relevant parameters on the optimal retailprice for each model are reported in Figs. 3 and 4. The behavior ofthe two prices in terms of each model parameter are similar, butthe values again are quite different. In general, the optimal retail

price of the TDB model is higher than that of the TIB model. Thereason behind this is that the negative effect of demand varianceis not substantial in the TIB model.14 The variance of the demandover the protection interval is s


DðpÞ. So, while optimizingthe decision variables in the TIB model, the intent of the model isto increase the review length and to decrease the retail price(which increases demand). This action allows it to reduce thereview cost as well as increase revenue. But, the TDB modelcannot do so since low price (and consequent high demand) willmake the demand highly variable resulting in more (and lengthy)backorders (and so high backordering cost). The model then triesto generate revenue by charging high prices.15

When s is low, the impact of the backordering cost structure inthe TIB model is minor and the optimal decision values are closeto the corresponding ones of the exact model. The optimal retailprice is increasing in terms of both s and L for both models. Thiscan be explained in an intuitive way: the increase of p offsets theincrease in the demand variance as a result of higher s and L.

For both models the optimal retail price p is increasing inreview cost K, the backordering cost rate b, and holding cost rateh, respectively. But, the optimal retail price is decreasing in termsof the price elasticity of the product and this is obvious: higher

14 Since even if there are a large number of backorders, their lengths do not

matter.15 From Figs. 1–4 also note that, as we indicated in our sensitivity analysis

results for the two models, the behavior of p* and t* with respect to b is the same

for the TIB model, but not for the TDB model.

price sensitivity of the product, lower retail price to thecustomers.

5.4. Optimal base stock level

About the effects of the relevant model parameters on theoptimal base stock level, we would like to point out that for all thetest scenarios, we noted that the TIB model holds much lower safety

stock than the TDB model. As explained before, the cost ofbackorders is underestimated in the TIB model (so the need forsafety stock is less). The aim of the optimization in that case is toreduce the holding cost by keeping less safety stock. Obviously,the TDB model needs to keep higher safety stock in order toreduce backordering costs. This means that using the TIB modelcan indeed increase the risk of shortages in real-life. However,note that the optimal base stock level is determined by thefollowing three factors: safety stock level, retail price, and thereview length. Specifically, the base stock level can be calculatedbased on the following expression: S¼ ½ðtþLÞþzs


�DðpÞ

(refer to (1)), where zsffiffiffiffiffiffiffiffiffitþLp

DðpÞ represents the safety stock part.Although zs


DðpÞ part is significantly lower for the TIBmodel, (t+L)D(p) is actually higher for that model. This is sobecause the optimal review length is higher (refer to Section 5.2)and so is the demand (since optimal retail price is lower, refer toSection 5.3). Consequently, the overall optimal base stock levelcan be higher in either model.16 For example, as far as the effect ofs is concerned, it seems that the optimal base stock level isdominated by the safety stock part for higher s values, while forlower s values it is the non-safety stock part which is moreimportant.

5.5. Optimal profits

In order to understand the implications on profit as a resultof using the TIB model if the backordering cost is time depen-dent, we proceed as follows. We first substitute the optimalreview length, optimal retail price, and the optimal stocking

16 Although we do not present it here, we noted this in our numerical

experiments.

TDBTIB

TDBTIB

TDBTIB

TDBTIB

TDBTIB

TDBTIB

0

2

4

6

8

3

3.5

4

4.5

5

5.5

3

3.5

4

4.5

5

5.5

22.53

3.54

4.55

5.5

0

5

10

15

20

33.5

44.5

55.5

6

1 10 1.11.5 2 2.5 3 3.5 4 4.5 5 5.5 20 30 40 50 60 70 80 90100 1.3 1.5 1.7 1.9

2 0.08 0.0066 10 14 18 0.1 0.12 0.14 0.16 0.18 0.008 0.01 0.012 0.014

σ L g

bK h

Fig. 3. Impact on optimal retail prices for D(p)¼ap�g.

TDBTIB

TDBTIB

TDBTIB

TDBTIB

TDBTIB

TDBTIB

7

7.5

8

8.5

9

7.5

7.7

7.9

8.1

8.3

8.5

88.18.28.38.48.5

77.37.67.98.28.58.8

4

6

8

10

12

14

8

8.1

8.2

8.3

8.4

8.5

σ L g

bK h

1.2 10 1.21.6 2 2.4 2.8 20 30 40 50 60 70 80 90100 1.6 2 2.4 2.8

2 0.08 0.0066 10 14 18 0.1 0.12 0.14 0.16 0.18 0.008 0.01 0.012 0.014

Fig. 4. Impact on optimal retail prices for D(p)¼A�gp.

17 Note that in Figs. 7 and 8, we change both b and L simultaneously where b is

from 0.08 to 0.17 with step of 0.01 and L is from 10 to 100 with step of 10.18 Notice that if only L or b is low, then the differences between A* and E* are

low, but the decision variable values for the two models can be quite different.


factor values of the TIB model into the average profit functionE(t, p, z) of the TDB model (we denote this profit by A*), andthen compare A* with the optimal profit value E* of the TDBmodel (E* is obtained by substituting the three optimal decisionvariable values of the TDB model in E(t,p,z)). The effects of allthe system parameters on the comparison are reported inFigs. 5 and 6.

It is obvious that A�oE� for all scenarios. Before analyzing themodels, our conjecture was that for most scenarios the absolutedifference between A* and E* would be low (because the TIB modelhas been discussed and analyzed in the literature for a long time).In other words, the performance of the TIB model should be goodenough under most business settings. In fact, this is not the case.Rather, in general, the penalty of using the optimal decisions from

the TIB model is quite substantial for most market/operational

settings. The only exceptions are the following scenarios—(i)when the demand uncertainty s is low (see Figs. 1–6), and (ii)

when the total backordering cost is low, e.g., when both lead timeL and backordering cost rate b are low (see Figs. 7 and 8).17 Only inthose cases the optimal decision variable values for both modelsare quite similar and the differences between A* and E* are quitelow (since the backordering costs are then relatively lessimportant).18 Our analysis clearly shows that the review lengthplays a significant role in determining the expected profit of theTDB model, and any deviation from the true optimal can beharmful. More importantly, we establish that managers shouldbe cautious about using the TIB model for their operations/

pricing decisions if the backordering cost is time dependent. Inmost real-life settings such an action can be highly detrimental

E*A*

E*A*

E*A*

E*A*

E*A*

E*A*

-40

-20

0

20

40

60

80

20

30

40

50

60

70

01020304050607080

010203040506070

-40

0

40

80

120

10

20

30

40

50

60

70

σ

L

g

bK h

1 1.5 2 2.5 3 3.5 4 4.5 5 5.5

10 20 30 40 50 60 70 80 90 100

1.1 1.3 1.5 1.7 1.9

2 0.08 0.0066 10 14 18 0.1 0.12 0.14 0.16 0.18 0.008 0.01 0.012 0.014

Fig. 5. Impact on optimal profits for D(p)¼ap�g.

E*A*

E*A*

E*A*

E*A*

E*A*

E*A*

σ L g

4050

607080

90100

70

75

80

85

90

405060708090100

70

75

80

85

90

95

30507090110130150170

65707580859095

K b h

1.2 10 1.21.6 2 2.4 2.8 20 30 40 50 60 70 80 90100 1.6 2 2.4 2.8

2 0.08 0.0066 10 14 18 0.1 0.12 0.14 0.16 0.18 0.008 0.01 0.012 0.014

Fig. 6. Impact on optimal profits for D(p)¼A�gp.


for the firm. Managers should be most careful when dealingwith high-tech products or innovative ones or products in theincubation/growth phases of the lifecycle. Such products arenormally characterized by high demand uncertainty and highbackordering costs, e.g., fashion apparel, laptops, i-Pods,semiconductors, telecom equipments (Ray et al., 2005). Onlyfor functional/commodity products and/or products in thematurity stage of the lifecycle, e.g., basic apparel, grocery,for which b and s are low, using the TIB model might beacceptable.

General comment: We would like to point out that, althoughthe values of the profits and decision variables for the twodemand forms (iso-elastic and linear) are different, their behaviorare almost the same for both models. In that sense, most of ourabove insights are quite robust.

6. Concluding remarks

In this paper we analyzed the joint operations-marketingdecision-making for a firm dealing with an uncertain, price-sensitive environment and employing periodic review inventorymanagement policy. The objective of the firm was to simulta-neously decide on the review length of each period, the base stocklevel at each review time point, and the retail price of the product.We discussed two models—TIB (time-independent-backordering)and TDB (time-dependent-backordering)—in this context whichdiffers mainly with regards to whether or not the length for whichexcess demands that are backordered persist are taken intoconsideration. Our main concern was to compare the performanceof the two models under different business conditions, andprovide managerial suggestions. First, we showed that the

0

5

10

15

20

25

TDBTIB

TDBTIB

0

1

2

3

4

5

6

0

10

20

30

40

50

60

70

80

E*A*

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

Profitt* p*

Fig. 7. Impact on optimal decisions and profits when both b and L are low (D(p)¼ap�g).

0

5

10

15

20

25

TDBTIB

TDBTIB

7.5

7.7

7.9

8.1

8.3

8.5

6065707580859095100

E*A*

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

Profitt* p*

Fig. 8. Impact on optimal decisions and profits when both b and L are low (D(p)¼A�gp).


optimization for both models can be reduced to one-variableproblems. In this context, our characterization of the profitfunction of the TIB model is especially noteworthy because suchanalysis has not been done in the literature before. Second, weprovided (partial) analytical sensitivity results for the two models,which indicated the difference in the behavior between the twomodels. Lastly, we carried out an extensive numerical study andshowed that using the decision variables from the TIB model asapproximations in the TDB model is not advisable for managers,irrespective of the form of the demand function. Specifically,because the TIB model underestimates the backordering costs, ittends to set longer-than-optimal review periods, charge lower-than-optimal retail prices and hold less-than-optimal safety stock.The differences in the decisions between the two models areusually quite substantial. Consequently, using the decisions fromthe TIB model results is significant profit penalty for most marketand operating conditions. Only when the demand uncertainty inthe market is quite low, using the TIB model might be anacceptable alternative. Keeping in mind the growing competitionand volatility in the market (which implies high demanduncertainty and high backordering costs), we would stronglysuggest that managers be extremely cautious about using the TIBmodel for decision-making in real-life in spite of its existence andpopularity in the literature.

There are two possible extensions of this paper which areworthwhile endeavors. First, an extension to the random lead timecase might be interesting. While we think that our qualitativeinsights will not change for such a case, the model will becomeanalytically much more difficult to handle. Another possiblegeneralization is to assume that excess demands are lost, ratherthan backordered. Particularly for retail industry, the lost salesassumption might be more realistic. To the best of our knowledge,even for inventory models which do not consider pricing or reviewlength decisions, there are very few publications in the literature

dealing with positive lead time and lost sales. These two possibleextensions might be valuable for both academic and practitionercommunities, and we leave them for future research.

Appendix

Proof of Lemma 1. First, it is obvious that Gðt,z,pÞ is concave interms of z by checking the second partial derivative of@2Gðt,z,pÞ=@z2. Thus, if tZb=2h, the unique maximizer z(t) ofGðt,z,pÞ in terms of z is 0; Otherwise, the unique maximizer z(t)satisfies the first order condition, i.e., z(t) satisfies 1�FðzÞ ¼ ðh=bÞt

from (5). Differentiation of h¼ ðb=tÞ½1�FðzÞ� yields zuðtÞ ¼�ðh=bÞ1=f ðzðtÞÞo0. &

Proof of Lemma 2. The corresponding average total profit can beexpressed as Gðt,pÞ ¼ ðp�cÞDðpÞ�BðtÞDðpÞ�K=t where BðtÞ ¼ ht=2þbsð


=tÞf ðzðtÞÞ. Taking the first partial derivative with respectto p, we get @Gðt,pÞ=@p¼Du½pþD=Du�c�BðtÞ�. Based on Assump-tion 1, Gðt,pÞ is unimodal in terms of p for tAð0,b=2hÞ. &

Proof of Theorem 1. For any tA ½b=2h,þ1Þ, by Lemma 2, theaverage total profit can be expressed as

GðtÞ ¼ ½pðtÞ�c�DðpðtÞÞ�BðtÞDðpðtÞÞ�K

t,

where BðtÞ ¼ ðht=2ÞþbsðffiffiffiffiffiffiffiffiffitþLp

=tÞf ð0Þ and p(t) is the uniquesolution of pþDðpÞ=DuðpÞ ¼ cþBðtÞ, i.e.,

pðtÞ ¼c�k1

1þk2þ

1

1þk2BðtÞ: ð12Þ

Differentiation on both sides with respect to t, we getpuðtÞ ¼ BuðtÞ=ð1þk2Þ. In order to characterize the behavior of GðtÞ, by


taking the first and second derivatives of GðtÞwith respect to t, we get

dGðtÞdt¼�BuðtÞDðpðtÞÞþ

K

t2

and

d2GðtÞdt2

¼�B00ðtÞDðpðtÞÞ�BuðtÞ2

1þk2Du�

2K

t3:

From dGðtÞ=dt¼ 0, we get BuðtÞDðpðtÞÞ ¼ K=t240. Note that B(t) ispositive and convex since BuðtÞ ¼ ðh=2Þ�bsððtþ2LÞ=ð2t2


ÞÞf ð0Þand B00ðtÞ ¼ bsðð3t2þ12tLþ8L2Þ=ð4t3ðtþLÞ3=2

ÞÞf ð0Þ. Hence, near anymaximizer of GðtÞ, B(t) is strictly increasing and convex and p(t) is alsostrictly increasing. Without loss of generality, in order to characterizethe behavior of GðtÞ on ½b=2h,þ1Þ, we only need to focus ontAðmaxfb=2h,tming,þ1Þ where tmin is the minimizer of B(t) on½b=2h,þ1Þ.

As DðpÞ=DuðpÞ ¼ ðk1þk2cÞ=ð1þk2Þþk2=ð1þk2ÞBðtÞ, we get

d2GðtÞdt2

fdGðtÞ=dt ¼ 0g

¼ �Du B00D

Duþ

Bu2

1þk2þ

2Bu

t

D

Du

( )

¼�Bu2Du

ð1þk2Þ

B00

Bu2þ

2

tBu

� �½k1þk2cþk2B�þ1

� �:

The last equality is obtained as DðpÞ=DuðpÞ ¼ k1þk2pðtÞ ¼

1=ð1þk2Þ½k1þk2cþk2B�. Note that both B00=Bu2 and 1=tBu are

decreasing in terms of t (this can be verified easily by taking the

first derivatives of both functions). Let GðtÞ ¼ ½B00=Bu2þ2=tBu�

½k1þk2cþk2B� and we need to analyze the behavior of G(t) on

½b=2h,þ1Þ. We divide the remaining proof into two cases based

on the value of k2.

�
Case 1 (k2Z0): In this case, it is obvious thatf½B00=Bu2þ2=tBu�½k1þk2cþk2B�þ1g is increasing in terms of t. � Case 2 (k2o0): If we can show that ½B00=Bu2þ2=tBu�B is also
strictly decreasing, then G(t) is increasing. Thus, it is sufficientto show that

LðtÞ ¼B00

Bu2þ

2

tBu

� �B¼

tB00

Buþ2

� �B

tBu

is strictly decreasing. Based on the expression of B00, we can seethat tB00 is positive and decreasing; On the other hand, weknow that Bu is positive and increasing. Thus, tB00=Bu isdecreasing. The remaining task is to show that B=tBu is strictlydecreasing and this is obvious as

B

tBu�1¼ bsf ð0Þ

3tþ4L

2tffiffiffiffiffiffiffiffiffitþLp

h

2t�bsf ð0Þ

tþ2L

2tffiffiffiffiffiffiffiffiffitþLp

� ��1

¼ bsf ð0Þ ht2


3tþ4L�bsf ð0Þ

tþ2L

3tþ4L

� ��1

:

Finally, as both tB00=Bu and B=tBu are decreasing, G(t) is strictlyincreasing.

Therefore, for both cases, G(t) is strictly increasing. Let t0 be the

minimal value on ½maxfb=2h,tming,þ1Þ such that f½B00=Bu2þ2=tBu�

½k1þk2cþk2B�þ1g40. Hence, GðtÞ is quasi-concave on ½maxfb=

2h,tming,t0� and quasi-convex on ½t0,þ1Þ. Note that Both B(t) and

p(t) tend to infinity while t tends to infinity. Hence, GðtÞ is non-

positive while t is very large and there does not exist any local

maximizer of GðtÞ on ðt0,þ1Þ. Regarding the behavior of GðtÞ on

½b=2h,maxfb=2h,tmingÞ, it is strictly increasing based on the

expression of GðtÞu and the convexity of B(t) on ½b=2h,þ1Þ. Thus,

the local maximizer of GðtÞ, if any, must be in ½maxfb=2h,tming,t0�

and GðtÞ is quasi-concave on this interval. We complete the

proof. &

Proof of Lemma 3. It is well known that r(z) is strictly increasing(recall that our f and F represents standard normal). HencegðzÞ40 for any zAð0,þ1Þ. In order to show that gðzÞo1 for anyzAð0,þ1Þ, it is equivalent to show that zf ðzÞ�z2ð1�FðzÞÞ�

ð1�FðzÞÞo0. As the left hand of this inequality tends to zero as z

tends to infinity, it is sufficient to show the first derivative of theleft hand is positive, i.e., 2½f ðzÞ�zð1�FðzÞÞ�40 and this is obviousas ru40. Thus, gðzÞo1 for any zAð0,þ1Þ. Now we are ready toshow the unimodality of g(z). Note that ru¼ rðr�zÞ. Then, takingthe first and the second derivatives with respect to z, we get

guðzÞ ¼ r�2zþzrðr�zÞ

and

g00ðzÞ ¼ 2rðr�zÞ�2þzrðr�zÞ2þzr½rðr�zÞ�1�:

From gu¼ 0, we get zr(r�z)¼2z�r. Substitution of this into g00ðzÞ,we get

g00ðzÞjfgu ¼ 0g ¼ 2rðr�zÞ�2þðr�zÞð2z�rÞ�zrþrð2z�rÞ

¼ 2½zðr�zÞ�1�o0:

The last inequality is because gðzÞo1 for any zA ð0,þ1Þ.

In the following, we only need to show that r(z) is convex. As

ru¼ ððf ðzÞ�zð1�FðzÞÞÞ=ð1�FðzÞÞÞf ðzÞ=ð1�FðzÞÞ, in order to show that

r0040, it is equivalent to show that f ðzÞ½f ðzÞ�zð1�FðzÞÞ�þ

½f ðzÞ�zð1�FðzÞÞ�2�ð1�FðzÞÞ240. As the left hand tends to zero as

z tends to infinity, in order to show the above inequality, it is

sufficient to the first derivative of the left hand is negative, i.e.,

D¼�zf ðzÞ½f ðzÞ�zð1�FðzÞÞ��2½f ðzÞ�zð1�FðzÞÞ�

ð1�FðzÞÞþ f ðzÞð1�FðzÞÞo0. Note that D can be rewritten as

D¼ ð1�FðzÞÞ½zð1�FðzÞÞ�f ðzÞ�þzfð1�FðzÞÞ2�f ðzÞ½f ðzÞ�zð1�FðzÞÞ�g

¼ �½gðzÞ�u

ð1�FðzÞÞ2:

Hence, it is sufficient to show that g(z) is increasing. Note that

limz-0 gðzÞ ¼ 0, limz-1 gðzÞ ¼ 1, the value of 1 is the upper bound

of g(z), and g(z) is unimodal in terms of z on ð0,þ1Þ, hence g(z) is

strictly increasing in terms of z on ð0,1Þ. &

Proof of Lemma 4. By rewriting the expression of B00ðtÞ in (10),we get

B00ðtÞ ¼bsffiffiffiffiffiffiffiffiffitþLp

1

f ðzÞ

3t2þ12tLþ8L2

4t3ðtþLÞf ðzÞ2�

tþ2L

t2

h

bzf ðzÞ�

h

b

� �2 tþL

t

( )

¼h

b

� �2 bs4tðtþLÞ3=2

1

f ðzÞfð3t2þ12tLþ8L2Þr2

�4ðtþLÞðtþ2LÞzr�4ðtþLÞ2g,

where r¼ f ðzÞ=ð1�FðzÞÞ. The last equality is obtained becauseðh=bÞt¼ ½1�F� by the definition of z(t). Again, by 1�F ¼ ðh=bÞt,we get

zu¼�h

b

1

f ðzÞ¼�

1

trand ru¼ r2�zr:

Taking the third derivative of B(t) and utilizing the above twoexpressions, we get

B000ðtÞjfB00 ðtÞ ¼ 0g ¼h

b


1

f ðzÞ�r2 12tLþ16L2

t

�

þzr2t2þ24tLþ24L2

t


�z2 4ðtþLÞðtþ2LÞ

t�

4t2�4tL�8L2

t

�:

From B00ðtÞ ¼ 0, we get zr¼ ðð3t2þ12tLþ8L2Þ= 4ðtþLÞðtþ2LÞÞr2�

ðtþLÞ=ðtþ2LÞ and z¼ ðð3t2þ12tLþ8L2Þ=4ðtþLÞ ðtþ2LÞÞr�ððtþLÞ=

ðtþ2LÞÞ1=r. Substitution of these two expressions into B000ðtÞ,we get

B000ðtÞjfB00 ðtÞ ¼ 0g ¼h

b


1

f ðzÞ�r2 3tðt2þ8tLþ8L2Þ

4ðtþLÞ

�

�1

r

� �2 4ðtþLÞ3

tþ

8L2ðtþLÞ

t

):

If tZL, it is easy to check that B000ðtÞjfB00 ðtÞ ¼ 0go0.

If toL, then B00ðtÞ can be rewritten as (note that ru¼ r2�zr)

B00ðtÞ ¼h

b


1

f ðzÞfð3t2þ12tLþ8L2Þr2

�4ðtþLÞðtþ2LÞzr�4ðtþLÞ2g

¼h

b


1

f ðzÞf�t2r2þ4ðtþLÞðtþ2LÞru�4ðtþLÞ2g:

Taking the third derivative of B(t), we get

B000ðtÞjfB00 ¼ 0g ¼h

b


1

f ðzÞf�2tr2þð10tþ12LÞru

�8ðtþLÞþ4ðtþLÞðtþ2LÞr00zug

oh

b


1

f ðzÞf�2tr2þð10tþ12LÞru�8ðtþLÞg:

The last inequality is obtained since r0040 and zuo0. From B00ðtÞ ¼ 0,

we get �tr2 ¼�ð4ðtþLÞðtþ2LÞ=tÞruþ4ðtþLÞ2=t. Substitution of this

expression into the right hand side of B000ðtÞjfB00 ¼ 0g, we get

B000ðtÞjfB00 ¼ 0goh

b


1

f ðzÞ

2

tfruðt2�6tL�8L2Þþ4LðtþLÞg:

If tAð0,LÞ, then B000ðtÞjfB00 ¼ 0go ðh=bÞ2ðbs=4tðtþLÞ3=2Þð1=f ðzÞÞð2=tÞfð2=

pÞðt2�6tL�8L2Þþ4LðtþLÞgo0. &

Proof of Proposition 1. For simplicity, in this proof we denotethe optimal decision variable values (t*, p*, z*) by (t, p, z). We knowthat the optimal retail price p and review cycle length t satisfy

K

t2¼DðpÞBuðtÞ and p¼

c�k1

1þk2þ

1

1þk2BðtÞ,

where BðtÞ ¼ ðh=2ÞtþbsðffiffiffiffiffiffiffiffiffitþLp

=tÞf ðzÞ and BuðtÞ ¼ ðh=2Þ�bs½ððtþ2LÞ=

ð2ðtÞ2ffiffiffiffiffiffiffiffiffitþLp

ÞÞf ðzÞ�ðh=bÞðffiffiffiffiffiffiffiffiffitþLp

=tÞz�.

(1)
We first consider the impact of b. Differentiating both sides ofthe above second equation with respect to b, we getð1þk2Þ@p=@b¼ sð

=tÞf ðzÞþBuðtÞ@t=@b. From this fact, weknow that if @t=@b40, then @p=@b40. On the other hand,differentiation on both sides of K

t2 ¼DðpÞBuðtÞ with respect to b,we get

�2K

t3�DB00

� �@t

@b¼ BuDu

@p

@b�sD

tþ2L

2ðtÞ2ffiffiffiffiffiffiffiffiffitþLp f ðzÞ

" #:

Note that at the optimality, we must have d2GðtÞ=dt2r0.Hence, we get ð�ð2K=t3Þ�DB00Þo0. Thus, if @p=@b40, then weget @t=@b40.

(2)
Differentiating both sides of p¼ ðc�k1Þ=ð1þk2Þþð1=ð1þk2ÞÞBðtÞ with respect to K, we get @p=@K ¼ ð1=ð1þk2ÞÞ
BuðtÞ@t=@K (note that at the optimality we always haveBuðtÞ40). Hence, we see that @t�=@K40 if and only if@p�=@K40.

(3)
Differentiating both sides of K=t2 ¼DðpÞBuðtÞ with respect to s,we get
�2K

t3�DB00

� �@t

@s¼DuBu

@p

@s�bD

tþ2L

2t2ffiffiffiffiffiffiffiffiffitþLp f ðzÞ:

Hence, if @t=@s40, then @[email protected] the other hand, differentiating both sides ofK=t2 ¼DðpÞBuðtÞ with respect to s, we get

�2K

t3�DB00

� �@t

@s ¼DuBu@p

@s�bDtþ2L

2t2ffiffiffiffiffiffiffiffiffitþLp f ðzÞ:

Hence, if @p=@s40, then @t=@s40.
(4) Differentiation on both sides of p¼ ðc�k1Þ=ð1þk2Þþ
ð1=ð1þk2ÞÞBðtÞ with respect to L yields

ð1þk2Þ@p

@L¼ Bu

@t

@Lþbs 1

2tffiffiffiffiffiffiffiffiffitþLp f ðzÞ:

Hence, if @t=@L40, then @[email protected] the other hand, differentiation on both sides ofK=t2 ¼DðpÞBuðtÞ with respect to L yields

�2K

t3�DB00

� �@t

@L¼DuBu

@p

@L�bs f ðzÞ½3tþ2L��2ðtþLÞz½1�FðzÞ�

4t2ðtþLÞ1:5:

Note that f ðzÞ=ð1�FðzÞÞ in terms of z is strictly increasing,hence, f ðzÞ�z½1�FðzÞ�40 for any z40. Thus, if @p=@L40, then@t=@L40. &

Proof of Lemma 5. Let HðtÞ ¼R t

0½b�ðhþbÞFððt�tÞ=ðsffiffiffiffiffiffiffiffiffiffitþLp

ÞÞ�dt.First, we need to show that H(t) is concave in terms of t onð0,þ1Þ. Differentiation with respect to t, we get

HuðtÞ ¼b�h

2�ðbþhÞ

Z t

0f

t�tsffiffiffiffiffiffiffiffiffiffitþLp

� �1

sffiffiffiffiffiffiffiffiffiffitþLp dt:

Note that

dt�t

sffiffiffiffiffiffiffiffiffiffitþLp

� �¼�

1


tþ2Lþt

2ðtþLÞ

� �dt:

Differentiation further for H(t) with respect to t, we get

H00ðtÞ ¼ �ðbþhÞ f ð0Þ1

sffiffiffiffiffiffiffiffiffitþLp �

Z t

0f


� �t�t


1

s2ðtþLÞdt

� �

¼�ðbþhÞ f ð0Þ1

sffiffiffiffiffiffiffiffiffitþLp �

Z t

0

2ffiffiffiffiffiffiffiffiffiffitþLp

sðtþ2LþtÞdf


� ��

¼�ðbþhÞ2ffiffiffiLp

sð2LþtÞf

t

sffiffiffiLp

� �þ

1

s

Z t

0f


� �t�tffiffiffiffiffiffiffiffiffiffi

tþtp

ðtþ2LþtÞdt

( ):

Now, it is clear that H00ðtÞo0. Hence, H(t) is concave in terms of t

on ð0,þ1Þ. Also note that when t tends to 0 from the right handside of 0, we have H(0+)¼0 and Huð0þ Þ ¼ ðb�hÞ=240. Thus, thereexists a positive t0 such that HðtÞr0 for any tA ½t0,þ1Þ andHðtÞ40 for any tAð0,t0Þ. We complete the proof. &

Proof of Proposition 2. For simplicity, in this proof we denotethe optimal decision variable values (t*, p*, z*) by (t, p, z). The twofirst order conditions are

p¼�k1

1þk2þ

cþrðtÞ1þk2

andK

t2¼ ruðtÞDðpÞ:

(1)
Differentiating both sides of the above two first orderconditions with respect to h, we get
@p

@b¼

ru1þk2

@t

@bþ

1

ð1þk2ÞðhþbÞrðtÞ

and

�2K

t3

@t

@b¼

ruDbþh

þr00D @t

@bþruDu

@p

@b:


Combining these two equations together, we get

�2K

t3�r00D� ðruÞ

2Du

1þk2

" #@t

@b¼

ruDu

ð1þk2ÞðbþhÞr0:

From this, we can see that the results hold true. Similarly, we
can show the results with respect to h. We complete the prooffor parts (1) and (2).
(3)
Differentiating both sides of the two first order conditionswith respect to K, we get
�2K

t3�r00D

� �@t

@K¼�

1

t2þruDu

@p

@K

and

@p

@K¼

ru1þk2

@t

@K:

Combining them together, we get

�2K

t3�r00D�ðruÞ

2Du

1þk2

!@t

@K¼�

1

t2:

Hence, we see that @t=@K40 and @p=@K40.
(4) Differentiating both sides of p¼�ðk1=ð1þk2ÞÞþðcþrðtÞÞ=ð1þk2Þ with respect to s, we get
@p

@s ¼1

1þk2

@t

@s þðhþbÞ

t

Z t

0


pf ðdðt,tÞÞdt

þðhþbÞs

t

Z t

0f ðdðt,tÞÞdðt,tÞ t�t

s2dt:

Thus, it is obvious that if @t�=@s40, then @p�=@s40. &

References

Axsaster, S., 2000. Inventory Control. Kluwer Academic Publishers, Boston.Bresnahan, T.F., Reiss, P.C., 1985. Dealer and manufacturer margins. RAND Journal

of Economics 16 (2), 253–268.Bylka, S., 2005. Turnpike policies for periodic review inventory model

with emergency orders. International Journal of Production Economics 93–94,357–374.

Chen, X., Simchi-Levi, D., 2004. Coordinating inventory control and pricingstrategies with random demand and fixed ordering cost: the finite horizoncase. Operations Research 52 (6), 887–896.

Chiang, C., 2008. Periodic review inventory models with stochastic supplier’s visitintervals. International Journal of Production Economics 115 (2), 433–438.

Chiu, H.N., 1995. A heuristic (R, T) periodic review perishable inventorymodel with lead times. International Journal of Production Economics 42 (1),1–15.

Eynan, A., Kropp, D.H., 1998. Periodic review and joint replenishment in stochasticdemand environments. IIE Transactions 30, 1025–1033.

Eynan, A., Kropp, D.H., 2007. Effective and simple EOQ-like solutions for stochasticdemand periodic review systems. European Journal of Operational Research180 (3), 1135–1143.

Flynn, J., 2000. Selecting T for a periodic review inventory model with staggereddeliveries. Naval Research Logistics 47, 329–352.

Gavirneni, S., 2004. Periodic review inventory control with fluctuating purchasingcosts. Operations Research Letters 32, 374–379.

Granot, D., Yin, S., 2005. On the effectiveness of returns policies in the price-dependent newsvendor model. Naval Research Logistics 52, 765–779.

Gupta, D., Wang, L., 2007. Capacity management for contract manufacturing.Operations Research 55 (2), 367–377.

Hadley, G., Whitin, T.M., 1963. Analysis of Inventory Systems. Prentice-Hall.Huh, W., Janakiraman, G., 2008. (s, S) optimality in joint inventory-pricing control:

an alternate approach. Operations Research 56 (3), 783–790.Jose, S.L.A., Sicilia, J., Garcıa-Laguna, J., 2006. Analysis of an inventory system with

exponential partial backordering. International Journal of Production Econom-ics 100 (1), 76–86.

Lau, S., Lau, H., 2003. Non-robustness of the normal approximation of lead-timedemand in a (Q, R) system. Naval Research Logistics 50, 149–166.

Petruzzi, N.C., Dada, M., 1999. Pricing and the newsvendor problem: a review withextensions. Operations Research 47 (2), 183–194.

Polatoglu, H., Sahin, I., 2000. Optimal procurement policies under price-dependentdemand. International Journal of Production Economics 65 (2), 141–171.

Rao, U., 2003. Convexity and sensitivity properties of the (R, T) inventory controlpolicy for stochastic demand. Manufacturing & Service Operations Manage-ment 5, 37–53.

Ray, S., Li, S., Song, Y., 2005. Tailored supply chain decision-making under price-sensitive stochastic demand and delivery uncertainty. Management Science 51(12), 1873–1891.

Ritz, R., 2005. Strategic Incentives for Market Share. Working Paper, University ofOxford, Department of Economics.

Silver, E.A., Robb, D.J., 2008. Some insights regarding the optimal reorder period inperiodic review inventory systems. International Journal of ProductionEconomics 112 (1), 354–366.

Silver, E.A., Pyke, D.F., Peterson, R., 1998. Inventory Management and ProductionPlanning and Scheduling, third ed. John Wiley & Sons.

Song, Y., Ray, S., Li, S., 2008. Structural properties of Buyback contracts for price-setting newsvendors. Manufacturing & Service Operations Management 10 (1),1–18.

Song, Y., Ray, S., Boyaci, T., 2009. Note: optimal dynamic joint inventory-pricingcontrol for multiplicative demand with fixed setup costs and lost sales.Operations Research 57 (1), 245–250.

Stern, L.W., El-Ansary, A.I., 1992. Marketing Channels, fourth ed. Prentice-Hall,Englewood Cliffs, NJ.

Tyworth, J.E., O’Neill, L., 1997. Robustness of the normal approximation of lead-time demand in a distribution setting. Naval Research Logistics 44, 165–186.

Wang, Y., Jiang, L., Shen, Z.J., 2004. Channel performance under consignmentcontract with revenue sharing. Management Science 50, 34–47.

Wang, Y., Gerchak, Y., 1996. Periodic review production models with variable capacity,random yield, and uncertain demand. Management Science 42, 130–139.

Yano, C., Gilbert, S.M., 2003. Coordinated pricing and production/procurementdecisions: a review. In: Chakravarty, A., Eliashberg, J. (Eds.), Managing BusinessInterfaces: Marketing, Engineering, and Manufacturing Perspectives. KluwerAcademic Publishers, Boston, MA, pp. 65–103.

Zipkin, P., 2000. Foundations of Inventory Management. McGraw-Hill HigherEducation, USA.

int. j. production economics - dost sci-net phil

Documents