Research Article Dynamic Inventory and Pricing Policy in a ...downloads.hindawi.com/journals/mpe/2015/269695.pdfDynamic Inventory and Pricing Policy in a Periodic-Review Inventory

Research ArticleDynamic Inventory and Pricing Policy ina Periodic-Review Inventory System with Finite OrderingCapacity and Price Adjustment Cost

Baimei Yang1 Chunyan Gao2 Na Liu3 and Liang Xu4

1School of Business Shanghai Dianji University Shanghai 201306 China2Department of Management Science Southwestern University of Finance and Economics Chengdu 611130 China3Business School Beijing Institute of Fashion Technology Beijing 100029 China4Department of Logistics Management Southwestern University of Finance and Economics Chengdu 611130 China

Correspondence should be addressed to Na Liu nanalaufashiongmailcom

Received 23 April 2015 Accepted 21 September 2015

Academic Editor Ruihua Liu

Copyright copy 2015 Baimei Yang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We consider a dynamic inventory control and pricing optimization problem in a periodic-review inventory system with priceadjustment cost Each order occurswith a fixed ordering cost the ordering quantity is capacitatedWe consider a sequential decisionproblem where the firm first chooses the ordering quantity and then the sale price tomaximize the expected total discounted profitover the sale horizon We show that the optimal inventory control is partially characterized by a (119904 1199041015840 119901) policy in four regionsand the optimal pricing policy is dependent on the inventory level after the replenishment decision We present some numericalexamples to explore the effects of various parameters on the optimal pricing and replenishment policy

1 Introduction

Traditional literature on the multistage inventory systemmainly focuses on replenishment decision with or withoutsetup cost The well-known result is that the order-up-topolicy is optimal for the systems without setup cost andthe (119904 119878) policy is optimal for the systems with setup costIncreasing researchers are devoted to the study of joint priceand inventory control in the multistage inventory systemOur paper belongs to this stream but our paper considers asequential decision problem in a periodic-review inventorysystem with fixed ordering cost and price adjustment costThe ordering quantity is capacitated this may be limitedby the storage capacity or the supply capability The firmfirst decides its inventory level and then chooses a sale priceto maximize its long-run profit Our result shows that theoptimal inventory and pricing decision still preservers athreshold-type structure

Our paper is related to literature on the optimal controlof a single product system with finite capacity and setup cost

Several studies have been devoted to this area Shaoxiang andLambrecht [1] obtain the generally known result that is theoptimal policy can only be partially characterized in the formof119883-119884 bands In particular when the inventory level is belowthe first band 119883 then produceorder the capacity and whenthe inventory level is over the second band 119884 produceordernothing If the inventory level is between the two bands theordering policy is complicated and depends on the instanceGallego and Scheller-Wolf [2] extend their work They derivethe structure of the policy between the bands The optimalpolicy is characterized by two numbers 119904 and 119904

1015840 which dividethe state space into four possible regions However noneof them have studied the pricing problem in the inventorycontrol problem Zhang et al [3] consider a single-itemfinite-horizon periodic-review coordinated decision modelon pricing and inventory control with capacity constraintsand fixed ordering cost They show that the profit-to-gofunction is strongly 119862119870-concave and the optimal policy hasan (119904 119878 119875)-like structure However the price adjustment costhas not been addressed Chao et al [4] recently consider

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 269695 8 pageshttpdxdoiorg1011552015269695

2 Mathematical Problems in Engineering

the joint pricing and inventory decisions They study a peri-odic-review inventory system with setup cost and finiteordering capacity in each periodThey show that the optimalinventory control is characterized by an (119904 119904

1015840 119901) policy infour regions of the starting inventory level However in theirpaper the selling price can be adjusted without any cost

In reality changing price is costly and incurs a priceadjustment cost In the economics literature there are twomajor types of price adjustment costs the managerial costsand the physical costs Rotemberg [5] Levy et al [6] Sladeand Groupe de Recherche en Economie Quantitative drsquoAix-Marseille [7] Aguirregabiria [8] Bergen et al [9] andZbaracki et al [10] have stated that both types of costs aresignificant in retailing and other industries According tothese empirical studies Chen et al [11] consider a periodic-review inventorymodel with price adjustment costThe priceadjustment cost consists of both fixed and variable com-ponents They develop the general model and characterizethe optimal policies for two special scenarios a model withinventory carryover and no fixed price-change costs anda model with fixed price-change costs and no inventorycarryover Although there is price adjustment cost they donot consider the finite ordering capacity

Under the assumption of random additive demandmodel our paper tries to investigate the structure of theoptimal inventory control and pricing policy in each periodWe show that the optimal inventory policy is partiallycharacterized by an (119904 119904

1015840 119901) policy on four regions in two ofthese regions the optimal policy is completely specified whilein the other two it is partially specified More specificallythe optimal ordering quantity in the first region is the fullcapacity while in the last region it is optimal to order nothingin the two middle regions the optimal decision is eitherto order to the maximum capacity to order to at least aprespecified level 1199041015840 or to order nothingThe optimal pricingpolicy119901(119910) in each period is dependent on the inventory levelafter the replenishment decision 119910 which is in general not amonotone function The key concept utilized is strong 119862119870-concavity which is an extension of119870-concavity and was firstintroduced by Gallego and Scheller-Wolf [2]

The rest of this paper is organized as follows In Section 2we induce themodel descriptionThe structural properties ofthe optimal inventory and pricing policy are characterized inSection 3 We present some numerical examples to show theeffects of various parameters on the optimal control policyin Section 4 Finally we conclude with some future researchdirection in Section 5

2 The Model

Consider a periodic-review inventory system with finiteordering capacity and price adjustment cost There are 119873

periods with the first period being 1 and last period being119873In each period the sequence of events is given as follows (1)inventory level is reviewed and replenishment order is placed(2) replenishment order arrives (3) a selling price is set (4)random demand is realized and (5) all costs are computed

In period 119899 the selling price is 119901119899 which is taken in

interval [119901119897 119901ℎ] and the demand is 119863

119899 We assume that the

demand 119863119899is sensitive to the selling price 119901

119899 Moreover we

consider an additive demand functionThe demand functionis 119863119899(119901119899) = 119889(119901

119899) + 120598119899 119899 = 1 119873 where 120598

119899is a random

variable with mean zero and 119889(119901119899) is the average demand

Furthermore 119889(119901119899) is a decreasing linear function of 119901

119899

When the selling price 119901119899increases from 119901

119897to 119901ℎ the average

demand decreases from 119889ℎto 119889119897 that is 119889

ℎ= 119889(119901

119897) and

119889119897= 119889(119901

ℎ) Each demand arrives requiring only one unit of

product and is satisfied from inventory if any If the demandcannot be satisfied from the on-hand inventory immediatelythen it is backlogged and incurs a backorder cost Thestructure of demand function indicates that determining theselling price 119901

119899is equivalent to setting the average demand

119889119899Each replenishment incurs a fixed ordering cost 119870 and

the variable unit ordering cost 119888 There is a finite orderingcapacity 119862 for each period which means the orderingquantity in each period cannot exceed 119862 where 119862 gt 0 If119862 is sufficiently large it generalizes to the incapacitated caseLet 119909119899be the inventory level at the beginning of period 119899

before placing an order and let 119910119899be the inventory level after

the order delivered At the end of each period the demandis realized and a revenue is received The expected revenue isgiven by 119903

119899(119889) = 119889

119899sdot119901119899(119889119899) which is assumed to be a concave

function Meanwhile an inventory holding and shortage costoccurs denoted by ℎ(119910

119899minus 119889119899) If 119909 ge 0 ℎ(119909) represents the

holding cost if 119909 lt 0 ℎ(119909) represents the shortage cost Forease of presentation we let 119866(119910) = 119864[ℎ(119910 minus 120598

119899)] Therefore

given that the inventory level after replenishment is 119910119899and

the expected demand for period 119899 is 119889119899 the expected holding

and shortage cost is 119866(119910119899minus 119889119899)

We assume that there is a fixed guide price1199010for deciding

the selling price 119901119899 Price changing from the guide price

to the actual price is costly The cost of a price adjustmentfrom guide price to the actual selling price in period 119899

is denoted by 119880119899(119901119899

minus 1199010) Zbaracki et al [10] and Chen

et al [11] pointed out that as the price adjustment costbecomes larger it would cost more on decision and internalcommunication Here we assume that the variable cost 119880

119899(sdot)

is convex and increaseswith |119901119899minus1199010|The forms of119880

119899(sdot) could

be either piecewise linear functions or quadratic functionsThe ordering quantity in period 119899 is119910

119899minus119909119899 therefore we have

119909119899le 119910119899le 119909119899+ 119862 due to the capacitated ordering quantity 119862

Therefore the expected total cost incurs in period 119899 includingsetup cost ordering cost holding and shortage cost and priceadjustment price is given by

1198701 [119910119899gt 119909119899] + 119888 (119910

119899minus 119909119899) + 119866 (119910

119899minus 119889119899)

+ 119880119899(119901119899minus 1199010)

(1)

where 1[119860] is the indicating function taking value 1 ifstatement 119860 is true and zero otherwise

We aim to obtain the optimal pricing and inventorydecisions in each period to maximize the expected totaldiscounted profit over the 119899 periods Let 119881

119899(119909) denote the

maximum expected total discounted profit from period 119899 to

Mathematical Problems in Engineering 3

the end of the planning horizon with the starting inventorylevel (before ordering decision) 119909 The optimality equation is

119881119899(119909) = max

119909le119910le119909+119862

max119901isin[119901119897119901ℎ]

minus1198701 [119910 gt 119909] + 119903 (119901)

minus 119888 (119910 minus 119909) minus 119866 (119910 minus 119889) minus 119880 (119901 minus 1199010)

+ 120572119864 [119881119899+1

(119910 minus 119889 (119901) minus 120598119899)]

(2)

where 120572 is the one-period discount factor 120572 isin [0 1] Theterminal condition is119881

119873+1(119909) equiv 0 Note that the price 119901

119899can

be indicated in the form of demand 119889119899by the inverse demand

function that is 119901119899

= 119901119899(119889119899) and the price adjustment

cost can be written in the form of 119889119899instead of 119901

119899 that

is 119880119899(119889119899minus 1199010) such that optimizing over the selling price

119901119899is equivalent to optimizing over the average demand 119889

119899

Therefore the optimality equation is rewritten as follows

119881119899(119909) = max

119909le119910le119909+119862

max119889isin[119889119897 119889ℎ]

minus1198701 [119910 gt 119909] + 119903 (119889)


+ 120572119864 [119881119899+1

(119910 minus 119889 minus 120598119899)]

(3)

For notation convenience we define another function

119882119899(119910) = minus119866 (119910) + 120572119864 [119881

119899+1(119910 minus 120598

119899)] (4)

Then the optimality equation is further simplified to

119881119899(119909) = 119888119909 + max

119909le119910le119909+119862

max119889isin[119889119897 119889ℎ]

minus1198701 [119910 gt 119909] + 119903119899(119889) minus 119888119910 minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889) = 119888119909

+ max119909le119910le119909+119862

minus1198701 [119910 gt 119909] minus 119888119910 + max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(5)

3 The Optimal Policy

In order to characterize the structural properties of theoptimal replenishment and pricing policy we first introducethe definition of strongly 119862119870-concave and properties of 119862119870-concave functions as well which is defined in Chao et al[4] This definition and the properties are very important instudying inventorymodels with finite capacity and setup cost

Definition 1 A function 119892(sdot) 119877 rarr 119877 is strongly 119862119870-concave if for all 119886 ge 0 119887 gt 0 and 119911 isin [0 119862] we have

119911

119887119892 (119910 minus 119886) + 119892 (119910) ge

119911

119887119892 (119910 minus 119886 minus 119887) + 119892 (119910 + 119911) minus 119870 (6)

The structure of strong 119862119870-concave function is shownin Figure 1 If 119866(119909) is strong 119862119870-concave it implies that theslope of the line made of points (119909 119866(119909)) and (119909 + 119911 119866(119909 +

119911)minus119870) is smaller than the slope of the linemade of points (119909minus

119886 minus 119887 119866(119909 minus 119886 minus 119887)) and (119909 minus 119886 119866(119909 minus 119886))Chao et al [4] also pointed out that the strongly 119862119870-

concave function possesses some additional properties asfollows

(1) If 119866 is strongly 119862119870-concave then it is also strongly119863119871-concave for 0 le 119863 le 119862 and 119871 ge 119870

(2) If 119866 is concave it is also strongly 119862119870-concave for anynonnegative 119862 and 119870

(3) If1198661is strongly119862119870

1-concave and119866

2is strongly119862119870

2-

concave then for 120572 120573 ge 0 1205721198661+ 1205731198662is strongly

119862(1205721198701+ 1205731198702)-concave

(4) If 119866 is strongly 119862119870-concave and 119883 is a randomvariable such that 119864[|119866(119910 minus 119883)|] lt infin then 119864[119866(119910 minus

119883)] is also strongly 119862119870-concave

In the following we aim to show that 119881119899(119909) preservers

the property of strong 119862119870-concavity Before going further

we first show that each term on the right hand side of (3)possesses some certain properties which will facilitate ouranalysis of objective function 119881

119899(119909)

Lemma 2 119880119899(119889 minus 119901

0) is convex in 119889

Proof Considering that119880119899(119889minus1199010) is continuous and second-

order derivable the convexity of119880119899(119889 minus 119901

0) can be proved by

its second derivative We have

119889119880119899(119889 minus 119901

0)

119889119889=

119889119880119899(119901 minus 119901

0)

119889119901

119889119901 (119889)

119889119889

1198892119880119899(119889 minus 119901

0)

1198891198892=

1198892119880119899(119901 minus 119901

0)

1198891199012(119889119901 (119889)

119889119889)

2

+119889119880119899(119901 minus 119901

0)

119889119901

1198892119901 (119889)

1198891198892

(7)

Since119889(119901119899) is linear anddecreasing on119901

119899 whichmeans119901(119889

119899)

is also linear and decreasing on 119889119899 then 1198892119901(119889)1198891198892 = 0

At the same time due to the convexity of 119880119899(sdot) 1198892119880

119899(119901 minus

1199010)1198891199012 ge 0 Therefore

1198892119880119899(119889 minus 119901

0)

1198891198892=

1198892119880119899(119901 minus 119901

0)

1198891199012(119889119901 (119889)

119889119889)

2

ge 0 (8)

which indicates that 119880119899(119889 minus 119901

0) is convex in 119889 Lemma 2 is

proved

Lemma 3 Let 119889119899(119910) be the maximizer of 119903

119899(119889) minus119880

119899(119889minus119901

0) +

119882119899(119910 minus 119889) then 119910 minus 119889

119899(119910) is increasing in 119910

Proof Due to the concavity of 119903(sdot) and Lemma 2 Lemma 3can be conducted directly by using the properties of super-modularity


20

18

16

14

12

10

8

6

4

2

00 1 2 3 4 5

y

x

K

G(x)

G(x minus a)

G(x minus a minus b)

G(x + z) minus K

G(x + z)

minus3 minus2 minus1

Figure 1 119862119870-concave function

Lemma 4 If 119882119899(119910) is strongly 119862119870-concave then

119892 (119910) = max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889) (9)

is also strongly 119862119870-concave

The proof of Lemma 4 is similar to that in Chao et al [4]We omit it for simplicity

Lemma 5 119881119899(119909) is strongly 119862119870-concave

Proof Lemma 5 can be proved by inductionWhen 119899 = 119873+1we have 119881

119873+1equiv 0 such that 119881

119873+1is strongly 119862119870-concave

Now suppose that 119881119899+1

(119909) is strongly 119862119870-concave then weproceed to prove that 119881

119899(119909) is also strongly 119862119870-concave

Due to the property of strong 119862119870-concavity we obtainthat120572119864[119881

119899+1(119910minus120598119899) is strongly119862(120572119870)-concave Consider that

minus119866119899(119910) is concave then 119882

119899(119910) is strongly 119862(120572119870)-concave

which is also strongly119862119870-concave Lemma 4 shows that 119892(119910)

is also strongly 119862119870-concave Therefore

119881119899(119909) = 119888119909 + max

119909le119910le119909+119862

minus1198701 [119910 gt 119909] minus 119888119910 + 119892 (119910) (10)

is also strongly 119862119870-concave Readers are referred to Gallegoand Scheller-Wolf [2] formore details Lemma 5 is concluded

The strong119862119870-concavity of119881119899(119909) characterizes the struc-

tural properties of the optimal inventory and pricing policyfor each period as given in the following theorem

Theorem6 Suppose that119909 is the starting inventory level at thebeginning of period 119899The optimal inventory policy is threshold-type policy which is characterized by two numbers 119904

119899and 1199041015840119899

where 119904119899

le 1199041015840119899 Furthermore the optimal inventory policy

possesses the following additional properties

(a) If 1199041015840119899minus 119862 le 119904

119899 then the optimal ordering policy is

(i) order capacity 119862 if 119909 lt 1199041015840119899minus 119862

(ii) order at least up to 1199041015840

119899if 1199041015840119899minus 119862 le 119909 lt 119904

119899

(iii) either order nothing or order at least up to 1199041015840119899if

119904119899le 119909 lt 1199041015840

119899

(iv) order nothing if 119909 ge 1199041015840119899

(b) If 1199041015840119899minus 119862 gt 119904


(i) order capacity 119862 if 119909 lt 119904119899

(ii) either order nothing or order119862 if 119904119899le 119909 lt 1199041015840

119899minus119862


1199041015840119899minus 119862 le 119909 lt 1199041015840

119899


The optimal pricing decision is characterized by 119901lowast119899(119910)

which depends on the postorder inventory position 119910 Further-more the optimal pricing decision 119901

lowast

119899(119910) as well as the optimal

average demand 119889lowast119899(119910) is in general not monotone in 119910

Proof Suppose

119867119899(119910)

= minus119888119910

+ max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(11)

Define 119904119899 119878119899 and 1199041015840

119899by

119878119899= inf 119910 isin 119877 | 119866 (119910) = sup

119910isin119877

119867119899(119910)

119904119899= inf 119909 | minus119870 + sup

119909le119910le119909+119862

119867119899(119910) le 119867

119899(119909)

1199041015840

119899= max119909 le 119878

119899| minus119870 + sup

119909le119910le119909+119862

119867119899(119910) ge 119867

119899(119909)

(12)

Obviously 119904119899le 1199041015840119899

The optimal pricing decision is determined by the maxi-mizer in Lemma 4 Let

119889lowast

119899(119910)

= arg max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(13)

which means that the optimal average demand in period 119899

depends on the replenished inventory level 119910 Since 119901 = 119901(119889)

is the inverse function of 119889 = 119889119899(119901) then we will obtain that

the optimal pricing decision is

119901lowast

119899(119910) = 119901 (119889

lowast

119899(119910)) (14)

when the replenished inventory level is 119910Therefore the opti-mal selling price in period 119899 also depends on the replenishedinventory level 119910 However 119901lowast

119899(119910) is not monotone in 119910 We

will give one example in Section 4The proof ofTheorem 6 isconcluded


x

sn + C

s998400n

sn

s998400n minus C

Order nothing

Order up to s998400n or order nothing

Order up to s998400n

Order C

(a) 1199041015840119899minus 119862 le 119904

119899

x

sn + C

s998400n

sn

s998400n minus C

Order nothing


Order C

Order C or order nothing

(b) 1199041015840119899minus 119862 gt 119904

119899

Figure 2 The structure of the optimal replenishment policy

The structure of the optimal inventory policy is presentedin Figure 2

Our results are similar to Gallego and Scheller-Wolf [2] inthat the optimal inventory policy can only be partially char-acterized When the inventory level before replenishment 119909is less than min1199041015840

119899minus 119862 119904

119899 the optimal ordering policy is to

order the full capacity When 119909 is larger than 1199041015840119899 the optimal

ordering policy is no order When min1199041015840119899minus 119862 119904

119899 le 119909 lt 1199041015840

119899

the optimal strategy is complicated When max1199041015840119899minus 119862 119904

119899 le

119909 lt 1199041015840119899 the optimal strategy may be to either order nothing or

order at least up to 1199041015840119899 When min1199041015840

119899minus 119862 119904

119899 le 119909 lt max1199041015840

119899minus

119862 119904119899 there would be two possibilities If 1199041015840

119899minus 119862 le 119904

119899 the

optimal policy is to order at least up to 1199041015840119899 If 1199041015840119899

minus 119862 gt 119904119899 the

optimal policy is no order or ordering full capacityMoreoverthe optimal pricing decision depends on the inventory levelafter replenishment

4 Numerical Tests

In order to explore the effects of the setup cost the orderingcapacity the guide price and the adjustment cost functionon the optimal control policy we conduct several numericalexperiments for a simple inventory problem with 119873 = 4

periods In the subsequent numerical experiments we usethe following basic settings the discount factor is 120572 = 09purchasing unit cost 119888 = 3 guide price 119901

0= 3 ordering

capacity 119862 = 10 and setup cost 119870 = 10 We adopt ℎ(119909) =

ℎmax0 119909 + 119887max0 minus119909 as the holding shortage cost ratefunction where ℎ = 2 and 119887 = 4 Suppose that 119880

119899(sdot) is

piecewise linear which means that119880119899(119901 minus119901

0) = 1198861max119901

0minus

119901 0 +1198862max119901minus119901

0 0 where 119886

1= 1198862= 05 The demand in

period 119899 is 119863119899(119901119899) = 119889(119901

119899) + 120598119899 where 119889(119901

119899) = 10 minus 119901

119899and

random error 120598119899

isin minus1 1 with probability mass function119875120598119899

= 1 = 119875120598119899

= minus1 = 05 Here 119901119899takes values in

0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

K = 5

K = 10

K = 15

minus5 0 5 10 15minus10

Inventory level x

Figure 3 Optimal replenished inventory level 119910 for different 119870

[1 9] Thus as 119901119899increases from 1 to 9 the average demand

decreases from 9 to 1

41 Effect of Setup Cost We study the effect of the setup cost119870 on the optimal inventory and pricing policyThe results areshown in Figures 3 and 4 In Figure 3 the 119909-axis representsthe inventory level before ordering 119909 and 119910-axis representsthe inventory level after ordering 119910 The value of 119909 goes fromminus10 to 15 with the increment of 1 In Figure 4 the 119909-axisrepresents the inventory level before ordering 119909 and 119910-axisrepresents the optimal selling price119901The value of 119909 also goes


K = 5

K = 10

K = 15


Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

Figure 4 Optimal selling price 119901 for different 119870

from minus10 to 15 with the increment of 1 Here we consider119870 = 5 119870 = 10 and 119870 = 15 separately

Figure 3 shows that the higher setup cost119870 implies lowerinventory level at which the optimal ordering policy changesfrom ordering to not ordering while Figure 4 shows that thehigher setup cost indicates higher optimal selling price Theresults are intuitive The trade-off is the setup cost holdingcost and sale revenue When the setup cost is high we willdecrease the replenishment frequency in order to reducethe ordering cost Hence we would place no order at lowinventory level and order up to a higher inventory level if weplace an order The other alternative way is to increase theselling price to reduce the demand in the purpose of savingsetup cost

Observation 1 The optimal selling price 119901 is not alwaysmonotonic in 119910

When 119870 = 5 and the inventory level before ordering 119909

is no less than 2 the optimal replenished inventory level 119910 isequal to 119909 Furthermore in Figure 4 when 119870 = 5 and 119909 ge 2the optimal selling price 119901 is not monotonic in 119909 in otherwords 119901 is not monotonic in 119910

42 Effect of Ordering Capacity The effects of orderingcapacity 119870 on the optimal inventory and pricing policy areshown in Figures 5 and 6 Higher ordering capacity meansthat we may order more every time without increasing costParticularly when the inventory level is high enough theoptimal policy is not to orderThen the ordering capacity hasno effect on the optimal ordering policy and selling priceWhen the inventory level is small higher ordering capacityindicates higher optimal replenished inventory level andlower optimal selling price which induces higher demand

0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

minus5minus5

0 5 10 15minus10

Inventory level x

C = 5

C = 7

C = 10



Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

C = 5

C = 7

C = 10


43 Effect of Guide Price The effects of guide price 1199010on the

optimal inventory and pricing policy are shown in Figures7 and 8 Compared with no guide price the existence ofthe guide price indicates higher inventory level at whichthe optimal ordering policy changes from ordering to notorderingThe guide price has no obvious effect on the optimalreplenishment inventory level but it influences the optimalselling price The optimal selling price would be closer to theguide price compared with the initial optimal selling pricewithout guide price For instance when the guide price is 5the optimal selling price would be lower than the initial oneunder small inventory level while the optimal selling price


0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

Without p0p0 = 5

p0 = 7



Inventory level x

Without p0p0 = 5

p0 = 7

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p


would be higher than the initial one under high inventorylevel

In Figure 8 when the inventory level is 15 the optimalselling price is 5 when the guide price is 5 while the optimalselling price is 4 when the guide price is 7 It leads to thefollowing observation

Observation 2 Higher guide price does not indicate higheroptimal selling price

44 Effect of Price Adjustment Cost Function The effects ofprice adjustment cost on the optimal ordering and pricingpolicy are shown in Figures 9 and 10We consider three cases

0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1

Figure 9 Optimal replenished inventory level 119910 for different 1198861and

1198862


Inventory level x

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1

Figure 10 Optimal selling price 119901 for different 1198861and 119886

2

1198861= 1 1198862= 9 1198861= 1198862= 5 and 119886

1= 9 1198862= 1 From Figure 9

we find that the price adjustment cost function has no obviouseffect on the optimal replenishment inventory level howeverit influences the optimal selling price obviously 119886

1lt 1198862

implies that it would be more costly when the selling priceis higher than the guide price 119886

1gt 1198862implies that it would

bemore costly when the selling price is smaller than the guideprice 119886

1= 1198862implies that it would be more costly when

the selling price is not equal to the guide price Hence inFigure 10 under the same inventory level the optimal sellingprice is the highest when 119886

1gt 1198862 while the optimal selling

price is the smallest when 1198861lt 1198862


5 Conclusions

In this paper we consider a dynamic inventory control andpricing optimization problem in a periodic-review inventorysystem with fixed ordering cost and price adjustment costAt the same time the ordering quantity is limited Here weassume that the price adjustment cost functions are piecewiselinear We show that the optimal inventory control similarto Chao et al [4] is also partially characterized by (119904 119904

1015840 119901)

policy in four regions and the optimal pricing policy isdependent on the inventory level after the replenishmentdecision From the numerical tests we present some statis-tical analysis to study the effects of various parameters on theoptimal control policy For example the higher setup cost 119870implies lower inventory level at which the optimal orderingpolicy changes from ordering to not ordering and higheroptimal selling price When the inventory level is smallhigher ordering capacity indicates high optimal replenishedinventory level and lower optimal selling price When theinventory level is high enough the optimal ordering policyand selling price are the same under different inventorylevel Optimal selling price would be closer to the guideprice compared with the initial optimal selling price withoutguide price while higher guide price does not indicate higheroptimal selling price Under the same inventory level theoptimal selling price is the highest when it would be morecostly when the selling price is smaller than the guide pricewhile the optimal selling price is the smallest when it wouldbe more costly when the selling price is larger than the guideprice

There are still many interesting issues worth studyingin the future research Our paper studied increasing convexprice adjustment cost exploring price adjustment cost func-tion with more complicated form may be one of potentialresearch directions In our paper the decision sequence isfirst inventory decision and then price decision but in realitythe firm may first set price to serve the target market andthen build up the inventory In this case what is the optimalpricing and replenishment policy Does the optimal controlpolicy still possess the similar structure

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported in part by the National NaturalScience Foundation of China under Grants nos 71201128 and71201127 Project for Training and Supporting YoungTeachersin Shanghai Universities (no ZZSDJ13007) and LeadingAcademic Discipline Project of Shanghai Dianji University(no 10XKJ01)

References

[1] C Shaoxiang andM Lambrecht ldquoX-Y band andmodified (s S)policyrdquo Operations Research vol 44 no 6 pp 1013ndash1019 1996

[2] G Gallego and A Scheller-Wolf ldquoCapacitated inventory prob-lems with fixed order costs some optimal policy structurerdquoEuropean Journal of Operational Research vol 126 no 3 pp603ndash613 2000

[3] J-L Zhang J Chen and C-Y Lee ldquoCoordinated pricing andinventory control problems with capacity constraints and fixedordering costrdquo Naval Research Logistics vol 59 no 5 pp 376ndash383 2012

[4] X Chao B Yang and Y Xu ldquoDynamic inventory and pricingpolicy in a capacitated stochastic inventory system with fixedordering costrdquo Operations Research Letters vol 40 no 2 pp99ndash107 2012

[5] J J Rotemberg ldquoSticky prices in the United Statesrdquo Journal ofPolitical Economy vol 90 no 6 pp 1187ndash1211 1982

[6] D Levy M Bergen S Dutta and R Venable ldquoThe magnitudeof menu costs direct evidence from large U S supermarketchainsrdquo The Quarterly Journal of Economics vol 112 no 3 pp790ndash825 1997

[7] M E Slade and Groupe de Recherche en Economie Quantita-tive drsquoAix-Marseille ldquoOptimal pricing with costly adjustmentevidence from retail-grocery pricesrdquo The Review of EconomicStudies vol 65 no 1 pp 87ndash107 1998

[8] V Aguirregabiria ldquoThe dynamics of markups and inventoriesin retailing firmsrdquo The Review of Economic Studies vol 66 no2 pp 275ndash308 1999

[9] M Bergen M Ritson S Dutta D Levy and M ZbarachildquoShattering the myth of costless price changesrdquo EuropeanManagement Journal vol 21 no 6 pp 663ndash669 2003

[10] M J Zbaracki M Ritson D Levy S Dutta and M BergenldquoManagerial and customer costs of price adjustment directevidence from industrialmarketsrdquoTheReview of Economics andStatistics vol 86 no 2 pp 514ndash533 2004

[11] X Chen S X Zhou and Y Chen ldquoIntegration of inventoryand pricing decisions with costly price adjustmentsrdquoOperationsResearch vol 59 no 5 pp 1144ndash1158 2011

Submit your manuscripts athttpwwwhindawicom

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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


the joint pricing and inventory decisions They study a peri-odic-review inventory system with setup cost and finiteordering capacity in each periodThey show that the optimalinventory control is characterized by an (119904 119904

1015840 119901) policy infour regions of the starting inventory level However in theirpaper the selling price can be adjusted without any cost

In reality changing price is costly and incurs a priceadjustment cost In the economics literature there are twomajor types of price adjustment costs the managerial costsand the physical costs Rotemberg [5] Levy et al [6] Sladeand Groupe de Recherche en Economie Quantitative drsquoAix-Marseille [7] Aguirregabiria [8] Bergen et al [9] andZbaracki et al [10] have stated that both types of costs aresignificant in retailing and other industries According tothese empirical studies Chen et al [11] consider a periodic-review inventorymodel with price adjustment costThe priceadjustment cost consists of both fixed and variable com-ponents They develop the general model and characterizethe optimal policies for two special scenarios a model withinventory carryover and no fixed price-change costs anda model with fixed price-change costs and no inventorycarryover Although there is price adjustment cost they donot consider the finite ordering capacity

Under the assumption of random additive demandmodel our paper tries to investigate the structure of theoptimal inventory control and pricing policy in each periodWe show that the optimal inventory policy is partiallycharacterized by an (119904 119904

1015840 119901) policy on four regions in two ofthese regions the optimal policy is completely specified whilein the other two it is partially specified More specificallythe optimal ordering quantity in the first region is the fullcapacity while in the last region it is optimal to order nothingin the two middle regions the optimal decision is eitherto order to the maximum capacity to order to at least aprespecified level 1199041015840 or to order nothingThe optimal pricingpolicy119901(119910) in each period is dependent on the inventory levelafter the replenishment decision 119910 which is in general not amonotone function The key concept utilized is strong 119862119870-concavity which is an extension of119870-concavity and was firstintroduced by Gallego and Scheller-Wolf [2]

The rest of this paper is organized as follows In Section 2we induce themodel descriptionThe structural properties ofthe optimal inventory and pricing policy are characterized inSection 3 We present some numerical examples to show theeffects of various parameters on the optimal control policyin Section 4 Finally we conclude with some future researchdirection in Section 5

2 The Model

Consider a periodic-review inventory system with finiteordering capacity and price adjustment cost There are 119873

periods with the first period being 1 and last period being119873In each period the sequence of events is given as follows (1)inventory level is reviewed and replenishment order is placed(2) replenishment order arrives (3) a selling price is set (4)random demand is realized and (5) all costs are computed

In period 119899 the selling price is 119901119899 which is taken in

interval [119901119897 119901ℎ] and the demand is 119863

119899 We assume that the

demand 119863119899is sensitive to the selling price 119901

119899 Moreover we

consider an additive demand functionThe demand functionis 119863119899(119901119899) = 119889(119901

119899) + 120598119899 119899 = 1 119873 where 120598

119899is a random

variable with mean zero and 119889(119901119899) is the average demand

Furthermore 119889(119901119899) is a decreasing linear function of 119901

119899

When the selling price 119901119899increases from 119901

119897to 119901ℎ the average

demand decreases from 119889ℎto 119889119897 that is 119889

ℎ= 119889(119901

119897) and

119889119897= 119889(119901

ℎ) Each demand arrives requiring only one unit of

product and is satisfied from inventory if any If the demandcannot be satisfied from the on-hand inventory immediatelythen it is backlogged and incurs a backorder cost Thestructure of demand function indicates that determining theselling price 119901

119899is equivalent to setting the average demand

119889119899Each replenishment incurs a fixed ordering cost 119870 and

the variable unit ordering cost 119888 There is a finite orderingcapacity 119862 for each period which means the orderingquantity in each period cannot exceed 119862 where 119862 gt 0 If119862 is sufficiently large it generalizes to the incapacitated caseLet 119909119899be the inventory level at the beginning of period 119899

before placing an order and let 119910119899be the inventory level after

the order delivered At the end of each period the demandis realized and a revenue is received The expected revenue isgiven by 119903

119899(119889) = 119889

119899sdot119901119899(119889119899) which is assumed to be a concave

function Meanwhile an inventory holding and shortage costoccurs denoted by ℎ(119910

119899minus 119889119899) If 119909 ge 0 ℎ(119909) represents the

holding cost if 119909 lt 0 ℎ(119909) represents the shortage cost Forease of presentation we let 119866(119910) = 119864[ℎ(119910 minus 120598

119899)] Therefore

given that the inventory level after replenishment is 119910119899and

the expected demand for period 119899 is 119889119899 the expected holding

and shortage cost is 119866(119910119899minus 119889119899)

We assume that there is a fixed guide price1199010for deciding

the selling price 119901119899 Price changing from the guide price

to the actual price is costly The cost of a price adjustmentfrom guide price to the actual selling price in period 119899

is denoted by 119880119899(119901119899

minus 1199010) Zbaracki et al [10] and Chen

et al [11] pointed out that as the price adjustment costbecomes larger it would cost more on decision and internalcommunication Here we assume that the variable cost 119880

119899(sdot)

is convex and increaseswith |119901119899minus1199010|The forms of119880

119899(sdot) could

be either piecewise linear functions or quadratic functionsThe ordering quantity in period 119899 is119910

119899minus119909119899 therefore we have

119909119899le 119910119899le 119909119899+ 119862 due to the capacitated ordering quantity 119862

Therefore the expected total cost incurs in period 119899 includingsetup cost ordering cost holding and shortage cost and priceadjustment price is given by

1198701 [119910119899gt 119909119899] + 119888 (119910

119899minus 119909119899) + 119866 (119910

119899minus 119889119899)

+ 119880119899(119901119899minus 1199010)

(1)

where 1[119860] is the indicating function taking value 1 ifstatement 119860 is true and zero otherwise

We aim to obtain the optimal pricing and inventorydecisions in each period to maximize the expected totaldiscounted profit over the 119899 periods Let 119881

119899(119909) denote the

maximum expected total discounted profit from period 119899 to



119881119899(119909) = max

119909le119910le119909+119862

max119901isin[119901119897119901ℎ]

minus1198701 [119910 gt 119909] + 119903 (119901)


+ 120572119864 [119881119899+1

(119910 minus 119889 (119901) minus 120598119899)]

(2)



119899can





119899 that



119899


119881119899(119909) = max

119909le119910le119909+119862

max119889isin[119889119897 119889ℎ]

minus1198701 [119910 gt 119909] + 119903 (119889)


+ 120572119864 [119881119899+1

(119910 minus 119889 minus 120598119899)]

(3)


119882119899(119910) = minus119866 (119910) + 120572119864 [119881

119899+1(119910 minus 120598

119899)] (4)


119881119899(119909) = 119888119909 + max

119909le119910le119909+119862

max119889isin[119889119897 119889ℎ]


119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889) = 119888119909

+ max119909le119910le119909+119862


119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(5)




119911

119887119892 (119910 minus 119886) + 119892 (119910) ge

119911








(3) If1198661is strongly119862119870

1-concave and119866


2-


119862(1205721198701+ 1205731198702)-concave






119899(119909)

Lemma 2 119880119899(119889 minus 119901




0) can be proved by


119889119880119899(119889 minus 119901

0)

119889119889=

119889119880119899(119901 minus 119901

0)

119889119901

119889119901 (119889)

119889119889

1198892119880119899(119889 minus 119901

0)

1198891198892=

1198892119880119899(119901 minus 119901

0)

1198891199012(119889119901 (119889)

119889119889)

2

+119889119880119899(119901 minus 119901

0)

119889119901

1198892119901 (119889)

1198891198892

(7)


119899 whichmeans119901(119889

119899)



119899(119901 minus

1199010)1198891199012 ge 0 Therefore

1198892119880119899(119889 minus 119901

0)

1198891198892=

1198892119880119899(119901 minus 119901

0)

1198891199012(119889119901 (119889)

119889119889)

2

ge 0 (8)



proved


119899(119889) minus119880

119899(119889minus119901

0) +

119882119899(119910 minus 119889) then 119910 minus 119889

119899(119910) is increasing in 119910



20

18

16

14

12

10

8

6

4

2

00 1 2 3 4 5

y

x

K

G(x)

G(x minus a)


G(x + z) minus K

G(x + z)




119892 (119910) = max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889) (9)
















119881119899(119909) = 119888119909 + max

119909le119910le119909+119862

minus1198701 [119910 gt 119909] minus 119888119910 + 119892 (119910) (10)





119899and 1199041015840119899

where 119904119899



(a) If 1199041015840119899minus 119862 le 119904




119899if 1199041015840119899minus 119862 le 119909 lt 119904

119899


119904119899le 119909 lt 1199041015840

119899


(b) If 1199041015840119899minus 119862 gt 119904




119899minus119862


1199041015840119899minus 119862 le 119909 lt 1199041015840

119899




lowast



Proof Suppose

119867119899(119910)

= minus119888119910

+ max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(11)

Define 119904119899 119878119899 and 1199041015840

119899by

119878119899= inf 119910 isin 119877 | 119866 (119910) = sup

119910isin119877

119867119899(119910)

119904119899= inf 119909 | minus119870 + sup

119909le119910le119909+119862

119867119899(119910) le 119867

119899(119909)

1199041015840

119899= max119909 le 119878

119899| minus119870 + sup

119909le119910le119909+119862

119867119899(119910) ge 119867

119899(119909)

(12)

Obviously 119904119899le 1199041015840119899


119889lowast

119899(119910)

= arg max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(13)





119901lowast

119899(119910) = 119901 (119889

lowast

119899(119910)) (14)





x

sn + C

s998400n

sn

s998400n minus C

Order nothing



Order C

(a) 1199041015840119899minus 119862 le 119904

119899

x

sn + C

s998400n

sn

s998400n minus C

Order nothing


Order C


(b) 1199041015840119899minus 119862 gt 119904

119899




119899minus 119862 119904




119899 le 119909 lt 1199041015840

119899


119899 le



119899minus 119862 119904

119899 le 119909 lt max1199041015840

119899minus


119899minus 119862 le 119904

119899 the


minus 119862 gt 119904119899 the


4 Numerical Tests



0= 3 ordering



119899(sdot) is


0) = 1198861max119901

0minus

119901 0 +1198862max119901minus119901

0 0 where 119886

1= 1198862= 05 The demand in

period 119899 is 119863119899(119901119899) = 119889(119901

119899) + 120598119899 where 119889(119901

119899) = 10 minus 119901

119899and



= 1 = 119875120598119899


0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

K = 5

K = 10

K = 15


Inventory level x






K = 5

K = 10

K = 15


Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p








0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

minus5minus5

0 5 10 15minus10

Inventory level x

C = 5

C = 7

C = 10



Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

C = 5

C = 7

C = 10





0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

Without p0p0 = 5

p0 = 7



Inventory level x

Without p0p0 = 5

p0 = 7

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p






0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


1198862


Inventory level x

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


2

1198861= 1 1198862= 9 1198861= 1198862= 5 and 119886

1= 9 1198862= 1 From Figure 9


1lt 1198862









5 Conclusions


1015840 119901)





Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119881119899(119909) = max

119909le119910le119909+119862

max119901isin[119901119897119901ℎ]

minus1198701 [119910 gt 119909] + 119903 (119901)


+ 120572119864 [119881119899+1

(119910 minus 119889 (119901) minus 120598119899)]

(2)



119899can





119899 that



119899


119881119899(119909) = max

119909le119910le119909+119862

max119889isin[119889119897 119889ℎ]

minus1198701 [119910 gt 119909] + 119903 (119889)


+ 120572119864 [119881119899+1

(119910 minus 119889 minus 120598119899)]

(3)


119882119899(119910) = minus119866 (119910) + 120572119864 [119881

119899+1(119910 minus 120598

119899)] (4)


119881119899(119909) = 119888119909 + max

119909le119910le119909+119862

max119889isin[119889119897 119889ℎ]


119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889) = 119888119909

+ max119909le119910le119909+119862


119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(5)




119911

119887119892 (119910 minus 119886) + 119892 (119910) ge

119911








(3) If1198661is strongly119862119870

1-concave and119866


2-


119862(1205721198701+ 1205731198702)-concave






119899(119909)

Lemma 2 119880119899(119889 minus 119901




0) can be proved by


119889119880119899(119889 minus 119901

0)

119889119889=

119889119880119899(119901 minus 119901

0)

119889119901

119889119901 (119889)

119889119889

1198892119880119899(119889 minus 119901

0)

1198891198892=

1198892119880119899(119901 minus 119901

0)

1198891199012(119889119901 (119889)

119889119889)

2

+119889119880119899(119901 minus 119901

0)

119889119901

1198892119901 (119889)

1198891198892

(7)


119899 whichmeans119901(119889

119899)



119899(119901 minus

1199010)1198891199012 ge 0 Therefore

1198892119880119899(119889 minus 119901

0)

1198891198892=

1198892119880119899(119901 minus 119901

0)

1198891199012(119889119901 (119889)

119889119889)

2

ge 0 (8)



proved


119899(119889) minus119880

119899(119889minus119901

0) +

119882119899(119910 minus 119889) then 119910 minus 119889

119899(119910) is increasing in 119910



20

18

16

14

12

10

8

6

4

2

00 1 2 3 4 5

y

x

K

G(x)

G(x minus a)


G(x + z) minus K

G(x + z)




119892 (119910) = max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889) (9)
















119881119899(119909) = 119888119909 + max

119909le119910le119909+119862

minus1198701 [119910 gt 119909] minus 119888119910 + 119892 (119910) (10)





119899and 1199041015840119899

where 119904119899



(a) If 1199041015840119899minus 119862 le 119904




119899if 1199041015840119899minus 119862 le 119909 lt 119904

119899


119904119899le 119909 lt 1199041015840

119899


(b) If 1199041015840119899minus 119862 gt 119904




119899minus119862


1199041015840119899minus 119862 le 119909 lt 1199041015840

119899




lowast



Proof Suppose

119867119899(119910)

= minus119888119910

+ max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(11)

Define 119904119899 119878119899 and 1199041015840

119899by

119878119899= inf 119910 isin 119877 | 119866 (119910) = sup

119910isin119877

119867119899(119910)

119904119899= inf 119909 | minus119870 + sup

119909le119910le119909+119862

119867119899(119910) le 119867

119899(119909)

1199041015840

119899= max119909 le 119878

119899| minus119870 + sup

119909le119910le119909+119862

119867119899(119910) ge 119867

119899(119909)

(12)

Obviously 119904119899le 1199041015840119899


119889lowast

119899(119910)

= arg max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(13)





119901lowast

119899(119910) = 119901 (119889

lowast

119899(119910)) (14)





x

sn + C

s998400n

sn

s998400n minus C

Order nothing



Order C

(a) 1199041015840119899minus 119862 le 119904

119899

x

sn + C

s998400n

sn

s998400n minus C

Order nothing


Order C


(b) 1199041015840119899minus 119862 gt 119904

119899




119899minus 119862 119904




119899 le 119909 lt 1199041015840

119899


119899 le



119899minus 119862 119904

119899 le 119909 lt max1199041015840

119899minus


119899minus 119862 le 119904

119899 the


minus 119862 gt 119904119899 the


4 Numerical Tests



0= 3 ordering



119899(sdot) is


0) = 1198861max119901

0minus

119901 0 +1198862max119901minus119901

0 0 where 119886

1= 1198862= 05 The demand in

period 119899 is 119863119899(119901119899) = 119889(119901

119899) + 120598119899 where 119889(119901

119899) = 10 minus 119901

119899and



= 1 = 119875120598119899


0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

K = 5

K = 10

K = 15


Inventory level x






K = 5

K = 10

K = 15


Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p








0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

minus5minus5

0 5 10 15minus10

Inventory level x

C = 5

C = 7

C = 10



Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

C = 5

C = 7

C = 10





0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

Without p0p0 = 5

p0 = 7



Inventory level x

Without p0p0 = 5

p0 = 7

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p






0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


1198862


Inventory level x

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


2

1198861= 1 1198862= 9 1198861= 1198862= 5 and 119886

1= 9 1198862= 1 From Figure 9


1lt 1198862









5 Conclusions


1015840 119901)





Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










20

18

16

14

12

10

8

6

4

2

00 1 2 3 4 5

y

x

K

G(x)

G(x minus a)


G(x + z) minus K

G(x + z)




119892 (119910) = max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889) (9)
















119881119899(119909) = 119888119909 + max

119909le119910le119909+119862

minus1198701 [119910 gt 119909] minus 119888119910 + 119892 (119910) (10)





119899and 1199041015840119899

where 119904119899



(a) If 1199041015840119899minus 119862 le 119904




119899if 1199041015840119899minus 119862 le 119909 lt 119904

119899


119904119899le 119909 lt 1199041015840

119899


(b) If 1199041015840119899minus 119862 gt 119904




119899minus119862


1199041015840119899minus 119862 le 119909 lt 1199041015840

119899




lowast



Proof Suppose

119867119899(119910)

= minus119888119910

+ max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(11)

Define 119904119899 119878119899 and 1199041015840

119899by

119878119899= inf 119910 isin 119877 | 119866 (119910) = sup

119910isin119877

119867119899(119910)

119904119899= inf 119909 | minus119870 + sup

119909le119910le119909+119862

119867119899(119910) le 119867

119899(119909)

1199041015840

119899= max119909 le 119878

119899| minus119870 + sup

119909le119910le119909+119862

119867119899(119910) ge 119867

119899(119909)

(12)

Obviously 119904119899le 1199041015840119899


119889lowast

119899(119910)

= arg max119889isin[119889119897119889ℎ]

119903119899(119889) minus 119880

119899(119889 minus 119901

0) + 119882

119899(119910 minus 119889)

(13)





119901lowast

119899(119910) = 119901 (119889

lowast

119899(119910)) (14)





x

sn + C

s998400n

sn

s998400n minus C

Order nothing



Order C

(a) 1199041015840119899minus 119862 le 119904

119899

x

sn + C

s998400n

sn

s998400n minus C

Order nothing


Order C


(b) 1199041015840119899minus 119862 gt 119904

119899




119899minus 119862 119904




119899 le 119909 lt 1199041015840

119899


119899 le



119899minus 119862 119904

119899 le 119909 lt max1199041015840

119899minus


119899minus 119862 le 119904

119899 the


minus 119862 gt 119904119899 the


4 Numerical Tests



0= 3 ordering



119899(sdot) is


0) = 1198861max119901

0minus

119901 0 +1198862max119901minus119901

0 0 where 119886

1= 1198862= 05 The demand in

period 119899 is 119863119899(119901119899) = 119889(119901

119899) + 120598119899 where 119889(119901

119899) = 10 minus 119901

119899and



= 1 = 119875120598119899


0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

K = 5

K = 10

K = 15


Inventory level x






K = 5

K = 10

K = 15


Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p








0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

minus5minus5

0 5 10 15minus10

Inventory level x

C = 5

C = 7

C = 10



Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

C = 5

C = 7

C = 10





0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

Without p0p0 = 5

p0 = 7



Inventory level x

Without p0p0 = 5

p0 = 7

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p






0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


1198862


Inventory level x

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


2

1198861= 1 1198862= 9 1198861= 1198862= 5 and 119886

1= 9 1198862= 1 From Figure 9


1lt 1198862









5 Conclusions


1015840 119901)





Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










x

sn + C

s998400n

sn

s998400n minus C

Order nothing



Order C

(a) 1199041015840119899minus 119862 le 119904

119899

x

sn + C

s998400n

sn

s998400n minus C

Order nothing


Order C


(b) 1199041015840119899minus 119862 gt 119904

119899




119899minus 119862 119904




119899 le 119909 lt 1199041015840

119899


119899 le



119899minus 119862 119904

119899 le 119909 lt max1199041015840

119899minus


119899minus 119862 le 119904

119899 the


minus 119862 gt 119904119899 the


4 Numerical Tests



0= 3 ordering



119899(sdot) is


0) = 1198861max119901

0minus

119901 0 +1198862max119901minus119901

0 0 where 119886

1= 1198862= 05 The demand in

period 119899 is 119863119899(119901119899) = 119889(119901

119899) + 120598119899 where 119889(119901

119899) = 10 minus 119901

119899and



= 1 = 119875120598119899


0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

K = 5

K = 10

K = 15


Inventory level x






K = 5

K = 10

K = 15


Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p








0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

minus5minus5

0 5 10 15minus10

Inventory level x

C = 5

C = 7

C = 10



Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

C = 5

C = 7

C = 10





0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

Without p0p0 = 5

p0 = 7



Inventory level x

Without p0p0 = 5

p0 = 7

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p






0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


1198862


Inventory level x

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


2

1198861= 1 1198862= 9 1198861= 1198862= 5 and 119886

1= 9 1198862= 1 From Figure 9


1lt 1198862









5 Conclusions


1015840 119901)





Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










K = 5

K = 10

K = 15


Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p








0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y

minus5minus5

0 5 10 15minus10

Inventory level x

C = 5

C = 7

C = 10



Inventory level x

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

C = 5

C = 7

C = 10





0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

Without p0p0 = 5

p0 = 7



Inventory level x

Without p0p0 = 5

p0 = 7

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p






0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


1198862


Inventory level x

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


2

1198861= 1 1198862= 9 1198861= 1198862= 5 and 119886

1= 9 1198862= 1 From Figure 9


1lt 1198862









5 Conclusions


1015840 119901)





Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

Without p0p0 = 5

p0 = 7



Inventory level x

Without p0p0 = 5

p0 = 7

0

1

2

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p






0

5

10

15

Opt

imal

repl

enish

ed in

vent

ory

leve

l y


Inventory level x

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


1198862


Inventory level x

3

4

5

6

7

8

9

10

Opt

imal

selli

ng p

rice p

a1 = 1 a2 = 9

a1 = a2 = 5

a1 = 9 a2 = 1


2

1198861= 1 1198862= 9 1198861= 1198862= 5 and 119886

1= 9 1198862= 1 From Figure 9


1lt 1198862









5 Conclusions


1015840 119901)





Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5 Conclusions


1015840 119901)





Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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