European Journal of Operational Research 203 (2010) 593–600

Contents lists available at ScienceDirect

European Journal of Operational Research

journal homepage: www.elsevier .com/locate /e jor

Production, Manufacturing and Logistics

An optimal production-inventory model for deteriorating items withmultiple-market demand

Yong He a,b,*, Shou-Yang Wang c, K.K. Lai b

a School of Economics and Management, Southeast University, Nanjing 210096, Chinab Department of Management Sciences, City University of Hong Kong, Tat Chee Avenue, Kowloon, Hong Kongc Institute of Systems Science, Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100080, China

a r t i c l e i n f o a b s t r a c t

Article history:Received 28 August 2008Accepted 3 September 2009Available online 8 September 2009

Keywords:Supply chain managementDeteriorating itemsMultiple-marketInventory management

0377-2217/$ - see front matter � 2009 Elsevier B.V. Adoi:10.1016/j.ejor.2009.09.003

* Corresponding author. Address: School of EcSoutheast University, Nanjing 210096, China. Tel./fax

E-mail address: heyong@126.com (Y. He).

The global markets of today offer more selling opportunities to the deteriorating items’ manufacturers,but also pose new challenges in production and inventory planning. From a production managementstandpoint, opportunities to exploit the difference in the timing of the selling season between geograph-ically dispersed markets for deteriorating items are important to improving a firm’s profitability. In thispaper, we examined the above issue with an insightful production-inventory model of a deterioratingitems manufacturer selling goods to multiple-markets with different selling seasons. We also provideda solution procedure to find the optimal replenishment schedule for raw materials and the optimal pro-duction plan for finished products. A numerical example was then used to illustrate the model and thesolution procedure. Finally, sensitivity analysis of the optimal solution with respect to major parameterswas carried out.

� 2009 Elsevier B.V. All rights reserved.

1. Introduction

In real life, the effect of decay and deterioration is very impor-tant in many inventory systems. In general, deterioration is definedby Wee (1993) as decay, damage, spoilage, evaporation, obsoles-cence, pilferage, loss of utility or loss of marginal value of a com-modity that results in decreasing usefulness. Most of the physicalgoods undergo decay or deterioration over time, the examplesbeing medicine, volatile liquids, blood banks, and others. Conse-quently, the production and inventory problem of deterioratingitems has been extensively studied by researchers. Ghare andSchrader (1963) were the first to consider ongoing deteriorationof inventory with constant demand. As time progressed, severalresearchers developed inventory models by assuming eitherinstantaneous or finite production with different assumptions onthe patterns of deterioration. In this connection, researchers mayrefer to work by Chakrabarti et al. (1998), Covert and Philip(1973), Dave (1979), Elsayed and Terasi (1983), Kang and Kim(1983) and Mishra (1975). Also, some researchers (Yang andWee, 2000; Wee and Jong, 1998) have studied the possibility andeffect of integration and co-operation between the buyer and theproducer of deteriorating items. Interested readers may refer to re-

ll rights reserved.

onomics and Management,: +86 25 83358312.

view articles by Raafat (1991) and Goyal and Giri (2001). Recently,several related articles were presented, dealing with such inven-tory problems (Chung and Liao, 2006; Maity et al., 2007; Ronget al., 2008; Chern et al., 2008; Ouyang et al., 2009).

For a manufacturer, the most important thing is to decide theoptimal replenishment schedule for raw materials and the optimalproduction plan for finished products. The main method to findingthe optimal production plan is to find an optimal production time,so that the production quantity for finished products should satisfythe following two requirements: one, it should meet demand anddeterioration; second, all products should be sold out in each cycle,that is, at the end of each cycle, the inventory should decrease tozero. This method can be seen in Lee and Hsu (2009), Liao (2007)and Yang and Wee (2002), and others. In this paper, we also usedthis method to find the optimal production plan. After deciding theproduction plan, the next step is to find the optimal replenishmentschedule for raw materials. Since the raw materials are also dete-riorating items, the replenishment schedule for raw materialsshould meet production and deterioration. Some researchers (Yangand Wee, 2003; Rau et al., 2003) have studied the deterioratingraw materials under the premise that the manufacturer sells hisproducts in one market.

The global markets of today offer to the manufacturer moreselling opportunities, as well as production location choices. Geo-graphically dispersed markets for deteriorating items providemultiple selling opportunities to the manufacturer. Anotherimportant advantage is the opportunity to exploit the different

1T

1d

2d

md

1md +

1md −

nd

1nd −

0 2T 1mT − mT 1mT + 1nT − nTT

Demand

Time

Fig. 1. Time versus demand.

594 Y. He et al. / European Journal of Operational Research 203 (2010) 593–600

timing of the selling season at the various markets (Kouvelis andGutierrez, 1997). Kouvelis and Gutierrez (1997) and Khouja(2001) both gave examples from the US garment industry. Theyfound that a US garment manufacturer could sell his summer fash-ion items to an Australian clothing retailer after US’s. In othercases, the difference in the timing of the selling seasons repre-sented a time lag in the phases of the product life cycle of thesame brand (or model) in different geographic markets. For exam-ple, athletic shoes traditionally have a six-month time lag betweenthe product life cycles of various models in the North Americanand European markets (New Balance Athletic Shoes, 1980). Some-times, because of limitations of capacity or capital, the manufac-turer will supply to the different market with a different sellingseason in order to satisfy the demands of all markets. For example,when demand increase suddenly because of political events ornatural disasters, the manufacturer often first offers the productsto the most important market, then to the secondary market,and so on.

The presence of geographically dispersed markets for deterio-rating items creates profitable opportunities in multiple-marketsfor a single location manufacturer. At the same time, because ofthe different selling seasons in different markets, the manufac-turer’s production planning and inventory problems become morecomplicated. Under the circumstances, the manufacturer will facea fluctuating demand, which is a discontinuous function. Someresearchers (Balkhi and Benkherouf, 1996; Papachristos andSkouri, 2000) have presented a method for finding ‘‘the optimalreplenishment of inventory” system model for deteriorating items,where demand is allowed to vary with time in an arbitrary way.However, there were also some studies (Chung et al., 2000; Giriet al., 1996; Giri and Chaudhuri, 1998; Hou, 2006; Wu et al.,2006) which have discussed inventory models for deterioratingitems where the demand rate is a function of the on-hand inven-tory. Other authors such as Covert and Philip (1973), Goswamiand Chaudhuri (1991), Manna and Chaudhuri (2006) and Changet al. (2006) developed inventory models for deteriorating itemswith time-dependent demand. The common characteristic of theabove papers was that the demand function is continuous and dif-ferentiable. Yang and Wee (2002) developed an integrated eco-nomic ordering policy for deteriorating items for a single vendorand multiple buyers. However, in this model, all buyers were as-sumed to be operating in the same market and for the manufac-turer, the demand rate was a constant.

However, with increasing global co-operation and competition,more and more manufacturers sell their products to different mar-kets worldwide. Since the each market has a different sellingseason, the demand function will be not continuous. In fact, the de-mand function will become a piecewise function with time. Thus,the traditional inventory models for deteriorating items will beineffective. Meanwhile, the piecewise demand function makes itmore difficult for the manufacturer to manage the supply of rawmaterials.

Therefore, in this paper, we developed a production-inventorymodel for deteriorating items with multiple-market demand,where each market has a different selling season and a differentconstant demand rate. To foster additional managerial insights,we performed extensive sensitivity analyses and illustrated our re-sults with a simulation study.

The paper was organized as follows: Section 2 is concerned withthe mathematical development and the method for finding theoptimal solutions. In Section 3, we presented a simulation studyto illustrate the model. In Section 4, sensitivity analysis was carriedout to identify the most sensitive parameters in the system. In Sec-tion 5, conclusions and topics for further research were presented.Mathematical references and derivations are contained in theAppendix.

2. Mathematical modeling and analysis

Suppose a manufacturer produces certain products and sellsthem in different markets, which have different selling seasons.All items are produced and sold in each cycle. Each cycle starts withthe first opening market and ends with the last closing market. De-mand rate for each market is constant and is different for each mar-ket. At the beginning of each cycle, the manufacturer piles up all themarkets’ demands in time order. Since each market has a differentdemand rate and selling season, the demand that the manufacturerfaces is a piecewise-constant function. We assumed that the func-tion has a number ðnÞ of time intervals and demand rates corre-spondingly. Let Ti ði ¼ 1;2; . . . ;nÞ be the critical time point wheredemand rate will change, di be the demand rate over the interval½Ti�1; Ti� (Fig. 1). Production starts at the every beginning of the cy-cle. As production continues, inventory begins to pile up continu-ously after meeting demand and deterioration. Production stopsat time T. The optimal production time T is the key point; if the pro-duction stops at this point the production quantity can meet de-mand and deterioration, and no overstock at the end of each cycle.

The following assumptions were used to formulate theproblem:

(a) A single product, a single manufacturer and multiple-mar-kets were assumed.

(b) Production rate is deterministic and constant.(c) Production rate is greater than any demand rate di.(d) Lead time was assumed to be negligible.(e) Deterioration rates of the materials and finished products

are deterministic and constant.(f) Shortages were not allowed.(g) Time horizon is finite.(h) There is no repair or replacement of deteriorated units dur-

ing the planning horizon.

Let the parameters of raw materials’ costs incurred by the man-ufacturer be as follows:sr ordering costcr unit price of raw materialshr holding cost of raw materials per unit time for the manu-

facturerf unit usage of raw materials per finished producthr deterioration rate of raw materialsIrðtÞ inventory level of raw materialsqr lot-size per delivery from supplier to manufacturernr number of raw material’s deliveries from supplier to man-

ufacturer per T

The manufacturer’s cost parameters are as follows:K0 setup cost

Y. He et al. / European Journal of Operational Research 203 (2010) 593–600 595

p production ratecp unit production cost of deteriorating itemshp unit holding cost of finished products per unit timeh constant deterioration rate of finished productsIiðtÞ inventory level in the ith interval ði ¼ 1;2; . . . ;nÞ

The following assumptions apply to the study:Fixed quantities of raw materials are delivered to the manufac-

turer at a fixed-time interval. The manufacturer produces enoughquantities of finished products to meet demand and deterioration.At the end of each cycle, the inventory level of raw materials and fin-ished products is zero. Our study may be precisely stated as follows:

Given a piecewise-constant function in demand, developing amathematical model to obtain the optimal number of deliveriesand order lot-size of raw materials and the optimal productiontime, when the cost of the inventory system is minimized.

2.1. Manufacturer’s finished products inventory model

We suppose the optimal production time T lies in the interval½Tm�1; Tm�, where demand rate is dm. The manufacturer’s finishedproducts inventory model for deteriorating items with multiple-market demand can be depicted as Fig. 1.

Let I�mðtÞ be the inventory level at time t 2 ½Tm�1; T�; IþmðtÞ be theinventory level at time t 2 ½T; Tm�. The instantaneous finished prod-ucts inventory level at any time t 2 ½0; T� is governed by the follow-ing differential equations:

dIkðtÞdtþ hIkðtÞ ¼ p� dk; Tk�1 6 t 6 Tk; k ¼ 1;2; . . . ;m� 1 ð1Þ

dI�mðtÞdt

þ hI�mðtÞ ¼ p� dm; Tm�1 6 t 6 T ð2Þ

Using various boundary conditions I1ð0Þ ¼ 0; IkðTk�1Þ ¼Ik�1ðTk�1Þ and I�mðTm�1Þ ¼ Im�1ðTm�1Þ, the solutions of above differ-ential equations are (see Appendix A)

IkðtÞ ¼pð1� e�htÞ � dk

hþXk

i¼1

di � di�1

he�hðt�Ti�1Þ; Tk�1 6 t

6 Tk; k ¼ 1;2; . . . ;m� 1 ð3Þ

I�mðtÞ ¼pð1� e�htÞ � dm

hþXm

i¼1

di � di�1

he�hðt�Ti�1Þ; Tm�1 6 t 6 T

ð4Þ

where T0 ¼ 0 and d0 ¼ 0.The instantaneous finished products inventory level at time

t 2 ½T; Tn� is governed by the following differential equations:

dIþmðtÞdt

þ hIþmðtÞ ¼ �dm; T 6 t 6 Tm ð5Þ

dIjðtÞdtþ hIjðtÞ ¼ �dj; Tj�1 6 t 6 Tj; j ¼ mþ 1;mþ 2; . . . ;n ð6Þ

Using various boundary conditions InðTnÞ ¼ 0 and Ij�1ðTj�1Þ ¼IjðTj�1Þ, the solutions of above differential equations are (similarto Appendix A):

IjðtÞ ¼ �dj

hþ dn

he�hðt�TnÞ �

Xn

i¼jþ1

di � di�1

he�hðt�Ti�1Þ; Tj�1 6 t

6 Tj; j ¼ mþ 1;mþ 2; . . . ;n ð7Þ

IþmðtÞ ¼ �dm

hþ dn

he�hðt�TnÞ �

Xn

i¼mþ1

di � di�1

he�hðt�Ti�1Þ; T 6 t 6 Tm

ð8Þ

The condition I�mðTÞ ¼ IþmðTÞ yields

pð1� e�hTÞ � dm

hþXm

i¼1

di � di�1

he�hðT�Ti�1Þ

¼ � dm

hþ dn

he�hðT�TnÞ �

Xn

i¼mþ1

di � di�1

he�hðT�Ti�1Þ ð9Þ

From (9), the optimal production time T satisfies the followingequation:

T ¼ 1h

lndnehTn �

Pni¼1ðdi � di�1ÞehTi�1

pþ 1

� �ð10Þ

From (10), we can find that T has no relation with m. Therefore, ifTi and di are known, we can directly get the value of T by using(10).

The inventory in the time interval ½Tk�1; Tk�; k ¼ 1;2; . . . ;m� 1is

Ik ¼Z Tk

Tk�1

IkðtÞdt

¼Z Tk

Tk�1

pð1� e�htÞ � dk

hþXk

i¼1

di � di�1

he�hðt�Ti�1Þ

" #dt

¼ p� dk

hðTk � Tk�1Þ þ

p

h2 e�hTk � e�hTk�1� �

�Xk

i¼1

� di � di�1

h2 e�hðTk�Ti�1Þ � e�hðTk�1�Ti�1Þ� �

ð11Þ

The inventory in ½Tm�1; T� is

I�m ¼Z T

Tm�1

I�mðtÞdt

¼Z T

Tm�1

pð1� e�htÞ � dm

hþXm

i¼1

di � di�1

he�hðt�Ti�1Þ

" #dt

¼ p� dm

hðT � Tm�1Þ þ

p

h2 e�hT � e�hTm�1� �

�Xm

i¼1

� di � di�1

h2 e�hðT�Ti�1Þ � e�hðTm�1�Ti�1Þ� �

ð12Þ

The inventory in ½T; Tm� is

Iþm ¼Z Tm

TIþmðtÞdt

¼Z T

Tm�1

� dm

hþ dn

he�hðt�TnÞ �

Xn

i¼mþ1

di � di�1

he�hðt�Ti�1Þ

" #dt

¼ �dm

hðTm � TÞ � dn

h2 e�hðTm�TnÞ � e�hðT�TnÞ� �

þXn

i¼mþ1

� di � di�1

h2 e�hðTm�Ti�1Þ � e�hðT�Ti�1Þ� �

ð13Þ

The inventory in the time interval ½Tj�1; Tj�; j ¼ mþ 1;mþ 2; . . . ;n is

Ij ¼Z Tj

Tj�1

IjðtÞdt

¼Z Tj

Tj�1

� dj

hþ dn

he�hðt�TnÞ �

Xn

i¼jþ1

di � di�1

he�hðt�Ti�1Þ

" #dt

¼ � dj

hðTj � Tj�1Þ �

dn

h2 e�hðTj�TnÞ � e�hðTj�1�TnÞ� �

þXn

i¼jþ1

� di � di�1

h2 e�hðTj�Ti�1Þ � e�hðTj�1�Ti�1Þ� �

ð14Þ

The holding cost of finished products, THp is (see Appendix B):

596 Y. He et al. / European Journal of Operational Research 203 (2010) 593–600

THp ¼ hp

Xm�1

k¼1

Ik þ I�m þ Iþm þXn

j¼mþ1

Ij

!

¼ hppT �

Pnk¼1dkðTk � Tk�1Þ

hð15Þ

The deterioration cost of finished products, TDp is:

TDp ¼ cp pT �Xn

k¼1

dkðTk � Tk�1Þ !

ð16Þ

The total cost of finished products, TCp is

TCp ¼ K0 þ THp þ TDp

¼ K0 þhp

hþ cp

� pT �

Xn

k¼1

dkðTk � Tk�1Þ !

ð17Þ

2.2. Manufacturer’s warehouse raw materials inventory model

The inventory system of raw materials is illustrated in Fig. 2.The replenishment lot-size is replenished at t ¼ 0. During the

periods T=nr , the inventory levels of raw materials decrease dueto demand and deterioration until they are zero at t ¼ T=nr . Theinstantaneous raw materials inventory level at time t can thereforebe represented by the following differential equation:

dIrðtÞdtþ hrIrðtÞ ¼ �fp; 0 6 t 6

Tnr

ð18Þ

The first-order differential equations can be solved by using theboundary conditions IrðT=nrÞ ¼ 0; we have

IrðtÞ ¼fphr

e�hr t� Tnrð Þ � 1

�; 0 6 t 6

Tnr

ð19Þ

From Fig. 2, Irð0Þ ¼ qr; hence from (19),

qr ¼fphr

ehr T=nr�1� �ð20Þ

It can be derived from (19) that the holding cost of raw materials,THr is

THr ¼ nrhr

Z T=nr

0IrðtÞdt ¼ nrhrfp

h2r

ehr T=nr � nrhrfpþ hrhrfpT

h2r

ð21Þ

The deterioration cost of raw materials, TDr is

TDr ¼ crðnrqr � fpTÞ ð22Þ

The total cost of raw materials, TCr is

TCr ¼ srnr þ THr þ TDr

¼ srnr þ nrfphr

hrþ cr

� ehr T=nr � 1

hr� T

nr

� ð23Þ

0 / rT n 2 / rT n … T

( )rI t

rq

Time

Fig. 2. Raw material’s inventory system.

2.3. The integrated inventory model and its solution

The integrated total cost of raw materials and finished products,TC is

TC ¼ TCr þ TCp ð24Þ

Observing (24), the objective of the study is to determine the valueof nr that minimizes TC, where nr is a discrete variable.

From (24), we get

@TC@nr¼ sr þ fp

hr

hrþ cr

� ehr T=nr � 1

hr� Tehr T=nr

nr

!ð25Þ

and

@2TC

ð@nrÞ2¼ fp

hr

hrþ cr

� T2hrehr T=nr

n3r

ð26Þ

It is obvious that @2TCð@nr Þ2

> 0. So, TC is convex in nr . Since the number of

deliveries, nr , is a discrete variable, one can find out the value of nr

from the following procedure:

(1) Set @TC@nr¼ 0; we obtain the approximate solution n̂r .

(2) Use v ¼ Intðn̂rÞ to represent the integer part of n̂r; the opti-mal value of nr must satisfy the following condition:

TC� n�r� �

¼minw

TC�ðwÞ; 8w 2 ½v ;v þ 1� ð27Þ

From above analyses, the optimal solution of T; qr and nr , denotedby T�; q�r and n�r , can be derived from (10), (20) and (27),respectively.

3. Numerical example

As an illustration, the case of a two-markets situation is dis-cussed. The related data were as follows:

The deterioration rate of raw materials and finished products isassumed to be 0.02 and 0.03 units per week, respectively. The unitusage of raw materials is 1.2 units per unit of finished product. Theproduction rate is 350 units per week. The demand rate of marketone and market two is 160 and 120 units per week, respectively.The selling season of market one is from week 1 to week 20. Theselling season of market two is from week 8 to week 30. The order-ing cost of raw materials is $400 per order. The set-up cost for pro-duction is $600. The unit price of raw materials is $5. The unitproduction cost is $10. The holding costs for raw materials and fin-ished products are $0.1 and $0.15 per week, respectively.

First, by piling up the two markets’ demands in time order wecan obtain the demand function that the manufacturer faces. Agraphical representation showing how to get the demand functionis given in Fig. 3.

From Fig. 3, we can get n ¼ 3; d1 ¼ 160; d2 ¼ 280;d3 ¼ 120; T1 ¼ 8; T2 ¼ 20; T3 ¼ 30.

By applying the solution procedure in Section 2, we haveT� ¼ 19:3; n̂r ¼ 6:4 and v ¼ 6. Then the optimum solution isshown in Table 1. A graphical representation showing the convexof the TC function is given in Fig. 4.

4. Sensitivity analysis

The change in the values of parameters may happen due touncertainties in any decision-making situation. In order to exam-ine the implications of these changes, a sensitivity analysis willbe of great help in decision-making. Using the numerical example

0

Demand (each market)

160

120

Time (week)

Market One

Market Two

20 30 8

0

Demand (for manufacturer)

160

120

Time (week)

1 8T =

280

1d

2d

3d

2 20T = 3 30T =

Fig. 3. A graphical representation showing the demand function.

Fig. 4. A graphical representation showing the convexity of the TC function withrespect to nr .

Y. He et al. / European Journal of Operational Research 203 (2010) 593–600 597

given in the preceding section, we performed the sensitivity anal-ysis on the optimal solution of the model with respect to majorparameters, such as p; di; h; hr and f, by changing each of the param-eters by +20%, +10%, �10% and �20%, taking one parameter at atime and keeping the remaining parameters unchanged. The re-sults of the sensitivity analysis are shown in Table 2. The percent-age change of these values is shown with respect to the base valuesused in Table 1. The following points are observed:

(1) When the production rate ðpÞ decreases or increases, q�r ; TC�pand TC� will also decrease or increase. But this trend isreversed for the optimal solution n�r ; T

� and TC�r . Intuitively,improving the production rate can increase the manufac-turer’s profit. Our analyses have shown that this intuitionis sometimes wrong. In this example, although improvingthe production rate can shorten the production time anddecrease the cost of raw materials, it also increases the costof finished products, and finally increases the total cost forthe manufacturer. That is, TC�r is slightly sensitive to changesin p, while TC�p and TC� are quite sensitive to changes in p.The main reason is that the deterioration rate of finishedproducts is higher than raw materials. Therefore, the pro-duction rate ðpÞ should be determined carefully in order toreduce the total cost effectively.

(2) When all diði ¼ 1;2;3Þ decrease, n�r ; T� and TC�r tend to

decrease while TC�p and TC� tend to increase. These resultsare opposite to that influenced by the production rate ðpÞ.Hence, once di decrease, the manufacturer should lower pro-duction rate purposely to mitigate the increase in total cost.

Table 1The optimal values of nr ; qr and T.

nr qr T TCr TC

6� 1394:0� 19:3� 5059� 19287�

7 1189.3 19.3 5072 19300

(3) T�; TC�r ; TC�p, and TC� decrease with decrease in the value ofparameter h. When h varies, TC�p and TC� are more sensitiveto h; but T� and TC�r are less sensitive to h. The main reasonis that if h decreases, the deterioration quantity of finishedproducts will decrease. Therefore, the manufacturer canreduce the production quantity of finished products andthe demand of raw materials. Accordingly, this will decreasethe costs of finished products and raw materials. Hence, try-ing to find a new technique to decreasing h is an effectiveway to decrease the total cost TC�.

(4) When hr and f decrease, TC�r and TC� will decrease. It is seenthat TC�r is more sensitive than TC�when hr or f varies. Thereason is that in this case TC�r just takes up a small propor-tion (about 25%) in TC�. Therefore, even if TC�r undergoes alarge change, it does not have much effect on the total costðTC�Þ. Also, we find that T� and TC�p have no relations withhr and f. Hence, in order to reduce the total cost TC�, themanufacturer also should make an effort to decrease hr

and f if the cost of raw materials is very large.(5) The changes of q�r are complicated. The reason is that qr has

negative relationship with nr . Sometimes the di decrease, butthe qr increase. For example, when di changes by �20%, nr

will decrease from 6 to 5. This will lead qr to increase by0.96%.

5. Conclusion

Global markets offer selling opportunities and pose productionmanagement challenges for manufacturers of deteriorating items.Exploiting the difference in timing of the selling season of the dete-riorating items at different markets is a unique opportunity to im-prove the profitability of a deteriorating items’ manufacturer. Inthis paper, we have suggested a method for finding the optimalproduction and inventory schedule for manufacturers of deterio-rating items. Here, we assume the manufacturer produces in onelocation and sells in different markets that have different sellingseasons. We have showed that our method helps minimize costs.

In this study, the proposed model considered the demand ratein each market as constant. In real life, we may consider the de-mand rate as a function of time, selling price, stock, etc. This willbe done in our future research.

Table 2Effect of changes in various parameters of the production-inventory model.

Parameter Percentage of change (%) % Change in

n�r q�r T� TC�r TC�p TC�

p 20 0.00 3.79 �13.14 �5.11 30.08 20.8510 0.00 2.02 �7.04 �2.68 16.01 11.11�10 16.67 �16.70 8.25 2.78 �18.40 �12.85�20 16.67 �19.05 18.02 5.55 �39.78 �27.89

di 20 16.67 �1.87 14.56 14.45 �19.58 �10.6510 16.67 �8.15 7.43 7.26 �8.70 �4.51�10 0.00 �8.00 �7.77 �7.92 6.32 2.59�20 �16.67 0.96 �15.90 �15.84 10.04 3.25

h 20 16.67 �12.65 2.32 2.38 10.20 8.1510 0.00 1.21 1.18 1.25 5.18 4.15�10 0.00 �1.24 �1.20 �1.27 �5.35 �4.28�20 0.00 �2.52 �2.44 �2.56 �10.86 �8.68

hr 20 16.67 �14.20 0.00 4.94 0.00 1.2910 16.67 �14.44 0.00 2.59 0.00 0.68�10 0.00 �0.32 0.00 �2.73 0.00 �0.72�20 0.00 �0.65 0.00 �5.46 0.00 �1.43

f 20 16.67 2.38 0.00 9.24 0.00 2.4210 16.67 �6.15 0.00 4.75 0.00 1.24�10 0.00 �10.00 0.00 �5.25 0.00 �1.38�20 0.00 �20.00 0.00 �10.51 0.00 �2.76

598 Y. He et al. / European Journal of Operational Research 203 (2010) 593–600

Acknowledgements

The authors thank the valuable comments of the anonymousreferees for an earlier version of this paper. Their comments havesignificantly improved the paper.

This research is supported by a grant from the Ph.D. ProgramsFoundation of Ministry of Education of China (No.200802861030). Also, this research is partly supported by the Min-istry of Education of China: Grant-in-aid for Humanity and SocialScience Research (No. 06JA630012).

Appendix A

Using Spiegel (1960), the solutions of (1) is

IkðtÞ ¼p� dk

hþ cke�ht ; Tk�1 6 t 6 Tk; k ¼ 1;2; . . . ;m� 1

where ck is a constant of integration.If k ¼ 1, from the boundary condition I1ð0Þ ¼ 0, we get

c1 ¼ �p� d1

h

If 2 6 k 6 m� 1, from the boundary condition IkðTk�1Þ ¼ Ik�1ðTk�1Þ,we have

p� dk

hþ cke�hTk�1 ¼ p� dk�1

hþ ck�1e�hTk�1

or

ck � ck�1 ¼dk � dk�1

hehTk�1

Therefore, we get the following relations:

c2 � c1 ¼d2 � d1

hehT1

c3 � c2 ¼d3 � d2

hehT2

c4 � c3 ¼d4 � d3

hehT3

..

.

ck � ck�1 ¼dk � dk�1

hehTk�1

adding up the above equations, we get

ck � c1 ¼Xk

i¼2

di � di�1

hehTi�1 ; 2 6 k 6 m� 1

or

ck ¼ �p� d1

hþXk

i¼2

di � di�1

hehTi�1 ; 2 6 k 6 m� 1;

Supposing d0 ¼ 0 and T0 ¼ 0, we get

ck ¼ �phþXk

i¼1

di � di�1

hehTi�1 ; k ¼ 1;2; . . . ;m� 1:

Therefore, the solution of (1) is

IkðtÞ ¼pð1� e�htÞ � dk

hþXk

i¼1

di � di�1

he�hðt�Ti�1Þ; Tk�1 6 t

6 Tk; k ¼ 1;2; . . . ;m� 1

Let k ¼ m, the solution of (2) is

I�mðtÞ ¼pð1� e�htÞ � dm

hþXm

i¼1

di � di�1

he�hðt�Ti�1Þ; Tm�1 6 t 6 T

Y. He et al. / European Journal of Operational Research 203 (2010) 593–600 599

Appendix B

We have

Xm�1

k¼1

Ik ¼Xm�1

k¼1

p� dk

hTk � Tk�1ð Þ þ p

h2 e�hTk � e�hTk�1� ��

�Xk

i¼1

di � di�1

h2 e�h Tk�Ti�1ð Þ � e�hðTk�1�Ti�1Þ� �#

¼Xm�1

k¼1

p� dk

hTk � Tk�1ð Þ þ p

h2 e�hTk � e�hTk�1� ��

�dk � dk�1

h2 e�h Tm�1�Tk�1ð Þ � 1� �

¼ ph

Tm�1 þp

h2 ðe�hTm�1 � 1Þ

�Xm�1

k¼1

dk

hTk � Tk�1ð Þ þ dk � dk�1

h2 e�h Tm�1�Tk�1ð Þ � 1� ��

I�m ¼Z T

Tm�1

I�mðtÞdt

¼Z T

Tm�1

p 1� e�htð Þ � dm

hþXm

i¼1

di � di�1

he�hðt�Ti�1Þ

" #dt

¼ p� dm

hðT � Tm�1Þ þ

p

h2 e�hT � e�hTm�1� �

�Xm

i¼1

� di � di�1

h2 e�hðT�Ti�1Þ � e�hðTm�1�Ti�1Þ� �

Iþm ¼Z Tm

TIþmðtÞdt

¼Z T

Tm�1

� dm

hþ dn

he�hðt�TnÞ �

Xn

i¼mþ1

di � di�1

he�hðt�Ti�1Þ

" #dt

¼ � dm

hðTm � TÞ � dn

h2 e�hðTm�TnÞ � e�hðT�TnÞ� �

þXn

i¼mþ1

� di � di�1

h2 e�hðTm�Ti�1Þ � e�hðT�Ti�1Þ� �

Xn

j¼mþ1

Ij ¼Xn

j¼mþ1

� dj

hðTj � Tj�1Þ �

dn

h2 e�hðTj�TnÞ � e�hðTj�1�TnÞ� ��

þXn

i¼jþ1

di � di�1

h2 e�hðTj�Ti�1Þ � e�hðTj�1�Ti�1Þ� �

g

¼ � dn

h2 1� e�hðTm�TnÞ� �

�Xn

j¼mþ1

dj

hðTj � Tj�1Þ

þXn

j¼mþ2

dj � dj�1

h2 1� e�hðTm�Tj�1Þ� �

THp ¼ hp

Xm�1

k¼1

Ik þ I�m þ Iþm þXn

j¼mþ1

Ij

!

¼ hp

hpT �

Xn

i¼1

di Ti � Ti�1ð Þ" #

� hp

h2 p 1� e�hT� �

þ dn 1� e�hðT�TnÞ� ��

þXn

i¼1

ðdi � di�1Þ e�hðT�Ti�1Þ � 1� �#

Using the condition (9), we have

THp ¼ hppT �

Pni¼1diðTi � Ti�1Þ

h

References

Balkhi, Z.T., Benkherouf, L., 1996. A production lot size inventory model fordeteriorating items and arbitrary production and demand rates. EuropeanJournal of Operational Research 92, 302–309.

Chakrabarti, T., Giri, B.C., Chaudhuri, K.S., 1998. An EOQ model for items Weibulldistribution deterioration shortages and trended demand – An extension ofPhilip’s model. Computers and Operations Research 25, 649–657.

Chang, H.J., Teng, J.T., Oyang, L.Y., Dye, C.Y., 2006. Retailer’s optimal pricing and lot-sizing policies for deteriorating items with partial backlogging. EuropeanJournal of Operational Research 168, 51–64.

Chern, M.S., Yang, H.L., Teng, J.T., Papachristos, S., 2008. Partial backlogginginventory lot-size models for deteriorating items with fluctuating demandunder inflation. European Journal of Operational Research 191, 127–141.

Chung, K.J., Liao, J.J., 2006. The optimal ordering policy in a DCF analysis fordeteriorating items when trade credit depends on the order quantity.International Journal of Production Economics 100, 116–130.

Chung, K.J., Chu, P., Lan, S.H., 2000. A note on EOQ models for deteriorating itemsunder stock dependent selling rate. European Journal of Operational Research124, 550–559.

Covert, R.P., Philip, G.C., 1973. An EOQ model for items with weibull distributiondeterioration. AIIE Transactions 5, 323–326.

Dave, U., 1979. On a discrete-in-time order-level inventory model for deterioratingitems. Operational Research 30, 349–354.

Elsayed, E.A., Terasi, C., 1983. Analysis of inventory systems with deterioratingitems. International Journal of Production Research 21, 449–460.

Ghare, P.M., Schrader, S.F., 1963. A model for exponentially decaying inventory.Journal of Industrial Engineering 14, 238–243.

Giri, B.C., Chaudhuri, K.S., 1998. Deterministic models of perishable inventory withstock-dependent demand rate and nonlinear holding cost. European Journal ofOperational Research 105, 467–474.

Giri, B.C., Pal, S., Goswami, A., Chaudhuri, K.S., 1996. An inventory model fordeteriorating items with stock-dependent demand rate. European Journal ofOperational Research 95, 604–610.

Goswami, A., Chaudhuri, K.S., 1991. An EOQ model for deteriorating items withshortages and a linear trend in demand. Journal of the Operational ResearchSociety 42, 1105–1110.

Goyal, S.K., Giri, B.C., 2001. Recent trends in modeling of deteriorating inventory.European Journal of Operational Research 134, 1–16.

Hou, K.L., 2006. An inventory model for deteriorating items with stock-dependentconsumption rate and shortages under inflation and time discounting.European Journal of Operational Research 168, 463–474.

Kang, S., Kim, I., 1983. A study on the price and production level of the deterioratinginventory system. International Journal of Production Research 21, 899–908.

Khouja, M., 2001. The effect of large order quantities on expected profit in thesingle-period model. International Journal of Production Economics 72, 227–235.

Kouvelis, P., Gutierrez, G.J., 1997. The newsvendor problem in a global market:Optimal centralized and decentralized control policies for a two-marketstochastic inventory system. Management Science 43, 571–585.

Lee, C.C., Hsu, S.L., 2009. A two-warehouse production model for deterioratinginventory items with time-dependent demands. European Journal ofOperational Research 194, 700–710.

Liao, J.J., 2007. On an EPQ model for deteriorating items under permissible delay inpayments. Applied Mathematical Modelling 31, 393–403.

Maity, A.K., Maity, K., Mondal, S., Maiti, M., 2007. A Chebyshev approximation forsolving the optimal production inventory problem of deteriorating multi-item.Mathematical and Computer Modelling 45, 149–161.

Manna, S.K., Chaudhuri, K.S., 2006. An EOQ model with ramp type demand rate,time dependent deterioration rate, unit production cost and shortages.European Journal of Operational Research 171, 557–566.

Mishra, R.B., 1975. Optimum production lot-size model for a system withdeteriorating inventory. International Journal of Production Research 13, 495–505.

New Balance Athletic Shoes, HBS Case Services, Harvard Business School, Boston,MA, 1980.

Ouyang, L.Y., Teng, J.T., Goyal, S.K., Yang, C.T., 2009. An economic order quantitymodel for deteriorating items with partially permissible delay in paymentslinked to order quantity. European Journal of Operational Research 194, 418–431.

Papachristos, S., Skouri, K., 2000. An optimal replenishment policy for deterioratingitems with time-varying demand and partial-exponential. Operations ResearchLetters 27, 175–184.

Raafat, F., 1991. Survey of literature on continuously deteriorating inventorymodels. Journal of the Operational Research Society 42, 27–37.

Rau, H., Wu, M.Y., Wee, H.M., 2003. Integrated inventory model for deterioratingitems under a multi-echelon supply chain environment. International Journal ofProduction Economics 86, 155–168.

600 Y. He et al. / European Journal of Operational Research 203 (2010) 593–600

Rong, M., Mahapatra, N.K., Maiti, M., 2008. A two warehouse inventory model for adeteriorating item with partially/fully backlogged shortage and fuzzy lead time.European Journal of Operational Research 189, 59–75.

Spiegel, M.R., 1960. Applied Differential Equations. Prentice-Hall, Englewood Cliffs,NJ.

Wee, H.M., 1993. Economic production lot size model for deteriorating itemswith partial back-ordering. Computers and Industrial Engineering 24, 449–458.

Wee, H.M., Jong, J.F., 1998. An integrated multi-lot-size production inventory modelfor deteriorating items. Management and Systems 5, 97–114.

Wu, K.S., Ouyang, L.Y., Yang, C.T., 2006. An optimal replenishment policy for non-instantaneous deteriorating items with stock-dependent demand and partialbacklogging. International Journal of Production Economics 101, 369–384.

Yang, P.C., Wee, H.M., 2000. Economic order policy of deteriorated item for vendor andbuyer: An integrated approach. Production Planning and Control 11, 474–480.

Yang, P.C., Wee, H.M., 2002. A single-vendor and multiple-buyers production-inventory policy for a deteriorating item. European Journal of OperationalResearch 143, 570–581.

Yang, P.C., Wee, H.M., 2003. An integrated multi-lot-size production inventorymodel for deteriorating item. Computers and Operations Research 30, 671–682.