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Int. J. Production Economics 103 (2006) 527–534

www.elsevier.com/locate/ijpe

Optimal ordering policy in a distribution system

Jing-An Lia, Yue Wub, Kin Keung Laia,c,�, Ke Liud

aDepartment of Management Sciences, City University of Hong Kong, Hong Kong, PR ChinabSchool of Management, University of Southampton, UK

cCollege of Business Administration, Hunan University, Changsha, 410082 Hunan, PR ChinadInstitute of Applied mathematics, Academy of Mathematics and System Sciences, Chinese Academy of Sciences, Beijing, PR China

Received 24 September 2003; accepted 11 November 2005

Available online 3 March 2006

Abstract

In conventional inventory management, the retailers monitor their own inventory levels and place orders at

the distributor when they think it is the appropriate time to reorder. The distributor receives these orders from the

retailers, prepares the product for delivery. Similarly, the distributor will place an order at the manufacturer at the

appropriate time.

Generally, the order that the distributor places at the manufacturer is larger than that the retailer places at the

distributor. In order to afford this large order, there should exist a long-term supply contract between the manufacturer

and distributor that can guarantee a stationary supply to the distributor. This paper discusses this case, and derives the

optimal stationary supply, that is, the optimal ordering policy of the distributor. Also computational results are presented.

r 2006 Elsevier B.V. All rights reserved.

Keywords: Inventory; Production; Order-up-to policies

1. Introduction

In conventional inventory management, theretailers monitor their own inventory levels, andwhen a retailer thinks that it is time to reorder, anorder for a quantity of the product is placed at thedistributor. The distributor receives these ordersfrom the retailers, prepares the product for delivery.Similarly, when the distributor thinks it is time toreorder, an order for a quantity of the product isplaced at the manufacturer. Even in the recentvendor managed inventory replenishment that

front matter r 2006 Elsevier B.V. All rights reserved

e.2005.11.007

ng author. Department of Management Sciences,

of Hong Kong, Hong Kong, PR China.

8563; fax: +852 2788 8560.

ss: mskklai@cityu.edu.hk (K.K. Lai).

refers to the situation in which a distributormonitors the inventory levels at its retailers anddecides when and how much inventory to replenishat each retailer, the distributor should place anorder at the manufacturer at the appropriate timetoo. Generally, the order that the distributor placesat the manufacturer is larger than that the retailerplaces at the distributor. In order to afford thislarge order, there should exist a long-term supplycontract between the manufacturer and distributorthat can guarantee a stationary supply to thedistributor. As we know, a long-term stationarysupply not only benefits the distributor, but alsodecreases the uncertainty of the supply of themanufacturer, which will benefit the manufacturertoo. Then, how many stationary supply will beoptimal?

.

ARTICLE IN PRESSJ.-A. Li et al. / Int. J. Production Economics 103 (2006) 527–534528

For example, the gas company in Hong Kong hastwo production plants, where gas is made from highquality naphtha instead of heavy fuel oil. Thecompany has settled facilities and manpower so thatshe can ensure a stationary gas production to satisfythe demand through their own network. Because thedemand is uncertain, e.g., high during dinning timeand low otherwise, the gas consumed will beuncertain too. If the company’s production fits thedemand like a glove, the company may alwaysregulate her facilities and manpower. However, thisregulation may cause unknown large costs. Then,how to use the stationary production to satisfy theuncertain demand so that total cost can beminimized will be an important problem. And thiscase is similar to the general one described above.

In general terms, when the demand is determinis-tic, this problem can always be solved by using theclassical economic-order-quantity (EOQ) model,where the ordering policy is to place an order fora fixed quantity of the product at each period. Andthere are much literature discussing this model,e.g., Zipkin, 2000, etc. Of course, more and morecomplex EOQ cases are presented. For example, theEOQ model with deteriorating items (Chang, 2004),the EOQ model with process reliability considera-tions (Tripathy et al., 2003) etc.

In recent years, more literature discusses the casewhere the demand is stochastic. Considering thecontinuous review inventory model, Archibald (1981)discusses the continuous review ðs;SÞ policies with lostsales, and develops a method to calculate an ðs;SÞpolicy that minimizes the average stationary cost in aninventory system. Wang and Gerchak (1996) analyzethe effects of variable capacity on optimal lot sizing incontinuous review environments, and obtain theoptimal conditions for generally distributed variablecapacity and the optimal order quantities. Mohebbiand Posner (1998) derive the stationary distribution ofthe inventory level in a continuous-review inventorysystem, and Mohebbi and Posner (2002) present thestationary distribution of the on-hand inventory usinga level-crossing methodology for a continuous reviewinventory system with lost sales, nonunit size, multiplereplenishment orders outstanding, and split deliveries.Also there are many inventory books, such as Axsater(2000), Bartmann and Beckmann (1992), Zipkin(2000), Heyman and Sobel (1984), discussing thecontinuous review model with ðr;QÞ or ðs;SÞ policies.

It is worth noting that there is more literaturefocusing on the period review inventory model.Considering the stochastic single product multi-

period inventory model, most of existing literaturefocuses on the optimal ðs;SÞ policy, which is toorder up to S when the inventory level is less than s,to do noting otherwise. When there is no setup costper any order, s will equal S and this policy isalways called myopic policy.

The optimal ðsn;SnÞ policies for n periods arepresented by Arrow et al. (1951), and provedtheoretically by Scarf (1960) using the k-convexfunction. For infinite horizon models with station-ary data, Iglehart (1963) shows that the ðs;SÞ policyis optimal too under the assumption of a convexholding and shortage cost function. Thenceforth,more researchers focus on the computation andgeneralization of the optimal ðs;SÞ policy, forexample, Johnson (1968), Federgruen and Zipkin(1984) and Zheng (1991), etc. Recently, moreliterature adds more restrictions to the multi-periodinventory problems and studies whether the ðs;SÞpolicy is optimal or not. Federgruen and Zipkin(1986) have shown that the optimal policy to thecapacitated problem is just a simple modification ofthe optimal base-stock policy to the uncapacitatedproblem in case of the setup cost K ¼ 0. Chen andLambrecht (1996) find the examples of finitehorizon where the simple structure fails to hold,and they obtain that the optimal policy does exhibita systematic pattern of what is called X–Y band:when the inventory level drops below X, order up tocapacity; when the inventory level is above Y, donothing; and when the inventory level is between X

and Y, the ordering pattern is not known. Based ona concept called ðC;KÞ-convexity, Chen (2004)shows that the X–Y band is no more than onecapacity of width, and presents a linear program-ming to find the optimal policy.

Our paper discusses the inventory problem facingthe distributor where there exists an stationary supplyto satisfy its stochastic demand. Considering thatthere exists a long-term stationary supply, and theextra order placed by the distributor can be satisfiedwith higher price by the manufacturer, and thestochastic orders placed by the retailers should besatisfied at once and backlogging and shortage arenot allowed, we form this problem to be a singleproduct, multi-period inventory problem. And weanalyze the cost function that contains the orderingcost, holding cost and extra shortage cost. Compar-ing with the aforementioned literature, the differenceand contributions are as follows. First, we prove thatthe cost function is a convex function of the amountof the stationary supply, so that the distributor can

ARTICLE IN PRESS

Table 1

General ordering policy

n 1 2 3 4 � � �

rn b�D1 b�D2 b�D3 b�D4 � � �

zn D1 D2 D3 D4 � � �

an b b b b � � �

J.-A. Li et al. / Int. J. Production Economics 103 (2006) 527–534 529

make the optimal ordering policy to minimize itstotal cost. Second, we provide a simulation methodto compute the optimal ordering policy.

This paper is organized as follows. In Section 2we present the general ordering policy. Section 3explores the stationary supply case and derive thecost function. Also, the comparison is shown at theend of Section 3. Section 4 provides our algorithmto compute the optimal stationary supply, and thecomputational result is presented. Conclusions andfuture work are listed in Section 5.

2. General ordering policy

This section discusses the general ordering policyof the distributor.

Suppose the demands D1;D2; . . . in successiveperiods are independent and identically distributednonnegative random variables with distributionfunction F ðxÞ and density function f ðxÞ in ½a; b�,respectively. Let rn denote the inventory on hand atthe end of period n. And an order zn is placed andwill be received at the beginning of period nþ 1. Weknow the inventory at the beginning of period nþ 1will be anþ1 ¼ rn þ zn. Let c be the purchasingcost per unit, ch be the holding cost per unit,ðxÞþ ¼ maxð0;xÞ, and a be the discount factor.

Because the demand should be satisfied at once,that is, backlogging is not allowed, if the shortage incase anoDn can be satisfied instantaneously withhigher price c0 per unit, we know rnþ1 ¼ ðrn þ zn�

Dnþ1Þþ. Let a1 be the initial inventory. The infinite

discounted cost will be as follows:

V ¼X1n¼1

an�1½czn þ chðan �DnÞþþ c0ðDn � anÞ

þ�.

(1)

Because zn ¼ anþ1 � ðan �DnÞþ, we know

V ¼ � ðc� chÞða1 �D1Þþþ c0ðD1 � a1Þ

þ

þX1n¼2

an�1½acan � ðc� chÞðan �DnÞþ

þ c0ðDn � anÞþ�.

Define wðan;DnÞ ¼ acan�ðc�chÞðan�DnÞþþc0ðDn�

anÞþ, and

GðanÞ ¼ Ewðan;DnÞ

¼ � ðc0 � acÞan þ ðc0 þ ch � cÞ

�

Z an

a

F ðxÞdxþ c0ED1.

Taking the derivative of GðanÞ twice, we haveG00ðanÞ ¼ ðc0 þ ch � cÞf ðanÞX0 when anX0. Weknow GðanÞ is a convex function of an so that thereexists a�, F ða�Þ ¼ ðc0 � acÞ=ðc0 � cþ chÞ, satisfyingGðanÞXGða�Þ, 8an 2 ½a; b�. When a1pa�, orderingup to a� during period n ¼ 1; 2; . . . will minimizethe expected value of V. Also, when a14a�, an ¼

minða�; rnÞ during period n ¼ 1; 2; . . . will minimizethe expected value of V too. At the same time, thedistributor should order ðDn � anÞ

þ with price c0 incase anoDn.

However, when the shortage in case anoDn

cannot be satisfied instantaneously by an extraorder, the distributor should hold extra b� a�

inventory for untimely needs. In this case, a� willbe b. The ordering policy will be as follows (seeTable 1).

The total cost will be V ¼P1

n¼1 an�1ðczn þ chrnÞ ¼

chb=ð1� aÞ þ ðc� chÞP1

n¼1 an�1Dn. And the expec-

ted cost is EV ¼ ðchbþ ðc� chÞED1Þ= ð1� aÞ. Also,the mean and variance of the order are ED1 andEðD1 � ED1Þ

2, respectively.

3. Long-term stationary supply

In this section, we focus on the long-termstationary supply.

If the demand dðtÞ, tX0 of the distributor iscontinuous and the stationary supply is qp units pertime unit, the inventory process can be drawn asFig. 1.

In Fig. 1, the supply is larger than the demandfrom time 0 to time t1, and at that time the excesssupply will be accumulated at the distributor. Fromtime t1 to time t2 the supply is less than the demand,and the inventory will be used to satisfy the excessdemand so that the inventory will be decreased. Ifthe inventory is used up, an extra order, if the ordercan be received instantaneously, will be used tosatisfy the excess part. As the demand exceeds or isexceeded by the supply, the inventory will bedecreases or increases, respectively. Let q1

hðtÞ; tX0be the amount of inventory at time t, and q2

hðtÞ; tX0

ARTICLE IN PRESS

Fig. 1. Demand and supply.

Fig. 2. Demand and supply.

J.-A. Li et al. / Int. J. Production Economics 103 (2006) 527–534530

be the amount of the extra order when q1hðtÞ ¼ 0 at

time t, and t0 ¼ 0. Fig. 1 shows that t 2 ½t2n; t2nþ1� ifqpXdðtÞ, otherwise t 2 ½t2nþ1; t2ðnþ1Þ�, n ¼ 0; 1; . . ..There will be

q1hðt̄Þ ¼

q1hðt2nÞ þ

R t̄

t2nðqp � dðtÞÞdt; t2npt̄pt2nþ1; n ¼ 0; 1; . . . ;

maxð0; q1hðt2nþ1Þ �

R t̄

t2nþ1ðdðtÞ � qpÞdtÞ; t2nþ1pt̄pt2nþ2; n ¼ 0; 1; . . . :

8<:

q2hðt̄Þ ¼

0; t2npt̄pt2nþ1; n ¼ 0; 1; . . . ;

�minð0; q1hðt2nþ1Þ �

R t̄

t2nþ1ðdðtÞ � qpÞdtÞ; t2nþ1pt̄pt2nþ2; n ¼ 0; 1; . . . :

(

Then, we can derive the discounted cost over time T.

V ðqpÞ ¼

Z T

0

e�at½cqp þ chq1hðtÞ þ c0q

2hðtÞ�dt. (2)

If the demand is stochastic, dðtÞ, tX0, will be astochastic process. Then, it will be very difficult tosolve (2). And we always use the discrete case that isdrawn as Fig. 2 instead of the continuous case.

Let qp be the amount of the stationary supply thatcan be received at the beginning of each period n,and Dn be the demand of period n; n ¼ 1; 2; . . ., asdefined in Section 2. In this section we discuss twoscenarios: one is that the extra order can be received

instantaneously; the other one is that the extra orderwill be received one period late.

3.1. The first case

In this case the extra order can be receivedinstantaneously. Supposing V nðx; qpÞ is the totalexpected discounted cost over n periods, where x isthe initial inventory and V 0ðx; qpÞ ¼ 0, V nðx; qpÞ canbe expressed by the following formula:

Vnðx; qpÞ ¼ cqp þ chEðxþ qp �DÞþ

þ c0EðD� x� qpÞþ

þ a½1� F ðxþ qpÞ�V n�1ð0; qpÞ

þ aZ xþqp

a

V n�1ðxþ qp � y; qpÞf ðyÞdy

¼ cqp þ ðch þ c0Þ

Z xþqp

a

F ðyÞdy

þ c0ED� c0ðxþ qpÞ

þ a½1� F ðxþ qpÞ�V n�1ð0; qpÞ

þ aZ xþqp

a

V n�1ðxþ qp � y; qpÞf ðyÞdy.

ð3Þ

Proposition 3.1. V nðx; qpÞ is convex in x and qp.

Proof. The proof will be by induction. When n ¼ 1,

q2V 1ðx; qpÞ

qx2¼ ðch þ c0Þf ðxþ qpÞX0,

q2V 1ðx; qpÞ

qq2p

¼ ðch þ c0Þf ðxþ qpÞX0,

aqV 1ð�; qpÞ

qxþ ch þ c0 ¼ aðch þ c0ÞF ðxþ qpÞ þ ch40,

q2V 0ðx; qpÞ

qxqqp

¼q2V 0ðx; qpÞ

qqpqx¼ ðch þ c0Þf ðxþ qpÞX0.

ARTICLE IN PRESSJ.-A. Li et al. / Int. J. Production Economics 103 (2006) 527–534 531

We know V1ðx; qpÞ is convex in x and qp. SupposeV nðx; qpÞ is convex in x and qp for nX1. In thefollowing, we always use qV nð�; �Þ=qx as qV nðx1; �Þ=qx1 and qV nð�; �Þ=qqp as qVnð�;x2Þ=qx2.

qVnþ1ðx; qpÞ

qx

¼ ðch þ c0ÞF ðxþ qpÞ � c0

þ aZ xþqp

a

qVnðxþ qp � y; qpÞ

qxf ðyÞdyX� c0

) aqV nþ1ð�; qpÞ

qxþ ch þ c040,

q2V nþ1ðx; qpÞ

qxqqp

¼q2Vnþ1ðx; qpÞ

qqpqx

¼ ðch þ c0Þf ðxþ qpÞ þ aqV nð0; qpÞ

qxf ðxþ qpÞ

þ aZ xþqp

a

q2Vnðxþ qp � y; qpÞ

qx2

"

þq2V nðxþ qp � y; qpÞ

qxqqp

#f ðyÞdyX0,

q2V nþ1ðx; qpÞ

qx2

¼ ðch þ c0Þf ðxþ qpÞ þ aqV nð0; qpÞ

qxf ðxþ qpÞ

þ aZ xþqp

a

q2V nðxþ qp � y; qpÞ

qx2f ðyÞdyX0,

qV nþ1ðx; qpÞ

qqp

¼ cþ ðch þ c0ÞF ðxþ qpÞ � c0

þ að1� F ðxþ qpÞÞqV nð0; qpÞ

qqp

þ aZ xþqp

a

qVnðxþ qp � y; qpÞ

qx

"

þqVnðxþ qp � y; qpÞ

qqp

#f ðyÞdy,

q2V nþ1ðx; qpÞ

qq2p

¼ ðch þ c0Þf ðxþ qpÞ þ a½1� F ðxþ qpÞ�

�q2V nð0; qpÞ

qq2p

þ aqVnð0; qpÞ

qxf ðxþ qpÞ

þ aZ xþqp

a

q2Vnðxþ qp � y; qpÞ

qx2

"

þq2Vnðxþ qp � y; qpÞ

qxqqp

þq2Vnðxþ qp � y; qpÞ

qqpqx

þq2V nðxþ qp � y; qpÞ

qq2p

#f ðyÞdyX0.

We know Vnþ1ðx; qpÞ is convex in x and qp. &

Then, we have the following theorem.

Theorem 3.1. The optimal stationary supply of each

period over N periods is q�pob,

V Nðx; q�pÞ ¼ min

qp

V Nðx; qpÞ.

Proof. From Proposition 3.1, we know, there existsq�p satisfying

V nþ1ðx; q�pÞ ¼ min

qp

V nþ1ðx; qpÞ.

Inserting qp ¼ b into the formulae of qVnðx; qpÞ=qx and qV nðx; qpÞ=qx, n ¼ 1; 2; . . ., we knowqV nðx; bÞ=qxXch and qV nðx; bÞ=qqpXcþ ch so thatq�pob. &

3.2. The second case

In this case we assume that any order will bereceived one period late, except the stationarysupply qp each period. In order to avoid theshortage, the available inventory at the beginningof each period should not be less than b.

For example, the initial inventory should beb� qp at least. After receiving qp, the inventory willbe b at least. After satisfying the demand D1, theinventory will be b�D1. When b�D1Xb� qp,the available inventory for next period will beb�D1 þ qpXb, so that no extra order should beplaced. However, when b�D1ob� qp, the avail-able inventory for next period will be b�D1 þ

qpob if no order is placed. In this case, shortagemay be caused. Then, an order b� qp � ðb�D1Þ ¼

D1 � qp should be ordered at this period. Then,the available inventory for next period will beb�D1 þD1 � qp ¼ b� qp.

Suppose that xi þ qp is available for period i. Weknow xi þ qpXb. If xi þ qp �DiXb� qp, no extraorder should be placed. However, if xi þ qp �Dio

ARTICLE IN PRESSJ.-A. Li et al. / Int. J. Production Economics 103 (2006) 527–534532

b� qp, the order ðb� qpÞ � ðxi þ qp �DiÞ should beplaced.

Let V nðx; qpÞ be the total expected discounted costover n periods where x is the initial inventory andV0ðx; qpÞ ¼ 0, V nðx; qpÞ can be expressed by thefollowing formula:

Vnðx; qpÞ

¼ cqp þ c0E½maxðb� qp;xþ qp �DÞ

� ðxþ qp �DÞ� þ chEðxþ qp �DÞ

þ a½1� F ðxþ 2qp � bÞ�Vn�1ðb� qp; qpÞ

þ aZ xþ2qp�b

a

Vn�1ðxþ qp � y; qpÞf ðyÞdy

¼ ðcþ ch � 2c0Þqp þ c0bþ ðch � c0Þx

þ c0

Z xþ2qp�b

a

F ðyÞdy� ðch � c0ÞED

þ a½1� F ðxþ 2qp � bÞ�Vn�1ðb� qp; qpÞ

þ aZ xþ2qp�b

a

Vn�1ðxþ qp � y; qpÞf ðyÞdy. ð4Þ

Proposition 3.2. V nðx; qpÞ is convex in x and qp.

Proof. The proof will be by induction. When n ¼ 1,

q2V 1ðx; qpÞ

qx2¼ c0f ðxþ 2qp � bÞX0,

q2V 1ðx; qpÞ

qq2p

¼ 4c0f ðxþ 2qp � bÞX0,

q2V 1ðx; qpÞ

qxqqp

¼ 2c0f ðxþ 2qp � bÞX0,

q2V 1ðx; qpÞ

qqpqx¼ 2c0f ðxþ 2qp � bÞX0,

q2V1ðx; qpÞ

qx2þ

q2V 1ðx; qpÞ

qq2p

�q2V1ðx; qpÞ

qxqqp

�q2V 1ðx; qpÞ

qqpqx¼ c0f ðxþ 2qp � bÞX0,

aqV 1ðx; qpÞ

qxþ c0 ¼ c0F ðxþ 2qp � bÞ þ c0ð1� aÞX0.

We know V1ðx; qpÞ is convex in x and qp. Suppos-ing Vnðx; qpÞ is convex in x and qp for nX1, andwe always use qV nð�; �Þ=qx as qVnðx1; �Þ=qx1

and qVnð�; �Þ=qqp as qV nð�; x2Þ=qx2.

q2Vnþ1ðx; qpÞ

qx2

¼ c0f ðxþ 2qp � bÞ

þ aqVnðb� qp; qpÞ

qxf ðxþ 2qp � bÞ

þ aZ xþ2qp�b

a

q2V nðxþ qp � y; qpÞ

qx2f ðyÞdyX0,

q2Vnþ1ðx; qpÞ

qxqqp

¼ 2c0f ðxþ 2qp � bÞ

þ 2aqVnðb� qp; qpÞ

qxf ðxþ 2qp � bÞ

þ aZ xþ2qp�b

a

q2V nðxþ qp � y; qpÞ

qx2

"

þq2Vnðxþ qp � y; qpÞ

qxqqp

#f ðyÞdyX0,

q2Vnþ1ðx; qpÞ

qqpqx

¼ 2c0f ðxþ 2qp � bÞ

þ 2aqVnðb� qp; qpÞ

qxf ðxþ 2qp � bÞ

þ aZ xþ2qp�b

a

q2V nðxþ qp � y; qpÞ

qx2

"

þq2Vnðxþ qp � y; qpÞ

qqpqx

#f ðyÞdyX0,

q2V nþ1ðx; qpÞ

qq2p

¼ 4c0f ðxþ 2qp � bÞ

þ 4aqV nðb� qp; qpÞ

qxf ðxþ 2qp � bÞ

þ a½1� F ðxþ 2qp � bÞ�

�q2Vnðb� qp; qpÞ

qx2þ

q2V nðb� qp; qpÞ

qq2p

"

�q2V nðb� qp; qpÞ

qxqqp

�q2V nðb� qp; qpÞ

qqpqx

#

ARTICLE IN PRESSJ.-A. Li et al. / Int. J. Production Economics 103 (2006) 527–534 533

þ aZ xþ2qp�b

a

q2V nðxþ qp � y; qpÞ

qx2

"

þq2V nðxþ qp � y; qpÞ

qxqqp

þq2V nðxþ qp � y; qpÞ

qqpqx

þq2V nðxþ qp � y; qpÞ

qq2p

#f ðyÞdyX0.

We know Vnþ1ðx; qpÞ is convex in x and qp. &

Then we can present the following theorem.

Theorem 3.2. The optimal stationary supply of each

period over N periods is q�pob, and VN ðx; q�pÞ ¼minqp

V Nðx; qpÞ:

Proof. From Proposition 3.2, we know there existsq�p satisfying

V nþ1ðx; q�pÞ ¼ min

qp

V nþ1ðx; qpÞ.

Inserting qp ¼ b into qV0ðx; qpÞ=qx, qV0ðx; qpÞ=qqp,n¼1; 2; . . ., we know qVnðx; bÞ=qqp40) q�pob. &

4. A numerical example

In this section, we compute different total costsaccording to different ordering policies discussed inthe previous sections.

Suppose c ¼ 100, ch ¼ 20, c0 ¼ 350, a1 ¼ 0, a ¼0:99, a ¼ 40, b ¼ 100. And the demand is astochastic variable with the normal distributionwhose mean and standard deviation are 70 and 10,respectively. Given 100 000 periods, we can geta� ¼ 84:6449, and the cost using the order-up-to a�

policy is 736081.05, while the cost of the retailerusing order-up-to b policy is 761198.15.

Now we will compute the optimal stationarysupply. From the theorems, we know the optimalstationary supply is q�p. And q�pob shows thatwe can search q�p from b. The following is oursimulation method.

Step 1: Given a step length D, N ¼ 100 000periods and n ¼ 1, i ¼ 0, qp ¼ b� iD, the initialinventory a1

1 ¼ 0 and V1N ðqpÞ ¼ 0 for Case 1, a1

2 ¼

b� qp and V 2NðqpÞ ¼ c0a1

2 for Case 2, 0oap1, �40;

Step 2: Generating the random demand Dn;In Case 1 the extra order can be received

instantaneously with the cost c0 per unit,

V 1NðqpÞ þ an�1ðcqp þ chða

11 þ qp �DnÞ

þ

þ c0ðDn � a11 � qpÞ

þÞ ) V 1

NðqpÞ; ða11 þ qp �DnÞ

þ

) a11;

In Case 2 the extra order should be placed oneperiod advance,

V 2NðqpÞ þ an�1ðcqp þ chða

12 þ qp �DnÞ

þ c0ðmaxðb� qp; a12 þ qp �DnÞ � ða

12 þ qp �DnÞÞÞ

) V 2NðqpÞ; a

12 ¼ maxðb� qp; a

12 þ qp �DnÞ;

If noN, nþ 1) n, continue this step;Step 3: If there exists i�, V �Nðb� ði

� þ 1ÞDÞ4V �Nðb� i�DÞ and V �Nðb� i�DÞoV �N ðb� ði

� � 1ÞDÞ,

I

f jV �N ðb� ði� þ 1ÞDÞ � V �Nðb� ði

� � 1ÞDÞjo�,

q�p ¼ b� i�D, Stop;

Else, b� ði� þ 1ÞD) a, b� ði� � 1ÞD) b,D2) D,

go to step 1;Else, i þ 1) i, n ¼ 1, qp ¼ b� iD, go to Step 2 to

compute V �N ðqpÞ.

Because of the stochastic property of the demand,the optimal q�p may vary around the optimal value.So we should choose the average value of severaliterations.

Then we obtain the optimal stationary ordering

policy q�p. Here in Case 1 q�p ¼ 68:11 and V1N ¼

787214:68, in Case 2 q�p ¼ 68:11 and V2N ¼

850307:56. The costs according to different qp are

drawn in Fig. 3, respectively.Fig. 3 shows that, the convexity of the discounted

expected cost based on the simulation data is notstrict, and approximating the minimal expected costneeds a large number of samples.

From the result 850307:564787214:684761198:154736081:05, we know that, if there doesnot exist a long-term stationary supply as describedin this paper, the ordering policies of the long-termstationary supply are not better than the order-up-to policies. However, if the cost incurred from theadjusted production is considered, the long-termstationary supply may be better, which depends onthe cost comparison.

ARTICLE IN PRESS

Fig. 3. Cost using different qp in Case 1 and Case 2.

J.-A. Li et al. / Int. J. Production Economics 103 (2006) 527–534534

Another interesting aspect of the stationarysupply is the length of the decision periods N.When N is large enough, the optimal stationarysupply in Cases 1 and 2 changes little around 68:11.Then, we may confirm that keeping the stationarysupply q�p ¼ 68:11 is the optimal ordering policy insuch distribution systems.

5. Conclusion

This paper discusses the optimal ordering policyin a distribution system where there exists one long-term supply contract between the manufacturer andthe distributor so that the distributor can bereplenished with a long-term stationary supply tosatisfy its stochastic demand. And we obtain thatthe cost is a convex function of the amount of thestationary supply, which exists in two cases. Usingthe simulation method, we can compute the optimalordering policy.

In future work, the different lead-time may beconsidered, and the variance of the stationarysupply plus the extra order may be very general.Also, the issue of contract pricing (time-based orquantity based) in the manufacturer–distributorrelationship will be very interesting.

Acknowledgements

This work was partially supported by the AnnualGrant of the University of Southampton, UK. Theauthors express deep gratitude to the anonymousreferees for their tireless efforts in improving thispaper.

References

Archibald, B.C., 1981. Continuous review (s,S) policies with lost

sales. Management Science 27, 1171–1177.

Arrow, K.J., Harris, T., Marschak, J., 1951. Optimal inventory

policy. Econometrica 19 (3), 250–272.

Axsater, S., 2000. Inventory Control. Kluwer Academic,

Dordrecht.

Bartmann, D., Beckmann, M.J., 1992. Inventory Control:

Models and Methods. Springer, Berlin.

Chang, C.T., 2004. An EOQ model with deteriorating items

under inflation when supplier credits linked to order quantity.

International Journal of Production Economics 88 (3),

307–316.

Chen, S.X., 2004. The infinite horizon periodic review problem

with setup costs and capacity constraints: a partial character-

ization of the optimal policy. Operations Research 52 (3),

409–421.

Chen, S.X., Lambrecht, M., 1996. X–Y Band and modified (s,S)

policy. Operations Research 44 (6), 1013–1019.

Federgruen, A., Zipkin, P., 1984. An efficient algorithm for

computing (s,S) policies. Operations Research 32 (6),

1268–1285.

Federgruen, A., Zipkin, P., 1986. Inventory model with limited

production capacity and uncertain demands I. The average-

cost criterion. II. The discounted-cost criterion. Mathematics

of Operations Research 11 (2), 193–215.

Heyman, D.P., Sobel, M.J., 1984. Stochastic Models in Opera-

tions Research, vol. II. McGraw-Hill Book Company, New

York.

Iglehart, D.L., 1963. Optimality of (s,S) policies in the infinite

horizon dynamic inventory problem. Management Science 9

(2), 259–267.

Johnson, E.L., 1968. On (s,S) policies. Management Science 15

(1), 80–101.

Mohebbi, E., Posner, M.J.M., 1998. A continuous-review

inventory system with lost sales and variable lead time. Naval

Research Logistics 45, 259–278.

Mohebbi, E., Posner, M.J.M., 2002. Multiple replenishment

orders in a continuous-review inventory system with lost

sales. Operations research Letters 30, 117–129.

Scarf, H., 1960. The optimal of (s,S) policies in the dynamic

inventory problem. In: Arrow, K.J., Karlin, S., Suppes, P.

(Eds.), Mathematical methods in the social sciences, 1959.

Proceeding of the First Stanford Symposium (Stanford),

pp. 196–202.

Tripathy, P.K., Wee, W.M., Majhi, P.R., 2003. An EOQ model

with process reliability considerations. Journal of the Opera-

tional Research Society 54 (5), 549–554.

Wang, Y., Gerchak, Y., 1996. Continuous review inventory

control when capacity is variable. International Journal of

Production Economics 45, 381–388.

Zheng, Y.S., 1991. A simple proof for optimality of (s,S) policies

in infinite-horizon inventory systems. Journal of Applied

Probability 28, 802–810.

Zipkin, P.H., 2000. Foundations of Inventory Management.

McGraw-Hill, New York.