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Int. J. Production Economics 103 (2006) 199–208

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Joint economic procurement—production–delivery policy formultiple items in a single-manufacturer, multiple-retailer system

Taebok Kim, Yushin Honga,�, Soo Young Changb

aDivision of Mechanical and Industrial Engineering, Pohang University of Science and Technology (POSTECH), San 31, Hyoja,

Pohang, Republic of KoreabDivision of Electronic and Computer Engineering, Pohang University of Science and Technology (POSTECH), San 31, Hyoja,

Pohang, Republic of Korea

Received 13 October 2003; accepted 3 June 2005

Available online 30 September 2005

Abstract

This paper proposes an analytical model to effectively integrate and synchronize the procurement, production and

delivery activities in a supply chain consisting of a single raw material supplier, a single manufacturer and multiple

retailers. The manufacturer produces multiple items on a single facility utilizing common raw material under a common

rotation cycle policy of not incurring shortages at any retailers. This problem is formulated as a variant of the classical

economic lot scheduling problem. The objective is to find the production sequence of multiple items, the common

production cycle length, and the delivery frequencies and quantities that minimize the average total cost. An efficient

heuristic algorithm is presented to solve the proposed problem. Through numerical tests, we show that the proposed

heuristic gives quite satisfactory solutions.

r 2005 Elsevier B.V. All rights reserved.

Keywords: Supply chain management; Joint procurement–production–delivery policy; Production sequence; Common rotation cycle;

Economic lot scheduling problem

1. Introduction

Consider a supply chain in which a single manufacturer procures common raw materials from an outsidesupplier, produces multiple items on a single production facility, and delivers the items to the correspondingretailers. In steel industries, for example, iron ores are divergently transformed into several hundreds offinished items through steel making processes in accordance with the specifications requested by retailers.Similar patterns of production can also be observed in many petrochemical industries where multiple items areproduced in batches through refining processes of crude oil. In these supply chains, accordingly, it is one of thekey management issues for the manufacturer to effectively synchronize and integrate activities ofprocurement, production, and delivery for the enhancement of the system-wide productivity as a whole.Therefore, by viewing the supply chain system as an integrated whole, the manufacturer should resolve (1) the

front matter r 2005 Elsevier B.V. All rights reserved.

e.2005.06.005

ng author. Tel.: +8254 279 2194; fax: +82 54 279 2870.

ss: yhong@postech.ac.kr (Y. Hong).

ARTICLE IN PRESST. Kim et al. / Int. J. Production Economics 103 (2006) 199–208200

scheduling issue accommodating multiple items on a single production facility, (2) the production and deliverylot sizing issues for the procurement of raw material, and (3) the delivery issue of finished items to the retailers.

An economic lot scheduling problem (ELSP) concerning how to schedule multiple items on a singleproduction facility is embedded in the problem to be solved. Two typical approaches have been presented tosolve the ELSPs: one is ‘common cycle’ approach presented by Hanssmann (1962) and the other is ‘basic

period’ approach of Bomberger (1966). In the common cycle approach, the manufacturer simply sets up andproduces all the items one by one in every production cycle under the assumption that all items have anidentical production cycle length. Further extensions of the common cycle approach were also presented byincorporating operational characteristics such as a multi-stage production system (Hsu and El-Najdawi,1990), flexible production rate (Khouja 1997), insufficient production capacity (Khoury et al., 2001), andsequence-dependent setup times (Wagner and Davis, 2002).

On the other hand, the basic period approach determines the basic period length as well as the integermultiple for each item simultaneously postulating that production cycles of respective items are integermultiples of the basic period, and Hsu (1983) showed that the general ELSP is NP-hard. Therefore, heuristicapproaches such as the power-of-two policy (Yao and Elmaghraby, 2001; Bertazzi, 2003) and the two-groupmethod (Ouenniche and Boctor, 2001) were introduced. In addition, efficient searching algorithms werealso proposed (Jensen and Khouja, 2003; Grznar and Riggle, 1997). However, all these papers treat the ELSPsas single-stage models without incorporating the delivery issue. Hahm (1990) extended the ELSP as atwo-stage model under the assumption that all components have an identical delivery cycle length. In hismodel, multiple components are produced on a single facility and accumulated until all the components arecompleted at the first stage, and then these components are delivered to the assembly stage at one time. Hahmdeveloped a solution procedure determining the lot size, the production sequence, and the common deliverycycle length.

Another issue to be addressed is a joint economic lot sizing (JELS) problem which determines productionand delivery lot sizes simultaneously in a manufacturer–retailer supply chain. As a seminal research for theJELS problem, Goyal (1976) derived an optimal lot size in a single-manufacturer–single-retailer systemassuming the production lot size is equal to the delivery lot size. Lu (1995) proposed an optimal solutionprocedure determining production and delivery lot sizes simultaneously in a single-manufacturer–single-retailer system under an assumption that the production lot size is an integer multiple of the delivery lot size.In contrast, Goyal (1995) and Hill (1997, 1999) proposed unequal-sized delivery lot policies postulating thattime intervals between successive deliveries and delivery quantities may vary according to the predetermineddelivery pattern. For multiple retailers ordering a common finished item, Banerjee and Burton (1994)developed a coordinated policy determining the production and delivery cycles satisfying the condition that allthe retailers have an identical delivery cycle. Kim and Ha (2003) proposed a model considering total relevantcosts for manufacturer and retailer and determined the optimal order quantity, the number of deliveries/setups, and the shipping quantities in a simple JIT single-manufacturer–single-retailer structure. Also, Kelleet al. (2003) and David and Eben-Chaime (2003) explored the partnership and the negotiation mechanismbetween the manufacturer and the retailer in terms of lot sizing and delivery scheduling in the same supplychain structure. In the case where multiple items are produced in a single facility, Kim et al. (2003) proposed ajoint production–delivery policy that determines an optimal common production cycle and delivery lot sizesunder the assumption that the production lot size of each item is an integer multiple of its correspondingdelivery lot size.

Several papers have been published to incorporate procurement activity for raw materials into theaforementioned joint production–delivery supply chain system. However, all these papers assume a fixeddelivery lot size to the retailer at a fixed interval of time without considering relevant costs incurred at theretailer. Sarker and Parija (1996) suggested a multi-order policy for raw materials to meet the requirements ona production facility and devised a solution procedure for simultaneously determining the number of ordersfor raw material and number of deliveries for the finished items in a cycle. Parija and Sarker (1999) extendedthis problem to the case where a manufacturer delivers a single item to multiple retailers. Nori and Sarker(1996) investigated the case where multiple items are produced on a single facility by the common cycleapproach and delivered to a single retailer assuming that delivery lot sizes of all items are equal and fixedin advance.

ARTICLE IN PRESS

Supplier Manufacturer Retailers

Procurement

Production(Single Facility)

Lumpy Deliveries

M

R1

R2

Ri

Rn

……

…… Common

Raw Materials

Multiple items

S

Fig. 1. Structure of the supply chain system.

T. Kim et al. / Int. J. Production Economics 103 (2006) 199–208 201

The problem discussed in this paper extends the model proposed by Nori and Sarker (1996), and deals witha more generalized procurement–production–delivery policy in a supply chain depicted in Fig. 1. An efficientsolution procedure is developed for determining the production sequence and common cycle length at themanufacturer as well as the respective delivery lot sizes to the corresponding retailers simultaneously. Incontrast to the model of Nori and Sarker (1996), in the proposed model, the production sequence at themanufacturer and the respective delivery lot sizes to the retailers are set as decision variables. Detaileddescription of the proposed problem, the relevant assumptions, and the mathematical formulation areelaborated in Section 2. Section 3 explains the solution procedure and its effectiveness. The final conclusionsare given in Section 4.

2. Model

The supply chain discussed in this paper consists of a single raw-material supplier, a single manufacturer,and multiple retailers as in Fig. 1. Demands at the retailers are assumed to be deterministic and constant overtime. Each retailer purchases its own specialized item from the manufacturer and sells it to the customers. Atthe beginning of each production cycle, the manufacturer purchases common raw material from the supplier,produces all ordered items in a single facility, and periodically delivers ordered quantities of the items to therespective retailers. The necessary parameters describing the supply chain system are listed below.

i: item index, i ¼ 1; 2; . . . ; n.Pi: production rate for item i in units per year.Di: demand rate for item i in units per year.AR: manufacturer’s ordering cost for raw material.HR: manufacturer’s holding cost for raw material per unit per year.si: manufacturer’s setup time for producing item i.Si: manufacturer’s setup cost for item i.Hi: manufacturer’s holding cost for item i per unit per year.Ai: retailer’s ordering and delivery cost for item i at each delivery.hi: retailer’s holding cost for item i per unit per year.

In the definitions above, note thatPn

i¼1ðDi=PiÞp1 must hold for the relevance of the proposed model.Each retailer places orders of its own item based on the EOQ-like policy, in other words, requests the

manufacturer to periodically deliver equal-sized quantity (i.e., equal delivery cycle). In a production run ofeach ordered item at the manufacturer, the production batch is set as an integer multiple of the ordered

ARTICLE IN PRESST. Kim et al. / Int. J. Production Economics 103 (2006) 199–208202

quantity from the retailer. However, if the retailers determine their order quantities independently without anysynchronization with the manufacturer, there are always possibilities that the ordered quantities from someretailers are not delivered in time, and that shortages may incur at those retailers. To avoid these shortages, weassume that the manufacturer adopts a common rotation cycle policy in which all items have the sameproduction cycle length, and produces them one by one in a fixed sequence in every rotation cycle.

The manufacturer places an order for raw material required to produce a single batch of all items in arotation cycle. The raw material for producing a batch is delivered to the manufacturer at the beginning of theproduction cycle, and the consumption rates of raw material are the same as the production rates ofcorresponding items. Fig. 2 shows sample inventory trajectories for the raw material and the finished items atthe manufacturer and the two retailers. Decision variables for the problem are listed below.

mi: delivery frequency for a single production batch of item i, m ¼ ðm1;m2; . . . ;mnÞ.qi: delivery quantity for item i at each delivery.T: common rotation cycle length.Z: production sequence of multiple items in a cycle.[j]: index for item at jth position on production schedule Z.

As shown in Fig. 2, the average inventory level for raw material depends on the production sequence formultiple items. Hence, for a given production sequence Z, the average inventory level for raw material at themanufacturer can be given by

IRðZÞ ¼ TXn

i¼1

D2i

2Pi

þXn�1i¼1

D i½ �

P i½ �

Xn

j¼iþ1

D j½ �

!þXn

i¼1

s i½ �

Xn

j¼i

D j½ �. (1)

In Eq. (1), the first and second terms represent the average inventory level during the production periods andthe setup periods respectively.

Now, the average total cost for the manufacturer and the retailers can be described as follows:

TRCðm;T ;ZÞ ¼AR þ

Pni¼1ðSi þ AimiÞ

� �T

þ T HRIRðZÞ þXn

i¼1

Di

2mi

Hi ð1�Di=PiÞmi � ð1� 2Di=PiÞ� �

þ hi

� �" #.

(2)

Item-2

Manufacturer

Item-1

Retailer-2

m1=5

m2=3

….

….….….

Raw material

Finished items

Retailer-1

…. ….

….….

P2

P1− P1−

−

T

q1

q2

Fig. 2. Inventory trajectories for the manufacturer and the retailers (n ¼ 2).

ARTICLE IN PRESST. Kim et al. / Int. J. Production Economics 103 (2006) 199–208 203

In Eq. (2), the first term includes the average ordering cost for raw material, the average setup cost for themanufacturer, and the ordering cost for the retailers. The second term is the holding costs for the raw materialand the finished items at the manufacturer and the retailers. Substituting IRðZÞ into Eq. (2) and rearrangingEq. (2) gives

TRCðm;T ;ZÞ ¼AR þ

Pni¼1ðSi þ AimiÞ

� �T

þ TXn

i¼i

ðai � giÞ=mi þ bi

� �þ d1

( )þ d2, (3)

where

ai ¼Dihi

2; bi ¼

DiHið1�Di=PiÞ

2; gi ¼

DiHið1� 2Di=PiÞ

2,

d1 ¼ HR

Xn

i¼1

D2i

2Pi

� �þXn�1i¼1

D i½ �

P i½ �

Xn

j¼iþ1

D j½ �

!( ); d2 ¼ HR

Xn

i¼1

s i½ �

Xn

j¼i

D j½ �

!.

Our objective is to determine the optimal delivery frequencies (m*), the optimal common cycle length (T*), andthe optimal production sequence (Z*) minimizing the average total cost given in Eq. (3).

First, given m and Z, it is clear that TRC(T|m, Z) is strictly convex with respect to T. Thus, the optimalcycle length for the given m and Z can be derived by solving dTRC(T|m,Z)/dT ¼ 0, and it is given by

T�ðm;ZÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAR þ

Pni¼1ðSi þ AimiÞPn

i¼1 ðai � giÞ=mi þ bi

� �þ d1

s. (4)

Substituting T�ðm;ZÞ into Eq. (3), the average total cost can be expressed as a function of m and Z asfollows:

TCðm;ZÞ � TRCðm;T�ðm;ZÞ;ZÞ ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAR þ

Xn

i¼1

ðSi þ AimiÞ

! Xn

i¼1

ðai � giÞ=mi þ bi

� �þ d1

( )vuut þ d2.

(5)

For a given Z, let

f ðmÞ ¼ AR þXn

i¼1

ðSi þ AimiÞ

! Xn

i¼1

ðai � gi=miÞ þ bi

� �þ d1

( )(6)

and assume that m is a positive real vector. Differentiating f(m) with respect to mi, we get

qf ðmÞ=qmi ¼ � ðai � giÞ�

m2i

� �AR þ

Xn

i¼1

ðSi þ AimiÞ

!þ Ai

Xn

i¼1

ðai � giÞ=mi þ bi

� �þ d1

( ); 8i. (7)

From Eq. (7), we can show that Eq. (8) holds at a stationary point m0 ¼ ðm01;m

02; . . . ;m

0nÞ satisfying

qf ðmÞ=qmi ¼ 0; 8i.

Ai

Xn

i¼1

ðai � giÞ=m0i þ bi

� �þ d1

( )¼ ðai � giÞ=ðm

0i Þ

2� �

AR þXn

i¼1

ðSi þ Aim0i Þ

!; 8i. (8)

In Eq. (8), let

l ¼Xn

i¼1

ðai � giÞ=m0i þ bi

� �þ d1

( ),AR þ

Xn

i¼1

ðSi þ Aim0i Þ

!,

then

ai � gi

� �=Aiðm

0i Þ

2¼ l; 8i. (9)

ARTICLE IN PRESST. Kim et al. / Int. J. Production Economics 103 (2006) 199–208204

Substituting Eq. (9) into Eq. (8) and rearranging Eq. (8) gives

l ¼Xn

i¼1

bi þ d1

!,AR þ

Xn

i¼1

Si

!. (10)

From Eqs. (9) and (10), a unique stationary point m0 minimizing TCðm;ZÞ for a given Z can be obtained as

m0 ¼ ðm01;m

02; . . . ;m

0nÞ; m0

i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðai � giÞ=Ai

� �AR þ

Xn

i¼1

Si

!, Xn

i¼1

bi þ d1

! !vuut ; 8i. (11)

It can be proven that the unique stationary point in Eq. (11) is a global optimal solution minimizing TC(m,Z)for a given Z, and proof is given in the Appendix. Substituting m0 ¼ ðm0

1;m02; . . . ;m

0nÞ into Eq. (4), the optimal

common cycle length (T�) can be obtained as in Eq. (12).

T�ðm0;ZÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiAR þ

Xn

i¼1

Si

!, Xn

i¼1

bi þ d1

!vuut . (12)

Before we adopt T�ðm0; ZÞ as the optimal cycle length, we must consider the setup times during theproduction cycle. Since the total setup time plus the total production time per cycle must be no longer than thecycle length, we have the following constraint on the cycle time:

TX

Xn

i¼1

si

,1�

Xn

i¼1

Di=Pi

!� Tmin. (13)

Since TRCðm; T ; ZÞ for a given Z is convex in T, the optimal cycle length is determined as the maximum ofT�ðm0; ZÞ and Tmin. Hence, if T�ðm0; ZÞoTmin, Tmin is set as the optimal cycle length, and the correspondingoptimal delivery frequency can be obtained as

m0 ¼ ðm01;m

02; . . . ;m

0nÞ; m0

i ¼ Tmin

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðai � giÞ=Ai

p; 8i. (14)

If m0i ði ¼ 1; . . . ; nÞ in Eq. (11) or Eq. (14), are all integers, the optimal delivery frequencies are determined as

m�i ¼ m0i ði ¼ 1; . . . ; nÞ, and the optimal cycle length can be obtained accordingly. However, if m0

i ði ¼ 1; . . . ; nÞare not integers, we can take the nearest integer values of m0

i , m0i

and m0

i

� �as possible candidates for the

integer-valued optimal frequency for item i. Consequently, a total of 2n candidate vectors for the optimalsolution are available. Evaluating the average total cost with these candidates, the one that gives the lowestaverage total cost is adopted as the optimal delivery frequencies. After obtaining m� ¼ ðm�1;m

�2; . . . ;m

�nÞ,

T�ðm�; ZÞ can be determined by Eq. (4), and the optimal delivery quantities can be calculated accordingly asin Eq. (15).

q�i ¼ T�ðm�; ZÞðDi=m�i Þ; 8i. (15)

Now, we have to consider the production sequence Z. Since ðd1T þ d2Þ is the only sequence-dependent termembedded in TRCðm; T ; ZÞ given in Eq. (3), a production sequence minimizing ðd1T þ d2Þ is also an optimalproduction sequence minimizing TRCðm; T ; ZÞ for given T. Since our problem is equivalent to a single-machine weighted completion time problem, the WSPT (weighted shortest processing time first) rule gives anoptimal solution (Pinedo, 1995). Thus, for given T, an optimal sequence minimizing ðd1T þ d2Þ can beobtained by arranging the items in non-increasing order of Di

�ðsi þDiT=PiÞ.

3. Solution

As explained in Section 2, it is mathematically intractable to derive optimal values, m*, T* and Z*

simultaneously. Therefore, a heuristic algorithm is developed by iteratively obtaining the optimal production

ARTICLE IN PRESST. Kim et al. / Int. J. Production Economics 103 (2006) 199–208 205

sequence and the optimal cycle length as well as the optimal delivery frequencies. Our proposed heuristic has asimilar structure to the solution procedure presented by Hahm (1990) to solve the ELDSP (economic lot anddelivery scheduling problem). The detailed procedure of the proposed heuristic is described below.

Solution procedure:

Step 0:

(Initialization)k ¼ 1.Determine the sequence Z(1) in non-increasing order of Pi.

Step 1:

For the sequence Z(k), calculate d1, d2, m0 and T�ðm0; ZðkÞÞ.

Step 2:

k ¼ k+1.

Determine the sequence Z(k) in non-increasing order of Di

�si þ DiT

�ðm0; Zðk�1ÞÞ�

Pi

� �� �.

If Zðk�1Þ ¼ ZðkÞ, Z� ¼ ZðkÞ and go to Step 3. Otherwise, go to Step 1.

Step 3:

Calculate Tmin and determine the optimal cycle length by T� ¼Max Tmin;T�ðm0; Z�Þ

� �.

Calculate m0 ¼ ðm01;m

02; . . . ;m

0nÞ, where m0

i ¼ T�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðai � giÞ=Ai

p; 8i.

Step 4:

If m0i ði ¼ 1; � � � ; nÞ are all integers, m� ¼ m0 and go to Step 6.

Step 5:

If T� ¼ T�ðm0;Z�Þ,m� ¼ ArgmMin TCðm; Z�Þj T�ðm; Z�ÞXTmin;m : 2n integer vectors

� �.

T� ¼ T�ðm�; Z�Þ. Go to Step 6.Otherwise

m� ¼ ArgmMin TRCðm;Tmin;Z�Þjm : 2n integer vectors

� �.

Step 6:

q�i ¼ DiT��

m�i ; 8i.

Numerical experiment is carried out to evaluate the performance of the proposed algorithm. In theexperiment, our first concern is to see how effective the proposed algorithm is in determining the productionsequence Z as compared with the optimal production sequence obtained by exhaustive enumeration. Toexecute the experiment for a broad range of the system parameters, randomized numeric values of theparameters are generated as follows: Pi�Uð12000; 15000Þ, Di�Uð2000; 4000Þ, Ai�Uð10; 25Þ, Si�Uð200; 500Þ,AR�Uð100; 250Þ, hi�Uð6; 9Þ, Hi�Uð4; 5Þ, and HR�Uð2; 4Þ, where U(a, b) denotes a uniform random variablebetween a and b. In addition, a total of 15 scenarios with 3 levels of n and 5 levels of si are prepared since theseparameters are expected to have significant effects in determining the production sequence. Note that the timeunit for setup times in the experiment is given in days. For each scenario, 1000 problem instances aregenerated. The average total costs for each problem instance obtained by the proposed algorithm, TRC(Heuristic), are calculated and compared with the average total costs by the exhaustive enumerations, TRC(Enumeration), respectively. Error ratios in percentage, i.e.,

e% ¼TRCðHeuristicÞ � TRCðEnumerationÞ

TRCðEnumerationÞ

� �� 100ð%Þ

are computed for 1000 problem instances in each scenario. Table 1 summarizes the result of the experiment.The first and second columns show values of n and si, respectively, and the third column represents the numberof problem instances in which the heuristic and the enumeration yield exactly the same solutions. Meansvalues, maximum values, and standard deviations of e% in 1000 problem instances are listed in the fourth,fifth, and sixth columns, respectively. As seen in the table, even the worst maximum value of e% in all scenariosis still less than 1%. Note that our heuristic has a similar structure to the one by Hahm (1990) as explainedabove. Jensen and Khouja (2003) proposed the polynomial time algorithm using bisectional search for theHahm’s problem, and concluded that there was no significant difference in performance between their solutionprocedure and the one by Hahm (1990). In addition, our heuristic generates the solutions within one second inthe average in cases of 30 retailers. From all these findings, we conclude that the proposed heuristic algorithm

ARTICLE IN PRESS

Table 1

Performance analysis of the proposed algorithm

Scenarios Number of instances e% ¼ 0.0 e%

n si Mean Max Standard deviation

3 U(0.0,1.0) 1000 0 0 0

U(1.0,2.0) 1000 0 0 0

U(2.0,3.0) 1000 0 0 0

U(3.0,4.0) 1000 0 0 0

U(0.0,4.0) 1000 0 0 0

4 U(0.0,1.0) 993 0.0002 0.0937 0.0037

U(1.0,2.0) 938 0.0018 0.1234 0.0092

U(2.0,3.0) 860 0.0046 0.2609 0.0173

U(3.0,4.0) 787 0.0102 0.1931 0.0273

U(0.0,4.0) 833 0.0149 0.5675 0.0527

5 U(0.0,1.0) 864 0.0022 0.0883 0.0081

U(1.0,2.0) 664 0.0080 0.1828 0.0188

U(2.0,3.0) 526 0.0165 0.1924 0.0296

U(3.0,4.0) 376 0.0314 0.3720 0.0487

U(0.0,4.0) 412 0.0586 0.6910 0.0911

T. Kim et al. / Int. J. Production Economics 103 (2006) 199–208206

performs quite effectively and efficiently. Furthermore, as expected, we can observe that e% tends to be highfor (1) large number of items, (2) longer production setup times, and (3) high variation among setup times.Another observation from the experiment is that the production rate is the key parameter in determining theproduction sequence when the setup times are relatively negligible. However, as the setup times increase,trade-offs between production rates and setup times may bring about combinatorial complexities and result inrelatively high error ratios.

4. Conclusions

This paper analyzes a supply chain with a single manufacturer and multiple retailers, and develops anefficient solution procedure for determining a joint procurement–production–delivery policy. Themanufacturer procures common raw material, produces multiple items on a single production facility basedon the common rotation cycle policy, and delivers the items to the corresponding retailers. An efficient andeffective solution procedure is explained to derive production sequence and common cycle length for themanufacturer, and delivery lot sizes for the multiple retailers minimizing the average total cost. It is shownthrough numerical experiments that the proposed algorithm performs quite satisfactorily. The proposedmodel can readily be applied to many practical manufacturing systems such as chemical and petrochemicalindustries.

Further research is needed to analyze more generalized case of n items and m retailers, where any retailercould order any number of the n items. The approach developed in this paper may be extended to the case thatany retailer could order multiple items and any single item could be ordered by one and only one retailer. Also,negotiation and/or coordination mechanisms may be worthy of future study through investigating thenegotiation mechanism in which anticipated losses are compensated for the parties involved in the supplychain caused by accepting the joint procurement–production–delivery policy.

Acknowledgement

This research was supported by the Advanced Product & Production Technology Center at POSTECH.

ARTICLE IN PRESST. Kim et al. / Int. J. Production Economics 103 (2006) 199–208 207

Appendix A

A.1. Global optimality of m0

Let m ¼ m0+d, where d ¼ ðd1; d2; . . . ; . . . ; dnÞ is an arbitrary displacement vector, and definegðm;ZÞ ¼ f ðm;ZÞ � f ðm0;ZÞ. Then g(m,Z) can be expressed by

gðm;ZÞ ¼ SXn

i¼1

ai � gi

mi

�ai � gi

m0i

� �þ b

Xn

i¼1

Aiðmi �m0i Þ

!þ

Xn

i¼1

Aimi

!Xn

i¼1

ai � gi

mi

� �(

�Xn

i¼1

Aim0i

!Xn

i¼1

ai � gi

m0i

� �)

¼ �SXn

i¼1

di

ai � gi

mim0i

� �þ b

Xn

i¼1

Aidi þXn

i¼1

Aimi

Xn

i¼1

aj � gj

mj

!� Aim

0i

Xn

i¼1

aj � gj

m0j

!( ), ðA:1Þ

where S ¼ AR þPni¼1

Si; b ¼Pni¼1

bi þ d1.

Substituting Eq. (11) into Eq. (A.1) and going through additional algebraic manipulation, Eq. (A.1) can bearranged as in Eq. (A.2).

gðm;ZÞ ¼ � bXn

i¼1

diAiðm0i Þ

2

mim0i

þ bXn

i¼1

Aidi þbS

Xn

i¼1

Aimi

! Xn

i¼1

Aiðm0i Þ

2

mi

!�

bS

Xn

i¼1

Aim0i

! Xn

i¼1

Aiðm0i Þ

2

m0i

!

¼ bXn

i¼1

Aid2i

mi

þbS

Xn

i¼1

Aimi

Xn

i¼1

Aiðm0i Þ

2

mi

�Xn

i¼1

Aim0i

Xn

i¼1

Aiðm0i Þ

2

m0i

!

¼ bXn

i¼1

Aid2i

mi

þbS

Xn

i¼1

Aimi

Xn

j¼1;jai

Ajðm0j Þ

2

mj

� Aim0i

Xn

j¼1;jai

Ajm0j

( )

¼ bXn

i¼1

Aid2i

mi

þbS

Xn�1i¼1

Xn

j¼iþ1

AimiAj

ðm0j Þ

2

mj

þ AjmjAi

ðm0i Þ

2

mi

� 2AiAjm0i m0

j

!

¼ bXn

i¼1

Aid2i

mi

þbS

Xn�1i¼1

Xn

j¼iþ1

AiAj m0j

ffiffiffiffiffiffiffiffiffiffiffiffiffimi

�mj

q�m0

i

ffiffiffiffiffiffiffiffiffiffiffiffiffimj

�mi

q� �2

. ðA:2Þ

From Eq. (A.2), gðm;ZÞ ¼ f ðm;ZÞ � f ðm0;ZÞ40 holds for any arbitrary vector mðam0Þ. Therefore, we canconclude that a unique stationary point m0 given by Eq. (11) is globally minimum point.

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