An Integrated Inventory Model Involving Manufacturing

International Journal of Production ResearchVol. 49, No. 4, 15 February 2011, 1219–1228

An integrated inventory model involving manufacturing

setup cost reduction in compound Poisson process

Chao-Kuei Huanga*, T.L. Chenga, T.C. Kaoa and S.K. Goyalb

aDepartment of Industrial Engineering and Management, Cheng Shiu University,Kaohsiung, Taiwan, ROC; bDepartment of Decision Sciences & M.I.S., John Molson

School of Business, Concordia University, 1455 De Maisonneuve Blvd. West,Montreal, Quebec, H3G 1M8, Canada

(Received 4 September 2009; final version received 5 January 2010)

The collaboration of vendor and buyer is one of the key factors for successfulsupply chain management. The most common strategy for a cooperative system isto propose an integrated replenishment plan. Almost all inventory models assumethat setup cost is deterministic and is not subject to control. However, in manypractical situations, setup cost can be reduced at an added investment. The paperassumes that setup cost can be reduced at an added investment and shortageis permitted during the lead time. This article relaxes the assumption that thedemand of lead time is deterministic and is assumed to be a compound Poissonprocess. A model is derived to determine an optimal integrated inventory policywith controllable setup cost. The expected annual integrated total cost functionis derived and a solution procedure is established to find out the optimal solution.Finally, a numerical example is provided to illustrate the solution procedure.

Keywords: economic order quantity; setup cost; compound Poisson process

1. Introduction

Just-in-time (JIT) manufacturing focuses mainly on purchasing and manufacturing therequired items for immediate consumption. JIT requires a new spirit of cooperationbetween the vendor and the buyer to be successfully implemented. Once a buyer’spurchasing decision is determined in cooperation with the vendor, the overall joint costcan be minimised. Therefore, an integrated inventory policy is helpful to determineeconomic order quantity and shipment policy. The vendor and buyer usually setup a longterm of production-purchasing agreement before any action is taken, and work togetherin a cooperative manner. Simultaneously, an optimal number of deliveries must bedetermined based on their integrated total cost function. This idea of optimising thebuyer’s and the vendor’s total functions jointly was initiated by Goyal (1976). Banerjee(1986) develops a joint economic-lot-size model for a special case where a vendor producesto order for a purchaser on a lot-for-lot basis. A review of related literature is givenby Goyal and Gupta (1989). Lu (1995) considers heuristics for the single-vendor

*Corresponding author. Email: [email protected]

ISSN 0020–7543 print/ISSN 1366–588X online

� 2011 Taylor & Francis

DOI: 10.1080/00207541003610270

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multi-buyer problem. Hill (1997) considers successive shipment sizes that increase by ageneral fixed factor, ranging from one to the ratio between the production rate and thedemand rate. Hill (1999) derives a globally optimal batching and shipping policy for theintegrated production-inventory problem. In addition, Hill and Omar (2006) develop anoptimal production and shipment policy assuming that unit stockholding costs decrease asthe stock moves down the chain. Hoque (2009) develops an alternative optimal solutiontechnique for a single-vendor single-buyer integrated production inventory model.A theoretical comparative analysis of the Hoque methods with Hill (1999), and Hill andOmar (2006) is carried out to show the Hoque solution technique is straightforwardand faster. In practice, some researchers discuss the equal shipment size policy. The equalshipment size policy is discussed by Aderohunmu et al. (1995). Affrisco et al. (1993)consider policies for a single vendor and a number of non-identical purchasers. Ha andKim (1997) study the integrated decision of the buyer and the vendor using geometricprogramming. Huang (2002) extends the Ha and Kim model to flawed items. Huang(2004) considers the effects of an unreliable process on a joint economic lot-sizing problemand establishes a solution procedure to determine an optimal solution. Chung (2008)provides a necessary and sufficient condition for the optimal solution that improved thesolution procedure in the Huang (2004) model. Chung and Wee (2008) investigate anintegrated production inventory deteriorating model considering the pricing policy,the imperfect production, the inspection planning, the warranty-period and thestock-level-dependant demand with the Weibull deterioration, partial backorder andinflation. Furthermore, Wee and Chung (2009) propose an integrated deterioratingproduction inventory model considering green component design, remanufacturing andJIT deliveries. Lin (2009) develops an integrated vendor-buyer inventory system withcontrollable lead time, backorder price discount, and effectively increases investment toreduce the ordering cost. Jaber and Goyal (2009) investigate an inventory model forcoordinating a four-level supply chain of a product with a vendor, multiple buyers andtier-1 and tier-2 suppliers. Pan and Yang (2002) extend the Ha and Kim (1997) model toincorporate controllable lead time. They assume that the demand of lead time L has acumulative distribution with finite mean uL and standard deviation �

ffiffiffiffiLp

. However, ingeneral, customers leave a supermarket in accordance with a Poisson process, NðtÞ,the total number of customers leave by time t. If Yi, the amount spent by the ith customer,i ¼ 1, 2, . . ., are independent and identically distributed, then fXðtÞ ¼

PNðtÞi¼1 Yi, t � 0g is a

compound Poisson process when XðtÞ denotes the total amount of money spent by time t(Ross 1993). Therefore, the demand of the lead time is a compound Poisson process andthe standard deviation is

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið�2 þ u2ÞL

p. To remedy the above-mentioned drawback,

a different viewpoint is proposed in this article. The demand of lead time is assumed to bea compound Poisson process in the paper.

In the above mentioned cases, they assume setup cost is prescribed, that is, it is notcontrollable. Actually, setup cost can be reduced by an additional investment, in otherwords, it can be controlled. Porteus (1985) investigates the impact of capital investment inreducing setup costs on the classical EOQ (economic order quantity) model for the firsttime. Billington (1987) presents an EPQ (economic production quantity) model with thesetup cost parameter replaced by a function of capital investment. Several relationshipsbetween the amount of capital investment and the setup cost level have been reported byPaknejad et al. (1990), Kim et al. (1992), and Nori and Sarker (1996). A commonassumption in the previous research has been that there is a continuous relationshipbetween capital investment in setup cost reduction and the setup cost level. In case studies

1220 C.-K. Huang et al.

of industrial situations, Trevino et al. (1993) indicate that there are only a finite number ofinvestment possibilities to reduce setup cost. For example, let us assume that new fixturesto facilitate lower setup will cost X dollars. Investing 0.9X dollars may accomplish nothingin setup cost reduction and spending 1.1X dollars will not help reduce setup costs anymorethan spending X dollars. Sarker and Coates (1997) develop a method to determine theoptimal amount of capital investment to reduce setup costs when there is a discontinuousinvestment-setup cost relationship. Therefore, the capital investment-setup cost levelfunction may in reality be discontinuous, the discontinuous case will also be investigated.In this paper, the setup cost reduction is considered and the demand of lead time isassumed to be a compound Poisson process in the integrated inventory system. This articlederives the expected annual integrated inventory cost function and develops a solutionprocedure for finding the optimal policy. Further, a numerical example is used to illustratethe benefits of integration.

2. Notation and assumptions

The following notation and assumptions are used throughout to develop the proposedintegrated vendor-buyer inventory model.

Notation:

R lot size per production run for the vendor;m the total number of shipments per lot from the vendor to the buyer, a positive

integer;Q the size of each shipment from the vendor to the buyer, Q ¼ R=m;D demand rate of the buyer;P production rate of the vendor, P4D;T cycle time, T ¼ R=D ¼ mQ=D;S0 original setup cost per production run;K investment per production run required to achieve setup cost S;S setup cost per production run, which is a strictly decreasing function of K, with

SðKÞ ¼ S0e�rK, where r is a known parameter, it can be estimated using

experienced data;h inventory holding cost per dollar per year invested in stocks;L length of lead time;cV unit production cost paid by the vendor;cB unit purchase cost paid by the buyer;� unit shortage cost.

Assumptions:

(1) There is only single-vendor and single-buyer for a single product.(2) Let NðLÞ denote the total number of customers during lead time L, and NðLÞ has a

Poisson distribution with mean �L.Yi, the demand quantity purchased by theith demand, i ¼ 1, 2, . . . , are independent and identically normal distributed withmean � and standard deviation �, then fXðLÞ ¼

PNðLÞi¼1 Yi,L � 0g is a compound

Poisson process when XðLÞ denotes the total amount purchased by time L.(3) The reorder point equals the sum of the expected demand during lead time and safety

stock.(4) Shortages are allowed and fully backordered.

International Journal of Production Research 1221

(5) There is a discontinuous investment-setup cost relationship in the model,SiðKiÞ ¼ S0e

�rKi , where i ¼ 0, 1, . . . , n and K0 ¼ 0.

3. Mathematical model

This section describes the mathematical model and presents an exact derivation of theintegrated total cost for the vendor and buyer. Figure 1 and Figure 2 depict the behaviourof inventory levels for both the vendor and the buyer according to the notation andassumptions outlined above. The expected annual total cost for both the buyer and thevendor consists of the buyer’s annual total cost and the vendor’s annual total cost.

(1) The buyer orders a lot of size R and the vendor manufactures R with a finiteproduction rate P(P4D) at one setup but ships in quantity Q to the buyer overm times. Therefore, the vendor reduced its setup cost, and the inventory cost assoon as the buyer’s lot size Q is produced, the lot is delivered to the buyer.By Assumption (2), the expected lead time demand is:

EðXðLÞÞ ¼ EðX1 þ X2 þ � � �XNðLÞÞ ¼ EðNðLÞÞEðXiÞ ¼ �Lu

and the variance is:

VarðXÞ ¼ ½EðXiÞ�2VarðNðLÞÞ þ EðNðLÞÞVarðXiÞ

¼ u2�Lþ �L�2

¼ �Lðu2 þ �2Þ:

Therefore, the reorder point is R ¼ �uLþ k�ffiffiffiffiffiffi�Lp

, where �2 ¼ u2 þ �2 and k isknown as the safety factor. By the Ouyang et al. (1996) paper, the expected shortageper replenishment cycle is given BðrÞ ¼ �

ffiffiffiffiffiffi�Lp

�ðkÞ, where �ðkÞ ¼ �ðkÞ � k½1�(ðkÞ�, and�, ( denote the standard normal p.d.f. and c.d.f., respectively. The average onhand inventory for the buyer is given by ðQ=2Þ þ k�


. Hence, the expected holdingcost per year is hcBððQ=2Þ þ k�

ffiffiffiffiffiffi�LpÞ. Therefore, the total expected annual cost for the

buyer is given by:

TcBðQ,LÞ ¼ hcBQ

2þ k�


� �þ�D

QBðrÞ: ð1Þ

Inventory level

Time

(Buyer)

mQ/D

Q

QQ

Figure 1. The inventory level for the buyer.


Figure 2. The accumulation and consumption of inventory for the vendor.


(2) The accumulation and depletion process of the vendor’s inventory for eachproduction cycle are shown in Figure 2, according to the Ha and Kim (1997) modeland the Woo et al. (1998) model. Figure 2 shows that the vendor’s holding cost peryear can be obtained as:

Holding cost per year¼½bold area� shaded area��cVh

mQ=D

¼mQ Q

Pþðm�1ÞQD� �

�mQðmQ=PÞ

2

h i�T Qþ2Qþ�� þ ðm�1ÞQ½ �

n ocVh

mD=Q

¼Q

2þðm�2ÞQ

21�

D

P

� �� cVh

After adding the setup cost and investment cost, the vendor’s annual total cost is given by:

TCVðm,Q,K Þ ¼S0e�rKD

mQþQ

21þ ðm� 2Þ 1�

D

P

� �� cVhþ

DK

Q: ð2Þ

Equations (1) and (2) confirming the annual total costs for the buyer and the vendor,therefore, the joint total expected annual cost is given by:

TCðm,Q,K Þ ¼D

Q

S0e�rK

mþ Kþ ��


�ðkÞ

þQ

2h 1þ ðm� 2Þð1�D=PÞ½ �cV þ cB� �

þ hcBk�ffiffiffiffiffiffi�Lp

ð3Þ

4. Methodology and the numerical example

For fixed Ki, taking the partial derivatives of TCðm,Q,KiÞ with respect to m and Q, weobtain:

@TCðm,Q,KiÞ

@m¼ �

DS0e�rKi

Qm2þQh

2ð1�D=PÞcv ð4Þ

and

@TCðm,Q,KiÞ

@Q¼ �

D

Q2

S0e�rKi

mþ Ki þ ��


�ðkÞ

þh

21þ ðm� 2Þð1�D=PÞ½ �cv þ cB

� �ð5Þ

For given m and Ki, TCðm,Q,KiÞ is a convex function in Q for Q4 0, because:

@2TCðm,Q,KiÞ

@Q2¼

2D

Q3

S0e�rKi

mþ Ki þ ��


�ðkÞ

4 0: ð6Þ

So it has a unique optimal solution. The minimum value of TCðm,Q,KiÞ will occur at thepoint Q which satisfies @TCðm,Q,KiÞ=@Q ¼ 0. Solving the equation, we have the optimalsolution at Q�i :

Q�i ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2D S0e�rKi=mþ Ki þ ��


�ðkÞ �

h ð1þ ðm� 2Þð1�D=PÞÞcV þ cB½ �

s: ð7Þ


Substituting Equation (7) into Equation (3) and rearranging the result leads to:

TCðm,KiÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2Dh

S0e�rKi

mþ Ki þ ��


�ðkÞ

ð1þ ðm� 2Þð1�D=PÞÞcV þ cB½ �

s

þ hcBk�ffiffiffiffiffiffi�Lp

:

ð8Þ

We may ignore the terms that are independent of m, so the minimisation of the problemcan be reduced to that of minimising:

ZðmÞ ¼ m 1�D

P

� �cV Ki þ ��


�ðkÞh i

þS0e�rKi cB � ð1� 2D=PÞcV½ �

mð9Þ

which is a convex function in m4 0. Since m is a positive integer, let m� denotethe minimum point of ZðmÞ. Using Zðm�Þ � Zðm� � 1Þ and Zðm�Þ � Zðm� þ 1Þ, directcomputation yields:

ðm� � 1Þm� �S0e�rKi cB � ð1� 2D=PÞcV½ �

1� DP

� �cV Ki þ ��


�ðkÞ � � m�ðm� þ 1Þ: ð10Þ

Using Equation (10) find out the value of m�. Substitute m ¼ m� into Equation (7) andcalculate the optimal value of Qi. Therefore, the following simple solution procedure isused to determine the optimal m�, Q� and K�.

Solution procedure:

Step 1: For each Ki, i ¼ 0, 1, 2, . . . , n, compute mi using Equation (10).

Step 2: Substitute m ¼ mi into Equation (7) and calculate the value of Qi.

Step 3: For each (mi,Qi,Ki), compute TCðmi,Qi,KiÞ, for i ¼ 0, 1, 2, . . . , n.

Step 4: Set TCðm�,Q�,K�Þ ¼ mini¼0,1,2,...,n fTCðmi,Qi,KiÞg, then ðm�,Q�,K�Þ is theoptimal solution.

4.1 Numerical example

The proposed analytic solution procedure is applied to solve the following numericalexample. Consider an inventory system with the following characteristics: D¼ 1000 units/year, P¼ 3200 unit/year, S0¼ $1000 per setup, cV¼ $20 per unit, cB¼ $25 per unit, � ¼ $5per unit, �¼ 2, L¼ 4 weeks, h¼ 0.2, k¼ 0.845, � ¼ 7 units/week, r ¼ 0.01.The setup cost can be reduced at an added investment cost and the setup cost

Table 1. Setup cost-investment data.

Project i Investment Ki Setup cost Si

1 50 6062 100 3683 400 18


reduction-investment has three situations with data shown in Table 1. Applying the aboveprocedure yields optimal solutions with K� ¼ 100, m� ¼ 2, and Q� ¼ 254. The expectedintegrated annual total cost is $2418. The solution procedure is summarised in Table 2.The (2705� 2418)¼ $287 displays the result in saving cost compared to the total costwithout crashing.

5. Conclusions

This article considered the single-vendor single-buyer integrated production inventoryproblem. Traditional integrated models consider that setup cost is not controllable and thedemand of lead time is deterministic. However, in many real situations, setup cost can bereduced at an added investment and the demand of the lead time is a compoundPoisson process. In this paper, the demand of lead time was assumed to be a compoundPoisson process. The study also assumed that shortage is permitted during the lead time,and setup cost can be reduced at an added investment cost. Therefore, the inventory modelof this paper is general. Further, the expected annual integrated total cost function hasbeen derived herein. The convex nature of this cost function helps to derive an analytic andsimple solution procedure to determine the optimal setup cost, economic order quantity,and the number of deliveries from vendor to buyer. Finally, the numerical exampleillustrates the solution technique. It indicates that the developed model produces asignificant amount of savings.

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An Integrated Inventory Model Involving Manufacturing