A dynamic model for the optimization of decoupling point and production planning in a supply chain

Int. J. Production Economics 131 (2011) 561–567

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Int. J. Production Economics

0925-52

doi:10.1

n Tel.:

E-m

journal homepage: www.elsevier.com/locate/ijpe

A dynamic model for the optimization of decoupling pointand production planning in a supply chain

In-Jae Jeong n

Department of Industrial Engineering, Hanyang University, 17 Haengdang-dong, Seongdong-gu, Seoul 133-791, Republic of Korea

a r t i c l e i n f o

Article history:

Received 22 February 2010

Accepted 21 January 2011Available online 18 February 2011

Keywords:

Optimal control theory

Production smoothing

Zero-inventory

Forecast error

73/$ - see front matter & 2011 Elsevier B.V. A

016/j.ijpe.2011.02.001

+82 2 2220 0412.

ail address: [email protected]

a b s t r a c t

In this paper, we propose a dynamic model to simultaneously determine the optimal position of

the decoupling point and production–inventory plan in a supply chain such that the total cost of the

deviation from the target production rate and the target inventory level is minimized. Using the

optimal control theory, we derive the closed form of the optimal solution when the production

smoothing policy and the zero-inventory policy are applied. The result indicates that under the

production smoothing policy, the overestimation of demand rate during the pre-decoupling stage

guarantees the existence of the optimal decoupling point; meanwhile the optimal decoupling point

exists under zero-inventory policy when the demand rate is underestimated. Also we perform

mathematical analysis on the behavior of the optimal production rate and the inventory level and

the effect of problem parameters such as the length of the product life cycle and the forecast error on

the performance.

& 2011 Elsevier B.V. All rights reserved.

1. Introduction

A decoupling point is a push–pull boundary in the supplychain. From the upstream of the supply chain (i.e., the rawmaterial supplier) to the decoupling point, the supply chain planis scheduled based on the demand forecast which is a pushstrategy; meanwhile, from the decoupling point to the down-stream of the supply chain (i.e., the end customer), the supplychain operations are driven by the customer orders rather thanforecasts which is a pull strategy. In terms of the product deliverystrategy, it is also known as a make-to-stock (MTS) (a make-to-order (MTO)) strategy in the push (pull) range of the supply chain.The appropriate positioning of the decoupling point is an impor-tant design issue in the realm of the supply chain management.As shown in Fig. 1, the candidate positions of the decoupling pointexist on any stage of the supply chain.

As Olhager (2003) mentioned, there are many reasons andeffects of shifting the decoupling point forward or backward tothe end customers. For example, if the decoupling point movesbackward to the customers, the response time to customer isincreased and the manufacturing efficiency improves; meanwhilethe forward shifting will cause the increase of WIP and thereduction in the product customization. The advantages of theforward shifting are the disadvantages of the backward shifting ofthe decoupling point and vice versa.

ll rights reserved.

There exist two different approaches in determining theposition of the decoupling point, strategic approaches and analy-tic approaches. The strategic approaches intend to provide guide-lines using knowledge-based systems or conceptual models forselecting decoupling point. The analytic approaches use mathe-matical models or simulation models to find an optimal positionof the decoupling point. Most of the mathematical model-basedapproaches tried to minimize the manufacturing related costssubject to satisfy the certain level of customer response time.

Most of the previous mathematical models assume that thedecoupling point is the unique decision variable. However theproduction planning, inventory policy and operational decisionssuch as scheduling and sequencing also affect to the performanceof the supply chain. It is obvious that the supply chain can beoptimized if all these issues can be handled simultaneously if theproblem complexity is not considered. In addition, previousmodels analyze either a static or steady state equilibrium ofsupply chains considering the minimization of average cost withthe infinite planning horizon. However, the nature of the problemis dynamic and the planning horizon is finite. The consideringproblem is dynamic since the true demand is realized in the post-decoupling point, thus the operations (e.g., production rate orinventory level) may be adjusted using the available demandinformation. Today it is widely accepted that the product marketundergo a life cycle of introduction, growth, maturity and even-tual decline not entering toward a stationary state. Therefore theassumption of the finite planning horizon is more realistic in realworld scenario.

In this paper, we consider a problem of determining theposition of the decoupling point and the production planning

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Fig. 1. Candidate positions of a decoupling point.

I.-J. Jeong / Int. J. Production Economics 131 (2011) 561–567562

and inventory strategy simultaneously under different planningpolicies. The proposed model is a continuous and dynamic modelsuch that the objective function is the minimization of the totalcost of the deviation from the target production rate and thetarget inventory level. The optimal position of the decouplingpoint, the optimal path of the production rate and the inventorylevel are the decision variables. The problem is solved usingoptimal control theory. Also we will examine the effect of theaccuracy of the forecasting error in pre-decoupling point, theeffect of the assumption of the finite planning horizon and thesensitivity analysis of the problem parameters.

The paper is organized as follows: Section 2 reviews therelevant literature. In Section 3, we describe the consideringproblem and propose dynamic models. The closed form ofoptimal solutions are presented when target parameters are givenin Section 4. We present some theoretical results for the produc-tion smoothing policy in Section 5. Section 6 represents results forthe zero-inventory policy. Finally some concluding remarks aregiven in Section 7.

2. Literature survey

For strategic positioning of a decoupling point, Olhager (2003)proposed a P/D (production lead time/delivery lead time) ratioand the relative demand volatility. MTO corresponds to a P/D ratiothat is less than one and MTS is a viable option for low relativedemand volatility. For low relative demand volatility, we mayemploy an MTS policy for the economies of scale even in therange of P/Do1. Naylor et al. (1999) combined the lean thinkingand agile manufacturing paradigms with the positioning of thedecoupling point. Authors insisted that the lean paradigm (agileparadigm) can be applied to the pre-decoupling point (post-decoupling point). Kundu et al. (2008) proposed a knowledge-based approach in determining the position of the decouplingpoint. They considered the tradeoff between the physical effi-ciency of the supply chain and the market responsiveness. Theresearch suggested that as the order volume and the product lifecycle increase and the unit transportation cost decreases, thedecoupling point must be shifted backward to the upstream. VanDonk (2001) proposed a guideline for choosing the position of thedecoupling point for food processing industries.

Yanez et al. (2009) proposed a simulation platform built on anagent-based planning system in order to determine an appro-priate position of the decoupling point in the timber industry. Theperformances of the different decoupling points are measuredusing the daily average inventory WIP and the weighted fill rateof demands. Jammernegg and Reiner (2007) considered thetradeoff between the inventory reduction and the increase inorder fulfillment cycle time. They studied a three-stage suppliernetwork using a stochastic simulation. Another type of simulationis based on generalized stochastic Petri nets by Viswanadham andRaghavan (2000). The objective of the Petri nets model is theminimization of the sum of inventory carrying cost and thedelayed delivery cost.

For the mathematical model-based approaches, Gupta andBenjaafar (2004) considered a model such that the objectivefunction is the minimization of the sum of inventory holding costand the product/process redesign cost subject to a service-levelconstraint. The problem is solved by the queuing theory. Sun et al.

(2008) considered the problem of positioning multiple decouplingpoints based on the bill of material of a product in a supplynetwork. They proposed a mathematical model with the mini-mization of supply chain cost which is the sum of the setup,inventory, stock-out and asset specificity cost subject to thedelivery time constraint required by the customer. The model is0–1 integer programming which can be solved by a commercialsoftware package.

Soman et al. (2004) pointed out that the decoupling pointdecision is closely related to the production planning, inventorypolicy and operational decisions. In a hybrid MTO–MTS system,important questions are as follows: the capacity allocation amongMTO and MTS products, the determination of the safety stock forMTS products, the order acceptance/rejection decision for MTOproducts and the scheduling and sequencing of products. How-ever the handling of the whole issues along with the problem ofthe positioning the decoupling point is very complex. Thus Somanet al. (2004) proposed a hierarchical planning framework for foodproduction system.

In this paper, we attack the problem of the determining theposition of the decoupling point, the production planning andinventory strategy simultaneously. It is also seen that the pro-posed solution is based on the mathematical model that can beoptimally solved using the optimal control theory. In addition, weaddress the issue of the effect of the forecasting error and thelength of product life cycle on the performance measure.

3. Problem description

Notations:

T decoupling point in a supply chainP0 estimated demand rate (i.e., production rate) during the

pre-decoupling pointP(t) production rate at time t during the post-

decoupling pointT0 length of product life cycleF(t) cumulative customer demands at time t where F(T0)¼N

D(t)¼F0(t) demand rate at time t

O constant target production rate during post-decoupling point

I(t) inventory level at time t during post-decoupling pointy constant target inventory level during post-

decoupling pointK constant cost per unit deviation from target

production rateh constant cost per unit deviation from target

inventory level

Consider time t over a finite product life cycle of a singleproduct. At t¼0, the production planner estimates the demandrate of the product, P0. The production starts with the productionrate P(t)¼P0 from the raw material procurement at t¼0 to thedecoupling point at t¼T. At the decoupling point, the truedemand rate D(t) is realized during the product life cycle fromtime t¼T to T+T0. When the production during the pre-decou-pling point is finished, the inventory level would be I(L)¼P0T

where L is the throughput time of the supply chain.

Fig. 2. Inventory level for production smoothing policy.

Fig. 3. Inventory level for zero-inventory policy.

I.-J. Jeong / Int. J. Production Economics 131 (2011) 561–567 563

Without loss of generosity, we can shift the time axis so thatt¼L becomes t¼0. Then the problem transforms to the optimiza-tion during the planning horizon, tA[0,T0] with the initial inven-tory condition, I(0)¼P0T.

The objective of this paper is to provide appropriate produc-tion plan so that it can balance between the frequent change ofproduction rate and the inventory holding cost for the giventarget production rate and the target inventory level. We canclassify the inventory management into two classes, the mini-mizing inventory cost and the maintaining inventory level. If weconsider a time-varying demand, we may develop a precisemathematical model in determining the exact inventory level atany time. Thus in this model, it would be appropriate to considerminimizing inventory cost (e.g., inventory holding cost and back-log cost) even though we have a complicated model. However theexact total inventory (i.e., the integration of inventory levelduring a certain period of time) may not be calculated in a closedform for a general function of inventory level. In this case, we mayachieve the minimization of inventory cost via an indirectapproach. That is, the maintaining of the target inventory levelcan be regarded as the minimization of inventory cost.

The general dynamic model (GEN) can be described as follows:

GENmin

Z T0

0fK PðtÞ�Oð Þ

2þh IðtÞ�yð Þ

2g dt

st: _IðtÞ ¼ PðtÞ�FuðtÞ

The objective of the model is to determine the production rateP(t) such that the deviation from the target production rate O, i.e.,K(P(t)�O)2 is minimized. Also the deviation of the inventory levelI(t) from the target inventory level, y, i.e., h(I(t)�y)2 must beminimized. The objective function is widely used quadraticobjective function of Holt et al. (1960) called HMMS model. Theconstraint implies that the inventory rate is equal to the produc-tion rate minus demand rate. For the proposed GEN model, wehave two main issues, the determination of the target productionrate and the target inventory level and the product life cyclerepresented as F(t).

3.1. Target settings

An important issue in the proposed GEN model is the deter-mination of the target production rate and the target inventorylevel which are highly dependent on the planning policy. Fig. 2shows the cumulative production quantity under the productionsmoothing policy. As shown in the figure, the production smooth-ing policy tries to keep the constant rate of production. Thesmoothed production rate may cause unnecessary inventoryand shortage if the target production rate is not appropriatelydetermined.

Fig. 3 shows the production plan and the inventory level of thezero-inventory policy. The zero-inventory policy is one of themost important philosophies in Toyota Production System (TPS)along with the Just In Time (JIT) production. In order not to incurunnecessary inventory, the production does not start until theinitial inventory is exhausted which was produced during thepre-decoupling stage. When the inventory level drops to zero, theproduction rate follows the exact form of the demand rate whichwill cause no inventory. However, the zero-inventory will causethe continuous changes of production rate in order to meetthe exact customer demands which may be technologicallyimpossible.

One possible setting of the target production rate and thetarget inventory level for production smoothing policy is asfollows:

O¼ P0 and y40 ð1Þ

Since the initial forecast of the demand rate is P0, keeping theinitial production rate during the product life cycle is importantfor the production smoothing policy. Under the productionsmoothing policy, we need safety stocks (i.e., y40) in order toprevent the shortage due to the lack of production if the demandis underestimated.

Another possible setting for zero-inventory policy is as fol-lows:

O¼ FuðtÞ and y¼ 0 ð2Þ

The ideal production rate is the demand rate so that it cannotproduce unnecessary inventory. The target inventory level couldbe zero which is the ideal inventory level.

3.2. Product life cycle

The following is the logistic model which is widely used for thedescription of a product life cycle:

FðtÞ ¼a

ð1þae�btÞþb ð3Þ

where a and b are parameters. a and b are constants such thatF(0)¼0 and F(T0)¼N are satisfied. However it is not easy to solvethe GEN model with the logistic function as a product life cycle.It is relatively easy to plan production in traditional staticinventory problems with a constant demand for example, EOQtype problems. However, in some cases, it is still difficult to solve


the problem even with a constant demand. One example is SingleWarehouse Multiple Retailer (SWMR) problem. Even though thedemand is constant, most studies have restricted their attentionto stationary and nested policies (Schwarz and Schrage, 1975,Graves and Schwarz, 1977; Maxwell and Muckstadt, 1985). Undera stationary and nested policy, the inventory level can be clarifiedand the optimization procedure can be applied.

Also we have the same difficulty in clarifying the inventorylevel when we deal with a dynamic problem. The derivation of theoptimal time path of the production plan and the inventory levelare not easy tasks in a dynamic model even if we have a constantdemand.

Therefore we consider a special model of GEN (SPC) withconstant demand as follows:

SPCmin

Z T0

0fK PðtÞ�Oð Þ

2þh IðtÞ�yð Þ

2gdt

st: _IðtÞ ¼ PðtÞ�D

In this model, we assume the demand rate is constant that isF(t)¼Dt such that F0(t)¼D(t)¼D. For the proposed model, SPC,we can apply optimal control theory. In addition, the SPC modelcan be used as a building block to solve GEN model. The logisticmodel of F(t) can be linearly approximated with constant demandrates for the separate ranges of introduction, growth, maturityand decline stages of the product life cycle.

The proposed model is a dynamic since the decision variablesare the optimal time paths of the production rate and theinventory level. The model assumes that the position of thedecoupling point is a continuous variable. There exist couples ofpapers dealing with the application of dynamic optimization tothe production–inventory models. Choi and Hwang (1986) stu-died the quadratic penalty between the actual production rate-inventory level and target production rate-inventory level for thedecaying item. Khemlnitsky and Gerchak (2002) considered theinventory holding cost, the shortage cost and the production costfor inventory-dependent demands. Benhadid et al. (2008)extended the quadratic penalty of production and inventory to atime-varying penalty costs.

4. Optimal solution for SPC model for given target parameters

In this section, we derive the optimal time path of theproduction rate, inventory level and the optimal position of thedecoupling point in closed forms for SPC model when the targetproduction rate and the target inventory level are given.

Using Pontryagin’s maximum principle of the optimal controltheory, the optimality condition for the inventory level (i.e., astate variable) is as follows:

_lðtÞ ¼ 2h IðtÞ�yð Þ ð4Þ

The optimality condition for the production rate (i.e., a controlvariable) is as follows:

lðtÞ ¼ 2KðPðtÞ�OÞ ð5Þ

The first derivative of (5) on l(t) is

_lðtÞ ¼ 2K _PðtÞ ð6Þ

From Eqs. (4) and (6),

IðtÞ ¼K

h_PðtÞþy ð7Þ

The first derivative of (7) on t is

_IðtÞ ¼K

h€PðtÞ ð8Þ

From the constraint of the model and (8),

K

h€PðtÞ ¼ PðtÞ�D ð9Þ

By solving the second order differential Eq. (9), the generalsolution of the production path can be found as follows:

PðtÞ ¼ c1effiffiffiffiffiffiffiffiffiðh=KÞp

tþc2e�ffiffiffiffiffiffiffiffiffiðh=KÞp

tþD ð10Þ

From (7), the inventory level can be determined as follows:

IðtÞ ¼

ffiffiffiffiK

h

rc1e

ffiffiffiffiffiffiffiffiffiðh=KÞp

t�c2e�ffiffiffiffiffiffiffiffiffiðh=KÞp

t� �

þy ð11Þ

The boundary conditions are as follows:

Pð0Þ ¼ P0 ð12Þ

Ið0Þ ¼ P0T ð13Þ

Eq. (12) means that the planning horizon start with the produc-tion rate with the forecasted demand rate P0. Eq. (13) implies thatthe initial inventory level at the beginning of planning horizon isequal to the production quantity at the decoupling point T withthe estimated demand rate P0.

Replacing (12) and (13) with (10) and (11), we obtain thefollowing equations:

c1þc2 ¼ P0�D ð14Þ

ffiffiffiffiK

h

rc1�c2ð Þ ¼ P0T�y ð15Þ

By solving Eqs. (14) and (15), we can calculate c1 and c2 asfollows:

c1 ¼1

2P0�D�y

ffiffiffiffiK

h

r !þ

1

2

ffiffiffiffiK

h

rP0T ð16Þ

c2 ¼1

2P0�Dþy

ffiffiffiffiK

h

r !�

1

2

ffiffiffiffiK

h

rP0T ð17Þ

Let a1 ¼ ð1=2ÞðP0�D�yffiffiffiffiffiffiffiffiffiffiffiffiðK=hÞ

pÞ, a2 ¼ ð1=2ÞðP0�Dþy

ffiffiffiffiffiffiffiffiffiffiffiffiðK=hÞ

pÞ

and b¼ ð1=2ÞffiffiffiffiffiffiffiffiffiffiffiffiðK=hÞ

pP0 then the optimal production path and

inventory level are as follows:

PðtÞ ¼ a1þbTð Þeffiffiffiffiffiffiffiffiffiðh=KÞp

tþ a2�bTð Þe�ffiffiffiffiffiffiffiffiffiðh=KÞp

tþD ð18Þ

IðtÞ ¼

ffiffiffiffiK

h

ra1þbTð Þe


t� a2�bTð Þe�ffiffiffiffiffiffiffiffiffiðh=KÞp

t� �

þy ð19Þ

Let f be the objective function value of SPC model then f can beformalized as follows:

f ¼ K

Z T0

0ða1þbTÞe


tþða2�bTÞe�ffiffiffiffiffiffiffiffiffiðh=KÞp

t�ðO�DÞ� �2�

þ ða1þbTÞeffiffiffiffiffiffiffiffiffiðh=KÞp

t� a2�bTð Þe�ffiffiffiffiffiffiffiffiffiðh=KÞp

t� �2

�dt

¼ K

ffiffiffiffiK

h

rða1þbTÞ2 e2


T0�1� �

�ða2�bTÞ2 e�2ffiffiffiffiffiffiffiffiffiðh=KÞp

T0�1� �n

þðO�DÞ2T0 �2ðO�DÞða1þbTÞ effiffiffiffiffiffiffiffiffiðh=KÞp

T0�1� �

þ2ðO�DÞða2�bTÞ e�ffiffiffiffiffiffiffiffiffiðh=KÞp

T0�1� �o

ð20Þ

The optimal decoupling point can be found using the firstderivative of (20) on T as follows:

@f

@T¼ 2

ffiffiffiffiffiffiKhp

b a1e2ffiffiffiffiffiffiffiffiffiðh=KÞp

T0þa2e�2ffiffiffiffiffiffiffiffiffiðh=KÞp

T0

� �nþb e2


T0�e�2ffiffiffiffiffiffiffiffiffiðh=KÞp

T0

� �T

�ðO�DÞ effiffiffiffiffiffiffiffiffiðh=KÞp

T0þe�ffiffiffiffiffiffiffiffiffiðh=KÞp

T0�2� �

�ða1þa2Þ

o


T� ¼ðO�DÞ e


T0þe�ffiffiffiffiffiffiffiffiffiðh=KÞp

T0�1� �

� a1e2ffiffiffiffiffiffiffiffiffiðh=KÞp

T0þa2e�2ffiffiffiffiffiffiffiffiffiðh=KÞp

T0

� �n ob e2



T0

� �ð21Þ

Theorem 1. Tn satisfying (21) is global optimal if and only if h/Ka0.

Proof. The following inequality holds if and only if h/Ka0 sincethe planning horizon T0 is strict positive:

@2f

@T2¼ 2

ffiffiffiffiffiffiKhp

b2 e2ffiffiffiffiffiffiffiffiffiðh=KÞp


T0

� �40 &

4.1. Effect of the length of product life cycle, T0

In SPC model, we assume the finite length of product life cycle.For infinite length of product life cycle, the following theoremholds:

Theorem 2. When T0-N, the optimal Tn is as follows:

T� ¼y� P0�Dð Þ


p� �P0

ð22Þ

Proof. In Eq. (21), divide the numerator and the denominator bye2


T0 then

T� ¼O�Dð Þ e�


T0þe�3ffiffiffiffiffiffiffiffiffiðh=KÞp


T0

� �� a1þa2e�4


T0

� �n ob 1�e�4


T0

� �ð23Þ

Take T0-N in Eq. (23) then Tn¼(�a1/b) which is equivalent to

Eq. (22). &

Theorem 2 implies when T0-N, the optimal decoupling pointis determined without the consideration of the targetproduction rate.

5. Optimal solution for SPC model under the productionsmoothing policy

In this section, we derive optimal decoupling point andproduction plan of SPC under the production smoothing policy.

Under the production smoothing policy, O¼P0 and the produc-tion rate must be maintained equal to the target production rateduring the whole planning, that is P(t)¼P0. Thus the penalty term forthe deviation from the target production rate is eliminated from theSPC model and a new dynamic model can be developed as follows:

min

Z T0

0h IðtÞ�yð Þ

2 dt

st: _IðtÞ ¼ PðtÞ�D

Using the same logic through (4)–(13), the optimal inventorylevel can be determined as follows:

IðtÞ ¼ ðP0�DÞtþP0T ð24Þ

Then the objective function value is calculated as follows:

f ¼h

3 P0�Dð ÞP0�Dð ÞþP0T�yð Þ

3� P0T�yð Þ

3n o

The first derivate of f on T becomes

@f

@T¼ P0hT0 P0�Dð ÞT0þ2 P0T�yð Þ

� �The optimal the coupling point can be determined as follows:

T� ¼2y� P0�Dð ÞT0ð Þ

2P0ð25Þ

Theorem 3. Under the production smoothing policy, Tn satisfy-

ing (25) is global optimal if and only if P04D.

Proof. ð@2f=@T2Þ ¼ 2P0T0 P0�Dð Þ40 if and only if P04D. &

Theorem 3 implies that the condition of the existence of theoptimal decoupling point under the production smoothing policyis the overestimation of the demand rate, P0 during the pre-decoupling stage.

6. Optimal solution for SPC model under zero-inventorypolicy

In this section, we derive the optimal time path of theproduction rate, inventory level and the optimal position of thedecoupling point in closed forms under zero-inventory strategy,that is O¼F0(t)¼D and y¼0.

Theorem 4. Under the zero-inventory policy, Tn exists if and only if

P0oD.

Proof. Using (21), O¼F0(t)¼D and y¼0, the optimal decouplingpoint can be derived from (21) as follows:

T� ¼�ð1=2Þ P0�Dð Þ e2


T0þe�2ffiffiffiffiffiffiffiffiffiðh=KÞp

T0

� �ffiffiffiKh

qP0 e2



T0

� � & ð26Þ

Contrary to the implication of Theorem 3, Theorem 4 impliesthat the optimal decoupling point exists under zero-inventorypolicy when the demand rate is underestimated during the pre-decoupling stage that is P0oD.

6.1. Behavior of the optimal P(t) and I(t)

Using (21), O¼F0(t)¼D and y¼0, we can calculate (a1+bTn)and (a2�bTn) as follows:

a1þbT� ¼� P0�Dð Þe�2


T0

e2ffiffiffiffiffiffiffiffiffiðh=KÞp


T0

� � ð27Þ

a2�bT� ¼P0�Dð Þe2


T0

e2ffiffiffiffiffiffiffiffiffiðh=KÞp


T0

� � ð28Þ

The first and the second derivative of P(t) and I(t) for optimalTn in (21) can be calculated as follows:

@PðtÞ

@t

T ¼ T�

¼

ffiffiffiffih

K

ra1þbT�ð Þe


t� a2�bT�ð Þe�ffiffiffiffiffiffiffiffiffiðh=KÞp

tn o

ð29Þ

@2PðtÞ

@t2

T ¼ T�

¼h

Ka1þbT�ð Þe


tþ a2�bT�ð Þe�ffiffiffiffiffiffiffiffiffiðh=KÞp

tn o

ð30Þ

@IðtÞ

@t

T ¼ T�

¼ a1þbT�ð Þeffiffiffiffiffiffiffiffiffiðh=KÞp

tþ a2�bT�ð Þe�ffiffiffiffiffiffiffiffiffiðh=KÞp

tn o

ð31Þ

@2IðtÞ

@t2

T ¼ T�

¼

ffiffiffiffih

K

ra1þbT�ð Þe


t� a2�bT�ð Þe�ffiffiffiffiffiffiffiffiffiðh=KÞp

tn o

ð32Þ

Because of Theorem 4, P0oD. In this case, P(t) is an increasingfunction since (31) is strict positive and I(t) is a convex functionsince (32) is strict positive. These results imply if we under-estimate demand during the pre-decoupling point, then theoptimal production rate increases and the corresponding optimalinventory level forms a convex function during the post-decou-pling point.

Fig. 4. Effect of e on PI.


6.2. Effect of the length of planning horizon, T0

For infinite length of the product life cycle under zero-inventory policy, the optimal decoupling point can be found fromthe Theorem 1 with y¼0 as follows:

T� ¼� P0�Dð Þ


pP0

ð33Þ

The results also imply that the optimal decoupling point existsunder zero-inventory policy when the demand rate is under-estimated during the pre-decoupling stage.

6.3. Effect of forecast error

For the infinite length of the product life cycle under zero-inventory policy, when we take T0-N for Eqs. (27) and (28),following equations hold:

a1þbT� ¼ 0 ð34Þ

a2�bT� ¼ P0�D ð35Þ

The optimal objective function value in equation (20) can bemodified as follows:

f ¼ K

Z 10

2 P0�Dð Þ2e�2


t dt¼ �K

ffiffiffiffiK

h

rP0�Dð Þ

2e�2ffiffiffiffiffiffiffiffiffiðh=KÞp

t

" #10

¼ K

ffiffiffiffiK

h

rP0�Dð Þ

2ð36Þ

Since the optimal decoupling point exist when P0oD underzero-inventory policy, we say that the forecast error is a ifP0¼(1+a)D where �1rar0.

Theorem 5. For the infinite length of the product life cycle under

zero-inventory policy, if the forecast error is improved from a to a0where �1rara0r0 then the percent improvement of the objec-

tive function value is as follows:

a2�ðauÞ2

ðauÞ2

!� 100 ð37Þ

Proof. Let f(a) and f(a0) be the objective function value when theforecast error is a and a0, respectively. Then f ðaÞ ¼ K


pðaDÞ2

and f ðauÞ ¼ KffiffiffiffiffiffiffiffiffiffiffiffiðK=hÞ

pðauDÞ2 due to (36). Therefore the percent

improvement (PI) of the objective function value by the improvedforecast error is PI¼ ðf ðaÞ�f ðauÞÞ=ðf ðauÞÞ � 100¼ ðða2�ðauÞ2Þ=ðauÞ2Þ�100: &

Let a0 ¼ea where e�100(0rer1) is the PI of forecast errorfrom a. Fig. 4 shows the behavior of PI on e. The Figure impliesthat the reduction of forecast error exponentially improves theobjective function value of the model.

7. Conclusion

In this paper, we proposed a dynamic model to simultaneouslydetermine the optimal position of the decoupling point andproduction–inventory plan. We considered a special case ofconstant demand rate of a product life cycle. Using the optimalcontrol theory, we derived the closed form of the optimal solutionunder the production smoothing policy and the zero-inventorypolicy.

The analysis showed that under the production smoothingpolicy, the overestimation of demand rate guarantees the exis-tence of the optimal decoupling point; meanwhile the optimaldecoupling point exists under zero-inventory policy when thedemand rate is underestimated during the pre-decoupling stage.Also we performed analysis on the behavior of the optimalproduction rate and the inventory level and the effect of problemparameters such as the length of the product life cycle and theforecast error on the performance.

It is possible to extend the results of this research in severaldifferent directions. For instance, we can consider the applicationof logistic model to represent the product life cycle for GENproblem. In this case, we believe that a heuristic approach such asa linear approximation of the logistic model may be required.Another issue is the development of a tradeoff policy between theproduction smoothing policy and the zero-inventory policy forSPC problem. The determination of an appropriate target produc-tion rate and a target inventory level is an important issue.Finally, the inclusion of more realistic assumption is necessarysuch as the time-varying penalty costs and the delivery lead timeto customer.

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A dynamic model for the optimization of decoupling point and production planning in a supply chain