A continuous review inventory model with order expediting

Int. J. Production Economics 87 (2004) 157–169

A continuous review inventory model with order expediting

Alfonso Dur!an*, Gil Guti!errez, R !omulo I. Zequeira!Area de Ingenier!ıa de Organizaci !on, Departamento de Ingenier!ıa Mec !anica, Universidad Carlos III de Madrid,

Ave. Universidad 30, Legan!es 28911, Madrid, Spain

Received 28 July 2002; accepted 19 March 2003

Abstract

In this paper we consider a continuous review inventory system in which the lead time L includes a part of fixed

duration, T ; and a part whose duration can be either Ta or Te; where Ta > Te: When the net inventory reaches the

reorder point r; an order of size Q is released. If the net inventory at time T after the order release is equal to or smaller

than re; the order is expedited at a cost comprising a fixed cost and a cost per unit and the lead time is L ¼ T þ Te;otherwise the lead time is L ¼ T þ Ta: Therefore, the decision variables considered are the order quantity Q; the reorderpoint r and the order expediting point re: Our aim is to find the inventory policy variables Q; r and re that minimize the

average cost rate. We present an algorithm to obtain the policy variables with global minimal costs when the inventory

policy decision variables are integers. We also discuss the case in which the decision variables are real valued.

r 2003 Elsevier Science B.V. All rights reserved.

Keywords: Continuous review; Lead-time; Emergency order; Order expediting; Reorder-point policy

1. Introduction

When lead times are not negligible, randomdemand during lead time might lead to shortage.If shortage implies high costs, stockouts can bereduced by issuing emergency orders.

Inventory models allowing for emergency ordershave been developed under various assumptions.Moinzadeh and Nahmias (1988) and Tagaras andVlachos (2001) have performed a thorough biblio-graphic review of these models. Moinzadeh andNahmias (1988) present a continuous reviewinventory model to find the optimal reorder point

and order quantity for the normal and emergencyreplenishments. This model was analyzed from adifferent perspective by Johansen and Thorstenson(1998). They present an inventory model withnormal and emergency orders where normalorders are managed through a ðQ; rÞ policy whileemergency orders are controlled by a reorder pointand an order-up-to level which depend on the timeremaining until the normal order is received.Vlachos and Tagaras (2001) analyse a periodicreview inventory system with a main and anemergency supply mode.

The issuance of an emergency order in a cycleleads to a smaller lead time for the first delivery inthe cycle. This effect can also be achieved byexpediting the normal order so that it is deliveredearlier. Lawson and Porteus (2000) consider order

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*Corresponding author. Tel.: +34-916-249-921; fax:+34-

916-249-430.

E-mail address: [email protected] (A. Dur!an).

0925-5273/03/$ - see front matter r 2003 Elsevier Science B.V. All rights reserved.

doi:10.1016/S0925-5273(03)00100-2

expediting in a multistage inventory model, inwhich the delivery of expedited items is madeinstantaneously. Minner (2003) reviews inventorymodels with multiple supply options and presentsrelated inventory problems from the field of multi-echelon systems.

Lead time usually comprises components suchas order preparation, order delivery, manufactur-ing and transportation (Tersine, 1994). In somecases, options exist for reducing the duration ofsome of these components (see Liao and Shyu,1991; Ryu and Lee, 2003). For example, there arecases in which transportation can be carried out ineither a slower or a faster mode (such as by air andby truck). In most cases achieving the shorter leadtime implies a cost premium.

This paper is based on the Moinzadeh andNahmias (1988) model. However, rather thanconsidering a second (emergency) order we analyzethe option of expediting the delivery of theoutstanding order at an additional cost. Weconsider a continuous review inventory system inwhich the lead time L includes a part of fixedduration, T ; and a part whose duration can beeither Ta or Te; where Ta > Te: When the netinventory reaches the reorder point r; an order ofsize Q is released. If the net inventory at time T

after the order release is equal to or smaller thanre; the order is expedited at a cost comprising afixed cost Ke and a cost per unit ce and the leadtime is L ¼ T þ Te; otherwise the lead time is L ¼T þ Ta: Therefore, the decision variables consid-ered are the order quantity Q; the reorder point r

and the order expediting point re: Our aim is tofind the inventory policy variables Q; r and re thatminimize the average cost rate. We present analgorithm to obtain the policy variables withglobal minimal costs when the inventory policydecision variables are integers. We also discuss thecase in which the decision variables are realvalued.

2. Notation

Q order quantityKa fixed ordering cost per orderKe fixed expediting cost per order

ca acquisition cost per unitce expediting cost per unith inventory holding cost per unit per time

unitp penalty cost per unit shortB expected number of units backordered in a

cyclem demand rate (units demanded per unit of

time)H expected on hand inventory per cycle in

units times time units.Ha expected on hand inventory in cycles

without order expeditingHe expected on hand inventory in cycles with

order expeditingp probability of expediting an orderIðtÞ net inventory at time t after the release of

the normal order of the cycleX ðtÞ demand accumulated up to time t

f ðx; tÞ probability density function of the de-mand x during a time interval of size t

(note that f ðx; tÞ ¼ 0 whenever xo0)

3. The model

We will adapt the model used by Moinzadehand Nahmias (1988) to obtain the average costrate in a system that allows order expediting.Similarly to Hadley and Whitin (1963) we assumethat there is never more than a single orderoutstanding. We will also assume that the demandprocess X ðtÞ; tX0; has independent increments.

If IðTÞpre; i.e., the inventory level at the end ofthe constant part of the lead time is smaller orequal than the order expediting point re; then at anadditional cost the order is expedited, to bedelivered at T þ Te: Otherwise the order isdelivered at T þ Ta: We will state the problem infunction of R ¼ r � re and r instead of in functionof re and r: We will assume rX0 and rXre; thusRX0: The decision of whether to expedite theorder or not can be expressed in terms of R bystating that if the demand during the fixedduration part of the lead time (T) is smaller thanR; then the order is not expedited, otherwise theorder is expedited.

ARTICLE IN PRESS

A. Dur !an et al. / Int. J. Production Economics 87 (2004) 157–169158

The probability of expediting an order is there-fore given by

p ¼PðIðTÞpreÞ

¼PðX ðTÞXr � reÞ

¼Z

N

R

f ðx;TÞ dx:

We will analyze first the expected number ofunits backordered per cycle B: To obtain B we willcondition first on the demand x during the firstcomponent of the lead time, of length T :

BðxÞ ¼

RN

y¼r�x½y � ðr � xÞ� f ðy;TaÞ dy; xoR;R

N

y¼r�x½y � ðr � xÞ� f ðy;TeÞ dy; xXR:

(

Then the expected number of units backorderedin a cycle B is

B ¼Z

N

x¼0

BðxÞf ðx;TÞ dx ¼ Ba þ Be;

where

Ba ¼Z R

x¼0

ZN

y¼r�x

½y � ðr � xÞ� f ðy;TaÞ dy f ðx;TÞ dx:

Be ¼Z

N

x¼R

ZN

y¼r�x

½y � ðr � xÞ� f ðy;TeÞ dy f ðx;TÞ dx:

We will now consider the determination of theexpected inventory in cycles with and withoutexpediting. Like Moinzadeh and Nahmias (1988),we will use Hadley and Whitin (1963, p. 165)approximation for the ðQ;RÞ model. Lau and Lau(2002) present a comparison of different methodsfor estimating the average inventory level in aðQ;RÞ system with backorders.

We will now consider the determination of theexpected inventory in cycles with and withoutexpediting. Like Moinzadeh and Nahmias (1988),we will use Hadley and Whitin (1963, p. 165)approximation for the ðQ;RÞ model.

We need to calculate the average demand rateduring the fixed duration part of the lead time (T)in the cases of expediting (le) and no expediting(la).

le ¼1

TE½X ðTÞjX ðTÞXR�

¼1

pT

ZN

R

xf ðx;TÞ dx;

la ¼1

TE½X ðTÞjX ðTÞoR�

¼1

ð1� pÞT

Z R

0

xf ðx;TÞ dx:

The following equivalence must hold:

ple þ ð1� pÞla ¼ m:

To obtain the expected on hand inventory percycle in the case of no expediting Ha we calculatethe areas under the trapezoids in Fig. 1. Hence weobtain

Ha ¼Q2

2m� QTa þ

Qr

mþ rT þ

1

2ml2aT2

� laTQ þ r

mþ

T

2

� �:

Similarly we find the expected on hand inven-tory per cycle in the order expediting case:

He ¼Q2

2m� QTe þ

Qr

mþ rT þ

1

2ml2eT2

� leTQ þ r

mþ

T

2

� �:

Then the expected on hand inventory per cycle is

H ¼ pHe þ ð1� pÞHa

¼Q2

2m� Q Ta þ pðTe � TaÞ �

r

mþ T

� �þ W ;

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Fig. 1. Expected inventory levels in cycles without order

expediting.

A. Dur !an et al. / Int. J. Production Economics 87 (2004) 157–169 159

where

W ¼T2

2m½l2a þ pðl2e � l2aÞ� �

mT2

2:

The expected length of the cycle is

Tt ¼T þ ð1� pÞ Ta þ1

m½Q � ðlaT þ mTaÞ�

� �

þ p Te þ1

m½Q � ðleT þ mTeÞ�

� �¼

Q

m:

The expected cost in a cycle TCðQ; r;RÞ is givenby the sum of the normal ordering cost Ka; the costof the normal order caQ; the expected cost ofexpediting pðKe þ ceQÞ; the expected inventorycarrying cost hH and the expected cost of back-ordering pB:

TCðQ; r;RÞ ¼ Ka þ caQ þ pðKe þ ceQÞ þ hH þ pB:

Then we should minimize the average cost rate(ECðQ; r;RÞ), where

ECðQ; r;RÞ ¼Ka

mQþ camþ pKe

mQþ pcem

þ hHmQþ pB

mQ:

We will consider first the partial derivative ofthe average cost rate with respect to Q:

@

@QECðQ; r;RÞ ¼ �

1

Q2mV þ

1

2h �

1

Q2mhW ;

where

V ¼ Ka þ pKe þ pB:

It can be seen that WX0 always holds. HenceV þ hWX0 since VX0 and hX0 and therefore theaverage cost rate is convex with respect to Q andthere is an optimal solution Qn for Q for fixed r

and R; which is given by the following equation:

Qn ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2m

V

hþ W

� �s: ð1Þ

We will now consider the derivatives withrespect to r:

@ECðQ; r;RÞ@r

¼ hmQ

@H

@rþ p

mQ

@Ba

@rþ

@Be

@r

� �:

Since

@H

@r¼

Q

m;

then

@ECðQ; r;RÞ@r

¼ h þ pmQ

@Ba

@rþ

@Be

@r

� �;

where

@Ba

@r¼ �

Z R

0

ZN

r�x

f ðy;TaÞ dy f ðx;TÞ dx;

@Be

@r¼ �

ZN

R

ZN

r�x

f ðy;TeÞ dy f ðx;TÞ dx:

Note that

@2Ba

@r2¼

Z R

0

f ðr � x;TaÞf ðx;TÞ dxX0;

@2Be

@r2¼

ZN

R

f ðr � x;TeÞf ðx;TÞ dxX0:

Then for fixed Q and R the problem is convex withrespect to r:

We can find, for fixed R and Q; the optimalvalue of r by solving the following equation:

h � pmQ

Z R

0

ZN

r�x

f ðy;TaÞ dy f ðx;TÞ dx

�

þZ

N

R

ZN

r�x

f ðy;TeÞ dy f ðx;TÞ dx

�¼ 0: ð2Þ

We will now consider the derivatives withrespect to R:

@ECðQ; r;RÞ@R

¼ mKe

Qþ ce

� �@p

@Rþ h

mQ

@H

@R

þ pmQ

@Ba

@Rþ

@Be

@R

�;

where

@H

@R¼ f ðR;TÞf�QðTa � TeÞ

þT

2mðle � laÞ½leT þ laT � 2R�g;

@Ba

@R¼

ZN

r�R

½x � ðr � RÞ� f ðx;TaÞ dx f ðR;TÞ;

@Be

@R¼ �

ZN

r�R

½x � ðr � RÞ� f ðx;TeÞ dx f ðR;TÞ:

ARTICLE IN PRESS


The first order condition for R to be optimal forfixed Q and r is given by the following equation:

�Ke

Qþ ce

� �þ

h

Q�QðTa � TeÞ þ

T

2mðle � laÞ

�

� ðleT þ laT � 2RÞ�þ

pQ

ZN

r�R

½x � ðr � RÞ�

� ½f ðx;TaÞ � f ðx;TeÞ� dx ¼ 0: ð3Þ

Note that for the cases R ¼ 0 and R-N weobtain, including a component for the backorder-ing cost, the EOQ model with lead time equal toT þ Te and T þ Ta; respectively. When R ¼ 0 weobtain p ¼ 1; le ¼ m; la ¼ 0; W ¼ 0 and B ¼ Be:When R-N we obtain p-0; le-N; la-m andB-Ba: When R ¼ N we have W ¼ 0:

It is easy to find a set of data for which theaverage cost rate for fixed r and Q is not convexwith respect to R; as indicated in the numericalexample. Our computational experience with themodel suggests that generally there is at most oneroot of Eq. (3) for fixed r and Q and that if we fix r

and Q the average cost rate attains its globalminimum at this root. It seems difficult todemonstrate this. For that reason we propose tolook for the optimal R in a set of integer values forR; from 0 to a value Rmax: Since the probability ofexpediting decreases with R; a natural way ofestablishing Rmax is to find the value of R forwhich the probability of expediting results negli-gible. That is, Rmax is the maximum integer valueR for which the following condition holds:Z

N

R

f ðx;TÞ dx Xg; ð4Þ

where g is the minimum expediting probability tobe considered in the model. Note that for practicalpurposes g can be chosen as a small value not veryfar from zero, for example 0.001.

We propose the following algorithm to findvalues of policy variables that are optimal or give afirst approximation to the optimal policy. Let theoptimal values of r and Q for a given R be denotedby rR and QR; respectively.

1. Set a value for g and using Eq. (4) find thecorresponding value for Rmax; restricted topositive integers.

2. For R ¼ 0; 1; :::;Rmax repeat steps 3 to 7.3. Set an initial value for Q (using for example

Eq. (1) with V ¼ Ka and W ¼ 0).4. Repeat steps 5 and 6 until no changes occur in

Q and r:5. Find r from Eq. (2).6. Find Q from Eq. (1).7. Set QR ¼ Q and rR ¼ r:8. Find Rn such that ECðQRn ; rRn ;RnÞ ¼

minRECðQR; rR;RÞ; then QRn ; rRn ;Rn is thesolution of the algorithm.

We now discuss the optimality of the solution ofthe algorithm.

One case of practical interest is when the policyvariables Q; r and R are restricted to integers. Forexample this is the case when the demand isgenerated by a Poisson process. In that case theprevious algorithm renders the optimal policy. Ifthe policy decision variables are not restricted tobe integers, then we can use the previous algorithmrestricted to integer values of R to give a firstapproximation of the optimal policy. Anotherapproach could be to use Eq. (3) to iterate with thefirst order conditions for r and Q to find the globaloptimum, based on our conjecture (validated bycomputational experiments) that there is at mostone zero of the first order condition for R:

4. Validation of the model and numerical examples

The objective of this section is to validate ourmodel by simulation and by comparison with thesolutions of the algorithm of Federgruen andZheng (1992) for the ðr;QÞ model. We will alsonumerically analyze the behavior of the modelwith respect to variations in its parameters. In thissection we will set g ¼ 0:001:

We will consider in this numerical examplePoisson demand. As in Johansen and Thorstenson(1998), to simulate the inventory system we haveused the regenerative method. To evaluate the costof combinations of the decision variables we havesimulated 10 000 cycles, where a cycle is defined asthe time period between two consecutive instantsin which the net inventory falls to the reorderpoint. We have assumed that there can be more

ARTICLE IN PRESS


than one order outstanding nevertheless for all thesimulations that we carried out the probability ofmore than one order outstanding was negligible.We have computed the average cost rate and itsstandard deviation using the classical approach(Law and Kelton, 1991).

We will consider a set of data similar toMoinzadeh and Nahmias (1988) for the demandrate, normal lead time and normal ordering,holding and backordering costs. It can be expectedthat the expediting costs Ke and ce in cases whereour model is applicable would not be larger thanthe expediting costs when the model of Moinzadehand Nahmias (1988) is applicable. That could, forexample, be the case if T is the manufacturingcomponent of the lead time and Ta or Te is thetransportation component of the lead time. There-fore, in our numerical examples we will considerexpediting costs that are smaller than the corre-sponding expediting costs in Moinzadeh andNahmias (1988).

From the first derivatives of the average costrate with respect to the policy variables Q; r and R

it can be seen that the optimal policy does notdepend on ca: Hence in this section we considerca ¼ 0:

As it can be seen in the graphical representationof the Poisson demand, if we assume that the onhand inventory decreases linearly we underesti-mate the holding cost in the case of Poissondemand. We have corrected this bias in allcalculations of the holding cost in the numericalexamples by adding the holding cost of half a unitto the holding cost obtained by assuming a lineardecrease of the on hand inventory. We havetested this correction with different values of theholding costs and it always improves the inven-tory holding cost approximation. It should benoted that, since the correction is constant, notintroducing it would not affect the stated aim ofthe model, which is to determine the set ofpolicy variables that minimize the average costrate.

Tables 1 and 2 present the optimal policyvariables r and Q for integer values of R forp ¼ 100 and p ¼ 500; respectively. Since T ¼ 0:5;for g ¼ 0:001 Rmax ¼ 9: We have also included thevalue R ¼ N: In Tables 1 and 2 we present theresults of the simulation, i.e. average cost rate andits standard deviation, for each set of policyvariables r; Q and R: Additionally, for the extremecases of expediting in all cycles or never expediting

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Table 1

Comparison of analytical and simulated results

Data m 5 T 0.5

Ka 25 Ta 0.5

Ke 5 Te 0.1

ce 0.2 h 1

p 100

F&Ze F&Zne

R 0 0 1 2 3 4 5 6 7 8 9 N N

p for that R 1.000 1.000 0.918 0.713 0.456 0.242 0.109 0.042 0.014 0.004 0.001 0 0

Optimal r 6 6 6 7 7 8 8 9 9 9 9 9 9

Optimal Q 19 19 19 18 18 17 17 17 17 17 17 17 17

Model average cost rate 23.23 23.23 22.93 22.21 21.54 21.22 21.28 21.36 21.56 21.76 21.88 21.94 21.94

Cost relative reduction �5.88 �4.51 �1.23 1.82 3.28 3.01 2.64 1.73 0.82 0.27 0 0

Simulation av. cost rate 23.15 22.95 22.18 21.48 21.16 21.15 21.24 21.61 21.72 22.03 21.83

Standard deviation 0.08 0.09 0.06 0.08 0.06 0.08 0.06 0.08 0.09 0.10 0.10

Percent difference 0.35 �0.09 0.14 0.28 0.28 0.61 0.56 �0.23 0.18 �0.68 0.50

F&Ze: optimal policy if all orders are expedited, according to the algorithm of Federgruen and Zheng (1992). F&Zne: optimal policy if

no orders are expedited, according to the algorithm of Federgruen and Zheng (1992). Cost relative reduction: reduction in average cost

rate with respect to Z&Fne. Standard deviation: simulation average cost rate standard deviation. Percent difference: difference between

model and simulation average cost rates, as percentage of the simulation average cost rate.


we present the results of the algorithm ofFedergruen and Zheng (1992). Note that, as itcould be expected, when the probability p ofexpediting in a cycle increases, the optimalreorder point decreases. Note that, in general,the difference between the model average costrate and the simulated average cost rate is less than1% of the simulated average cost rate, thusindicating that the model is accurate. Positiveand negative differences occur in comparableproportions, thus not suggesting the existence ofa bias in the cost model. It is worth highlightingthat the algorithm of Federgruen and Zheng(1992) leads to the same policy variables andaverage cost rates for the cases R ¼ 0 and R ¼ N:For p ¼ 100 the optimal policy variablesare R ¼ 4; r ¼ 8 and Q ¼ 17: That implies aprobability of expediting in a cycle of 0.242 anda reduction of 3.28% in the average cost ratewith respect to the optimal policy with R ¼ N:As it could be expected, for p ¼ 500 the percentageof reduction increases, to 4.36%. Neverthelessthe optimal probability of expediting in a cycledecreases with respect to the case p ¼ 100 becausethe optimal reorder point increases in twounits.

Our numerical analysis shows that for r ¼ 10and Q ¼ 17 and the data of Table 2, the averagecost rate is not convex with respect to R:

Table 3 contains the optimal inventory policyfor different values of Ke and ce: As it could beexpected, when the expediting costs increase, thepercentage reduction in the average cost rate incomparison with the policy of never expeditingdecreases. Table 3 also shows that, for a givenvalue of ce; when Ke increases the optimal R

increases or does not change.In the rest of this section we will study the

behavior of our model with respect to variations inthe lead time components T ; Te and Ta: Weconsider first the behavior of the model withrespect to the variation of Te; while the other leadtime components are held constant. Note thatwhen Te decreases the expediting option becomesincreasingly attractive. This behavior can be seenin Table 4. Nevertheless, for this set of data, giventhe significant expediting costs, even when Te

becomes very small the expediting option isselected with a probability of only 0.242.

In Table 5 we analyze the dependency of theoptimal policy variables on Ta: For small valuesof Ta the optimal policy implies almost never

ARTICLE IN PRESS

Table 2

Comparison of analytical and simulated results for a different back order cost rate

Data m 5 T 0.5

Ka 25 Ta 0.5

Ke 5 Te 0.1

ce 0.2 h 1

p 500

F&Ze F&Zne

R 0 0 1 2 3 4 5 6 7 8 9 N N

p for that R 1.000 1.000 0.918 0.713 0.456 0.242 0.109 0.042 0.014 0.004 0.001 0 0

Optimal r 8 8 8 8 9 9 10 10 11 11 11 11 11

Optimal Q 18 18 18 18 17 17 17 17 17 17 17 17 17

Model average cost rate 24.57 24.57 24.24 23.52 22.93 22.59 22.57 22.87 23.06 23.26 23.43 23.60 23.60

Cost relative reduction �4.11 �2.71 0.34 2.84 4.28 4.36 3.09 2.29 1.44 0.72 0

Simulation av. cost rate 24.62 24.31 23.47 22.96 22.74 22.55 22.71 23.20 23.11 23.11 23.65

Standard deviation 0.16 0.14 0.13 0.12 0.19 0.15 0.17 0.16 0.14 0.15 0.21

Percent difference �0.20 �0.29 0.21 �0.13 �0.66 0.09 0.70 �0.60 0.65 1.38 �0.21

F&Ze: optimal policy if all orders are expedited, according to the algorithm of Federgruen and Zheng (1992). F&Zne: optimal policy if

no orders are expedited, according to the algorithm of Federgruen and Zheng (1992). Cost relative reduction: reduction in average cost

rate with respect to Z&Fne. Standard deviation: simulation average cost rate standard deviation. Percent difference: difference between

model and simulation average cost rates, as percentage of the average simulation cost rate.


ARTIC

LEIN

PRES

S

Table 3

Optimal inventory policies for different expediting costs

Data m 5 T 0.5

Ka 25 Ta 0.5

h 1 Te 0.1

p 500

Ke 5 10 15

ce 0.0 0.2 0.4 0.6 0.8 1 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

Optimal R 4 5 5 5 5 5 6 5 5 5 5 6 6 6 5 5 5 6 6 6 7

Optimal r 9 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 11

Optimal Q 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17 17

Model average cost rate 22.35 22.57 22.68 22.79 22.90 23.00 23.10 22.62 22.73 22.84 22.95 23.05 23.10 23.14 22.78 22.88 23.00 23.07 23.12 23.16 23.18

Cost relative reduction 5.30 4.36 3.90 3.43 2.97 2.54 2.12 4.15 3.69 3.22 2.75 2.33 2.12 1.95 3.47 3.05 2.54 2.25 2.03 1.86 1.78

Simulation av. cost rate 22.47 22.55 22.76 22.76 23.01 22.99 22.92 22.43 22.65 22.81 22.95 22.83 23.02 23.32 22.63 22.87 22.96 23.29 23.09 23.23 23.32

Standard deviation 0.17 0.15 0.14 0.12 0.17 0.13 0.16 0.11 0.14 0.12 0.13 0.16 0.17 0.22 0.12 0.12 0.15 0.22 0.19 0.2 0.15

Percent difference �0.12 0.02 �0.08 0.03 �0.11 0.01 0.18 0.19 0.08 0.03 0 0.22 0.08 �0.18 0.15 0.01 0.04 �0.22 0.03 �0.07 �0.14

Cost relative reduction: reduction in average cost rate with respect to the optimal policy if no orders are expedited, according to the algorithm of Federgruen and Zheng

(1992). Standard deviation: simulation average cost rate standard deviation. Percent difference: difference between model and simulation cost rates, as percentage of the

simulation cost rate.

A.

Du

r!an

eta

l./

Int.

J.

Pro

du

ction

Eco

no

mics

87

(2

00

4)

15

7–

16

9164

choosing the expediting option, since the value ofTa will not be different enough from Te as tochoose the expediting option frequently. When Ta

increases, the optimal R initially decreases, sincethe larger attainable reduction in lead time Ta � Te

makes expediting more attractive. However, theoptimal reorder point r also increases when Ta

increases, thus making expediting less necessary.In this example, when Ta changes from 0.6 to 0.8, r

increases by two units, from 10 to 12. That surge in

ARTICLE IN PRESS

Table 4

Optimal inventory policies for different values of Te while holding T and Ta constant

Data m 5 h 1

Ka 25 p 500

Ke 5 T 0.5

ce 0.2 Ta 0.5

Second component of the lead time, Te 0.02 0.04 0.06 0.08

Optimal R 4 4 4 5

p for optimal R 0.242 0.242 0.242 0.109

Optimal r 9 9 9 10

Optimal order quantity 17 17 17 17

Model average cost rate 22.54 22.54 22.55 22.57

Cost relative reduction 4.49 4.49 4.45 4.36

Simulation average cost rate 22.61 22.60 22.60 22.54

Simulation standard deviation 0.18 0.18 0.17 0.12

Percent difference �0.31 �0.27 �0.22 0.13

Cost relative reduction: reduction in average cost rate with respect to the optimal policy if no orders are expedited, according to the

algorithm of Federgruen and Zheng (1992). Percent difference: difference between model and simulation average cost rates, as

percentage of the simulation average cost rate.

Table 5

Optimal inventory policies for different values of Ta while holding T and Te constant

Data m 5 h 1

Ka 25 p 500

Ke 5 T 0.5

ce 0.2 Te 0.1

Second component of the lead time, Ta 0.2 0.4 0.6 0.8 1.0 1.2

Optimal R 6 5 4 5 5 0

p for optimal R 0.042 0.109 0.242 0.109 0.109 1.000

Optimal r 8 9 10 12 14 8

Optimal Q 17 17 17 17 17 18

Model average cost rate 22.09 22.23 22.95 23.78 24.50 24.57

Z&Fne 22.55 23.36 23.89 24.55 25.00 25.58

Cost relative reduction 2.02 4.84 3.92 3.13 2.03 3.95

Simulation average cost rate 22.12 22.18 23.2 23.72 24.53 24.48

Simulation standard deviation 0.16 0.15 0.17 0.19 0.17 0.12

Percent difference �0:11 0.23 �1:09 0.26 �0:13 �0:36

F&Zne: optimal policy if no orders are expedited, according to the algorithm of Federgruen and Zheng (1992). Cost relative reduction:

reduction in average cost rate with respect to the optimal policy if no orders are expedited, according to the algorithm of Federgruen

and Zheng (1992). Percent difference: difference between model and simulation average cost rates, as percentage of the simulation

average cost rate.


r more than offsets the impact of Ta on R; thus theoptimal R increases again from 4 to 5.

This example also highlights that, for certainvalues of the parameters, an extreme policy, i.e.never expedite or always expedite, could beoptimal. When Ta ¼ 1:2; the savings in holdingcosts that can be attained by shortening the leadtime through expediting make it optimal toexpedite every order, i.e. R ¼ 0:

These results can be illustrated using Fig. 2,depicting the minimum average cost rate for 11integer values of R ranging from 0 to 10 and 40values of Ta ranging from 0.2 to 1.2. Theexpediting probability for each value of R isshown within brackets. When R ¼ 0 the minimalaverage cost rate does not change with Ta; which isobviously correct since when R ¼ 0 the optimalpolicy does not depend on Ta:

When Ta is small, and therefore potentialsavings in holding costs are also small, the optimalpolicy, represented in the figure by the set ofdecision variables that render the minimal averagecost rate for a given Ta; implies an R > 0: Forlarger values of Ta; higher holding costs pushaverage cost rates upwards, to the point wherealways expediting, R ¼ 0; becomes optimal.

It is worth highlighting, in Fig. 2, howdependent the average cost rate is on the choice

of R when Ta ¼ 1:2; as illustrated by the differencein the average cost rate between R ¼ 0 and R ¼ 2:That can be attributed to how r is chosen by thealgorithm, for a given R; in order to minimize theaverage cost rate. For Ta ¼ 1:2; when R ¼ 0the inventory policy is focused on reducingholding costs by reducing the reorder point r to8, which it can do since lead times will always besmall. When R becomes 1 or 2, r can no longer bekept that low, since that would lead to significantbackordering penalties, thus holding cost savingsare limited; therefore, for R ¼ 1 the optimal r

becomes 10, and for R ¼ 1 it becomes 12.However, the expediting probability, and thereforethe expediting costs, are still high. For values of R

greater than 2, in this example, the reduction inexpediting costs leads to smaller average cost rates.

The shape of Fig. 2 for Ta ¼ 1:2 should not beinterpreted as a counterexample, for the case inwhich the inventory policy decision variables areintegers, of our statement above that ‘‘Ourcomputational experience with the model suggeststhat generally there is at most one root of Eq. (3)for fixed r and Q and that if we fix r and Q theaverage cost rate attains its global minimum at thisroot’’. Indeed, Fig. 2 shows that, for Ta ¼ 1:2;average cost rate first increases with R; thenreaches a maximum, then decreases, then increases

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Fig. 2. Minimum average cost rates for different values of R and Ta:


again. As explained above, however, that is not forconstant r and Q; but rather for the optimal valuesof r and Q for each R: In fact, for the set ofparameters used in Fig. 2, for Ta ¼ 1:2; if r and Q

are held constant and equal to 8 and 18,respectively, the average cost rates that correspondto the integer values of R between 0 and 10 are:24.57, 27.52, 42.16, 74.49, 117.87, 158.23, 186.27,201.51, 208.19, 210.54 and 211.19, respectively.For Ta ¼ 1:0; if r ¼ 14 and Q ¼ 17; the averagecost rates that correspond to the integer values ofR between 0 and 10 are: 29.82, 29.27, 27.89, 26.23,25.00, 24.50, 24.55, 24.84, 25.14, 25.34 and 25.45,respectively. Therefore, for these cases our con-jecture also holds.

We will now analyze the simultaneous variationof several components of the lead time. Table 6shows the optimal policy variables and associatedaverage cost rates for different values of T ; whenboth T þ Ta and the ratio Ta=Te are held constant.The optimal expediting probability (which, in thiscase, is not directly linked to R; since T is notconstant) initially increases with T ; then itdecreases. That highlights the two conditions thatan expediting decision must meet in order tojustify its costs: substantial lead time reduction

and good information to decide which orders toexpedite. Given the existence of expediting costs,expediting an order must, to be worthwhile, causea significant decrease in the lead time. Thatexplains why, for T ¼ 0:6; it is less likely thatit is worthwhile to expedite an order than forT ¼ 0:4; since the attainable lead time reduction issmaller. That is even more the case for T ¼ 0:8: Onthe other hand, the reason why the probability ofexpediting is lower for T ¼ 0:2 than for T ¼ 0:4 isthat, after only 20% of the total lead time haselapsed, there is little information to take thedecision of which orders are more likely to cause astockout and are therefore candidates for expedit-ing. When T increases, the proportion of the totallead time demand that takes place in T (deviationsin which, therefore, are known when the expedit-ing decision is taken) also increases.

Table 7 shows the optimal policy variables andassociated average cost rates for different values ofTa; when both T þ Ta and Te are held constant,for two different values of the expediting costs Ke:When Ta is small, such as when Ta ¼ 0:2; thedifference with Te is small, thus leading to a smallprobability of acceleration, both for Ke ¼ 5 andfor Ke ¼ 15: When Ta increases, for example to

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Table 6

Optimal inventory policies for different values of T while holding T þ Ta and Ta=Te constant

Data m 5 h 1

Ka 25 p 500

Ke 5 T þTa

1

ce 0.2 Ta=Te 5

First component T of the lead time 0.2 0.4 0.6 0.8

Optimal R 4 4 6 8

p for optimal R 0.019 0.143 0.084 0.051

Optimal r 11 10 10 10

Optimal Q 17 17 17 17

Model average cost rate 23.39 22.82 22.43 22.73

Cost relative reduction 0.89 3.31 4.96 3.67

Simulation av. cost rate 23.12 22.83 22.33 22.52

Standard deviation 0.11 0.15 0.12 0.16

Percent difference 1.17 �0.04 0.45 0.93

Cost relative reduction: reduction in cost rate with respect to the optimal policy if no orders are expedited, according to the algorithm

of Federgruen and Zheng (1992). Percent difference: difference between model and simulation average cost rates, as percentage of the

simulation average cost rate.


Ta ¼ 0:4; the probability increases. However,when Ta increases even further, to Ta ¼ 0:6 orTa ¼ 0:8; since T þ Ta is being held constant andequal to 1, T becomes small. As discussed in theprevious paragraph, the small proportion of thetotal lead time demand that has already takenplace when the expediting decision is takenprovides only limited information on which tobase a decision on which orders are likely to leadto a stockout and should therefore be expeditedand which orders should not be expedited. There-fore, depending on the specific cost parameters, theoptimal policy will converge towards either alwaysexpediting or never expediting. In this example, forKe ¼ 5 the expediting probability will rise untilalways expediting becomes the optimal policy.Conversely, for Ke ¼ 15 the expediting probabilitywill decrease when Ta increases to Ta ¼ 0:6 andTa ¼ 0:8:

It is worth highlighting that the optimal orderquantity in all these numerical examples is alwaysbetween 17 and 19 units. This can be explained bythe fact that we have held constant the normalordering cost, holding costs and demand rate, aswell as by the well established robustness of theEOQ formula.

5. Conclusions

In this paper we have developed a model to findthe optimal inventory policy (or a policy approx-imating the optimal one) when there is an expeditingoption. We have presented an algorithm to obtainthe policy variables that attain a global minimalaverage cost rate when the inventory policy decisionvariables are integers. We have also discussed thecase when the decision variables are real valued.

We have verified extensively the model bysimulation. In the numerical examples we havestudied the behavior of our model with respect tothe variation of some parameters. That analysishighlights how the proposed model could be used,not only to establish the inventory managementpolicy for a given set of parameters (costs, leadtimes, etc.) but also to understand how the systemwould be affected by changes in these parameters,once the inventory policy is adjusted accordingly.For example, we have observed that the minimalaverage cost rate can be strongly dependent on the(normal) second component of the lead time, atleast for some combinations of parameters. Thatwould support the interest of the analysis of theoptimal determination of the second component of

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Table 7

Optimal inventory policies for different values of Ta while holding T þ Ta and Te constant, for two values of Ke

Data m 5 h 1

Ka 25 p 500

ce 0.2 T þ Ta 1

Te 0.1

Ke 5 15

Second component of the lead time, Ta 0.2 0.4 0.6 0.8 0.2 0.4 0.6 0.8

Optimal R 9 6 4 0 9 6 5 4

p for optimal R 0.021 0.084 0.143 1.000 0.021 0.084 0.053 0.019

Optimal r 11 10 10 5 11 10 10 11

Optimal Q 17 17 17 18 17 17 18 17

Model average cost rate 23.10 22.46 22.84 23.1 23.17 22.71 23.20 23.46

Cost relative reduction 2.12 4.83 3.22 2.12 1.82 3.77 5.93 0.59

Simulation average cost rate 23.05 22.33 22.91 22.86 23.14 22.77 23.13 23.75

Simulation standard deviation 0.13 0.12 0.16 0.1 0.14 0.15 0.18 0.19

Percent difference 0.22 0.58 �0.31 1.05 0.13 �0.26 �0.30 �1.22

Cost relative reduction: reduction in average cost rate with respect to the optimal policy if no orders are expedited, according to the

algorithm of Federgruen and Zheng (1992). Percent difference: difference between model and simulation average cost rates, as

percentage of the average simulation cost rate.


the lead time when it is controllable (see Liao andShyu, 1991).

To simplify the determination of the optimalpolicy in our model we have not included in themodel a backorder cost rate. As noted byJohansen and Thorstenson (1998) if there is nota backorder cost rate in addition to the penaltycost p per unit backordered there is no economicincentive in the model to reduce backorders oncethey have occurred. However, in this model, onceinventory is depleted and backordering starts, theonly way to reduce the number of additionalbackorders is to expedite the arrival of theoutstanding order, therefore the penalty cost perunit backordered induces a somehow similarbehavior to the backorder cost rate.

Further research could be done around theoption of expediting only a fraction of the orderquantity Q; which might be relevant when theexpediting cost per unit ce is high.

Acknowledgements

The authors are very grateful to the referees fortheir fast revision and useful comments whichimproved the original version of the paper.

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A continuous review inventory model with order expediting