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Int. J. Production Economics 97 (2005) 121–129

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Optimal pricing and ordering policy under permissible delayin payments

Jinn-Tsair Tenga,�, Chun-Tao Changb, Suresh Kumar Goyalc

aDepartment of Marketing and Management Sciences, William Paterson University of New Jersey, Wayne, NJ 07470-2103, USAbDepartment of Statistics, Tamkang University, Tamsui, Taipei, Taiwan 25137, Republic of China

cDepartment of Decision Sciences & MIS, Concordia University, Montreal, Quebec, Canada H3G1M8

Received 1 March 2003; accepted 1 April 2004

Available online 1 September 2004

Abstract

Many researchers have assumed that the selling price is the same as the purchase cost, and developed various EOQ

models for a retailer when the supplier offers a permissible delay in payments. In this paper, we complement the

shortcoming of their models by considering the difference between the selling price and the purchase cost. We then

develop an algorithm for a retailer to determine its optimal price and lot size simultaneously when the supplier offers a

permissible delay in payments. Our theoretical result contradicts to their conclusion that the economic replenishment

interval and order quantity generally increases marginally under the permissible delay in payments.

r 2004 Elsevier B.V. All rights reserved.

Keywords: Pricing; Inventory; Permissible delay; Deteriorating items; Finance

1. Introduction

In the traditional economic order quantity(EOQ) model, it assumes that the buyer mustpay for the items purchased as soon as the itemsare received. Typically, a supplier permits thebuyer a period of time, say 30 days, to settle thetotal amount owed to him. Usually, interest is notcharged for the outstanding amount if it is paidwithin the permissible delay period. This credit

e front matter r 2004 Elsevier B.V. All rights reserve

e.2004.04.010

ng author. Fax: 1-973-720-2809.

ss: tengj@wpunj.edu (J.-T. Teng).

term in financial management is denoted as ‘‘net30’’ (e.g., see Brigham, 1995). However, if thepayment is not paid within the permissible delayperiod, then interest is charged on the outstandingamount under the previously agreed terms andconditions. Therefore, a buyer can earn theinterest on the accumulated revenue received,and delay the payment up to the last moment ofthe permissible period allowed by the supplier. Thepermissible delay in payments reduces the buyer’scost of holding stock because it reduces theamount of capital invested in stock for theduration of the permissible period. Hence, it is a

d.

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marketing strategy for the supplier to attract newcustomers who consider it to be a type of cost (orprice) reduction. However, the strategy of grantingcredit terms adds not only an additional cost to thesupplier but also an additional dimension ofdefault risk to the supplier.Goyal (1985) developed an EOQ model under

conditions of permissible delay in payments. Heignored the difference between the selling price andthe purchase cost, and concluded that the eco-nomic replenishment interval and order quantitygenerally increases marginally under the permis-sible delay in payments. Although Dave (1985)corrected Goyal’s model by assuming the fact thatthe selling price is necessarily higher than itspurchase price, his viewpoint did not draw muchattention to the recent researchers. Aggarwal andJaggi (1995) then extended Goyal’s model fordeteriorating items. Jamal et al. (1997) furthergeneralized the model to allow for shortages anddeterioration. Hwang and Shinn (1997) developedthe optimal pricing and lot sizing for the retailerunder the condition of permissible delay inpayments. Liao et al. (2000) developed an inven-tory model for stock-depend demand rate when adelay in payment is permissible. Recently, Changand Dye (2001) extended the model by Jamal et al.to allow for not only a varying deterioration rateof time but also the backlogging rate to beinversely proportional to the waiting time. Allabove models (except Dave, 1985) ignored thedifference between unit price and unit cost, andobtained the same conclusion as in Goyal (1985).In contrast, Jamal et al. (2000) and Sharker et al.(2000) amended Goyal’s model by considering thedifference between unit price and unit cost, andconcluded from computational results that theretailer should settle his account relatively sooneras the unit selling price increases relative to theunit cost. Recently, Teng (2002) provided analternative conclusion from Goyal (1985), andmathematically proved that it makes economicsense for a well-established buyer to order lessquantity and take the benefits of the permissibledelay more frequently. Chang et al. (2003) thenextended Teng’s model, and established an EOQmodel for deteriorating items in which the supplierprovides a permissible delay to the purchaser if the

order quantity is greater than or equal to apredetermined quantity.In this paper, by assuming the selling price is

necessarily higher than the purchase cost, weestablish an appropriate model for a retailer todetermine its optimal price and lot size simulta-neously when the supplier offers a permissibledelay in payments. As we know, demand is afunction of price. Therefore, the retailer firstdecides upon the unit selling price and determinesthe expected demand, and then determines the lotsize based on the expected demand. However, tosolve the problem, we must resort to backwardinduction. Consequently, we first derive theoptimal lot size for a given price, and then findthe optimal price that maximizes the total profit.In contrast to the previous results by many otherresearchers (e.g., Liao et al., 2000; Jamal et al.,1997; Aggarwal and Jaggi, 1995; Goyal, 1985), ourtheoretical result (i.e., Theorem 2) shows that theeconomic replenishment interval and order quan-tity generally decreases under the permissible delayin payments. Finally, our computational resultsreveal that a higher value of permissible delayperiod causes a lower unit selling price but a higherprofit. In short, when a retailer negotiates a longerpermissible delay period from the supplier, theretailer can pass some of the cost savings to thecustomers by lowering the selling price, whileincreasing demand and profit.

2. Assumptions and notation

To develop the mathematical model, the follow-ing assumptions are being made:

(1)

The demand for the item is a downwardsloping function of the price. There are threemajor popular demand functions: constantelasticity, negative exponential, and linear.For simplicity, we assume that demand is aconstant elasticity function of the price.

(2)

Shortages are not allowed. (3) In reality, the retailer has numerous ways to

spend the profit from sales, such as expansion,R/D, new product development, hardware andsoftware upgrade, etc. For simplicity, we

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assume here that the retailer spends the profitin other activities than paying off the loan.During the time the account is not settled,generated sales revenue is deposited in aninterest bearing account. At the end of thisperiod, the retailer pays off all units sold, keepsthe rest for the use of the other activities, andstarts paying for the interest charges on theitems in stocks. In the meantime, the retailerstarts accumulating profit for the use of theother activities. As a matter of fact, the readercan easily develop a similar model in which theretailer pays off the loan whenever he/she hasmoney, such as in Jamal et al. (2000) andSharker et al. (2000).

(4)

Time horizon is infinite.

In addition, the following notation is usedthroughout this paper:

h the unit holding cost per year excludinginterest charges

p the selling price per unitc the unit purchasing cost, with cop

Ic the interest charged per $ in stocks peryear by the supplier

Id the interest earned per $ per years the ordering cost per orderm the period of permissible delay in settling

account; that is, the trade credit periodQ the order quantityy the constant deterioration rate, where

0pyoo1I(t) the level of inventory at time t, 0ptpT

T the replenishment time intervalD the annual demand, as a decreasing

function of price; we set D(p)=ap�b,where a40 and b41

Z(T, p) the total annual profit

The total annual profit consists of: (a) the salesrevenue, (b) cost of placing orders, (c) cost ofpurchasing, (d) cost of carrying inventory (exclud-ing interest charges), (e) cost of interest payable foritems unsold after the permissible delay m (notethat this cost occurs only if T4m), and (f) interestearned from sales revenue during the permissibleperiod.

3. Mathematical formulation

The level of inventory I(t) gradually decreasesmainly to meet demands and partly due todeterioration. Hence, the variation of inventorywith respect to time can be described by thefollowing differential equations:

dIðtÞ

dtþ yIðtÞ ¼ �D; 0ptpT ; ð1Þ

with the boundary condition I(T)=0. Conse-quently, the solution of (1) is given by

IðtÞ ¼D

y½eyðT�tÞ � 1�; 0ptpT ; ð2Þ

and the order quantity is

Q ¼ Ið0Þ ¼D

yðeyT � 1Þ: ð3Þ

The total annual profit consists of the following:

ðaÞ Sales revenue ¼ D p; ð4Þ

ðbÞ Cost of placing orders ¼ s=T ; ð5Þ

ðcÞ Cost of purchasing ¼ cQ=T ¼cD

yTðeyT

� 1Þ;

ð6Þ

ðdÞ Cost of carrying inventory ¼ h

Z T

0

IðtÞ dt=T

¼hD

y2TðeyT � 1Þ �

hD

y: ð7Þ

Regarding interests payable and earned (i.e.,costs of (e) and (f)), we have the following twopossible cases based on the values of T and m.These two cases are depicted graphically in Fig. 1.

Case 1: Tpm

In this case, the customer sells DT units in totalby the end of the replenishment cycle time T, andhas cDT to pay the supplier in full by the end ofthe credit period m. Consequently, there is nointerest payable. However, the interest earned peryear is

pId

Z T

0

Dtdt þ DTðm � TÞ

� �=T ¼ pIdDðm � T=2Þ:

ð8Þ

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Q Q

0 T m 0 m T

Inventory Level Inventory Level

Time Time

Case 1. T ≤ m Case 2. T ≥ m

Fig. 1. Graphical representation of two inventory systems.

J.-T. Teng et al. / Int. J. Production Economics 97 (2005) 121–129124

As a result, the total annual profit Z1(T, p) is

Z1ðT ; pÞ ¼ pD �s

T�

Dðh þ cyÞ

y2TðeyT � 1Þ þ

hD

yþ pIdDðm � T=2Þ: ð9Þ

Case 2: TXm

As stated in Assumption 3 above, the buyer sellsDm units in total by the end of the permissibledelay m, and has cDm to pay the supplier. Theitems in stock are charged at interest rate Ic by thesupplier starting at time m. Thereafter, the buyergradually reduces the amount of financed loanfrom the supplier due to constant sales andrevenue received. As a result, the interest payableper year is

cI c

Z T

m

IðtÞ dt=T ¼cI cD

y2T½eyðT�mÞ � 1�

�cI cD

yTðT � mÞ: ð10Þ

Next, during the permissible delay period, thebuyer sells products and deposits the revenue intoan account that earns Id per dollar per year.Therefore, the interest earned per year is

pId

Z m

0

Dtdt=T ¼pIdD

2Tm2: ð11Þ

Hence, the total annual profit Z2(T, p) is

Z2ðT ; pÞ ¼ pD �s

T�

Dðh þ cyÞ

y2TðeyT � 1Þ

þhD

y�

cI cD

y2T½eyðT�mÞ � 1�

þcI cD

yTðT � mÞ þ

pIdD

2Tm2: ð12Þ

Note that there are many different ways tocalculate the interest payable as well as interestearned, such as Goyal (1985), Aggarwal and Jaggi(1995), and Teng (2002). For simplicity, we useGoyal’s approach throughout this paper.Hence, the total annual profit Z(T, p) is written

as

ZðT ; pÞ ¼Z1ðT ; pÞ for Tpm;

Z2ðT ; pÞ for TXm:

�

Although Z1(m, p)=Z2(m, p), Z(T, p) is a con-tinuous function of T either in (0, m) or in (m, N),but not in both.

4. Determination of the optimal replenishment time

for any given price

For low deterioration rates, we can assume

eyT � 1þ yT þ ðyTÞ2=2: ð13Þ

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Hence, the total annual profit will be given by

Z1ðT ; pÞ � AZ1ðT ; pÞ ¼ Dp½1þ Idðm � T=2Þ�

�s

T� Dc �

DT

2ðh þ cyÞ ð14Þ

and

Z2ðT ; pÞ � AZ2ðT ; pÞ ¼ D½p � cð1� I cmÞ�

�1

Ts þ

Dm2ðcI c � pIdÞ

2

� ��

DT

2ðh þ cyþ cI cÞ:

ð15Þ

That is, the approximation of total annual profitAZ(T, p) is written as

AZðT ; pÞ ¼AZ1ðT ; pÞ for Tpm;

AZ2ðT ; pÞ for TXm:

�

Although AZ1(m, p)=AZ2(m, p), AZ(T, p) is acontinuous function of T either in (0, m) or in(m, N). However, we know from Theorem 1below that AZ(T, p) is not continuous in (0, N)because only one case of AZi (T, p) can occur. Forexample, h=$0.65/unit/year, Ic=0.09/$/year,Id=0.06/$/year, c=$5.0 per unit, y=0.05,p=$10 per unit, m=0.1 year, s=50, a=1000000and b=2.0, we obtain from Theorem 1 belowthat 2spðh þ cyþ pIdÞDm2 ¼ 150; AZ(T, p)=AZ1(T, p), and the optimal T*=0.081650om, asshown in Fig. 2. For an example of Case 2 (i.e.,AZ(T, p)=AZ2(T, p)), we assume that h=$0.65/unit/year, Ic=0.09/$/year, Id=0.06/$/year, c=$5.0per unit, y=0.05, p=$10 per unit, m=0.1 year,s=100, a=1000000 and b=2.0. Then we obtain from

Fig. 2. Graph of AZ(T, p)=AZ1(T, p) with Tpm.

Theorem 1 that 2sX(h+cy+pId)Dm2=150, AZ(T,p)=AZ2(T,p), and the optimal T*=0.1170634m, asshown in Fig. 3.Note that the purpose of this approximation is

to obtain the unique closed-form solution for theoptimal T. By taking the first- and second-orderderivatives of AZi (T, p), for i=1 and 2, withrespect to T, we obtain

qAZ1ðT ; pÞ

qT¼

1

T2s �

D

2ðh þ cyþ pIdÞ; ð16Þ

qAZ2ðT ; pÞ

qT¼

1

T2s þ

Dm2ðcI c � pIdÞ

2

� �

�D

2ðh þ cyþ cI cÞ; ð17Þ

q2AZ1ðT ; pÞ

qT2¼ �2

1

T3so0 ð18Þ

and

q2AZ2ðT ; pÞ

qT2¼ �2

1

T3s þ

Dm2ðcI c � pIdÞ

2

� �: ð19Þ

Consequently, for a fixed p, AZ1(T, p) is astrictly concave function of T. Thus, there exists aunique value of T1 which maximizes AZ1(T, p)as

T1 ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2s=Dðh þ cyþ pIdÞ

p¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2s=Dg1

p; ð20Þ

where

g1 ¼ h þ cyþ pId: ð21Þ

Fig. 3. Graph of AZ(T, p)=AZ2(T, p) with TXm.

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To ensure T1pm, we substitute (20) intoinequality T1pm, and obtain that

if and only if 2spðh þ cyþ pIdÞDm2; then T1pm:

ð22Þ

Hence, substituting (20) into (3), the optimalEOQ for Case 1 (i.e., T1pm) is

QnðT1Þ ¼D

yðeyT1 � 1Þ � DðT1 þ yT2

1=2Þ

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2sD=ðh þ cyþ pIdÞ

pþ ys=ðh þ cyþ pIdÞ:

ð23Þ

Substituting (20) into (14), we obtain

AZ1ðpÞ ¼ AZ1ðT1ðpÞ; pÞ

¼ D½pð1þ IdmÞ � c� �ffiffiffiffiffiffiffiffiffiffiffiffiffi2g1Ds

p: ð24Þ

From (22), we know that T2Xm implies2sX(h+cy+pId)Dm2. Consequently, we obtain:2s+Dm2(cIc�pId)X(h+cy+cIc)Dm240, and

q2AZ2ðT ; pÞ

qT2¼ �½2s þ Dm2ðcI c � pIdÞ�=T3o0:

ð25Þ

Therefore, for a fixed p, AZ2(T, p) is also astrictly concave function of T. Likewise, we obtainthe optimal solution to AZ2(T, p) as

T2 ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½2s þ Dm2ðcI c � pIdÞ�=½Dðh þ cyþ cI cÞ�

q

¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2s1=Dg2

p; ð26Þ

where

s1 ¼ s þDm2ðcI c � pIdÞ

2and g2 ¼ h þ cyþ cI c:

ð27Þ

To ensure T2 is existent and T2Xm, wesubstitute (26) into inequality T2Xm, and obtainthat

if and only if 2sXðh þ cyþ pIdÞDm2; then T2Xm:

ð28Þ

Hence, the optimal EOQ for Case 2 (i.e., T2Xm) is

QnðT2Þ � DðT2 þ yT22=2Þ

¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½2sD þ D2m2ðcI c � pIdÞ�=½h þ cyþ cI c�

q

þ y½2sþ Dm2ðcI c � pIdÞ�=f2½h þ cyþ cI c�g:

ð29Þ

Substituting (26) into (15), we obtain

AZ2ðpÞ ¼ AZ2ðT2ðpÞ; pÞ

¼ D½p � cð1� I cmÞ� �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g2Ds1

p: ð30Þ

In the classical EOQ model, the supplier must bepaid for the items as soon as the customer receivesthem. Therefore, it is a special case of Case 2 withm=0. As a result, the classical optimal EOQ is

Qn ¼D

yðeyTn � 1Þ �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2sD=ðh þ cyþ cI cÞ

pþ ½ys=ðh þ cyþ cI cÞ�: ð31Þ

By comparing (22) and (28), we have the followingtheorem.

Theorem 1. For low deterioration rates, we can

obtain the following results.

ð1Þ If 2soðh þ cyþ pIdÞDm2; then Tn ¼ T1:

ð2Þ If 2s4ðh þ cyþ pIdÞDm2; then Tn ¼ T2:

ð3Þ If 2s ¼ ðh þ cyþ pIdÞDm2; then Tn ¼ m:

Proof. It immediately follows from (22) and (28).

Similarly, from (23), (29) and (31), we have thefollowing theorem.

Theorem 2. For low deterioration rates, we get the

following results:

ðaÞ If cI copId; then Q�ðT1Þ and Q�ðT2ÞoQ�:

ðbÞ If cI c4pId; then Q�ðT1Þ and Q�ðT2Þ4Q�:

ðcÞ If cI c ¼ pId; then Q�ðT1Þ and Q�ðT2Þ ¼ Q�:

Proof. It is obvious from (23), (29) and (31).

Note that Theorems 1 and 2 here are a general-ization of the corresponding Theorems 1 and 2 ofTeng (2002), in which the deterioration rate iszero. By assuming that p=c and Ic4Id (i.e., Part(b) of Theorem 2), many recent researchers (e.g.,Liao et al., 2000; Jamal et al., 1997; Aggarwaland Jaggi, 1995) concluded that Q*(T1) andQ*(T2)4Q*. However, in reality, pId is in generalgreater than cIc. Consequently, we know from Part(a) of Theorem 2 that Q*(T1) and Q*(T2)oQ*.

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5. Determination of the optimal price

Taking the first derivative of (h+cy+pId)D(p)m2 with respect to p, we obtain

IdDðpÞm2 þ ðh þ cyþ pIdÞD0ðpÞm2

¼ m2½ðh þ cyÞD0ðpÞ � Idðb� 1ÞDðpÞ�o0: ð32Þ

Hence, (h+cy+pId)D(p)m2 is a strictly decreas-ing function of p. Using the facts in (22) and (28),we set P0 such that

2s ¼ ðh þ cyþ p0IdÞDðp0Þm2: ð33Þ

Consequently, we know from (22) and (28) that

AZðpÞ ¼AZ1ðpÞ ¼ AZ1ðT1ðpÞ; pÞ for ppp0;

AZ2ðpÞ ¼ AZ2ðT2ðpÞ; pÞ for pXp0:

�

ð34Þ

To obtain the optimal price, taking the firstderivative of (24) with respect to p and setting theresult to be zero, we have

dAZ1ðpÞ

dp¼ D½ð1þ IdmÞð1� bÞ þ p�1bc

þ ð2g1DsÞ�1=2sðg1p�1b� IdÞ� ¼ 0:

ð35Þ

Next, we need to check the second-ordercondition for concavity. That is

d2AZ1ðpÞ

dp2¼ Df�p�2bc þ ð2g1Þ

�3=2s1=2

½g21p�2bðb� 2Þ þ I2d �go0: ð36Þ

Likewise, from (30), we obtain the first-ordercondition for AZ2(p) as

dAZ2ðpÞ

dp¼ D½ð1� bÞ þ p�1bcð1� I cmÞ

� g2ð2g2Ds1Þ�1=2

ð�bp�1s1 þ s01Þ�

¼ Dfð1� bÞ þ p�1bcð1� I cmÞ

� g2ð2g2Ds1Þ�1=2

ð�bp�1s1

þDm2

2ðIdðb� 1Þ � p�1bcI cÞ�g ¼ 0:

ð37Þ

The second-order condition for concavity is

d2AZ2ðpÞ

dp2¼ Df�p�2bcð1� I cmÞ

þ ð2s1Þ�3=2

ðg2Þ1=2

ðDÞ�1=2

½p�2s21b2þ s0

21

� 2bp�1s1s01 � 2s1ðp

�2s1b� 2p�1bs01

þDm2

2p�1bcI c�g

¼ Df�p�2bcð1� I cmÞ

þ ð2s1Þ�3=2

ðg2Þ1=2

ðDÞ�1=2

½ðbp�1s1 þ s01Þ2

� 2p�1s1bðs1 þDm2

2cI cÞ�go0: ð38Þ

6. An algorithm

Based on the above discussion, we develop thefollowing solution algorithm to determine anoptimal solution for the approximate model.

Step 1: Determine p0 by solving (33).Step 2: If there exists a p1 such that p1pp0, and

p1 satisfies both the first-order condition as in (35)and the second-order condition for concavity as in(36), then we determine T1(p1) by (20) andAZ1ðT1ðp1Þ; p1Þ by (24). Otherwise, we setAZ1ðT1ðp1Þ; p1Þ ¼ 0:

Step 3: If there exists a p2 such that p2Xp0, andp2 satisfies both the first-order condition as in (37)and the second-order condition for concavityas in (38), then we calculate T2(p2) by (26)and AZ2ðT2ðp2Þ; p2Þ by (30). Otherwise, we setAZ2ðT2ðp2Þ; p2Þ ¼ 0:

Step 4: If AZ1ðT1ðp1Þ; p1ÞXAZ2ðT2ðp2Þ; p2Þ; thenthe optimal total annual profit is AZ�ðT�ðp�Þ; p�Þ ¼

AZ1ðT1ðp1Þ; p1Þ; and stop. Otherwise, AZ�ðT�ðp�Þ;p�Þ¼ AZ2ðT2ðp2Þ; p2Þ; and stop.

7. Numerical examples

Example 1. For generality, we use the followingexample in which c Icop*Id. Given h=$0.5/unit/year, Ic=0.09/$/year, Id=0.06/$/year, c=$4.5 perunit, y=0.05, s=$10/per order, a=100000 andb=1.5. Using the above solution algorithm, we

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Table 1

Optimal solutions for different trade credit period m

m(days) p0 p* T* Q(T*) AZ* Z*

5 0.8076 p2=13.6432 T2=0.094077 187.1249 17943.671 17943.529

10 2.1765 p2=13.6258 T2=0.092887 185.1077 17957.162 17957.039

15 4.0594 p2=13.6079 T2=0.090938 181.5699 17972.387 17972.280

20 6.5569 p2=13.5897 T2=0.088184 176.4136 17989.461 17989.361

25 9.8363 p2=13.5712 T2=0.084555 169.4845 18008.552 18008.467

30 14.1329 p1=13.5535 T1=0.080547 161.1747 18029.911 18030.279

40 27.1234 p1=13.5312 T1=0.080482 162.0156 18074.581 18075.204

50 49.1789 p1=13.5090 T1=0.080418 162.2895 18119.287 18120.214

60 85.5835 p1=13.4869 T1=0.080354 162.5574 18164.029 18164.276

70 143.1669 p1=13.4648 T1=0.080290 162.8307 18208.809 18209.456

J.-T. Teng et al. / Int. J. Production Economics 97 (2005) 121–129128

obtain the computational results for various valuesof m as shown in Table 1. Table 1 reveals that (1)the difference between AZ* and Z* is negligible,(2) a higher value of m causes a higher value of Z*,but lower values of p* and T*, (3) if T*Xm, then ahigher value of m causes a lower value of Q(T*),(4) if T*pm, then a higher value of m causes ahigher value of Q(T*), and (5) we know fromEq. (31) that the classical optimal EOQQ*=187.6761, which confirms the result in Part(a) of Theorem 2 (i.e., Q*(T1) and Q*(T2)oQ*,if c Icop*Id).

8. Conclusions and future research

In this paper, we develop an appropriate pricingand lot-sizing model for a retailer when thesupplier provides a permissible delay in payments.We then establish the necessary and sufficientconditions for the unique optimal replenishmentinterval, and use Taylor’s series approximation toobtain the explicit closed-form optimal solution.Next, we derive the first and second-order condi-tions for finding the optimal price, and thendevelop an algorithm to solve the problem.Furthermore, we establish Theorem 1, whichprovides us a simple way to obtain the optimalreplenishment interval by examining the explicitconditions. We then compare the classical EOQwith the proposed model here, and obtainTheorem 2. Finally, our numerical example revealsthat a higher value of the permissible delay m

causes a higher value of the unit profit Z*, butlower values of the selling price p* and thereplenishment cycle time T*.The model proposed in this paper can be

extended in several ways. For instance, we mayextend the constant deterioration rate to a two-parameter Weibull distribution. Also, we couldconsider the demand as a function of quality aswell as time varying. Finally, we could generalizethe model to allow for shortages, quantitydiscounts, discount and inflation rates, and others.

Acknowledgements

This research was partially supported by theAssigned Released Time for Research and aSummer Research Funding from the WilliamPaterson University of New Jersey.

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