optimal retailer's ordering policy with deteriorating items under twolevel trade credit

15
This article was downloaded by: [New York University] On: 10 October 2014, At: 00:33 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Journal of Information and Optimization Sciences Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tios20 Optimal retailer's ordering policy with deteriorating items under twolevel trade credit Yung-Fu Huang a , Ming-Hon Hwang a , Yu-Cheng Tu a & Hao-Wei Yang a a Department of Marketing and Logistics Management , Chaoyang University of Technology , Taichung , Taiwan, R.O.C. Published online: 18 Jun 2013. To cite this article: Yung-Fu Huang , Ming-Hon Hwang , Yu-Cheng Tu & Hao-Wei Yang (2010) Optimal retailer's ordering policy with deteriorating items under twolevel trade credit, Journal of Information and Optimization Sciences, 31:4, 941-954, DOI: 10.1080/02522667.2010.10700004 To link to this article: http://dx.doi.org/10.1080/02522667.2010.10700004 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

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Page 1: Optimal retailer's ordering policy with deteriorating items under twolevel trade credit

This article was downloaded by: [New York University]On: 10 October 2014, At: 00:33Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: MortimerHouse, 37-41 Mortimer Street, London W1T 3JH, UK

Journal of Information and Optimization SciencesPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/tios20

Optimal retailer's ordering policy with deterioratingitems under twolevel trade creditYung-Fu Huang a , Ming-Hon Hwang a , Yu-Cheng Tu a & Hao-Wei Yang aa Department of Marketing and Logistics Management , Chaoyang University ofTechnology , Taichung , Taiwan, R.O.C.Published online: 18 Jun 2013.

To cite this article: Yung-Fu Huang , Ming-Hon Hwang , Yu-Cheng Tu & Hao-Wei Yang (2010) Optimal retailer's orderingpolicy with deteriorating items under twolevel trade credit, Journal of Information and Optimization Sciences, 31:4,941-954, DOI: 10.1080/02522667.2010.10700004

To link to this article: http://dx.doi.org/10.1080/02522667.2010.10700004

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose ofthe Content. Any opinions and views expressed in this publication are the opinions and views of the authors,and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be reliedupon and should be independently verified with primary sources of information. Taylor and Francis shallnot be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to orarising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: Optimal retailer's ordering policy with deteriorating items under twolevel trade credit

Optimal retailer’s ordering policy with deteriorating items under two-level trade credit2

Yung-Fu Huang ∗

Ming-Hon Hwang4

Yu-Cheng TuHao-Wei Yang6

Department of Marketing and Logistics ManagementChaoyang University of Technology8

TaichungTaiwan, R.O.C.10

AbstractIn practice, in the supply chain there are two levels of trade credit: one is offered to the12

retailer from its supplier; the other is offered to the customer from its retailer. Moreover, thecommodity under consideration usually deteriorates over time. The present study therefore14

proposes a retailer’s EOQ (economic order quantity) model by considering deteriorating itemsunder two-level trade credit. The proposed model takes into account both levels of trade16

credit. In the first place, the authors model the retailer’s inventory decision as a costminimization problem. Secondly, the authors prove the convexity of the inventory function18

in terms of relevant annual costs. The authors then move on to construct an easy-to-usetheorem to efficiently determine the optimal replenishment cycle, hence, the optimal order20

quantity. Finally, the authors provide several numerical examples to illustrate the theoremand to conduct a sensitivity analysis. Based on the proposed model, the authors conclude that22

a longer replenishment cycle (a larger order quantity) is directly related to a higher orderingcost, and, is inversely related to a higher deteriorating rate.24

Keywords and phrases : Economic order quantity, inventory, optimization, sensitivity analysis,

deteriorating item.26

1. Introduction

In the classical economic order quantity (EOQ) model, it was tacitly as-28

sumed that the purchaser must pay for the items purchased in the momentthe items are received. However, in practice, the supplier frequently offers30

∗Corresponding author’s address: No. 168, Jifong E. Rd., Wufong Township, TaichungCounty 41349, Taiwan, R.O.C. E-mail: [email protected]——————————–Journal of Information & Optimization SciencesVol. 31 (2010), No. 4, pp. 941–954c© Taru Publications

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Page 3: Optimal retailer's ordering policy with deteriorating items under twolevel trade credit

942 Y. F. HUANG ET AL.

the trade credit, i.e., permissible delay in payments, to its purchasers.Goyal [7] established a single-item inventory model under permissible2

delay in payments. Chung [4] developed an efficient decision procedureto determine the economic order quantity under condition of permissible4

delay in payments. Teng [17] assumed that the selling price was not equalto the purchasing price to modify Goyal’s model [7]. Chung and Huang [5]6

investigated this issue within EPQ framework and developed an efficientsolving procedure to determine the optimal replenishment cycle for the8

retailer. Huang and Chung [10] investigated the inventory policy undercash discount and trade credit. Chung and Liao [6] adopted alternative10

payment rules to develop the inventory model and obtain different results.Recently, Huang [9] adopted the payment rule discussed in Chung and12

Liao [6], and, assumed finite replenishment rate, to investigate the buyer’sinventory problem.14

All above published papers modeling trade credit implicitly assumedthat the supplier would offer the retailer a delay payment period but16

the retailer would not offer a delay payment period to its customers.In other words, all previously published papers assume there is only18

one level of trade credit. We know that this assumption is debatablein the practical business situations. Recently, Huang [8] modified this20

assumption to assume that the retailer would adopt a similar trade creditpolicy to stimulate its customers’ demand in developing the retailer’s22

replenishment model. This viewpoint that there are two levels of tradecredit corresponds, more closely to the supply chain models in practice.24

But Huang [8] ignored the deteriorating effect of items in developinghis model. However, in real-life situations, there is an inventory loss26

by deterioration in categories of commodity such as volatile liquids,blood banks, medicines, electronic components, and fashion goods. Many28

articles have studied inventory models for deteriorating items. Someof the prominent papers are discussed below. Aggarwal and Jaggi [1]30

considered the inventory model with an exponential deterioration rateunder the condition of permissible delay in payments. Hwang and Shinn32

[11] modeled an inventory system for retailer’s pricing and lot sizingpolicy for exponentially deteriorating products under the condition of34

permissible delay in payment. Liao et al. [14] and Sarker et al. [15]investigated this topic with inflation. Jamal et al. [12], and Chang and36

Dye [2] examined this issue with allowable shortage. Jamal et al. [13], and

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OPTIMAL RETAILER’S ORDERING POLICY 943

Sarker et al. [16] addressed the optimal payment time under permissibledelay in payment with deterioration. Chang et al. [3] discussed this2

topic with deteriorating items under time-value of money and finite timehorizon. Therefore, this paper is purported to extend Huang’s model [8]4

by considering deteriorating items. Specifically, the authors model theretailer’s inventory decision as a cost minimization problem to determine6

the retailer’s optimal replenishment cycle and order quantity. The rest ofthis paper is organized in the following manner. In Section 2, the authors8

describe the notation and assumptions used throughout this paper. InSection 3, the mathematical models are derived. In Section 4, the authors10

prove the convexity of the inventory model in terms of relevant annualcost functions. In Section 5, the authors search for an approximately12

closed-form solution to the optimal replenishment cycle problem by useof Taylor’s series, and obtain an ease-to-use theorem to determine the14

optimal replenishment cycle. In Section 6, several numerical examples aregiven to illustrate the theorem so established in the present paper and to16

conduct a sensitivity analysis. Finally, in Section 7, the authors present theconclusions.18

2. Notation and assumptions

The following notation and assumptions are used throughout this20

paper.

Notation:22

D = demand rate per year;A = ordering cost per order;c = unit purchasing price per item;h = unit stock holding cost per item per year excluding interest

charges;Ie = interest earned per dollar per year;Ik = interest charged per dollar in stocks per year by the supplier;Θ = the deterioration rate of the inventory;M = the retailer’s trade credit period offered by the supplier in years;N = the customer’s trade credit period offered by the retailer in years;Q = the order quantity;I(t) = the level of inventory at time t, 0 ≤ t ≤ T;T = the replenishment cycle time in years;

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944 Y. F. HUANG ET AL.

TVC(T) = the annual total relevant cost, which is a function of T;T∗ = the optimal replenishment cycle time which minimizes

TVC(T) given T > 0;Q∗ = the optimal order quantity = DT∗.

Assumptions:2

(A) Demand rate, D , is known and remains constant.(B) Shortages are not allowed.4

(C) Time horizon is infinite.(D) Replenishment is instantaneous.6

(E) There is no repair or replacement of the deteriorated inventory duringa given cycle.8

(F) The constant fraction θ of on hand inventory gets deteriorated pertime unit.10

(G) Ik ≥ Ie , M ≥ N .(H) When T > M , the account is settled at T = M and the retailer starts12

paying for the interest charges due on the items in stock with rate Ik .When T ≤ M , the account is settled at T = M and the retailer does14

not need to pay any interest charges.(I) The retailer can accumulate revenue and earn interest after its cus-16

tomer pays for the amount of purchasing cost to the retailer until theend of the trade credit period offered by the supplier. That is, the18

retailer can accumulate revenue and earn interest during the periodfrom N (after the expiry of trade credit offered to the customers by the20

retailer) to M (before the expiry of trade credit offered to the retailerby the supplier) with rate Ie under the condition of trade credit.22

3. Mathematical formulation

The level of inventory I(t) is depleted by the effects of demand24

and deterioration, then the differential equation which describes theinstantaneous states of I(t) over (0, T) is given as:26

dI(t)dt

+θI(t) = −D, where 0 ≤ t ≤ T

with the boundary condition I(T) = 0 . The solution of above equation is28

given by

I(t) =Dθ[eθ(T−1) − 1], where 0 ≤ t ≤ T .30

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OPTIMAL RETAILER’S ORDERING POLICY 945

Noting that I(0) = Q , the quantity ordered in each replenishment cycle is

Q =Dθ(eθT − 1) .2

Furthermore, the annual total relevant cost function consists of the or-dering cost, inventory holding cost, cost of deteriorated units and the4

opportunity cost of the capital. From now on, the individual cost isevaluated before they are grouped together.6

(i) Annual ordering cost = AT .

(ii) Annual inventory holding cost (excluding the opportunity cost of8

the capital) = hT∫ T

0 I(t)dt = hDθ2T (e

θT −θT − 1) .

(iii) Annual cost of deteriorated units = c(Q−DT)T = cD

θT (eθT −θT − 1) .10

(iv) From assumptions (H) and (I), there are three cases in terms ofannual opportunity cost of the capital.12

Case I: T ≥ M

the annual opportunity cost of capital14

=cIk

∫ TM I(t)dt − cIe

∫ MN Dtdt

T

=cIkDθ2T

[eθ(T−M) −θ(T − M)− 1]− cIeD(M2 − N2)

2T16

Case II: N ≤ T < M

the annual opportunity cost of capital18

= − cIe[∫ T

N Dtdt + DT(M − T)]T

= − cIeD2T

[2MT − N2 − T2]20

Case III: T < N

the annual opportunity cost of capital22

= − cIe∫ M

N DTdtT

= cIeD(M − N)

According to the above conditions, we have24

TVC(T) =

TVC1(T) if M ≤ T (1a)

TVC2(T) if N ≤ T < M (1b)

TVC3(T) if 0 < T < N (1c)

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Page 7: Optimal retailer's ordering policy with deteriorating items under twolevel trade credit

946 Y. F. HUANG ET AL.

where

TVC1(T) =AT+

Dθ2T

(cθ+ h)(eθT −θT − 1)2

+cIkDθ2T

[eθ(T−M) −θ(T − M)− 1]

− cIeD(M2 − N2)

2Tif T > 0, (2)4

TVC2(T) =AT+

Dθ2T

(cθ+ h)(eθT −θT − 1)

− cIeD2T

(2MT − N2 − T2) if T > 0 (3)6

and

TVC3(T) =AT+

Dθ2T

(cθ+ h)(eθT −θT − 1)− cIeD(M − N)8

if T > 0. (4)

Because TVC1(M) = TVC2(M) at T = M and TVC2(N) = TVC3(N) at10

T = N , TVC(T) is continuous and well defined.

4. The convexity12

In this section, the authors demonstrate that three inventory functionsdescribed in Section 3 are convex on their appropriate domains.14

Theorem 1.

(i) TVC1(T) is convex on [M, ∞) .16

(ii) TVC2(T) is convex on (0, ∞) .

(iii) TVC3(T) is convex on (0, ∞) .18

(iv) TVC(T) is convex on (0, ∞) .

Before proving Theorem 1 , we need the following lemma.20

Lemma 1.

eθ(T−M) − 1 −θTeθ(T−M) +θ2T2

2eθ(T−M) +θM − θ2(M2 − N2)

2> 022

if T ≥ M .

Proof. Let g(T) = eθ(T−M) − 1 − θTeθ(T−M) + θ2T2

2 eθ(T−M) + θM −24

θ2(M2−N2)2 , then we have g′(T) = θ3T2

2 eθ(T−M) . So g(T) is increasing

on (M, ∞) and g(T) > g(M) = θ2 N2

2 > 0 if T > M . Consequently,26

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OPTIMAL RETAILER’S ORDERING POLICY 947

eθ(T−M) − 1 −θTeθ(T−M) + θ2T2

2 eθ(T−M) +θM − θ2(M2−N2)2 > 0 if T ≥ M .

This completes the proof of Lemma 1. ¤2

Proof of Theorem 1. (i): Form equation (2), the derivatives yield

TVC′1(T) = − A

T2 +D

θ2T2 (cθ+ h)(θTeθT − eθT + 1)4

+cIkDθ2T2 [θTeθ(T−M) − eθ(T−M) + 1 −θM]

+cIeD(M2 − N2)

2T2 (5)6

and

TVC′′1 (T) =

2AT3 +

2D(cθ+ h)θ2T3

[eθT

(1 −θT +

θ2T2

2

)− 1

]8

+2cIkDθ2T3

[eθ(T−M) − 1 −θTeθ(T−M)

+θ2T2

2eθ(T−M) +θM

]− cIeD(M2 − N2)

T310

≥ 2AT3 +

2D(cθ+ h)θ2T3 [(eθT · e−θT)− 1]

+2cIkDθ2T3

[eθ(T−M) − 1 −θTeθ(T−M)

12

+θ2T2

2eθ(T−M) +θM − θ2(M2 − N2)

2

]

=2AT3 +

2cIkDθ2T3

[eθ(T−M) − 1 −θTeθ(T−M)

14

+θ2T2

2eθ(T−M) +θM − θ2(M2 − N2)

2

]. (6)

Lemma 1 imply that d2TVC1(T)dT2 > 0 if T ≥ M . Therefore, TVC1(T) is16

convex on [M, ∞) .

(ii) and (iii): From equations (3) and (4), the derivatives yield18

TVC′2(T) = − A

T2 +D(cθ+ h)

θ2T2 (θTeθT − eθT + 1)

− cIeD2T2 (N2 − T2), (7)20

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948 Y. F. HUANG ET AL.2

TVC′′2 (T) =

2AT3 +

2D(cθ+ h)θ2T3

[eθT

(1 −θT +

12θ2T2

)− 1

]

+cIeDN2

T34

>2AT3 +

2D(cθ+ h)θ2T3 [eθT · e−θT − 1] +

cIeDN2

T3

=2AT3 +

cIeDN2

T3 > 0, (8)6

TVC′3(T) = − A

T2 +D(cθ+ h)

θ2T2 (θTeθT − eθT + 1) (9)

and8

TVC′′3 (T) =

2AT3 +

2D(cθ+ h)θ2T3

[eθT(1 −θT +

θ2T2

2)− 1

]

>2AT3 +

2D(cθ+ h)θ2T3 [eθT · e−θT − 1]10

=2AT3 > 0. (10)

Therefore, TVC2(T) and TVC3(T) is convex on (0, ∞) , respectively.12

(iv) Case I implies that TVC′1(T) is increasing on [M, ∞) . Cases II and III

implies that TVC′2(T) and TVC′

3(T) is increasing on (0, M] . Since14

TVC′1(M) = TVC′

2(M) and TVC′2(N) = TVC′

3(N) , then TVC′(T) isincreasing on T > 0 . Consequently TVC(T) is convex on T > 0 .16

Combing the above arguments, we have completed the proof. ¤

5. Theoretical results18

Since all TVCi(T) (i = 1, 2, 3) are convex on their appropriatedomains. We are in a position to consider the following equations, which20

are the necessary and sufficient conditions for obtaining the optimalsolution.22

First order conditions:

TVC′1(T) = 0, (11)24

TVC′2(T) = 0 (12)

and26

TVC′3(T) = 0. (13)

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OPTIMAL RETAILER’S ORDERING POLICY 949

We can simplify equations (11)-(13) to obtain

−A +Dθ2 (cθ+ h)(θTeθT − eθT + 1)2

+cIkDθ2 [θTeθ(T−M) − eθ(T−M) + 1 −θM]

+cIeD(M2 − N2)

2= 0 , (14)4

− A +D(cθ+ h)

θ2 (θTeθT − eθT + 1)− cIeD2

(N2 − T2) = 0 (15)

and6

− A +D(cθ+ h)

θ2 (θTeθT − eθT + 1) = 0. (16)

From equations (14)-(16), we cannot obtain the explicit closed-form solu-8

tion of the optimal cycle time, T∗i (i = 1, 2, 3) . In reality, the value for

the deterioration rate θ is usually very small. For simplicity, using the10

truncated Taylor’s series expansion for the exponential term, we have

eθT ≈ 1 +θT + (θT)2/2 as θT is small.12

Then we obtain

θTeθT − eθT + 1 =(θT)2

2+ o(θ2) ≈ (θT)2

2(17)14

and

θTeθ(T−M) − eθ(T−M) + 1 −θM =(θT)2

2− (θM)2

2+ o(θ2)16

≈ (θT)2

2− (θM)2

2. (18)

Substituting equations (17) and (18) into equations (14)-(16), we can get18

T∗1 ≈

√2A + cD[M2(Ik − Ie) + N2 Ie]

D[h + c(θ+ Ik)], (19)

T∗2 ≈

√2A + cDN2 Ie

D[h + c(θ+ Ie)](20)20

and

T∗3 ≈

√2A

D(h + cθ). (21)22

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950 Y. F. HUANG ET AL.

To ensure T∗1 ≥ M , we substitute equation (19) into inequality T∗

1 ≥ M ,and obtain that2

if − 2A + DM2[h + c(θ+ Ie)]− cDN2 Ie ≤ 0, then T∗1 ≥ M. (22)

Similarly, we substitute equation (20) into inequality N ≤ T∗2 < M , and4

obtain that

if − 2A + DM2[h + c(θ+ Ie)]− cDN2 Ie > 06

and − 2A + DN2(h + cθ) ≤ 0, then N ≤ T∗2 < M. (23)

Finally, we substitute equation (21) into inequality T∗3 < N , and obtain8

that

if − 2A + DN2(h + cθ) > 0, then T∗3 < N. (24)10

For convenience, we define

∆1 = −2A + DM2[h + c(θ+ Ie)]− cDN2 Ie (25)12

and

∆2 = −2A + DN2(h + cθ). (26)14

Since M ≥ N , we can easily obtain ∆1 ≥ ∆2 . From equations (19)-(26),we know that a higher value of A causes a longer replenishment cycle,16

and vice versa. In contrast, a higher value of h or θ causes a shorterreplenishment cycle, and vice versa. Putting all together, we have the18

following theorem to determine the optimal replenishment cycle time T∗ .

Theorem 2.20

(i) If ∆2 > 0 , then TVC(T∗) = TVC3(T∗3 ) . Hence, T∗ is equal to T∗

3 .

(ii) If ∆1 > 0 and ∆2 ≤ 0 , then TVC(T∗) = TVC2(T∗2 ) . Hence, T∗ is22

equal to T∗2 .

(iii) If ∆1 ≤ 0 , then TVC(T∗) = TVC1(T∗1 ) . Hence, T∗ is equal to T∗

1 .24

Proof. It immediately follows from equations (22)-(26).

6. Numerical examples26

In this section, the authors provide several numerical examples toillustrate the theoretical results contained in Theorem 2 as reported in28

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OPTIMAL RETAILER’S ORDERING POLICY 951

Section 5. At the same time, the authors also carry out a sensitivity analysisconcerning the impact of a change in the parametric value on the optimal2

replenishment cycle and optimal order quantity.

Example 1 (Optimal solution for T∗3 ). Given A = $50 /order, D = 70004

units/year, M = 0.1 year, N = 0.05 year, c = $100 /unit, Ik =

$0.15 /$/year, Ie = $0.12 /$/year, θ = 0.01 and h = $5 /unit/year.6

Then, we can obtain ∆1 = 950 > 0 and ∆2 = 5 > 0 . Using Theorem 2(i),we get T∗ = T∗

3 = 0.048795 year. The corresponding optimal order8

quantity will be Q∗ = DT∗3 = 341.565 units.

Example 2 (Optimal solution for T∗2 ). Given A = $100 /order, D = 200010

units/year, M = 0.1 year, N = 0.05 year, c = $100 /unit, Ik =

$0.15 /$/year, Ie = $0.12 /$/year, θ = 0.01 and h = $5 /unit/year.12

Then, we can obtain ∆1 = 100 > 0 and ∆2 = −170 < 0 . UsingTheorem 2(ii), we get T∗ = T∗

2 = 0.084984 year. The corresponding14

optimal order quantity will be Q∗ = DT∗2 = 169.968 units.

Example 3 (Optimal solution for T∗1 ). Given A = $100 /order, D = 100016

units/year, M = 0.1 year, N = 0.05 year, c = $100 /unit, Ik =

$0.15 /$/year, Ie = $0.12 /$/year, θ = 0.01 and h = $5 /unit/year.18

Then, we can obtain ∆1 = −50 < 0 and ∆2 = −185 < 0 . UsingTheorem 2(iii), we get T∗ = T∗

1 = 0.11127 year. The corresponding20

optimal order quantity will be Q∗ = DT∗1 = 111.27 units.

Example 4 (Sensitivity analysis on A ). Given D = 5000 units/year,22

M = 0.1 year, N = 0.05 year, c = $100 /unit, Ik = $0.15 /$/year,Ie = $0.12 /$/year, 0 = 0.01 and h = $5 /unit/year. If A = $30 , 60 or 9024

per order, then from Theorem 2 we can easily obtain the optimal solutionsas shown in Table 1. It indicates that a higher value of ordering costal26

implies a longer replenishment cycle T∗ and a larger order quantity Q∗ .

Example 5 (Sensitivity analysis on θ ). Given A = $60 /order, D =28

7000 units/year, M = 0.1 year, N = 0.05 year, c = $100 /unit,Ik = $0.15 /$/year, Ie = $0.12 /$/year and h = $5 /unit/year. If30

θ = 0.01 , 0.02 or 0.03, then from Theorem 2 we can easily obtain theoptimal solutions as shown in Table 2. It indicates that a higher value of32

deteriorating rate θ implies a shorter of replenishment cycle T∗ and asmaller order quantity Q∗ .34

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952 Y. F. HUANG ET AL.

Table 1Sensitivity analysis on A

Ordering ∆1 ∆2 Theorem 2 Replenishment Economic order

cost A cycle T∗ quantity Q∗

30 > 0 > 0 (i) T∗3 = 0.044721 223.6068

60 > 0 < 0 (ii) T∗2 = 0.054772 273.8613

90 > 0 < 0 (ii) T∗2 = 0.060553 302.765

2

Table 2Sensitivity analysis on θ

Deterioration ∆1 ∆2 Theorem 2 Replenishment Economic order

rate θ cycleT∗ quantity Q∗

0.01 > 0 < 0 (ii) T∗2 = 0.051177 358.2364

0.02 > 0 > 0 (i) T∗3 = 0.049487 346.4102

0.03 > 0 > 0 (i) T∗3 = 0.046291 324.037

4

7. Conclusions

In reality, we have witnessed two levels of trade credit: one is offered6

to the retailer from its supplier; the other is offered to the customersfrom its retailer. Moreover, it is usually the case that the commodity8

under consideration will deteriorate over time. The authors are thereforemotivated to develop a retailer’s EOQ model for the deteriorating item in10

the presence of two-level trade credit.

In order to obtain the explicit closed-form solutions to the proposed12

model, the authors use Taylor’s series approximation. Through suchprocedure, the authors establish an easy-to-use Theorem 2 from the14

retailer’s perspective. Finally, some numerical examples are provided toillustrate this Theorem, and to obtain the following managerial insights:16

(1) a higher value of ordering cost brights about longer replenishmentcycle and a larger order quantity; (2) a higher value of deterioration rate18

brights about shorter replenishment cycle and a smaller order quantity.

Acknowledgements. This paper is supported by NSC Taiwan, number20

NSC 96-2221-E-324-007-MY3 and CYUT.

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OPTIMAL RETAILER’S ORDERING POLICY 953

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Received November, 200914

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