The optimal ordering policy of the EOQ model under trade credit depending on the ordering quantity from the DCF approach

European Journal of Operational Research 196 (2009) 563–568

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European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor

Production, Manufacturing and Logistics

The optimal ordering policy of the EOQ model under trade credit depending on the ordering quantity from the DCF approach Kun-Jen Chung a,*, Jui-Jung Liao b a b

College of Business, Chung Yuan Christian University, Chung Li, Taiwan, ROC Department of Business Administration, Chihlee Institute of Technology, Taipei, Taiwan, ROC

a r t i c l e

i n f o

Article history: Received 27 March 2007 Accepted 1 April 2008 Available online 22 April 2008 Keywords: Discounted cash-flows (DCF) Trade credit Inventory

a b s t r a c t This paper discusses the optimum order quantity of the EOQ model that is not only dependent on the inventory policy but also on firm’ credit policy. Here, the conditions of using a discounted cash-flows (DCF) approach and trade credit depending on the quantity ordered are discussed. We consider that if the order quantity is less than at which the delay in payments is permitted, the payment for the item must be made immediately. Otherwise, the fixed trade credit period is permitted. This paper incorporates all concepts of a discounted cash-flows (DCF) approach, trade credit and the quantity ordered and develops a new inventory model to generalize Chung [Chung, K.H., 1989. Inventory control and trade credit revisited, Journal of the Operational Research Society 40, 495–498]. Ó 2008 Elsevier B.V. All rights reserved.

1. Introduction In recent years, the literature has contained a great number of decision models for both credit decisions and inventory management, as indicated by Goyal (1985), Chu et al. (1998) and Jamal et al. (1997). However, almost all of these models suffer from the following defects: They ignore the timing and magnitude of cashflows. In fact, it is well settled in corporate finance that the value of a firm is the present value of its cash-flow stream. The appeal of the discounted cash-flow approach to inventory modeling is that it frames the purchasing and stockholding decisions precisely. Specifically, Chung (1989) presents the economic ordering policies in the presence of trade credit using a discounted cash-flows (DCF) approach. He divides the study into four cases: (a) Instantaneous cash-flows (the case of the basic EOQ model). (b) Credit only on units in stock when T 6 M (where T denotes the inventory cycle time and M denotes the credit period). (c) Credit only on units in stock when T P M. (d) Fixed credit. In this regard, this paper is only concerned with cases (a)–(c). On the other hand, the credit policy may seem as an alternative to price discounts because such policies are not thought to provoke competitors to reduce their prices and thus introduce lasting price reductions, or because such policies are traditional in the firm’s industry. Furthermore, Khouja and Mehrez (1996) investigate the

* Corresponding author. Tel.: +886 3 2655708; fax: +886 3 2655099. E-mail addresses: [email protected], [email protected] (K.-J. Chung). 0377-2217/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2008.04.018

effect of supplier credit policies on the optimal order quantity within the economic order quantity framework. The supplier credit policies addressed in Khouja and Mehrez (1996) fall into two categories. One is that supplier credit policies where terms are independent of the order quantity and the other is that supplier credit policies where credit terms are linked to the order quantity. In the latter case, suppliers use favorable credit terms to encourage customers to order large quantities. In other words, the favorable credit terms apply only at large order quantities and are used in place of quantity discounts. In this regard, this paper is only concerned with the latter case. Combining Chung (1989) and Khouja and Mehrez (1996), we can unify cases (a)–(c) in Chung (1989). Consequently, this paper generalizes Chung (1989). 2. The model This section analyzes the trade credit problem using DCF approach to fully recognize the time value of money in determining the optimal policy where trade credit depends on the ordering quantity. The following notations and assumptions will be used throughout: 2.1. Notations T = the inventory cycle time, which is a decision variable; C = the purchase cost per unit; D = the demand rate per unit time; h = the out-of-pocket inventory-carrying costs as a proportion of the value of inventory per unit time; r = the opportunity cost (i.e. the discount rate) per unit time;

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K = the ordering cost; M = the credit period; W = quantity at which the delay in payments is permitted;

Furthermore, the present value of all future cash-flows in this case is PV 1 ðTÞ ¼

ðr þ hÞCDT þ rK hCD  2 rð1  erT Þ r hCDT þ rK ðr  hÞCD PV 2 ðTÞ ¼ þ rð1  erT Þ r2

PV 1 ðTÞ ¼

if T > 0; if T > 0;

PV 3 ðTÞ ¼

hCDT þ rK þ ðT  MÞrCDerM þ CDð1  erM Þ hCD  2 rð1  erT Þ r

PV 1 ðTÞ ¼ the present 8 > PV 1 ðTÞ W < M> ¼ PV 2 ðTÞ > D : PV 3 ðTÞ

if T > 0;

if 0 < T <

;

if W 6 T < M; D if M 6 T;

PV 1 ðTÞ ¼ the present value of all future cash-flow cost when ( ; PV 1 ðTÞ if 0 < T < W W D ¼ M6 D 6 T; PV 3 ðTÞ if W D T* = the optimal cycle time of PV1(T) when T > 0. 2.2. Assumptions (1) (2) (3) (4) (5)

The demand for the item is constant with time. The ordering lead time is zero. Shortage are not allowed. Time period is infinite. If Q < W, the delay in payment is not permitted. Otherwise, certain fixed trade credit period M is permitted. That is, . Q < W holds if and only if T < W D (6) During the credit period, the firm makes payment to the supplier immediately after use of the materials. On the last day of the credit period, the firm pays the remain balance.

The above assumption (5) lead to the two cases to discuss: (A) and (B) M 6 W . M>W D D Additionally, this paper is concerned with cases (1)–(3) in Chung (1989). The present value of cash-flows in above three cases are discussed as follows: Case 1: Instantaneous cash-flows (the case of the basic EOQ model) Case 1 presents the DCF approach for the EOQ model under the assumption of instantaneous inventory holding costs, which implies the simultaneity of the arrival of an order and the capital investment in inventories. Therefore, the present value of the ordering cost can be shown as K þ KerT þ Ke2rT þ    ¼

K 1  erT

the present value of the purchase cost can be shown as CDT þ CDTerT þ CDTe2rT þ    ¼

CDT 1  erT

and the present value of the out-of-pocket inventory-carrying cost can be shown as Z T Z T hC DðT  tÞert dt þ DðT  tÞerðTþtÞ dt 0

þ

C

Z

T

Dert dt þ

0

T 0

 DðT  tÞerð2TþtÞ dt þ    ¼

hCDT hCD  2 : rð1  erT Þ r

Z

T

DerðTþtÞ dt þ

0

Z

T

0

 CD Derð2TþtÞ dt þ    ¼ : r

Furthermore, the present value of all future cash-flows in this case is PV 2 ðTÞ ¼

hCDT þ rK ðr  hÞCD þ rð1  erT Þ r2

if T > 0:

Case 3: Credit only on units in stock when T P M Case 3 deals with a similar situation to case 2. During the credit period, the firm makes payment to the supplier immediately after the use of the materials. On the last day of the credit period, the firm pays the remaining balance. Additionally, the credit period is shorter than the inventory length in this case. The present value of the purchase cost can be shown as Z M Dert dt þ DðT  MÞerM C 0

þ

Z

M

 DerðTþtÞ dt þ DðT  MÞerðTþMÞ þ   

0

CD½ðT  MÞrerM þ ð1  erM Þ : ¼ rð1  erT Þ Furthermore, the present value of all future cash-flows in this case is hCDT þ rK þ ðT  MÞrCDerM þ CDð1  erM Þ hCD  2 rð1  erT Þ r if T > 0:

PV 3 ðTÞ ¼

Our objective is to minimize the present value of all future cashflow cost PV1(T). That is, Minimize

PV 1 ðTÞ

Subjective to

T > 0:

From now on, we will discuss the situations of the two cases, respectively. (A) Suppose that M > W . D For this case, combining the cases discussed in Chung (1989), we have 8 W > < PV 1 ðTÞ if 0 < T < D ; ðaÞ W ð1Þ PV 1 ðTÞ ¼ PV 2 ðTÞ if D 6 T < M; ðbÞ > : ðcÞ PV 3 ðTÞ if M 6 T; where PV 1 ðTÞ ¼

ðr þ hÞCDT þ rK hCD  2 rð1  erT Þ r

if T > 0;

ð2Þ

PV 2 ðTÞ ¼

hCDT þ rK ðr  hÞCD þ rð1  erT Þ r2

if T > 0

ð3Þ

0

Z

if T > 0:

Case 2: Credit only on units in stock when T 6 M Case 2 assumes the existence of the credit period M. During the credit period, the firm makes payment to the supplier immediately after the use of the materials. On the last day of the credit period, firm pays the remaining balance. Additionally, the credit period is greater than the inventory cycle length in this case. The present value of the purchase cost can be shown as

value of all future cash-flows cost when W D

ðr þ hÞCDT þ rK hCD  2 rð1  erT Þ r

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and

PV 02

rM

PV 3 ðTÞ ¼

hCDT þ rK þ ðT  MÞrCDe þ CDð1  erM Þ rð1  erT Þ hCD if T > 0:  2 r

ð4Þ

From the following lemma, we shall find that PV1(T) > PV2(T) if T > 0. rT

      W W erð D Þ W W hCD erð D Þ  1  r ¼   r2 K 2 W D D r 1  erð D Þ

and PV 02 ðMÞ ¼ PV 03 ðMÞ ¼

rT

Lemma 1. rTe  e + 1 > 0 if rT > 0. 0

Proof. Let h(x) = xex  ex + 1 for x P 0. Then we have h (x) = xex > 0 for x > 0. So h(x) is increasing on x P 0. We get h(x) > h(0) = 0 for x > 0. Let x = rT. We get rTerT  erT + 1 > 0 if rT > 0. We have completed the proof. h From Eqs. (2) and (3), we have PV 1 ðTÞ  PV 2 ðTÞ ¼

rð1  erM Þ2

fhCDðerM  1  rMÞ  r2 Kg:

Furthermore, we let    W W D1 ¼ ðr þ hÞCD erð D Þ  1  r  r2 K; D    W W D2 ¼ hCD erð D Þ  1  r  r2 K D

ð10Þ

ð11Þ ð12Þ

D3 ¼ hCDðerM  1  rMÞ  r2 K:

ð13Þ PV 02 ðTÞ

if T > 0:

ð5Þ W 

W 

Since PV2(M) = PV3(M) and PV 1 D > PV 2 D , PV1(T) is continuous . except T ¼ W D (B) Suppose that M 6 W . D , PV1(T) can be expressed as follows: When M 6 W D ( PV 1 ðTÞ if 0 < T < W ; ðaÞ D ð6Þ PV 1 ðTÞ ¼ 6 T: ðbÞ PV 2 ðTÞ if W D From the following lemma, we get that PV1(T) > PV3(T) if T P M. Lemma 2. (1  erM)rT  erM(erM  rM  1) > 0 if T P M. Proof. Let k1(T) = (1  erM)rT  erM(erM  rM  1) for T > 0, then 0 we have k1 ðTÞ ¼ ð1  erM Þr > 0: Hence k1(T) is increasing on T > 0. However, k1(M) = erM + rM  1 > (1  rM) + rM  1 = 0, we get that k1(T) P k1 (M) > 0 for T P M. Consequently, (1  erM)rT  erM(erM  rM  1) > 0 if T P M. This completes the proof. h Form Eqs. (2) and (4), we have PV 1 ðTÞ  PV 3 ðTÞ ¼

erM

and

CDðrTerT  erT þ 1Þ : rðerT  1Þ

Lemma 1 implies that PV 1 ðTÞ > PV 2 ðTÞ

ð9Þ

CD½ð1  e

rM

rM

rM

ÞrT  e ðe rð1  erT Þ

 rM  1Þ

:

Lemma 2 implies that PV1(T) > PV3(T) if T P M. Further, PV1(T) is continuous except T ¼ W . D

and the following proposition implies D1 > D2. Proposition 1. Let P(x) = ex  1  x for x > 0. Then P(x) > 0 for x > 0. 2

n

Proof. From ex ¼ 1 þ x þ x2! þ    þ xn! þ   , we have PðxÞ ¼ ex  1 P xk x¼ 1 kP2 k! > 0 for all x > 0. We have completed the proof. h By Proposition 1, it is easy to show that D1 > D2. Eqs. (11)–(13) yield   W W < 0 if and only if T 1 > ; D1 < 0 if and only if PV 01 D D   W W D2 < 0 if and only if PV 02 < 0 if and only if T 2 > ; D D D3 < 0

if and only if

D3 < 0

if and only if

PV 02 ðMÞ PV 03 ðMÞ

< 0 if and only if < 0 if and only if

T 2 T 3

ð14Þ ð15Þ

> M;

ð16Þ

> M:

ð17Þ

Therefore, we have the following results. Theorem 1

3. A theorem (A) Suppose that M >

Since PV2(T) is convex on T > 0, which implies that is increas , so we get that D3 > D2 if ing on T > 0. We have PV 02 ðMÞ > PV 02 W D . M>W D From Eqs. (8) and (9), we have    

   W W W erð D Þ W r ðW DÞ  r CD e  PV 02 ¼  1 PV 01 2 W D D D 1  erð D Þ

W . D

Eqs. (2)–(4) are coincide with those of Eqs. (6), (12) and (16) in Chung (1989). Moreover, Chung and Huang (2000) has proved that PV1(T), PV2(T) and PV3(T) are convex on (0, 1). dPV ðT  Þ

Let T i be such that dTi i ¼ 0 ði ¼ 1; 2; 3Þ. By convexity of PVi(T) (i = 1, 2, 3), we have 8  > < 0 if 0 < T < T i ; ðaÞ dPV i ðTÞ <  ð7Þ ðbÞ ¼ 0 if T ¼ T i ; dT > : ðcÞ > 0 if T i < T: Eq. (7) imply that PVi(T) is decreasing on ð0; T i  and increasing on ½T i ; 1Þ ði ¼ 1; 2; 3Þ. Eqs. (2)–(4) yield

      W W erð D Þ W r ðW 2 DÞ  1  r PV 01 ðr þ hÞCD e K ; ¼   r 2 W D D r 1  erð D Þ ð8Þ

(1) If D1 > 0, D2 P 0 and D3 > 0, then PV 1 ðT  Þ ¼ minfPV 1 ðT 1 Þ; associated with the least cost. g. Hence T* is T 1 or W PV 1 W D D (2) If D1 > 0, D2 < 0 and D3 > 0, then PV 1 ðT  Þ ¼ PV 1 ðT 2 Þ. Hence T* is T 2 . (3) If D1 > 0, D2 < 0 and D3 6 0, then PV 1 ðT  Þ ¼ PV 1 ðT 3 Þ. Hence T* is T 3 . (4) If D1 6 0, D2 < 0 and D3 > 0, then PV 1 ðT  Þ ¼ PV 1 ðT 2 Þ. Hence T* is T 2 . (5) If D1 6 0, D2 < 0 and D3 6 0, then PV 1 ðT  Þ ¼ PV 1 ðT 3 Þ. Hence T* is T 3 .

Proof   > 0, (1) If D1 > 0, D2 P 0 and D3 > 0, which imply that PV 01 W D   0 W 0 0 PV 2 D P 0, PV 2 ðMÞ > 0 and PV 3 ðMÞ > 0. Eqs. (14)–(17) , T 2 6 W , T 2 < M and T 3 < M, respectively. imply that T 1 < W D D Furthermore, Eq. (7a–c) implies that (i) PV3(T) is increasing on [M, 1). ; MÞ. (ii) PV2(T) is increasing on ½W D

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PV1(T) is decreasing on ð0; T 1  and increasing on Þ. ½T 1 ; W D Combining (i), (ii) and (iii), we conclude that PV1(T) has the Þ and PV1(T) has the minminimum value at T ¼ T 1 on ð0; W D on ½W ; 1Þ. Hence PV 1 ðT  Þ ¼ imum value at  T¼ W D D  W associminfPV 1 ðT 1 Þ; PV 1 D g. Consequently, T* is T 1 or W D ated with the least cost.   > 0, If D1 > 0, D2 < 0 and D3 > 0, which imply that PV 01 W D   0 W 0 0 PV 2 D < 0, PV 2 ðMÞ > 0 and PV 3 ðMÞ > 0. Eqs. (14)–(17) , T 2 > W , T 2 < M and T 3 < M, respectively. imply that T 1 < W D D Furthermore, Eq. (7a–c) implies that (i) PV3(T) is increasing on [M, 1). ; T 2  and increasing on (ii) PV2(T) is decreasing on ½W D  ½T 2 ; M. Þ. (iii) PV1(T) is decreasing on ð0; T 1  and increasing on ½T 1 ; W D Combining (i), (ii) and (iii), we conclude that PV1(T) has Þ and PV1(T) has the the minimum value at T ¼ T 1 on ð0; W D ; 1Þ. Since PV1(T) > PV2(T) if minimum value at T ¼ T 2 on ½W D T > 0, then PV 1 ðT  Þ ¼ PV 1 ðT 2 Þ. Consequently, T* is T 2 .   If D1 >0, D2 < 0 and D3 6 0, which imply that PV 01 W > 0, D 0 W 0 0 PV 2 D < 0, PV 2 ðMÞ 6 0 and PV 3 ðMÞ 6 0. Eqs. (14)–(17) , T 2 > W , T 2 P M and T 3 P M, respectively. imply that T 1 < W D D Furthermore, Eq. (7a–c) implies that (i) PV3(T) is decreasing on ½M; T 3  and increasing on ½T 3 ; 1. ; M. (ii) PV2(T) is decreasing on ½W D Þ. (iii) PV1(T) is decreasing on ð0; T 1  and increasing on ½T 1 ; W D Combining (i), (ii) and (iii), we conclude that PV1(T) has the Þ and PV1(T) has the minminimum value at T ¼ T 1 on ð0; W D ; 1Þ. Since PV2(T) is decreasing imum value at T ¼ T 3 on ½W D and T 2 P M > W , we have PV 1 ðT 1 Þ > PV 2 ðT 1 Þ, ð0; T 2 , T 1 < W D D  PV 2 ðT 1 Þ > PV 2 ðMÞ and PV 2 ðMÞ ¼ PV 3 ðMÞ > PV 3 ðT 3 Þ. Hence we conclude that PV1(T) has the minimum value at T ¼ T 3 on (0, 1). Consequently, T* is T 3 .   6 0, If D1 6 0, D2 < 0 and D3 > 0, which imply that PV 01 W D   6 0, PV 02 ðMÞ > 0 and PV 03 ðMÞ > 0. Eqs. (14)–(17) PV 01 W D , T 2 > W , T 2 < M and T 3 < M. Furthermore, imply that T 1 P W D D Eq. (7a–c) implies that (i) PV3(T) is increasing on [M, 1). ; T 2  and increasing on (ii) PV2(T) is decreasing on ½W D  ½T 2 ; M. Þ. (iii) PV1(T) is decreasing on ð0; W   W  WD  > PV and PV Since PV 1 W 2 D 2 D > PV 2 ðT 2 Þ, combining (i), D (ii) and (iii), we conclude that PV1(T) has the minimum value at T ¼ T 2 on (0, 1). Consequently, T* is T 2 .   6 0, If D1 6 0, D2 < 0 and D3 6 0, which imply that PV 01 W D   0 W 0 0 PV 2 D < 0, PV 2 ðMÞ 6 0 and PV 3 ðMÞ 6 0. Eqs. (14)–(17) , T 2 > W , T 2 P M and T 3 P M, respectively. imply that T 1 P W D D Furthermore, Eq. (7a–c) implies that (i) PV3(T) is decreasing on ½M; T 3  and increasing on ½T 3 ; 1Þ. ; M. (ii) PV2(T) is decreasing on ½W D Þ. (iii) PV1(T) is decreasing on ð0; W D     Since PV 1 W > PV 2 W , combining (i), (ii) and (iii), we conD D clude that PV1(T) has the minimum value at T ¼ T 3 on (0, 1). Consequently, T* is T 3 . (iii)

(2)

(3)

(4)

(5)

Combining the above arguments, we have completed the proof of Theorem 1. h . (B) Suppose that M 6 W D , PV1(T) can be expressed as follows: When M 6 W D ( PV 1 ðTÞ ¼

PV 1 ðTÞ

if 0 < T < W ; D

PV 2 ðTÞ

if

W D

6 T:

Eq. (4) yield that (      W W erð D Þ W r ðW rM DÞ  1  r CDðh þ re PV 03 Þ e ¼ W D D rð1  erð D Þ Þ2 )  rCDð1  erM  rMerM Þ  r 2 k :

ð18Þ

We let    W W D4 ¼ CDðh þ rerM Þ erð D Þ  1  r  rCDð1  erM  rMerM Þ  r 2 k: D ð19Þ From Eqs. (11), (19) and Proposition 1, we have       W W 1 þerM ðerM rM 1Þ >0: D1 D4 ¼CDr ð1erM Þ erð D Þ r D Therefore, we get that D1 > D4. From Eq. (19), we also find that   W W D4 < 0 if and only if PV 03 < 0 if and only if T 3 > : D D

ð20Þ

Then, we have the following result. Theorem 2   g. (1) If D1 > 0 and D4 P 0, then PV 1 ðT  Þ ¼ minfPV 1 ðT 1 Þ; PV 1 W D  associated with the least cost. Hence T* is T 1 or W D (2) If D1 > 0 and D4 < 0, then PV 1 ðT  Þ ¼ minfPV 1 ðT 1 Þ; PV 1 ðT 3 Þg. Hence T* is T 1 or T 3 associated with the least cost. (3) If D1 6 0 and D4 < 0, then PV 1 ðT  Þ ¼ PV 1 ðT 3 Þ. Hence T* is T 3 .

Proof   > 0 and (1) If D1 > 0 and D4 P 0, which imply that PV 01 W D 0 W and PV 3 D P 0. Eqs. (14) and (20) imply that T 1 < W D . Furthermore, Eq. (7a–c) implies that T 3 6 W D ; 1Þ. (i) PV3(T) is increasing on ½W D Þ. (ii) PV1(T) is decreasing on ð0; T 1  and increasing on ½T 1 ; W D Combining (i) and (ii), we conclude that PV1(T) has the minÞ and PV1 (T) has the minimum imum value at T ¼ T 1 on ð0; W D on ½W ; 1Þ. Hence PV 1 ðT  Þ ¼ minfPV 1 ðT 1 Þ; value at T ¼ W D D   associated with the PV 1 W g. Consequently, T* is T 1 or W D D least cost.   > 0 and (2) If D1 > 0 and D4 < 0, which imply that PV 01 W D 0 W and PV 3 D < 0. Eqs. (14) and (20) imply that T 1 < W D . Furthermore, Eq. (7a–c) implies that T 3 > W D ; T 3  and increasing on (i) PV3(T) is decreasing on ½W D  ½T 3 ; 1Þ. Þ. (ii) PV1(T) is decreasing on ð0; T 1  and increasing on ½T 1 ; W D Combining (i) and (ii), we conclude that PV1(T) has the minÞ and PV1 (T) has the minimum imum value at T ¼ T 1 on ð0; W D ; 1Þ. Hence PV 1 ðT  Þ ¼ minfPV 1 ðT 1 Þ; value at T ¼ T 3 on ½W D PV 1 ðT 3 Þg. Consequently, T* is T 1 or T 3 associated with the least cost.   6 0 and (3) If D1 6 0 and D4 < 0, which imply that PV 01 W D   0 W and PV 3 D < 0. Eqs. (14) and (20) imply that T 1 P W D . Furthermore, Eq. (7a–c) implies that T 3 > W D ; T 3  and increasing on (i) PV3(T) is decreasing on ½W D  ½T 3 ; 1Þ. Þ. (ii) PV1(T) is decreasing on ð0; W D Combining (i) and (ii), we conclude that PV1(T) is decreasing Þ and PV1(T) has the minimum value at T ¼ T 3 on on ð0; W D     ; 1Þ. Since PV 1 W ½W > PV 3 W , we conclude that PV1(T) D D D has the minimum value at T ¼ T 3 on (0, 1). Consequently, T* is T 3 .

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Combining the above arguments, we have completed the proof of Theorem 2.

Example 8. If r = $0.01/$, M = 2 and W = 33, then W ¼ 2:2 > M, D D1 = 977560 < 0, and D4 = 72270 < 0. Using Step 3 (3), we get T  ¼ T 3 ¼ 2:53 and PV1(T*) = 1891.5.

4. The algorithm In this section, we shall combine Section 3 to outline the algorithm to help us to decide the optimal cycle time and optimal order quantity. The algorithm Step 1: If M 6 W , then go to Step 3. Otherwise, go to Step2. D Step 2: associ(1) If D1 > 0, D2 P 0 and D3 > 0, then T* is T 1 or W D ated with the least cost. (2) If D1 > 0, D2 < 0 and D3 > 0, then T* is T 2 . (3) If D1 > 0, D2 < 0 and D3 6 0, then T* is T 3 . (4) If D1 6 0, D2 < 0 and D3 > 0, then T* is T 2 . (5) If D1 6 0, D2 < 0 and D3 6 0, then T* is T 3 . Step 3: (1) (2) (3)

6. Special cases , PV1(T) can be expressed as follows: When M > W 8 D W PV > < 1 ðTÞ if 0 < T < D ; PV 1 ðTÞ ¼ PV 2 ðTÞ if W 6 T < M; D > : PV 3 ðTÞ if M 6 T: On the other hand, when M 6 W , PV1(T) can be expressed as D follows: ( PV 1 ðTÞ if 0 < T < W ; D PV 1 ðTÞ ¼ 6 T: PV 3 ðTÞ if W D There are the following cases to occur: (a) When M = 0, then we have PV 1 ðTÞ ¼ PV 1 ðTÞ for all T > 0:

associated with If D1 > 0 and D4 P 0, then T* is T 1 or W D the least cost. If D1 > 0 and D4 < 0, then T* is T 1 or T 3 associated with the least cost. If D1 6 0 and D4 < 0, then T* is T 3 .

According to the Intermediate Value Theorem Thomas and Finney (1996) and Chung (1999), the bisection method can be locate T i ði ¼ 1; 2; 3Þ. 5. Numerical examples To illustrate the results. Let us apply the proposed method to solve the following numerical examples. The following parameters K = $5/order, C = $1, D = 15units and h = 0.1/unit are used from Examples 1–8. In addition, r = $0.3/$ is used from Examples 1–5. Example 1. If M = 30 and W = 100, then W D ¼ 6:6667 < M, D1 = 25.8844 > 0, D2 = 6.1336 > 0 and D3 = 12139 > 0. Using Step 2 * (1), we get T  ¼ W D ¼ 6:6667 and PV(T ) = 77.6665. ¼ 2 < M, D1 = 0.8827 > 0, Example 2. If M = 30 and W = 30, then W D D2 = 0.1168 < 0 and D3 = 12139 > 0. Using Step 2 (2), we get T  ¼ T 2 ¼ 2:29 and PV1(T*) = 66.4375. ¼ 2 < M, D1 = 0.8827 > 0, Example 3. If M = 2 and W = 30, then W D D2 = 0.1168 < 0 and D3 = 0.1168 < 0. Using Step 2 (3), we get T  ¼ T 3 ¼ 2:11 and PV1(T*) = 66.5202. ¼ 0:6667 < M, Example 4. If M = 30 and W = 10, then W D D1 = 0.3216 < 0, D2 = 0.4179 < 0 and D3 = 12139 > 0. Using Step 2 (4), we get T  ¼ T 2 ¼ 2:2867 and PV1(T*) = 66.4374. ¼ 0:6667 < M, Example 5. If M = 2 and W = 10, then W D D1 = 0.3216 < 0, D2 = 0.4179 < 0 and D3 = 0.1168 < 0. Using Step 2 (5), we get T  ¼ T 3 ¼ 2:11 and PV1(T*) = 66.5202. ¼ 3:3333 > M, Example 6. If r = $0.1/$, M = 2 and W = 50, then W D D1 = 0.1368 > 0, and D4 = 0.0936 > 0. Using Step 3 (1), we get T* = 2.28 and PV1 (T*) = 192.5079. ¼ 2:2 > M, Example 7. If r = $0.1/$, M = 2 and W = 33, then W D D1 = 0.0282 > 0, and D4 = 0.0051 < 0. Using Step 3 (2), we get T  ¼ T 3 ¼ 2:28 and PV1(T*) = 192.5079.

ð21Þ

Then Eq. (21) is consistent with Eq. (6) in Chung (1989). (b) When W = 1, then we have PV 1 ðTÞ ¼ PV 1 ðTÞ for all T > 0:

ð22Þ

Then Eq. (22) is consistent with Eq. (6) in Chung (1989) as well. (c) When W = 0, then we have

PV 2 ðTÞ if 0 < T < M; ðaÞ ð23Þ PV 1 ðTÞ ¼ ðbÞ PV 3 ðTÞ if T P M: Eqs. (23a,b) are consistent with Eqs. (12) and (16) in Chung (1989), respectively. Combining the above arguments, the inventory system in this paper generalizes Chung (1989). This case has been discussed in Chung and Huang (2000). Therefore, we have the following result. Theorem 3. Suppose that W = 0. (1) PV1(T) is convex on T > 0. (2) If T 2 > M, then T  ¼ T 3 . (3) If T 2 6 M, then T  ¼ T 2 . Theorem 3 has been discussed in Chung and Huang (2000). 7. Summary This paper presents the discounted cash-flows (DCF) approach for the analysis of the optimal inventory policy in the presence of trade credit depending on the ordering quantity. If Q < W, the delay in payment is not permitted. Otherwise, the fixed trade period M is and (2) M 6 W to be expermitted. There are two cases (1) M > W D D plored. Theorem 1 gives the solution procedure to find T* when . Theorem 2 gives the solution procedure to find T* when M>W D W M 6 D. Numerical examples are given to illustrate Theorems 1 and 2. Furthermore, an algorithm to find the optimal replenishment cycle time is presented. Finally, there are three special cases are discussed in this paper. Finally, this paper generalizes Chung (1989). References Chu, P., Chung, K.J., Lan, S.P., 1998. Economic order quantity of deteriorating items under permissible delay in payments. Computers and Operations Research 25, 817–824. Chung, K.H., 1989. Inventory control and trade credit revisited. Journal of the Operational Research Society 40, 495–498.

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Chung, K.J., 1999. Bounds on the optimum order quantity in a DCF analysis of the EOQ model under trade credit. Journal of Information and Optimization Sciences 20, 137–148. Chung, K.J., Huang, C.K., 2000. The convexities of the present value functions of all future cash outflows in inventory and credit. Production Planning and Control 11, 133–140. Goyal, S.K., 1985. Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society 36, 335–338.

Jamal, A.M.M., Sarker, B.R., Wang, S., 1997. An ordering policy for deteriorating items with allowable shortages and permissible delay in payment. Journal of the Operational Research Society 48, 826–833. Khouja, M., Mehrez, A., 1996. Optimal inventory policy under different supplier credits. Journal of Manufacturing Systems 15, 334–339. Thomas, G.B., Finney, R.L., 1996. Calculus with Analytic Geometry, ninth ed. Addison-Wesley Publishing Company Inc.