Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging

Int. J. Production Economics 144 (2013) 610–617

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Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging Liang-Yuh Ouyang a, Chun-Tao Chang b,n a b

Department of Management Sciences, Tamkang University, Tamsui, Taipei 25137, Taiwan, ROC Department of Statistics, Tamkang University, Tamsui, Taipei 25137, Taiwan, ROC

art ic l e i nf o

a b s t r a c t

Article history: Received 27 March 2012 Accepted 10 April 2013 Available online 25 April 2013

The traditional economic production quantity (EPQ) model assumes that the production products are all perfect. It is not always true in the real production system, due to imperfect production process or other factors, imperfect quality items may be produced. Furthermore, it is well-known that the total production-inventory costs can be reduced by reworking the imperfect quality items produced with a relatively smaller additional reworking and holding costs. In addition, the permissible delay in payments offered by the supplier is widely adopted in the practical business market. In this study, we explore the effects of the reworking imperfect quality items and trade credit on the EPQ model with imperfect production processes and complete backlogging. A mathematical model which includes the reworking and shortage costs, interest earned and interest charged is presented. Besides, an arithmetic-geometric mean inequality approach is employed and an algorithm is developed to find the optimal production policy. Furthermore, some numerical examples and sensitivity analysis are provided to demonstrate the proposed model. & 2013 Published by Elsevier B.V.

Keywords: Inventory Imperfect production process Complete backlogging Permissible delay in payments Arithmetic–geometric mean inequality

1. Introduction In the classical EPQ model, it is implicitly assumed that the production products are all perfect. However, in the real manufacturing circumstance, due to imperfect production process or other factors, the defective items may be produced. Several scholars have developed various analytical models to study the EPQ model with defective items. Rosenblatt and Lee (1986) were one of the early researchers who studied the effects of an imperfect production process on the optimal production cycle time for the classical economic manufacturing quantity (EMQ) model. Porteus (1986) introduced a relationship between process quality control and lot sizing. Zhang and Gerchak (1990) presented joint lot sizing and inspection policy in an economic order quantity (EOQ) model with random yield. Cheng (1991) developed an EOQ model with demand-dependent unit production cost and imperfect production processes. Ben-Daya (2002) formulated an integrated model with joint determination of EPQ and preventive maintenance level under an imperfect process. Lin et al. (2003) examined an integrated production-inventory model for imperfect production processes under inspection schedules. Recently, Sana (2010) developed a production-inventory model in an imperfect production process. Many related articles in EPQ models with

n

Corresponding author. Tel.: +886 2 26215656x2632; fax: +886 2 26209732. E-mail address: [email protected] (C.-T. Chang).

0925-5273/$ - see front matter & 2013 Published by Elsevier B.V. http://dx.doi.org/10.1016/j.ijpe.2013.04.027

imperfect quality items can be found such as Salameh and Jaber (2000), Sana et al. (2007), Yoo et al. (2009), Sana (2011), Sarker et al. (2010), Sarker and Moon (2011), Sarker (2012), Yoo et al. (2012) and their references. All of the above studies about EPQ models with imperfect production processes focused on determining the optimal production lot size. The issue that the imperfect quality items can be reworked was ignored. It is well-known that the total productioninventory costs can be reduced by reworking the imperfect quality items produced with a relatively smaller additional reworking and holding costs. Numerous studies on the problems of EPQ model with rework process have been discussed by Liu and Yang (1996), Hayek and Salameh (2001), Chiu (2003, 2008), Chiu et al. (2004, 2010), Jamal et al. (2004), Chiu and Chiu (2006), Taleizadeh et al. (2010) and their references. Actually, today trade credit is widespread and represents an important proportion of company finance. Businesses, especially small businesses, with limited financing opportunities, may be financed by their suppliers rather than by financial institutions (Petersen and Rajan, 1997). On the other hand, offering trade credit to retailers may encourage the supplier sales and reduce the onhand stock level (Emery, 1987). Goyal (1985) was the first to establish an EOQ model with a constant demand rate under the condition of a permissible delay in payments. Teng (2002) modified Goyal’s (1985) model by considering the difference between the selling price and purchase cost, and found that the economic replenishment interval and order quantity decrease under the

L.-Y. Ouyang, C.-T. Chang / Int. J. Production Economics 144 (2013) 610–617

permissible delay in payments in certain cases. Chang et al. (2003) developed an EOQ model with deteriorating items under supplier’s credits linked to ordering quantity. Tsao et al. (2011) proposed a production model with reworking imperfect items and trade credit. Numerous interesting and relevant paper related to trade credits such as Aggarwal and Jaggi (1995), Jamal et al. (1997), Chang (2004), Ouyang et al. (2005), Teng et al. (2005), Goyal et al. (2007), Liao (2008), Teng and Chang (2009), Chang et al. (2010), Musa and Sani (2012), Su (2012) and so on. Because trade credit is a widespread and popular payment method, thus, in order to respond to the real business behavior, we study an EPQ model with imperfect quality and complete backlogging when the supplier offers a permissible delay in payments. Some simple algebraic manipulations and an arithmetic–geometric mean inequality approach are employed to determine the optimal production lot size and backorder level. Besides, an algorithm is developed to find the optimal solution. Finally, some numerical examples and sensitivity analysis are presented to illustrate the proposed model.

2. Notation and assumptions The following notation and assumptions will be adopted in this article. Notation P λ K v c

production rate demand rate setup cost for each production run purchasing cost of raw material per unit production cost per item including purchasing cost and inspecting cost, c 4v s selling price per unit, s 4c h holding cost per item per unit time, excluding the interest charge b shortage cost per item per unit time x the proportion of imperfect quality items produced, where 0 ox o1 d the production rate of imperfect quality regular production process per unit time, where d ¼Px P1 the rate of reworking of imperfect quality items cR reworking cost for each imperfect quality item Q production lot size for each cycle B allowable backorder level T production cycle length H1 maximum level of on-hand inventory when regular production process stops H maximum level of on-hand inventory in units, when the reworking ends M permissible delay period offered by the supplier Ic the interest charged per dollar per unit time in stocks by the supplier Ie the interest earned per dollar per unit time TCi(Q, B) inventory total cost per cycle for case i, i¼1, 2, 3 TCUi(Q,B) inventory total cost per unit time for case i, i.e., TCUi(Q, B) ¼TCi(Q,B)/T, i¼1, 2, 3.

(5) The supplier provides the manufacturer a permissible delay in payments. During the trade credit period the account is not settled, generated sales revenue is deposited in an interest bearing account with interest rate I e . At the end of the permissible delay, the manufacturer pays off all units ordered, and starts paying for the interest charges on the raw material in stocks with interest rate I c .

3. Mathematical formulation First, a short problem description is provided. The manufacturer buys all raw materials Q units per order from the supplier to product and the unit purchasing price of raw material is v. The supplier offers the manufacturer a permissible delay period M. That is, the manufacturer buys raw materials at time zero and must pay the purchasing cost vQ at time M. The unit production cost is c and the unit selling price of the perfect items is s. The production process starts at time zero. A constant product rate P is considered during the regular production uptime. The process may generate x percent of imperfect quality items at a production rate d¼ Px. Thus, the produced items fall into two groups, the perfect and the imperfect. The production rate for perfect item ð1−xÞP is larger than the demand rate λ. All imperfect quality items are assumed to be reworkable at a rate of P1, and rework process starts when regular production process ends. All reworked items are assumed to become perfect items after rework process. Shortages are allowed and completely backlogged. In this situation, the production-inventory system follows the pattern depicted in Fig. 1. From Fig. 1, the expressions of production uptime t1 and t2, reworking time t3, production downtime t4, shortage permitted time t5, the maximum levels of on-hand inventory H1 and H, and the cycle length T are as follows: t1 ¼

B P−d−λ

ð1Þ

t2 ¼

H1 P−d−λ

ð2Þ

t3 ¼

xQ dQ ¼ P1 P1 P

ð3Þ

t4 ¼

H λ

ð4Þ

t5 ¼

B λ

ð5Þ

Assumptions (1) Each product is made by a raw material. (2) Production rate for perfect items is larger than demand rate, i. e., (1−x)P 4λ. (3) All imperfect quality items can be reworked and become perfect items. (4) Shortages are allowed and completely backlogged.

611

Fig. 1. Graphical representation of the inventory system.

612

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H 1 ¼ ðP−d−λÞ

Q −B P

ð6Þ


 λ dλ −B H ¼ H 1 þ ðP 1 −λÞ t 3 ¼ Q 1− − P P1 P

ð7Þ

and T ¼ t1 þ t2 þ t3 þ t4 þ t5 ¼

Q λ

ð8Þ

In addition, for convenience, we let t a ≡t 1 , t b ≡t 1 þ t 2 ¼ Q =P, t c ≡t 1 þ t 2 þ t 3 ¼ ðQ =PÞ þ ðxQ =P 1 Þ, and t d ≡t 1 þ t 2 þ t 3 þ t 4 ¼ ðQ −BÞ=λ. The inventory total cost per cycle consists of the following components: (a) (b) (c) (d)

The production cost is cQ. The repair cost is cR xQ . The setup cost is K. The holding cost is  
 H1 H1 þ H H dðt 1 þ t 2 Þ dðt 1 þ t 2 Þ t 3 t3 þ t4 þ h ðt 1 þ t 2 Þ þ t2 þ h 2 2 2 2 2

 h 1−x h λ 2 2 h B − QB þ 1− Q : ¼ 2λ 1−x−λ=P λ 2λ P

Fig. 3. Graphical representation of interest earned and interest charged for t a ≤M o t d .

(e) The shortage cost is ðb=2ÞBðt 1 þ t 5 Þ ¼ ðbB2 =2Þð1=ðPð1−xÞ−λÞþ 1=λÞ. (f) Interest earned and interest charged Based on the values of M, ta, and td, we have the following three possible cases: (1) M o t a , (2) t a ≤M o t d , and (3) M≥t d . These three cases are presented in Figs. 2–4. Case 1. M ot a In this case, the manufacturer starts production and replenishing shortage at time 0. As a result, the manufacturer accumulates revenue in an account that earns I e per dollar per year starting from 0 to M. The interest earned per cycle is sIe multiplied by the sum of the areas of two triangles BDC and BAC as shown in Fig. 2. Hence, the interest earned per cycle is " # λM 2 ðP−d−λÞM 2 ðP−dÞM 2 þ ¼ sI e : sI e 2 2 2 On the other hand, the manufacturer pays off all units sold by M at time M, keeps the profits, and starts paying for the interest charges on the items sold after M. The interest charged per cycle is vI c times the sum of the areas of two triangles MGA and DEF as shown in Fig. 2. Therefore, the interest charged per cycle is given

Fig. 4. Graphical representation of interest earned for M≥t d .

by vI c

"

# ðP−d−λÞðt a −MÞ2 λ ðt d −MÞ2 þ 2 2 " # ðP−dÞB2 Q2 QB ðP−dÞM 2 −Q M þ þ − : ¼ vI c λ 2λðP−d−λÞ 2λ 2

Case 2. t a ≤M o t d Since t a ≤M o t d , the interest earned per cycle is sI e multiplied by the sum of the areas of triangle BDC and trapezoid BAMC which are shown in Fig. 3. Hence, the interest earned per cycle is ( ) " # λM 2 ½ðM−t a Þ þ MB λM 2 B2 sIe þ þ MB− ¼ sI e : 2 2 2 2ðP−d−λÞ On the other hand, the interest charged per cycle is vI c times the area of the triangle DEF shown in Fig. 3. Therefore, the interest charged per cycle is given by " #   λ Q 2 B2 λ M 2 Q B −MQ þ MB : vI c ðt d −MÞ2 ¼ vI c þ þ − 2 λ 2λ 2λ 2 Case 3. M≥t d

Fig. 2. Graphical representation of interest earned and interest charged forM o t a .

In this case, the manufacturer receives the total revenue at time t d , and is able to pay the supplier the total purchase cost at time M. Since t d is shorter than or equal to the credit period M, the manufacturer faces no interest charged. On the other hand, the interest earned per cycle is sIe multiplied by the sum of the areas of two trapezoids BAMC and BEFC as shown in Fig. 4. As a result,

L.-Y. Ouyang, C.-T. Chang / Int. J. Production Economics 144 (2013) 610–617

the interest earned per cycle is   ½ðM−t a Þ þ MB ½ðM−t d Þ þ Mλ t d þ sI e 2 2 " # 2 2 ðP−dÞB Q QB þ MQ : ¼ sI e − − þ λ 2λ ðP−d−λÞ 2λ According to the above arguments, we can obtain the inventory total cost per cycle as follows: TCi(Q, B)¼production cost+repair cost+setup cost+holding cost +shortage cost+interest charged−interest earned, i¼ 1, 2, 3. Case 1 M o t a ðP−dÞM 2 TC 1 ðQ ; BÞ ¼ cQ þ cR xQ −vI c MQ þ K þ ðvI c −sIe Þ 2 

1 1−x h vI c ðb þ h þ vI c ÞB2 − þ QB þ 2λ 1−x−λ=P λ λ  
 h λ vI c 2 þ 1− þ Q : 2λ P 2λ

For convenience, we let G2 ¼ G1 ≡c þ cR x−vI c M

 1−x λsI e 1 þ vI c þ 40; U 2 ≡ðb þ hÞ 1−x−λ=P P 1−x−λ=P
 λ V 2 ¼ V 1 ≡h 1− þ vI c 4 0 and W 2 ¼ W 1 ≡h þ vI c 4 0: P Then, Eq. (13) can be rewritten as (
 1 W2 ðvI c −sI e ÞλM 2 U 2 B− Qþ TCU 2 ðQ ; BÞ ¼ 2Q U2 U2
 ðvI c −sI e ÞλM þ2 λG2 þ W 2 Q U2 ! )
 vI c −sI e W 22 2 2 þðvI c −sIe Þλ M 1− þ 2Kλ þ V 2 − Q2 : U2 U2 ð14Þ Case 3 M≥t d

ð9Þ

Hence, the inventory total cost per unit time is TCU 1 ðQ ; BÞ ¼ TC 1 ðQ ; BÞ=T 1  2λQ ðc þ cR x−vI c MÞ ¼ 2Q

TC 3 ðQ ; BÞ ¼ cQ þ cR xQ −sI e MQ þ K

 1 1−x h sIe ðb þ h þ s I e ÞB2 − þ QB þ 2λ 1−x−λ=P λ λ  
 h λ sI e 2 þ 1− þ Q : 2λ P 2λ

ð15Þ

Hence, the inventory total cost per unit time is

þ2λK þ λðvI c −sIe ÞPð1−xÞM 2 −2ðh þ vI c ÞQ B
  
  1−x λ B2 þ h 1− þ vI c Q 2 : þðb þ h þ vI c Þ 1−x−λ=P P ð10Þ For convenience, we let G1 ≡c þ cR x−vI c M;

 1−x ð1−xÞP ¼ ðb þ h þ vI c Þ 4 0; U 1 ≡ðb þ h þ vI c Þ 1−x−λ=P ð1−xÞP−λ
 λ þ vI c 4 0 and W 1 ≡h þ vI c 40: V 1 ≡h 1− P Then, Eq. (10) can be rewritten as (
2 1 W1 U 1 B− Q þ 2λG1 Q þ 2Kλ TCU 1 ðQ ; BÞ ¼ 2Q U1 ! ) W2 þλðvI c −sIe ÞPð1−xÞM 2 þ V 1 − 1 Q 2 U1

613

TCU 3 ðQ ; BÞ ¼ TC 3 ðQ ; BÞ=T 1 n 2λQ ðc−sI e M þ cR xÞ þ 2λK−2ðh þ sI e ÞQ B ¼ 2Q
  
  1−x λ B2 þ h 1− þ sIe Q 2 : þðb þ h þ sI e Þ 1−x−λ=P P ð16Þ For convenience, we let G3 ≡c−sI e M þ cR x
 1−x U 3 ≡ðb þ h þ sI e Þ 1−x−λ=P
 ð1−xÞP 4 0; ¼ ðb þ h þ sIe Þ ð1−xÞP−λ
 λ þ sI e 4 0 and W 3 ≡h þ sI e 4 0: V 3 ≡h 1− P

ð11Þ

Then, Eq. (16) can be rewritten as TCU 3 ðQ ; BÞ ¼

1 2Q

(


2 W3 U 3 B− Q þ 2λG3 Q þ 2Kλ þ U3

V 3−

! ) W 23 Q2 : U3

ð17Þ

Case 2 t a ≤ M o t d TC 2 ðQ ; BÞ ¼ cQ þ cR xQ −vI c MQ þ K þ ðvI c −sIe ÞλM 2 =2
   1 1−x 1 λ sIe ðb þ hÞB2 þ vIc þ B2 þ 2λ 1−x−λ=P 2λ Pð1−x−λ=PÞ  
 h vI c h λ vI c 2 1− þ Q : −ð þ ÞQ B þ ðvI c −sI e Þ M B þ λ 2λ P λ 2λ ð12Þ

4. Theoretical results In this section, simple algebraic manipulations and an arithmetic-geometric mean inequality approach are used to find the optimal production lot size and backorder level. The arithmetic-geometric mean inequality is: if a 40 and b 40, then pffiffiffiffiffiffi ða þ bÞ=2≥ ab, and the inequality holds when a ¼b. Case 1. M o t a

Hence, the inventory total cost per unit time is TCU 2 ðQ ; BÞ ¼ TC 2 ðQ ; BÞ=T 1 n 2λQ ðc þ cR x−vI c MÞ þ 2Kλ þ ðvI c −sIe Þλ2 M 2 ¼ 2Q þ2KλðvI c −sI e Þλ2 M 2 −2ðh þ vI c ÞQ B þ2ðvI c −sI e Þλ MB

  1−x λsIe 1 þvI c þ B2 þ ðb þ hÞ 1−x−λ=P P 1−x−λ=P  
 o λ þ vI c Q 2 : þ h 1− ð13Þ P

For minimizing TCU1(Q, B) in Eq. (11), we let
2 W1 B− Q ¼ 0; then U1 W1 Q B¼ U1

ð18Þ

Substituting Eq. (18) into Eq. (11), we get TCU 1 ðQ Þ≡TCU 1 ðQ ; BÞ ¼ λG1 þ

2Kλ þ λðvI c −sI e ÞPð1−xÞM 2 ðU 1 V 1 −W 21 ÞQ þ 2U 1 2Q

ð19Þ

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It can be shown that U 1 V 1 −W 21 40 (see Appendix A for proof). When 2Kλ þ λ ðvI c −sIe ÞPð1−xÞM 2 4 0, the three conditions proposed by Cardenas-Barron (2010) can be verified. Therefore, the arithmetic–geometric mean inequality can be used as optimization method to minimize the inventory total cost per unit time. That is, we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCU 1 ðQ Þ≥λG1 þ ½2Kλ þ λ ðvI c −sIe ÞPð1−xÞM 2  ðV 1 U 1 −W 21 Þ=U 1 ; and the equality holds when 2Kλ þ λðvI c −sI e ÞPð1−xÞM 2 ðU 1 V 1 −W 21 Þ Q ¼ 2U 1 2Q This implies the optimal production lot size (say Q 1 ) is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 1 ½2Kλ þ λ ðvI c −sI e ÞPð1−xÞM 2  ð20Þ Q1 ¼ U 1 V 1 −W 21

Hence, TCU1(Q) is a strictly increasing function on the open interval ðU 1 M½Pð1−xÞ−λ=W 1 ; ∞Þ. Consequently, the value of Q which minimizes TCU1(Q) does not exist. Based on the above results, the following lemma is obtained. Lemma 1. (1) if 2Kλ4Δ1 , then Q ¼ Q 1 is the optimal value which minimizes TCU1(Q). (2) if 2Kλ ≤Δ1 , then the value of Q which minimizes TCU1(Q) does not exist. Case 2. t a ≤M o t d

Similarly, for minimizing TCU2(Q, B) in Eq. (14), we let 2 ðvI c −sI e ÞλM 2 B− W ¼ 0; U2 Q þ U2

Therefore, the optimal backorder level (say B1) can be obtained as W1 W1 Q ¼ B1 ¼ U1 1 U1

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 1 ½2Kλ þ λðvI c −sIe ÞPð1−xÞM 2  U 1 V 1 −W 21

then B ¼

TCU 2 ðQ Þ≡TCU 2 ðQ ; BÞ 
 1 ðvI c −sI e ÞλM 2 λG2 þ W 2 Q ¼ 2Q U2
 vI c −sI e þðvI c −sI e Þλ2 M 2 1− U2 ! ) W 22 Q2 ; þ2Kλ þ V 2 − U2 ¼ λG2 þ W 2 ðvI c −sI e ÞλM=U 2 þ

Note that, when 2Kλ 4Δ1 , we have 2Kλ þ λ ðvI c −sI e ÞPð1−xÞM 2 " # ! V 1 U 21 4 ½Pð1−xÞ−λ2 −U −sI ÞPð1−xÞ M2 −λðvI c e 1 W 21

W 21

4 0:

Hence, Q 1 in Eq. (20) and B1 in Eq. (21) are well-defined. As a result, the corresponding minimum inventory total cost per unit time is TCU 1 ðQ 1 ; B1 Þ ¼ λG1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ ½2Kλ þ λðvI c −sIe ÞPð1−xÞM 2  ðV 1 U 1 −W 21 Þ=U 1 : Conversely, if 2Kλ ≤Δ1 , then U 1 M½Pð1−xÞ−λ=W 1 4 0, we obtain

for

any

given

TCU 1 ðq2 Þ−TCU 1 ðq1 Þ i
1 1  1h 2Kλ þ λðvI c −sI e ÞPð1−xÞM 2 ¼ − 2 q2 q1 1 ðU 1 V 1 −W 21 Þ þ ðq2 −q1 Þ þ 2U 1 " ! # 1 q2 −q1 W 21 2 ¼ V 1− q1 q2 −2Kλ−λðvI c −sI e ÞPð1−xÞM 2 q1 q2 U1 " ! 1 q2 −q1 W 2 U 21 2 V 1− 1 M ½Pð1−xÞ−λ2 4 2 q1 q2 U 1 W 21 i −2Kλ−λðvI c −sI e ÞPð1−xÞM 2 " ! 1 q2 −q1 V 1 U 21 ½Pð1−xÞ−λ2 −U ¼ M 2 −2Kλ 1 2 q1 q2 W 21 i −λðvI c −sI e ÞPð1−xÞM 2 ≥0:

2Kλ þ ðvI c −sIe Þλ2 M 2 ðU 2 −vI c þ sI e Þ=U 2 ðU 2 V 2 −W 22 ÞQ þ : 2U 2 2Q ð25Þ

It can be shown that U 2 V 2 −W 22 40 (the proof is similar to Appendix A, it is omitted here). When 2Kλ þ ðvI c −sIe Þλ2 M 2 ðU 2 −vI c þ sI e Þ=U 2 4 0, similar to the arguments as in Case 1, the arithmetic-geometric mean inequality can be used as optimization method to minimize the inventory total cost per unit time. That is, we obtain

þλ ðvI c −sI e ÞPð1−xÞM 2 U 1 M2

ð24Þ

Substituting Eq. (24) into Eq. (14), we get ð21Þ

To ensure M o t a , using t a ≡t 1 , Eq. (1), d ¼ Px and B1 ¼ W 1 Q 1 =U 1 , which is equivalent to Q 1 4U 1 M½Pð1−xÞ−λ=W 1 . Substituting Eq. (20) into this inequality, we obtain that if and only if 2Kλ 4 Δ1 , then M o t a , (22) where " # ! 2 2 V 1 U1 Δ1 ¼ ½Pð1−xÞ−λ −U 1 −λðvI c −sIe ÞPð1−xÞ M 2 : W 21

¼ ½Pð1−xÞ−λ2 ðU 1 V 1 −W 21 Þ

W2 ðvI c −sI e ÞλM Q− U2 U2

ð23Þ q2 4 q1 4

TCU 2 ðQ Þ≥λG2 þ W 2 ðvI c −sI e ÞλM=U 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 2Kλ þ ðvI c −sIe Þλ2 M 2 ½1−ðvI c −sI e Þ=U 2 ðU 2 V 2 −W 22 Þ=U 2 and the equality holds when 2Kλ þ ðvI c −sI e Þλ2 M 2 ½1−ðvI c −sIe Þ=U 2  ðU 2 V 2 −W 22 Þ Q ¼ : 2Q 2U 2 This implies the optimal production lot size (say Q 2 ) is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KλU 2 þ ðvI c −sI e Þλ2 M 2 ðU 2 −vI c þ sI e Þ Q2 ¼ : ð26Þ U 2 V 2 −W 22 Therefore, the optimal backorder level (say B2) can be obtained as W2 ðvI c −sIe ÞλM Q − U2 U2 2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W 2 2KλU 2 þ ðvI c −sI e Þλ2 M 2 ðU 2 −vI c þ sI e Þ ¼ U2 U 2 V 2 −W 22

B2 ¼

−

ðvI c −sI e ÞλM : U2

ð27Þ

To ensure t a ≤M o t d , using t a ≡t 1 , t d ≡t 1 þ t 2 þ t 3 þ t 4 ¼ ðQ 2 −B2 Þ=λ, Eq. (1), d ¼ Px and B2 ¼ W 2 Q 2 =U 2 −ðvI c −sI e ÞλM=U 2 , which is equivalent to   λM½U 2 −ðvI c −sI e Þ U2 ðvI c −sI e ÞλM ½Pð1−xÞ−λM þ oQ2 ≤ : U 2 −W 2 U2 W2

L.-Y. Ouyang, C.-T. Chang / Int. J. Production Economics 144 (2013) 610–617

Substituting Eq. (26) into this inequality, we obtain that if and only if Δ2 o 2Kλ ≤Δ3 ; then t a ≤M ot d ; where ( Δ2 ¼

and

ð28Þ

 ) ðU 2 þ sI e −vI c Þ2 λ2 ðU 2 V 2 −W 22 Þ 2 U 2 −vI c þ sI e ⋅ −ðvI −sI Þλ M2 ; c e U2 U2 ðU 2 −W 2 Þ2

½ðP−d−λÞU 2 þ ðvI c −sIe Þλ2 ðU 2 V 2 −W 22 Þ ⋅ U2 W 22   U 2 −vI c þ sI e −ðvI c −sIe Þλ2 M2 : U2

TCU 2 ðQ 2 ; B2 Þ ¼ λG2 þ W 2 ðvI c −sI e ÞλM=U 2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n o þ 2Kλ þ ðvI c −sI e Þλ2 M 2 ½1−ðvI c −sI e Þ=U 2  ðU 2 V 2 −W 22 Þ=U 2 :

ð29Þ Conversely, if 2Kλ 4 Δ3 , then for any given λM½U 2 −ðvI c −sI e Þ=  ðU 2 −W 2 Þ oq1 oq2 ≤ðU 2 =W 2 Þ ½Pð1−xÞ−λM þ ðvI c −sI e ÞλM=U 2 , we can show that TCU2(q2 ) −TCU 2 ðq1 Þ o0 (the proof is similar to Case 1, it is omitted here). Hence, TCU2(Q) is a strictly decreasing function on the half-closed interval (λM½U 2 −ðvI c −sI e Þ=ðU 2 −W 2 Þ,ðU 2 =W 2 Þ  ½Pð1−xÞ−λM þ ðvI c −sIe ÞλM=U 2 ]. Consequently, TCU2(Q) has a minimum value at the boundary  point Q ¼ ðU 2 =W 2 Þ ½Pð1−xÞ−λM þ ðvI c −sI e ÞλM=U 2 . Likewise, if 2Kλ ≤Δ2 , then for any given λM½U 2 −ðvI c −sI e Þ=  ðU 2 −W 2 Þ oq1 o q2 ≤ðU 2 =W 2 Þ ½Pð1−xÞ−λM þ ðvI c −sIe ÞλM=U 2 , we have TCU 2 ðq2 Þ−TCU 2 ðq1 Þ 4 0. Hence, TCU2(Q) is a strictly increasing function on the half-closed interval (λM½U 2 −ðvI c −sI e Þ=ðU 2 −W 2 Þ,  ðU 2 =W 2 Þ ½Pð1−xÞ−λM þ ðvIc −sI e ÞλM=U 2 ]. Consequently, the value of Q which minimizes TCU2(Q) does not exist. Based on the above results, the following lemma is obtained. Lemma 2. (1) if Δ2 o 2Kλ ≤Δ3 , then Q ¼ Q 2 is the optimal value which minimizes TCU2(Q).  (2) if 2Kλ 4 Δ3 , then Q ¼ ðU 2 =W 2 Þ ½Pð1−xÞ−λM þ ðvI c −sI e ÞλM=U 2 is the optimal value which minimizes TCU2(Q). (3) if 2Kλ ≤Δ2 , then the value of Q which minimizes TCU2(Q) does not exist. Case 3. M≥t d Likewise, for minimizing TCU3(Q, B) in Eq. (17), we let B−

B¼

W3 Q U3

2 ¼ 0; then

W3 Q: U3

ð30Þ

Substituting Eq. (30) into Eq. (17), it gets TCU 3 ðQ Þ≡TCU 3 ðQ ; BÞ ¼ λG3 þ

Kλ ðU 3 V 3 −W 23 ÞQ þ : Q 2U 3

Kλ ðU 3 V 3 −W 23 Þ Q ¼ : Q 2U 3 This implies the optimal production lot size (say Q 3 ) is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2KλU 3 Q3 ¼ : ð32Þ U 3 V 3 −W 23 Therefore, the optimal backorder level (say B3 ) can be obtained as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi W3 W3 2KλU 3 B3 ¼ : ð33Þ Q3 ¼ U3 U 3 U 3 V 3 −W 23

Note that, when 2Kλ4Δ2 , similar to the arguments as in Case 1, it can be shown that Q 2 in Eq. (26) and B2 in Eq. (27) are welldefined. As a result, the corresponding minimum inventory total cost per unit time is




geometric mean inequality, we obtain qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCU 3 ðQ Þ≥λG3 þ 2KλðU 3 V 3 −W 23 Þ=U 3 and the equality holds when

(

Δ3 ¼

615

ð31Þ

It can be shown that U 3 V 3 −W 23 40 (the proof is similar to Appendix A, it is omitted here), hence, by using the arithmetic–

To ensure M≥t d , using t d ≡t 1 þ t 2 þ t 3 þ t 4 ¼ðQ 3 −B3 Þ=λ and B3 ¼ W 3 Q 3 =U 3 , which is equivalent to Q 3 ≤U 3 λM=ðU 3 −W 3 Þ. Substituting Eq. (32) into this inequality, we obtain that if and only if 2Kλ ≤Δ4 ; then M≥t d ;

ð34Þ

where λ2 U 3 ðU 3 V 3 −W 23 Þ 2 Δ4 ¼ M : ðU 3 −W 3 Þ2 Therefore, when 2Kλ ≤Δ4 , the corresponding minimum inventory total cost per unit time is qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi TCU 3 ðQ 3 ; B3 Þ ¼ λG3 þ 2KλðU 3 V 3 −W 23 Þ=U 3 : ð35Þ then for any given 0 Conversely, if 2Kλ 4 Δ4 , oq1 o q2 ≤ U 3 λM=ðU 3 −W 3 Þ, we can show that TCU3(q2 )− TCU 3 ðq1 Þ o 0. Hence, TCU3(Q) is a strictly decreasing function on the half-closed interval ð0; U 3 λM=ðU 3 −W 3 Þ. Consequently, TCU3(Q) has a minimum value at the boundary point Q ¼ U 3 λM=ðU 3 −W 3 Þ. Based on the above results, the following lemma is obtained. Lemma 3. (1) if 2Kλ ≤ Δ4 , then Q ¼ Q 3 is the optimal value which minimizes TCU3(Q). (2) if 2Kλ 4 Δ4 , then Q ¼ U 3 λM=ðU 3 −W 3 Þis the optimal value which minimizes TCU3(Q). In order to determine the optimal values of Q and B (say Q n and Bn ) simultaneously, we develop the following algorithm based on the above lemmas. Algorithm. Step 1: Determine Δ1 , Δ2 , Δ3 and Δ4 . Step 2: Find the optimal solutions for Case 1. If 2Kλ 4Δ1 , then the optimal solutions Q 1 and B1 can be determined by Eqs. (20) and (21). The corresponding minimum inventory total cost per unit time TCU1(Q 1 ,B1 ) is obtained from Eq. (23). Otherwise, set TCU1(Q 1 ,B1 )¼∞. Step 3: Find the optimal solutions for Case 2. (1) If Δ2 o 2Kλ ≤ Δ3 , the optimal solutions Q 2 and B2 can be determined by Eqs. (26) and (27). The corresponding minimum inventory total cost per unit time TCU2(Q 2 ,B2 ) is obtained from Eq. (29). (2) If 2Kλ 4 Δ3 , then the optimal solution Q 2 ¼ ðU 2 =W 2 Þ ½Pð1−xÞ−λM +ðvIc −sIe ÞλM=U 2 . Substituting Q 2 into Eq. (24), the optimal solution B2 can be determined. The corresponding minimum inventory total cost per unit time TCU2(Q 2 ,B2 ) is obtained from Eq. (25). (3) If 2Kλ ≤Δ2 , set TCU2(Q 2 ,B2 )¼ ∞.

616

L.-Y. Ouyang, C.-T. Chang / Int. J. Production Economics 144 (2013) 610–617

Step 4: Find the optimal solutions for Case 3. If 2Kλ ≤ Δ4 , then the optimal solutions Q 3 and B3 can be determined by Eqs. (32) and (33). The corresponding minimum inventory total cost per unit time TCU3(Q 3 ,B3 ) is obtained from Eq. (35). Otherwise, the optimal solution Q 3 ¼ U 3 λM=ðU 3 −W 3 Þ. Substituting Q 3 into Eq. (30), the optimal solution B3 can be determined. The corresponding minimum inventory total cost per unit time TCU3(Q 3 ,B3 ) is obtained from Eq. (31). Step 5: Find min {TCU1(Q 1 ,B1 ), TCU2(Q 2 ,B2 ), TCU3(Q 3 ,B3 )}. Set TCU(Q n ,Bn )¼min {TCU1(Q 1 ,B1 ), TCU2(Q 2 ,B2 ), TCU3(Q 3 ,B3 )}, then (Q n ,Bn ) is the optimal solution. Once the optimal solution ðQ n ; Bn Þis obtained, the optimal production cycle length T n ¼ Q n =λ follows.

Table 1 Optimal solutions for different values of M, I c and I e in Example 2. M

Ic

Ie

Tn

Qn

Bn

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.3

0.09

0.05

0.03 0.05 0.07 0.09 0.11 0.13 0.09

0.05

T 1 ¼ 2.281048 T 1 ¼2.266860 T 2 ¼ 2.249101 T 2 ¼ 2.241618 T 2 ¼ 2.231961 T 2 ¼2.220100 T 2 ¼ 2.206001 T 2 ¼ 2.522292 T 2 ¼ 2.411330 T 2 ¼ 2.322284 T 2 ¼ 2.249101 T 2 ¼ 2.187801 T 2 ¼2.135652 T 2 ¼2.294350 T 2 ¼ 2.249101 T 2 ¼ 2.205096 T 2 ¼2.162075 T 2 ¼2.119819

Q 1 ¼ 4562 Q 1 ¼ 4534 Q 2 ¼4498 Q 2 ¼4483 Q 2 ¼4464 Q 2 ¼ 4440 Q 2 ¼4412 Q 2 ¼5044 Q 2 ¼4823 Q 2 ¼4645 Q 2 ¼4498 Q 2 ¼4376 Q 2 ¼ 4271 Q 2 ¼ 4589 Q 2 ¼4498 Q 2 ¼ 4410 Q 2 ¼ 4324 Q 2 ¼ 4240

B1 ¼2062 B1 ¼ 2050 B2 ¼ 2039 B2 ¼ 2049 B2 ¼ 2058 B2 ¼ 2064 B2 ¼2069 B2 ¼1988 B2 ¼ 2007 B2 ¼ 2023 B2 ¼ 2039 B2 ¼ 2053 B2 ¼2065 B2 ¼ 2074 B2 ¼ 2039 B2 ¼ 2007 B2 ¼1977 B2 ¼1950

0.3

5. Numerical examples Example 1. Consider a production system with the following data: P ¼10,000, λ¼ 2000, P 1 ¼600, K ¼750, v ¼1.5, c ¼2, cR ¼ 0.5, s¼ 4, b ¼0.25, h¼ 0.2, x¼ 0.05, M ¼0.3, I c ¼ 0.09 and I e ¼0.05 in appropriate units. Using the proposed algorithm above, the optimal solutions are obtained as follows: optimal production cycle length T n ¼ T 2 ¼2.249101, optimal production lot size n Q ¼Q 2 ¼4498, optimal backorder level Bn ¼B2 ¼2039, and the optimal manufacturer’s inventory total cost TCU(Q n ,Bn )¼4613.06. Example 2. We now study the effects of permissible delay period M, interest charged I c and interest earned I e on the optimal production cycle length T n , the optimal production lot size Q n , the optimal backorder quantity Bn and the optimal manufacturer’s inventory total cost TCU(Q n ,Bn ). Using the same data as in Example 1, by the proposed algorithm, we obtain the computational results for different values of M∈{0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7}, I c ∈{0.03, 0.05, 0.07, 0.09, 0.11, 0.13} and I e ∈{0.03, 0.05, 0.07, 0.09, 0.11} as shown in Table 1. From Table 1, we get the following findings: (1) For fixed I c and I e , the value of M offered by the supplier increases, the values of T n , Q n and TCU(Q n ,Bn ) decrease. It means that if the length of permissible delay period increases, then the manufacturer’s optimal production cycle length, production lot size and inventory total cost per unit time will be decreased. Besides, there are two possible situations for the optimal Bn : (i) The value of M ( ≤0.3) increases, the value of Bn decreases; it implies that if the length of permissible delay period is shorter, the influence of it on the optimal backorder level is negative. (ii) The value of M (≥0.3) increases, the value of Bn increases; it shows that if the length of permissible delay period is longer, the influence of it on the optimal backorder level is positive. (2) For fixed M and I e , a larger value of I c causes larger values of Bn and TCU(Q n ,Bn ), but smaller values of T n and Q n . It means that if the interest charged increases, then the optimal backorder level and manufacturer’s inventory total cost per unit time will increase, but the optimal production cycle length and production lot size will decrease. (3) For fixed M and I c , a larger value of I e causes lower values of T n , Q n ,Bn and TCU (Q n ,Bn ). It means that if the interest earned increases, then the optimal production cycle length, production lot size, backorder level and manufacturer’s inventory total cost per unit time will decrease.

6. Conclusions In order to reflect the real manufacturing circumstance and the practical business behavior, firstly, we establish a mathematical

0.03 0.05 0.07 0.09 0.11

TCU(Q n ,Bn ) 4677.89 4646.81 4613.06 4578.11 4542.53 4506.30 4469.41 4569.98 4586.53 4600.72 4613.06 4623.89 4633.49 4628.02 4613.06 4597.82 4582.32 4566.55

model to study the optimal production policy for an EPQ inventory system with imperfect production processes under permissible delay in payments and complete backlogging. Next, a simple algebraic manipulation and an arithmetic–geometric mean inequality method are employed to determine the optimal production lot size and backorder level. Besides, an algorithm is developed to find the optimal solution. Finally, a numerical example is given to illustrate the theoretical results and the sensitivity analysis of key model parameters is also examined. Some managerial insights are obtained as follows: (1) a larger value of the length of permissible delay period causes smaller values of manufacturer’s optimal production cycle length, production lot size and inventory total cost per unit time; (2) a higher value of interest charged results in higher values of the backorder level and manufacturer’s inventory total cost per unit time, but lower values of the production cycle length and production lot size; and (3) a higher value of interest earned cause lower values of production cycle length, production lot size, backorder level and manufacturer's inventory total cost per unit time.

Acknowledgments The authors are grateful to anonymous referees for their encouragement and constructive comments. The work of the second author was partially supported by the National Science Council of ROC Grant NSC 99-2410-H-032-056. Appendix A: the proof of U 1 V 1 4 W 21 Because U 1 ≡ðb þ h þ vI c Þ




 ð1−xÞP 4 0; ð1−xÞP−λ


 λ V 1 ≡h 1− þ vI c 4 0; P and W1 ≡ h þ vI c 40, we can get     
  ð1−xÞP ð1−xÞP λ þ ðh þ vI c Þ h 1− þ vI c Þ U1 V 1 ¼ b ð1−xÞP−λ ð1−xÞP−λ P    ð1−xÞP λ h þ vI c −h ≥ðh þ vIc Þ ð1−xÞP−λ P   ðh þ vI c Þð1−xÞðP−λÞ vI c ð1−xÞλ þ ¼ ðh þ vI c Þ ð1−xÞP−λ ð1−xÞP−λ

L.-Y. Ouyang, C.-T. Chang / Int. J. Production Economics 144 (2013) 610–617

≥ðh þ vI c Þ2

ð1−xÞðP−λÞ ≥ðh þ vI c Þ2 ¼ W 21 : ð1−xÞP−λ

References Aggarwal, S.P., Jaggi, C.K., 1995. Ordering policies of deteriorating items under permissible delay in payments. Journal of the Operational Research Society 46, 658–662. Ben-Daya, M., 2002. The economic production lot-sizing problem with imperfect production processes and imperfect maintenance. International Journal of Production Economics 76, 257–264. Cardenas-Barron, L.E., 2010. A simple method to compute economic order quantities: some observations. Applied Mathematical Modelling 34, 1684–1688. Chang, C.T., 2004. An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. International Journal of Production Economics 88, 307–316. Chang, C.T., Ouyang, L.Y., Teng, J.T., 2003. An EOQ model for deteriorating items under supplier credits linked to ordering quantity. Applied Mathematical Modelling 27, 983–996. Chang, C.T., Teng, J.T., Chern, M.S., 2010. Optimal manufacturer’s replenishment policies for deteriorating items in a supply chain with up-stream and downstream trade credits. International Journal of Production Economics 127, 197–202. Cheng, T.C.E., 1991. An economic order quantity model with demand-dependent unit production cost and imperfect production processes. IIE Transaction 23, 23–28. Chiu, Y.P., 2003. Determining the optimal lot size for the finite production model with random defective rate, the rework process, and backlogging. Engineering Optimization 35, 427–437. Chiu, S.W., 2008. Production lot size problem with failure in repair and backlogging derived without derivatives. European Journal of Operational Research 188, 610–615. Chiu, S.W., Cheng, C.B., Wu, M.F., Yang, J.C., 2010. An algebraic approach for determining the optimal lot size for EPQ model with rework process. Mathematical and Computational Applications 15, 364–370. Chiu, S.W., Chiu, Y.P., 2006. Mathematical modelling for production system with backlogging and failure in repair. Journal of Scientific and Industrial Research 65, 499–506. Chiu, S.W., Gong, D.C., Wee, H.M., 2004. The effects of the random defective rate and the imperfect rework process on the economic production quantity model. Japan Journal of Industrial and Applied Mathematics 21, 375–389. Emery, G..W., 1987. An optimal financial response to variable demand. Journal of Financial and Quantitative Analysis 22, 209–225. Goyal, S.K., 1985. Economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society 36, 335–338. Goyal, S.K., Teng, J.T., Chang, C.T., 2007. Optimal ordering policies when the supplier provides a progressive interest scheme. European Journal of Operational Research 179, 404–413. Hayek, P.A., Salameh, M.K., 2001. Production lot sizing with the reworking of imperfect quality items produced. Production Planning and Control 12, 584–590. Jamal, A.M.M., Sarker, B.R., Mondal, S., 2004. Optimal manufacturing batch size with rework process at a single-stage production system. Computers & Industrial Engineering 47, 77–89. Jamal, A.M., Sarker, B.R., Wang, S., 1997. An ordering policy for deteriorating items with allowable shortage and permissible delay in payment. Journal of the Operational Research Society 48, 826–833. Liao, J.J., 2008. An EOQ model with noninstantaneous receipt and exponentially deteriorating items under two-level trade credit. International Journal of Production Economics 113, 852–861.

617

Lin, C.S., Chen, C.H., Kroll, D.E., 2003. Integrated production-inventory models for imperfect production processes under inspection schedules. Computers & Industrial Engineering 44, 633–650. Liu, J.J., Yang, P., 1996. Optimal lot-sizing in an imperfect production system with homogeneous reworkable jobs. European Journal of Operational Research 91, 517–527. Musa, A., Sani, B., 2012. Inventory ordering policies of delayed deteriorating items under permissible delay in payments. International Journal of Production Economics 136, 75–83. Ouyang, L.Y., Chang, C.T., Teng, J.T., 2005. An EOQ model for deteriorating items under trade credits. Journal of the Operational Research Society 56, 719–726. Petersen, M.A., Rajan, R.G., 1997. Trade credit: theories and evidence. The Review of Financial Studies 10, 661–691. Porteus, E.L., 1986. Optimal lot sizing, process quality improvement and setup cost reduction. Operations Research 34, 137–144. Rosenblatt, M.J., Lee, H.L., 1986. Economic production cycles with imperfect production processes. IIE Transactions 18, 48–55. Salameh, M.K., Jaber, M.Y., 2000. Economic production quantity model for items with imperfect quality. International Journal of Production Economics 64, 59–64. Sana, S.S., 2010. A production inventory model in an imperfect production process. European Journal of Operational Research 200, 451–464. Sana, S.S., 2011. A production-inventory model of imperfect quality products in a three-layer supply chain. Decision Support Systems 50, 539–547. Sana, S.S., Goyal, S.K., Chauhuri, K.S., 2007. An imperfect production process in a volume flexible inventory model. International Journal of Production Economics 105, 548–559. Sarker, B., 2012. An inventory model with reliability in an imperfect production process. Applied Mathematics and Computation 218, 4881–4891. Sarker, B., Moon, I., 2011. An EPQ model with inflation in an imperfect production system. Applied Mathematics and Computation 217, 6159–6167. Sarker, B., Sana, S.S., Chaudhuri, K.S., 2010. Optimal reliability, production lot size and safety stock in an imperfect production system. International Journal of Mathematics and Operational Research 2, 467–490. Taleizadeh, A.A., Wee, H.M., Sadjadi, S.J., 2010. Multi-product production quantity model with repair failure and partial backordering. Computers and Industrial Engineering 59, 45–54. Su, C.H., 2012. Optimal replenishment policy for an integrated inventory system with defective items and allowable shortage under trade credit. International Journal of Production Economics 139, 247–256. Teng, J.T., 2002. On economic order quantity under conditions of permissible delay in payments. Journal of the Operational Research Society 53, 915–918. Teng, J.T., Chang, C.T., 2009. Optimal manufacturer’s replenishment policies in the EPQ model under two levels of trade credit policy. European Journal of Operational Research 196, 177–185. Teng, J.T., Chang, C.T., Goyal, S.K., 2005. Optimal pricing and ordering policy under permissible delay in payments. International Journal of Production Economics 97, 121–129. Tsao, Y.C., Che, T.H., Huang, S.M., 2011. A production policy considering reworking of imperfect items and trade credit. Flexible Services and Manufacturing Journal 23, 48–63. Yoo, S.H., Kim, D., Park, M.S., 2009. Economic production quantity model with imperfect-quality items, two-way imperfect inspection and sales return. International Journal of Production Economics 121, 255–265. Yoo, S.H., Kim, D., Park, M.S., 2012. Lot sizing and quality investment with quality cost analyses for imperfect production and inspection processes with commercial return. International Journal of Production Economics 140, 922–933. Zhang, X., Gerchak, Y., 1990. Joint lot sizing and inspection policy in an EOQ model with random yield. IIE Transaction 22, 41–47.