Optimal replenishment policy for an integrated inventory system with defective items and allowable shortage under trade credit

Int. J. Production Economics 139 (2012) 247–256

Contents lists available at SciVerse ScienceDirect

Int. J. Production Economics journal homepage: www.elsevier.com/locate/ijpe

Optimal replenishment policy for an integrated inventory system with defective items and allowable shortage under trade credit Chia-Hsien Su n Department of Business Administration, Tungnan University, No. 152 Sec. 3, PeiShen RD., ShenKeng, Taipei 222, Taiwan

a r t i c l e i n f o

abstract

Article history: Received 31 August 2009 Accepted 20 March 2012 Available online 22 May 2012

This study proposes a single-supplier, single-retailer integrated inventory model that accounts for defective items that arrive in a retailer’s order under a full-lot inspection policy. All defective items are returned to the supplier in next delivery. After receipt of the returned items, the supplier will classify them into two types: items that still have some worth and waste items. For those items that still have some worth, the supplier will offer customers a discount in order to minimize losses arising from these defective items. The supplier needs to pay a disposal fee for those items classified as waste items. Shortages are allowed and are fully backlogged. To encourage sales the supplier offers trade credit to the retailer. A two-echelon inventory model is established, and the decision variables include: replenishment cycle time, the time taken to run out of stock and the number of lots delivered from the supplier to the retailer. An algorithm is developed to determine the optimal supply chain strategy and numerical examples are provided to show the solution procedure. Also, a sensitivity analysis is conducted on the main parameters of the model. & 2012 Elsevier B.V. All rights reserved.

Keywords: Supply chain Defective items Return policy Complete backlogging Trade credit

1. Introduction The ultimate objective of effective supply chain management is the reduction of costs, improvement of cash flow and increased operational efficiency across the entire business through connecting inventory control, purchasing coordination and sales order processing with market demand. In a competitive business environment the ability to integrate one’s supply chain is essential for company success. The joint optimization concept for the supplier and retailer was initiated by Goyal (1976). Banerjee (1986) extended Goyal’s (1976) model and assumed that the supplier followed a lot-for-lot shipment policy with respect to a retailer. Later, Goyal (1988) relaxed the lot-for-lot policy and illustrated that the inventory cost could be significantly reduced if the supplier’s economic production quantity (EPQ) was an integer multiple of the retailer’s purchase quantity. Lu (1995) then generalized Goyal’s (1988) model by relaxing the assumption that the supplier could supply the retailer only after completing the entire lot size. Many researchers (Goyal, 1995, 2000; Ha and Kim, 1997; Viswanathan, 1998; Hill, 1999; Goyal and Nebebe, 2000; Woo et al., 2001; Pan and Yang, 2002; Khan and Sarker, 2002; Kim and Ha, 2003; Kelle et al., 2003; Yao and Chiou, 2004; Siajadi et al., 2005; Hoque, 2008; Sarker and Diponegoro, 2009; Glock, 2011, 2012) continued to propose more batching and shipping policies for integrated inventory models.

n

Fax: þ11 886 2 86625984. E-mail address: [email protected]

0925-5273/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ijpe.2012.05.001

The above inventory integration models commonly adopt the unrealistic assumption that all units received by a retailer are of good quality when studying strategic production/transportation decisions. In reality, imperfect production process, flaws in the goods transportation process and many other factors inevitably lead to a certain proportion of defective items being received during a production run. Over the years, several different economic order quantity (EOQ) models that account for defective items have been developed. For example, Rosenblatt and Lee (1986) and Porteus (1986) initially considered the effects of an imperfect production process on quality imperfection and on lot size. Salameh and Jaber (2000) extended the traditional EPQ/EOQ model by accounting for imperfect quality items. They considered the issue of poor-quality items being sold as a single batch by the end of a 100% screening process. Wu and Ouyang (2000) considered the potential for an arrival order lot to contain some defective items and the number of defective items in a sampled sub-lot to be a random variable. Currently, several relevant papers exist that study EPQ models for items with imperfect quality such as Ouyang et al. (2002), Chiu (2003), Chang (2003), Balkhi (2004), Hou and Lin (2004) and Papachristos and Konstantaras (2006). These models determine an optimal policy from the perspective of either the retailer or the supplier only. Integrated vendor–buyer models that consider defective items have also been presented (see, for example Affisco et al., 2002; Singer et al., 2003; Comeaux and Sarker, 2005; Huang, 2004; Lo et al., 2007; Chung and Wee, 2008; Maddah and Jaber, 2008; Chiu et al., 2011; Khan et al., 2011). Enterprises and academic bent on the improvement of production process to eliminate defectives and reduce waste. However, in

248

C.-H. Su / Int. J. Production Economics 139 (2012) 247–256

practice defective is inevitable (Schwaller, 1988). Facing growing levels of competitive and economic pressures, more and more enterprises have begun to view the defective treatment as a process that may be used to manage costs and drive additional revenues. Also, the growing global concern for the environment has led to increased interest in the treatment of defective. In this paper, a common defective treatment for manufactures of clothing, shoe, accessory, furniture, electronics, toys and bedding is involved. For example, a famous clothing company produces middle and top grade women’s apparel. Their brands are sold in China’s major cities more than 300 stores. All returns from retailers will be check and classed. While the returned clothes with minor production defects, such as stain, skipped or dropped stitches, wavy bottom hem, sewing thread not matching, etc., will be cut label and sold to wholesalers with discount price. If the defects affecting the usability and salability, such as: fabric hole, shading among panel, wrong measurement, dye patches etc., then the returns will be scraped. The above integrated inventory models for items with imperfect quality failed to account for the effect of trade credit on optimal policies. Trade credit is a widespread tool 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). Furthermore, offering trade credit to retailers may improve supplier sales and reduce on-hand stock levels (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. Aggarwal and Jaggi (1995) extended Goyal’s (1985) model to include deteriorating items. Jamal et al. (1997) further generalized this issue with allowable shortages. Kim et al. (1995) examined the effects of a credit period on the ordering policies from the supplier’s viewpoint. 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 decreased under the permissible delay in payments in certain cases. Numerous relevant papers have been produced relating to trade credit such as Huang (2003), Ouyang et al. (2005, 2009), Teng et al. (2005), Su et al. (2007), Chen and Kang (2010a) and Yu (in press). By taking the considerations of imperfect-item and trade credit as described above, Li et al. (2009) developed a model to determine the retailer’s optimal replenishment policy with defective items under conditions of permissible delay of payments. Further, Chen and Kang (2010b) investigated the issue of defective items with a permissible delay in payment from the perspective of both the vendor and buyer. However, in their models the occurrence of shortage in the inventory system is overlooked. In real life, many famous products or modern goods, for example Apple’s iPad and iPhone, may cause a situation in which customers may prefer to wait for back orders while shortages occur. Inventory shortage problems can interfere with a company’s profits and customer service. Therefore, for inventory managers of manufacturing and retail organizations how to control inventory in the supply chain that enable them to minimize inventory costs and meet customer demand is worth discussing. This paper proposes a single-supplier, single-retailer integrated inventory model that accounts for defective items that arrive in a retailer’s order under a full-lot inspection policy. All defective items are returned to the supplier and classified into two types: items that still useful and waste items. The former are sold to customers in a discounted price and the later cost a disposal fee. Shortages are allowed and are fully backlogged. For the retailer, trade credit is permissible. A two-echelon inventory model is established and the decision variables include: replenishment cycle time, the time taken to run out of stock and the number of lots delivered from the supplier to the retailer. An algorithm is developed to determine the optimal

strategy and numerical examples are provided to show the solution procedure. Furthermore, a sensitivity analysis is conducted on the main parameters of the model. Finally, conclusions and possible future research topics are provided.

2. Notation and assumptions The following notation and assumptions are used throughout this paper. 2.1. Notation D A F f

g b hb1 hb2 c1 c2 c3 Ie Ip p Q t T q

l w R S v1 v2 v3 b hv Iv m M

retailer’s demand rate per unit time retailer’s ordering cost per order retailer’s freight cost fix freight cost per delivery value related freight cost freight cost per unit retailer’s unit stock holding cost of good quality items per unit time excluding interest charges retailer’s unit stock holding cost of defective items per unit time excluding interest charges, where hb2 rhb1 retailer’s unit purchasing price retailer’s unit inspecting cost retailer’s unit backlogging cost per unit time retailer’s interest earned per dollar per unit time retailer’s interest charged per dollar in stocks per unit time retailer’s unit selling price for items of good quality retailer’s order quantity of good quality items per order the length of stock-end cycle of the retailer (decision variable) the length of replenishment cycle of the retailer (decision variable) supply quantity per delivery from the supplier to the retailer in a production batch percentage of defective items in each deliver, lA[0,1) percentage of disposal items in each return, wA[0,1) supplier’s production rate supplier’s setup cost per setup supplier’s unit production cost supplier’s unit inspection cost of returned items supplier’s unit disposal cost supplier’s unit clearing price of useable defective items supplier’s unit stock holding cost per unit time supplier’s capital opportunity cost per dollar per unit time number of shipments from supplier to retailer per batch production run, a positive integer (decision variable) the length of the trade credit period offered by the supplier.

2.2. Assumptions 1. There is a single-supplier and a single-retailer for a single product in this model. 2. Replenishments are instantaneous and the lead time is zero. 3. Shortages are allowed and these are fully backlogged. 4. Each batch is dispatched to the retailer in m equal-sized shipments, where m is a positive integer. 5. An arriving lot q contains some defective items with defective rate l. 6. The retailer orders a lot of size Q which is the sum of good quality items in m equally-sized shipments, i.e., Q¼m(1  l)q.

C.-H. Su / Int. J. Production Economics 139 (2012) 247–256

7. As the retailer orders a lot of size Q, the supplier produces a batch quantity Q/(1  l) at one set-up, with a finite production rate R, and (1  l)R4D. 8. As soon as the lot comes into the warehouse of the retailer, a rapid, no damage and 100% inspection process of the lot is conducted, with a fixed cost c2 per unit. Inspection times are assumed to be negligible. 9. Defective items in each lot will be discovered and returned to the supplier at the time of delivery of the next lot. Hence, the number of good quality items (1 l)q, and the length of each shipping cycle for the retailer is T¼(1  l)q/D. 10. As soon as the return defective items are taken back to the supplier, a rapid, no damage and 100% inspection process of the lot is conducted, with a fixed cost v2 per unit. Inspection times are assumed to be negligible. 11. Among all return defective items w percent are totally useless, therefore the supplier need spend v3 per unit to dispose them. On the other hand, (1  w) percent return defective items can still use. The supplier sells these poor quality items at a discount price b (i.e., bov1) to contracted wholesalers. To save storage costs all returns handling is completed at the end of the 100% inspection process. 12. The freight cost F contains a fix cost f per shipment and a quantity-dependent cost bq, i.e. F(q)¼fþ bq. 13. The supplier propose a certain credit period and the retailer’s sales revenue generated during the credit period is deposited in an interest bearing account with rate Ie. At the end of the period, the credit is settled and the retailer starts paying the capital opportunity cost for the items in stock with rate Ip. 14. In every replenishment cycle, the supplier incurs a opportunity cost with a finance rate Iv for offering trade credit. 3. Model formulation

249

will check and classify them into two types immediately: the still worthy and the wastes. The first can be sold as a single batch at a discounted price and the wastes need to be clear away by paid. To save storage costs all returns handling is completed at the end of the check process. Moreover, to encourage sales and promote market share, the supplier often offers trade credit to retailers. The relationship between the supplier, the retailer and the customers is illustrated in Fig. 1. To formulate the integrated inventory model, the supplier’s total cost per unit time is discussed first. Then the retailer’s total cost per unit time is discussed. 3.1. Supplier’s total cost per unit time The supplier’s total relevant cost per production run consists of the following elements: (a) Cost of set up: The set-up cost per production run is given by S. (b) Cost of production: The supplier spends the production cost v1 of each product, therefore in a production cycle the total production cost is v1Q/(1  l) ¼v1mq ¼v1mDT/(1  l). (c) Cost of carrying inventory: When the supplier produces the first q units, he/she will deliver them to the retailer, after that the supplier will make the deliver on every T¼(1 l)q/D unit of time until the inventory level falls to zero (see Fig. 2). With unit stock holding cost hv per unit time, the holding cost per production run can be calculated as follows:    q ðm1Þð1lÞq 1 mq  mq hv mq þ R D 2 R  ð1lÞq ½1 þ2 þ    þ ðm1Þq D hv m½ðm1Þð1lÞRDðm2Þq2 2DR   2 hv mD D T ðm1Þð1lÞ ðm2Þ : ¼ 2 R 1l

¼ A supply-chain system discussed in this paper consists of a single supplier who makes a single type of finished products, stores them in a finished goods inventory, then sells and delivers them to the retailer. In a production cycle, the supplier produces a batch quantity of Q/(1 l) and delivers them in m shipments of equal size q. An arriving lot q contains some defective items with defective rate l. To guarantee the quality, as soon as the lot comes into the warehouse, a 100% inspection process of the lot by the retailer is conducted, with a fixed cost c2 per unit. In inspecting, the product out of the specification limits will be detected and returned to the supplier in next delivery. After receiving the returns, the supplier

ð1Þ

In addition, all returns handling is completed at the end of the check process with no warehousing. Thus the supplier does not incur holding cost associated with the return defective items. (d) Treatment cost of defective items: For an arriving lot with defective rate l, there are lq units defective items which will be returned. To identify the value of return items, there is an inspection process for all return items

Fig. 1. The relationship among the members in this model.

250

C.-H. Su / Int. J. Production Economics 139 (2012) 247–256

with unit inspection cost v2. Thus the inspection cost of defective items is v2mlq¼v2mlDT/(1 l) in a production cycle. Furthermore, among all return defective items w percent are totally useless, with unit disposal cost v3 the supplier need spend v3mwlq¼wv3mlDT/(1 l) to deal with these. On the other side, (1w) percent return defective items can still use. The supplier sells these poor quality items at a price bper unit, so the revenue from selling poor quality items is bm(1 w)lq¼ (1 w)bmlDT/ (1 l). As a result, in a production cycle the total cost dealing with the defective items is v2 mlDT wv3 mlDT ð1wÞbmlDT ½v2 þ wv3 ð1wÞbmlDT þ  ¼ : 1l 1l 1l 1l

(e) Opportunity cost for offering a trade credit: Following the contract, the supplier offers a fixed credit period M to the retailer. Without trade credit, a simultaneous exchange of goods for cash may happen at the start of every order period. The supplier will not receive a payment until M. With a finance rate Iv, in every production cycle the supplier incurs a lost-opportunity cost of c1Ivm(1  l)qM¼ c1IvmMDT for the same funds if invested elsewhere. Consequently, the supplier total cost per unit time can be expressed as (  2 1 v1 mDT hv mD T Sþ ATC v ðmÞ ¼ þ mT 1l 2 1l   D ðm1Þð1lÞ ðm2Þ R

quantity mq R

accumulated inventory for the manufacturer mq accumulated inventory for the retailer

q R

time

(m − 1)(1 − )q D

þ

 ½v2 þ wv3 ð1wÞbmlDT þ c1 Iv mMDT : 1l

3.2. Retailer’s total cost per unit time The retailer’s total relevant cost per replenishment cycle consists of the following elements: (a) Cost of placing orders: The retailer orders Q quantity with an ordering cost A and receives his/her order in m delivery, so the ordering cost per replenishment cycle is A/m. (b) Cost of freight: The freight cost is F(q)¼fþ bq¼fþ bDT/(1 l) in a replenishment cycle. (c) Cost of inspection: Since the supplier each shipment size is q, and the unit inspecting cost for items is c2, the retailer’s inspecting cost per replenishment cycle is c2q ¼c2DT/(1  l). (d) Cost of carrying inventory (excluding interest charges): Due to each arriving lot contains a fixed percentage of defective, l, the maximum amount of good quality items is Dt, with unit holding cost of good quality items per unit time hb1, the retailer’s holding cost of good quality items per replenishment cycle is hb1Dt2/2. Similarly, the amount of defective items in each shipment is lq, the replenishment cycle is T and the retailer’s unit holding cost of defective items per unit time is hb2. Hence, the retailer’s holding cost of defective items per replenishment cycle is hb2lqT ¼hb2lDT2/ (1 l). Therefore, the retailer’s total holding cost of good quality and defective items per replenishment cycle is hb1Dt2/ 2þhb2lDT2/(1  l). (e) Cost of backlogging: Shortages in inventory are allowed and completely backlogged. With unit backlogging cost c3, the backlogging cost R Tt per replenishment cycle is 0 c3 Dy dy ¼ c3 DðTtÞ2 =2. (f) Cost of interest charges for unsold items after the permissible delay period M and interest earned from sales revenue during the permissible period: Base on the values of M, t and T, three cases are considered: (i) Mrt rT (ii) trM rT (iii) t rTrM. Case 1. Mrt rT (see Fig. 3) When retailer’s permissible payment time expires on or before the inventory is depleted completely, the retailer can sell the items and earn interest with rate Ie until the end of the credit

Fig. 2. The supplier’s inventory pattern.

quality

Dt DM

(1−)q

time

M T

t (1−)

ð2Þ

q −t D

Fig. 3. Retailer’s accumulated interest eared and opportunity cost when Mr t r T.

C.-H. Su / Int. J. Production Economics 139 (2012) 247–256

period M. Thus, the interest earned per replenishment cycle is   Z M  M þ Tt , Ie pDydy þ Ie p ð1lÞqDt M ¼ Ie pDM 2 0 where Iep[(1 l)q Dt]M is the earn interest of backorder quantity. On the other hand, the retailer still has some inventory on hand when paying the total purchasing amount to the supplier. Hence, for the items still in stock, retailer endures a capital opportunity cost at a rate of Ip; the opportunity cost per replenishment cycle for unsold items is obtained by Z t Ip c1 DðtMÞ2 : Ip c1 DðtyÞdy ¼ 2 M Case 2. t rMrT (see Fig. 4) As the credit period expires on or after the inventory is depleted completely, the retailer pays no opportunity cost for the purchase items. Through the credit period, retailer sells the products and uses the sales revenue to earn interest at a rate of Ie. Thus, the interest earned per replenishment cycle is Z t  Ie pDydy þIe pDtðMtÞ þ Ie p ð1lÞqDt M 0

¼

  Ie pDt2 t2 þIe pDtðMtÞ þ Ie pDMðTtÞ ¼ Ie pD TM : 2 2

Therefore, the total cost per unit time for the retailer can be expressed as follows: 8 > < ATC 1 ðt,TÞ, M r t rT, ATC b ðt,TÞ ¼ ATC 2 ðt,TÞ, t rM rT, > : ATC ðt,TÞ, t rT rM, 3

where ATC 1 ðt,TÞ ¼

A þmf Dðb þ c2 þ hb2 lTÞ D½c3 ðTtÞ2 þhb1 t 2  þ þ mT 1l 2T þ

Dc1 Ip ðtMÞ2 DMpIe ½M þ 2ðTtÞ ,  2T 2T

ð3Þ

and ATC 2 ðt,TÞ ¼ ATC 3 ðt,TÞ ¼

A þ mf Dðb þ c2 þ hb2 lTÞ D½c3 ðTtÞ2 þ hb1 t 2  þ þ mT 1l 2T 

DpIe ð2MTt 2 Þ : 2T

ð4Þ

3.3. Channel’s total cost per unit time

Case 3. t rTrM(see Fig. 5) When t rTrM, the interest earned and opportunity cost per replenishment cycle are same as Case 2.

Once the supplier and retailer have form a strategic alliance in order to minimize their cost, then trading parties will collaborate and share information to achieve improved benefits. Under this circumstance, the joint total cost per unit time for the supplier

quantity

Dt (1−)q

time T M t (1−)q −t D Fig. 4. Retailer’s accumulated interest eared when t r Mr T.

quality

(1−)q

251

Dt

time T t M

(1−λ)q −t D

Fig. 5. Retailer’s accumulated interest eared when t rT rM.

252

C.-H. Su / Int. J. Production Economics 139 (2012) 247–256

and retailer is 8 > < JATC 1 ðm,t,TÞ, JATCðm,t,TÞ ¼ JATC 2 ðm,t,TÞ, > : JATC ðm,t,TÞ, 3

By solving Eqs. (8) and (9), the unique solution of (t,T) (denoted by (t1,T1)) can be found and as follows: " sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1lÞ2 ½2Kðc3 þ c1 Ip þ hb1 Þ þ DM 2 ðc1 Ip pIe Þðc3 þ pIe þ hb1 Þ t1 ¼ c3 Df2½W þ lð1lÞhb2 ðc3 þc1 Ip þ hb1 Þ þ c3 ð1lÞ2 ðc1 Ip þ hb1 Þg

M r t rT, t rM rT, t rT rM,

þ ðc1 Ip pIe ÞM 

where JATC 1 ðm,t,TÞ ¼ ATC v ðmÞ þATC 1 ðt,TÞ ¼

DMpIe ½M þ2ðTtÞ Dc1 Ip ðtMÞ2 þ , 2T 2T

T1 ¼

ð6Þ

ð7Þ

ð8Þ

Furthermore, for a fixed m, we can show that JACT1(m,t,T) is a convex function at point (t,T)¼(t1,T1) (the proof see Appendix B). Therefore, for a fixed m, (t1,T1) is the optimal solution which minimizes JACT1(m,t,T). Similarly, by solving Eqs.(10) and (11), we can get a unique solution of (t,T) (for JACTi(m,t,T), denoted as (ti,Ti), i¼ 2,3) and are given by c3 t2 ¼ t3 ¼ c3 þpIe þhb1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1lÞ2 ½2Kðc3 þ pIe þ hb1 Þ ,  Df2½W þ lð1lÞhb2 ðc3 þpIe þhb1 Þ þ c3 ð1lÞ2 ðpIe þ hb1 Þg ð15Þ and T2 ¼ T3 ¼ ð1lÞ2 ½2Kðc3 þ pIe þhb1 Þ Df2½W þ lð1lÞhb2 ðc3 þpIe þ hb1 Þ þ c3 ð1lÞ2 ðpIe þ hb1 Þg

ð9Þ

:

ð16Þ

It is obvious that Ti 4ti, for i¼2,3. Furthermore, to insure ti rM, i¼2,3, we submit (15) into inequality t rM and obtain if D r0 then t i rM,

  2DT 2 W þ lð1lÞhb2

ð1lÞ2 f2K þD½c3 ðt 2 T 2 Þ  pIe MðM2tÞ þ c1 Ip ðtMÞ2 þ t 2 hb1 g ¼ 0,

For detailed proof, please see Appendix A. Note that when

D Z0 holds, then T1 Zt1 40.

JATC(m,t,T) is a convex function of m. Therefore, the search for the optimal shipment number, m, is reduced to find a local optimal solution. Next, in order to obtain the solutions for minimum joint total cost function, JACTi(m,t,T), i¼1,2,3, for fixed m, the following conditions are necessary:

@JATC 1 ðm,t,TÞ 1 ¼ @T 2½ð1lÞT2

ð14Þ

D ¼ 2K½c3 ð1lÞ2 DM 2 ðc3 þ pIe þ hb1 Þ f2½W þ lð1lÞhb2 ðc3 þ pIe þhb1 Þ þ c3 ð1lÞ2 ðpIe þ hb1 Þg:

To find the optimal solution, say (mn,tn,Tn), that minimizes the above integrated total cost, the following procedures are taken. First, for fixed t and T, check the effect of m on the joint total cost per unit time JATC(m,t,T). With the fact

@JATC 1 ðm,t,TÞ D ¼ c3 ðtTÞ þpIe M þ c1 Ip ðtMÞ þ thb1 ¼ 0, @t T

To insure M rt1 rT1, substituting (12) and (13) into inequality Mrt rT, results in

where

4. Theoretical results

i ¼ 1, 2, 3:

Df2½W þ lð1lÞhb2 ðc3 þc1 Ip þ hb1 Þ þ c3 ð1lÞ2 ðc1 Ip þ hb1 Þg

if and only if D Z 0, then M rt 1 rT 1 ,

and j [v2 þwv3  (1 w)b]l þc1MIv(1  l) þv1 þ b þc2.

@2 JATCðm,t,TÞ @2 JATC i ðm,t,TÞ 2ðA þ SÞ 40, ¼ ¼ @m2 @m2 m3 T

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð1lÞ2 ½2Kðc3 þ c1 Ip þ hb1 Þ þ DM 2 ðc1 Ip pIe Þðc3 þ pIe þ hb1 Þ ð13Þ

ð5Þ

JATC 2 ðm,t,TÞ ¼ JATC 3 ðm,t,TÞ ¼ ATC v ðmÞ þ ATC 2 ðt,TÞ h i 2 2 Aþ S þ mf Dðj þ hb2 lTÞ D c3 ðTtÞ þhb1 t þ þ ¼ mT 1l 2T    2 T hv D D DpIe ð2MTt 2 Þ ðm1Þð1lÞ ðm2Þ  , þ 1l 2T R 2T

ð12Þ

and

Aþ S þ mf Dðj þ hb2 lTÞ D½c3 ðTtÞ2 þ hb1 t 2  þ þ mT 1l 2T    2 T hv D D ðm1Þð1lÞ ðm2Þ þ 1l 2T R



1 , c3 þ c1 Ip þhb1

for i ¼ 2, 3:

ð17Þ

Besides, by using the similar arguments as that in Appendix B, we can easily to show that, for a fixed m, JACTi(m,t,T) has a minimum value at point (t,T)¼ (ti,Ti), i¼2,3. Therefore, the following result is obtained. Lemma 1. For any given m,

@JATC 2 ðm,t,TÞ @JATC 3 ðm,t,TÞ D ¼ ¼ c3 ðtTÞ þ tðpIe þhb1 Þ ¼ 0, @t @t T ð10Þ and   @JATC 2 ðm,t,TÞ @JATC 3 ðm,t,TÞ 1 ¼ ¼ 2DT 2 W þ lð1lÞhb2 @T @T 2½ð1lÞT2  ð11Þ ð1lÞ2 f2K þD½c3 ðt 2 T 2 Þ þ t 2 ðpIe þhb1 Þg ¼ 0 where K

A þ S þmf m

and

W

  hv D ðm1Þð1lÞ ðm2Þ : R 2

(i) if D Z0, then the length of stock-end cycle is t1 and the optimal replenishment cycle length is T1. (ii) if D r0 and M rT2, then the length of stock-end cycle is t2 and the optimal replenishment cycle length is T2. (iii) if D r0 and M ZT3, then the length of stock-end cycle is t3 and the optimal replenishment cycle length is T3. Proof. The proof immediately follows from (14) and (17). Summarizing the above results, we can establish the following algorithm to find the optimal solution (mn,tn,Tn). Algorithm. Step 1. Set m¼1.

C.-H. Su / Int. J. Production Economics 139 (2012) 247–256

Step 2. Determine t1 and T1 by Eqs. (12) and (13). If D Z0, then substituting (t1,T1) into Eq. (5) to get JACT1(m,t1,T1); otherwise, let JACT1(m,t1,T1)¼N. Step 3. Determine t2 and T2 by Eqs. (15) and (16). If D r0 and Mr T2, then substituting (t2,T2) into Eq. (6) to get JATC2(m,t2,T2); otherwise, let JATC2(m,t2,T2)¼N. Step 4. Determine t3 and T3 by Eqs. (15) and (16). If D r0 and MZ T3, then substituting (t3,T3) into Eq. (6) to get JATC3(m,t3,T3); otherwise, let JATC3(m,t3,T3)¼N. Step 5. Find min{JATC1(m,t1,T1), JATC2(m,t2,T2), JATC3(m,t3,T3)}. Set JATC(m,t(m),T(m)) ¼min{JATC1(m,t1,T1), JATC2(m,t2,T2), JATC3(m,t3,T3)}, then, (t(m),T(m)) is the optimal solution for given m Step 6. Let m¼ mþ1, repeat Steps 2–5 to find JATC(m,t(m),T(m)). Step 7. If JATC(m,t(m),T(m))rJATC(m 1,t(m  1),T(m  1)), go to Step 6. Otherwise, go to Step 8. Step 8. Set (mn,tn,Tn)¼(m 1,t(m  1),T(m  1)), and hence (mn,tn,Tn) is the optimal solution. Once the optimal solution (mn,tn,Tn) is obtained, the optimal delivery quantity per cycle qn ¼DTn/(1  l) and order quantity Qn ¼mn(1  l)qn follows. 5. Numerical examples and discussion To illustrate the results obtained under the proposed model, several examples are considered with the parameter values given in Table 1. Table 1 Parameter values for numerical examples. D ¼ 4500 units/year A ¼ $600/order f ¼ $277.2/delivery b ¼$0.5/unit hb1 ¼ $11/unit/year hb2 ¼ $7/unit/year Ie ¼ $0.1/dollar/year Ip ¼ $0.12/dollar/year

v1 ¼ $5/unit v2 ¼ $2/unit v3 ¼ $1/unit R¼ 10,000 units/year hv ¼ $3/unit/year Iv ¼$0.08/dollar/year b ¼$2.5/unit

l ¼0.2 w¼ 0.8 c1 ¼ $15/unit c2 ¼ $1.6/unit c3 ¼ $12/unit M ¼ 30 days S ¼$1200/setup

Table 2 The solving process. m

D

(t(m), T(m))a

JATC(m, t(m), T(m))

1 2 3 4 5 6 7 8

2265 213  640  1192  1625  1997  2336  2652

(t1,T1) ¼(46.30, 101.18) (t1,T1) ¼(31.64, 70.88) (t2,T2)¼ (25.68, 57.78) (t2,T2) ¼(22.24, 50.04) (t2,T2)¼ (19.89, 44.75) (t2,T2) ¼(18.16, 40.85) (t2,T2)¼ (16.81, 37.82) (t2,T2)¼ (15.72, 35.38)

50,419.26 47,444.09 46,384.38 45,911.17 45,695.89 45,616.20 45,615.76b 45,664.91

a b

The values of t(m) and T(m) is presented by days. The optimal solution for Example 1.

253

Example 1. By using the algorithm proposed in Section 4 and software MATHEMATICA 4.0, the solution procedure for the optimal (mn,tn,Tn) along with minimum total cost per year is shown in Table 2. Using the algorithm the optimal shipment number from the supplier to the retailer is mn ¼7, the optimal stock-end cycle length for the retailer is tn ¼t2 ¼16.81 days, and the optimal replenishment cycle length is Tn ¼T2 ¼37.82 days. Under such conditions, the supply quantity from the supplier to the retailer in a production batch is qn ¼ DTn/(1  l) ¼583 units per delivery and the retailer’s order quantity of good quality items is Qn ¼ m (1 l)qn ¼3264 units per order. The corresponding minimum total cost of the system is JATC(mn,tn,Tn) ¼JATC2(7,16.81,37.82)¼ $45615.76. Example 2. The effect of credit terms on performance are analyzed using the data in Table 1, MA{7, 14, 30, 45, 60} and applying the above solution algorithm. Following this, the optimal results are obtained for various values of M as shown in Table 3. Furthermore, the optimal solution in the case of no trade credit (i.e., M¼ 0) is listed in the first line of Table 3 to show the effect of trade credit strategy. From Table 3 it can be seen that when the supplier refrains from offering trade credit to the buyer (i.e., M¼0), the entire supply chain incurs the highest cost. The minimum total cost of the system JATC(mn,tn,Tn) decreases as the credit period increases. The computational results demonstrate that the buyer can always take advantage of trade credit. Table 3 also shows that when the other parameters are kept constant as the parameter M increases, the optimal replenishment cycle length Tn, the optimal length of stock-end cycle of the retailer tn and the optimal supply quantity qn per delivery from the supplier to the retailer will decrease. The optimal order quantity Qn will first decrease and then increase. Once the credit period M reaches 30 day or more, the parameters mn, tn,Tn, qn and Qn stay constant, while ATCv increases and both ATCb and JATC(mn,tn,Tn) decrease. The above results illustrate that when the supplier extended his/her credit terms from 0 to 30 day, the retailer will shorten the replenishment cycle length Tn to take advantage of the trade credit more frequently. Here, the optimal order quantity Qn first decreases then increases. It is noted that if the credit period M offered by the supplier is too short (e.g., 7 days or 14 days) to encourage the retailer to increase order quantity, the trade credit strategy will fail. As a result, the supplier will suffer losses for a strategic mismatch. On the other hand, when the incentive provided by the supplier is long enough, it is found that the retailer’s order quantity increases from 3057 units to 3264 units as M increases from 0 to 30 days. Therefore, not only the supplier’s total costs but his/her total profits increase. When the supplier’s allowable delay in payment time is too long (i.e., M430 days), as M increases the retailer’s total cost decreases as a result of increasing interest earned and the absence of

Table 3 Performances of supply chain for various credit terms. Ma

mn

tna

Tna

qn

Qn

ATCv

ATCb

JATC(mn,tn,Tn)

Supplier’s profitb ($/year)

0 7 14 30 45 60

6 6 6 7 7 7

20.00 19.34 18.62 16.81 16.81 16.81

T1 ¼ 41.33 T1 ¼ 41.26 T1 ¼ 41.05 T2 ¼ 37.82 T3 ¼ 37.82 T3 ¼ 37.82

637 636 633 583 583 583

3057 3052 3036 3264 3264 3264

29481.17 29583.25 29682.44 29972.50 30194.42 30416.34

17050.81 16781.43 16479.74 15643.25 14903.53 14163.80

46,531.98 46,364.69 46,162.17 45,615.76 45,097.95 44,580.14

16,380.12 16,199.23 15,862.79 18,986.22 18,764.30 18,542.38

a b

The values of M,tn and Tn is presented by days. Supplier’s profit¼ c1Q ATCv.

254

C.-H. Su / Int. J. Production Economics 139 (2012) 247–256

opportunity cost for the purchase items. To avoid high transportation and inventory holding costs the retailer maintains a small lot size and a correspondingly short replenishment cycle length as the credit period M is extended. It is seen the retailer’s order quantity keeps (Qn ¼3264 units) when M increases. To illustrate the effects of credit terms on the supplier’s performance, the supplier’s profits are listed in the last column of Table 3. Table 3 shows though the total cost increases as M increases, the maximum profit occurs when M¼30 for the supplier. These results reveal that to avoid a strategic mismatch and unnecessary losses the supplier should evaluate his/her credit terms comprehensively before adopting it. Example 3. A sensitivity analysis is conducted to study the effect of changes in model parameters on the optimal solution and the joint total cost JATC. To carry out the analysis the value of one of the parameters (D, R, A, S, f, b, l, p, v1, v2, v3, b, c1, c2, c3, hv, hb1, hb2, Iv, Ie, Ip, w) is changed by  40%,  20%,  10%, þ10%, þ20%, and þ40%, while all the other parameters remain unchanged. The results of the sensitivity analysis are shown in Table 4. The percentage cost savings (PCS) is derived as follows: PCS¼ JATC JATCn/JATCn  100%, where JATCn is the optimal minimum total cost in Example 1. The results in Table 4 can be summarized as follows: (a) The sensitivity of JATC to parameter changes: Highly sensitive—D, v1, l and c2. Moderately sensitive—R, f, hv and w. Slightly sensitive—A, S, b, p, v2, v3, b, c1, c3, hb1, hb2, Iv and Ie. (b) The relationship between JATC and model parameters: Positively related—D, R, A, S, f, b, l, v1, v2, v3, c1, c2, c3, hv, hb1, hb2, Iv and w. Negatively related—p, b and Ie. Unrelated—Ip. Note that in this example the relationship between M and the optimal solution (Tn,tn) can be expressed as tn rM rTn. In other words, as the permissible payment time M expires on or after the inventory is depleted completely at time tn, the retailer pays no

opportunity cost for the purchase items. Therefore, the minimum joint total cost JATC(mn,tn,Tn) will not vary as the parameter Ip changes. The value of JATC(mn,tn,Tn) is most sensitive to market demand. In addition, the results demonstrate that the parameters l, v1 and c2 are important influences on the minimum joint total cost. The results indicate that to improve the system effectively, members of the supply chain must work carefully to improve market demand through better product quality level via economies of production.

6. Conclusion In this study an integrated inventory model is proposed that accounts for defective items in a retailer’s arrival order lot under a full-lot inspection policy and also accounts for shortages. It is assumed that retailers are permitted trade credit. All defective items delivered by the supplier will be returned and classified into two categories: those that still have some worth and those that are waste items. For study purposes, a two-echelon inventory model is built and the average cost function is formulated. By employing a novel algorithm, the optimal duration of the replenishment cycle time, the time taken to run out of stock and the number of lots delivered from the supplier to the retailer is determined. Additionally, a sensitivity analysis is conducted on the main parameters of the model. Through mathematical analysis it is shown that whilst the extension of supplier credit terms will allow retailers to take advantage of trade credit more frequently, the retailer may order a smaller quantity to shorten the replenishment cycle length. Conversely, if the allowable delay in payment time is sufficiently long, then the retailer is unlikely to reduce their order quantity because there is a trade-off with transportation costs. A shorter replenishment cycle length resulting from a smaller order size incurs higher transportation costs which the retailer will try to avoid. This study provides a useful managerial insight that suppliers should critically evaluate their credit terms carefully in order to avoid a strategic mismatch and unnecessary losses. The results illustrate that to improve the system effectively

Table 4 Results of the sensitivity analysis on different parameters.  40 n

D R A S f

b l p v1 v2 v3 b c1 c2 c3 hv hb1 hb2 Iv Ie Ip w

 20 n

n

 10 n

n

þ10 n

n

þ20 n

n

þ40 n

(t , T )

PCS (%)

(t , T )

PCS (%)

(t , T )

PCS (%)

(t , T )

PCS (%)

(t , T )

PCS (%)

(tn, Tn)

PCS (%)

(24.11,54.25) (15.59,35.07) (17.52,39.41) (16.85,37.91) (14.07,31.66) (16.81,37.82) (19.03,42.81) (17.82,37.72) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (13.87,42.78) (17.90,40.28) (22.26,41.93) (18.81,42.32) (16.81,37.82) (19.46,41.19) (16.81,37.82) (16.81,37.82)

 34.77  5.64%  0.80  1.62  2.28  1.97  5.41 0.90  19.73  1.97  0.79 0.49  0.39  7.89  1.02  2.66  0.58  0.79  0.39 1.11 0.00  2.76

(19.51,43.90) (16.89,38.00) (17.84,40.14) (17.52,39.41) (16.00,36.01) (16.81,37.82) (18.59,41.82) (17.30,37.77) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.27,41.69) (17.80,40.05) (20.00,41.33) (18.47,41.57) (16.81,37.82) (18.79,41.01) (16.81,37.82) (16.81,37.82)

 11.13  1.46  0.39  0.80  1.08  0.99  2.82 0.45  9.87  0.99  0.39 0.25  0.19  3.95  0.45  1.26  0.26  0.39  0.19 0.56 0.00  1.38

(18.76,42.20) (17.27,38.86) (18.0040.50) (17.84,40.14) (16.41,36.92) (16.81,37.82) (18.37,41.34) (17.05,37.79) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (17.26,41.24) (17.28,38.89) (19.03,41.08) (18.31,41.21) (16.81,37.82) (18.47,40.93) (16.81,37.82) (16.81,37.82)

 8.46  0.61  0.20  0.39  0.54  0.49  1.44 0.23  4.93  0.49  0.20 0.12  0.10  1.97  0.21  0.62  0.12  0.19  0.10 0.28 0.00  0.69

(16.42,36.95) (17.83,40.11) (16.94,38.12) (17.08,38.42) (18.55,41.73) (16.81,37.82) (16.63,37.42) (17.90,40.87) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (17.56,37.52) (17.72,39.86) (16.08,37.64) (16.68,37.52) (16.81,37.82) (16.53,37.75) (16.81,37.82) (16.81,37.82)

8.27 0.42 0.18 0.36 0.49 0.49 1.48  0.23 4.93 0.49 0.20  0.12 0.10 1.97 0.18 0.56 0.11 0.18 0.10  0.28 0.00 0.69

(15.18,34.17) (17.56,39.52) (17.08,38.42) (17.34,39.01) (18.93,42.58) (16.81,37.82) (16.45,37.02) (17.65,40.89) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (18.25,37.25) (17.31,38.94) (15.40,37.48) (16.55,37.24) (16.81,37.82) (16.27,37.69) (16.81,37.82) (16.81,37.82)

16.39 0.76 0.36 0.71 0.96 0.99 3.02  0.46 9.87 0.99 0.39  0.25 0.19 3.95 0.35 1.11 0.20 0.35 0.19  0.57 0.00 1.38

(13.72,30.87) (19.00,42.75) (17.34,39.01) (17.85,40.17) (19.67,44.25) (16.81,37.82) (16.11,36.24) (17.18,40.94) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (16.81,37.82) (19.44,36.79) (16.56,37.27) (14.22,37.20) (16.30,36.68) (16.81,37.82) (15.76,37.57) (16.81,37.82) (16.81,37.82)

32.05 1.24 0.71 1.41 1.88 1.97 6.33  0.92 19.73 1.97 0.79  0.49 0.39 7.89 0.63 2.17 0.38 0.70 0.39  1.15 0.00% 2.76

C.-H. Su / Int. J. Production Economics 139 (2012) 247–256

members of the supply chain are required to work carefully to improve market demand through increased product quality. Future research could modify or extend the present model in several ways. First, in this study a single-supplier and a singleretailer for a single product were considered. This foundation could be further extended to more practical situations, such as considering multiple retailers or multiple items, or taking raw material supply into account. Other possible extensions are to formulate the demand as a function of the length of the allowable payment delay time and the other factors, or to examine the interactions between the supplier and the retailer from the perspective of game theory. Finally, it would be of significant interest to relax the deterministic assumptions, such as demand rate, defective rate, and backorder rate; and thus extend the methodology to operate in an uncertain or stochastic environment.

Acknowledgments

255

or equivalently (c3 þc1Ip þhb1)T1 Zc3T1 þ(c1Ip pIe)M, that is T1 Z

c3 T 1 þðc1 Ip pIe ÞM ¼ t1 : c3 þ c1 Ip þ hb1

ðA6Þ

As a result, from (A5) and (A6), we have Mr t1 rT1. The proof is completed. Appendix B. JACT1(m,t,T) is a convex function at the point (t,T) ¼(t1,T1) for a given m. Proof. For fixed m, taking the second order partial derivatives of JACT1(T,t,m) with respect to t and T, then substituting (t,T)¼(t1,T1) into these equations, we obtain  Dðc3 þ c1 Ip þhb1 Þ @2 JATC 1 ðm,t,TÞ ¼ 40, ðB1Þ   T1 @t 2 ðt 1 ,T 1 Þ

The author is indebted to the anonymous referees for providing valuable comments and suggestions. This research was supported by the National Science Council of the Republic of China under Grant NSC 98-2410-H-236-002.

Appendix A. If and only if D Z0, then Mrt1 rT1, where D be defined as in (14)

 @2 JATC 1 ðm,t,TÞ   @T 2 ðt

¼ 1 ,T 1 Þ

D½2W þ ð1lÞ2 c3 þ2ð1lÞlhb2  ð1lÞ2 T 1

4 0,

ðB2Þ

and  @2 JATC 1 ðm,t,TÞ   @T@t

ðt 1 ,T 1 Þ

 @2 JATC 1 ðT,t,mÞ ¼   @t@T

¼

ðt1 ,T 1 Þ

Dc3 : T1

ðB3Þ

Thus, Proof. From (12) and (13), we have t1 ¼

c3 T 1 þ ðc1 Ip pIe ÞM : c3 þ c1 Ip þhb1

ðA1Þ

Assuming M rt1 rT1 holds, that is t1 ¼

c3 T 1 þ ðc1 Ip pIe ÞM ZM c3 þ c1 Ip þhb1

and

T 1 Zt 1 ¼

c3 T 1 þ ðc1 Ip pIe ÞM , c3 þc1 Ip þ hb1

which implies T1 Z

c3 þ pIe þhb1 M c3

and

T1 Z

c1 Ip pIe M: c1 Ip þ hb1

ðA2Þ

 @2 JATC 1 ðm,t,TÞ   @t 2 ðt

1 ,T 1 Þ

 @2 JATC 1 ðT,t,mÞ   @T 2

ðt1 ,T 1 Þ

2

 @2 JATC 1 ðT,t,mÞ 4   @T@t

32 5

ðt 1 ,T 1 Þ

n  o ¼ 2 W þ lð1lÞhb2 ðc3 þ cIp þ hb1 Þ þ ð1lÞ2 c3 ðc1 Ip þ hb1 Þ  2 D 4 0: ðB4Þ ð1lÞT 1 Since (t1,T1) is the unique solution which satisfy @JATC1(m,t,T)/ @t ¼0 and @JATC1(m,t,T)/@T¼0, therefore JACT1(m,t,T) is a convex function at point (t,T)¼ (t1,T1). The proof is completed. References

Because ðpIe þ hb1 Þðc3 þ c1 Ip þhb1 Þ c3 þ pIe þ hb1 c1 Ip pIe 40,  ¼ c3 ðc1 Ip þ hb1 Þ c3 c1 Ip þ hb1

ðA3Þ

thus, from (A2), we obtain T1 Zc3 þpIe þhb1/c3M. Substituting (13) into this inequality, after some algebra operation, we obtain 2K½c3 ð1lÞ2 Z DM2 ðc3 þ pIe þhb1 Þ 2½W þ lð1lÞhb2 ðc3 þpIe þ hb1 Þ þ c3 ð1lÞ2 ðpIe þhb1 Þ that is, D Z0. Conversely, if D Z0, we can show that T1 Z

c3 þ pIe þhb1 M, c3

ðA4Þ

or equivalently c3T1 þ(c1Ip  pIe)M Z(c3 þc1Ip þhb1)M, that is t1 ¼

c3 T 1 þ ðc1 Ip pIe ÞM Z M: c3 þ c1 Ip þhb1

ðA5Þ

On the other hand from (A3) and (A4), we have T1 Zc1Ip pIe/ cIp þhb1M,

Affisco, J.F., Paknejad, M.J., Nasri, F., 2002. Quality improvement and setup reduction in the joint economic lot size model. European Journal of Operational Research 142, 497–508. Aggarwal, S.P., Jaggi, C.K., 1995. Ordering policies of deteriorating items under permissible delay in payments. Journal of the Operational Research Society 46 (5), 658–662. Balkhi, Z.T., 2004. An optimal solution of a general lot size inventory model with deteriorated and imperfect products, taking into account inflation and time value of money. International Journal of Systems Science 35 (2), 87–96. Banerjee, A., 1986. A joint economic-lot-size model for purchaser and vendor. Decision Sciences 17 (3), 292–311. Chang, H.C., 2003. Fuzzy opportunity cost for EOQ model with quality improvement investment. International Journal of Systems Science 34 (6), 395–402. Chen, L., Kang, F., 2010a. Integrated inventory models considering the two-level trade credit policy and a price-negotiation scheme. European Journal of Operational Research 205, 47–58. Chen, L., Kang, F., 2010b. Coordination between vendor and buyer considering trade credit and items of imperfect quality. International Journal of Production Economics 123 (1), 52–61. 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 (4), 427–437. Chiu, Y.S., Liu, S.C., Chiu, C.L., Chang, H.H., 2011. Mathematical modeling for determining the replenishment policy for EMQ model with rework and multiple shipments. Mathematical and Computer Modelling 54, 2165–2174. Chung, C.J., Wee, H.W., 2008. Scheduling and replenishment plan for an integrated deteriorating inventory model with stock-dependent selling rate. The International Journal of Advanced Manufacturing Technology 35 (7), 665–679.

256

C.-H. Su / Int. J. Production Economics 139 (2012) 247–256

Comeaux, E.J., Sarker, B.R., 2005. Joint optimization of process improvement investments for manufacture-buyer cooperative commerce. Journal of the Operational Research Society 56, 1310–1324. Emery, G., 1987. An optimal financial response to variable demand. Journal of Financial and Quantitative Analysis 22 (2), 209–225. Glock, C.H., 2012. Coordination of a production net work with a single buyer and multiple vendors. International Journal of Production Economics 135, 771–780. Glock, C.H., 2011. A multiple-vendor single-buyer integrated inventory model with a variable number of vendors. Computers & Industrial Engineering 60, 173–182. Goyal, S.K., 1976. An integrated inventory model for a single manufacture-single customer problem. International Journal of Production Research 15 (1), 107–111. Goyal, S.K., 1985. Economic order quantity under conditions of permissible delay in payment. Journal of the Operational Research Society 36, 335–338. Goyal, S.K., 1988. A joint economic lot size model for purchaser and vendor: a comment. Decision Sciences 19, 236–241. Goyal, S.K., 1995. A one-vendor multi-buyer integrated inventory model: a comment. European Journal of Operational Research 82 (1), 209–210. Goyal, S.K., 2000. On improving the single-vendor single-buyer integrated production inventory model with a generalized policy. European Journal of Operational Research 125, 429–430. Goyal, S.K., Nebebe, F., 2000. Determination of economic production-shipment policy for a single-vendor-single-buyer system. European Journal of Operational Research 121, 175–178. Ha, D., Kim, S.L., 1997. Implementation of JIT purchasing: an integrated approach. Production Planning & Control 8 (2), 152–157. Hill, R.M., 1999. The optimal production and shipment policy for the single-vendor single-buyer integrated production-inventory problem. International Journal of Production Research 37 (11), 2463–2475. Hoque, M.A., 2008. Synchronization in the single-manufacturer multi-buyer integrated inventory supply chain. European Journal of Operational Research 188, 811–825. Hou, K.L., Lin, L.C., 2004. Optimal production run length and capital investment in quality improvement with an imperfect production process. International Journal of Systems Science 35 (2), 133–137. Huang, C.K., 2004. An optimal policy for a single-vendor single-buyer integrated production-inventory problem with process unreliability consideration. International Journal of Production Economics 91, 91–98. Huang, Y.F., 2003. Optimal retailer’s ordering policies in the EOQ model under trade credit financing. Journal of the Operational Research Society 54 (9), 1011–1015. 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 (8), 826–833. Kelle, P., Al-khateeb, F., Miller, P.A., 2003. Partnership and negotiation support by joint optimal ordering/setup policies for JIT. International Journal of Production Economics 81–82, 431–441. Khan, M., Jaber, M.Y., Guiffrida, A.L., Zolfaghari, S., 2011. A review of the extensions of a modified EOQ model for imperfect quality items. International Journal of Production Economics 132 (1), 1–12. Khan, L.R., Sarker, R.A., 2002. An optimal batch size for a JIT manufacturing system. Computers and industrial engineering 42, 127–136. Kim, J.S., Hwang, H., Shinn, S.W., 1995. An optimal credit policy to increase wholesaler’s profit with price dependent demand function. Production Planning & Control 6, 45–50. Kim, S.L., Ha, D., 2003. A JIT lot-splitting model for supply chain management: Enhancing buyer–supplier linkage. International. Journal of Production Economics 86, 1–10. Li, J., Feng, H., Wang, M., 2009. A replenishment policy with defective products, backlog and delay of payments. Journal of Industrial and Management Optimization 5 (4), 867–880.

Lo, S.T., Wee, H.M., Huang, W.C., 2007. An integrated production-inventory model with imperfect production processes and Weibull distribution deterioration under inflation. International Journal of Production Economics 106 (1), 248–260. Lu, L., 1995. A one-vendor multi-buyer integrated inventory model. European Journal of Operational Research 81, 312–323. Maddah, B., Jaber, M.Y., 2008. Economic order quantity for items with imperfect quality: revisited. International Journal of Production Economics 112, 808–815. Ouyang, L.Y., Chen, C.K., Chang, H.C., 2002. Quality improvement, setup cost and lead-time reductions in lot size reorder point models with an imperfect production process. Computers and Operations Research 29, 1701–1717. Ouyang, L.Y., Ho, C.H., Su, C.H., 2005. Optimal strategy for the integrated vendorbuyer inventory model with adjustable production rate and trade credit. International Journal of Information and Management Sciences 16 (4), 19–37. Ouyang, L.Y., Ho, C.H., Su, C.H., 2009. An optimization approach for joint pricing and ordering problem in an integrated inventory system with order-size dependent trade credit. Computers & Industrial Engineering 57, 920–930. Pan, J.C.H., Yang, J.S., 2002. A study of an integrated inventory with controllable lead time. International Journal of Production Research 40 (5), 1263–1273. Papachristos, S., Konstantaras, I., 2006. Economic ordering quantity models for items with imperfect quality. International Journal of Production Economics 100, 148–154. Petersen, M.A., Rajan, R.G., 1997. Trade credit: theories and evidence. Review of financial studies 10 (3), 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 process. 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. Sarker, B.R., Diponegoro, A., 2009. Optimal production plans and shipment schedules in a supply-chain system with multiple suppliers and multiple buyers. European Journal of Operational Research 194, 753–773. Schwaller, R.L., 1988. EOQ under inspection costs. Production and Inventory Management Journal 29 (3), 22–24. Siajadi, H., Ibrahim, R.N., Lochert, P.B., 2005. Joint economic lot size in distribution system with multiple shipment policy. International Journal of Production Economics 102, 302–316. Singer, M., Donoso, P., Traverso, P., 2003. Quality strategies in supply chain alliances of disposable items. Omega 31, 499–509. Su, C.H., Ouyang, L.Y., Ho, C.H., Chang, C.T., 2007. Retailer’s inventory policy and manufacture’s delivery policy under two-level trade credit strategy. AsiaPacific Journal of Operational Research 24 (5), 613–630. Teng, J.T., 2002. On the 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., Goyal, S.K., 2005. Optimal pricing and ordering policy under permissible delay in payments. International Journal of Production Economics 97, 121–129. Viswanathan, S., 1998. Optimal strategy for the integrated vendor–buyer inventory model. European Journal of Operational Research 105, 38–42. Woo, Y.Y., Hsu, S.L., Wu, S., 2001. An integrated inventory model for a single vendor and multiple buyers with ordering cost reduction. International Journal of Production Economics 73, 203–215. Wu, K.S., Ouyang, L.Y., 2000. Defective units in (Q,r,L) inventory model with sublot sampling inspection. Production Planning & Control 11 (2), 179–186. Yao, M.J., Chiou, C.C., 2004. On a replenishment coordination model in an integrated supply chain with one vendor and multiple buyers. European Journal of Operational Research 159 (2), 406–419. Yu, J.C.P. A collaborative strategy for deteriorating inventory system with imperfect items and supplier credits. International Journal of Production Economics, http://dx.doi.org/10.1016/j.ijpe.2011.11.018, in press.