a joint optimal ordering and delivery policy for an integrated supplier–retailer inventory model...

Applied Mathematics and Computation 218 (2012) 7498–7514

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate /amc

A joint optimal ordering and delivery policy for an integratedsupplier–retailer inventory model with trade credit and defective items

Yu-Jen Lin a,⇑, Liang-Yuh Ouyang b, Ya-Fang Dang c

a Department of Industrial Engineering and Management, St. John’s University, Tamsui, New Taipei City, Taiwanb Department of Management Sciences, Tamkang University, Tamsui, New Taipei City, Taiwanc Transaction Management Department, Taiwan Life Insurance Co., Ltd., Taipei, Taiwan

a r t i c l e i n f o

Keywords:ProductionIntegrated inventory modelDelay in paymentDefective items

0096-3003/$ - see front matter � 2012 Elsevier Incdoi:10.1016/j.amc.2012.01.016

⇑ Corresponding author. Address: Department of ITaipei City 25135, Taiwan.

E-mail address: [email protected] (Y.-J. Lin).

a b s t r a c t

In this paper, we propose an integrated supplier–retailer inventory model in which bothsupplier and retailer have adopted trade credit policies, and the retailer receives an arrivinglot containing some defective items. The customer’s market demand rate depends on thelength of the credit period offered by retailer. Our objective is to determine the retailer’soptimal order cycle length, the order quantity, and the optimal number of shipments perproduction run from the supplier to the retailer so that the entire supply system has max-imum profit. We develop an algorithm to find the optimal solution for the supply chain.Several numerical examples are provided to illustrate the theoretical results, and sensitiv-ity analysis of major parameters including the defective rate in a production batch, theretailer’s trade credit period and the customer’s trade credit period in the model arepresented.

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1. Introduction

In most of the early literature dealing with inventory problems, emphases were placed on either the retailer’s or the sup-plier’s perspective to minimize the cost and/or to maximize the profit. Recently, the integrated inventory models have be-come more and more important, because the supplier and the retailer can increase their mutual benefit through strategiccooperation. Goyal [1] developed an integrated model for a single supplier-single retailer system to find the optimal orderquantity of the retailer so that the total cost at the system is minimized. Monahan [2] examined the quantity discount prob-lem from the supplier point of view and obtained the minimum cost of the entire supply chain. Banerjee [3] presented a jointeconomic-lot-size model where a supplier produces for a retailer to order on a lot-for-lot basis. Goyal [4] generalized Baner-jee’s [3] model by relaxing the assumption of the lot-for-lot policy of the supplier and illustrated that the inventory cost canbe reduced significantly if the supplier’s economic production quantity is a positive integer multiple of the retailer’s pur-chase quantity. Lu [5] assumed that the supplier’s production rate is greater than the demand rate, and the delivery startsas soon as the quantity ordered by the retailer is produced, and later on goods are delivered on a lot-for-lot basis. Goyal [6]relaxed the lot-for-lot policy and assumed that if the demand is constant, shipment sizes will increase according to the ratioof production rate and demand rate. Goyal and Nebebe [7] proposed the first shipment to be smaller and is followed by ship-ments of equal size. Recently, Ouyang et al. [8] proposed an integrated inventory model with quality improvement and lead

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ndustrial Engineering and Management, St. John’s University, 499, Sec. 4, Tam King Road, Tamsui, New

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Y.-J. Lin et al. / Applied Mathematics and Computation 218 (2012) 7498–7514 7499

time reduction. Other related studies of the integrated inventory model include Yang and Wee [9], Yang et al. [10], Wee andChung [11], Teng et al. [12], and so on.

In the traditional inventory models, the theme of defective items is always ignored. However, defective items can bycaused by the incomplete production process and/or damage in transit. And the number of defective items will influencethe on-hand level and the number of orders in the inventory system. In addition, if the retailer sells defective items withoutinspection, the customers will complain, return the goods, or even never come back. In all cases, substantial costs are in-curred. Already there are some scholars who have studied and developed various analytical inventory models about defec-tive items. Porteus [13] and Rosenblatt and Lee [14] are among the first ones who analyzed a significant relationshipbetween quality imperfection and lot size. Next, Paknejad et al. [15] proposed a modified EOQ model with stochastic de-mand, and the model included the number of defective items in a lot as a random variable. In each delivery, the defectiveitems will be found in each lot and sent back to the supplier in the delivery time of the next batch. Salameh and Jaber[16] presented an EPQ model with defective items, and they assumed that the production rate for the non-defective itemsis greater than the demand rate. Ouyang and Chang [17] presented an investment in quality improvement inventory modelinvolving defective items production process with controllable lead time. There are more papers related to this issue ofdefective items such as Chung and Hou [18], Hou [19], Rahim and Al-Hajailan [20], Lin [21], Wee et al. [22], Sarkar [23],and Barzoki et al. [24], etc.

Furthermore, in practical situations, in order to motivate retailer to increase order quantity and market share, the supplieroften offers a trade credit to the retailer, that is, the retailer may receive goods or services without having to pay until some-time later. Haley and Higgins [25] first presented an inventory model with the permissible delay in payments. Ferris [26]derived a transactions theory of trade credit use from the motives of trading partners to economize on the joint costs of ex-change. Kingsman [27] considered the effects of different ways of payment on ordering and stocking. Goyal [28] establishedan EOQ inventory model with interest earned and paid under the condition of permissible delay in payments. Aggarwal andJaggi [29] extended Goyal’s [28] model to include deteriorating items. Jamal et al. [30] further generalized this issue withallowable shortages. Buzacott and Zhang [31] proposed an inventory management to incorporate asset-based financing intoproduction decisions. In their paper, the retailers buy a product from the suppliers and then sell it to the customers in whichthe retailers require asset-based financing by bank to purchase product from the suppliers. Among other relative inventory-financing issues studies were Hill and Riener [32], Abad and Jaggi [33], Chen and Kang [34], Huang and Hsu [35], Ho et al.[36] and Thangam and Uthayakumar [37].

Because of changing of the business environment, the delay payments of trade credit change with each passing day. Thereexist numerous interesting and relevant papers related to trade credits, but most assume that the supplier offers a tradecredit to the retailer. However, the retailer wishes to motivate the customer’s demand rate and to reduce the on-hand stockcost, and offers a trade credit to the customers. Huang [38] considered an EOQ inventory model in which both supplier andretailer have adopted trade credit policies. Su et al. [39] developed an integrated supplier-retailer inventory model in whichthe customer’s demand for goods is positively correlated to the credit period offered by the retailer. They discussed how toobtain optimal order quantity, shipping, and inventory policy.

In this paper, we considered the integrated supplier-retailer inventory model in which both supplier and retailer haveadopted a trade credit strategy, involving defective items, and the demand rate varies according to the length of the cus-tomer’s trade credit period. This study aims to find the optimal retailer’s replenishment cycle length, the order quantity,and the optimal number of shipments per production run from the supplier to the retailer, so that the entire supply chainhas maximum profit. An algorithm is developed to determine the optimal solution. Finally, numerical examples are pre-sented to illustrate the solution procedure, and sensitivity analysis of major parameters involved in the model is also made.

2. Notation and assumptions

In this paper, the mathematical model is developed on the basis of the following notation and assumptions.Notation:

P
The supplier’s production rate. D The retailer’s demand rate. K The supplier’s setup cost per order. A The retailer’s ordering cost per unit ordered. F Transportation cost per delivery. hv The supplier’s holding cost per item per unit time. hb1
The retailer’s holding cost per non-defective item per unit time, excluding interest charges.
hb2
The retailer’s holding cost per defective item per unit time (including treatment cost), excluding interestcharges, hb2

6 hb1.

s
The retailer’s unit screening cost. x The retailer’s screening rate per order. c The supplier’s defective rate in a production batch, is given, 0 6 c < 1.

7500 Y.-J. Lin et al. / Applied Mathematics and Computation 218 (2012) 7498–7514

w
The supplier’s unit treatment cost of defective items. c The supplier’s production cost per unit. v Unit price charged by the supplier to the retailer. p Unit retail price charged by the retailer to the customers. M The retailer’s trade credit period offered by the supplier per order. N The customer’s trade credit period offered by the retailer per order, where N 6M. IBe The retailer’s interest earned per dollar per unit time. IBp The retailer’s capital opportunity cost per dollar per unit time. Ivp The supplier’s capital opportunity cost per dollar per unit time. Q The retailer’s order quantity per order (non-defective items). q Size of the shipments from the supplier to the retailer in a production batch. n Number of shipments from the supplier to the retailer per production run, a positive integer. n⁄ Optimal number of shipments from the supplier to the retailer per production run. T The retailer’s replenishment cycle length. T(n) The retailer’s replenishment cycle length for a fixed n. T⁄ The retailer’s optimal replenishment cycle length. JTP(n,T) The channel’s total profit per unit time, with number of shipments nand cycle length T. JTP⁄ The optimal channel’s total profit per unit time, i.e., JTP⁄ = JTP(n⁄,T⁄).
Assumptions

1. There is single-supplier, single-retailer for a single product, and the inventory system deals with only one type of item.2. For each transportation cost F, there is no influence for each shipment quantity.3. Shortages are not allowed.4. The retailer orders quantity Q (non-defective items) per order and each batch is dispatched to the retailer in n equally

sized shipments. The size of each batch delivered to retailer is q, therefore, the supplier manufactures in batches of sizenq, where n is a positive integer.

5. The retailer receives an arriving lot containing some defective items with defective rate c. Upon arrival of an order, allthe items in the lot are inspected with screen ratex. The defective items will be found in each lot and sent back to thesupplier in the delivery time of the next batch. Therefore, the retailer receives an arriving lot containing (1 � c)q non-defective items and the length of replenishment cycle is T = (1 � c)q/D, and order quantity Q is the total of non-defec-tive items in the n shipments, i.e., Q = n(1 � c)q = nDT.

6. Since shortages are not allowed, the supplier’s production rate for the non-defective items is greater than the retailer’sdemand rate, i.e., (1 � c)P > D.

7. The relationship between the supplier’s production cost (c), the retailer’s purchase cost (v) and retail price (p) isp > v > c.

8. We assume that both supplier and retailer have adopted a trade credit policy: supplier offers a credit period M to theretailer, and the retailer offers a credit period N to each customer, where M and N are given and N 6M.

9. The retailer sells the item and uses the sales revenue to earn interest at a rate IBe in the time interval [N,M]. At the endof the permissible delay period M, the retailer pays off all purchasing cost to the supplier and incurs a capital oppor-tunity cost at rate of IBp for the items still in stock and for the items sold but which have not yet been paid by the cus-tomers. Similarly, during the credit period, [0,N], the customer does not pay the payment until the end of the creditperiod offered by the retailer.

10. Since the supplier offers retailer a trade credit strategy, the supplier cannot receive the payment immediately afterdelivery of the items and therefore has to incur an opportunity cost at a rate of Ivp.

11. As stated Jaggi et al. [40], ‘‘it is observed that credit period offered by the retailer to its customers has a positive impacton demand of an item.’’ For simplicity, we assume that the market demand rate for the item increases exponentiallywith the customer’s credit period N and is given by D(N) = D0ehN, where D0 > 0 is the market demand rate when theretailer does not offer a credit period to the customers, and h P 0 is a constant factor (this demand function can befound in Su et al. [39] and Teng and Lou [41]). For notational simplicity, D(N) and D will be used interchangeablythroughout this paper.

3. The basic model

3.1. Retailer’s total profit per unit time

The retailer’s total profit per unit time included the sales revenue, the interest earned, the ordering cost, the transporta-tion cost, the holding cost, the screening cost, and the opportunity cost. These components are calculated as follows:


(a) Sales revenueFor each order quantity Q, the retailer is charged vQ from the supplier, and pQ is received from the customer. There-fore, the retailer’s sales revenue per unit time is

ðp� vÞQ=ðnTÞ ¼ Dðp� vÞ:

(b) Ordering costThe retailer’s ordering cost per unit ordered is A. Each batch is dispatched to retailer in n equally sized shipments, andeach replenishment cycle length is T, hence, the ordering cost is A/(nT) per unit time.

(c) Transportation costTransportation cost per delivery is F, hence, transportation cost is F/T per unit time.

(d) Holding costThe retailer’s inventory pattern is shown in Fig. 1. Upon arrival of an order, all the items are inspected with screeningrate x by the retailer. Hence, the retailer’s holding cost can be split into two components: non-defective holding costand defective holding cost. The replenishment cycle length is T, the number of non-defective items is (1 � c)q = DT,and the holding cost per non-defective items per unit time is hb1 . Hence, the retailer’s holding cost of non-defectiveitems and undetected defective items per unit time is

hb1

TDT2

2þ cq2

2x

!¼ hb1

DT2

1þ cD

xð1� cÞ2

" #:

Moreover, the number of defective items in each received lot is c q, the screening rate is x, the duration of the screening per-iod is q/x, and the holding cost per defective items per unit time is hb2 . Hence, the retailer’s holding cost of defective items perunit time is

hb2

TcqT � cq2

2x

� �¼ hb2

cDT1� c

1� D2xð1� cÞ

� �:

Therefore, the total holding cost per unit time is

hb1DT

21þ cD

xð1� cÞ2

" #þ hb2

cDT1� c

1� D2xð1� cÞ

� �:

(e) Screening costThe supplier delivers batches of size q to the retailer, and the retailer’s unit screening cost is s. Therefore, the retailer’sscreening cost per unit time is sq/T = sD/(1 � c).

(f) Opportunity cost and interest earnedThe retailer has to bear both the opportunity costs for items kept in stock when the payment is paid and for items soldwithout being paid for. On the other hand, the retailer’s interest is earned during the permissible settlement period.

'sRetailer inventory level

q

DT=

Time/q x

(1 ) /T q Dγ= −

/Q D nT=

(1 )qγ−

Fig. 1. Inventory pattern of the retailer.


That is, the retailer uses the sales revenue to earn interest before the due date, M. At the end of this period, the retaileris no longer to accumulate the interest earned, and the retailer starts paying for the interest charges on the items instock.

The replenishment cycle length is T. The supplier offers trade credit period of length M to the retailer, and the retaileroffers the customers a trade credit period N. The retailer sells out the stock at time T and collects all returns at time in everycycle. Considering the relationship between T, M, and T + N, there are three possible cases as: (i) T + N 6M, (ii) T 6M 6 T + Nand (iii) T P M. These cases are depicted in Fig. 2.

3.1.1. Case 1. T + N 6M(i.e., T 6M � N)In this case, the retailer has received all the payment of sales goods from the customers at time T + N, but he/she does not

pay the supplier until the end of the credit period, M. Thus, the retailer has to pay neither opportunity cost for items kept instock nor for items sold without being paid for. On the other hand, the retailer uses the sales revenue to earn interest at a rateof IBe, and thus the retailer’s interest earned per unit time is

pIBe

T

Z T

0Dtdt þ DTðM � T � NÞ

� �¼ pIBeD M � N � T

2

� �:

3.1.2. Case 2. T 6M 6 T + N (i.e., M � N 6 T 6M)In this case, the retailer pays to the supplier at M which is after the time that the retailer sold all items and before the time

that the retailer collects all returns. That is, the retailer cannot receive the payment immediately after delivery all of theitems to the customers, but pays off the supplier at the due date, M, the retailer has to endure a capital opportunity cost dur-ing the time interval [M,T + N] at a rate of IBp, and thus the retailer’s capital opportunity cost per unit time is

TimeT TimeT

TimeT

.

.

.

1.Case T N M+ ≤

Re 'tailer s inventory level Re 'tailer s inventory level

2.Case T M T N≤ ≤ +

DT

N T N+ M

DT

( )D M N−

N

N

M

M

T N+

T N+

Re 'tailer s inventory level

3.Case T M≥

DT

:Note

Denotes the accumulate interest earned

cosDenotes the capital opportunity t for

the items sold but have not yet been paid

cosDenotes the capital opportunity t for

the items on hand

Fig. 2. Retailer’s inventory level and accumulated interest earned.


pIBpD2TðT þ N �MÞ2 ¼ pIBpD N �M þ T

2þ ðN �MÞ2

2T

" #:

On the other hand, during the interval [N,M], retailer uses the sales revenue to earn interest at a rate of IBe. Thus, the retailer’sinterest earned per unit time is pIBeD(M � N)2/(2T).

3.1.3. Case 3. T P MIn this case, the retailer pays to the supplier at the end of the credit period, M, which is before the inventory is depleted

completely. Hence, the retailer still has some stock on hand during the time interval [M,T]. Therefore, the retailer incurs acapital opportunity cost per unit time is

vIBp

T

Z T

MDðT � tÞdt ¼ vIBpDðT �MÞ2

2T¼ vIBpD

T2þM2

2T�M

!:

Furthermore, for the items have already sold but have not yet been paid during the time interval [T,T + N], thus, the retailerbears an opportunity cost per unit time which is

pIBpD2T½ðT þ N �MÞ2 � ðT �MÞ2� ¼ pIBpDN 1þ N � 2M

2T

� �:

On the other hand, during the trade credit period, the retailer uses the sales revenue to earn interest during the time interval[N,M] at a rate of IBe, thus, the retailer’s interest earned per unit time is

pIBe

T

Z M

NDðt � NÞdt ¼ pIBeDðM � NÞ2

2T:

Summarizing the above cases, the total profit per unit time for the retailer is as follows:

TBPðTÞ ¼TBP1ðTÞ; if T 6 M � N;

TBP2ðTÞ; if M � N 6 T 6 M;

TBP3ðTÞ; if T P M;

8><>:

where

TBP1ðTÞ ¼ sales revenue� ordering cost� transportation cost� holding cost� screening cost

þ interest earned

¼ Dðp� vÞ � AnT� F

T� hb1

DT2

1þ cD

xð1� cÞ2

" #� hb2

cDT1� c

1� D2xð1� cÞ

� �� sD

1� cþ pIBeD M � N � T

2

� �; ð1Þ


� capital opportunity costþ interest earned


T� hb1 DT

21þ cD

xð1� cÞ2

" #� hb2cDT

1� c1� D

2xð1� cÞ

� �� sD

1� c

� pIBpD N �M þ T2þ ðN �MÞ2

2T

" #þ pIBeDðM � NÞ2

2T; ð2Þ


� capital opportunity costþ interest earned


T� hb1

DT2

1þ cD

xð1� cÞ2

" #� hb2

cDT1� c

1� D2xð1� cÞ

� �� sD

1� c� vIBpD

T2þM2

2T�M

!

� pIBpDN 1þ N � 2M2T

� �þ pIBeDðM � NÞ2

2T: ð3Þ

3.2. Supplier’s total profit per unit time

The supplier’s total profit per unit time is sales revenue minus setup cost, treatment cost of defective items, holding cost,opportunity cost (supplier cannot receive the payment immediately after delivery of the items), each element can be calcu-lated as follows:


(a) Sales revenueThe supplier produces one unit product for c and sells to the retailer for v. Hence, the supplier’s sales revenue per unittime is D(v � c).

(b) Setup costThe supplier incurs a batch setup cost K, and each production cycle length is nT. Therefore, the supplier’s setup cost perunit time is K/(nT).

(c) Treatment cost of defective itemsThe number of defective items in each production run is ncq, and the treatment cost per defective items is w. There-fore, the treatment cost of defective items per unit time is wncq/(nT) = wcD/(1 � c).

(d) Holding costSince the supplier’s production rate for the perfect items is greater than the retailer’s demand rate, the supplier’sinventory level will grow up. When the total required amount nq is accomplished, the supplier stops production.Moreover, each shipment quantity of the supplier is q, and makes continuous delivery on average every (1 � c)q/Dunits of time until the supplier’s inventory level falls to zero. Therefore, the supplier’s inventory per cycle can beobtained by subtracting the retailer’s accumulated inventory level from the supplier’s accumulated inventory level(see, Fig. 3), that is,

nqqPþ ðn� 1Þð1� cÞq

D

� �� nq

2� nq

P� ½1þ 2þ � � � ðn� 1Þ�q � ð1� cÞq

D¼ nq2

Pþ nðn� 1Þq2ð1� cÞ

2D� n2q2

2P

¼ nT2D2

Pð1� cÞ2þ nðn� 1ÞT2D

2ð1� cÞ �n2T2D2

2Pð1� cÞ2:

Hence, the supplier’s holding cost per unit time is

hv

nTnT2D2

Pð1� cÞ2þ nðn� 1ÞT2D

2ð1� cÞ �n2T2D2

2Pð1� cÞ2

" #¼ hvDT

1� cD

Pð1� cÞ þn� 1

2� nD

2Pð1� cÞ

� �:

(e) Opportunity costDue to the supplier offers credit period M to the retailer, supplier will not receive the payment until M. Hence, theopportunity cost per unit time for the supplier is vIVp(1 � c)qM/T = vIVpMD.Aforementioned, for fixed payment date M, the total profit per unit time for the supplier can be expressed as

Inventory level

nq

nq

P

q

P

Time

( 1)(1 ) /n q Dγ− −

Accumulated inventory

for the supplier

Accumulated inventory

for the retailer

Fig. 3. Inventory pattern of the supplier.


TVPðnÞ ¼ sales revenue� setup cost� treatment cost of defective items� holding cost� opportunity cost

¼ Dðv � cÞ � KnT� wcD

1� c� hvDT

1� cD

Pð1� cÞ þn� 1

2� nD

2Pð1� cÞ

� �� vIVpMD: ð4Þ

3.3. The supply chain total profit per unit time

In practice, the formation of strategic alliances between supplier-retailer is all over the industries. Traditionally, the sup-plier-retailer relationship, we know that the variations in demand to supplier from the retailer are far greater than the var-iation in demand seen by the retailer. Besides, the supplier has far better experience to understand his lead times andproduction capacities than the retailer does. Therefore, as margins downgrade and the customer satisfaction becomes moreand more important, it makes sense to establish cooperative efforts between the supplier and the retailer in order to leveragethe knowledge of both parties.

Integrated supplier-retailer inventory model is a set of approaches utilized to efficiently integrate the supplier and theretailer, so that the supplier and retailer’s profit gain and cash flow management to achieve optimal situation.

Base on the description given above, we consider the situation in which the supplier and the retailer have established along-term strategic partnership and contracted to commit to the relationship, and they will jointly determine the best policyfor the whole supply chain. Under this circumstance, supply chain total profit per unit time is given by:

JTPðn; TÞ ¼JTP1ðn; TÞ; if T 6 M � N;

JTP2ðn; TÞ; if M � N 6 T 6 M;

JTP3ðn; TÞ; if T P M;

8><>: ð5Þ

where

JTP1ðn; TÞ ¼ TBP1ðTÞ þ TVPðnÞ

¼ Dðp� cÞ � DðsþwcÞ1� c

� vIVpMDþ pIBeDðM � NÞ � Aþ K þ nFnT

� DT2

hb1þ hb1

cD

xð1� cÞ2þ pIBe

" #

� DT1� c

hb2c�hb2

cD2xð1� cÞ þ

hvDPð1� cÞ þ

hvðn� 1Þ2

� nhvD2Pð1� cÞ

� �; ð6Þ



� vIVpMD� pIBpDðN �MÞ � Aþ K þ nFnT

� pðIBp � IBeÞDðM � NÞ2

2T

� DT2

hb1þ hb1

cD

xð1� cÞ2þ pIBp

" #� DT

1� chb2

c� hb2cD

2xð1� cÞ þhvD

Pð1� cÞ þhvðn� 1Þ

2� nhvD

2Pð1� cÞ

� �: ð7Þ



� pIBpDN � vMDðIVp � IBpÞ �Aþ K þ nF

nT

� D2T

vIBpM2 þ pIBpNðN � 2MÞ � pIBeðM � NÞ2h i

� DT2

hb1þ hb1

cD

xð1� cÞ2þ vIBp

" #

� DT1� c

hb2c� hb2cD

2xð1� cÞ þhvD


2� nhvD

2Pð1� cÞ

� �: ð8Þ

Note that, JTP1 (n,M � N) = JTP2(n,M � N) and JTP2(n,M) = JTP3 (n,M). Therefore, for a fixed n, JTP(n,T) is a continuous func-tion of T. Our objective is to determine the optimal replenishment cycle length T⁄ and the optimal number of shipments n⁄, sothat the supply chain total profit per unit time has maximum value.

4. Solution procedure

First, for a given T, to understand the effect of shipment number n on the total profit per unit time, we temporarily relaxesthe integer requirement on n, and taking the second partial derivative of JTP(n,T) with respect to n, it gets

@2JTPðn; TÞ@n2 ¼ @

2JTPiðn; TÞ@n2 ¼ �2ðAþ KÞ

n3T< 0; i ¼ 1;2;3::

Thus, for fixed T, JTP(n,T) is a concave function of n. Hence, for a given T, the search for the optimal number of shipments, n⁄,is reduced to find a local optimal solution.


Next, we want to determine the optimal replenishment cycle length, T(n), for any given n. We consider the following threecases:

4.1. Case 1. T 6M � N

For a given n, we want to find a value of T which maximizes JTP1(n,T). Taking the first partial derivative of JTP1(n,T) withrespect to T and setting the result equals to zero, that is,

@JTP1ðn; TÞ@T

¼ Aþ K þ nF

nT2 � D2

hb1 þhb1

cD

xð1� cÞ2þ pIBe

" #�u ¼ 0: ð9Þ

where

u ¼ D1� c

hb2c� hb2cD

2xð1� cÞ þhvD


2� nhvD

2Pð1� cÞ

� �> 0:

Next, motivated by Eq. (9), we let

f1ðTÞ ¼Aþ K þ nF

nT2 � D2

hb1þ hb1cD

xð1� cÞ2þ pIBe

" #�u; T 2 ð0;M � N�: ð10Þ

Taking derivative of f1(T) with respect to T, we get df1ðTÞdT ¼ �

2ðAþKþnFÞnT3 < 0.

Thus f1(T) is strictly decreasing on (0,M � N]. Since limT!0þ f1ðTÞ ¼ 1 and

f1ðM � NÞ ¼ Aþ K þ nF

nðM � NÞ2� D

2hb1 þ

hb1cD

xð1� cÞ2þ pIBe

" #�u ¼ A� D1

nðM � NÞ2;

where

D1 ¼ nðM � NÞ2 D2

hb1þ hb1cD

xð1� cÞ2þ pIBe

" #þu

( )� K � nF:

Thus we have the following results:

Corollary 1. For a given n,

(a) if A 6D1, then the solution of T 2 (0,M � N] in Eq. (9) (say TðnÞ1 Þ not only exists but also is unique.(b) if A > D1, then the solution of T 2 (0,M � N] in Eq. (9) does not exists.

Proof. See the Appendix A. h

Corollary 2. For a given n,

(a) if A 6 D1, then the total profit per unit time JTP1(n,T) has the global maximum value at the point T ¼ TðnÞ1 , whereTðnÞ1 2 ð0; M � N� and satisfies Eq. (9).

(b) if A > D1, then the total profit per unit time JTP1(n,T) has a maximum value at the upper boundary point T = M � N.

Proof. See the Appendix B. h

Remark. For given n, when the retailer’s optimal replenishment cycle length T ¼ T ðnÞ1 , from Eq. (9), it can be found that thevalue of TðnÞ1 is independent of Mbut dependent on N because D = D(N) = D0ehN. Thus, the corresponding optimal order quan-tity Q⁄ is also independent of M but dependent on N.

4.2. Case 2. M � N 6 T 6M

Similarly, for a given n, the necessary condition for the total profit per unit time JTP2(n,T) in Eq. (7) to be maximum is@JTP2ðn;TÞ

@T ¼ 0, which gives

Aþ K þ nF

nT2 þ pðIBp � IBeÞDðM � NÞ2

2T2 � D2

hb1þ hb1cD

xð1� cÞ2þ pIBp

" #�u ¼ 0; ð11Þ


where u is defined as above.Let


nT2 þ pðIBp � IBeÞDðM � NÞ2

2T2 � D2

hb1þ hb1cD

xð1� cÞ2þ pIBp

" #�u; T 2 ½M � N;M�: ð12Þ

Taking derivative of f2(T) with respect to T, we get

df2ðTÞdT

¼ �2ðAþ K þ nFÞnT3 � pðIBp � IBeÞDðM � NÞ2

T3 :

For notational simplicity, let S1 � 2(A + K + nF) + np(IBp � IBe)D(M � N)2. We now consider the following two situations: (1)S1 > 0 and (2) S1 6 0.

4.2.1. Case 2.1. S1 > 0In this situation, df2ðTÞ

dT ¼ �S1

nT3 < 0, for all T 2 (M � N,M). Hence, f2(T) is a strictly decreasing function on T 2 [M � N,M].Furthermore,

f2ðM � NÞ ¼ Aþ K þ nF

nðM � NÞ2� D

2hb1þ hb1cD

xð1� cÞ2þ pIBe

" #�u ¼ A� D1

nðM � NÞ2

and

f2ðMÞ ¼Aþ K þ nF

nM2 þ pðIBp � IBeÞDðM � NÞ2

2M2 � D2

hb1þ hb1

cD

xð1� cÞ2þ pIBp

" #�u ¼ A� D2

nM2 ;

where D1 is defined as above, and

D2 ¼ nM2 D2

hb1 þhb1

cD

xð1� cÞ2þ pIBp

" #þu

( )� npðIBp � IBeÞDðM � NÞ2

2� K � nF:

Because D2 P D1, we have the following results:

Corollary 3. For a given n, when S1 > 0 and

(a) if D1 6 A 6 D2, then the solution of T 2 [M � N,M] in Eq. (11) (say T ðnÞ2 Þ not only exists but also is unique.(b) if A < D1 or A > D2, then the solution of T 2 [M � N,M] in Eq. (11) does not exist.

Proof. The proof is similar to that of Corollary 1, we omit it here. h

Corollary 4. For a given n, when S1 > 0 and

(a) if D1 6 A 6D2, then the total profit per unit time JTP2(n,T) has the global maximum value at the point T ¼ T ðnÞ2 , whereTðnÞ2 2 ½M � N;M� and satisfies Eq. (11).

(b) if A < D1, then the total profit per unit time JTP2(n,T) has a maximum value at the lower boundary point T = M � N.(c) if A > D2, then the total profit per unit time JTP2(n,T) has a maximum value at the upper boundary point T = M.

Proof. The proof is similar to that of Corollary 2, we omit it here. h

4.2.2. Case 2.2. S1 6 0In this situation, the left hand side of Eq. (11) is negative; therefore, we can obtain the following result.

Corollary 5. For a given n, if S1 6 0, then the total profit per unit time JTP2(n,T) has a maximum value at the lower boundary pointT = M � N.

Proof. If S1 6 0, then

@JTP2ðn; TÞ@T

¼ S1

2nT2 �D2

hb1 þhb1

cD

xð1� cÞ2þ pIBp

" #�u < 0;


for all T 2 (M � N,M). Thus, JTP2(n,T) is a strictly decreasing function of T in [M � N,M]. Therefore, the total profit per unittime JTP2(n,T) has a maximum value at the lower boundary point T = M � N. h

4.3. Case 3. T P M

Likewise, for a given n, the necessary condition for the total profit per unit time JTP3(n,T) in Eq. (8) to be maximum is

@JTP3ðn; TÞ@T

¼ Aþ K þ nF

nT2 þ D

2T2 vIBpM2 þ pIBpNðN � 2MÞ � pIBeðM � NÞ2h i

� D2

hb1 þhb1

cD

xð1� cÞ2þ vIBp

" #�u ¼ 0; ð13Þ

where u is defined as above.Let


nT2 þ D

2T2 vIBpM2 þ pIBpNðN � 2MÞ � pIBeðM � NÞ2h i

� D2

hb1þ hb1

cD

xð1� cÞ2þ vIBp

" #�u; ð14Þ

Taking derivative of f3(T) with respect to T, we get

df3ðTÞdT

¼ �2ðAþ K þ nFÞnT3 � D

T3 ½vIBpM2 þ pIBpNðN � 2MÞ � pIBeðM � NÞ2�:

For convenience, let S2 � 2(A + K + nF) + nD[vIBpM2 + pIBpN(N � 2M) � pIBe(M � N)2].By using similar arguments as that in Case 2, we can obtain the following results.

Corollary 6. For a given n, when

(i) S2 > 0 and
(a) if A P D2, then the total profit per unit time JTP3(n,T) has the global maximum value at the point T ¼ T ðnÞ3 , where
T ðnÞ3 2 ½M;1Þ and satisfies Eq. (13).(b) if A < D2, then the total profit per unit time JTP3(n,T) has a maximum value at the lower boundary point T = M.

(ii) S2 6 0, then the total profit per unit time JTP3 (n,T) has a maximum value at the lower boundary point T = M.

Since

S2 � S1 ¼ nD½vIBpM2 þ pIBpNðN � 2MÞ � pIBeðM � NÞ2 � pðIBp � IBeÞðM � NÞ2�

¼ nDfvIBpM2 þ pIBp½NðN � 2MÞ � ðM � NÞ2�g ¼ nDIBpðv � pÞM2 < 0;

that is, S2 < S1, we make a summary of results of Corollaries 2,4,5 and 6, and obtain the following theorems:

Theorem 1. For a given n, when 0 < S2 < S1, we have

(a) if A 6D1, and� �
(i) JTP1 n; TðnÞ1 P JTP3ðn;MÞ, then T ðnÞ ¼ TðnÞ1 .
(ii) JTP1 n; TðnÞ1

� �< JTP3ðn;MÞ, then T(n) = M.

(b) if D1 < A < D2, then TðnÞ ¼ T ðnÞ2 .(c) if A P D2, and � �

(i) JTP1ðn;M � NÞP JTP3 n; TðnÞ3 , then T(n) = M � N.

(ii) JTP1ðn;M � NÞ < JTP3 n; TðnÞ3 ��

, then TðnÞ ¼ TðnÞ3 .

Proof. It immediately follows from Corollaries 2,4 and Corollaries 6(i),

JTP1ðn;M � NÞ ¼ JTP2ðn;M � NÞ and JTP2ðn;MÞ ¼ JTP3ðn;MÞ: �

Theorem 2. For a given n, when S2 6 0 < S1, we have

(a) if A 6D1, and� �
(i) JTP1 n; TðnÞ1 P JTP3ðn;MÞ, then T ðnÞ ¼ TðnÞ1 .
(ii) JTP1 n; TðnÞ1


(b) if D1 < A < D2, then TðnÞ ¼ T ðnÞ2 .(c) if A P D2, and


(i) JTP1(n,M � N) P JTP2(n,M) = JTP3(n,M), then T(n) = M � N.(ii) JTP1(n,M � N) < JTP2(n,M) = JTP3(n,M), then T(n) = M.

Proof. It immediately follows from Corollaries 2,4, and Corollaries 6(ii),


Theorem 3. or a given n, when S2 < S1 6 0, we have

(a) if A 6 D1, and� �
(i) JTP1 n; T ðnÞ1 P JTP3ðn;MÞ, then T ðnÞ ¼ TðnÞ1 .
(ii) JTP1 n; T ðnÞ1


(b) if A > D1, and
(i) JTP1(n, M � N) = JTP2(n, M � N) P JTP3(n, M), then T(n) = M � N.(ii) JTP1(n, M � N) = JTP2(n, M � N) < JTP3(n, M), then T(n) = M.
Proof. Corollaries It immediately follows from 2,5 and 6(ii),


Now, we can establish the following algorithm to obtain the optimal solution (n⁄,T⁄).

Algorithm

Step 1. Set n = 1.Step 2. Compute

S1 ¼ 2ðAþ K þ nFÞ þ npðIBp � IBeÞDðM � NÞ2;

S2 ¼ 2ðAþ K þ nFÞ þ nD½vIBpM2 þ pIBpNðN � 2MÞ � pIBeðM � NÞ2�;

D1 ¼ nðM � NÞ2 D2

hb1 þhb1

cD

xð1� cÞ2þ pIBe

" #þu

( )� K � nF;

D2 ¼ nM2 D2

hb1 þhb1

cD

xð1� cÞ2þ pIBp

" #þu

( )� npðIBp � IBeÞDðM � NÞ2

2� K � nF:

Step 3.(i) If 0 < S2 < S1, then go to Step 4.(ii) If S2 6 0 < S1, then go to Step 5.(iii) If S2 < S1 6 0, then go to Step 6.

Step 4. � �
(i) When A 6D1, we find the value TðnÞ1 from the Eq. (9), and then use Eqs. (6) and (8) to determine JTP1 n; T ðnÞ1
and JTP3(n,M), respectively. Next, let JTPðn; TðnÞÞ � max JTP1 n; TðnÞ1

� �; JTP3ðn;MÞ

n o, and go to Step 7.

(ii) When D1 < A < D2, we find the value T ðnÞ2 from the Eq. (11), and then use Eq. (7) to determine JTP2 n; TðnÞ2

� �.

Next, let JTPðn; T ðnÞÞ � JTP2 n; TðnÞ2

� �, and go to Step 7.

(iii)When A P D2, we find the value T ðnÞ3 from the Eq. (13), and then use Eqs. (6) and (8) to determine

JTP1(n,M � N) and JTP3 n; TðnÞ3

� �, respectively. Next, let JTPðn; T ðnÞÞ �max JTP1ðn;M � NÞ; JTP3 n; TðnÞ3

� �n o, and go

to Step 7.
Step 5. � �
(i) When A 6D1, we find the value TðnÞ1 from the Eq. (9), and then use Eqs. (6) and (8) to determine JTP1 n; T ðnÞ1

and JTP3(n,M), respectively. Next, let JTPðn; TðnÞÞ � max JTP1 n; TðnÞ1



(ii) When D1 < A < D2, we find the value T ðnÞ2 from the Eq. (11), and then use Eq. (7) to determine JTP2 n; TðnÞ2

� �.

Next, let JTPðn; T ðnÞÞ � JTP2 n; TðnÞ2

� �, and go to Step 7.

(iii) When A P D2, we use Eqs. (6)–(8) to determine JTP1(n,M � N), JTP2(n,M) andJTP3(n, M), respectively. Then let JTP(n,T(n)) �max{JTP1(n,M � N), JTP2(n,M) = JTP3(n,M)}, and go to Step 7.


Step 6.(i) When A 6D1, we find the value TðnÞ1 from the Eq. (9), and then use Eqs. (6) and (8) to determine JTP1 n; T ðnÞ1

� �and JTP3(n,M), respectively. Next, let JTPðn; TðnÞÞ �max JTP1 n; TðnÞ1



(ii) When A > D1, we use Eqs. (6)–(8) to determine JTP1(n,M � N), JTP2(n,M � N) and JTP3(n,M), respectively. Next,let JTP(n,T(n)) �max{JTP1(n,M � N) = JTP2(n,M � N), JTP3(n,M)}, and go to Step 7.

Step 7. Set n = n + 1, and repeat Steps 2–6 to get JTP(n,T(n)).Step 8. If JTP(n,T(n)) P JTP(n � 1,T(n�1)), then go to Step 7, otherwise setting JTP(n⁄,T⁄) = JTP(n � 1,T(n�1)), then (n⁄,T⁄)is the optimal solution.

Once the optimal solution (n⁄,T⁄) is obtained, the optimal order quantity for the retailer Q⁄ = n⁄DT⁄ follows.

5. Numerical examples

Example 1. In order to illustrate the solution procedure of the model developed in this paper, we applied the followingnumerical example, which is devised partially with the data from Su et al. [39]: c = $11/unit, v = $20/unit, p = $25/unit,hv = $0.2/unit/year, hb1

¼ $0:2=unit=year, M = 30/365 year, N = 10/365 year and h 2 {0,1,1.5,2}. Besides, given the followingdata: P = 15,000 units/year, D0 = 3000 units/year, K = $400/setup, A = $100/order, hb2

¼ $0:1=unit=year, IVp = 0.05, IBe = 0.025,IBp = 0.035, F = $25/shipment, w = $5/unit, s = $0.5/unit, x = 175,200 units/year and c = 0.01.

Applying the Algorithm procedure, we obtain the optimal results for various values of h as shown in Table 1. Meanwhile,for comparing the difference in performances for the retailer with/without offering trade credit to the customers, we also listthe optimal solution of no trade credit in the first row of Table 1 (i.e., N = 0).

From Table 1, we observe that when both supplier and retailer offer trade credit, if h = 0, that is, when the customer is notsensitive to the credit period N, then the retailer, the supplier, and the entire supply chain have the lowest profit, sometimeseven lower than that gained when the retailer has not offered the customer trade credit. In this situation, because the de-mand rate is not sensitive to the trade credit strategy, and the retailer has to bear the opportunity cost for items sold withoutbeing paid for, thus the retailer has a lower profit. Hence, when the customer is not sensitive to the length of trade credit (i.e.,when the demand rate does not increase with the length of the trade credit), it is better for the retailer not to offer tradecredit to the customers.

In addition, we can observe that when the constant factor h increases, the market demand rate, the optimal order quan-tity, and all of the retailer, the supplier, the entire supply chain total profit are increasing, while the optimal replenishmentcycle length T⁄ decreases. This means the retailer will shorter the replenishment cycle to increase the number of orders andto take advantage of the trade credit more frequently. Therefore, when the customer is sensitive to the credit period, theretailer, the supplier, and the entire supply chain can profit from the trade credit strategy.

Example 2. In this example, to understand the effect of various values of rate of defective item, c, on the entire supply chain.For different c 2 {0.01,0.03,0.05,0.08,0.1}, using the same parameter values as in Example 1, and setting the value h = 1 (i.e.,D = 3083 units/year), we apply the Algorithm to obtain the optimal solutions, the results are shown in Table 2.

From Table 2, we observe that, with a fixed market demand, when the rate of defective items increases, the profits of theretailer, the supplier, and the entire supply chain decrease, which means the profit and the rate of defective items go in oppo-site directions. In this situation, the profit of supplier will fall greater than that of the retailer. Therefore, the supplier has toreduce the rate of defective items so as to increase his/her profits. In addition, when the rate of defective items increases, theoptimal order quantity, Q⁄, and the optimal replenishment cycle length, T⁄, decrease. In other words, due to the increasedrate of defective items, the retailer will order a smaller quantity and shorten the replenishment cycle length.

Example 3. In this example, we try to study the influence of the various supply chain parameters of M and N on the optimalsolutions. We consider different M and N, using the same parameter values as in Example 1, and setting the value h = 1 and

Table 1The summary of optimal solutions for Example 1.

M(year)

N(year)

h D S1 S2 D1 D2 n⁄ T⁄ (year) Q⁄ (units/order)

Profit ($/year)

Retailer Supplier Channel

30/365 0 1.0 3000 1550.66 1515.19 �415.03 2491.58 10 T2 = 0.1418 4255.60 13206.84 26002.52 39209.36

30/365 10/365 0 3000 1574.77 1535.75 �552.14 3014.84 11 T2 = 0.1322 4363.04 13148.72 25999.07 39147.791.0 3083 1575.45 1535.36 �549.25 3103.11 11 T2 = 0.1306 4432.48 13520.52 26731.59 40252.111.5 3125 1575.80 1535.15 �547.78 3147.87 11 T2 = 0.1299 4467.72 13710.31 27105.51 40815.822.0 3168 1576.16 1534.95 �546.30 3193.03 11 T2 = 0.1291 4503.31 13902.74 27484.63 41387.37


c S1 S2 D1 D2 n⁄ T⁄ (year) Q⁄ (units/order) Profit ($/year)


0.01 1575.45 1535.36 �549.25 3103.11 11 T2 = 0.1306 4432.48 13520.52 26731.59 40252.110.03 1575.45 1535.36 �547.71 3145.11 11 T2 = 0.1299 4408.13 13487.28 26405.59 39892.870.05 1575.45 1535.36 �546.13 3188.40 11 T2 = 0.1292 4383.44 13452.63 26065.98 39518.610.08 1575.45 1535.36 �543.67 3255.87 11 T2 = 0.1281 4345.78 13397.84 25529.07 38926.910.10 1575.45 1535.36 �541.96 3302.63 11 T2 = 0.1273 4320.24 13359.28 25151.40 38510.68


c = 0.01. We apply the Algorithm to obtain the optimal solutions, and the results are shown in Table 3. Simultaneously, whenthe retailer does not offer trade credit strategy to the customers (i.e., N = 0), the optimal solutions are also shown in Table 3.

From Table 3, we observe that when the retailer does not offer trade credit to the customers (i.e., N = 0, and henceD = 3000), the supplier offers a longer length of credit period M to retailer, then the retailer’s profit will increase, but the prof-its of supplier and the profits of entire supply chain decrease. This means when the supplier offers longer credit period to theretailer, the opportunity cost for the supplier will increase because of the delayed payment. On the other hand, due to thelonger credit period, the retailer earns more interest. The results illustrate that when the retailer does not offer trade credit tothe customers, while the supplier offers longer the credit period to the retailer, the supplier and the entire supply chain profitdrops. Furthermore, when N(>0) is fixed, and when the value of M increases, the retailer, the supplier, and the entire supplychain will have the same results in profits too.

In addition, when M is fixed, and when the value of N increases, the retailer, the supplier, and the entire supply chain haveincreased profits, and the optimal replenishment cycle length T⁄ decreases. This means that the longer length of credit periodoffered by the retailer to the customers may better motivate the market demand. Hence, the retailer, the supplier, and theentire supply chain profit enjoy an increase in profit.

Example 4. When the supplier and the retailer do not cooperate with each other, both them will determine their own policyindependently. First, the supplier makes his/her own individual optimal decision. In response, the retailer formulates his/herown optimal policy. To explore the advantage of coordination among the integrated model, we use the same data in Example1, except for setting h = 1.5. The optimal solutions for the supplier and the retailer can be obtained, and the results are shownin Table 4.

The computation result presents that when the retailer makes his/her decision independently, the optimal order quantityQ⁄ = 4,878.12 units and the optimal replenishment cycle length T⁄ = 0.1561 years, and the retailer’s maximum annual totalprofit is TBP(T⁄) = $13,711.68. Using the retailer optimal decision, we computed results that the optimal number of ship-ments from the supplier to the retailer per production run n⁄ = 10, the supplier’s optimal production quantity per productionrun is n�q ¼ Q�

1�c ¼ 4;927:39 units, and the supplier’s annual total profit is TVP(n⁄) = $27,093.59. Therefore, when the supplier

and the retailer do not cooperate with each other, the channel annual total profit is TBP(T⁄) + TVP(n⁄) = $13,711.68 +$27,093.59 = $40,805.27. To explore the advantage of coordination throughout the supply chain, using the same data as


M (year) N (year) D S1 S2 D1 D2 n⁄ T⁄ (year) Q⁄ (units/order) Profit ($/year)


30/365 0 3000 1550.66 1515.19 �415.03 2491.58 10 T2 = 0.1418 4255.60 13206.84 26002.52 39209.3610/365 3083 1575.45 1535.36 �549.25 3103.11 11 T2 = 0.1306 4432.48 13520.52 26731.59 40252.1115/365 3126 1564.51 1523.86 �603.44 3153.51 11 T2 = 0.1294 4451.68 13676.61 27105.70 40782.3130/365 3257 1550.00 1507.64 �675.00 3298.05 11 T2 = 0.1266 4536.40 14148.69 28259.21 42407.90

60/365 0 3000 1632.39 1504.71 162.05 4626.50 9 T1 = 0.1577 4259.79 13368.46 25758.76 39127.2210/365 3083 1644.64 1498.84 18.17 5764.45 10 T2 = 0.1443 4450.98 13696.73 26480.33 40177.0615/365 3126 1618.78 1470.96 �102.41 5854.97 10 T2 = 0.1423 4450.21 13860.56 26851.16 40711.7230/365 3257 1610.50 1441.08 �378.70 7240.86 11 T2 = 0.1290 4624.10 14362.43 27990.49 42352.9245/365 3394 1565.76 1389.23 �598.34 7545.26 11 T2 = 0.1250 4668.56 14809.35 29182.11 44051.4660/365 3536 1550.00 1366.06 �675.00 7842.60 11 T2 = 0.1222 4756.92 15382.50 30423.60 45806.10

90/365 0 3000 1860.39 1573.11 1145.87 7183.85 9 T1 = 0.1577 4259.79 13522.57 25512.19 39034.7610/365 3083 1783.27 1488.01 807.78 7417.87 9 T1 = 0.1559 4326.51 13851.37 26231.00 40082.3715/365 3126 1746.95 1447.62 649.24 7535.21 9 T1 = 0.1549 4360.36 14018.67 26597.92 40616.5930/365 3257 1720.02 1373.48 358.04 9442.31 10 T1 = 0.1417 4618.11 14536.82 27725.25 42262.0745/365 3394 1691.85 1294.66 14.89 11596.28 11 T2 = 0.1300 4852.90 15070.87 28900.78 43971.6560/365 3530 1615.69 1201.84 �357.80 12068.56 11 T2 = 0.1248 4856.67 15613.73 30131.88 45745.6190/365 3839 1550.00 1100.69 �675.00 13002.81 11 T2 = 0.1181 4991.34 16722.92 32750.34 49473.26

Table 4Comparison between the independent and integrated polices.

Independent Integrated Allocated

Optimal order quantity (Q⁄) 4,878.12 4,467.72Optimal length of replenishment cycle (T⁄) 0.1561 0.1299Optimal number of shipments from vendor to buyer in one production run (n⁄) 10 11Supplier production quantity per production run (Q⁄/(1 � c)) 4,927.39 4,512.84Supplier’s total profit $27,093.59 $27,105.51 $27,100.61a

Retailer’s total profit $13,711.68 $13,710.31 $13,715.21b

Sum of total profit $40,805.27 $40,815.82 $40,815.82

a 40;815:82 � 27;093:5940;805:27 ¼ 27;100:61.

b 40;815:82 � 13;711:6840;805:27 ¼ 13;715:21.


above and applying similar procedures, we can obtain the optimal solutions for the integrated inventory model, the results ofwhich are also presented in Table 4. Table 4 reveal the maximum joint annual total for the supply chain profitJTP(n⁄,T⁄) = $40,815.82 is larger than the channel annual total profit TBP(T⁄) + TVP(n⁄) = $40,805.27. As result, it shows thatadopting an integrated policy can improve the total profit of the supply chain performance. To encourage the retailer tocooperate with the supplier, we applied the compensation method suggested by Goyal [1] so that the long term partnershipbetween the supplier and the retailer can remain intact. We evaluate the allocation of JTP(n⁄,T⁄) as the following equations:

the retailer : JTPðn�; T�Þ � TBPðT�ÞTBPðT�Þ þ TVPðn�Þ and

the supplier : JTPðn�; T�Þ � TVPðn�ÞTBPðT�Þ þ TVPðn�Þ :

The results of allocated annual total profit are listed in the last column of Table 4.

6. Concluding remarks

In this paper, based on our analysis, it is found that when both supplier and retailer adopt a trade credit strategy, if thecustomers are sensitive to the length of credit period offered by the retailer, then the trade credit strategy is effective inincreasing the profits of the entire supply chain. However, when the customers are not sensitive to the length of credit periodoffered by the retailer, the retailer should not adopt the trade credit strategy. Therefore, before adopting a trade credit strat-egy, the retailer should have a complete understanding of the preference of customers to trade credit in order to avoid a stra-tegic mismatch and unnecessary losses.

In addition, when the rate of defective items increases, the profit of the entire supply chain will decrease. Therefore, thesupplier should endeavor to enhance the production quality to reduce defective rate.

Furthermore, if only the supplier offers trade credit to the retailer, the retailer will gain in profit with or without offeringtrade credit to the customers, but the supplier’s profit falls. If both supplier and retailer offer trade credit, then both they andthe entire supply chain have better benefit than when the retailer does not offer trade credit. Therefore, for a win–win resultfor the whole supply chain, it is suggested that the retailer offer trade credit to the customer.

In practical situation, a business’s credit period is dictated by an industry standard or by its competition. However, a busi-ness may find it is necessary to adjust its trade credit policy with the prevailing circumstances. Therefore, in future researchon this problem, it would be interesting to allow that the credit period treated as one of the decision.

Acknowledgements

The authors thank Editor Melvin Scott and the anonymous referees for several valuable and helpful suggestions on anearlier version of the paper. This research was supported by National Science Council of the Republic of China under GrantNSC- 97-2410-H-129-002.

Appendix A. Proof of Corollary 1

Proof of part (a). If A 6 D1, then f1(M � N) 6 0. Since f1(T) is strictly decreasing on (0,M � N], and limT!0þ f1ðTÞ ¼ 1, by the

intermediate value Theorem, there exists a unique TðnÞ1 2 ð0;M � N� such that f1 T ðnÞ1

� �¼ 0.

Proof of part (b). If A > D1, then f1(M � N) > 0, which implies f1(T) > 0, for all T 2 (0,M � N]. Thus there is not exists a valueT 2 (0,M � N] such that f1(T) = 0.


Appendix B. Proof of Corollary 2

Proof of part (a). For a given n, if A 6 D1, from Corollary 1(a), it can be seen that TðnÞ1 2 ð0; M � N� is the unique solution ofEq. (9). Now, taking the second partial derivative of JTP1(n,T) with respect to T, and then finding the value of the function at

point T ðnÞ1 , we obtain @2 JTP1ðn;TÞ@T2

��T¼TðnÞ1

¼ � 2ðAþKþnFÞnðTðnÞ1 Þ

3 < 0. Thus TðnÞ1 is the global maximum point of JTP1(n, T).

Proof of part (b). For a given n, if A > D1, by the proof process in Corollary 1(b), we have f1(T) > 0 for all T 2 (0,M � N].Thus @JTP1ðn;TÞ

@T ¼ f1ðTÞ > 0; T 2 ð0;M � NÞ, which implies that JTP1(n,T) is a strictly increasing function of T in (0,M � N].Hence, JTP1(n,T) has a maximum value at the upper boundary point T = M � N.

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a joint optimal ordering and delivery policy for an integrated supplier–retailer inventory model...

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