Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers

Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Contents lists available at ScienceDirect

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

Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers Jiang Wu a, Ya-Lan Chan b,n a b

School of Statistics, Southwestern University of Finance & Economics, Chengdu 611130, China Department of International Business, Asia University, Taichung, Taiwan 41354, ROC

art ic l e i nf o

a b s t r a c t

Article history: Received 29 December 2013 Accepted 31 March 2014

In practice, a credit-worthy retailer frequently receives a permissible delay on the entire purchase amount without collateral deposits from his/her supplier (i.e., an up-stream full trade credit). By contrast, a retailer usually requests his/her credit-risk customers to pay a fraction of the purchase amount at the time of placing an order, and then grants a permissible delay on the remaining balance (i.e., a down-stream partial trade credit). In addition, many products such as blood banks, pharmaceuticals, fruits, vegetables, volatile liquids, and others deteriorate constantly and have their expiration dates. However, not many researchers have taken the expiration date of a deteriorating item into consideration. The purpose of this paper is to establish optimal lot-sizing policies for a retailer who sells a deteriorating item to credit-risk customers by offering partial trade credit to reduce his/her risk. The proposed model is a generalized case of many previous models. By applying theorems in pseudo-convex fractional functions, we can easily prove that the optimal solution not only exists but is also unique. Moreover, we propose three discrimination terms, which can easily identify the optimal solution among all possible alternatives. Finally, some numerical examples are presented to highlight the theoretical results and managerial insights. & 2014 Elsevier B.V. All rights reserved.

Keywords: EOQ Trade credit Deteriorating item Expiration date Credit-risk customer

1. Introduction Harris (1913) established the classical economic order quantity (EOQ) model based on the assumptions that the purchase items are non-perishable and can be sold indefinitely, and the retailer must pay for the entire purchase cost as soon as the purchase items are received. In reality, many products (e.g., fruits, vegetables, medicines, volatile liquids, blood banks and others) not only deteriorate continuously (e. g., evaporation, obsolescence, and spoilage) but also have their expiration dates. Ghare and Schrader (1963) derived a revised form of the economic order quantity (hereafter EOQ) model by assuming exponential decay. Then Covert and Philip (1973) extended Ghare and Schrader's constant deterioration rate to a two-parameter Weibull distribution. Dave and Patel (1981) considered an EOQ model for deteriorating items with time-proportional demand when shortages were prohibited. Sachan (1984) further extended the model to allow for shortages. Goswami and Chaudhuri (1991) generalized an EOQ model for deteriorating items from a constant demand pattern to a linear trend in demand. Hariga (1996) established optimal EOQ models for deteriorating items with time-varying demand. Goyal and Giri (2001) provided a survey on the recent trends in modeling of deteriorating

n

Corresponding author. Tel.: þ 886 4 2332 3456x48038. E-mail address: [email protected] (Y.-L. Chan).

inventory. Skouri et al. (2009) considered inventory models with ramp-type demand rate and Weibull deterioration rate. Skouri et al. (2011) further generalized the model to add a permissible delay in payments under consideration. Dye (2013) provided some results on finding the optimal replenishment and preservation technology strategies for a non-instantaneous deteriorating inventory model. Recently, Chen et al. (2013b) proposed economic production quantity (EPQ) models for deteriorating items. All the above mentioned papers did not consider the fact that deteriorating items have their expiration dates. In fact, the study of deteriorating items with expiration dates has received a relatively little attention in the literature. Currently, Bakker et al. (2012) provided an excellent review of inventory systems with deterioration since 2001. In practice, a seller frequently offers his/her buyers a permissible delay in payment (i.e., trade credit) for settling the purchase amount. Usually, there is no interest charge if the outstanding amount is paid within the permissible delay period. However, if the payment is not paid in full by the end of the permissible delay period, then interest is charged on the outstanding amount. Goyal (1985) proposed an EOQ model under conditions of permissible delay in payments. Aggarwal and Jaggi (1995) extended Goyal's model for deteriorating items. Jamal et al. (1997) further generalized Aggarwal and Jaggi's model to allow for shortages. Chang et al. (2003) developed an EOQ model for deteriorating items under supplier credits linked to ordering quantity. Huang (2003) proposed an EOQ model in which the supplier offers

http://dx.doi.org/10.1016/j.ijpe.2014.03.023 0925-5273/& 2014 Elsevier B.V. All rights reserved.

Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i

J. Wu, Y.-L. Chan / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

2

the retailer a permissible delay and the retailer in turn provides his/ her customers another permissible delay to stimulate demand. Ouyang et al. (2006) established an EOQ model for deteriorating items to allow for partial backlogging under trade credit financing. Liao (2007) presented an EPQ model for deteriorating items under permissible delay in payments. Teng (2009) developed ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers. Hu and Liu (2010) presented an EPQ model with permissible delay in payments and allowable shortages. Teng et al. (2011) extended an EOQ model for stock-dependent demand to supplier's trade credit with a progressive payment scheme. Teng et al. (2012) generalized traditional constant demand to non-decreasing demand. Lou and Wang (2013) proposed an integrated inventory model with trade credit financing in which the vendor decides his/her production lot size while the buyer determines his/her expenditure. Lately, Chen et al. (2013a) built up an EOQ model when conditionally permissible delay links to order quantity. Most of the above mentioned articles assumed that buyers are good-credit customers and receive full trade credit. Hence, the use of other trade credit strategies to reduce default risks with credit-risk customers has received a relatively little attention in the literature. Recently, Seifert et al. (2013) presented an excellent review of trade credit financing. Some relevantly recent articles in trade credit financing were developed by Chern et al. (2013), Taleizadeh (2014), Wu et al. (2014), and Yang et al. (2013). In this paper, an EOQ model for deteriorating items with expiration dates was developed in a supply chain with up-stream full trade credit and down-stream partial trade credit financing. The proposed model is a generalized case of Goyal (1985), Teng (2002), Huang (2003), Teng and Goyal (2007), Teng (2009), and Chen and Teng (2014). Further, by applying the existing theoretical results in pseudoconvex fractional functions, we can easily prove that the optimal solution not only exists but also is unique. Moreover, we propose three discrimination terms to identify the optimal solution among possible alternatives. Finally, some numerical examples are presented to highlight the theoretical results and managerial insights.

T Q TRC (T) T* TRC*

2.2. Assumptions Next, the following assumptions are made to establish the mathematical inventory model. 1. All deteriorating items have their expiration rates. The physical significance of the deterioration rate is the rate to be closed to 1 when time is approaching to the maximum lifetime m. To make the problem tractable, one can assume the same as in Wang et al. (2014) that the deterioration rate is θðtÞ ¼

2. 3.

2. Notation and assumptions 4. For simplicity, the notation and the assumptions used through the paper are presented below. 2.1. Notation o c p h Ic Ie t IðtÞ θðtÞ m S R α D

the retailer's ordering cost per order in dollars the retailer's purchasing cost per unit in dollars the market price per unit in dollars (with p 4c) the retailer's average stock holding cost per unit per year in dollars the interest charged per dollar per year in stocks by the supplier the interest earned or return on investment per dollar per year the time in years the retailer's inventory level in units at time t the time-varying deterioration rate at time t, where 0 r θðtÞ r 1 the maximum lifetime of the deteriorating item in years the supplier's permissible delay period to the retailer in years the retailer's permissible delay period to his/her customers in years the fraction of the total purchase cost that a credit-risk customer must pay at the time of placing an order the retailer's annual constant demand rate in units

the retailer's replenishment cycle time in years (decision variable) the retailer's economic order quantity in units the retailer's total relevant cost per year in dollars the retailer's optimal replenishment cycle time in years the retailer's optimal total relevant cost per year in dollars

5.

1 ; 1þmt

0 r t rT r m:

ð1Þ

Note that it is obvious that the replenishment cycle time T is less than or equal to m, and Eq. (1) is a general case for nondeteriorating items, in which m-1 and θðtÞ-0. Replenishment rate is instantaneous and lead time is zero. The retailer receives a full trade credit period of S years from his/ her supplier (i.e., the retailer orders items at time 0, and must pay the supplier at time S without interest charges), and in turn provides a partial trade credit to his/her credit-risk customers who must pay α portion of the total purchasing cost at the time of placing an order as a collateral deposit, and then receive a permissible delay of R years on the outstanding amount (i.e., the customer orders items at time t, and must pay the delay payment at time tþR). Notice that to good-credit customers, the retailer may provides a full trade credit in which we simply set α¼0. Hence, the proposed model includes the special case in which the retailer offers a down-stream full trade credit to his/her customers. If S Z R, then the retailer deposits the sales revenue into an interest bearing account. If S Z T þ R (i.e., the permissible delay period is longer than the time at which the retailer receives the last payment from his/her customers), then the retailer receives all revenue and pays off the entire purchase cost at the end of the permissible delay S. Otherwise (if S o T þ R), the retailer pays the supplier the sum of money from all units sold by S  R and the collateral deposit received from t¼0 to S, keeps the profit for the use of the other activities, and starts paying for the interest charges on the items sold after S  R. If S rR, the retailer receives immediate payments from customers, and deposits the revenue into an interest bearing account until the end of the permissible delay S. As to delayed payment, the retailer must finance ð1 αÞcDT for delayed payment at t ¼S, and pay off the loan at t¼T þR.

3. Mathematical formulation of the model The retailer receives Q units at t¼0. Hence, the inventory starts with Q units at t¼0, and then gradually depletes to zero at t¼T due to the combination effect of demand and deterioration. Hence, the inventory level is governed by the following differential equation:

dIðtÞ 1 ¼ D IðtÞ; dt 1þmt

0 r t r T;

ð2Þ

Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i

J. Wu, Y.-L. Chan / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

with the boundary condition I(T)¼0. Solving the differential Eq. (2), we have Z T IðtÞ ¼ e  δðtÞ eδðuÞ Ddu; 0 r t r T; ð3Þ t

where Z δðtÞ ¼

0

t

  t du 1þ m ¼  lnð1þ m  uÞ0 ¼ lnð1 þmÞ  lnð1 þ m  tÞ ¼ ln : 1þmu 1 þm t

ð4Þ Substituting (4) into (3), we get the inventory level at time t as 1 þm t IðtÞ ¼ D 1 þm

Z

T t

  1 þm 1 þm t du ¼ Dð1 þm  tÞln ; 1 þ m u 1 þm  T

0 r t r T:

Consequently, the retailer's order quantity is   1 þm Q ¼ Ið0Þ ¼ Dð1 þ mÞln : 1þmT

ð5Þ

ð6Þ

3

On the other hand, the retailer grants his/her customers a permissible delay of R years, and receives the money from his/ her buyers from t¼R through t¼ Tþ R. Thus, at t¼S the retailer receives acDSdollars from immediate payment and ð1  aÞcd ðS  RÞ dollars from delayed payment, and pays his/her supplier acDS þ ð1  aÞcdðS RÞ dollars. Consequently, after at t¼ S the retailer must finance all items sold after t¼S for the portion of immediate payment, and all items sold after t¼S  R for the portion of delayed payment at an interest charged I c per dollar per year. As a result, the interest charged per cycle is (c/p)I c times the total area of the triangle ABC and the triangle A0 B0 C0 as shown in Fig. 1. Therefore, the annual interest charged is given by cI c D ½αðT  SÞ2 þ ð1  αÞðT þ R  SÞ2 : 2T

As a result the retailer's annual total relevant cost by using (7)–(11) is:

The retailer's annual total relevant cost consists of the following: TRC 1 ðTÞ ¼

( "     1 1þm ð1 þ mÞ2 1þm ln cDð1 þ mÞln þ o þhD 2 T 1 þ m T 1 þ m T

(a) the purchasing cost including the cost of deteriorating items   cQ cDð1 þ mÞ 1 þm ¼ ¼ ln ; ð7Þ T T 1þmT

# T 2 ð1 þ mÞT cI c D ½αðT  SÞ2 þ  :þ 2 2 4

(b)

þð1 αÞðT þ R  SÞ2  

o the ordering cost ¼ ; T

(c) the holding cost excluding interest charges per cycle   Z Z h T hD T 1 þ m t dt IðtÞdt ¼ ð1 þm  tÞln ¼ T 0 T 0 1þmT " #   hD ð1 þ mÞ2 1þm T 2 ð1 þ mÞT ¼ þ  ln ; T 1 þm  T 2 2 4

ð8Þ

ð9Þ

 pI e D 2 ½αS þ ð1  αÞðS  RÞ2 : 2

ð13Þ

Likewise, the retailer receives all immediate payment by time T (r S) so that there is no interest charged for the portion of immediate payment. However, the retailer must finance all items sold in time interval ½S  R; T. Therefore, the annual interest charged is cI c D ð1  αÞðT þ R  SÞ2 : 2T

ð14Þ

Consequently, from (7)–(9), (13) and (14) we know that the retailer's annual total relevant cost is

3.1. Annual total relevant cost for the case of S 4R Based on values of S, T, and T þ R (i.e. the time at which the retailer receives the payment from the last customer), three sub-cases can occur: (i) S rT, (ii) T rS r T þR, and (iii) T þ Rr S. Notice that the case of S ¼ T is applicable for both Cases (i) S rT, and (ii) T r S r T þ R. Similarly, the case of S ¼ T þ R is applicable for both Cases (ii) T r S r T þR, and (iii) T þ Rr S. For these three cases, we derive the annual interest earned and the annual interest charged accordingly. Sub-case 1.1. S r T In this sub-case, the retailer accumulates revenue and earns interest: from the portion of immediate payment starting time 0 through S, and from the portion of delayed payment starting time R through S. Hence, the interest earned per cycle is I e times the total area of the triangle OSA and the triangle RSA0 as shown in Fig. 1. Therefore, the annual interest earned is given by pI e D ½αS2 þ ð1  αÞðS  RÞ2 : 2T

ð12Þ

Sub-case 1.2. T r S r T þ R Again, the retailer accumulates revenue and earns interest from two accounts: the portion of immediate payment is starting t¼0 through t¼S, and the portion of delayed payment is starting time t¼R through t¼S. Hence, the retailer's annual interest earned as shown in Fig. 2 is pI e D 2 ½αT þ 2αTðS  TÞ þ ð1  αÞðS  RÞ2 : 2T

(d) the interest earned during the credit period, and (e) the interest charged by the supplier. The problem here is for the retailer to determine his/her optimal replenishment cycle time T* such that his/her annual total relevant cost is minimized. Nevertheless, for the derivation of the retailer's annual total relevant cost, the up-stream and the down-stream trade credits should be taken into consideration. From the values of S and R, there are two possible cases: S4R, and S r R. Then, these two cases are examined separately below.

ð11Þ

ð10Þ

TRC 2 ðTÞ ¼

   1 1 þm cDð1 þ mÞln þo T 1þmT " #   ð1 þmÞ2 1þm T 2 ð1 þ mÞT þ  ln þ hD 1þmT 2 2 4 þ

cI c D pI e D 2 ð1  αÞðT þ R  SÞ2  ½αT þ2αTðS  TÞ 2 2 o

þ ð1  αÞðS  RÞ2 :

ð15Þ

Sub-case 1.3. T þ R r S In this sub-case, the retailer receives the total revenue before the supplier's trade credit period S, and hence there is no interest charged. From Fig. 3, we know that the annual interest earned is: pI e D ½2S T  2ð1  αÞR: 2

ð16Þ

Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i

J. Wu, Y.-L. Chan / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

4

Revenue

α pDT

Revenue

B

C

A

0

B’

(1-α)pDT

C’

A’

Time

S T Immediate Payment

0

R

S T T+R Delayed Payment

Time

Fig. 1. S 4R and Sr T.

Revenue

Revenue

C’

(1-α)pDT

αpDT

A’

0

Time

T S Immediate Payment

0 R

B’

S T+R Delayed Payment

Time

Fig. 2. S 4R and T rS r T þR.

Hence, the retailer's annual total relevant cost is    1 1þm cDð1 þ mÞln þo TRC 3 ðTÞ ¼ T 1þmT " #   ð1 þmÞ2 1þm T 2 ð1 þ mÞT þ hD þ  ln 1þmT 2 2 4 ) 2 T  pI e D½ST   ð1  αÞRT : 2

immediate payment is pI e D 2 ðαS Þ: 2T

ð17Þ

Combining (12), (15), and (17), the retailer's annual total relevant cost is given as 8 > < TRC 1 ðTÞ; if S rT TRCðTÞ ¼ TRC 2 ðTÞ; if S R r T rS ð18Þ > : TRC ðTÞ; if T r S  R 3 It is clear from (12), (15), and (17) that TRC 1 ðSÞ ¼ TRC 2 ðSÞ; and TRC 2 ðS RÞ ¼ TRC 3 ðS  RÞ:

ð19Þ

Hence, TRC(T) is continuous in T Z 0: Then we proceed with the case of R Z S.

3.2. Annual total relevant cost for the case of R Z S Now based on values of S and T, the following two sub-cases can occur: (i) S r T, and (ii) S Z T. Let us discuss them accordingly. Sub-case 2.1. S rT From Fig. 4 we know the annual interest earned from the

ð20Þ

In this sub-case, for immediate payment the retailer must finance αcDðT SÞ at t ¼S, and pay off the loan at t ¼T. As to delayed payment, the retailer must finance ð1  αÞcDT for delayed payment at t¼S, and pay off the loan at t¼T þR. Therefore, the annual interest charged is o cI c Dn αðT  SÞ2 þ ð1  αÞT½T þ 2ðR  SÞ : 2T

ð21Þ

Consequently, the annual total relevant cost is

TRC 4 ðTÞ ¼

   1 1þm cDð1 þ mÞln þo T 1þmT " #   2 ð1 þmÞ 1þm T 2 ð1 þ mÞT þ hD þ  ln 1þmT 2 2 4  o pI D cI c Dn e αS2 : þ αðT  SÞ2 þ ð1  αÞT½T þ2ðR  SÞ  2 2

ð22Þ

We then discuss the last sub-case in which R Z S Z T. Sub-case 2.2. S Z T From Fig. 5 we know the annual interest earned from the immediate payment is pI e D α½T þ 2ðS  TÞg: 2

ð23Þ

Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i

J. Wu, Y.-L. Chan / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

In this sub-case, there is no interest charged for immediate payment. However, the retailer must finance ð1  αÞcDT for delayed payment at t ¼S, and pay off the loan at t¼ Tþ R. Therefore, the annual interest charged is cI c D ð1 αÞ½T þ 2ðR SÞ: 2

þ

" #   ð1 þmÞ2 1þm T 2 ð1 þ mÞT þ  ln 1þmT 2 2 4

 cI c D pI e D ð1 αÞT½T þ 2ðR  SÞ  αT½T þ 2ðS TÞ : 2 2

ð24Þ

ð25Þ Combining (22) and (25), we know that the retailer's annual total relevant cost is ( TRC 4 ðTÞ; if S r T TRCðTÞ ¼ TRC 5 ðTÞ; if S Z T

Consequently, the retailer's annual total relevant cost is TRC 5 ðTÞ ¼

þ hD

5

   1 1þm cDð1 þ mÞln þo T 1þmT

Revenue

Revenue

(1-α)pDT

αpDT

0

T M Immediate Payment

Time

0

R

T T+R S Delayed Payment

Time

Fig. 3. S4R and T þ Rr S.

Revenue

Revenue

C

αpDT

A

0

(1-α)pDT

Loan Amount

B

S T Immediate Payment

Time

0

S

R T T+R Delayed Payment

Time

Fig. 4. Sr R and Sr T.

Revenue

Revenue

Loan Amount (1-α)pDT

αpDT

0

T S Immediate Payment

Time

0 T S

R T+R Delayed Payment

Time

Fig. 5. Sr R and SZ T.

Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i

J. Wu, Y.-L. Chan / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

6

It is clear that TRC(T) is continuous in T, and has the following properties TRC 4 ðSÞ ¼ TRC 5 ðSÞ:

ð26Þ

Next, we discuss some special cases.

The maximum lifetime for non-deteriorating items is approaching infinity. Hence, our proposed model is a generalized model for nondeteriorating items, in which m is approaching infinity. Using Calculus, L'Hospital's Rule, and simplifying terms, we can simplify the problem for non-deteriorating items as shown below. For details, please see Appendix A. The retailer's order quantity per cycle in (6) becomes   1þm Q ¼ Ið0Þ ¼ Dð1 þ mÞln ¼ DT when m-1: ð27Þ 1 þ m T Similarly, the retailer's annual holding cost excluding interest charges in (9) is simplified to " #   hD ð1 þmÞ2 1þm T 2 ð1 þ mÞT hDT þ  : ð28Þ lim ln ¼ m-1 T 1þmT 2 2 2 4 Consequently, (12), (15), and (17) are reduced to ( 1 hDT 2 cI c D cDT þ o þ ½αðT  SÞ2 þ ð1  αÞðT þ R  SÞ2  þ TRC 1 ðTÞ ¼ T 2 2  pI e D 2  ½αS þ ð1  αÞðS  RÞ2  ; ð29Þ 2

( ) 1 hDT 2 T2 cDT þ o þ  pI e D½ST  ð1  αÞRT ; TRC 3 ðTÞ ¼ T 2 2

qðxÞ ¼

f ðxÞ gðxÞ

ð32Þ

is (strictly) pseudo-convex, if f(x) is non-negative, differentiable and (strictly) convex, and g(x) is positive, differentiable and concave. Let us apply the above theoretical results to obtain the optimal solution T* such that TRC i ðT n Þ for i ¼ 1; 2; :::; 5 is minimized.

3.3. A special case for non-deteriorating items

( 1 hDT 2 cI c D cDT þ o þ ð1  αÞðT þ R  SÞ2 TRC 2 ðTÞ ¼ þ T 2 2  pI e D 2 ½αT þ 2αTðS  TÞ þ ð1  αÞðS  RÞ2  ;  2

function

ð30Þ

ð31Þ

respectively. As a result, the proposed model here is a generalized model for numerous previous research models: (a) If m-1, α ¼ 0, R ¼ 0, p ¼ c, and I c 4 I e , then the proposed model is simplified to Chen and Teng (2014). (b) If m-1, α ¼ 0, and R ¼ 0, then the proposed model is reduced to Teng (2002). (c) If m-1, α ¼ 0, p ¼ c, and I c 4 I e , then the proposed model is simplified to Huang (2003). (d) If m-1, and α ¼ 0, then the proposed model is reduced to Teng and Goyal (2007). (e) If m-1, then the proposed model becomes Teng (2009). (f) If α ¼ 0, and R ¼ 0, then the proposed model is simplified to Chen and Teng (2014). Now, our aim is to determine the optimal replenishment cycle T* for Cases of S 4 R, and S r R.

4.1. Optimal replenishment cycle time for the case of S 4 R We separately minimize each of TRC i ðTÞ for i ¼ 1; 2;and 3. By applying the above mentioned theoretical results, we can obtain the following results. Theorem 1. (1) TRC 1 ðTÞ is a strictly pseudo-convex function in T, and hence exists a unique minimum solution T 1 n . (2) If S rT 1 n , then TRC 1 ðTÞ subject to S r T is minimized at T 1 n . (3) If S 4T 1 n , then TRC 1 ðTÞ subject to S r T is minimized at S. Proof. See Appendix B. To find T 1 n , taking the first-order derivative of TRC 1 ðTÞ, setting the result to zero, and re-arranging terms, we get h ih  i T 1þm o Dð1 þmÞ c þ hð1 2þ mÞ 1 þ m  T  ln 1 þ m  T þ

D½αS2 þ ð1  αÞðS  RÞ2 ðpIe  cI c Þ DT 2 ðh þ2cI c Þ þ ¼ 0: 2 4

ð33Þ

From Theorem 1, we know that (33) has a unique solution T 1 n . If T 1 n Z S, then TRC 1 ðTÞ is minimized at T 1 n . Otherwise, TRC 1 ðTÞ is minimized at S. By using the analogous argument, we have the following results. Theorem 2. (1) TRC 2 ðTÞ is a strictly pseudo-convex function in T, and hence exists a unique minimum solution T 2 n . (2) If S  Rr T 2 n rS, then TRC 2 ðTÞ subject to T r S rT þ R is minimized at T 2 n . (3) If T 2 n r S R, then TRC 2 ðTÞ subject to T r S r T þ R is minimized at S R. (4) If T 2 n Z S, then TRC 2 ðTÞ subject to T r S rT þ R is minimized at S. Proof. See Appendix C. To get T 2 n , taking the first-order derivative of TRC 2 ðTÞ, setting the result to zero, and re-arranging terms, we get h ih  i T 1þm o Dð1 þmÞ c þ hð1 2þ mÞ 1 þ m  T  ln 1 þ m  T þ

Dð1  αÞðS  RÞ2 ðpI e  cI c Þ DT 2 ½h þ 2αpI e þ 2ð1  αÞcI c  þ ¼0 2 4

ð34Þ

It is clear from Theorem 2 that (34) has a unique solution T 2 n . If S  R r T 2 n r S, then TRC 2 ðTÞ is minimized at T 2 n . If T 2 n r S  R, then TRC 2 ðTÞ is minimized at S R. If S rT 2 n , then TRC 2 ðTÞ is minimized at S. Finally, for the case of RoS, we have the following similar results for TRC 3 ðTÞ. Theorem 3.

4. Determination of the optimal replenishment cycle In this section, the necessary and sufficient conditions for the determination of the optimal replenishment cycle are presented for the case of S 4 R first, and then the case of S r R. According to Theorems 3.2.9, and 3.2.10 in Cambini and Martein (2009), the

(1) TRC 3 ðTÞ is a strictly pseudo-convex function in T, and hence exists a unique minimum solution T 3 n . (2) If T 3 n r S  R, then TRC 3 ðTÞ subject to T þR r S is minimized at T 3n. (3) If T 3 n Z S  R, then TRC 3 ðTÞ subject to T þR r S is minimized at S  R.

Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i

J. Wu, Y.-L. Chan / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Proof. See Appendix D. Similarly, taking the first-order derivative of TRC 3 ðTÞ, setting the result to zero, and re-arranging terms, we get

 

hð1 þ mÞ T 1þm DT 2 ðh þ 2pI e Þ Dð1 þ mÞ c þ  ln oþ ¼ 0: 2 1þmT 1þmT 4

ð35Þ From Theorem 3, we know that (35) has a unique solution T 3 n . If T 3 n r S  R, then TRC 3 ðTÞ is minimized at T 3 n . If T 3 n Z S  R, then TRC 3 ðTÞ is minimized at S R. For simplicity, let us define two discrimination terms.



hð1 þ mÞ SR 1 þm  ln o Δ1 ¼ Dð1 þ mÞ c þ 2 1 þm  ðS  RÞ 1 þ m  ðS  RÞ þ

DðS  RÞ2 ðh þ 2pI e Þ ; 4

ð36Þ

and

þ

2

D½αS þ ð1  αÞðS  RÞ ðpIe  cI c Þ DS ðh þ 2cI c Þ þ : 2 4

ð37Þ

Δ2 o 0, then TRCðTÞ is minimized at T 1 n . Δ2 ¼ 0, then TRCðTÞ is minimized at S. Δ1 o 0 and Δ2 4 0, TRCðTÞ is minimized at T 2 n . Δ1 ¼ 0, then TRCðTÞ is minimized at S–R. Δ1 4 0, then TRCðTÞ is minimized at T 3 n .

It is clear from Theorem 6 that (39) has a unique solution, T 5 n . If T 5 n r S, then TRC 5 ðTÞ is minimized at T 5 n . Otherwise, TRC 5 ðTÞ is minimized at S. Again, let us define the third discrimination term.

 

hð1 þ mÞ S 1þm  ln o Δ3 ¼ Dð1 þ mÞ c þ 2 1þmS 1þmS þ

½h þ 2αpI e þ 2ð1 αÞcI c DS2 : 4

ð40Þ

(1) If Δ3 o 0, then TRCðTÞ is minimized at T 4 n . (2) If Δ3 ¼ 0, then TRCðTÞ is minimized at S. (3) If Δ3 4 0, TRCðTÞ is minimized at T 5 n .

5. Numerical examples In this section, we provide some numerical examples to illustrate several distinct theoretical results as well as to gain some managerial insights. Example 1. Let us assume that D¼ 2000 units per year, S ¼0.16 years, R ¼0.08 years, m ¼1 year, p ¼$10, c ¼$5 per unit, o ¼$100 per order, h¼$2 per unit per year, α¼ 0.05, Ic ¼0.03 per dollar per year, and Ie ¼ 0.03 per dollar per year. Calculating two discrimination terms, we get

4.2. Optimal replenishment cycle time for the case of S r R Similar to the case of S 4 R, we have the following results.

Δ1 ¼ 68:0292;

Theorem 5. (1) TRC 4 ðTÞ is a strictly pseudo-convex function in T, and hence exists a unique minimum solution T 4 n . (2) If S r T 4 n , then TRC 4 ðTÞ subject to S r T is minimized at T 4 n . (3) If S 4 T 4 n , then TRC 4 ðTÞ subject to S r T is minimized at S. Proof. The proof is similar to that in Theorem 1, and hence is omitted. To find T 4 n , taking the first-order derivative of TRC 4 ðTÞ, setting the result to zero, and re-arranging terms, we derive h ih  i 2 T 1þm Dð1 þ mÞ c þ hð1 2þ mÞ 1 þ m  o þ αDS ðpI2e  cI c Þ  T  ln 1 þ m  T DT 2 ðh þ2cI c Þ ¼ 0: 4

ð39Þ

Proof. The proof is omitted.

Proof. See Appendix F.

þ

½h þ 2αpI e þ 2ð1  αÞcI c DT 2 ¼ 0: 4

Theorem 7.

Theorem 4. If If If If If

þ

2

Then we can prove that Δ1 o Δ2 (See Appendix E). Combining Theorems 1–3, we have the following result.

(1) (2) (3) (4) (5)

Proof. The proof is omitted. To get T 5 n , taking the first-order derivative of TRC 5 ðTÞ, setting the result to zero, and re-arranging terms, we derive h ih i T 1þm Dð1 þ mÞ c þ hð1 2þ mÞ 1 þ m  T  lnð1 þ m  T Þ o

We then have the following results.



 

hð1 þ mÞ S 1þm  ln o Δ2 ¼ Dð1 þ mÞ c þ 2 1 þm  S 1þmS 2

7

ð38Þ

It is obvious from Theorem 5 that (38) has a unique solution, T 4 n . If T 4 n Z S, then TRC 4 ðTÞ is minimized at T 4 n . Otherwise, TRC 4 ðTÞ is minimized at S. By using the analogous argument, we have the following results. Theorem 6. (1) TRC 5 ðTÞ is a strictly pseudo-convex function in T, and hence exists a unique minimum solution T 5 n . (2) If S Z T 5 n , then TRC 5 ðTÞ subject to S Z T is minimized at T 5 n . (3) If S r T 5 n , then TRC 5 ðTÞ subject to S Z T is minimized at S.

and

Δ2 ¼ 30:6415:

By using Theorem 4, we know that T n ¼ T 2 n . Solving (34), and substituting the result into (15), we obtain the following optimal solution T n ¼ T 2 n ¼ 0.1406, Q n ¼ DT 2 n ¼ 291.540, and TRC n ðT n Þ ¼11,356. 3473. Example 2. Using the same data as those in Example 1, we study the sensitivity analysis on the optimal solution with respect to each parameter. The computational results are shown in Table 1. The sensitivity analysis reveals that: (1) a higher value of α, p, or I e causes lower values of T*, Q*, and TRC*(T*); (2) by contrast a higher value of o or R causes higher values of T*, Q*, and TRC*(T*); (3) a higher value of D, c, h, or I c causes a lower value of T* while a higher value of TRC*(T*); and (4) conversely a higher value of m causes higher of T* while a lower value of TRC*(T*). Simple economic interpretations of the above results are as follows. (1) The retailer earns more interest from trade credit if the fraction of the immediate payment α (as well as p, or I e ) is higher. Hence, the retailer orders less quantity to take the benefit more often, and thus pays less the total relevant cost. (2) If the ordering cost o (as well as R) is higher, then the retailer pays more the total relevant cost and orders more quantity to reduce the number of orders. (3) If the holding cost h (as well as c, D, or I c ) is higher, then the retailer pays more the total relevant cost but orders less quantity to reduce holding cost. Finally, (4) If the maximum lifetime m is higher, the retailer pays less deterioration cost as well as the total

Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i

J. Wu, Y.-L. Chan / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

8

relevant cost, and orders more quantity than that the case of shorter m.

the supplier and the retailer) or a non-cooperative Nash or Stackelberg equilibrium solution for each player.

Acknowledgments 6. Conclusions and future research The use of a down-stream partial trade credit to reduce default risks with credit-risk customers has received a very little attention by the researchers. In this paper, we have built an EOQ model for deteriorating items with maximum lifetime in a supply chain in which the retailer receives an up-stream full trade credit from his/ her supplier while offers a down-stream partial trade credit to his/ her credit-risk customers. The proposed model has been a generalized model including Goyal (1985), Teng (2002), Huang (2003), Teng and Goyal (2007), Teng (2009), and Chen and Teng (2014) as special cases. By applying the theoretical results in convex fractional programs, we have obtained the necessary and sufficient conditions for finding a unique optimal solution. Furthermore, we have proposed three discrimination terms to identify the optimal solution among alternatives. Finally, we have used several numerical examples to show all possible alternatives. Following most traditional EOQ models, we have adopted the objective to minimize the total relevant cost. However, one of two anonymous referees suggests an excellent alternative in which the objective is to maximize the total profit by assuming that the ending inventory is not zero and can be sold at salvage value in order to reduce accelerating deterioration cost as time approaches to the expiration date. As a result, one may extend our EOQ model from zero ending-inventory to non-zero ending-inventory. Also, one could generalize the model to allow for shortages and partial backlogging. Finally, the proposed model with single player can be extended to an integrated cooperative model for both players (e.g., Table 1 Sensitivity analysis on parameters. Parameter

Δ1

Δ2

Tn

Qn

TRCn(Tn)

D ¼ 2000 D ¼ 2500 D ¼ 3000 R ¼0.04 R ¼0.08 R ¼0.12 S ¼0.16 S ¼0.20 S ¼0.25 c¼3 c¼5 c¼8 o ¼50 o ¼100 o ¼200 h¼1 h¼2 h¼4 m¼ 1 m¼ 2 m¼ 3 Ic ¼ 0.02 Ic ¼ 0.03 Ic ¼ 0.04 Ie ¼0.02 Ie ¼0.03 Ie ¼0.04 p ¼ 10 p ¼ 15 p ¼ 20 α ¼ 0.05 α ¼ 0.10 α ¼ 0.20

 68.0292  60.0365  52.0438  26.5573  68.0292  92.1672  68.0292  26.5573 51.3889  74.7866  68.0292  57.8931  18.0292  68.0292  168.029  74.6079  68.0292  54.8718  68.0292  73.9878  76.8872  68.0292  68.0292  68.0292  68.6692  68.0292  67.3892  68.0292  67.0692  66.1092  68.0292  68.0292  68.0292

30.6416 63.3019 95.9623 31.7816 30.6416 29.9576 30.6416 108.3690 237.5830 0.9479 30.6416 75.1821 80.6416 30.6416  69.3584 3.54191 30.6416 84.8409 30.6416 3.9740  8.6396 29.7296 30.6416 31.5536 29.9056 30.6416 31.3776 30.6416 31.7456 32.8496 30.6416 30.7856 31.0736

T2 ¼ 0.1406 T2 ¼ 0.1261 T2 ¼ 0.1153 T2 ¼ 0.1398 T2 ¼ 0.1406 T2 ¼ 0.1411 T2 ¼ 0.1406 T2 ¼ 0.1398 T3 ¼ 0.1393 T2 ¼ 0.1593 T2 ¼ 0.1406 T2 ¼ 0.1221 T2 ¼ 0.1000 T2 ¼ 0.1406 T1 ¼ 0.1962 T2 ¼ 0.1573 T2 ¼ 0.1406 T2 ¼ 0.1186 T2 ¼ 0.1406 T2 ¼ 0.1570 T1 ¼ 0.1673 T2 ¼ 0.1410 T2 ¼ 0.1406 T2 ¼ 0.1402 T2 ¼ 0.1411 T2 ¼ 0.1406 T2 ¼ 0.1401 T2 ¼ 0.1406 T2 ¼ 0.1399 T2 ¼ 0.1391 T2 ¼ 0.1406 T2 ¼ 0.1405 T2 ¼ 0.1404

291.540 325.614 356.308 289.86 291.54 292.543 291.54 289.86 288.803 331.934 291.540 252.033 205.259 291.540 413.113 327.749 291.540 244.626 291.540 322.439 341.881 292.480 291.540 290.614 292.577 291.540 290.500 291.540 289.980 288.416 291.540 291.393 291.099

11,356.3 14,008.0 16,643.9 11,336.8 11,356.3 11,372.6 11,356.3 11,335.6 11,337.3 7197.31 11,356.3 17,563.0 10,941.2 11,356.3 11,949.4 11,204.0 11356.3 11,619.5 11,356.3 11,219.1 11,145.6 11,355.1 11,356.3 11,357.6 11,361.6 11,356.3 11,351.1 11,356.3 11,348.5 11,340.6 11,356.3 11,354.1 11,349.7

The authors would like to thank Editor Peter Kelle, Main Guest Editor Cárdenas-Barrón, and two anonymous referees for their constructive comments and an excellent suggestion for future research.

Appendix A. Formulas for non-deteriorating items Using Calculus and simplifying terms, we have:

" #   d 1þm 1 þ m T 1 1þm T ¼ ln ¼  : 2 dm 1 þ m T 1 þ m 1 þ m  T ð1 þ m  TÞ ð1 þ mÞð1 þm  TÞ

ðA1Þ Using L'Hospital's Rule, we obtain:   1þm ðd=dmÞlnðð1 þ mÞ=ð1 þ m  TÞÞ ¼ lim lim ð1 þ mÞln m-1 m-1 1 þ m T ðd=dmÞð1=1 þ mÞ  T=ð1 þ mÞð1 þ m  TÞ ¼ lim m-1  1=ð1 þ mÞ2 Tð1 þmÞ ¼ T: ðA2Þ ¼ lim m-11 þ m  T Consequently, the seller's order quantity per cycle in (7) becomes   1þm ¼ DT when m-1 ð27Þ Q ¼ Ið0Þ ¼ Dð1 þmÞln 1þmT Similarly, we can get the following results: " #   ð1 þmÞ2 1þm ð1 þ mÞT lim  ln m-1 1þmT 2 2 " # 1 lnðð1 þ mÞ=ð1 þ m  TÞÞ  ðT=1 þ mÞ ¼ lim 2m-1 1=ð1 þ mÞ2 " # 1  T=ð1 þ m TÞð1 þ mÞ þ ðT=ð1 þ mÞ2 Þ ¼ lim 2m-1  2=ð1 þ mÞ3 " # 1 T 2 ð1 þ mÞ 1 T2 ¼ lim T 2 ¼ : ¼ lim 2m-1 2ð1 þ m  TÞ 4m-1 4

ðA3Þ

As a result, we know that the retailer's annual holding cost excluding interest charges in (10) is simplified to " #   hD ð1 þ mÞ2 1þm T 2 ð1 þ mÞT hDT þ  ð28Þ ln lim ¼ m-1 T 1þmT 2 2 2 4

Appendix B. Proof of Theorem 1 From (13), let

  1þm þo f 1 ðTÞ ¼ cDð1 þ mÞln 1þmT " #   2 ð1 þ mÞ 1þm T 2 ð1 þ mÞT þ  ln þ hD 1þmT 2 2 4 cI c D ½αðT  SÞ2 þ ð1  αÞðT þ R SÞ2  2 pI e D 2 ½αS þ ð1  αÞðS  RÞ2 ;  2

þ

ðB1Þ

and g 1 ðTÞ ¼ T 4 0:

ðB2Þ

Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i

J. Wu, Y.-L. Chan / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Taking the first-order and second-order derivatives of f 1 ðTÞ, we have: " # cDð1 þ mÞ hD ð1 þ mÞ2 0 þ f 1 ðTÞ ¼ þ T  ð1 þ mÞ þcI c D½T  S þ ð1  αÞR; 1þmT 2 1þmT ðB3Þ and ″

f 1 ðTÞ ¼

cDð1 þ mÞ ð1 þ m TÞ

þ 2

hD 2

"

1þm 1þmT

2

# þ 1 þ cI c D 4 0:

ðB4Þ

Therefore, q1 ðTÞ ¼ f 1 ðTÞ=g 1 ðTÞ ¼ TRC 1 ðTÞ is a strictly pseudo-convex function in T, which completes the proof of Part (1) of Theorem 1. The proof of Parts (2) and (3) immediately follows from Part (1) of Theorem 1. This completes the proof of Theorem 1.

Appendix C. Proof of Theorem 2 From (16), let

  1þm f 2 ðTÞ ¼ cDð1 þ mÞln þo 1 þ m T " #   ð1 þ mÞ2 1þm T 2 ð1 þmÞT þ  ln þhD 1þmT 2 2 4 cI c D pI e D ð1  αÞðT þ R SÞ2  ½αT 2 þ2αTðS  TÞ þ 2 2 þð1 αÞðS RÞ2 ;

0

ðC1Þ

" # cDð1 þ mÞ hD ð1 þ mÞ2 þ þ T  ð1 þ mÞ þcI c Dð1  αÞðT þ R  SÞ 1þmT 2 1þmT

and ″ f 2 ðTÞ ¼

cDð1 þ mÞ

hD þ ð1 þ m TÞ2 2

ðC3Þ "

1þm 1þmT

2

" # 2 hD 1þm þ 1 þ pI e D 4 0: 2 1 þ m T

ðD4Þ

Therefore, q3 ðTÞ ¼ f 3 ðTÞ=g 3 ðTÞ ¼ TRC 3 ðTÞ is a strictly pseudo-convex function in T, which completes the proof of Part (1) of Theorem 3. The proof of Parts (2) and (3) immediately follows from Part (1) of Theorem 3. This completes the proof of Theorem 3. Appendix E. Proof of Δ1 o Δ2 Since TRC 2 ðTÞ is a strictly pseudo-convex function in T, we know from (34) that 

 

d 1 hð1 þ mÞ T 1þm TRC 2 ðTÞ ¼ 2 Dð1 þmÞ c þ  ln o dT 2 1þmT 1þmT T ) Dð1  αÞðS  RÞ2 ðpI e cI c Þ DT 2 ½h þ 2αpI e þ 2ð1 αÞcI c  þ þ 2 4

ðE2Þ

Thus, Δ1 o Δ2 . This completes the proof. Appendix F. Proof of Theorem 4 From (33), we have 

 

d 1 hð1 þ mÞ T 1þm TRC 1 ðTÞ ¼ 2 Dð1 þmÞ c þ  ln o dT 2 1þmT 1þmT T ) D½αS2 þð1 αÞðS RÞ2 ðpI e cI c Þ DT 2 ðh þ 2cI c Þ þ þ ðF1Þ 2 4 If Δ2 o 0, then it is clear from (F1) that

# þ 1 þ ð1  αÞcI c D þ αpI e D 4 0: ðC4Þ

Therefore, q2 ðTÞ ¼ f 2 ðTÞ=g 2 ðTÞ ¼ TRC 2 ðTÞ is a strictly pseudo-convex function in T, which completes the proof of Part (1) of Theorem 2. The proof of Parts (2)–(4) immediately follows from Part (1) of Theorem 2. This completes the proof of Theorem 2.

lim

d

T-1dT

TRC 1 ðTÞ ¼ 1;

and

d Δ2 TRC 1 ðSÞ ¼ 2 o0: dT S

ðF2Þ

By applying the Mean Value Theorem and Theorem 1, we know that there exists a unique T 1 n A ðS; 1Þ such that ðd=dTÞTRC 1 ðT 1 n Þ ¼ 0. Hence, TRC 1 ðTÞ is minimizing at the unique point T 1 n , which satisfies (33). By using the analogous argument, we have d Δ1 d Δ2 TRC 2 ðS  RÞ ¼ o TRC 2 ðSÞ ¼ 2 o 0; dT ðS  RÞ2 dT S

ðF3Þ

which implies TRC 2 ðTÞ is minimizing at S. Likewise, we get

Appendix D. Proof of Theorem 3

lim

Again, by using (18), we let

d

ξ-0dT

"  1þm ð1 þ mÞ2 1þm þo þ hD Þ lnð f 3 ðTÞ ¼ cDð1 þ mÞln 1 þ m T 1þmT 2 # T 2 ð1 þ mÞT T2 þ  ðD1Þ  pI e D½ST  ð1  αÞRT; 2 4 2 

and g 3 ðTÞ ¼ T 4 0:

ð1 þ m  TÞ

þ 2

d Δ1 d Δ2 TRC 2 ðM  NÞ ¼ o TRC 2 ðMÞ ¼ 2 : dT ðM NÞ2 dT M

ðC2Þ

 pIe DαðS  TÞ;

cDð1 þmÞ

ðE1Þ

Taking the first-order and second-order derivatives of f 2 ðTÞ, we have: f 2 ðTÞ ¼

″

f 3 ðTÞ ¼

is an increasing function in T, and hence we get

and g 2 ðTÞ ¼ T 4 0:

and

9

ðD2Þ

Taking the first-order and second-order derivatives of f 3 ðTÞ, we have: " # cDð1 þ mÞ hD ð1 þ mÞ2 0 þ þ T  ð1 þ mÞ pI e D½S  T ð1 αÞR; f 3 ðTÞ ¼ 1þmT 2 1þmT ðD3Þ

TRC 3 ðξÞ o

d Δ1 TRC 3 ðS  RÞ ¼ o 0; dT ðS  RÞ2

ðF4Þ

which implies TRC 3 ðTÞ is minimizing at S R. Consequently, by using (19), (F3) and (F4), we obtain TRC 1 ðT 1 n Þ oTRC 1 ðSÞ ¼ TRC 2 ðSÞ o TRC 2 ðS  RÞ ¼ TRC 3 ðS  RÞ:

ðF5Þ

As a result, we complete the proof that if Δ2 o 0, then TRCðTÞ is minimized at T 1 n . By using the analogous argument, one can easily prove the rest of Theorem 4. References Aggarwal, S.P., Jaggi, C.K., 1995. Ordering policies of deteriorating items under permissible delay in payments. J. Oper. Res. Soc. 46 (5), 658–662. Bakker, M., Riezebos, J., Teunter, R.H., 2012. Review of inventory systems with deterioration since 2001. Eur. J. Oper. Res. 221 (2), 275–284. Cambini, A., Martein, L., 2009. Generalized Convexity and Optimization: Theory and Applications. Springer, Berlin.

Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i

10

J. Wu, Y.-L. Chan / Int. J. Production Economics ∎ (∎∎∎∎) ∎∎∎–∎∎∎

Chang, C.T., Ouyang, L.Y., Teng, J.T., 2003. An EOQ model for deteriorating items under supplier credits linked to ordering quantity. Appl. Math. Model. 27 (12), 983–996. Chen, S.C., Cárdenas-Barrón, L.E., Teng, J.T., 2013a. Retailer’s economic order quantity when the supplier offers conditionally permissible delay in payments link to order quantity. Int. J. Prod. Econ., http://dx.doi.org/10.1016/ j.ijpe.2013.05.032, in press. Chen, S.C., Teng, J.T., Skouri, K., 2013b. Economic production quantity models for deteriorating items with up-stream full trade credit and down-stream partial trade credits. Int. J. Prod. Econ., http://dx.doi.org/10.1016/j.ijpe.2013.07.024 in press. Chen, S.C., Teng, J.T., 2014. Retailer's optimal ordering policy for deteriorating items with maximum lifetime under supplier's trade credit financing. Appl. Math. Model., http://dx.doi.org/10.1016/j.apm.2013.11.056, in press. Chern, M.S., Pan, Q., Teng, J.T., Chan, Y.L., Chen, S.C., 2013. Stackelberg solution in a vendor-buyer supply chain model with permissible delay in payments. Int. J. Prod. Econ. 144 (1), 397–404. Covert, R.B., Philip, G.S., 1973. An EOQ model with Weibull distribution deterioration. AIIE Trans. 5 (4), 323–326. Dave, U., Patel, L.K., 1981. (T, Si) policy inventory model for deteriorating items with time proportional demand. J. Oper. Res. Soc. 32 (2), 137–142. Dye, C.Y., 2013. The effect of preservation technology investment on a noninstantaneous deteriorating inventory model. Omega 41 (5), 872–880. Ghare, P.M., Schrader, G.P., 1963. A model for an exponentially decaying inventory. J. Ind. Eng. 14 (5), 238–243. Goswami, A., Chaudhuri, K.S., 1991. An EOQ model for deteriorating items with shortages and a linear trend in demand. J. Oper. Res. Soc. 42 (12), 1105–1110. Goyal, S.K., 1985. Economic order quantity under conditions of permissible delay in payments. J. Oper. Res. Soc. 36 (4), 335–338. Goyal, S.K., Giri, B.C., 2001. Recent trends in modeling of deteriorating inventory. Eur. J. Oper. Res. 134 (1), 1–16. Hariga, M.A., 1996. Optimal EOQ models for deteriorating items with time-varying demand. J. Oper. Res. Soc. 47 (10), 1228–1246. Harris, F.W., 1913. How many parts to make at once. Factory, Mag. Manag. 10 (2), 135–136 (and 152). Hu, F., Liu, D., 2010. Optimal replenishment policy for the EPQ model with permissible delay in payments and allowable shortages. Appl. Math. Model. 34 (10), 3108–3117. Huang, Y.F., 2003. Optimal retailer's replenishment policy for the EPQ model under the supplier's trade credit policy. Prod. Plan. Control 15 (1), 27–33. Jamal, A.M.M., Sarker, B.R., Wang, S., 1997. An ordering policy for deteriorating items with allowable shortage and permissible delay in payment. J. Oper. Res. Soc. 48 (8), 826–833.

Liao, J.J., 2007. On an EPQ model for deteriorating items under permissible delay in payments. Appl. Math. Model. 31 (3), 393–403. Lou, K.R., Wang, W.C., 2013. A comprehensive extension of an integrated inventory model with ordering cost reduction and permissible delay in payments. Appl. Math. Model. 37 (7), 4709–4716. Ouyang, L.Y., Teng, J.T., Chen, L.H., 2006. Optimal ordering policy for deteriorating items with partial backlogging under permissible delay in payments. J. Glob. Optim. 34 (2), 245–271. Sachan, R.S., 1984. On (T, Si) policy inventory model for deteriorating items with time proportional demand. J. Oper. Res. Soc. 35 (11), 1013–1019. Seifert, D., Seifert, R.W., Protopappa-Sieke, M., 2013. A review of trade credit literature: opportunity for research in operations. Eur. J. Oper. Res. 231 (2), 245–256. Skouri, K., Konstantaras, I., Papachristos, S., Ganas, I., 2009. Inventory models with ramp type demand rate, partial backlogging and Weibull deterioration rate. Eur. J. Oper. Res. 192 (1), 79–92. Skouri, K., Konstantaras, I., Papachristos, S., Teng, J.T., 2011. Supply chain models for deteriorating products with ramp type demand rate under permissible delay in payments. Expert Syst. Appl. 38 (12), 14861–14869. Taleizadeh, A.A., 2014. An EOQ model with partial backordering and advance payments for an evaporating item. Int. J. Prod. Econ., http://dx.doi.org/10.1016/ j.ijpe.2014.01.023, in press. Teng, J.T., 2002. On the economic order quantity under conditions of permissible delay in payments. J. Oper. Res. Soc. 53 (8), 915–918. Teng, J.T., 2009. Optimal ordering policies for a retailer who offers distinct trade credits to its good and bad credit customers. Int. J. Prod. Econ. 119 (2), 415–423. Teng, J.T., Goyal, S.K., 2007. Optimal ordering policies for a retailer in a supply chain with up-stream and down-stream trade credits. J. Oper. Res. Soc. 58 (9), 1252–1255. Teng, J.T., Krommyda, I.P., Skouri, K., Lou, K.R., 2011. A comprehensive extension of optimal ordering policy for stock-dependent demand under progressive payment scheme. Eur. J. Oper. Res. 215 (1), 97–104. Teng, J.T., Min, J., Pan, Q., 2012. Economic order quantity model with trade credit financing for non-decreasing demand. Omega 40 (3), 328–335. Wang, W.C., Teng, J.T., Lou, K.R., 2014. Seller's optimal credit period and cycle time in a supply chain for deteriorating items with maximum lifetime. Eur. J. Oper. Res. 232 (2), 315–321. Wu, J., Skouri, K., Teng, J.T., Ouyang, L.Y., 2014. A note on “Optimal replenishment policies for non-instantaneous deteriorating items with price and stock sensitive demand under permissible delay in payment”. Int. J. Prod. Econ., http://dx.doi.org/ 10.1016/j.ijpe.2013.12.017, in press. Yang, S., Hong, K.S., Lee, C., 2013. Supply chain coordination with stock-dependent demand rate and credit incentives. Int. J. Prod. Econ., http://dx.doi.org/10.1016/j. ijpe.2013.06.014, in press.

Please cite this article as: Wu, J., Chan, Y.-L., Lot-sizing policies for deteriorating items with expiration dates and partial trade credit to credit-risk customers. International Journal of Production Economics (2014), http://dx.doi.org/10.1016/j.ijpe.2014.03.023i