INTRODUCTION
In the real world, the supplier often makes use of the permissible delay in payments policy to promote his/her commodities. Therefore, it makes economic sense for the retailer to delay the settlement of the replenishment account up to the last moment of the permissible period allowed by the supplier. From the viewpoint of the supplier, the supplier hopes that the payment is paid from retailer as soon as possible. It can avoid the possibility of resulting in bad debt. So, in most business transactions, the supplier will offer the credit terms mixing cash discount and trade credit to the retailer. The retailer can obtain the cash discount when the payment is paid before cash discount period offered by the supplier. Otherwise, the retailer will pay full payment within the trade credit period. Many articles related to the inventory policy under permissible delay in payments and cash discount can be found in Huang^{[1,2]} and Huang and Chung^{[3]}.
Recently, Chung and Huang^{[4]} investigated the topic of permissible delay in payments within the EPQ (economic production quantity) framework. Therefore, this study tries to incorporate both Chung and Huang^{[4]} and Huang and Chung^{[3]} to develop the retailer’s inventory model. That is, we want to investigate the retailer’s optimal replenishment policy under permissible delay in payments and cash discount within the EPQ framework. Then mathematical models have been derived for obtaining the optimal cycle time for item so that the annual total relevant cost is minimized.
Model formulation: For convenience, we adopt the same notation and assumptions as in Chung and Huang^{[4]} and Huang and Chung^{[3]}.
The annual total relevant cost for the retailer can be expressed as:
TVC (T) =ordering cost+stockholding cost+purchasing cost+interest payableinterest earned.
We show that the annual total relevant cost is given by:
Case 1: Payment is paid at time M_{1}
where:
and
Then, we find TVC_{11}(PM_{1}/D)=TVC_{12}(PM_{1}/D) and TVC_{12}(M_{1})= TVC_{13}(M_{1}). Hence TVC_{1}(T) is continuous and welldefined. All TVC_{11}(T), TVC_{12}(T), TVC_{13}(T) and TVC_{1}(T) are defined on T > 0.
Case 2: Payment is paid at time M_{2}
where:
and
Then, we find TVC_{21}(PM_{2}/D)=TVC_{22}(PM_{2}/D) and TVC_{22}(M_{2})= TVC_{23}(M_{2}). Hence TVC_{2}(T) is continuous and welldefined. All TVC_{21}(T), TVC_{22}(T), TVC_{23}(T) and TVC_{2}(T) are defined on T>0.
Decision rule of the optimal cycle time T*: The main purpose of this section is to develop a solution procedure to determine the optimal cycle time T*.
From equations (2)(4) and (6)(8)
yield:
and
Equations (14), (18) and (20)
imply that all TVC_{13}(T), TVC_{22}(T) and TVC_{23}(T)
are convex on T>0. However, equation (10) implies that
TVC_{11}(T) is convex on T>0 if 2AcM_{1}^{2}pI_{k}
(1r)+cDM_{1}^{2}[I_{k} (1r)I_{e}]>0; equation
(12) implies that TVC_{12}(T) is convex on T>0 if 2A+cM_{1}^{2}[I_{k}(1r)I_{e}]>0
and equation (16) implies that TVC_{21}(T) is convex
on T>0 if 2AcM_{2}^{2 } PI_{k}+cDM_{2}^{2}
(I_{k}I_{e}) >0. Furthermore, we have
Let TVC_{ij}’ (T)=0, for all i=1~2 and j=1~3. Then we can obtain
and
Equation (21) implies that the optimal value of T for the
case of T≥PM_{1}/D, that is T_{11}*≥PM_{1}/D.
We substitute equation (21) into T_{11}*≥ PM_{1}/D,
then we can obtain the optimal value of T if and only if
Similar disscussion, we can obain following results:
if and only if
and if and only if 2A+DM_{1}^{2} (hρ+cI_{e})
≤ 0.
T_{13}^{*} ≤ M_{1} if and only if 2A+DM_{1}^{2}
(hρ+cI_{e}) ≥ 0.
T_{21}^{*} ≥ PM_{2}/D if and only if
M_{2}≤T_{22}^{*}≤PM_{2}/D
if and only if
and if and only if 2A+DM_{2}^{2 }(hρ+cI_{e})
≤ 0.
T_{23}^{*}≤M_{2} if and only if 2A+DM_{2}^{2}
(hρ+cI_{e}) ≥ 0.
Let
and
From equations (27)(30), we can obtain
Δ_{3}>Δ_{1}>Δ_{2} and Δ_{3}>Δ_{4}>Δ_{2}.
Summarized above arguments, we can obtain following results.
Theorem 1:
(A) 
If Δ_{2}≥0, then TVC(T*)= min{TVC_{1}(T_{13}*),
TVC_{2} (T_{23}*)}. Hence T* is T_{13}* or T_{23}*
associated with the least cost. 
(B) 
If Δ_{1}≥0, Δ_{2}<0 and Δ_{4}≥0,
then TVC(T*)= min {TVC_{1}(T_{12}*), TVC_{2}(T_{23}*)}.
Hence T* is T_{12}* or T_{23}* associated with the least
cost. 
(C) 
If Δ_{1}≥0, Δ_{2}<0 and Δ_{4}<0,
then TVC(T*)= min {TVC_{1}(T_{12}*), TVC_{2}(T_{22}*)}.
Hence T* is T_{12}* or T_{22}* associated with the least
cost. 
(D) 
If Δ_{1}<0 and Δ_{4}≥0, then TVC(T*)=
min {TVC_{1}(T_{11}*), TVC_{2}(T_{23}*)}.
Hence T* is T_{11}* or T_{23}* associated with the least
cost. 
(E) 
If Δ_{1}<0, Δ_{3}>0 and Δ_{4}<0,
then TVC(T*)= min {TVC_{1}(T_{11}*), TVC_{2}(T_{22}*)}.
Hence T* is T_{11}* or T_{22}* associated with the least
cost. 
(F) 
If Δ_{3}≤0, then TVC(T*)= min {TVC_{1}(T_{11}*),
TVC_{2}(T_{21}*)}. Hence T* is T_{11}* or T_{21}*
associated with the least cost. 
Table 1: 
Optimal cycle time with various r 

Theorem 1 immediately determines the optimal cycle time T* after computing the numbers Δ_{1}, Δ_{2}, Δ_{3} and Δ_{4}. Theorem 1 is really very simple.
Numerical examples: To illustrate the results, let us apply the proposed method to solve the following numerical examples. The optimal cycle time is summarized in Table 1.
CONCLUSION
The supplier offers the permissible delay in payments policy to stimulate the demand of the retailer. However, the supplier can also use the cash discount policy to attract retailer to pay the full payment of the amount of purchasing cost to shorten the collection period. This study investigates the retailer’s replenishment policy under permissible delay in payments and cash discount within the EPQ framework and provides a very efficient solution procedure to determine the optimal cycle time T*.
ACKNOWLEDGMENT
The authors would like to thank the NSC in Taiwan to partly support this study. This project no. NSC 932213E324025.