
Research Article


Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity 

KuangHua Hsu,
HungFu Huang,
YuCheng Tu
and
YungFu Huang



ABSTRACT

In the previous related studies, the inventory replenishment problems under permissible delay in payments are independent of the order quantity. In this study, the restrictive assumption of the trade credit independent of the order quantity is relaxed. This study discusses the inventory policies under permissible delay in payments when a larger order quantity.





INTRODUCTION
In the classical EOQ model, it is tacitly assumed that the payment of an order is made on the receipt of items by the inventory system. In practice, however, this may not be true. Under certain conditions suppliers are known to offer their customers a delay in paying for an order of a particular commodity. Frequently, suppliers allow credit for some fixed time period for settling the payment for the goods and do not charge any interest from the buyer on the amount owed during this credit period. However, a higher interest is charged if the payment is not settled by the end of the credit period. The existence of credit period serves to reduce the cost of holding stock to the user, because it reduces the amount of capital invested in stock for the duration of the credit period. Recently, several researchers have developed analytical inventory models with consideration of permissible delay in payments.
Goyal (1985) established a singleitem inventory model under permissible delay in payments. Khouja and Mehrez (1996) investigated the effect of four different supplier credit policies on the optimal order quantity within the EOQ framework. Chung (1998) developed an efficient decision procedure to determine the economic order quantity under condition of permissible delay in payments. Teng (2002) assumed that the selling price was not equal to the purchasing price to modify Goyal’s model (1985). Chung and Huang (2003a) investigated this issue within EPQ (economic production quantity) framework and developed an efficient solving procedure to determine the optimal replenishment cycle for the retailer. Huang and Chung (2003) investigated the inventory policy under cash discount and trade credit. Chung and Huang (2003b) adopted alternative payment rules to develop the inventory model and obtain different results. Huang (2004) adopted the payment rule discussed in Chung and Huang (2003b) and assumed finite replenishment rate, to investigate the buyer’s inventory problem. Huang (2006) extended Huang (2003) to develop retailer’s inventory policy under retailer’s storage space limited. Recently, Huang (2007) incorporated Chung and Huang (2003a) and Huang (2003) to investigate retailer’s ordering policy.
This research combines the above both studies by Goyal (1985) and Khouja and Mehrez (1996) to discuss the inventory policies under permissible delay in payments when a larger order quantity. Finally, numerical examples are used to illustrate all theorems in this study.
MODEL FORMULATION
Notation:
Q 
= 
Order quantity 
D 
= 
Annual demand 
W 
= 
Quantity at which the delay in payments is permitted 
A 
= 
Cost of placing one order 
c 
= 
Unit purchasing price 
h 
= 
Unit stock holding cost per year excluding interest charges 
I_{e} 
= 
Interest which can be earned per $ per year 
I_{p} 
= 
Interest charges per $ investment in inventory per year 
M 
= 
Trade credit period 
T 
= 
The cycle time 
TVC(T) 
= 
The total relevant cost function per unit time 
T* 
= 
The optimal cycle time of TVC(T) 
Assumptions:
• 
Demand rate is known and constant 
• 
Shortages are not allowed 
• 
Time period is infinite 
• 
The lead time is zero 
• 
If Q<W, i.e., T<W/D, the delayed payment is not permitted. Otherwise,
fixed trade credit period M is permitted. Hence, if Q<W, pay cQ when
the order is received. If Q>=W, pay cQ M time periods after the order is
received. 
• 
During the time the account is not settled, generated sales revenue is
deposited in an interestbearing account. At the end of this period, the
account is settled and we start paying for the interest charges on the items
in stock 
• 
I_{p}>=I_{e} 
The annual total relevant cost consists of the following elements. There are
two cases to occur:
Case 1: Suppose that M>=W/D.
• 
Annual ordering cost = 
• 
Annual stock holding cost (excluding interest charges) = 
• 
There are three cases to occur in cost of interest charges for the items
kept in stock per year. 
• 
0<T<W/D 
Cost of interest charges for the items kept in stock per cycle =
Cost of interest charges for the items kept in stock per year =
In this case, no interest charges are paid for the items kept in stock.
Cost of interest charges for the items kept in stock per cycle =
Cost of interest charges for the items kept in stock per year =
• 
There are three cases to occur in interest earned per year. 
• 
0<T<W/D 
In this case, no interest earned because the delayed payment is not permitted.
From the above arguments, the annual total relevant cost for the retailer can
be expressed as:
TVC(T) = ordering cost + stockholding cost + interest payable  interest earned
We show that the annual total relevant cost, TVC(T), is given by:
Where:
and
Since TVC_{1}(W/D)>TVC_{2}(W/D), TVC_{2}(M) = TVC_{3}(M),
TVC(T) is continuous except T = W/D. Furthermore, we have TVC_{1}(T)>TVC_{2}(T)
for all T>0. Equation 2, 3 and 4
yield
and
Equation 6, 8 and 10 imply
that TVC_{1}(T), TVC_{2}(T) and TVC_{3}(T) are convex
on T > 0. Moreover, we have .
Case 2: Suppose that M<W/D
If M<W/D, Eq. 1a, b, c
will be modified as:
Since TVC_{1}(W/D)>TVC_{3}(W/D), TVC(T) is continuous except
T = W/D. Equation 6 and 10 imply that both
TVC_{1}(T) and TVC_{3}(T) are convex on T>0.
DECISION RULE OF THE OPTIMAL CYCLE TIME T* When M>=W/D
Recall
and
Then
We have T_{2}*>=T_{1}* and T_{3}*>=T_{1}*.
By the convexity of TVC_{i}(T) (i = 1, 2, 3), we see
and
Equation 14ac, 15ac
and 16ac imply that TVC_{i}(T)
is decreasing on (0, T_{i}* ] and increasing on [T_{i}*, ∞)
for all i = 1, 2, 3. Equation 5, 7 and 9
yield that:
and
Furthermore, we let
and
Equation 20, 21 and 22
yield that Δ_{1}>=Δ_{2} and Δ_{3}>=Δ_{2}.
Furthermore, we have
Δ_{1}>0 if and only if
T_{1}*<W/D 
(23) 
Δ_{2}>0 if and only if
T_{2}*<W/D 
(24) 
Δ_{3}>0 if and only if
T_{2}*<M 
(25) 
Δ_{3}>0 if and only if
T_{3}*<M 
(26) 
Therefore, the optimal cycle times can be obtained as follows:
Table 1: 
The optimal cycle time and optimal order quantity using Theorem
1 

Table 2: 
The optimal cycle time and optimal order quantity using Theorem
2 

Theorem 1:
• 
If Δ_{1}>0, Δ_{2}>0 and Δ_{3}>0,
then TVC(T*) = min {TVC_{1}(T_{1}*), TVC_{2}(W/D)
}. Hence T* is T_{1}* or W/D associated with the least cost. 
• 
If Δ_{1}>0, Δ_{2}<=0 and Δ_{3}>0,
then TVC(T*) = TVC_{2}(T_{2}*) and T* = T_{2}*. 
• 
If Δ_{1}>0, Δ_{2}<=0 and Δ_{3}<=0,
then TVC(T*) = min {TVC_{1}(T_{1}*), TVC_{3}(T_{3}*)}.
Hence T* is T_{1}* or T_{3}* associated with the least cost. 
• 
If Δ_{1}<=0, Δ_{2}<=0 and Δ_{3}>0,
then TVC(T*) = TVC_{2}(T_{2}*) and T* = T_{2}*. 
• 
If Δ_{1}<=0, Δ_{2}<=0 and Δ_{3}<=0,
then TVC(T*) = TVC_{3}(T_{3}*) and T* = T_{3}*. 
Proof: Appendix.
DECISION RULE OF THE OPTIMAL CYCLE TIME T* When M<W/D
In this section, we will discuss the condition of M<W/D. Equation
1a, b, c will be reduced to
Equation 5 and 9 yield that
and
Furthermore, we let
and
From Eq. 30 and 31, we can find that Δ_{1}>=Δ_{4}.
Furthermore, we have
Δ_{1}>0 if and only if
T_{1}*<W/D 
(32) 
Δ_{4}>0 if and only if
T_{3}*<W/D 
(33) 
Therefore, the optimal cycle times can be obtained as follows:
Theorem 2:
• 
If Δ_{1}>0 and Δ_{4}>=0, then
TVC(T*) = min {TVC_{1}(T_{1}*), TVC_{3}(W/D)}. Hence
T* is T_{1}* or W/D associated with the least cost. 
• 
If Δ_{1}<=0 and Δ_{4}<0, then TVC(t*) = TVC_{3}(T_{3}*)
and T* = T_{3}*. 
• 
If Δ_{1}>0 and Δ_{4}<0, then TVC(T*) =
min {TVC_{1}(T_{1}*), TVC_{3}(T_{3}*)}.
Hence T* is T_{1}* or T_{3}* associated with the least cost. 
Proof: Appendix.
NUMERICAL EXAMPLES
To illustrate all results, let us apply the proposed method to solve the following
numerical examples. The optimal cycle times and optimal order quantity are shown
in Table 1, 2, respectively.
CONCLUSIONS
The purpose of this paper is to investigate the effect of supplier credit policies
depending on the order quantity within the Economic Order Quantity (EOQ) framework.
Our inventory model generalizes Goyal (1985) and Khouja and Mehrez (1996). Theorem
1 gives the decision rule of the optimal cycle time when M >= W/D. However,
Theorem 2 does the decision rule of the optimal cycle time when M < W/D.
Finally, numerical examples are used to illustrate all results obtained by this
paper.
ACKNOWLEDGMENTS
This study is supported by NSC Taiwan, No. NSC 962221E324007MY3 and CYUT.
APPENDIX
Proof of Theorem 1:
• 
If Δ_{1}>0, Δ_{2}>0 and Δ_{3}>0,
then T_{1}*<W/D, T_{2}*<W/D, T_{2}*<M and
T_{3}*<M. So, we have 
Equation 14a c, 15a c
and 16a c imply that
(i) 
TVC_{3}(T) is increasing on [M, ∞). 
(ii) 
TVC_{2}(T) is increasing on [W/D, M]. 
(iii) 
TVC_{1}(T) is decreasing on (0, T_{1}*] and increasing
on [T_{1}*, W/D). 
Combining (i), (ii) and (iii), we have TVC(T*) = min {TVC _{1}(T _{1}*),
TVC _{2}(W/D)}. Hence, T* is T _{1}* or W/D associated with the
least cost.
• 
If Δ_{1}>0, Δ_{2}<=0 and Δ_{3}>0,
then T_{1}*<W/D, T_{2}*>=W/D, T_{2}*<M and
T_{3}*<M. So, we have 
Equation 14a c, 15a c
and 16a c imply that
(i) 
TVC_{3}(T) is increasing on [M, ∞). 
(ii) 
TVC_{2}(T) is decreasing on [W/D, T_{2}*] and increasing
on [T_{2}*, M]. 
(iii) 
TVC_{1}(T) is decreasing on (0, T_{1}*] and increasing
on [T_{1}*, W/D). 
Combining (i), (ii) and (iii), we have TVC(T*) = min {TVC _{1}(T _{1}*),
TVC _{2}(T _{2}*)}. Since TVC _{1}(T) >TVC _{2}(T)
for all T>0, we obtain TVC _{1}(T _{1}*)>TVC _{2}(T _{2}*).
Hence T* = T _{2}*.
• 
If Δ_{1}>0, Δ_{2}<=0 and Δ_{3}<=0,
then T_{1}*<W/D, T_{2}*>=W/D, T_{2}*>=M and T_{3}*>=M.
So, we have 
Equation 14a c, 15a c
and 16a c imply that
(i) 
TVC_{3}(T) is decreasing on [M, T_{3}*] and
increasing on [T_{3}*, ∞). 
(ii) 
TVC_{2}(T) is decreasing on [W/D, M]. 
(iii) 
TVC_{1}(T) is decreasing on (0, T_{1}*] and increasing
on [T_{1}*, W/D). 
Combining (i), (ii) and (iii), we have TVC(T*) = min {TVC _{1}(T _{1}*),
TVC _{3}(T _{3}*)}. Hence, T* is T _{1}* or T _{3}*
associated with the least cost.
• 
If Δ_{1}<=0, Δ_{2}<=0 and Δ_{3}>0,
then T_{1}*>=W/D, T_{2}*>=W/D, T_{2}*<M and T_{3}*<M.
So, we have 
Equation 14a c, 15a c
and 16a c imply that
(i) 
TVC_{3}(T) is increasing on [M, ∞). 
(ii) 
TVC_{2}(T) is decreasing on [W/D, T_{2}*] and increasing
on [T_{2}*, M]. 
(iii) 
TVC_{1}(T) is decreasing on (0, W/D). 
Combining (i), (ii) and (iii), we have TVC(T*) = TVC _{2}(T _{2}*).
Hence T* = T _{2}*.
• 
If Δ_{1}<=0, Δ_{2}<=0 and Δ_{3}<=0,
then T_{1}*>=W/D, T_{2}*>=W/D, T_{2}*>=M and T_{3}*>=M.
So, we have 
Equation 14a c, 15a c
and 16a c imply that
(i) 
TVC_{3}(T) is decreasing on [M, T_{3}*] and
increasing on [T_{3}*, ∞). 
(ii) 
TVC_{2}(T) is decreasing on [W/D, M]. 
(iii) 
TVC_{1}(T) is decreasing on (0, W/D). 
Combining (i), (ii) and (iii), we have TVC(T*) = TVC_{3}(T_{3}*).
Hence T* = T_{3}*.
Proof of Theorem 2:
• 
If Δ_{1}>0 and Δ_{4}>=0, then
T_{1}*<W/D and T_{3}*<=W/D. So, we have .
Equation 14ac and 16ac imply that 
(i) 
TVC_{3}(T) is increasing on [W/D, ∞). 
(ii) 
TVC_{1}(T) is decreasing on (0, T_{1}*] and increasing
on [T_{1}*, W/D). 
Combining (i) and (ii), we have TVC(T*) = min {TVC_{1}(T_{1}*),
TVC_{3}(W/D)}. Hence T* is T_{1}* or W/D associated with the
least cost.
• 
If Δ_{1}<=0 and Δ_{4}<0, then
T_{1}*>=W/D and T_{3}*>W/D. So, we have Equation
14ac and 16ac imply that 
(i) 
TVC_{3}(T) is decreasing on [W/D, T_{3}*]
and increasing on [T_{3}*, ∞). 
(ii) 
TVC_{1}(T) is decreasing on (0, W/D). 
Combining (i) and (ii), we have TVC(T*) = TVC_{3}(T_{3}*).
Hence T* = T_{3}*.
• 
If Δ_{1}>0 and Δ_{4}<0, then
T_{1}*<W/D and T_{3}*>W/D. So, we have Equation
14ac and 16ac imply that 
(i) 
TVC_{3}(T) is decreasing on [W/D, T_{3}*]
and increasing on [T_{3}*, ∞). 
(ii) 
TVC_{1}(T) is decreasing on (0, T_{1}*] and increasing
on [T_{1}*, W/D). 
Combining (i) and (ii), we have TVC(T*) = min {TVC_{1}(T_{1}*),
TVC_{3}(T_{3}*)}. Hence T* is T_{1}* or T_{3}*
associated with the least cost.

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