INTRODUCTION
In a real world, the supplier often makes use of the trade credit policy
to promote their commodities. Goyal (1985) is frequently cited when the
inventory systems under conditions of trade credit are discussed. 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 (2003) 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. Huang (2004) adopted alternative
payment rules 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 (2003) and Huang (2003)^{
}to investigate retailer`s ordering policy.
Goyal (1985) implicitly makes the following assumptions:
• 
Supplier credit policy offered to the retailer where
credit terms are independent of the order quantity. That is, whatever
the order quantity is small or large, the retailer can take the benefits
of payment delay. 
• 
The unit selling price and the unit purchasing price are assumed
to be equal. However, in practice, the unit selling price is not lower
than the unit purchasing price in general. 
According to the above arguments, this article will adopt the following
assumptions to modify the Goyal`s model (1985).
• 
To encourage retailer to order large quantity, the supplier
may give the longer trade credit period for a large order quantity. 
• 
The selling price per unit and the unit purchasing price are not
necessarily equal to match the practical situations. This viewpoint
can be found in Teng (2002). 
Hence, in this study, we not only incorporate the above assumptions (i)
and (ii) to modify the Goyal`s model (1985), but also provide an easytounderstand
and simpletoapply arithmeticgeometric mean inequality approach without
using derivatives to obtain the optimal replenishment cycle time. This
approach could therefore be used easily to introduce the basic inventory
theories to younger students who lack the knowledge of calculus. Finally,
numerical examples are given to illustrate the proposed model and its
optimal solution.
MODEL FORMULATION
Here, we want to develop the inventory model under trade credit period
to take the order quantity into account. The following notation and assumptions
will be used to develop our inventory model.
Notation:
Q 
= 
Order quantity 
D 
= 
Annual demand 
α 
= 
The fraction of trade credit period, 0 ≤α < 1 
A 
= 
Cost of placing one order 
c 
= 
Unit purchasing price 
s 
= 
Unit selling price 
h 
= 
Unit stock holding cost per year excluding interest charges 
I_{p} 
= 
Interest charges per $ investment in inventory per year 
I_{e} 
= 
Interest which can be earned per $ per year 
T 
= 
The cycle time in years 
M 
= 
The trade credit period in years depending on the order quantity
= α T 
TVC(T) 
= 
The annual total relevant cost when T > 0 
T* 
= 
The optimal cycle time of TVC(T) 
Q* 
= 
The optimal order quantity = DT*. 
Assumptions:
• 
Demand rate is known and constant 
• 
Shortages are not allowed 
• 
Time period is infinite 
• 
Replenishments are instantaneous 
• 
The trade credit period is dependent of the order quantity. That
is, M = α T (0 ≤α < 1) 
• 
During the time the account is not settled, generated sales revenue
is deposited in an interestbearing account. When the account is settled,
the retailer pays off all units sold and keeps his/her profits and
starts paying for the higher interest charges on the items in stock 
• 
s ≥ c 
The annual total relevant cost consists of the following elements.
• 
Annual ordering cost = A/T 
• 
Annual stock holding cost (excluding interest charges) = DTh/2 
• 
Cost of interest charges for the items kept in stock per year = 
• 
Interest earned per year = 
From the above arguments, the annual total relevant cost for the retailer
can be expressed as
TVC(T) 
= 
ordering cost + stockholding cost + interest payableinterest
earned 
The research show that the annual total relevant cost, TVC(T), is given
by
THEORETICAL RESULT
We here use an easy and simple arithmeticgeometricmeaninequality approach
(Horn and Johnson, 1985) to obtain the optimal cycle time that minimizes
the annual total relevant cost. The arithmeticgeometric mean inequality
is as follows. For any two real positive numbers, say a and b, the arithmetic
mean a+b/2 is always greater than or equal to the geometric mean .
Namely,
The equation holds only if a = b.
To minimize the annual total relevant cost, we can rewrite TVC(T) in
(1) as follows:
By using the arithmeticgeometric mean inequality, we can easily obtain
that
Table 1: 
The optimal cycle time and optimal order quantity with
various values of α and s 

When the equality
holds, TVC(T) has a minimum. Hence the optimal value of T for TVC(T)
(say T*) can be determined by (4), namely:
Therefore, the optimal order quantity Q* is
NUMERICAL EXAMPLES
To illustrate the theoretical result obtained in this article, let us apply
the proposed method to efficiently solve the following numerical examples. The
optimal cycle time and optimal order quantity are shown in Table
1.
CONCLUSIONS
The purpose of this study adopts an easy and simple arithmeticgeometric
mean inequality approach to investigate the effect of trade credit period
depending on the order quantity and the retailer`s unit selling price
not necessarily equaled to the purchasing price per unit within the Economic
Order Quantity (EOQ) framework. Using this approach presented in this
article, we can find the optimal cycle time and optimal order quantity
without using differential calculus.
From the final numerical examples, we can obtain following managerial
insights. The retailer will order more quantity to take the benefits of
the longer trade credit period as possible when the fraction of trade
credit period is higher. In addition, the retailer will order larger quantity
to take the benefits of the longer trade credit period when the larger
the differences between the unit selling price and the purchasing price
per item.
ACKNOWLEDGMENTS
The authors would like to thank anonymous referees for their valuable
and constructive comments and suggestions that have led to a significant
improvement on an earlier version of this paper. This paper is supported
by NSC Taiwan, no. NSC 962221E324007MY3 and CYUT.