INTRODUCTION
The traditional Economic Order Quantity (EOQ) model assumes that retailer’s capitals are adequate and must pay for the items as soon as the items are received. However, this may not be true. In practice, the supplier will offer the retailer a delay period, that is the trade credit period, in paying for the amount of purchase. Before the end of the trade credit period, the retailer can sell the goods and accumulate revenue and earn interest. A higher interest is charged if the payment is not settled by the end of the trade credit period. All previously published models discussed delay permitted assumed that the supplier would offer the retailer a delay period but the retailer would not offer the delay period to his customer. That is one level of delay permitted. Huang (2003) modified this assumption to assume that the retailer will adopt the delay permitted policy to stimulate his customer demand to develop the retailer’s replenishment model. That is two levels of delay permitted. This new viewpoint is more matched reallife situations in the supply chain model. Many studies have appeared in the literature that treat inventory problems with varying conditions under one level of delay permitted. Some of the prominent studies are discussed here.
Goyal (1985) established a singleitem inventory model under permissible delay in payments. 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. Chung et al. (2005) investigated retailer’s lotsizing policy under permissible delay in payments depending on the ordering quantity. 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.
Recently, Huang et al. (2007) extended Huang’s (2003) model to
investigate the situation in which the retailer has a powerful position. That
is, they assume that the retailer can obtain the full trade credit offered by
the suppliers and the retailer just offers the partial trade credit to his customers.
Under these conditions, the retailer can obtain the most benefits. In practice,
this model setting is more realistic. In the present study, we try to use the
more easily algebraic method to find the optimal solution in Huang et al.
(2007) model. In previous all published studies which have been derived using
differential calculus to find the optimal solution and the need to prove optimality
condition with secondorder derivatives. The mathematical methodology is difficult
to many younger students who lack the knowledge of calculus. In recent studies,
CárdenasBarrón (2001) and Grubbström and Erdem (1999) showed
that the formulae for the EOQ and EPQ with backlogging derived without differential
calculus. They mentioned that this approach must be considered as a pedagogical
advantage for explaining the basic inventory concepts to students that lack
knowledge of derivatives, simultaneous equations and the procedure to construct
and examine the Hessian matrix. This algebraic approach could be used easily
to introduce the basic inventory theories to younger students who lack the knowledge
of calculus.
Under these conditions, we model the retailer’s inventory decisionmaking as a cost minimization problem to determine the retailer’s optimal ordering policies.
MODEL FORMULATION
The following notation and assumptions will be used throughout the study:
Notation:
D 
= 
Demand rate per year 
A 
= 
Ordering cost per order 
c 
= 
Unit purchasing price 
h 
= 
Unit stock holding cost per year excluding interest charges 
α 
= 
The customer’s fraction of the total amount payable at the time of
placing an order within the delay period to the retailer, 0 ≥ α
≥ 1 
I_{e} 
= 
Interest earned per $ per year 
I_{k} 
= 
Interest charged per $ in stocks per year by the supplier 
M 
= 
The retailer’s trade credit period as measured by years offered by
the supplier 
N 
= 
The customer’s trade credit period as measured by years offered by
the retailer 
T 
= 
The cycle time in years 
TRC(T) 
= 
The annual total relevant cost, which is a function of T 
T* 
= 
The optimal cycle time of TRC(T) 
Q* 
= 
The optimal order quantity, also defined by DT* 
Assumptions:
• 
Demand rate, D, is known and constant 
• 
Shortages are not allowed 
• 
Time horizon is infinite 
• 
Replenishments are instantaneous 
• 
I_{k} ≤ I_{e}, M ≤ N 
• 
Since the supplier offers the full trade credit to the retailer. When
T ≤ M, the account is settled at T = M and the retailer starts paying
for the interest charges on the items in stock with rate I_{k}.
When T ≥ M, the account is settled at T = M and the retailer does not
need to pay any interest charge 
• 
Since the retailer just offers the partial trade credit to his customers.
Hence, his customers must make a partial payment to the retailer when the
item is received. Then his customers must pay off the remaining balance
at the end of the trade credit period offered by the retailer. That is,
the retailer can accumulate interest from his customer partial payment on
[0, N] and from the total amount of payment on [N, M] with rate I_{e}. 
The annual total relevant cost consists of the following elements.
• 
Annual ordering cost = 
• 
Annual stock holding cost (excluding interest charges) = 
• 
According to assumption (6), there are three cases to consider in costs
of interest charges for the items kept in stock per year. 
Case 1: M ≥ T
Annual interest payable =cI_{k} D(TM)^{2}/2T
Case 2: N ≥ T ≥ M
In this case, annual interest payable = 0
Case 3: T ≥ N
Similar to Case 2, annual interest payable = 0
According to assumption (7), there are three cases to consider in interest earned per year.
Case 1: M ≥ T, as shown in Fig. 1.
Case 2: N ≥ T ≥ M, as shown in Fig. 2.
Case 3: T ≥ N, as shown in Fig. 3.
From the above arguments, the annual total relevant cost for the retailer can
be expressed as:
TRC(T) = ordering cost + stockholding cost + interest payableinterest earned.
Where:
and
Since TRC_{1}(M) = TRC_{2}(M) and TRC_{2}(N) = TRC_{3}(N),
TRC(T) is continuous and welldefined. All TRC_{1}(T), TRC_{2}(T),
TRC_{3}(T) and TRC(T) are defined on T > 0.
Then, we can rewrite

Fig. 1: 
The total amount of interest earned when M ≥ T 

Fig. 2: 
The total amount of interest earned when N ≥ T ≥ M 

Fig. 3: 
The total amount of interest earned when T ≥ N 
From Eq. 5 the minimum of TRC_{1}(T) is obtained
when the quadratic nonnegative term, depending on T, is equal to zero. The
optimum value T_{1}* is
Therefore,
Similarly, we can derive TRC_{2}(T) without derivatives as follows.
From Eq. 8 the minimum of TRC_{2}(T) is obtained
when the quadratic nonnegative term, depending on T, is equal to zero. The
optimum value T_{2}* is
Therefore,
Likewise, we can derive TRC_{3}(T) algebraically as follows.
From Eq. 11 the minimum of TRC_{3}(T) is obtained
when the quadratic nonnegative term, depending on T, is equal to zero. The
optimum value T_{3}* is
Therefore,
DECISION RULE OF THE OPTIMAL CYCLE TIME T*
From Eq. 6 the optimal value of T for the case of T ≤
M is T_{1}* ≤ M. We can substitute Eq. 6 into T_{1}*
≤ M to obtain the optimal value of T
Similarly, from Eq. 9 the optimal value of T for the case
of N ≥ T ≥ M is N ≥ T_{2}* ≥ M. We can substitute Eq.
9 into N ≥ T_{2}* ≥ M to obtain the optimal value of T:
if and only if Δ_{1}= 2A+DM^{2}(h+cI_{e})cD(1α)N^{2}I_{e}
≤ 0 and
Finally, from Eq. 12 the optimal value of T for the case
of T ≥ N is T_{3}* ≥ N. We can substitute Eq. 12
into T_{3}* ≥ N to obtain the optimal value of T:
From above arguments, we see Δ_{1} ≤ Δ_{2} and
summarize following results.
Theorem 1:
(A) 
If Δ_{2} ≤ 0, then TRC(T*) = TRC(T_{3}*)
and T* = T_{3}*. 
(B) 
If Δ_{1} > 0 and Δ_{2} < 0, then TRC(T*)
= TRC(T_{2}*) and T* = T_{2}*. 
(C) 
If Δ_{1} ≥ 0, then TRC(T*) = TRC(T_{1}*) and T*
= T_{1}*. 
Theorem 1 immediately determines the optimal cycle time T* after computing
for the numbers Δ_{1} and Δ_{2}. Theorem 1 is an efficient
solution procedure.
NUMERICAL EXAMPLES
To illustrate the result developed in this study, let us apply the proposed
method to solve the following numerical examples. For convenience, the numerical
values of the parameters are selected randomly. The optimal solutions for different
parameters of α, N and c are shown in Table 1.
Table 1: 
Optimal solutions under various parametric values 

LetA = $80/order,D = 2000 units/year,h = $7/unit/year, I_{k}
= $0.15/$/year,I_{e} = $0.13/$/year,M = 0.1 year 
CONCLUSIONS
This study further relaxes the assumption of the twolevel trade credit policy in the previously published works to investigate the inventory problem in which the retailer maintains a powerful position derived without derivatives. Theorem 1 helps the retailer accurately and speedily determining the optimal ordering policy after computing for the numbers Δ_{1} and Δ_{2}. Finally, numerical examples are given to illustrate the result developed in this study.
ACKNOWLEDGMENTS
This study is supported by NSC Taiwan, No. NSC 962221E324007MY3 and CYUT.