INTRODUCTION
In most business transactions, the supplier will allow a specified credit period to the retailer for payment without penalty to stimulate the demand of his/her products. 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 trade credit period. Recently, several papers have appeared in the literature that treat inventory problems with varying conditions under the consideration of permissible delay in payments. Some of the prominent papers are discussed below.
Goyal^{[1]} established a singleitem inventory model under permissible delay in payments. Chung^{[2]} developed an alternative approach to determine the economic order quantity under condition of permissible delay in payments. Aggarwal and Jaggi^{[3]} considered the inventory model with an exponential deterioration rate under the condition of permissible delay in payments. Chang et al.^{[4]} extended this issue to the varying rate of deterioration. Liao et al.^{[5]} and Sarker et al.^{[6]} investigated this topic with inflation. Jamal et al.^{[7]} and Chang and Dye^{[8]} extended this issue with allowable shortage. Chang et al.^{[9]} extended this issue with linear trend demand. Chen and Chuang^{[10]} investigated light buyer’s inventory policy under trade credit by the concept of discounted cash flow. Hwang and Shinn^{[11]} modeled an inventory system for retailer’s pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payment. Jamal et al.^{[12]} and Sarker et al.^{[13]} addressed the optimal payment time under permissible delay in payment with deterioration. Teng^{[14]} assumed that the selling price not equal to the purchasing price to modify Goyal’s model^{[1]}. Chung et al.^{[15]} discussed this issue under the selling price not equal to the purchasing price and different payment rule. Khouja and Mehrez^{[16]} investigated the effect of four different supplier credit policies on the optimal order quantity within the EOQ framework. Shinn and Hwang^{[17]} determined the retailer’s optimal price and order size simultaneously under the condition of ordersizedependent delay in payments. They assumed that the length of the credit period is a function of the retailer’s order size and also the demand rate is a function of the selling price. Chung and Huang^{[18]} extended this problem within the EPQ framework and developed an efficient procedure to determine the retailer’s optimal ordering policy. Huang^{[19]} extended this issue under two levels of trade credit and developed an efficient solution procedure to determine the optimal lotsizing policy of the retailer. Huang and Chung^{[20]} extended Goyal’s model^{[1]} to cash discount policy for early payment.
In previous all published papers 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 papers,
Grubbström and Erdem^{[21]} and CárdenasBarrón^{[22]}
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 EOQ and EPQ concepts to students
that lack knowledge of derivatives, simultaneous equations and the procedure
to construct and examine the Hessian matrix. This algebraic approach could therefore
be used easily to introduce the basic inventory theories to younger students
who lack the knowledge of calculus.
Goyal^{[1]} is frequently cited when the inventory systems under conditions of permissible delay in payments are discussed. Then, Teng^{[14]} assumed that the selling price not equal to the purchasing price to modify Goyal’s model^{[1]}. Therefore, this paper tries to investigate Goyal’s model^{[1]} and Teng’s model^{[14]} using the algebraic method to determine the optimal cycle time. In addition, we summarize some previously published results of other researchers to determine the optimal cycle time.
Goyal’s model: For convenience, most notation and assumptions similar
to Goyal^{[1]} will be used in the present study.
Notation 
D 
= 
annual demand 
S 
= 
cost of placing one order 
c 
= 
unit purchasing price per item 
h 
= 
unit stock holding cost per item per year excluding interest charges 
I_{d} 
= 
interest which can be earned per $ per year 
I_{c} 
= 
interest charges per $ investment in inventory per year 
t 
= 
permissible delay period in years 
T 
= 
the cycle time in years 
Z(T) 
= 
the annual total relevant cost 
T* 
= 
the optimal cycle time of Z(T). 
Assumptions 
1. 
Demand rate is known and constant. 
2. 
Shortages are not allowed. 
3. 
Time period is infinite. 
4. 
I_{c}≥I_{d}. 
5. 
During the time the account is not settled, generated sales revenue is
deposited in an interestbearing account. When T≥t, the account is settled
at T=t and we start paying for the interest charges on the items in stock.
When T≤t, the account is settled at T=t and we do not need to pay any
interest charge. 
Algebraic modeling: The annual total relevant cost for the retailer can be expressed as Z(T)=ordering cost+stockholding cost+interest payableinterest earned. Based on the above notation and assumptions, Goyal^{[1]} showed that:
where
and
Since Z_{1}(t)=Z_{2}(t), Z(T) is continuous for T>0.
Then, we can rewrite
Equation (4) represents that the minimum of Z_{1}(T)
is obtained when the quadratic nonnegative term, depending on T, is made equal
to zero. Therefore, the optimum value T_{1}* is:
Therefore, equation (4) has a minimum value for the optimal
value of T_{1}* reducing Z_{1}(T) to
Similarly, we can derive Z_{2}(T) without derivatives as follows:
Equation (7) represents that the minimum of Z_{2}(T)
is obtained when the quadratic nonnegative term, depending on T, is made equal
to zero. Therefore, the optimum value T_{2}* is
Therefore, equation (7) has a minimum value for the optimal
value of T_{2}* reducing Z_{2}(T) to
Decision rule of the optimal cycle time T*: From above equation
(5) implies that the optimal value of T for the case of T≥t, that is
T_{1}*> t. We substitute equation (5) into T_{1}*>
t, then we can obtain that
Similarly, equation (8) implies that the optimal value of
T for the case of T≤t, that is T_{2}*< t. We substitute equation
(8) into T_{2}*< t, then we can obtain that
Of course, if T_{1}*= T_{2}*= t, we can obtain that
Furthermore, we let Δ=2S+Dt^{2}(h+cI_{d}). Then, we can
obtain following results.
Theorem 1 
A. 
If Δ>0, then T*=T_{2}*. 
B. 
If Δ<0, then T*=T_{1}*. 
C. 
If Δ=0, then T*= T_{1}*= T_{2}* =t. 
Theorem 1 has been discussed in Chung^{[2]}.
Teng’s model: We discuss Teng’s model^{[14]} which
modified the Goyal’s model^{[1]} by considering the difference
between unit price and unit cost. So, Teng^{[14]} define the extra notation
and modified above assumption (5) as follows:
Extra notation 
p 
= 
unit selling price per item, p>c 

= 
the optimal cycle time of 
Amended assumption: During the time the account is not settled, generated sales revenue is deposited in an interestbearing account. At the end of this period, the retailer pays off all units sold, keeps profits and starts paying for the interest charges on the items in stocks.
Algebraic modeling: Based on the above arguments and easy interpreting, we use following notation to express Teng’s model^{[14]}.
where
and
Z_{3}(t) and Z_{4}(t) are similar to Z_{1}(t) and Z_{2}(t)
in Teng ^{14]}, respectively. Since Z_{3}(t)=Z_{4}(t),
is continuous for T>0. Then, we can rewrite
Equation (16) represents that the minimum of Z_{3}(T)
is obtained when the quadratic nonnegative term, depending on T, is made equal
to zero. Therefore, the optimum value T_{3}* is
Therefore, equation (16) has a minimum value for the optimal
value of T_{3}* reducing Z_{3}(T) to
Similarly, we can derive Z_{4}(T) without derivatives as follows:
Equation (19) represents that the minimum of Z_{4}(T)
is obtained when the quadratic nonnegative term, depending on T, is made equal
to zero. Therefore, the optimum value T_{4}* is
Therefore, equation (19) has a minimum value for the optimal
value of T_{4}* reducing Z_{4}(T) to
T_{3}* and T_{4}* are similar to T_{1}* and T_{2}* in Teng ^{[14]}, respectively.
Decision rule of the optimal cycle time :
From above equation (17) implies that the optimal value of
T for the case of T≥t, that is T_{3}*>t. We substitute equation
(17) into T_{3}*>t, then we can obtain that
Similarly, equation (20) implies that the optimal value
of T for the case of T≤t, that is T_{4}*< t. We substitute equation
(20) into T_{4}*< t, then we can obtain that
Of course, if T_{3}*= T_{4}*= t, we can obtain that
From above arguments, we can summarize following results.
Theorem 2 has been discussed in Teng^{[14]}.
Furthermore, we let Δ_{l}=2S+Dt^{2}(h+pI_{d}).
Then, we can modify Theorem 2 to following results.
Theorem 3 
A. 
If Δ_{l}>0, then =T*_{4} 
B. 
If Δ_{l}<0, then =T*_{3} 
C. 
If Δ_{l}=0, then =T*_{3}=T*_{4}=t. 
Theorem 3 is an effective procedure to find the optimal cycle time
by easy judgment Δ_{1}. If p=c, the Δ_{1} will be
equal to Δ in Theorem 1. Then Theorem 3 will be reduced to Theorem 1. So,
Theorem 1 is a special case of Theorem 3.
In reallife business transactions, the supplier offers the trade credit policy
to stimulate the demand of the retailer. The present study adopts the easy algebraic
procedure to reinvestigate Goyal’s model^{[1]} and Teng’s
model^{[14]} to find the optimal cycle time under permissible delay
in payments. Using the algebraic approach is a more accessible approach to ease
the learning of basic inventory theories for younger students who lack the knowledge
of differential calculus.
The proposed model can be extended in several ways. For instance, we may generalize the model to allow for shortages, quantity discounts, time value of money, finite time horizon, finite replenishment rate and others.
ACKNOWLEDGMENTS
The author would like to thank to anonymous referees for their valuable and constructive comments that have led to a significant improvement in the original paper. Furthermore, we want to thank that this study is partly supported by NSC Taiwan, project no. NSC 922213E324014.