
Research Article


An EOQ Model Under Trade Credit Linked to Order Quantity Using Algebraic Method


YungFu Huang



ABSTRACT

In 1985, Goyal investigated the inventory problem under permissible delay in
payments independent of the order quantity. In the present study, the restrictive
assumption of the trade credit independent of the order quantity is relaxed.
The present study investigates the retailer`s
inventory problem under permissible delay in payments dependent of the order
quantity within the economic order quantity (EOQ) framework. In addition, we
adopt the algebraic method to determine the retailer`s
optimal ordering policy under minimizing the annual total relevant cost was
adopted. Then, two theorems are developed to efficiently determine the optimal
cycle time and optimal order quantity for the retailer. Finally, numerical examples
are given to illustrate these theorems obtained in the present study.





INTRODUCTION Several studies have carried out previously treat inventory problems with varying conditions under the consideration of permissible delay in payments. 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 varying rates of deterioration. Liao et al.^{[5]} and Sarker et al.^{[6]} investigated this topic in the presence of 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 a light buyer’s inventory policy under trade credit by the concept of discounted cash flows. 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 was not equal to the purchasing price to modify Goyal’s model^{[1]}. Chung et al.^{[15]} discussed this issue under the assumptions that the selling price is not equal to the purchasing price and different payment rules. 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]} examined 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. Goyal^{[1]} is well known when the inventory systems under conditions of permissible delay in payments and implicitly makes the following assumption. 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. Under this condition, the effect of stimulating the retailer’s demand may be reduced. So, the present study will adopt the following assumption to modify the Goyal’s model^{[1]}. To encourage retailer to order a large quantity, the supplier may give the trade credit period only for a large order quantity. In other words, the retailer requires immediate payment for a small order quantity. This viewpoint can be found in Khouja and Mehrez^{[16]} and Chang et al.^{[21]}. In addition, in previous study 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 study, Grubbström and Erdem^{[22]} and CárdenasBarrón^{[23]} showed that the formulae for the economic order quantity (EOQ) and economic production quality (EPQ) with backlogging could be 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. Therefore, we want to adopt the algebraic method to investigate the effect of trade credit policy depending on the order quantity within the economic order quantity (EOQ) framework. Then, two theorems are developed to efficiently determine the optimal cycle time and optimal order quantity for the retailer. Finally, numerical examples are given to illustrate these theorems obtained in the present study.
Algebraic model formulation 
Notation 
Q 
= 
Order quantity 
D 
= 
Annual demand 
W 
= 
Minimum order quantity at which the trade credit is permitted 
A 
= 
Cost of placing one order 
c 
= 
Unit purchasing price per item 
h 
= 
Unit stock holding cost per item 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 
M 
= 
The trade credit period 
T 
= 
The cycle time 
TRC(T) 
= 
The annual total relevant cost when T>0 
T* 
= 
The optimal cycle time of TRC(T) 
Q* 
= 
The optimal order quantity=DT* 
Assumptions 
1. 
Demand rate is known and constant 
2. 
Shortages are not allowed 
3. 
Time period is infinite 
4. 
Replenishments are instantaneous with a known and constant lead time 
5. 
If Q<W, i.e. T<W/D, the trade credit 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 
6. 
During the time the account is not settled, generated sales revenue is
deposited in an interestbearing account. When T≥M, the account is settled
at T=M, the retailer starts paying for the higher interest charges on the
items in stock. When T≤M, the account is settled at T=M and the buyer
does not need to pay any interest charge 
7. 
I_{p } ≥ I_{e} 
The annual total relevant cost consists of the following elements. Two situations
arise. (I) M≥W/D and (II) M<W/D
Case I: Suppose that M≥W/D 
1. 
Annual ordering cost = A/T 
2. 
Annual stock holding cost (excluding interest charges) = DTh/2 
3. 
There are three cases in terms of cost of interest charges for the items
kept in stock per year.
(I) : 0<T<W/D
In this case, the retailer must pay cDT when the order is received since
the trade credit is not permitted. Therefore, Cost of interest charges for
the items kept in stock per cycle = cI_{p}DT^{2}/2
Cost of interest charges for the items kept in stock per year = cI_{p}DT/2
(ii) : W/D≤T≤M
In this case, the fixed trade credit period M is permitted since Q≥W.
According to assumption (6), no interest charges are paid for the items
kept in stock
(iii) : M≤T
In this case, the fixed trade credit period M is permitted since Q≥W.
According to assumption (6), Cost of interest charges for the items kept
in stock per cycle = cI_{p}D(TM)^{2}/2
Cost of interest charges for the items kept in stock per year = cI_{p}D(TM)^{2}/2T

4. 
There are three cases in terms of interests earned per year.
(I) : 0<T<W/D
In this case, no interest earned since the trade credit is not permitted.
(ii) : W/D≤T≤M
In this case, the fixed trade credit period M is permitted since Q≥W.
According to assumption (6), Interests earned per cycle 





(iii) : M≤T
In this case, the fixed trade credit period M is permitted since Q≥W.
According to assumption (6), 




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. It is shown that the annual total relevant cost, TRC(T), is given by
where and Since TRC_{1}(W/D)>TRC_{2}(W/D), TRC_{2}(M)=TRC_{3}(M), TRC(T) is continuous except T=W/D. Then, we can rewrite
Equation (5) represents that the minimum of TRC_{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 (5) has a minimum value for the optimal
value of T_{1}* reducing TRC_{1}(T) to
Similarly, we can derive TRC_{2}(T) without derivatives as follows:
Equation (8) represents that the minimum of TRC_{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 (8) has a minimum value for the optimal
value of T_{2}* reducing TRC_{2}(T) to
Likewise, we can derive TRC_{3}(T) algebraically as follows:
Equation (11) represents that the minimum of TRC_{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 (11) has a minimum value for the optimal
value of T_{3}* reducing TRC_{3}(T) to
Case II: Suppose that M<W/D: If M<W/D, equations
1(a, b, c) will be modified as
Since TRC_{1}(W/D)>TRC_{3}(W/D), TRC(T) is continuous except T=W/D.
Determination of the optimal cycle time T*
(1) When M≥W/D: Above equation (6) implies that the
optimal value of T for the case of 0<T<W/D, that is 0<T_{1}*<W/D.
We substitute equation (6) into 0<T_{1}*<W/D,
then we can obtain the optimal value of T
Similarly, equation (9) implies that the optimal value of
T for the case of W/D≤T≤M, that is W/D≤T_{2}*≤M. We substitute
equation (9) into W/D≤T_{2}*≤M, then we can
obtain the optimal value of T
Finally, equation (12) implies that the optimal value of
T for the case of T≥M, that is T_{3}*≥M. We substitute equation
(12) into T_{3}*≥M, then we can obtain the optimal value of T
Furthermore, we let and
From equations (17), (18) and (19),
we can easily obtain Δ_{3 }≥ Δ_{2}. In addition,
we know TRC_{1}(W/D)>TRC_{2}(W/D), TRC_{2}(M)=TRC_{3}(M),
TRC(T) is continuous except T=W/D from equations (2), (3)
and (4). Then, we can summarize above arguments and obtain
following results:
Theorem 1 
1. 
If Δ_{1}>0, Δ_{2}>0 and Δ_{3}>0,
then TRC(T*)=min {TRC_{1}(T_{1}*), TRC_{2}(W/D)}.
Hence T* is T_{1}* or W/D associated with the least cost. 
2. 
If Δ_{1}>0, Δ_{2}≤0 and Δ_{3}>0,
then TRC(T*)=min {TRC_{1}(T_{1}*), TRC_{2}(T_{2}*)}.
Hence T* is T_{1}* or T_{2}* associated with the least cost. 
3. 
If Δ_{1}>0, Δ_{2}≤0 and Δ_{3}≤0,
then TRC(T*)=min {TRC_{1}(T_{1}*), TRC_{3} (T_{3}*)}.
Hence T* is T_{1}* or T_{3}* associated with the least cost. 
4. 
If Δ_{1}≤0, Δ_{2}>0 and Δ_{3}>0,
then TRC(T*)=TRC_{2}(W/D) and T*=W/D. 
5. 
If Δ_{1}≤0, Δ_{2}≤0 and Δ_{3}>0,
then TRC(T*)=TRC_{2} (T_{2}*) and T*=T_{2}*. 
6. 
If Δ_{1}≤0, Δ_{2}≤0 and Δ_{3}≤0,
then TRC(T*)=TRC_{3}(T_{3}*) and T*=T_{3}*. 
(2) When M<W/D: In another condition M<W/D, equations
1(a, b, c) will be modified as
Similarly, equation (6) implies that the optimal value of
T for the case of 0<T<W/D, that is 0<T_{1}*<W/D. We substitute
equation (6) into 0<T_{1}*<W/D, then we can
obtain the optimal value of T
Likewise, equation (12) implies that the optimal value of
T for the case of T≥W/D, that is T_{3}*≥W/D. We substitute equation
(12) into T_{3}*≥W/D, then we can obtain the optimal value of
T
Furthermore, we let and
We know TRC_{1}(W/D)>TRC_{3}(W/D), TRC(T) is continuous
except T=W/D from equations (2) and (4).
Then, we can summarize above arguments and obtain following results:
Theorem 2 
1. 
If Δ_{1} > 0 and Δ_{4} > 0,
then TRC(T*)=min {TRC_{1}(T_{1}*),TRC_{3}(W/D)}.
Hence T* is T_{1}* or W/D associated with the least cost. 
2. 
If Δ_{1} ≤ 0 and Δ_{4} ≤ 0, then TRC(T*)=TRC_{3}(T_{3}*)
and T*=T_{3}*. 
3. 
If Δ_{1} > 0 and Δ_{4} ≤ 0, then TRC(T*)=min
{TRC_{1}(T_{1}*),TRC_{3}(T_{3}*)}. Hence
T* is T_{1}* or T_{3}* associated with the least cost. 
4. 
If Δ_{1} ≤ 0 and Δ_{4} > 0, then TRC(T*)=TRC_{3}(W/D)
and T*=W/D. 
Numerical examples: To illustrate all results obtained in the present study, let us apply the proposed method to efficiently solve the following numerical examples.
Table 1: 
The optimal cycle time and optimal order quantity with various
values of W and c 

To study the effects of the minimum order quantity to obtain the permissible delay, W and unit purchasing price per item, c, on the optimal cycle time and optimal order quantity for the retailer derived by the proposed method, we solve the example in Table 1 with various values of W and c. The following inferences can be made based on Table 1. When W is increasing, the optimal cycle time and optimal order quantity for the retailer will not decrease. It implies that the retailer will order more quantity to take the benefits of trade credit as much as possible when the minimum order quantity to obtain the trade credit is higher. When c is increasing, the optimal cycle time and optimal order quantity for the retailer are not increasing. This result implies that the retailer will not order more quantity to take the benefits of the trade credit more frequently. The supplier offers the trade credit policy to stimulate the demand of the retailer in general. For reaching the effect of trade credit, the supplier may give the trade credit period only for a large order quantity. In other words, the retailer requires immediate payment for a small order quantity. This situation is very reasonable in the real business transactions. We develop the retailer’s inventory model in this situation. These results are very helpful to the inventory replenishment decisionmakers. Future study may further incorporate the proposed model into more realistic assumptions, such as allowable shortages, limited storage capacity and a finite rate of replenishment.
ACKNOWLEDGMENTS The author would like to thank that this study is partly supported by NSC Taiwan, project no. NSC 922213E324014.

REFERENCES 
Aggarwal, S.P. and C.K. Jaggi, 1995. Ordering policies of deteriorating items under permissible delay in payments. J. Operational Res. Soci., 46: 658662. Direct Link 
CardenasBarron, L.E., 2001. The economic production quantity (EPQ) with shortage derived algebraically. Int. J. Prod. Econ., 70: 289292. CrossRef 
Chang, C.T., L.Y. Ouyang and J.T. Teng, 2003. An EOQ model for deteriorating items under supplier credits linked to ordering quantity. Applied Math. Modell., 27: 983996. CrossRef  Direct Link 
Chang, H.J. and C.Y. Dye, 2001. An inventory model for deteriorating items with partial backlogging and permissible delay in payments. Int. J. Syst. Sci., 32: 345352. CrossRef  Direct Link 
Chang, H.J., C.H. Hung and C.Y. Dye, 2001. An inventory model for deteriorating items with linear trend demand under the condition of permissible delay in payments. Prod. Plant Control, 12: 274282.
Chang, H.J., C.Y. Dye and B.R. Chuang, 2002. An inventory model for deteriorating items under the condition of permissible delay in payments. Yugoslav J. Operat. Res., 12: 7384.
Chen, M.S. and C.C. Chuang, 1999. An analysis of light buyer`s economic order model under trade credit. AsiaPacific J. Operat. Res., 16: 2334.
Chung, K.J. and Y.F. Huang, 2003. The optimal cycle time for EPQ inventory model under permissible delay in payments. Int. J. Prod. Econ., 84: 307318. CrossRef  Direct Link 
Chung, K.J., 1998. A theorem on the determination of economic order quantity under conditions of permissible delay in payments. Comput. Operat. Res., 25: 4952. Direct Link 
Chung, K.J., Y.F. Huang and C.K. Huang, 2002. The replenishment decision for EOQ inventory model under permissible delay in payments. Opsearch, 39: 327340.
Goyal, S.K., 1985. Economic order quantity under conditions of permissible delay in payments. J. Operat. Res. Soc., 36: 3538.
Grubbstrom, R.W. and A. Erdem, 1999. The EOQ with backlogging derived without derivatives. Int. J. Prod. Econ., 59: 529530. CrossRef 
Huang, Y.F. and K.J. Chung, 2003. Optimal replenishment and payment policies in the EOQ model under cash discount and trade credit. Asia Pac. J. Operat. Res., 20: 177190. Direct Link 
Huang, Y.F., 2003. Optimal retailer`s ordering policies in the EOQ model under trade credit financing. J. Operat. Res. Soc., 54: 10111015.
Hwang, H. and S.W. Shinn, 1997. Retailer's pricing and lot sizing policy for exponentially deteriorating products under the condition of permissible delay in payments. Comput. Operat. Res., 24: 539547. CrossRef 
Jamal, A.A.M., B.R. Sarker and S. Wang, 1997. An ordering policy for deteriorating items with allowable shortage and permissible delay in payments. J. Operational Res. Soc., 48: 826833. Direct Link 
Jamal, A.M.M., B.R. Sarker and S. Wang, 2000. Optimal payment time for a retailer under permitted delay of payment by the wholesaler. Int. J. Prod. Econ., 66: 5966. CrossRef 
Khouja, M. and A. Mehrez, 1996. Optimal inventory policy under different supplier credit policies. J. Manuf. Syst., 15: 334349. Direct Link 
Liao, H.C., C.H. Tsai and C.T. Su, 2000. An inventory model with deteriorating items under inflation when a delay in payments is permissible. Int. J. Prod. Econ., 63: 207214.
Sarker, B.R., A.M.M. Jamal and S. Wang, 2000. Optimal payment time under permissible delay in payment for products with deterioration. Prod. Plant Control, 11: 380390.
Sarker, B.R., A.M.M. Jamal and S. Wnag, 2000. Supply chain model for perishable products under inflation and permissible delay in payment. Comput. Operat. Res., 27: 5975.
Shinn, S.W. and H. Hwang, 2003. Optimal pricing and ordering policies for retailers under ordersizedependent delay in payments. Comput. Operat. Res., 30: 3550.
Teng, J.T., 2002. On the economic order quantity under conditions of permissible delay in payments. J. Operat. Res. Soc., 53: 915918. CrossRef  Direct Link 



