Research Article
An Algebraically Derived Minimal Cost Solution Technique of the EOQ Model Under Conditional Trade Credit
Not AvailableDepartment of Business Administration, Chaoyang University of Technology, Taichung, Taiwan, R.O. C.
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. 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. |
• | At the end of the credit period, the account is settled. The retailer starts paying for higher interest charges on the items in stock and returns money of the remaining balance immediately when the items are sold. |
According to the above arguments, this study will adopt the following assumptions to modify the Goyals (1985) model.
• | To encourage retailer to order large quantity, the supplier may give the trade credit period only for large order quantity. In other words, the retailer requires immediate payment for small order quantity. This viewpoint can be found in Chang et al. (2003). |
• | 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) and Chung et al. (2002). |
• | The retailer needs cash for business transactions. At the end of the credit period, the retailer pays off all units sold and keeps his/her profits for business transactions or other investment use. This viewpoint also can be found in Teng (2002) and Chung et al. (2002). |
Hence, we want to incorporate the above assumptions (i), (ii) and (iii) to modify the Goyals model (1985). In addition, 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 second-order derivatives. The mathematical methodology is difficult to many younger students who lack the knowledge of calculus. In recent papers, Grubbström and Erdem (1999) and Cárdenas-Barrón (2001) showed that the formulae for the EOQ and EPQ with backlogging derived without differential calculus. This algebraic approach could therefore be used easily to introduce the basic inventory theories to younger students who lack the knowledge of calculus. Therefore, we want to adopt the algebraic procedure to investigate the effect of trade credit policy depending on the order quantity and the retailers unit selling price not necessarily equaled to the purchasing price per unit 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.
Algebraic model formulation: Here, we want to develop the inventory model under trade credit to take the order quantity into account. When the order quantity is less than the fixed quantity at which the delay in payments is permitted, the payment for the items must be made immediately. Otherwise, the fixed trade credit period is permitted. The following notation and assumptions will be used to develop our inventory model.
Notation:
Q | = | Order quantity |
D | = | Annual demand |
W | = | Minimum order quantity at which the delay in payments is permitted |
A | = | Cost of placing one order |
c | = | Unit purchasing price |
s | = | Unit selling price |
h | = | Unit stock holding cost per year excluding interest charges |
Ip | = | Interest charges per $ investment in inventory per year |
Ie | = | Interest which can be earned per $ per year |
M | = | The trade credit period in years |
T | = | The cycle time in years |
TVC(T) | = | The annual total variable cost when T>0 |
T* | = | The optimal cycle time of TVC(T) |
Q* | = | The optimal order quantity = DT* |
• | Demand rate is known and constant. |
• | Shortages are not allowed. |
• | Time period is infinite. |
• | Replenishments are instantaneous. |
• | 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 interest-bearing account. When T≥M, the account is settled at T = M, 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. When T≤M, the account is settled at T = M and the retailer does not need to pay any interest charge. |
• | s≥c, Ip≥Ie. |
The annual total variable cost consists of the following elements. There are two cases to occur: (1) M≥W/D and (2) M<W/D.
Case 1: Suppose that M≥W/D.• | Annual ordering cost = A/T |
• | Annual stock holding cost (excluding interest charges) = DTh/2 |
• | There are three sub-cases to occur in cost of interest charges for the items kept in stock per year. |
(i) |
In this case, the retailer must pay cDT when the order is received since the delayed payment is not permitted. Therefore,
Cost of interest charges for the items kept in stock per cycle = cIpDT2/2
Cost of interest charges for the items kept in stock per year = cIpDT/2
(ii) |
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) |
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 = cIpD(TM)2/2
Cost of interest charges for the items kept in stock per year = cIpD(TM)2/2T
• | There are three sub-cases to occur in interest earned per year. |
(i) |
In this case, no interest earned since the delayed payment is not permitted.
(ii) |
In this case, the fixed trade credit period M is permitted since Q ≥ W. According to assumption (6),
(iii) |
In this case, the fixed trade credit period M is permitted since Q ≥ W. According to assumption (6),
Interest earned per cycle =
Interest earned per year =
From the above arguments, the annual total variable cost for the retailer can be expressed as:
We show that the annual total variable cost, TVC(T), is given by:
Where,
(2) |
(3) |
and
(4) |
Since TVC1(W/D) > TVC2(W/D), TVC2(M) = TVC3(M), TVC(T) is continuous except T = W/D.
Then, we can rewrite:
(5) |
From Eq. 5 the minimum of TVC1(T) is obtained when the quadratic non-negative term, depending on T, is equal to zero. The optimum value T1* is:
(6) |
Therefore,
(7) |
Similarly, we can derive TVC2(T) without derivatives as follows:
(8) |
From Eq. 8 the minimum of TVC2(T) is obtained when the quadratic non-negative term, depending on T, is equal to zero. The optimum value T2* is
(9) |
Therefore,
(10) |
Likewise, we can derive TVC3(T) algebraically as follows.
(11) |
From Eq. 11 the minimum of TVC3(T) is obtained when the quadratic non-negative term, depending on T, is equal to zero. The optimum value T3* is:
(12) |
Therefore,
(13) |
Case 2: Suppose that M<W/D.
If M<W/D, equations 1(a, b, c) will be modified as:
Since TVC1(W/D)>TVC3(W/D), TVC(T) is continuous except T = W/D.
Decision rule of the optimal cycle time T*
Case 1: When M≥W/D
Equation 6 gives that the optimal value of T* for the case when 0<T<W/D so that 0< T1*< W/D. We substitute Eq. 6 into 0< T1*< W/D, then we can obtain that 0< T1*< W/D if and only if
(14) |
Similarly, Eq. 9 gives that the optimal value of T* for the case when W/D≤T≤M so that W/D≤T2*≤M. We substitute Eq. 9 into W/D≤T2*≤M, then we can obtain that W/D≤T2*≤M if and only if
(15) |
Finally, Eq. 12 gives that the optimal value of T* for the case when T≥M so that T3*≥M. We substitute Eq. 12 into T3*≥M, then we can obtain that T3*≥M if and only if
(16) |
Furthermore, we let:
(17) |
(18) |
and
(19) |
From Eq. 17, 18 and 19, we can easily obtain Δ3≥Δ2. In addition, we know TVC1(W/D)>TVC2(W/D), TVC2(M) = TVC3(M), TVC(T) is continuous except T = W/D from Eq. 2, 3 and 4. Then, we can summarize above arguments and obtain following results.
Theorem 1• | If Δ1>0, Δ2>0 and Δ3>0, then TVC(T*) = min{TVC1(T1*), TVC2(W/D) }. Hence T* is T1* or W/D associated with the least cost. |
• | If Δ1>0, Δ2≤0 and Δ3>0, then TVC(T*) = min{TVC1(T1*), TVC2(T2*) }. Hence T* is T1* or T2* associated with the least cost. |
• | If Δ1>0, Δ2≤0 and Δ3≤0, then TVC(T*) = min{TVC1(T1*), TVC3(T3*) }. Hence T* is T1* or T3* associated with the least cost. |
• | If Δ1≤0, Δ2>0 and Δ3>0, then TVC(T*) = TVC2(W/D) and T* = W/D. |
• | If Δ1≤0, Δ2≤0 and Δ3>0, then TVC(T*) = TVC2(T2*) and T* = T2*. |
• | If Δ1≤0, Δ2≤0 and Δ3≤0, then TVC(T*) = TVC3(T3*) and T* = T3*. |
Case 2: When M<W/D
In another condition M<W/D, equations 1(a, b, c) will be modified as:
Similarly, Eq. 6 gives that the optimal value of T* for the case when 0<T<W/D so that 0<T1*<W/D. We substitute Eq. 6 into 0<T1*< W/D, then we can obtain that 0<T1*< W/D if and only if
(20) |
Likewise, Eq. 12 gives that the optimal value of T* for the case when T≥W/D so that T3*≥W/D. We substitute Eq. 12 into T3*≥W/D, then we can obtain that T3*≥W/D if and only if
(21) |
Furthermore, we let:
(22) |
and
Table 1: | The optimal cycle time and optimal order quantity with various values of W and s |
Let A = $200/order, D = 5000 units/year, c = $30/unit, h = $5/unit/year, Ip = $0.15/$/year, Ie = $0.05/$/year and M = 0.1 year |
(23) |
We know TVC1(W/D)>TVC3(W/D), TVC(T) is continuous except T = W/D from Eq. (2) and (4). Then, we can summarize above arguments and obtain following results.
Theorem 2• | If Δ1>0 and Δ4>0, then TVC(T*) = min {TVC1(T1*), TVC3(W/D)}. Hence T* is T1* or W/D associated with the least cost. |
• | If Δ1≤0 and Δ4≤0, then TVC(T*) = TVC3(T3*) and T*=T3*. |
• | If Δ1>0 and Δ4≤0, then TVC(T*) = min {TVC1(T1*), TVC3(T3*)}. Hence T* is T1* or T3* associated with the least cost. |
• | If Δ1≤0 and Δ4>0, then TVC(T*) = TVC3(W/D) and T*=W/D. |
Numerical examples: To illustrate all results obtained in this research, let us apply the proposed method to efficiently solve the following numerical examples. The optimal cycle time and optimal order quantity are summarized in Table 1.
The purpose of this study adopts the algebraic procedure to investigate the effect of trade credit policy depending on the order quantity and the retailers 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 study, we can find the optimal cycle time and optimal order quantity without using differential calculus. Two ease-to-use theorems help the retailer accurately and quickly determining the optimal inventory policy under minimizing the annual total variable cost.
From the final numerical examples, we can obtain following managerial insights. The retailer will order more quantity to take the benefits of trade credit as possible when the minimum order quantity to obtain the permissible delay is higher. In addition, the retailer will not order too large quantity to pay higher holding cost for the item under the delayed payment is permitted. And last, the retailer will order less quantity to take the benefits of the trade credit more frequently when the larger the differences between the unit selling price and the purchasing price per item.
This study is partly supported by NSC Taiwan, project No. NSC 94-2416-H-324-003 and we also would like to thank the CYUT to finance this research.