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Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity



Kuang-Hua Hsu, Hung-Fu Huang, Yu-Cheng Tu and Yung-Fu Huang
 
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ABSTRACT

In the previous related studies, the inventory replenishment problems under permissible delay in payments are independent of the order quantity. In this study, the restrictive assumption of the trade credit independent of the order quantity is relaxed. This study discusses the inventory policies under permissible delay in payments when a larger order quantity.

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  How to cite this article:

Kuang-Hua Hsu, Hung-Fu Huang, Yu-Cheng Tu and Yung-Fu Huang, 2008. Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity. Journal of Applied Sciences, 8: 1049-1054.

DOI: 10.3923/jas.2008.1049.1054

URL: https://scialert.net/abstract/?doi=jas.2008.1049.1054

INTRODUCTION

In the classical EOQ model, it is tacitly assumed that the payment of an order is made on the receipt of items by the inventory system. In practice, however, this may not be true. Under certain conditions suppliers are known to offer their customers a delay in paying for an order of a particular commodity. Frequently, suppliers allow credit for some fixed time period for settling the payment for the goods and do not charge any interest from the buyer on the amount owed during this credit period. However, a higher interest is charged if the payment is not settled by the end of the credit period. The existence of credit period serves to reduce the cost of holding stock to the user, because it reduces the amount of capital invested in stock for the duration of the credit period. Recently, several researchers have developed analytical inventory models with consideration of permissible delay in payments.

Goyal (1985) established a single-item inventory model under permissible delay in payments. 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 (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. 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.

This research combines the above both studies by Goyal (1985) and Khouja and Mehrez (1996) to discuss the inventory policies under permissible delay in payments when a larger order quantity. Finally, numerical examples are used to illustrate all theorems in this study.

MODEL FORMULATION

Notation:

Q = Order quantity
D = Annual demand
W = Quantity at which the delay in payments is permitted
A = Cost of placing one order
c = Unit purchasing price
h = Unit stock holding cost per year excluding interest charges
Ie = Interest which can be earned per $ per year
Ip = Interest charges per $ investment in inventory per year
M = Trade credit period
T = The cycle time
TVC(T) = The total relevant cost function per unit time
T* = The optimal cycle time of TVC(T)

Assumptions:

Demand rate is known and constant
Shortages are not allowed
Time period is infinite
The lead time is zero
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. At the end of this period, the account is settled and we start paying for the interest charges on the items in stock
Ip>=Ie

The annual total relevant cost consists of the following elements. There are two cases to occur:

M>=W/D
M<W/D

Case 1: Suppose that M>=W/D.

Annual ordering cost = Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
Annual stock holding cost (excluding interest charges) = Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
There are three cases to occur in cost of interest charges for the items kept in stock per year.
0<T<W/D

Cost of interest charges for the items kept in stock per cycle = Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

Cost of interest charges for the items kept in stock per year = Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

W/D<=T<=M

In this case, no interest charges are paid for the items kept in stock.

M<=T

Cost of interest charges for the items kept in stock per cycle = Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

Cost of interest charges for the items kept in stock per year = Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

There are three cases to occur in interest earned per year.
0<T<W/D

In this case, no interest earned because the delayed payment is not permitted.

W/D<=T<=M

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

M<=T

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

From the above arguments, the annual total relevant cost for the retailer can be expressed as:

TVC(T) = ordering cost + stock-holding cost + interest payable - interest earned

We show that the annual total relevant cost, TVC(T), is given by:

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(1a)
(1b)
(1c)

Where:

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(2)

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(3)

and

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(4)

Since TVC1(W/D)>TVC2(W/D), TVC2(M) = TVC3(M), TVC(T) is continuous except T = W/D. Furthermore, we have TVC1(T)>TVC2(T) for all T>0. Equation 2, 3 and 4 yield

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(5)

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(6)

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(7)

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(8)

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(9)

and

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(10)

Equation 6, 8 and 10 imply that TVC1(T), TVC2(T) and TVC3(T) are convex on T > 0. Moreover, we have Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity.

Case 2: Suppose that M<W/D

If M<W/D, Eq. 1a, b, c will be modified as:

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

Since TVC1(W/D)>TVC3(W/D), TVC(T) is continuous except T = W/D. Equation 6 and 10 imply that both TVC1(T) and TVC3(T) are convex on T>0.

DECISION RULE OF THE OPTIMAL CYCLE TIME T* When M>=W/D

Recall

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(11)

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(12)

and

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(13)

Then

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

We have T2*>=T1* and T3*>=T1*. By the convexity of TVCi(T) (i = 1, 2, 3), we see

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(14a)
(14b)
(14c)

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(15a)
(15b)
(15c)

and

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(16a)
(16b)
(16c)

Equation 14a-c, 15a-c and 16a-c imply that TVCi(T) is decreasing on (0, Ti* ] and increasing on [Ti*, ∞) for all i = 1, 2, 3. Equation 5, 7 and 9 yield that:

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(17)

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(18)

and

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(19)

Furthermore, we let

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(20)

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(21)

and

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(22)

Equation 20, 21 and 22 yield that Δ1>=Δ2 and Δ3>=Δ2. Furthermore, we have

Δ1>0 if and only if T1*<W/D
(23)

Δ2>0 if and only if T2*<W/D
(24)

Δ3>0 if and only if T2*<M
(25)

Δ3>0 if and only if T3*<M
(26)

Therefore, the optimal cycle times can be obtained as follows:

Table 1: The optimal cycle time and optimal order quantity using Theorem 1
Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

Table 2: The optimal cycle time and optimal order quantity using Theorem 2
Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

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*) = TVC2(T2*) and T* = T2*.
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(T2*) and T* = T2*.
If Δ1<=0, Δ2<=0 and Δ3<=0, then TVC(T*) = TVC3(T3*) and T* = T3*.

Proof: Appendix.

DECISION RULE OF THE OPTIMAL CYCLE TIME T* When M<W/D

In this section, we will discuss the condition of M<W/D. Equation 1a, b, c will be reduced to

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(27a)
(27b)

Equation 5 and 9 yield that

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(28)

and

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(29)

Furthermore, we let

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(30)

and

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity
(31)

From Eq. 30 and 31, we can find that Δ1>=Δ4. Furthermore, we have

Δ1>0 if and only if T1*<W/D
(32)

Δ4>0 if and only if T3*<W/D
(33)

Therefore, the optimal cycle times can be obtained as follows:

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.

Proof: Appendix.

NUMERICAL EXAMPLES

To illustrate all results, let us apply the proposed method to solve the following numerical examples. The optimal cycle times and optimal order quantity are shown in Table 1, 2, respectively.

CONCLUSIONS

The purpose of this paper is to investigate the effect of supplier credit policies depending on the order quantity within the Economic Order Quantity (EOQ) framework. Our inventory model generalizes Goyal (1985) and Khouja and Mehrez (1996). Theorem 1 gives the decision rule of the optimal cycle time when M >= W/D. However, Theorem 2 does the decision rule of the optimal cycle time when M < W/D. Finally, numerical examples are used to illustrate all results obtained by this paper.

ACKNOWLEDGMENTS

This study is supported by NSC Taiwan, No. NSC 96-2221-E-324-007-MY3 and CYUT.

APPENDIX

Proof of Theorem 1:

If Δ1>0, Δ2>0 and Δ3>0, then T1*<W/D, T2*<W/D, T2*<M and T3*<M. So, we have

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

Equation 14a-c, 15a-c and 16a-c imply that
(i) TVC3(T) is increasing on [M, ∞).
(ii) TVC2(T) is increasing on [W/D, M].
(iii) TVC1(T) is decreasing on (0, T1*] and increasing on [T1*, W/D).

Combining (i), (ii) and (iii), we have 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 T1*<W/D, T2*>=W/D, T2*<M and T3*<M. So, we have

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

Equation 14a-c, 15a-c and 16a-c imply that
(i) TVC3(T) is increasing on [M, ∞).
(ii) TVC2(T) is decreasing on [W/D, T2*] and increasing on [T2*, M].
(iii) TVC1(T) is decreasing on (0, T1*] and increasing on [T1*, W/D).

Combining (i), (ii) and (iii), we have TVC(T*) = min {TVC1(T1*), TVC2(T2*)}. Since TVC1(T) >TVC2(T) for all T>0, we obtain TVC1(T1*)>TVC2(T2*). Hence T* = T2*.
If Δ1>0, Δ2<=0 and Δ3<=0, then T1*<W/D, T2*>=W/D, T2*>=M and T3*>=M. So, we have

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

Equation 14a-c, 15a-c and 16a-c imply that
(i) TVC3(T) is decreasing on [M, T3*] and increasing on [T3*, ∞).
(ii) TVC2(T) is decreasing on [W/D, M].
(iii) TVC1(T) is decreasing on (0, T1*] and increasing on [T1*, W/D).

Combining (i), (ii) and (iii), we have 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 T1*>=W/D, T2*>=W/D, T2*<M and T3*<M. So, we have

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

Equation 14a-c, 15a-c and 16a-c imply that
(i) TVC3(T) is increasing on [M, ∞).
(ii) TVC2(T) is decreasing on [W/D, T2*] and increasing on [T2*, M].
(iii) TVC1(T) is decreasing on (0, W/D).

Combining (i), (ii) and (iii), we have TVC(T*) = TVC2(T2*). Hence T* = T2*.
If Δ1<=0, Δ2<=0 and Δ3<=0, then T1*>=W/D, T2*>=W/D, T2*>=M and T3*>=M. So, we have

Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity

Equation 14a-c, 15a-c and 16a-c imply that
(i) TVC3(T) is decreasing on [M, T3*] and increasing on [T3*, ∞).
(ii) TVC2(T) is decreasing on [W/D, M].
(iii) TVC1(T) is decreasing on (0, W/D).

Combining (i), (ii) and (iii), we have TVC(T*) = TVC3(T3*). Hence T* = T3*.

Proof of Theorem 2:

If Δ1>0 and Δ4>=0, then T1*<W/D and T3*<=W/D. So, we have Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order Quantity. Equation 14a-c and 16a-c imply that

(i) TVC3(T) is increasing on [W/D, ∞).
(ii) TVC1(T) is decreasing on (0, T1*] and increasing on [T1*, W/D).

Combining (i) and (ii), we have 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 T1*>=W/D and T3*>W/D. So, we have Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order QuantityEquation 14a-c and 16a-c imply that

(i) TVC3(T) is decreasing on [W/D, T3*] and increasing on [T3*, ∞).
(ii) TVC1(T) is decreasing on (0, W/D).

Combining (i) and (ii), we have TVC(T*) = TVC3(T3*). Hence T* = T3*.

If Δ1>0 and Δ4<0, then T1*<W/D and T3*>W/D. So, we have Image for - Optimal Inventory Planning under Permissible Delay in Payments When a Larger Order QuantityEquation 14a-c and 16a-c imply that

(i) TVC3(T) is decreasing on [W/D, T3*] and increasing on [T3*, ∞).
(ii) TVC1(T) is decreasing on (0, T1*] and increasing on [T1*, W/D).

Combining (i) and (ii), we have TVC(T*) = min {TVC1(T1*), TVC3(T3*)}. Hence T* is T1* or T3* associated with the least cost.

REFERENCES
1:  Chung, K.J., 1998. A theorem on the determination of economic order quantity under conditions of permissible delay in payments. Comput. Operat. Res., 25: 49-52.
Direct Link  |  

2:  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: 307-318.
CrossRef  |  Direct Link  |  

3:  Chung, K.J. and Y.F. Huang, 2003. Economic ordering policies for items under permissible delay in payments. J. Inform. Optim. Sci., 24: 329-344.

4:  Goyal, S.K., 1985. Economic order quantity under conditions of permissible delay in payments. J. Operat. Res. Soc., 36: 335-338.
Direct Link  |  

5:  Huang, Y.F., 2003. Optimal retailer's ordering policies in the EOQ model under trade credit financing. J. Operat. Res. Soc., 54: 1011-1015.
Direct Link  |  

6:  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: 177-190.
Direct Link  |  

7:  Huang, Y.F., 2004. Optimal retailer's replenishment policy for the EPQ model under supplier's trade credit policy. Prod. Plan. Control, 15: 27-33.
CrossRef  |  Direct Link  |  

8:  Huang, Y.F., 2006. An inventory model under two levels of trade credit and limited storage space derived without derivatives. Applied Math. Model., 30: 418-436.
CrossRef  |  Direct Link  |  

9:  Huang, Y.F., 2007. Optimal retailer's replenishment decisions in the EPQ model under two levels of trade credit policy. Eur. J. Operat. Res., 176: 1575-1589.
CrossRef  |  Direct Link  |  

10:  Khouja, M. and A. Mehrez, 1996. Optimal inventory policy under different supplier credit policies. J. Manuf. Syst., 15: 334-349.
Direct Link  |  

11:  Teng, J.T., 2002. On the economic order quantity under conditions of permissible delay in payments. J. Operat. Res. Soc., 53: 915-918.
CrossRef  |  Direct Link  |  

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