Subscribe Now Subscribe Today
Research Article
 

A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method



Chih-Sung Lai, Yung-Fu Huang and Hung-Fu Huang
 
Facebook Twitter Digg Reddit Linkedin StumbleUpon E-mail
ABSTRACT

In 1996, Khouja and Mehrez investigated the effect of supplier credit policies depending on the order quantity. The authors assumed that the supplier offers the retailer fully permissible delay in payments if the retailer ordered a sufficient quantity. Otherwise, permissible delay in payments would not be permitted. However, in this article, we want to extend this case by assuming that the supplier would offer the retailer partially permissible delay in payments when the retailer ordered a sufficient quantity. Otherwise, permissible delay in payments would not be permitted. Then, we model the retailer`s inventory system and develop three theorems to efficiently determine the optimal lot-sizing decisions for the retailer.

Services
Related Articles in ASCI
Search in Google Scholar
View Citation
Report Citation

 
  How to cite this article:

Chih-Sung Lai, Yung-Fu Huang and Hung-Fu Huang , 2006. A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method. Journal of Applied Sciences, 6: 406-410.

DOI: 10.3923/jas.2006.406.410

URL: https://scialert.net/abstract/?doi=jas.2006.406.410

INTRODUCTION

To encourage customers to order a large quantity, the supplier may give the payments delay only for a large order quantity. In other words, the supplier requires immediate payments for a small order quantity. In 1985[1] considering the inventory replenishment problem under permissible delay in payments independent of the order quantity. In 1996[2] investigated the effect of supplier credit policies depending on the order quantity. Then, Chang et al.[3] and Chung and Liao[4] established an EOQ model for deteriorating items under supplier credits linked to order quantity. In this regard, Chang[5] extended Chung and Liao[4] by taking into account inf lation and finite time horizon. Recently, Chung et al.[6] investigated retailer’s lot-sizing policy under permissible delay in payments depending on the ordering quantity. Chang and Dye[7] investigated an inventory model for deteriorating items with time varying demand and deterioration rates when the credit period depends on the retailer’s ordering quantity.

However, all above published papers dealing with retailer’s lot-sizing policy in the presence of payments delay assumed that the supplier offers the retailer full payments delay when the retailer ordered a sufficient quantity. Otherwise, payments delay would not be permitted. That is, the retailer would obtain 100% payments delay if the retailer ordered a sufficient quantity. We know that this is just an extreme case. In reality, the supplier can relax this extreme case to offer the retailer partial payments delay. That is, the retailer must make a partial payment to the supplier when the order is received. Then, the retailer must pay off the remaining balances at the end of the permissible delay period. In other words, the supplier requires immediate full payments for a small order quantity. That is, payments delay would not be permitted.

Under this condition, we model the retailer’s inventory system and develop three theorems for efficiently determination the optimal lot-sizing decisions for the retailer.

Model formulation: The following notation and assumptions are used throughout this article.

Notation

D = demand rate per year
W = quantity at which the partially delay payments permitted per order
A = cost of placing one order
c = unit purchasing price
h = unit stock holding cost per year excluding interest charges
α = the fraction of the total amount owed payable at the time of placing an order, 0<α≤1
Ie = interest earned per $ per year
Ik = interest charges per $ investment in inventory per year
M = the length of the payments delay period in years
T = the cycle time in years
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

Demand rate is known and constant.
Shortages are not allowed.
Time horizon is infinite.
Replenishments are instantaneous.
Ik≥Ie.
If Q<W, i.e. T<W/D, the payments delay would not be permitted. Otherwise, partially delayed payment is permitted. Hence, if Q$W, the retailer must make a partial payment, αcDT, to the supplier. Then the retailer must pay off the remaining balances, (1-α)cDT, at the end of the trade credit period. Otherwise, as the order is filled, the retailer must pay the full payments immediately to the supplier.
During the time period that the account is not settled, generated sales revenue is deposited in an interest-bearing account.

The model: The annual total relevant cost consists of the following elements. There are three situations to occur: (I) M$W/D, (II) M<W/D#M/α and (III) M/α<W/D.

Case I: Suppose that M≥W/D

Annual ordering cost = A/T
Annual stock holding cost (excluding interest charges) = DTh/2.
From assumptions (6) and (7), there are four sub-cases in terms of annual opportunity cost of the capital.

Sub-case 1: M/α≤T

Annual opportunity cost of the capital

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

Sub-case 2: M≤T<M/α

Annual opportunity cost of the capital

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

Sub-case 3: W/D≤T<M

Annual opportunity cost of the capital

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

Sub-case 4: 0 <T<W/D

Annual opportunity cost of the capital = Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

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

Annual total relevant cost = ordering cost + stock-holding cost + opportunity cost of the capital.

We show that the annual total relevant cost is given by:

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

Where:

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(2)

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(3)

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(4)

and

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(5)

Since, Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method TRC3(W/D)< TRC4(W/D), TRC (T) is continuous except at T = W/D. All TRC1 (T), TRC2 (T), TRC3 (T) , TRC4 (T) and TRC (T) are defined on T>0. From Eq. 2-5, we can obtain TRC4 (T) >TRC1 (T), TRC4 (T) > TRC3 (T) for all T > 0 and TRC4 (T) > TRC2 (T) for M < T < M/α. That is, TRC4 (T) will be higher than all TRC1 (T), TRC2 (T) and TRC3 (T) on suitable domain. From now on, we can neglect TRC4 (T) when we want to develop the efficient procedure to determine the optimal lot-sizing decisions for the retailer.

Then, we can rewrite

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(6)

From Eq. 6 the minimum of TRC1 (T) is obtained when the quadratic non-negative term, depending on T, is equal to zero. The optimum value T1* is

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(7)

Therefore,

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(8)

Similarly, we can derive TRC2 (T) without derivatives as follows:

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(9)

From Eq. 9 the minimum of TRC2 (T) is obtained when the quadratic non-negative term, depending on T, is equal to zero. The optimum value T2* is

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(10)

Therefore,

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(11)

Likewise, we can derive TRC3 (T) algebraically as follows:

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(12)

From Eq. 12 the minimum of TRC3 (T) is obtained when the quadratic non-negative term, depending on T, is equal to zero. The optimum value T3* is

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(13)

Therefore,

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(14)

Case II: Suppose that M<W/D≤M/α: If M<W/D≤M/α, Eq. 1(a, b, c, d) will be modified as

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

Since TRC1 (M/α) = TRC2 (M/α), TRC2 (W/D)<TRC4 (W/D), TRC (T) is continuous except at T = W/D. All TRC1 (T), TRC2 (T), TRC4 (T) and TRC (T) are defined on T>0. Similar to Case I discussion, TRC4 (T) will be higher than both TRC1 (T) and TRC2 (T) on suitable domain. Hence, we can neglect TRC4 (T) when we want to develop the efficient procedure to determine the optimal lot-sizing decisions for the retailer.

Case III: Suppose that M/α<W/D: If M/α<W/D, Eq. 1 (a, b, c, d) will be modified as

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

Since TRC1 (W/D) < TRC4 (W/D), TRC (T) is continuous except at T = W/D. All TRC1 (T), TRC4 (T) and TRC (T) are defined on T>0. We can easily obtain that TRC4 (T)>TRC1 (T) for all T>0. Hence, we can neglect TRC4 (T) when we want to develop the efficient procedure to determine the optimal lot-sizing decisions for the retailer.

Determination of the optimal cycle time T*
Case I: Suppose that M≥W/D:
Equation 7 gives that the optimal value of T* for the case when T≥M/α so that T1*≥M/α. We substitute Eq. 7 into T1*≥M/α, then we can obtain that:

M/α ≤ T1* if and only if

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

Similar disscussion, we can obain following results:

M≤T2*< M/α if and only if

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

and

if and only if,

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

W/D≤T3*<M if and only if

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

and

if and only if

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

Furthermore, we let

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(17)

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(18)

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(19)

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(20)

and

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(21)

From Eq. 17-21, we can easily obtain Δ21, Δ2≥Δ3, Δ43 and Δ4≥Δ5. Summarized above arguments, the optimal cycle time T* can be obtained as follows.

Theorem 1

(A) If Δ1>0, Δ3>0 and Δ5>0, then TRC (T*) = TRC3 (W/D). Hence T* is W/D.
(B) If Δ1>0 Δ3>0 and Δ5≤0, then TRC (T*) = TRC3 (T3*). Hence T* is T3*.
(C) If Δ1>0, Δ3≤0 and Δ5>0, then TRC (T*) = TRC2 (T2*). Hence T* is T2*.
(D) If Δ1>0, Δ3≤0, Δ4>0 and Δ5≤0, then TRC ( T*) = min{TRC2 (T2*), TRC3 (T3*)}. Hence T* is T2* or T3* associated with the least cost.
(E) If Δ1>0 and Δ4≤0, then TRC (T*) = TRC2 (T2*). Hence T* is T2*.
(F) If Δ1≤0, Δ3>0 and Δ5>0, then TRC ( T*) = TRC1 (T1*). Hence T* is T1*.
(G) If Δ1≤0, Δ3>0 and Δ5≤0, then TRC (T*) = min{TRC1 (T1*), TRC3 (T3*)}. Hence T* is T1* or T3* associated with the least cost.
(H) If Δ1≤0, Δ2>0, Δ3≤0 and Δ5>0, then TRC( T*) = min{TRC1 (T1*), TRC2 (T2*)}. Hence T* is T1* or T2* associated with the least cost.
(I) If Δ1≤0, Δ2>0, Δ3≤0, Δ4>0 and Δ5≤0, then TRC (T*) = min{TRC1 (T1*), TRC2 (T2*), TRC3 (T3*)}. Hence T* is T1*, T2* or T3* associated with the least cost.
(J) If Δ1≤0, Δ2>0 and Δ4≤0, then TRC (T*) = min{TRC1 (T1*), TRC2 (T2*)}. Hence T* is T1* or T2* associated with the least cost.
(K) If Δ2≤0 and Δ5>0, then TRC (T*) = TRC1 (T1*). Hence T* is T1*.
(L) If Δ2≤0, Δ4>0 and Δ5≤0, then TRC (T*) = min{TRC1(T1*), TRC3 (T3*)}. Hence T* is T1* or T3* associated with the least cost.
(M) If Δ2≤0 and Δ4≤0, then TRC (T*) = TRC1(T1*). Hence T* is T1*.

Case II: Suppose that M<W/D≤M/α: If M<W/D≤M/α, Eq. 1 (a, b, c and d) will be modified as Eq. 15 (a, b and c). Similar to above Case I discussion, we can obtain following results:

M/α≤T1* if and only if

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

W/D≤T2*<M/α if and only if

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

and

if and only if

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

Furthermore, we let

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(22)

From Eq. 17, 18 and 22, we can easily obtain Δ21 and Δ26. Summarized above arguments, the optimal cycle time T* can be obtained as follows:

Theorem 2

(A) If Δ1>0 and Δ6>0, then TRC (T*) = TRC2 (W/D). Hence T* is W/D.
(B) If Δ1>0 and Δ6≤0, then TRC (T*) = TRC2 (T2*). Hence T* is T2*.
(C) If Δ1≤0 and Δ6>0, then TRC (T*) = TRC1 (T1*). Hence T* is T1*.
(D) If Δ1≤0, Δ2>0 and Δ6≤0, then TRC (T*) = min{TRC1 (T1*), TRC2 (T2*)}. Hence T* is T1* or T2* associated with the least cost.
(E) If Δ2≤0, then TRC (T*) = TRC1(T1*). Hence T* is T1*.

Case III: Suppose that M/α<W/D: If M/α<W/D, Eq. 1 (a, b, c and d) will be modified as Eq. 16 (a and b). Similar to above Case I and Case II discussions, we can obtain following results:

W/D≤T1* if and only if

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method

Furthermore, we let

Image for - A Deterministic Inventory Model under Quantity-depended Payments Delay Policy Using Algebraic Method
(23)

Summarized above arguments, the optimal cycle time T* can be obtained as follows:

Theorem 3

(A) If Δ7>0, then TRC (T*) = TRC1(W/D). Hence T* is W/D.
(B) If Δ7≤0, then TRC (T*) = TRC1 (T1*). Hence T* is T1*.

CONCLUSIONS

The assumption in previously published results that the full payments delay is permitted if the retailer ordered a sufficient quantity. We know 100% payments delay is just an extreme case. This article amends the assumption of the full payments delay to partial payments delay when the retailer ordered a sufficient quantity. We adopt the assumption to model the retailer’s inventory problem. In addition, we establish three easy-to-use theorems to help the retailer to find the optimal lot-sizing policy.

ACKNOWLEDGEMENTS

This article 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 article.

REFERENCES

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

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

3:  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: 983-996.
CrossRef  |  Direct Link  |  

4:  Chung, K.J. and J.J. Liao, 2004. Lot-sizing decisions under trade credit depending on the ordering quantity. Comp. Oper. Res., 31: 909-928.
Direct Link  |  

5:  Chang, C.T., 2004. An EOQ model with deteriorating items under inflation when supplier credits linked to order quantity. Int. J. Prod. Econ., 88: 307-316.
CrossRef  |  

6:  Chung, K.J., S.K. Goyal and Y.F. Huang, 2005. The optimal inventory policies under permissible delay in payments depending on the ordering quantity. Int. J. Prod. Econ., 95: 203-213.
Direct Link  |  

7:  Chang, H.J. and C.Y. Dye, 2005. The effect of credit period on the optimal lot size for deteriorating items with time varying demand and deterioration rates. Asia-Pacific J. Operat. Res., 22: 211-227.
Direct Link  |  

©  2021 Science Alert. All Rights Reserved