**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 |

I_{e} |
= |
interest earned per $ per year |

I_{k} |
= |
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. |

• |
I_{k}≥I_{e}. |

• |
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

**Sub-case 2: M≤T<M/α**

Annual opportunity cost of the capital

**Sub-case 3: W/D≤T<M**

Annual opportunity cost of the capital

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

Annual opportunity cost of the capital =

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:

Where:

and

Since,
TRC_{3}(W/D)< TRC_{4}(W/D), TRC (T) is continuous except
at T = W/D. All TRC_{1} (T), TRC_{2} (T), TRC_{3} (T)
, TRC_{4} (T) and TRC (T) are defined on T>0. From Eq.
2-5, we can obtain TRC_{4} (T) >TRC_{1}
(T), TRC_{4} (T) > TRC_{3} (T) for all T > 0 and TRC_{4}
(T) > TRC_{2} (T) for M < T < M/α. That is, TRC_{4}
(T) will be higher than all TRC_{1} (T), TRC_{2} (T) and TRC_{3}
(T) on suitable domain. From now on, we can neglect TRC_{4} (T) when
we want to develop the efficient procedure to determine the optimal lot-sizing
decisions for the retailer.

Then, we can rewrite

From Eq. 6 the minimum of TRC_{1} (T) is obtained
when the quadratic non-negative term, depending on T, is equal to zero. The
optimum value T_{1}* is

Therefore,

Similarly, we can derive TRC_{2} (T) without derivatives as follows:

From Eq. 9 the minimum of TRC_{2} (T) is obtained when the quadratic non-negative term, depending on T, is equal to zero. The optimum value T_{2}* is

Therefore,

Likewise, we can derive TRC_{3} (T) algebraically as follows:

From Eq. 12 the minimum of TRC_{3} (T) is obtained when the quadratic non-negative term, depending on T, is equal to zero. The optimum value T_{3}* is

Therefore,

**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

Since TRC_{1} (M/α) = TRC_{2} (M/α), TRC_{2} (W/D)<TRC_{4} (W/D), TRC (T) is continuous except at T = W/D. All TRC_{1} (T), TRC_{2} (T), TRC_{4} (T) and TRC (T) are defined on T>0. Similar to Case I discussion, TRC_{4} (T) will be higher than both TRC_{1} (T) and TRC_{2} (T) on suitable domain. Hence, we can neglect TRC_{4} (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

Since TRC_{1} (W/D) < TRC_{4} (W/D), TRC (T) is continuous except at T = W/D. All TRC_{1} (T), TRC_{4} (T) and TRC (T) are defined on T>0. We can easily obtain that TRC_{4} (T)>TRC_{1} (T) for all T>0. Hence, we can neglect TRC_{4} (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 T_{1}*≥M/α.
We substitute Eq. 7 into T_{1}*≥M/α, then
we can obtain that:

M/α ≤ T_{1}* if and only if

Similar disscussion, we can obain following results:

M≤T_{2}*< M/α if and only if

and

if and only if,

W/D≤T_{3}*<M if and only if

and

if and only if

Furthermore, we let

and

From Eq. 17-21, we can easily obtain Δ_{2}>Δ_{1}, Δ_{2}≥Δ_{3}, Δ_{4}>Δ_{3} and Δ_{4}≥Δ_{5}. Summarized above arguments, the optimal cycle time T* can be obtained as follows.

**Theorem 1**

(A) |
If Δ_{1}>0, Δ_{3}>0 and Δ_{5}>0,
then TRC (T*) = TRC_{3} (W/D). Hence T* is W/D. |

(B) |
If Δ_{1}>0 Δ_{3}>0 and Δ_{5}≤0,
then TRC (T*) = TRC_{3} (T_{3}*). Hence T* is T_{3}***. |

(C) |
If Δ_{1}>0, Δ_{3}≤0 and Δ_{5}>0,
then TRC (T*) = TRC_{2} (T_{2}*). Hence T* is T_{2}*. |

(D) |
If Δ_{1}>0, Δ_{3}≤0, Δ_{4}>0
and Δ_{5}≤0, then TRC ( T*) = min{TRC_{2} (T_{2}*),
TRC_{3} (T_{3}*)}. Hence T* is T_{2}** *or
T_{3}*** associated with the least cost. |

(E) |
If Δ_{1}>0 and Δ_{4}≤0, then TRC (T*)
= TRC_{2} (T_{2}**)*. Hence T* is T_{2}*. |

(F) |
If Δ_{1}≤0, Δ_{3}>0 and Δ_{5}>0,
then TRC ( T*) = TRC_{1} (T_{1}*). Hence T* is T_{1}*. |

(G) |
If Δ_{1}≤0, Δ_{3}>0 and Δ_{5}≤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. |

(H) |
If Δ_{1}≤0, Δ_{2}>0, Δ_{3}≤0
and Δ_{5}>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. |

(I) |
If Δ_{1}≤0, Δ_{2}>0, Δ_{3}≤0,
Δ_{4}>0 and Δ_{5}≤0, then TRC (T*) = min{TRC_{1}
(T_{1}**)*, TRC_{2} (T_{2}**)*, TRC_{3}
(T_{3}*)}. Hence T* is T_{1}*, T_{2}** *or
T_{3}* associated with the least cost. |

(J) |
If Δ_{1}≤0, Δ_{2}>0 and Δ_{4}≤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. |

(K) |
If Δ_{2}≤0 and Δ_{5}>0, then TRC (T*)
= TRC_{1} (T_{1}*). Hence T* is T_{1}*. |

(L) |
If Δ_{2}≤0, Δ_{4}>0 and Δ_{5}≤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. |

(M) |
If Δ_{2}≤0 and Δ_{4}≤0, then TRC (T*)
= TRC_{1}(T_{1}*). Hence T* is T_{1}*. |

**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≤T_{1}* if and only if

W/D≤T_{2}*<M/α if and only if

and

if and only if

Furthermore, we let

From Eq. 17, 18 and 22,
we can easily obtain Δ_{2}>Δ_{1} and Δ_{2}>Δ_{6}.
Summarized above arguments, the optimal cycle time T* can be obtained as follows:

**Theorem 2**

(A) |
If Δ_{1}>0 and Δ_{6}>0, then
TRC (T*) = TRC_{2} (W/D). Hence T* is W/D. |

(B) |
If Δ_{1}>0 and Δ_{6}≤0, then TRC (T*)
= TRC_{2} (T_{2}**)*. Hence T* is T_{2}*. |

(C) |
If Δ_{1}≤0 and Δ_{6}>0, then TRC (T*)
= TRC_{1} (T_{1}*). Hence T* is T_{1}*. |

(D) |
If Δ_{1}≤0, Δ_{2}>0 and Δ_{6}≤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. |

(E) |
If Δ_{2}≤0, then TRC (T*) = TRC_{1}(T_{1}*).
Hence T* is T_{1}*. |

**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≤T_{1}* if and only if

Furthermore, we let

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

**Theorem 3**

(A) |
If Δ_{7}>0, then TRC (T*) = TRC_{1}(W/D).
Hence T* is W/D. |

(B) |
If Δ_{7}≤0, then TRC (T*) = TRC_{1} (T_{1}*).
Hence T* is T_{1}*. |

**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.