INTRODUCTION
The traditional EOQ model assumes that retailer must pay for the items as soon
as the items are received. However, in reallife situations, the supplier may
offer the retailer a delay period, that is trade credit period, in paying for
the amount of purchasing cost. The retailer can sell the goods to accumulate
revenue and earn interest before the end of trade credit period. But if the
payment is delayed beyond that period, a higher interest will be charged. Such
a convenience is likely to motivate customer to order more quantities because
paying later indirectly reduces the purchase cost. Therefore, trade credit is
an important source of financing for intermediate purchasers of goods and services
and plays a large role in our economy. Recently, Hou and Lin
(2008) considered an ordering policy with a cost minimization procedure
for deterioration items under trade credit and time discounting. Other many
related articles can be found by Goyal (1985), Aggarwal
and Jaggi (1995), Jamal et al. (1997), Sarker
et al. (2000), Chung et al. (2005),
Chung and Liao (2006), Ouyang et
al. (2006) and Huang (2007). On the other hand,
many researchers have studied inventory models for deteriorating items such
as volatile liquids, blood banks, medicines, electronic components and fashion
goods. The analysis of deteriorating inventory problems began with Ghare
and Schrader (1963), who developed a simple Economic Order Quantity (EOQ)
model with a constant rate of deterioration. Since then, many related articles
could be found by Heng et al. (1991), Sarker
et al. (1997) and Balkhi and Benkherouf (2004).
While determining the optimal ordering policy, the effects of inflation and
time value of money cannot be ignored. The pioneer research in this direction
was Buzacott (1975), who developed an EOQ model with
inflation subject to different types of pricing policies. Other related articles
can be found in Misra (1979) and Ray
and Chaudhuri (1997), Liao et al. (2000) and
Chung and Lin (2001).
This study develops a deterministic inventory model when a delay in payments is permissible. The effects of the inflation, deterioration and delay in payments are discussed. Moreover, an algorithm is developed to obtain the optimal the number of replenishment, cycle time and order quantity.
NOTATION AND MODEL
The following notation is used throughout the study:
H 
: 
Length of planning horizon 
T 
: 
Replenishment cycle time 
n 
: 
No. of replenishment during the planning horizon; n = H/T 
Q 
: 
Order quantity, units/cycle 
D 
: 
Demand rate per unit time, units/unit time 
A 
: 
Ordering cost at time zero, $/order 
c 
: 
Per unit cost of the item, $/unit 
h 
: 
Inventory holding cost per unit per unit time excluding interest charges,
$/unit/unit time 
θ 
: 
Deterioration rate, a constant fraction of the onhand inventory, units/unit
time 
r 
: 
Discount rate represent the time value of money 
f 
: 
Inflation rate 
R 
: 
The net discount rate of inflation 
R 
: 
rf 
I_{e} 
: 
The interest earned per dollar per unit time 
I_{c} 
: 
The interest charged per dollar in stocks per unit time by the supplier,
I_{c}≥I_{e} 
M 
: 
The permissible delay in settling account (i.e., the trade credit period) 
Let I(t) be the inventory level during the first replenishment cycle. This
inventory level is depleted by the effects of demand and deterioration. So,
the variation of I(t) with respect to t is governed by the following differential
equation:
with the boundary condition I(T) = 0. The solution of (1) can be represented
by:
Consequently, initial inventory after replenishment becomes:
The present value of the total replenishment costs is given by:
and the present value of total purchasing costs is given by:
The present value of the total holding costs over the time horizon H is given
by:
Case I: M≤T = H/n
In this case, the present value of the interest payable during the first replenishment
cycle is:
hence, the present value of the total interest payable over the time horizon
H is:
Next, the present value of the interest earned during the first replenishment
cycle is:
hence, the present value of the total interest earned over the time horizon
H is:
Therefore, the total present value of the costs over the time horizon H is:
Case II: M > T = H/n
The interest earned in the first cycle is the interest earned during the time
period (0, T) plus the interest earned from the cash invested during the time
period (T, M) after the inventory is exhausted at time T and it is given by:
hence, the present value of the total interest earned over the time horizon
H is:
Since, the replenishment cost, purchasing cost and inventory holding cost over
the time horizon H are the same as Case I, the total present value of the costs,
TVC_{2}(s, n), is given by:
At M = T = H/n, we find TVC_{1}(n) = TVC_{2}(n). Consequently, we have:
where, TVC_{1}(n) and TVC_{2}(n) as expressed in Eq.
11 and 14, respectively.
ALGORITHM
The following algorithm is developed to derive the optimal n, T, Q and TVC(n) values:
Step 1: Start by choosing a discrete variable n, where n is any integer number equal or greater than 1.
Step 2: If T = H/n≥M for different integer n values, derive TVC_{1}(n)
from (11); If T = H/n≤M for different integer n values, derive TVC_{2}(n)
from Eq. 14.
Step 3: Repeat Step 1 and 2 for all possible n values with T = H/n≥M
until the minimum TVC_{1}(n) is found from Eq. 11
and let n_{1}* = n. For all possible n values with T = H/n≥M until
the minimum TVC_{2}(n) is found from Eq. 14 and
let n_{2}* = n. The n_{1}*, n_{2}*, TVC_{1}(n*)
and TVC_{2}(n*) values constitute the optimal solution and the satisfy
the following conditions:
Where:
and
Step 4: Select the optimal number of replenishment n* such that:
Hence, optimal order quantity Q^{*} is obtained by substituting n*
into (3) and optimal cycle time, T*, is T* = H/n*.
NUMERICAL RESULTS
An example is devised here to illustrate the results of the general model developed in this study with the following data.
The demand rate, D = 600 unit /year, the replenishment cost, A =$80/order,
the holding cost excluding interest charges, h = $2.4/unit/year, the per unit
item cost, c = $15/unit, the constant rate of deterioration, θ = 0.15,
the net discount rate of inflation, R = $0.12/$/year, the interest charged per
$ in stocks per year by the supplier, I_{c} = $0.18/$/year, the interest
earned per $ per year, I_{e} = $0.16/$/year and the planning horizon,
H, is 5 year. The permissible delay in settling account, M = 60 days = 60/360
years (assume 360 days per year). Using the solution procedure, we have the
computational results shown in Table 1. We find the Case I is optimal option
in credit policy. From the case, the minimum total present value of costs is
found when the number of replenishment, n, is 23. With 23 replenishments, the
optimal cycle time T is 0.217 year, the optimal order quantity, Q = 132.59 units
and the optimal total present value of costs, TVC = $36296.70.
Table 1: 
The numerical results 

*Optimal solution 
CONCLUSION
This study develops a deterministic inventory model for deteriorating items over a finite planning horizon when the supplier provides a permissible delay in payments. The model considers the effects of deterioration, inflation and permissible delay in payments. Based on the DCF approach we permit a proper recognition of the financial implication of the opportunity cost in inventory analysis. In addition, we have presented an optimal solution procedure to find the optimal number of replenishment, cycle time and order quantity to minimize the total present value of costs. Finally, a numerical example is given to illustrate the results. Further research can be done for case with stochastic market demand when the supplier provides a permissible delay in payments and cash discount.
ACKNOWLEDGMENT
This research was partially supported by the National Science Research Council of Taiwan under Grant NSC 962416H240002MY2.