= present value of the cash flows for the first inventory horizon,

= present value of the cash flows for the infinite planning horizon,

= the total relevant costs per unit time,

= the order quantity,

= the purchasing cost per unit,

= the ordering cost per order,

= the inventory carrying cost of raw material per unit per year,

= the inventory carrying cost of finished product per unit per year,

= the production rate unit time,

= the demand rate per unit time,

= the discount rate,

= the cycle length, and

= the optimal cycle length.

1) Production rate is greater than demand rate.

2) Production rate and demand rate is known and constant.

3) Shortage is not allowed.

4) A single item is considered.

5) Time horizon is infinite.

The traditional EPQ model assumes that the replenishments are instantaneous and the relevant costs only contain ordering cost, stock holding cost of products and cost of buying materials. In fact, it is different from the classical EPQ model. Because we need to buy the raw materials at the beginning of the cycle time for production and then sale the products. So, about the total relevant cost, we also must consider about the ordering and the stock-holding cost for the raw materials in stock (shown in Figure 1). On the other hand, we must take the time value of money into consideration to avoid consuming cost when a company makes a decision. Therefore, in this section we will calculate the total relevant costs per unit time of the modified EPQ model and the present value of modified EPQ model. Moreover, we can compute the optimal cycle time by the total relevant costs.

There are two models considered in this paper.

Case 1. The modified EPQ model.

The total relevant cost per unit time consists of the following elements:

1) The ordering cost per order =2) The purchasing cost per unit =

3) The stock carrying cost per unit =.

From the above elements, the total relevant cost per unit time can be expressed as

Taking the derivative of Equation (1) with respect to T

Derivative Equation (2) with respect to T,

Equations (3) imply that is convex on. We have that is decreasing on and increasing on.

Letting

and solving Equation (4), we obtain the optimal cycle time

and the optimal replenishment quantity

When,

From Case 1 we can found the purchasing cost per unit time is irrelevant to time and the optimal cycle time of total relevant cost not taking the time value into consideration is the same the optimal cycle time of classical EOQ model.

Case 2. The present value of modified EPQ model.

Case 2 presents the present value for the basic EPQ model under the assumption of adding the raw materials stock-holding cost.

For this case, the stock holding cost of raw materials and products are the same. We will divide into two parts to discuss the model in the following.

(A) The present value of cash flows for the first cycle time:

First, we discuss about the inventory level of raw materials. To satisfy the demand of customers, we need to buy the amount of in stock at the beginning of every cycle time. The level of inventory in raw materials at time is zero. Besides, we need to consider about the level of inventory in products. From the beginning to time, the products which supplier produces need to satisfy both the need of customers and accumulation of stock to sell from time to the end of cycle time. The annual rate of production is greater than the annual rate of demand. At time has the biggest amount of inventory in stock. Furthermore, the level of inventory of raw materials at time is zero. The inventory reaches the zero level at the end of the cycle time.

From the above arguments, we show that the present value of cash flows for the first inventory horizon, is given by

Equation (9) can be simplified as

(B) The present value of cash flows for the infinite planning horizon:

If the inventory horizon from the first cycle time up to the cycle time, then the present value is

Furthermore, the present value of infinite planning horizon, , is given by the following

From this procedure of calculating, we found two interesting results:

1) The present value of total stock-holding cost of raw materials plus products from the beginning to time t is the same as the stock-holding cost of classical EOQ model;

2) The present value of total relevant cost is independent of the production rate (if the production rate is greater than the demand rate).

On the other hand, we can obtain an exact optimal cycle time using the following proposition and theorem.

Proposition. has a unique local minimum on. Proof. Taking the first derivatives of Equation (12), is as follows

Let

Taking the derivative of Equation (13) with respect to,

and it can be simplified as

We have is a strictly increasing function since. Since and. There exists a solution such that. Therefore, is a unique global minimum on.

This completes the proof.

Theorem. The upper bound of optimal cycle time is, i.e..

Proof. From, we have. We known

Simplifying Equation (15) as follows

and approximating in Equation (16) by. Equation (16) implies that

Considering the following equation:

Let, then is the unique positive root of Equation (18). Equation (17) yields

and which implies that

Combining Equation (19) and Equation (20), we obtain

Since is a strictly increasing function, we have

This completes the proof.

In this case, we assume the lower bound of optimal cycle time is zero. Combining the above theorem and proposition, we can obtain the bounds for the optimal cycle length as follows:

In addition, we can compute an exact optimal cycle length using the logic of the following algorithm. This algorithm similar to that of Chung and Lin (cf. [7]).

The bisection algorithm ([10]):

Step 1 LetStep 2 Set and

Step 3 Set.

Step 4 If, go to step 6.

Otherwise go to step 5.

Step 5 If, set.

If, set.

Then go to Step 3.

Step 6. That is, is the exact optimal cycle length.