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We discuss five areas of inventory model, including reusable raw material, EPQ model, optimization, random planning horizon and present value. In the traditional EPQ model, the stock-holding cost of raw material was not counted as a part of relevant cost. We explored the possibility of reducing a company’s impact on the environment and increasing their competitiveness by recycling their repair and waste disposal. The products are manufactured with reusable raw material. Our analysis takes into account the time value, and the present value method is applied to determine the optimal inventory policies for reusable items with random planning horizon. Results show how the heuristic approach can achieve global optimum. Numerical examples are given to validate the proposed system.

Reuse of material and products is not a new subject. Using the repaired and new made products, Richter [

The EOQ (Economic Ordering Quantity) model was first proposed by Harris [

The mathematical models developed in this study are based on the following definitions and assumptions.

Definitions:

PVC(T): the present value of the cash flow for the first inventory horizonC(T): the expected present value of the cash flow for the random planning horizonTRC(T): the total relevant cost per unit timeQ: the order sizeS: the cost of placing an orderP: the production rateD: the demand rateC: the purchasing cost per unit item,

Assumptions:

1) Production rate is greater than demand rate.

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

3) Shortage is not allowed.

4) A single item is considered.

5) The time horizon is not infinite.

6) The stock holding cost of raw materials and products are the same.

7)

8) At the end of the planning horizon, the raw material is used up and the productions are sold out.

9) The random planning horizon time x follows an exponential distribution with parameter

Two models used in this analysis are illustrated below (see

Model 1: We use the annual total relevant cost to find the optimal solution.

The annual total relevant cost TRC(T) consists of the following elements:

The ordering cost per unit time =

The purchasing cost per order per unit time =

The stock holding cost of raw material per unit time =

The stock holding cost of products per unit time =

The total relevant cost per unit time can be expressed as TRC(T) = the ordering cost per unit time

+ the purchasing cost per order per unit time

+ the stock holding cost of raw material per unit time

+ the stock holding cost of products per unit time

=

The first and second derivatives of TRC(T) are

Set

The unique solution of above equation is

At

Model 2: Using the expected present value of total relevant cost for random planning horizon C(T) to find the optimal solution.

We assume that the random planning horizon x is located on the (k + 1)th cycle time. The present value of total relevant cost at the first cycle time PVC(T) consists of the following elements:

The ordering cost = S.

The purchasing cost =

The stock holding cost of raw material =

The stock holding cost of products =

The present value of total relevant cost at the first cycle time PVC(T) is as follows:

The present value of total relevant cost from the beginning of the first cycle time to the beginning of the (k + 1)th cycle time is as follows:

The present value (at time of kT) of total relevant cost at the (k + 1)th cycle time consists of the following elements:

The purchasing cost =

The stock holding cost of raw material =

The stock holding cost of products =

The present value of total relevant cost at the (k + 1)th cycle time is as follows:

Now, we want to find the expected present value of total relevant cost for random planning horizon C(T).

The present value of total relevant cost includes:

(A) The present value of total relevant cost from the beginning of the first cycle time to the beginning of the (k+1)th cycle time.

(B) The present value of total relevant cost at the (k + 1)th cycle time.

If we assume that the probability density function of x is

Let

where

Then we obtain the first derivative of g(T) as follows:

Since

From the calculations described above, we have proved that

Since

It is not easy to solve

We set the lower bound as

Setting

And from the above, we can now simplify the inequality to

The positive solution of

Theorem:

Proof: Since

By Equation (97),

From the above,

Now we can compute the exact optimal cycle length

Step 1: Let

Step 2: Set

where

Step 3: Set

Step 4: If

Otherwise go to Step 5.

Step 5: If

And if

Then go to Step 3.

Step 6:

Example 1: If we set the numbers as S = 1000, P = 2000, D = 1500, c = 10,

Example 2: If we set the numbers as S = 2000, P = 2000, D = 1500, c = 10,

Example 3: If we set the numbers as S = 1000, P = 2000, D = 1500, c = 20,

Example 4: If we set the numbers as S = 1000, P = 2000, D = 1500, c = 10,

Example 5: If we set the numbers as S = 1000, P = 2000, D = 1500, c = 10,

Example 6: If we set the numbers as S = 1000, P = 2000, D = 1500, c = 10,

The planning horizon is random and has an exponential distribution with a parameter that influences the optimal cycle time. The raw material is used up and the products are sold out at the end of the planning horizon. Thus, the results are more economical in this model. The use of reusable raw material is beneficial and worthwhile. The models in this research can be applied in companies that use reusable or new raw material. There are many interesting results derived from the numerical examples that future research should take into consideration. It is not necessary for the stocking holding cost of raw material and the stock holding cost of products to be the same.