The mathematical models developed in this study are based on the following definitions and assumptions.
: the annual total relevant cost per unit time;
: the annual total relevant cost containing reusable raw materials per unit time;
: the annual total relevant cost containing imperfect-quality items per unit time;
: the order size;
S: the setup cost;
: the production rate;
: the demand rate;
: the purchasing cost per unit raw materials;
: the reusable rate;
: the percentage of defective items in finished productions;
: the cycle length;
: the discount rate;
: the stock holding cost of raw materials per item per unit time;
: the stock holding cost of finished products per item per unit time;
: the screening rate;
: the unit screening cost;
: the selling price of unit imperfect-quality items;
: the screening time.
(1) Production rate P is greater than demand rate D(2) Production rate P and demand rate D are known and constant(3) Shortage is not allowed(4) A single item is considered,
(5)
,
(6)
(7) The imperfect-quality items are sold at the end of the screening time(8) 
and
.
There are three models considered in this paper.
Model 1: Using the total relevant cost per unit time to find the optimal cycle length (It is shown in
The annual total relevant cost TRC(T) consists of the following elements:
The ordering cost per unit time 
(1)
The total relevant cost per unit time can be expressed as follows:
The unique solution of above equation is
The optimal order quantity is
At time
, the 
has a global minimum on 
If
then 
and 
The optimal quantity of this model is the same as that of the traditional EOQ inventory model.
Model 2: Using the annual total relevant cost using 100% reusable raw materials to find the optimal solution (It is shown in
The topic of environmental protection in the world is enthusiastically concerned in recent years. So, by the recycle of the repair and waste disposal productions, we can reduce the impact to environment and increase the ability of competition in business. Also the amount of reused raw materials at the beginning is
, then the amount of 
of usable raw materials can be got.
The annual total relevant cost with 100% reusable raw materials 
consists of the following elements:
The setup cost per unit time 
The purchasing cost per order per unit time 
The stock holding cost of raw materials per unit time
The stock holding cost of products per unit time
The total relevant cost per unit time can be expressed as follows:
And the first and second derivatives of 
are
and
respectively Set
, then we have the following result:
The unique solution of above equation is
At time
, the 
has a global minimum on 
since 
for all
.
Model 3: Using the annual total relevant cost containing imperfect-quantity items to find the optimal cycle length (It is shown in
This is an EPQ model by accounting for fixed percentage imperfect-quality items and considering the issue that poor quality items were sold as a single batch with lower price than it of good quality items at the end of the screening time.
The annual total relevant costs 
consists of the following elements:
The ordering cost per unit time 
The purchasing cost per order per unit time 
The screening cost per unit time 
The stock holding cost of raw materials per unit time
The stock holding cost of products per unit time
The money earned from selling the imperfect-quality items per unit time
The total relevant cost per unit time can be expressed as follows:
And the first and second derivatives of 
are
.
for all
, respectively.
Set
, and set
then we have the following result:
The unique solution of above equation is
.
At the time of T***, the 
has a global minimum on 
since 
for all
.
