Fig.1Graphical demonstration of inventory control diagram
3.1. Fixed ordering cost

The fixed ordering cost in the length of a finite horizon 0, t_1

TC_A=A (5)

3.2. Purchasing cost

According to fig.1 of inventory level the purchasing cost of
?TC?_P=(CD(e^(??t?_1 )-e^((?t_1)/2)))/?-(?Q_1 t_1)/2 (6)

3.3. Holding cost excluding interest cost

We locate the average inventory quantity to obtain holding cost
?TC?_h=I_h ?_0^(t_1)??I(t)dt-I_(hQ_1 t_1 )/2=?_0^(t_1)??(D(e^(??t?_1 )-e^((?t_1)/2)))/?dt-I_(hQ_1 t_1 )/2=(I_h D)/?^2 (e^(?t_1 )-2e^((?t_1)/2)+1)-(???I?_h Q?_1 t_1)/2 ?? (7)

3.4. Optimal inventory level and optimal time

To obtain the EOQ by optimizing the total cost

TC=?TC?_A+?TC?_h+?TC?_P (8)

By subsisting Eq. (5, 6, 7) in Eq. (8)
Then
TC=A +(CD(e^(??t?_1 )-e^((?t_1)/2)))/?-(??Q?_1 t_1)/2+(I_h D)/?^2 (e^(?t_1 )-2e^((?t_1)/2)+1)-(??I_h Q?_1 t_1)/2 (9)

Deputy the t_1=(2Q_1)/(D-?Q_1 ) in Eq. (9)

TC=A +(CDQ_1/(D-??Q?_1 ))-C(?Q_1?^2/(D-??Q?_1 ))-(?I_h ?Q_1?^2)/((D-??Q?_1)) (10)

Deviating Eq. (10) with respect to Q_1

(d?TC?_1)/(dQ_1 )=CD(D/(D-??Q?_1 )^2 )-C( (2DQ_1-???Q?_1?^2)/(D-?Q_1 )^2 )-?I_h ((2DQ_1-???Q?_1?^2)/(D-??Q?_1 )^2 )=0

?Q_1?^*=D/? 1-(1-?C/(C+?I_h ))^(1/2)

Or
?Q_1?^*=D/? 1+(1-?C/(C+?I_h ))^(1/2)

Lemma (1):

a) If 0?k_1?1
i) ??D?Q?_1?^* If ?Q_1?^*=D/? 1-(1-?C/(C+?I_h ))^(1/2)

When k_1?? , 0?k_1?1
ii) ??D>Q?_1?^* If ?Q_1?^*=D/? 1-(1-?C/(C+?I_h ))^(1/2)

When k_1