Fig.1Graphical fig.1 of inventory level the purchasing cost

Fig.1Graphical demonstration of inventory control diagram3.1.

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