I(t)=? I?_0 (t) ?_t^(t_1)??(P-D) e^(?+?)u=? (P-D)/(?+?) (e^(?+?)(t_1-t) -1),?I(t)?_0=e^(-(?+?)t)

I(t)=? I?_0 (t) ?_t^(t_1)??(P-D)e^(?+?)u=? (P-D)/(?+?) (e^(?+?)(t_1-t) -1),?I(t)?_0=e^(-(?+?)t) (2)

According to Eq.2 the maximum inventory quantity at the begin each period is given as

Q=(P-D)/(?+?) (e^((?+?) t_1 )-1),t_1=Fb/N (3)

3.1. Fixed ordering cost
We assumed the number of replenishment is N so that the fixed ordering cost over the planning horizon under the inflation consideration is:

?TC?_A=?_(j=0)^N??A_j T=?_(j=0)^N?A? e^(-RT)=A((e^(-(N+1)RT)-1)/(e^(-RT)-1)),T=Fb/N (4)

?TC?_A=?_(j=0)^N??A_j T=?_(j=0)^N?A? e^(-RT)=A((e^((-(N+1)Rb)/N)-1)/(e^((-Rb)/N)-1)),T=Fb/N (5)

3.2. Holding cost excluding interest cost
We find the average inventory quantity to obtain holding cost
I ?=?_0^(t_1)??I(t)dt=?_0^(t_1)??(P-D)/(?+?)(e^(?+?)(t_1-t) dt= (P-D)/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1) ?? (6)
By using Eq. 7 we have obtained holding cost is as follows

?TC?_h=?_(j=0)^(N-1)??I_h C_j I ?=?_(j=0)^(N-1)??I_h Ce^(-R_j T)=(I_h C(P-D))/(?+?)^2 (e^((?+?) t_1 )-(?+?)?? t_1-1) (7)
Since T=b/N then equation number (8) it will be as
?TC?_h=(I_h C(P-D))/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1)(e^(-Rb)-1)/(e^(-Rb/N)-1) (8)

3.3. Purchasing cost

According to fig.1 of inventory level the purchasing cost of j^th cycle is calculated as
?CP?_j=C_j I_m=C_j (P-D)/(?+?)(e^((?+?) Fb/N)-1) (9)
The total purchasing cost over the planning horizon can be obtained as
Special case for total purchasing cost when
?TC?_P=?_(j=0)^(N-1)???CP?_j=C(P-D)/(?+?)(e^((?+?) Fb/N)-1)(? (e^(-Rb)-1)/(e^(-Rb/N)-1)) (10)


TC=A(e^(-(N+1)Rb/N)-1)/(e^(-Rb/N)-1)+ ?((C(P-D)+)/(?+?) (e^((?+?) t_1 )-1)[email protected]((P-D)I_h)/(?+?)^2 (e^((?+?) t_1 )-(?+?) t_1-1) )(e^(-Rb)-1)/(e^(- Rb/N)-1) (11)

3.5. Economic order quantity
3.5. 1.Economic order quantity
To find EOQ by minimizing the total cost function as the following

+(C(P-D)(e^((?+?) t_1 )-1))/(?+?)+((P-D)I_h)/((??+?)?^2 ) (e^(((?+?) t_1)/N)-(?+?) t_1-1) (e^(-Rb)-1)/(e^((-Rb)/N)-1) (12)
Since t_1=Fb/N
Q=((P-D))/(?+?) (e^((?+?) t_1 )-1) (13)
By substituting the Eq.13 in the equation Eq.12, then it can be rewritten as

TC=A(e^(-(N+1)Rb/N)-1)/(e^(-Rb/N)-1) +CQ+(I_h Q)/(?+?)-(I_h (P-D))/(?+?)^2 ln?((?+?)Q/(P-D)+1)(e^(-Rb)-1)/(e^(- Rb/N)-1) (14)
By taking derivate the Eq.14 With respect to to find out the minimum value of total cost function then
dTC/dQ=(C+I_h/(?+?)-((P-D)I_h)/(P-D+(?+?)Q))(e^(-Rb)-1)/(e^(- Rb/N)-1) =0

Here there two cases.

If Q>(P-D)/(?+?)

Q^*=(P-D)/(?+?)(I_h (?+?))/(I_h+(?+?)C)-1
This is an infeasible solution as economic order quantity.

(ii) If Q0
Since P > D, Q>0, ?+?>0
So that the total cost function has a minimum value at the point Q^*
The period of the first time run.
?t_1?^* =1/(?+?) ln?(((?+?)Q^*)/(P-D)+1) (17)