In this paper, the mathematical model is developed with the following assumptions

The planning horizon is finite.

Single item inventory control.

Demand and deterioration and break-ability rates are constant.

Production rate is constant.

Deterioration and break-ability occur as soon as the items are received into inventory.

There is no replacement or repair of breakable items during the period under consideration.

The shortage is not allowed.

The inflation rate is constant and the time value of money is considered.

The lead time is zero.

The inventory level at the end of the planning horizon will be zero.

The cost factors are deterministic.

The number of replenishment is restricted to one integer.

The total relevant cost consists of fixed ordering, purchasing, holding interest payable, interest earned from sales revenue during the permissible period.

The last order is only being placed to satisfy the balance in the stock of the last period.

2.3. Notation

C_P = The present value of holding cost during the first replenishment cycle.

Q = The order quantity in the replenishment.

TC_A = The total fixed ordering cost (0, b).

TC_h = The total holding cost (0, b)

TC_P = The total purchasing cost (0, b).

TC_s = The total shortage cost (0, b).

TC = The total relevant cost (0, b).

2.4. Parameters

The mathematical model is representing the following parameters

A = The fixed ordering cost per replenishment, $order.

C = The unit purchasing price at time zero, $order.

C (t) = The unit purchasing price at time t, C(t)=Ce^(-RT).

D = The constant demand rate per unit time.

p = The production rate per cycle.

b = The length of the finite planning horizon.

i = The constant inflation rate.

I(t) = The inventory level at time t.

I_C = The interest charged per $per year by the supplier.

r = The discount rate representing the time value of money.

R=r-1, representing the net constant discount rate of inflation.

T = The length of each replenishment cycle.

T_j = The total time that elapsed up to, including interest charges.

? t?_j = The time at which the inventory level in the j^th

V = The unit selling price at time t.

C_S=The shortage cost pre-unit time ,$unitunit time

C_S (t)=The shortage cost pre-unit time ,$unitunit time at time t ,C_s (t)=C_s e^(-RT).

V(t) = The selling price per unit at time t, V(t)= Ve^(-RT)

? = The constant deterioration rate, units/unit time.

? = The constant break-ability rate, units/unit time.

3. Mathematical model

Let is the inventory level at any time t, Depletion due to demand and deterioration, break-ability will occur simultaneously. The first order differential equation that describes the instantaneous state of over the open interval (0, b) is given by.

(dI(t))/dt+?I(t)+?I(t)=-(P-D),0?t?t_1,0???1,0???1, P>D (1)