2. Material and methods

2.2. Assumptions and notions

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2.2.1. Assumptions
In this paper, the mathematical model is developed with the following assumptions
The planning horizon is finite.
Replenishment rate is infinite.
Single item inventory control.
Demand and deterioration rate are constant.
Deteriorating occurs as soon as the items are received into inventory within 0,t_n.
The shortage is not allowed.
The leading time is nonzero.
The purchasing cost is more than the holding cost.
The inventory level is constant within the second component of first run time, non zero within the planning horizon.
The total relevant cost consists of fixed ordering, purchasing and holding cost.

2.3. Notation

D= The demand rate quantity in period 0,t_n.
Q_1= The quantity within A, B.
Q_M= The quantity within 0, A
t_1= The first component of the first runtime.
t_w1= The first leading time.
t_m1= The second component of the first runtime.
Q = The total quantity within 0, B.
TC_A = The total fixed ordering cost during 0, t_n.
I_h=The holding cost.
TC_h = The total holding cost during 0, t_1 .
TC_P = The total purchasing cost during 0,t_1.
TC= The total relevant cost during 0,t_1.

2.4. Parameters

T = The length of the finite planning horizon.
I_1 (t) = The inventory level at time 0,t_m.
I_2 (t) = The inventory level at time t_m ,t_1 .
? t?_1 = The first run time of replenishment.
? = The constant deteriorating rate units/unit time during 0,t_1 .

3. Mathematical model