INTRODUCTION
An important problem confronting inventory decision makers is the design of efficient replenishment policies to keep the cost of the inventory system as low as possible. In several existing models, it is assumed that products have infinite shelflife. However, it is known in practice that not all products (medicines, volatile liquids, blood banks, etc.) possess this characteristic. In many inventory systems, deterioration of goods in the form of direct spoilage or gradual physical decay in course of time is a realistic phenomenon and hence, it should be considered in inventory modeling. Hence, the need to study inventory systems with deterioration arises.
In this study, we are interested in finding the optimal replenishment schedule
for an inventory system with shortages, in which items are deteriorating at
a constant rate. The demand rates are increasing with time over a known and
finite planning horizon. For inventory systems, the optimal replenishment policy
usually depends on the setup cost, holding (carrying) cost, the backorder (shortage)
cost and the demand pattern. The classical approach in deterministic inventory
modeling is to assume a uniform demand rate. Much study has been carried out
to extend the EOQ model in order to accommodate timevarying demand patterns.
Silver and Meal (1969) gave a heuristic solution procedure
for the inventory model with time varying demand Donaldson
(1977) exhibited a very complicated solution procedure taking demand to
be linear. Ritchie (1980, 1984,
1985) obtained on exact solution for linearly trended
demand. Mitra developed a simple procedure for adjusting the economic order
quantity model for linearly increasing and decreasing demand.
Dave and Patel (1981) derived a lot size model for
constant deterioration of items with time proportional demand. Sachan
(1984) allowed shortages in Dave and Patel (1981)’s
model. Related study by Bahari Kashani (1989), Deb
and Chaudhuri (1987), Mudreshwar (1988), Goyal
(1986), Hariga (1994), Xu and
Wang (1991), Niketa and Shah (2006), Chung
and Ting (1993), Hariga (1994, 1995)
and Jalan et al. (1996) and their references.
Covert and Philip (1973) developed inventory model using
a two parameter Weibull distribution for deterioration of units. Philip
(1974) formulated inventory model when deterioration start after some time
and used a threeparameter Weibull distribution for deterioration of units.
Misra (1975) extended Covert and
Philip (1973)’s model for finite rate of replenishment. It is assumed
in all the earlier models that the holding cost per item/per unit time and the
setup cost per order are known and constant. But the holding cost and the set
up cost may not always be constant. In order to generalize EOQ models, various
functions describing holding cost were introduced by several researchers like
Muhlemann and Valtis Spanopulos. In this study, the researchers have developed
a generalized EOQ model for deteriorating items where the demand rate, deterioration
rate, holding cost and ordering cost are all expressed as linearly increasing
functions of time. Shortages in inventory are also allowed and are completely
backlogged.
The assumption of timedependent holding cost and ordering cost is justified when the price index increases with time. Deterioration rate obviously increases with the passage of time.
Model formulation
Assumptions: The model contains the following fundamental assumptions:
• 
An Inventory of a single item operates for a prescribed timehorizon
of length ‘H’ 

• 
The demand rate R (t) = a + bt, a>0, b>0, a>>b
is increasing function of time 

• 
Shortages are allowed and are completely backordered. Shortages
are not allowed in the last replenishment cycle. The shortage cost is C_{1}
per unit short per unit of time 

• 
C is the purchase cost per unit of the item in inventory 

• 
The holding cost, h (t) per item per unit of time, unit is
time dependent and its functional form is assumed as h (t) = h + αt,
h>0, α≥0 

• 
The ordering cost, K (t) depends on the total time elapsed
up to the beginning of each cycle and is taken as K (t) = K + βt, K>0,
β≥0 

• 
The units in an inventory deteriorate per unit time during
the period H. i.e., θ(t) = θ_{0}+γt, 0≤θ(t)<1
and γ≥0 

• 
The inventory level at the end of the timehorizon H is zero 

• 
Replenishment rate is infinite and lead time is zero 

• 
S_{j} is the time point at which the inventory level
in the jthreplenishment cycle drops to be zero, j = 1, 2, 3 ….., (m1).
For the last replenishment cycle, t_{m} = H 

• 
t_{j} = j(H/n) is the total time elapsed up to and
including the jth replenishment cycle, j = 0, 1, 2, … m 

Let Q_{j}(t) be the inventory level at any time t in the jth replenishment cycle (j = 1, 2, 3, ….. m) (Fig. 1) then the instantaneous states of Q_{j}(t) are described by the following differential equations. (This inventory is depleted by the combined effect of demand and deterioration). So Q_{j}(t) is given by:
 Fig. 1: 
Inventory level variation with time (for the case of n = 3) 

With the boundary condition, Q_{1j}(S_{j}) = 0 and
From Eq. 1, we get:
And the general solution for Eq. 2 is shown by:
The holding cost for the jth replenishment cycle is:
Now, the number of items deteriorated during the jth replenishment cycle is:
Putting the value of Q_{1j}(t) in D_{j} in the Eq. 6, we get:
Now, the ordering cost for the jth interval is:
The total shortage over the jth replenishment cycle is:
Therefore, the Total Cost (TC) of the inventory system over the time H is shown by:
COMPUTATIONAL ALGORITHIM
Researchers find optimal solution numerically using the following algorithm:
Step 1: a, b, C, C1, K, h, θ, α, β, γ and H. Set m = 2 and TC (1) =0.
Step 2: Set t_{j1 }= (j1) H/m for j = 1, 2, 3… m.
Step 3: For j = 1, 2, 3… m1, find S_{j} using Eq. 10.
Step 4: Calculate the Total Cost (TC (m)) of the inventory system using Eq. 9.
Step 5: If TC (m)<TC (m1), then set m = m + 1 and go to step 2, otherwise go to step 6.
Step 6: Set m* = m1, S_{j} * = Sj (m1), j = 1, 2, 3, …, m*1 and TC* = TC(m1).
Step 7: Stop
If C_{1} →∞ (i.e., shortages are not allowed) then S_{j} = t_{j} for j = 1, 2, 3… m. Then m* and TC* can be determined easily following the algorithm as stated above.
NUMERICAL EXAMPLE
Consider the parameters of the inventory system as:
K = 90 h = 4 a = 10 b = 2 C = 0.5 C1 = 1.0
θ = 0.1 α = 0.1 β = 0.15 H = 10 γ = 0.001
One root of the equation i.e., S_{j} = 0.4998 (Table 1) shows the effect of changes in various parameters on number of orders to be placed and total cost TC of an inventory system.
• 
An increase in a and decrease in m and increase in j gives
TC less after two points it becomes negative 

Table 1: 
The effect of changes in verious parameters on numbers of
orders to be placed and total cost TC of an inventory system 


• 
An increase in b, m and j, TC decreases and it becomes negative
after two points also 

• 
An increase in C, m and j, TC decreases and it becomes negative
after one point 

• 
An increase in C_{1}, m and j, TC decreases and it
becomes negative after one point 

• 
An increase in α, m and j, TC decreases and it becomes
negative after one point 

• 
An Increase in h, decrease in m and increase in j, TC decreases
and it becomes negative after second point 

• 
An increase in θ, m and j, TC decreases and it becomes
negative after second point 

CONCLUSION
In the present study, we have formulated an EOQ model for deteriorating items over a finite time horizon H. The concluding remarks of the model are.
The deterministic demand is decreasing with time. It is usually observed in the market of electronic components like television, DVD, Freezers etc. There are also shortages.
Since almost all items undergo either direct spoilage or physical decay in the course of time, deterioration factor has an important role to play in an inventory.