INTRODUCTION
The problem of inventory systems under inflationary conditions has received attention in recent years. Usually, the inflation rates have been assumed constant over the planning horizon. But, many economic factors may affect on the future changes in the inflation rate; such as changes in the world inflation rate, rate of investment, demand level, labor cost, cost of raw materials, rates of exchange, rate of unemployment, productivity level, tax, liquidity, etc. The current study considers timedependent inflation rates for deterioration items with shortage.
Inventoried goods can be broadly classified into four metacategories:
• 
Obsolescence: Refers to items that lose their value through time
due to rapid changes of technology or the introduction of a new product
by a competitor 
• 
Deterioration: Refers to the damage, spoilage, dryness, vaporization,
etc. of the products 
• 
Amelioration: Refers to items whose value or utility or quantity
increase with time 
• 
No obsolescence/deterioration/amelioration 
If the rate of obsolescence, deterioration or amelioration is not sufficiently
low, its impact on modeling of such an inventory system cannot be ignored. There
are a few studies for obsolescing and ameliorating items. Moon
et al. (2005) considered the ameliorating/deteriorating items on
an inventory model with timevarying demand pattern. Against obsolescing and
ameliorating items, the many researches has founded in the deteriorating inventory
area in recent years. Products such as vegetables, fish, medicine, blood, gasoline
and radioactive chemicals have finite shelf life and start to deteriorate once
they are produced. The deteriorating inventory models under inflationary conditions
are studied greatly. In a few of these works, deterioration rate is not constant.
For instance, Balkhi (2004a) presented a production
lot size inventory model that the production, demand and deterioration rates
are known, continuous and differentiable functions of time. Shortages are allowed,
but only a fraction of the stock out is backordered and the rest is lost. Lo
et al. (2007) developed an integrated productioninventory model
with assumptions of varying rate of deterioration, partial backordering, inflation,
imperfect production processes and multiple deliveries. The most of the inventory
systems for deteriorating items are considered a constant deterioration rate
which will state in continuance.
Some researches in inflationary inventory systems assumed timevarying demand
rate. Yang et al. (2001) extended the inventory
lotsize models to allow for inflation and fluctuating demand, which is more
general than constant, increasing, decreasing and logconcave demand patterns.
Other works is performed by Balkhi (2004a).
Several researchers have considered finite replenishment rate for inflationary
inventory systems. Wee and Law (1999) derived a deteriorating
inventory model under inflationary conditions for determining economic production
lot size when the demand rate is a linear decreasing function of the selling
price. Balkhi (2004b) proposed two flexible production
lot size inventory models for deteriorating items in which the production rate
at any instant depends on the demand and the onhand inventory level at that
instant. Another research is performed by Lo et al.
(2007).
The stockdependent demand rate models are prepared with some researchers.
Hou and Lin (2004) developed an inventory model under
inflation and time discounting for deteriorating items with stockdependent
selling rate. The selling rate is assumed to be a function of the current inventory
level and the rate of deterioration is assumed to be constant. Liao
and Chen (2003) surveyed a retailer's inventory control system for the optimal
delay in payment time for initial stockdependent consumption rate when a wholesaler
permits delay in payment. The effect of inflation rate, deterioration rate,
initial stockdependent consumption rate and a wholesaler's permissible delay
in payment is discussed. A deterministic Economic Order Quantity (EOQ) inventory
model taking into account inflation and time value of money developed for deteriorating
items with priceand stockdependent selling rates by Hou
and Lin (2006). An efficient solution procedure is presented to determine
the optimal number of replenishment, the cycle time and selling price. Hou
(2006) prepared an inventory model for deteriorating items with stockdependent
consumption rate. Maiti et al. (2006) proposed
an inventory model with stockdependent demand rate and two storage facilities
under inflation and time value of money where the planning horizon is stochastic
in nature and follows exponential distribution with a known mean. An inventory
model under inflation for deteriorating items with stockdependent consumption
rate and partial backlogging shortages proposed by Yang
et al. (2009).
Other efforts in inventory systems under inflationary conditions are performed
under the assumption of the permissible delay in payments. Chang
(2004) proposed an EOQ model for deteriorating items under inflation when
the supplier offers a permissible delay to the purchaser if the order quantity
is greater than or equal to a predetermined quantity. Shah
(2006) derived an inventory model by assuming constant rate of deterioration
of units in an inventory, time value of money under the conditions of permissible
delay in payments. Other models are prepared by Liao and
Chen (2003).
The inflationary inventory models with two warehouses are proposed previously.
Yang (2004) discussed the twowarehouse inventory problem
for deteriorating items with a constant demand rate and shortages. Yang
(2006) extended the models introduced in Yang (2004)
to incorporate partial backlogging and then compare the 2 twowarehouse inventory
models based on the minimum cost approach.
The mentioned studies have considered a constant and wellknown inflation rate
over the time horizon. Horowitz (2000) discussed a simple
EOQ model with a Normal distribution for the inflation rate and the firm’s
cost of capital. He showed the importance of taking into account the inflation
rate and time discounting, especially when the former is relatively high or
when there is considerable uncertainty as to either the inflation rate or the
marginal cost of capital. Mirzazadeh et al. (2009)
considered stochastic inflationary conditions with variable probability density
functions (pdfs) over the time horizon and the demand rate is dependent to the
inflation rates (any arbitrary pdfs can be used). The developed model, also,
implicates to finite replenishment rate, finite time horizon, deteriorating
items with shortages. The objective is minimization of the expected present
value of costs over the time horizon. The numerical example and case study have
been provided for evaluation and validation of the theoretical results and some
special cases of the model are discussed.
The existing inventory models under inflation and time value of money are numerous.
Most of the previous researches consider a constant inflation rate over time
and a few models consider stochastic inflationary conditions. Furthermore, in
some practical situations, the inflation rate can be well approximated by continuoustimedependent
function. Therefore, this study assumes the inflation is timedependent. Also,
the demand rate is assumed to be inflationproportional. A numerical example
is shown for explaining the theoretical results. Then, the particular cases,
which follow the main problem, are discussed corresponding to the situation
of:
Case (I) 
: 
Constant inflation rates 
Case (II) 
: 
Identical rate of inflation 
Case (III) 
: 
Without shortages 
Case (IV) 
: 
No deterioration 
Case (V) 
: 
Identical inflation rate, no shortages and no deterioration 
Case (VI) 
: 
Constant and identical inflation rate, no shortages and no deterioration 
THE ASSUMPTION, NOTATIONS AND DESCRIPTION OF THE MODEL
First, the assumptions and notations are explained and then the proposed inventory
system is described. The mathematical models in this study are developed based
on the following assumptions:
• 
A constant fraction of the onhand inventory deteriorates per unit time,
as soon as the item is received into inventory 
• 
The internal and external inflation rates are timevarying over the time
horizon 
• 
Shortages are allowed and fully backlogged, except for the final cycle 
• 
The replenishment is instantaneous and the replenishment cycle is the
same for each period 
• 
The initial inventory level is zero 
• 
The system brings about for prescribed timehorizon of length H 
The following notations are used:
i_{m} 
= 
The internal (for m = 1) and the external (for m = 2) inflation rates
which are linear timedependent: 
where, a_{m} and b_{m} are real number. If b_{m}<0
the inflation rate will decrease and if b_{m}>0 the inflation rate
will increase over time. If b_{m}=0 the inflation rate is stable that
will be discussed in the special cases
r 
= 
The interest rate 
R_{m} 
= 
The discounted rate net of inflation: R_{m} = ri_{m} 
D (i_{1},i_{2 }) 
= 
The demand rate per unit time is a function of inflation rates 
where, a_{0}, b_{0} and c_{0} are fixed real number
θ 
= 
The constant deterioration rate per unit time (0 <θ<1 ) 
p 
= 
The purchase cost at time zero 
S 
= 
The ordering cost per order at time zero 
c_{1m} 
= 
The internal (for m=1) and external (for m=2) inventory carrying cost
per unit per unit time at time zero 
c_{2m} 
= 
The internal (for m=1) and external (for m=2) shortages cost per unit
per unit time at time zero 
H 
= 
The finite time horizon 
T 
= 
The interval of time between replenishment 
k 
= 
The proportion of time in any given inventory cycle which orders can be
filled from the existing stock 
n 
= 
The number of replenishments during time horizon 
TVC (n,k) 
= 
The total present value of costs over the time horizon 
Other notations will be introduced later. The demand rate is dependent to inflation
rates and the inflation rates are timeproportional.

Fig. 1:  Graphical
representation of the inventory system 
Therefore, the demand rate can be explained as follow:
The graphical representation of the inventory system is shown in Fig. 1. The time horizon, H, is divided into n equal cycles each of length T so that T=H/n. Initial and final inventory levels are both zero. Each inventory cycle except the last cycle can be divided into two parts: [(j1)T, (j+k1)T] and [(j+k1)T,jT]. During the time interval [(j1)T,jT], the inventory level leads to zero and shortages occur at time (j+k1)T. Shortages are accumulated until jT before they are backordered and are not allowed in the last replenishment cycle. The optimal inventory policy yields the ordering and shortage points, which minimize the present value of the total inventory system costs over the time horizon.
THE MATHEMATICAL MODELING AND ANALYSIS
Each inventory cycle, except the last cycle, is divided to the two different parts. During the time interval [(j1)T, (j+k1)T], the level of inventory, I_{1j}(t_{1j}), gradually decreases mainly to meet demands and partly due to deterioration. Hence, the variation of inventory with respect to time can be described by the following differential equation:
The shortages occur at time (k+j1)T and accumulated until jT before they are backordered. The shortages level to be represented by I_{2j}(t_{2j}):
In the last cycle shortages are not allowed and the inventory level, I_{3}(t_{3}), is governed by the following differential equation:
The boundary conditions are:
The solution of Eq. 46 after apply the
boundary conditions and considering Eq. 3 are as follow:
The objective of the problem is minimization of the total present value of costs over the time horizon. Consider CP, CH, CS and CR as the present value of costs of purchasing, holding, shortages and replenishment respectively. The total present value of costs over the time horizon (TVC(n,k)) is:
The detailed analysis is given as follows:
The ordering costs: The replenishment cost occurs at the start of inventory cycle and therefore, the present value of the ordering cost for (j+1)th cycle is:
Therefore, the total ordering cost is:
The purchasing cost: During any given period, the order quantity consists of both demand and deterioration for the relevant period excluding shortage part of the period and the amount required to satisfy the demand during the shortage period in the preceding time interval. For the jth cycle (j = 1, 2, …, n1) the present value of the purchase cost can be formulated as follows:
In the last period shortages are not allowable, therefore, the present value of the purchase cost is:
The total purchasing cost over the time horizon would be:
The carrying cost: The present value of the inventory carrying cost for the jth cycle (j = 1, 2, …, n1) for the mth class (m = 1, 2) is:
In the last inventory cycle from similar machinations, we have:
Therefore, the total holding cost over the time horizon is:
The shortages cost: The shortages cost for jth cycle can be calculated as follow:
Using above equation, the total shortages cost is as follows (note that shortages
are not allowable in the last cycle):
So, the present value of the total costs of the inventory system over the time
horizon is obtained using Eq. 15, 18,
21 and 23 that is shown in Eq.
13. The problem is determining n and k so that minimize the total inventory
system cost. Optimal value of n and k can be obtained with setting number of
replenishments equal to 1 and increasing it. For a given value of n, the necessary
condition of optimality sets the first derivative of TVC(n,k) with respect to
k equals to zero. After several algebraic operations, the first derivative can
be calculated as follow:
By increasing n, the objective function leads to the minimum and then by increasing
n, the objective function increases monotonously. The iterative methods such
as Newton method can be used to solve the model. The secondorder condition
for a minimum is:
THE NUMERICAL EXAMPLE
Following example is providing according to the results. Let internal and external inflation rates as follow:
The demand rate, which is a linear function of the inflation rates, is:
Table 1:  The
optimal solution 

The
minimum cost over the time horizon is 67750.32* for n* = 21 and k* = 0.606786 
The company interest rate is 20% per annum, the deterioration rate of the onhand
inventory per unit time is 0.01 and the length of time horizon is 10 years.
r = 0.2; H=10years; θ = 0.01
The system costs at the beginning of time horizon are c_{11} = 0.2 ; c_{12} = 0.4; c_{21} = 0.8; c_{22} = 0.6; p = 5; S = 100
Using these parameter values, the optimal solution of the models is obtained and the results are shown in Table 1.
The minimum cost over the time horizon is 67750.32 for n* = 21 and k* = 0.606786. Optimal interval of time between replenishment, T*, equals to H/n* = 0.467 year. The shortages occur after elapsing 60.7% of the cycle time or 103 days.
SOME PARTICULAR CASES
An attempt has been made in this section to study six important special cases of the model.
Case (I): Constant inflation rates: Assume the inflation rates do not change over the time horizon, i.e.,
Therefore, the demand rate is:
The total present value of costs, TVC(n,k), can be obtained with placing Eq. 26 in Eq. 13. The optimal solution in this case, with considering the previous numerical example, is as follows: n* = 22, k* = 0.470016, TVC(n,k) = 59871.78 and T* = 0.454 year. We can see that k decreases considerably in comparison with the main model.
Case (II): Identical rate of inflation: If internal and external inflation
rates be identical to each other, the present value of the total cost TVC(n,k)
can be obtained by deleting
in Eq. 13 and substituting:
In the previous numerical example let:
Therefore, the optimal solution in this case will be: n* = 20, k* = 0.577967, TVC(n,k) = 64474.57 and T* = 0.5year. The number of replenishment, n, inventory system costs and k decrease.
Case (III): Without shortages: If shortages are not allowed, k = 1 can be substituted in expression Eq. 13 and the present value of the total variable cost, TVC(n), can be obtained. The minimum solution of TVC(n) for the discrete variable of n must satisfy the following equation:
where, ΔTVC(n) = TVC(n)TVC(n1). In the numerical example, using the above inequality, the following solution is obtained: n* = 26, TVC(n) = 68 543.95 and T* = 0.385year. It shows that n and TVC increase in this case in comparison with the main model.
Case (IV): No deterioration: If there has not been deterioration over
time for the available inventory, i.e., θ = 0, the cost function after
calculation, will be rewritten as follows:
where,
The total costs of the inventory system can be minimized by the explained methods.
The optimal solution is as follow: n* = 41, k* = 0.667940, TVC(n,k) = 44513.44
and T* = 0.244year. Thus, n and k are increased and TVC is decreased in comparison
with the main model. This is an expected phenomenon, because the goods do not
deteriorated and the inventory system manager can increases order quantity,
i.e., decreases n and increases the inventory level in warehouse (or increases
k).
Case (V): Identical inflation rate, no shortages and no deterioration: Now assume that the internal and external inflation rates be identical to each other, no shortages allowed and θ = 0. This can be solved by using Eq. 30, substituting k = 1 and considering Eq. 28. Therefore, the optimal solution is as follows: n* = 25, TVC(n,k) = 65 271.90 and T* = 0.4 year. In comparison with the main model, the number of replenishment increases and the total costs decreases.
Case (VI): Constant and identical inflation rate, no shortages and no deterioration:
The optimal order policy in this case will be obtained with using Eq.
30, substituting k = 1 and considering Eq. 26 and 28,
which is as follow: n* = 30, TVC(n,k) = 59 001.70 and T = 0.333 year. Similarly,
the previous case, the number of replenishment increases and the total costs
decrease.
CONCLUSIONS
In this study, an inventory model under inflationary conditions with shortages for deteriorating items has been proposed. The internal and external inflation rates are timedependent. Also, the inflationproportional demand rate has been considered. The objective is determining the optimal values of the time interval between replenishment and the time occurrence of shortages over the time horizon to minimize the total costs of the inventory systems. A numerical example has been given to illustrate the theoretical results. Finally, six special cases have been discussed. These special cases are compared with the main model through the numerical example.