
Research Article


Effects of Uncertain Inflationary Conditions on an Inventory Model for Deteriorating Items with Shortages 

Abolfazl Mirzazadeh



ABSTRACT

This study proposes an inventory model with stochastic internal and external inflation rates for deteriorating items and allowable shortages. The many economic, political, social and cultural variables affect the inflation rates. For instance, economic factors such as changes in the world inflation rate, demand level, labor cost, cost of raw materials, exchange rates, unemployment rate, productivity level, tax, liquidity, etc are effective in this direction. Therefore, the assumption of constant inflation rates is not valid, especially, when the time horizon is long. This model considers stochastic inflationary conditions. Numerical examples are used to illustrate the theoretical results, which are further clarified through a sensitivity analysis on the model parameters. It has been shown that the optimal solution is highly sensitive to considerable uncertainty of the inflation rates.





Received: July 26, 2010;
Accepted: September 03, 2010;
Published: October 11, 2010


INTRODUCTION
The classical inventory models have not been considered the inflation and time
value of money. However, consequence of high inflation, it is important to investigate
how timevalue of money influences various inventory policies. Since, 1975 a
series of related papers appeared that considered the effects of inflation on
the inventory system. Before the 1990s, the earlier efforts have been considered
simple situations. The first attempt in this field has been reported by Buzacott
(1975) that dealt with an Economic Order Quantity (EOQ) model with inflation
subject to different types of pricing policies. Misra (1979)
developed a discountedcost model and included internal (company) and external
(general economy) inflation rates for various costs associated with an inventory
system. Sarker and Pan (1994) surveyed the effects of
inflation and the time value of money on order quantity with finite replenishment
rate. Some efforts were extended to consider variable demand, such as Uthayakumar
and Geetha (2009), Maity (2010), Vrat
and Padmanabhan (1990), Datta and Pal (1991), Hariga
(1995), Hariga and BenDaya (1996) and Chung
(2003).
In above cases, it has been implicitly assumed that the rate of inflation is
known with certainty. Yet, inflation enters the inventory picture only because
it may have an impact on the future inventory costs and the future rate of inflation
is inherently uncertain and unstable. Horowitz (2000)
discussed an EOQ model with a normal distribution for the inflation rate.
Certain types of commodities either deteriorate or become obsolete throughout
course of time and hence are unstable. For example, the commonly used goods
like fruits, vegetables, meat, foodstuffs, perfumes, alcohol, gasoline, radioactive
substances, photographic films, electronic components, etc. where deterioration
is usually observed during their normal storage period. Inventoried goods can
be broadly classified into four metacategories (Goyal and
Giri, 2001):
• 
Obsolescence refers to items that lose their value through
time because of 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 
There are several studies of deteriorating inventory models under inflationary
conditions. Chung and Tsai (2001) presented an inventory
model for deteriorating items with the demand of linear trend considering the
timevalue of money. Wee and Law (2001) derived a deteriorating
inventory model under inflationary conditions when the demand rate is a linear
decreasing function of the selling price. Chen and Lin (2002)
discussed an inventory model for deteriorating items with a normally distributed
shelf life, continuous timevarying demand and shortages under an inflationary
and time discounting environment. Chang (2004) established
a deteriorating EOQ model when the supplier offers a permissible delay to the
purchaser if the order quantity is greater than or equal to a predetermined
quantity. Yang (2004) discussed the twowarehouse inventory
problem for deteriorating items with a constant demand rate and shortages. Moon
et al. (2005) considered ameliorating/deteriorating items with a
timevarying demand pattern. 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 the exponential distribution with a known
mean. 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. A Two storage inventory problem with dynamic demand and interval
valued leadtime over a finite time horizon under inflation and timevalue of
money considered by Dey et al. (2008). Maity
and Maiti (2008) developed a numerical approach to a multiobjective optimal
inventory control problem for deteriorating multiitems under fuzzy inflation
and discounting.
Mirzazadeh (2010) assumed the inflation is timedependent
and demand rate is assumed to be inflationproportional. Roy
et al. (2009) prepared an inventory model for a deteriorating item
with displayed stock dependent demand under fuzzy inflation and time discounting
over a random planning horizon. Another research has been performed by Ameli
et al. (2011) with considering an economic order quantity model for
imperfect items under fuzzy inflationary conditions. Other efforts on inflationary
inventory systems for deteriorating items have been made by Hsieh
and Dye (2010), Sana (2010), Su
et al. (1996), Chen (1998), Wee
and Law (1999), Sarker et al. (2000), Yang
et al. (2001, 2009), Liao
and Chen (2003), Balkhi (2004a, b),
Hou and Lin (2006), Shah (2006),
Hou (2006), Jaggi et al. (2006),
Yang (2006) and Chern et al.
(2008).
In this study, a detailed analysis has been done for surveying the effect of uncertain inflationary conditions on the optimal ordering policy under stochastic inflationary conditions and arbitrary probability density functions (pdfs) for the internal and external inflation rates. Deteriorating items and shortages have been considered. A numerical example and a sensitivity analysis are used to illustrate the model. This study concluded that the No. of replenishments and the expected value of cost are sensitive to the external inflation rate and the optimal solution is sensitive to the uncertainty of the inflation rates when the standard deviations of the inflation rates are sufficiently high. Particular cases of the problem, which follow the main problem and correspond to the situation of (1) a single inflation rate for all cost components, (2) no shortages, (3) no deterioration and (4) all the three previous cases together are discussed. THE MATHEMATICAL MODEL AND ANALYSIS
The following assumptions are used throughout this study:
• 
The internal and external inflation rates are random variables
with known distribution 
• 
The demand rate is known and constant 
• 
Shortages are allowed and fully backlogged except for the final cycle 
• 
The replenishment is instantaneous and lead time is zero 
• 
The system operates for prescribed timehorizon of length H 
• 
A constant fraction of the onhand inventory deteriorates per unit time 
The cost components may be divided into internal and external classes. Van
Hees and Monhemius (1972) have given the breakdown of the various costs
of inventory system. In the real world, internal and external costs exhibit
different behaviours, so that the internal cost changes by the current inflation
rate of the company and the external cost varies with the inflation rate of
the general economy (or of the supplier company). Therefore, two different pdfs
for the inflation rates can be used in this model. The notations described as
follows:
i_{m} 
: 
Internal (for m = 1) and external (for m = 2) inflation rates 
f(i_{m}) 
: 
The pdf of i_{m} 
r 
: 
The discount rate 
D 
: 
The demand rate per unit time 
A 
: 
The ordering cost per order at time zero 
c_{lm} 
: 
The internal (for m = 1) and external (for m = 2) inventory carrying cost
(for l = 1) and shortage cost (for l = 2) per unit per unit time at time
zero 
p 
: 
The external purchase cost at time zero 
θ 
: 
The constant deterioration rate 
M_{im}(Y) 
: 
The moment generating function of i_{m} for m = 1 and 2 
H 
: 
The fixed time horizon 
Other notations will be introduced later. It is assumed that the length of
the planning horizon is H = nT, where, n is an integer for the number of replenishments
to be made during period H and T is an interval of time between replenishment.
The unit of time can be considered as a year, a month, a week, etc. and k (0<k≤1)
is the proportion of time in any given inventory cycle which orders can be filled
from the existing stock. Thus, 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 total expected inventory cost over the time horizon.
The mathematical formulation: Let ECP, ECH, ECS and ECR denote the expected present values of the purchasing, carrying, shortage and replenishment costs, respectively. The detailed analysis of each cost function is given as follows: Expected present value of the purchasing cost: The expected present value of the purchasing cost for the jth period (j = 1, 2, …, n1), as shown in Appendix A, is equal to:


(1) 
and for the last period is given by Appendix A: Therefore, the total purchase cost for all cycles can be written as follows: Expected present value of the inventory cost: The inventory carrying cost is divided into internal (for m = 1) and external (for m = 2) classes. From Appendix B the expected present value of the inventory carrying cost for the jth cycle for the mth class can be written as: In the last period the inventory level comes to zero at the end of period. Therefore: The total internal and external carrying costs for all cycles can be given as follows: Expected present value of the shortages cost: The shortages cost may be divided to internal and external classes similar to the holding cost. The expected present value of the shortages cost for the jth cycle for the mth class can be formulated as follows (Appendix C): The total shortages cost during the entire planning horizon H can be written as follows: Expected present value of the ordering cost: The expected present value of the ordering cost for replenishment at time (j1)T for the jth cycle is: The total replenishment cost can be given as follows: Hence, the total expected inventory cost of the system during the entire planning horizon H is given by: THE SOLUTION PROCEDURE The problem is determining n and k to lead the minimum of the total expected inventory system cost. For a given value of n, the necessary condition of optimality is as follows: The iterative methods such as Newton method can be used for calculating k. By increasing n, the objective function decreases to lead to minimum. The secondorder condition for a minimum is: NUMERICAL EXAMPLE According to the results, the following example is providing. Let r = $0.2/$/year, D = 1000 units/year, A = $100/order, c_{11} = $0.2/unit/year, c_{12} = $0.4/unit/year, c_{21} = $0.8/unit/year, c_{22} = $0.6/unit/year, p = $5/unit, H = 10 years, θ = 0.01. The internal and external inflation rates have the normal distribution function with means of μ_{1} = 0.08 and μ_{2} = 0.14, standard deviations of σ_{1} = 0.04 and σ_{2} = 0.06, respectively. The results are shown in Table 1. The minimum expected cost over the planning horizon is 44 537.26 for n* = 41 and k* = 0.664623. Optimal interval of time between replenishment, T*, equals to H/n* = 0.244 year. SENSITIVITY ANALYSIS
To study the effects of system parameters changes D, H, θ, r, μ_{1},
μ_{2}, σ_{1}, σ_{2}, A, p, c_{11},
c_{12}, c_{21} and c_{22} on the optimal cost, the replenishment
time and k* which is derived by the proposed method, a sensitivity analysis
was performed. This fact is done by increasing the parameters by 20, 50, 100%
and decreasing the parameters to 20, 50, 90%, taking each one at a time and
keeping the remaining parameters at their original values. The following conclusion
can be derived from the sensitivity analysis based on Table 2:
• 
As the mean of the internal inflation rate increases, the
number of replenishments (n) decreases and k increases. By increasing the
mean of external inflation rate, increase the number of replenishments (n)
and k. The optimal expected present value of cost (ETVC) increases when
μ_{1} and μ_{2} increase but is highly sensitive
to μ_{2}. Induction of this result is the purchase cost increasing
by external inflation rate is more than other cost components, i.e., purchase
cost constitutes considerable portion of total cost. Hence, the total cost
has a similar role as the purchase cost 
• 
Table 2 shows that the optimal value of k and number
of replenishments (n) are insensitive to changes in the standard deviations
of inflation rates. But, the operating doctrine is highly sensitive, to
high values of σ_{1} and σ_{2}. For example, considering
σ_{1} = 0.04 and σ_{2} = 0.4, Table
3 shows that the optimal number of replenishments equals 1 and we have
no shortages (n* = k* = 1). The numerical example has solved by considering
σ_{1} = 0.04, 0.1, 0.2, 0.3, 0.4, 0.5 and σ_{2}
= 0.06, 0.1, 0.2, 0.3, 0.4 and 0.5. The results are shown in Table
3. The derived model is sufficiently sensitive if the uncertainty of
inflation rates is higher than a certain magnitude. In that situation, larger
order quantity should be placed 
• 
High uncertainty inflation rate is occurred in the real world, for instants,
international financial Statistical Yearbook, 2004, shows that the mean
change of wholesale prices indices in Russia in 19932001 was 1.91 and it’s
the standard deviation was 3.05. For another case, in Norway in 19972001
the mean change of wholesale prices indices were 0.08 and the standard deviation
was 0.2 
• 
The number of replenishments (n) is highly sensitive to the change of
the parameters D, A and H, is slightly sensitive to changes in c_{12}
and insensitive to changes in r, θ, p, c_{11}, c_{21}
and c_{22} 
• 
The optimal value of k is highly sensitive to the change of the parameters
c_{12}, c_{21} and c_{22}, is moderately sensitive
to r and c_{11} and is insensitive to D, θ, H, p and A 
• 
The total expected inventory cost of the system is highly sensitive to
the changes in the parameters D, r, H and p and insensitive to θ, A,
c_{11}, c_{12}, c_{21} and c_{22} 
Table 1: 
Optimal solution for numerical example 

Table 2: 
Effects of changes in model parameters on n, k and optimal
expected system cost 

Table 3: 
Effects of changes in the standard deviations of inflation
rates on n, k and optimal expected system cost 

SOME PARTICULAR CASES Here, an attempt has been made to study some important special cases of the model.
Case (I): If the internal and external inflation rates have the same
pdf, the expected present value of the total cost ETVC (n,k) can be obtained
by deleting: in
Eq. 11 and substituting:
The previous numerical example assumes that the inflation rate has the normal distribution function with the mean of μ = 0.11 and the standard deviation of σ = 0.05. The optimal solution in this case is as follows: n* = 38, k* = 0.666099, ETVC(n,k) = 39 296.36 and T* = 0.263 year. The number of replenishments and inventory system cost decrease and k increases. Case (II): If shortages are not allowed, k = 1 can be substituted in expression (11) and the expected present worth of the total variable cost ETVC(n) can be obtained. The minimum solution of ETVC(n) for the discrete variable of n must satisfy the following equation: where, ETVC(n) = ETVC(n)ETVC(n1). In the numerical example, using the above inequality, the following solution is obtained: n* = 50, ETVC(n) = 45 613.73 and T* = 0.2 year. It shows that n and ETVC increase in the without shortages case. Case (III): If available inventory has no deterioration (θ = 0) over time the cost function, after modeling, may be rewritten as follows: The cost function can be minimized by the methods indicated in this study. For a given value of n, the necessary condition of optimality is: and the sufficient condition is the second derivative is positive. In this case, the numerical result is obtained as follows: n* = 41, k* = 0.667940, ETVC(n,k) = 44 513.44 and T* = 0.244 year. Thus ETVC is decreased, k is increased and n is not changed in comparison to the main model.
Case (IV): Now assume that the internal and external inflation rates
have the same pdf, no shortages allowed and θ = 0. This may be solved by
using Eq. 16 and 17, substituting k =
1 and considering (14). Therefore, the optimal solution is as follows: n* =
46, ETVC(n,k) = 40 391.48 and T* = 0.217.
CONCLUSIONS Usually, in the inventory systems under inflationary conditions, it has been assumed that the inflation rates are constant over the planning horizon. But, many economic, political, social and cultural variables may also affect the future changes in the inflation rate. Therefore, assuming constant inflation rates is not valid, especially when the time horizon is more than two years. The current study considers stochastic inflation rates where any distribution function is applicable. This model incorporates some realistic features that are likely to be associated with the inventory of certain types of goods. First, deterioration over time is a natural feature for goods. Second, occurrence of inventory shortages is a natural real situation phenomenon. In the solution process of the problem, the subject of moment generating function has been used. The numerical examples have been given and sensitivity analysis has been conducted to illustrate the theoretical results. The results of sensitivity analysis indicate that which of the parameters are more sensitive to the optimal solution. Also, the results indicate the importance of taking into account stochastic inflation, especially when there is considerable uncertainty associated with the inflation rates. Finally, four special cases have been discussed: identical inflation rates, no shortages situation, no deterioration and considering all the three cases simultaneously. These cases are compared with the main model through the numerical example. APPENDIX A During any given period, the order quantity is consisting 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 expected present value of the purchase cost can be formulated as follows:
The above equation can be rewritten as Eq. 1. In the last period shortages are not allowable, therefore the expected present value of the purchase cost is:
where R_{2} = ri_{2}. It can similarly be rewritten as Eq.
2.
APPENDIX B The expected present value of the inventory carrying cost for the jth cycle (j = 1, 2, …, n1) for the mth class (m = 1,2) is:
where, R_{m} = r i_{m} and Eq. B 1 can be rewritten as Eq. 4. In the last period for the mth class (m = 1,2) from similar machinations we have:
APPENDIX C
The expected present value of the shortages cost for the jth cycle (j = 1, 2, …, n1) for the mth class (m = 1,2) can be computed as:
where, R_{m} = r i_{m} . It can be rewritten as Eq.
7.

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