# An EOQ Model for Perishable Items with Power Demand

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```							      An EOQ Model for Perishable Items with Power
Demand and Partial Backlogging
Tarun Jeet Singh
Department of Applied Science
D.S. Institute of Technology and Management
(tarunjeet20@yahoo.co.uk)
Volume 15, Number 1
March 2009, pp. 65-72                               Shiv Raj Singh
Department of Mathematics
D.N. College, Meerut, India
Rajul Dutt
Department of Management
JP Institute of Management, Meerut, India
(rajuldatt@rediffmail.com)

For fashionable commodities and other products with a short life cycle, the
willingness of a customer to wait for backlogging during a shortage period is
declining with the length of waiting time. Hence, the longer the waiting time,
the smaller the backlogging rate. Therefore, the backlogging rate should be a
variable and must be dependent on the waiting time for the next replenishment.
In the present work an inventory model is developed in which shortages are
allowed and partially backlogged. The backlogging rate is taken to be
inversely proportional to the waiting time for the next replenishment. Demand
follows power pattern. When fresh and new items arrive in stock they begin to
decay after a fixed time interval called the life period of items. Here, this
realistic concept is taken into consideration with variable rate of deterioration.
Finally, numerical examples are also used to study the behaviour of the model.
Keywords: Power Demand, Life Time, Deterioration, Partial Backlogging

1. Introduction
In recent years, mathematical ideas have been used in different areas in real life
problems, particularly for controlling inventory. One of the important concerns of the
management is to decide when and how much to order or to manufacture so that the
total cost associated with the inventory system should be minimum. This is
somewhat more important, when the inventory undergo decay or deterioration.
Deterioration is defined as change, damage, decay, spoilage obsolescence and loss of
utility or loss of original value in a commodity that results in the decreasing
usefulness from the original one. It is well known that certain products such as
vegetable, medicine, gasoline, blood and radioactive chemicals decrease under
deterioration during their normal storage period. As a result, while determining the
optimal inventory policy of that type of products, the loss due to deterioration cannot
be ignored. In the literature of inventory theory, the deteriorating inventory models
have been continually modified so as to accumulate more practical features of the
66                                                    AN EOQ MODEL FOR PERISHABLE ITEMS

real inventory systems. A number of authors have discussed inventory models for
non deteriorating items. However, there are certain substances in which deterioration
plays an important role and items cannot be stored for a long time. When the items of
the commodity are kept in stock as an inventory for fulfilling the future demand,
there may be the deterioration of items in the inventory system. Various types of
inventory models for items deteriorating at a constant rate were discussed by
Roychowdhury and Chaudhuri (1983), Padmanabhan and Vrat (1995), Balkhi and
Benkherouf (1996) and Yang (2005) etc. In practice it can be observed that constant
rate of deterioration occurs rarely. Most of the items deteriorate fast as the time
passes. Therefore, it is much more realistic to consider the variable deterioration rate.
In a realistic product life cycle, demand is increasing with time during the growth
phase. A power demand pattern inventory model for deteriorating items was
discussed by Dutta and Pal (1988). Chang and Dye (1999) developed an EOQ model
with power demand and partial backlogging.
Furthermore, when the shortage occurs, some customers are willing to wait for
back order and others would turn to buy from other sellers. Researchers such as Park
(1982), Hollier and Mak (1983) and Wee (1995) considered the constant partial
backlogging rates during the shortage period in their inventory models. In many
cases customers are conditioned to a shipping delay and may be willing to wait for a
short time in order to get their first choice. In some inventory systems, such as
fashionable commodities, the length of the waiting time for the next replenishment
would determine whether the backlogging will be accepted or not. Therefore, the
backlogging rate should be variable and dependent on the length of the waiting time
for the next replenishment. Abad (1996) investigated an EOQ model allowing
shortage and partial backlogging. Many researchers have modified inventory policies
by considering the “time proportional partial backlogging rate” such as Chang and
Dye (1999), Wang (2002), Teng and Yang (2004), Wu et al (2006), Singh and Singh
(2007), Dye et al. (2007), Singh et al. (2008) and so on.
In the present paper the deterministic inventory model with power demand pattern
is developed in which inventory is depleted not only by demand but also by
deterioration. Deterioration rate is assumed to be time dependent with the concept of
life time of items. Shortages are allowed and partially backlogged. The effect due to
change in various parameters have been considered in the model numerically.

2. Assumptions and Notation
In developing the mathematical model of the inventory system the following
assumptions and notations are being made:
•    I (t) is the inventory level at any time t, t ≥ 0 and S is the initial inventory level.
•    µ is the life time of items and θ (t ) is the variable deterioration rate s.t. θ(t) =
θt, 0<θ<<1.
•                                                            (         )
D(t ) is the demand rate at any time t s.t. D(t ) = dt (1−n ) / n / nT 1 / n . D is the fixed
quantity, n is the parameter of power demand pattern, the value of n may be any
positive number. T is the planning horizon.
SINGH, SINGH, DUTT                                                                                 67

•       C ,C1 ,C 2 ,C3 and C 4 denote the set up cost for each replenishment, inventory
carrying cost per unit time, deterioration cost per unit, shortage cost for
backlogged items and the unit cost of lost sales respectively. All of the cost
parameters are positive constants.
•       No replenishment or repair of deteriorated items is made during a given cycle.
•       A single item is considered over the fixed period T units of time, which is
subject to variable deterioration rate.
•       Deterioration of the items is considered only after the life time of items.
•       The replenishment occurs instantaneously at an infinite rate.

•       Shortages are allowed and backlogging rate is
(dt   (1− n ) / n
)/ nT   1/ n
, when inventory
1 + α (T − t )
is in shortage. The backlogging parameter α is positive constant and 0<α<<1.

3. Formulation and Solution of the Model
Let Q be the total amount of inventory produced or purchased at the beginning of
each cycle and after fulfilling backorders let us assume we get an amount S (>0) as
initial inventory. During the period (0, µ) the
Inventory Level

S

t1                         T
Time
0                       µ
Lost Sales

inventory level gradually diminishes due to market demand only. After life time
deterioration can take place, therefore during the period (µ, t1) the inventory level
decreases due to the market demand and deterioration of items and falls to zero at
time t1. The period (t1, T) is the period of shortage which is partially backlogged. The
depletion of inventory is given in the figure.
The differential equation governing the inventory level I(t) at any time t during the
cycle (0, T) are such as
I ′(t ) = − D (t ), 0≤t≤µ                                                        … (1)
68                                                                                     AN EOQ MODEL FOR PERISHABLE ITEMS

I ′(t ) + θ (t ) I (t ) = − D (t ),               µ ≤ t ≤ t1                                                                  … (2)
D (t )
I ′(t ) = −                     ,                 t1 ≤ t ≤ T                                                                  … (3)
1 + α (T − t )
The boundary conditions are                                       I (t ) = S when t = 0                                       … (4)
And I (t ) = 0                  when t = t1                                                                                   … (5)
The solutions of equation (1), (2) and (3) are given by
dt 1 / n
I (t ) = S −                ,                     0≤t≤µ                                                                       … (6)
T 1/ n
d  θt 2  1 / n 1 / n
I (t ) =     1 / n 
T 
1 −    t1 − t +
           ( θ
2(2n + 1)
t1        )
( 2 n +1) / n
(
− t ( 2 n+1) / n                       ) , µ ≤ t ≤ t
               1    … (7)
     2                                                                             


and I (t ) =
d 
1/ n 
T 
(1 − αT ) t1 − t 1 / n +
1/ n          α
(
n +1
t1
( n +1) / n
)            (
− t ( n +1) / n                  ) ,
          t1 ≤ t ≤ T        … (8)

The value of initial inventory level (S) is given by

S=
dµ 1 / n
1/ n
d  1/ n
T 
(
+ 1 / n  t1 − µ 1 / n +
θ
2(2n + 1)
t1    )
( 2 n +1) / n
(          
− µ ( 2 n+1) / n  e −θ µ / 2
2
)                         … (9)
T                                                                            
During period (0, T) total number of units holding (IH) can be obtained as
µ                 t1
I H = ∫ I (t )dt + ∫ I (t )dt
0                 µ

d  1 (3n+1) / n θ 3 1 / n        θ  (3n+1) / n 1 (3n+1) / n 
= 1/ n     t1       − µ t1 +              nµ       + t1                                                                  … (10)
T  n +1             3          (3n + 1)            3          
Total amount of deteriorated units (ID) during the period (0, T) is given by
dθ
(                                )        (                            )
t1
 1 ( 2 n+1) / n                n       ( 2 n +1) / n                   
I D = ∫ θ (t ) I (t )dt =                                      − µ 2 t1 −                           − µ ( 2 n +1) / n 
1/ n

T 1/ n    2 t1                        2n + 1
t1                                        … (11)
µ                                                                                                             
Total amount of shortage units (IS) during the period (0, T) is given as

T
d  n ( n+1) / n (1 − 2αT ) ( n+1) / n
I S = − ∫ I (t )dt =                             +                      + (αT − 1)t1 T
1/ n
1/ n 
T                  t1
t1                  T  n +1             n +1
2αn 2                        α        ( 2 n +1) / n 
T ( 2 n+1) / n +        t1                                                                                  … (12)
(n + 1)(2n + 1)                  2n + 1                  
Total amount of lost sales (IL) during the period (0, T) is given by
T
         1                   dα                           n ( n+1) / n     1 ( n+1) / n     1/ n 
I L = ∫ 1 −                 D(t )dt = 1 / n                        n +1T        +      t1        − t1 T                  … (13)
t1    1 + α (T − t )           T                                            n +1                    
Total average cost of the system per unit time is given by
SINGH, SINGH, DUTT                                                                          69

1
K=     [C ′ + C1 I H + C2 I D + C3 I S + C4 I L ]                        … (14)
T
To minimize total average cost per unit time (K), the optimal value of t1 can be
dK
obtained by solving the following equation             =0                … (15)
d t1

d 2K
provided this satisfies the following condition                >0                       … (16)
dt12
Equation (15) is equivalent to
θ           θC

(       )         (         )
C1 t1 + t13 − µ 3  + 2 t12 − µ 2 + (t1 − T ){C3 (1 + α (t1 − T ) ) + αC 4 } = 0       … (17)
    3             2
By solving equation (17), the value of t1 can be obtained and with the use of this
optimal value equation (14) provides minimum total average cost per unit time of the
system in consideration.

4. Numerical Illustration
To illustrate the model numerically, the following parameter values are considered:
d = 50 units, n = 2 units, α = 0.1 unit, θ =0.02 unit, µ = 0.4 year, T = 1 year, C =
Rs.200 per order, C1 = Rs.3.0 per unit per year, C2 = Rs.10.0 per unit, C3 = Rs.12.0
per unit per year, C4 = Rs.4.0 per unit, then for the minimization of total average
cost, optimal policy can be obtained such as t1 = 0.798349 year, S = 44.701076 units
and K = Rs.229.727292 per year.

Effects of Backlogging Parameter (α): The backlogging parameter (α) has initially
been taken as 0.1. Now, we vary backlogging parameter from 0.08 to 0.12 and
observe the effects it has over the solution.

Parameter Value        % Change             t1               S                  K
0.08                  -20       0.797928           44.689239      229.687594
0.09                  -10       0.798138           44.695144      229.707430
0.10                   0        0.798349           44.701076      229.727292
0.11                  +10       0.798560           44.707007      229.747148
0.12                  +20       0.798771           44.712938      229.766997

The study of above table reveals the following interesting facts with the increment in
backlogging parameter:
•    We notice an increase in the inventory period.
•    An increase in the initial inventory level is observed.
•    The value of total average cost of the system also keeps increasing.
70                                              AN EOQ MODEL FOR PERISHABLE ITEMS

Effects of Deterioration Parameter (θ): Initially, the deterioration parameter (θ)
has been taken as 0.02. We observe the following effects with the variation in
deterioration parameter from 0.016 to 0.024.

Parameter Value    % Change           t1            S               K
0.016           -20          0.798871      44.710567       229.683879
0.018             -10        0.798724      44.703975       229.709333
0.020              0         0.798349      44.701076       229.727292
0.022             +10        0.797975      44.693138       229.745263
0.024             +20        0.797601      44.685186       229.763212

From the above table, we observe some interesting facts, which are quite obvious
when considered in the light of reality. If we increase the value of deterioration
parameter then we notice that:
1. Period of inventory decreases.
2. The initial inventory decreases.
3. Total average cost of the system increases.

Effects of Life Time Parameter (µ): Initially, the life time parameter ( ) has been
taken as 0.40. We observe the following effects with the variation in this parameter
from 0.32 to 0.48.
Parameter Value    % Change           t1           S                K
0.32            -20          0.797926     44.697616        229.806831
0.36             -10        0.798125     44.698991        229.766853
0.40               0        0.798349     44.701076        229.727292
0.44             +10        0.798599     44.703976        229.689058
0.48             +20        0.798876     44.707791        229.653044

The following observations have been made on the basis of above table. If we
increase the value of this parameter then:
1. We notice an increment in the period in which inventory holds.
2. We notice an increment in the initial inventory level.
3. The total average cost of the system goes on decreasing progressively.

5. Conclusion
In this paper we have discussed the deteriorating inventory model with variable rate
of deterioration. The demand rate taken in this section is called power pattern
demand (PPD) and if it occurs, then stockiest can use a different policy other than
the conventional policy based on liner pattern. If the large portion of demand occurs
SINGH, SINGH, DUTT                                                                 71

at the beginning of the period, we use n >1 and if it occurs at the end of the period,
we use 0< n< 1. Constant demand rate corresponds to n =1 and n = ∞ corresponds to
instantaneous demand. Shortages are allowed and the backlogging rate is dependent
on the duration of waiting time and varies inversely. Model is illustrated
numerically. A future study will incorporate more realistic assumptions in the
proposed model such as stochastic nature of demand.

6. References
1.  Park, K. S. (1982): Inventory models with partial backorders. International
Journal of Systems Science, 13, 1313-1317.
2. Hollier, R.H. and Mak, K.L. (1983): Inventory replenishment policies for
deteriorating items in a declining market. International Journal of Production
Economics, 21, 813-826.
3. Roychowdhury, M. and Chaudhuri, K.S. (1983): An order level inventory
model for deteriorating items with finite rate of replenishment. Opsearch, 20,
99-106.
4. Dutta, T.K. and Pal, A.K. (1988): Order level inventory system with power
demand pattern for items with variable rate of deterioration. Indian Journal of
Pure and Applied Maths. 19(11), 1043-1053.
5. Padmanabhan, G. and Vrat, P. (1995): EOQ models for perishable items
under stock dependent selling rate. European Journal of Operational Research,
86, 281-292.
6. Wee, H.M. (1995): Joint pricing and replenishment policy for deteriorating
inventory with declining market. International Journal of Production
Economics, Volume 40, Issues 2-3, Pages 163-171.
7. Balkhi, Z.T. and Benkherouf, L. (1996): A production lot size inventory
model for deteriorating items and arbitrary production and demand rate.
European Journal of Operational Research, 92, 302-309.
8. Abad, P.L. (1996): Optimal pricing and lot sizing under conditions of
perishability and partial backlogging. Management Science, 42(8), 1093-1104.
9. Chang, H.J. and Dye, C.Y. (1999): An EOQ model for deteriorating items with
time varying demand and partial backlogging. Journal of the Operational
Research Society, 50, 1176-1182.
10. Chang, H.J. and Dye, C.Y. (1999): An EOQ model for deteriorating items with
time varying demand and partial backlogging. Journal of the Operational
Research Society, 50(11), 1176-1182.
11. Wang, S.P. (2002): An inventory replenishment policy for deteriorating items
with shortages and partial backlogging. Computers and Operational Research,
29, 2043-2051.
12. Teng, J.T. and Yang, H.L. (2004): Deterministic economic order quantity
models with partial backlogging when demand and cost are fluctuating with
time. Journal of the Operational Research Society, 55(5), 495-503.
72                                            AN EOQ MODEL FOR PERISHABLE ITEMS

13. Yang, H.L. (2005): A comparison among various partial backlogging inventory
lot-size models for deteriorating items on the basis of maximum profit.
International Journal of Production Economics, 96, 119-128.
14. Wu, K.S., Ouyang, L.Y. and Yang, C.T. (2006): An optimal replenishment
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15. Singh, S.R. and Singh, T.J. (2007): An EOQ inventory model with Weibull
distribution deterioration, ramp type demand and partial backlogging. Indian
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