Upload
others
View
7
Download
0
Embed Size (px)
Citation preview
5.1 INTRODUCTION
As inventory
various situations, especially inflation. In present times, inflation is a
global phenomenon. Inflation is defined as that state of disequilibrium in
which an expansion of purchasing power tends to cause or is the effect of
an increase in the price level. A period of prolonged, persistent and
continuous inflation results in the economic, political, social and moral
disruption of society. Inflation and time value of money have also
attracted attention of researchers. So, we have considered the concept of
inflation in an inventory system.
In recent years many researchers assuming that the production rate
of a manufacturing system is often assumed to be constant, but in fact
production rate is a variable under managerial control. Production rate
may be influenced due to demand.
An attempt has been made to study a more realistic situation, for
assuming that production rate is decision variable. Volume flexibility of a
manufacturing system is its ability to be operated profitably at different
overall output levels. Volume flexibility permits a manufacturing system
to adjust production upwards or downwards within wide frontier period to
Chapter 5 An EPQ Model for Decaying I
-141-
the start of production of a lot. It helps to reduce the rate of production to
avoid rapid accrual of inventories.
Demand is the most volatile of all the market forces, as it is least
controlled by management personnel. Even a slight change in the demand
pattern for any particular item causes a lot of havoc with the
manufacturing unit concerned. Overall, it means that every time the
demand for any commodity undergoes a noticeable change, the inventory
manager has to reformulate the complete logistics of management for that
item. In such a situation, the inventory manager has to look out for a new
method, by which he can easily assimilate the changed inclination of the
customers into the existing policy. It is perceived that the steady demand
after its increment with the time is never continued indefinitely. The
demand pattern for many products, which initially increases with time for
some period, becomes steady after some period rather than increasing.
Rather it would be followed by exponential decrement with respect to
time after some period of time and becomes asymptotic in nature. Thus,
the demand would be illustrated by three successive time period-
classified time dependent ramp type functions, in which in the first phase
the demand increases with time and after that it becomes steady and
towards the end in the final phase it decreases and becomes asymptotic.
Chapter 5 An EPQ Model for Decaying I
-142-
Most of the researches on real market oriented time dependent
demand is very restrictive. Goyal (1986) and Goswami and Chaudhuri
(1991) discussed different types of inventory models with linear trend in
demand. Hill (1995) considered a product subject to a period of
increasing demand, according to a general power law, followed by a
period of level demand. The characteristic of ramp type demand can be
found in Mandal and Pal (1998) who has taken order level inventory
system with ramp type demand rate for deterioration items. Wu et al.
(1999) developed an EOQ model with ramp type demand rate for items
with deterioration. Wu and Ouyang (2000) developed an order-level
inventory system for deteriorating items with a ramp type demand
function of time. It not only provides the solution procedure for the
problem, but also enumerate two possible shortage models. Wu (2001)
presented an EOQ inventory model which depleted not only by demand
but also by Weibull distribution deterioration, in which the demand rate is
assumed that with a ramp type function of time. In the model, shortages
are allowed partial backlogging and the backlogging rate is variable and
is dependent on waiting time for the next replenishment. Giri et al.
(2003) solved a single-item single-period Economic Order Quantity
model for deteriorating items with a ramp type demand and Weibull
deterioration distribution is considered. The shortages in inventory are
Chapter 5 An EPQ Model for Decaying I
-143-
allowed and backlogged completely. The model is developed over an
infinite planning horizon. Manna and Chaudhuri (2006) also solved an
order level inventory system for deteriorating items and developed with
demand rate as a ramp type function of time. The finite production rate is
proportional to the demand rate and deterioration rate is time
proportional. The unit production cost is inversely proportional to the
demand rate. Deng et al. (2007) considered the inventory models for
deteriorating items with ramp type demand rate. Singh and Singh (2007)
developed an EOQ inventory model with Weibull distribution
deterioration, ramp type demand and partial backlogging. A single item
order level inventory model for a seasonal product is discussed where the
demand rate is represented by a ramp-type time dependent function.
Shortages are not allowed over a finite time horizon by Panda et al.
(2008). Skouri et al. (2009) determined an inventory model with general
ramp type demand rate, time dependent (Weibull) deterioration rate and
partial backlogging of unsatisfied demand is considered.
An inventory model for decaying items with ramp type demand
rate under volume flexibility has been discussed. Inventory deteriorates
over time at a time dependent deterioration rate. Production rate is taken
as demand dependent. Holding cost is taken as variable and it is a linear
increasing function of time. Shortages in inventory are allowed and
Chapter 5 An EPQ Model for Decaying I
-144-
partially backlogged. The backlogging rate is an exponential decreasing
function of time. The effect of inflation has also been considered for
various costs associated with the inventory system. A numerical example
is presented to illustrate the model with cost minimization technique.
5.2 ASSUMPTIONS AND NOTATIONS
In developing the mathematical model of the inventory system the
following assumptions are being made:
1. A single item is considered over a prescribed period T units of
time.
2. Production is considered which depends on the demand,
P(D)=kD(t).
3. The unit production cost is a function of production rate.
4. Production rate is considered to be decision variable.
5. The demand rate D(t) at time t is deterministic and taken as a ramp
type function of time i.e. ,
where H(t- s function defined as :
0
1
, tH(t
, t
6. The replenishment rate is infinite and lead-time is zero.
Chapter 5 An EPQ Model for Decaying I
-145-
7. Holding cost is taken as linear.
8. When the demand for goods is more than the supply. Shortages
will occur. Customers encountering shortages will either wait for
the vendor to reorder (backlogging cost involved) or go to other
vendors (lost sales cost involved). In this model shortages are
allowed and the backlogging rate is exp(- t), when inventory is in
shortage. The backlogging parameter is a positive constant.
9. The variable rate of deterioration is taken as (t) = t, where 0 <
<< 1 and only applied to on hand inventory.
10. No replacement or repair of deteriorated items is made during a
given cycle.
Notations
I(t) : The inventory level at any time t, t 0.
T : Planning horizon.
r : Inflation rate.
(g+ht) : The holding cost per unit per unit time.
C2 : The deterioration cost per unit.
C3 : The shortage cost per unit per unit time.
Chapter 5 An EPQ Model for Decaying I
-146-
C4 : The opportunity cost due to lost sales.
C : The replenishment cost per order.
K : Production rate
The unit production cost is given by:
PG
C N H KK
, where N, G, H are all positive constants. This
cost function is based on the following factors:
The material cost N per unit item is fixed.
As the production rate increases, some costs like labour and energy
costs are equally distributed over a large no. of units. Hence, the unit
production cost decreases, because (G/K) decrease as the production
rate (K) increases.
The third term (HK) associated with tool or die costs is proportional to
the production rate.
5.3 FORMULATION AND SOLUTION OF THE MODEL
Initially, the inventory level is zero. The production starts at time t
= 0, when the inventory level become q deterioration can take place and
after time demand becomes steady, it reaches to maximum inventory
level S at time t1. After this production stops and at time t2, the inventory
Chapter 5 An EPQ Model for Decaying I
-147-
becomes zero. At this time shortage starts developing and at time t3. After
this time fresh production starts to clear the backlog by the time T. Here,
our aim is to find the optimum values of t1, t2, t3, t4 that minimize the total
average cost (K) over the time horizon (0, T). The depletion of inventory
is given in the figure 5.1
Fig. 5.1: Representation of production model.
The inventory system at any time t can be represented by the
following differential equations:
( 1)I (t) 0 t . (5.1)
( 1)I (t) 1 . 5.2)
I (t) 1 2 . 5.3)
Chapter 5 An EPQ Model for Decaying I
-148-
I (t) Ae e , . (5.4)
( 1)I (t) K Ae , 3 . 5.5)
with the boundary conditions,
I(0) =0, I(t1) = S, I(t2) = 0 and I(T) = 0. (5.6)
The solutions of the equations are given by:
2 22 2 2 2 2
3( 1) ( 1) 2 2 22
AI(t) K Se K e ,
22e ( , 0 t . 5.7)
and 2
3 321
1 6
(t t )I(t) Ae (t t) . 5.8)
3 2 31 1 1( ) [ ]
6 2 3rtI t t t t t t t e 1 2 5.9)
2A
I(t) e e e , . 5.10)
( 1)
( )K A
I(t) e e 3 . 5.11)
Due to continuity of I(t) at point t = , it follows from equations (5.7) and
(5.8), one has
Chapter 5 An EPQ Model for Decaying I
-149-
3 3
2 2 211 3
1(1 ) 2 2 2
6 2S A K (t
23
(1 )A
e K ( . 5.12)
The total average cost consists of following elements:
(i) Production cost per cycle (PC)
1
30( )[ ]
t Trt rt rt
t
GN HK Ke dt Ke dt Ke dt
K. 5.13)
(ii) Holding cost per cycle (CH)
1 2
10
[ ( ) ( ) ( ) ( ) ( ) ( ) ]t t
rt rt rt
t
g ht I t e dt g ht I t e dt g ht I t e dt .
5.14)
(iii) Cost of deteriorated units per cycle (CD)
1 2
1
20
[ ( ) ( ) ( ) ]t t
rt rt rt
t
C tI t e dt tI t e dt tI t e dt . 5.15)
(iv) Shortages cost per cycle (CS)
3
2 3
3 [ ( ( )) ( ( )) ]t T
rt rt
t t
C I t e dt I t e dt . 5.16)
(v) Opportunity cost due to lost sales per cycle (CLS)
Chapter 5 An EPQ Model for Decaying I
-150-
3
2
4 [ 1 ]t
t
C ( e ) Ae e dt . 5.17)
Therefore, the total average cost per unit time of our model is
obtained as follows
K(t1, t2, t3)=[Production cost + Holding cost + Deterioration cost +
Shortage cost + Opportunity cost]/T. (5.18)
To minimize the total cost per unit time, the optimal values of t1
and T can be obtained by solving the following equations simultaneously
1
0K
t. (5.19)
2
0K
t. (5.20)
And 3
0K
t. (5.20)
provided, they satisfy the following conditions :
2 2 2
2 2 21 2 3
0 0, 0K K K
,t t t
. 5.21)
Equations (5.19), (5.20) and (5.21) are highly nonlinear equations.
Therefore, numerical solution of these equations is obtained by using the
software.
Chapter 5 An EPQ Model for Decaying I
-151-
5.4 NUMERICAL ILLUSTRATION
To illustrate the model numerically the following parameter values
are considered:
K = 150 units, h= 0.4. r = 0.05 unit,
= 0.2 unit, C2 = Rs. 10.0 per unit, =0.002 unit,
= 0.2 year, C4 = Rs. 4.0 per unit, t2 =0.6,
t3 =0.8, = 0.1 unit, T = 1 year,
G=2000, N=150, H=0.01,
g= Rs. 3.0 per unit per year,
C3 = Rs. 12.0 per unit per year,
Chapter 5 An EPQ Model for Decaying I
-152-
Table 5.1: Effect of various parameters.
% Change t1 S K
-20 0.398067 38.597235 157.696583
-10 0.398901 38.597235 157.882913
0 0.399224 38.597235 158.115354
+10 0.400593 38.597235 158.272559
+20 0.401448 38.597235 158.524118
% Change t1 S K
-20 0.399333 38.600815 158.095851
-10 0.399279 38.599028 158.106061
0 0.399224 38.597235 158.115354
+10 0.399170 38.595447 158.124164
+20 0.399116 38.593661 158.132515
Effects of Ramp Parameter ( )
% Change t1 S K
-20 0.399032 38.826936 158.586542
-10 0.399133 38.710155 158.346963
0 0.399224 38.597235 158.115354
+10 0.399307 38.488159 157.886035
+20 0.399345 38.129344 156.782943
Chapter 5 An EPQ Model for Decaying I
-153-
Fig. 5.2: Variation in t1 with respect to .
Fig. 5.3 with respect to .
Chapter 5 An EPQ Model for Decaying I
-154-
Fig. 5.4 with respect to .
Fig. 5.5: Variation in t1 with respect to .
Chapter 5 An EPQ Model for Decaying I
-155-
Fig. 5.6 with respect to .
Fig. 5.7: Variation in Total with respect to .
Chapter 5 An EPQ Model for Decaying I
-156-
Fig. 5.8: Variation in t1 with respect to µ .
Fig. 5.9: Variation in with respect to µ .
Chapter 5 An EPQ Model for Decaying I
-157-
Fig. 5.10: Variation in with respect to µ .
5.5 OBSERVATIONS
From the numerical illustration of the model, it is observed that the
period in which inventory holds increases with the increment in
backlogging and ramp parameters while inventory period decreases with
the increment in deterioration. Initial inventory level decreases with the
increment in deterioration and ramp parameters while inventory level
increases with the increment in backlogging parameter. The total average
cost of the system goes on increasing with the increment in the
backlogging and deterioration parameters while it decreases with the
increment in ramp parameters.
Chapter 5 An EPQ Model for Decaying I
-158-
5.6 CONCLUSIONS
A production system for deteriorating items is developed with
ramp type demand rate. Demand rate is taken as exponential ramp type
function of time, in which demand decreases exponentially for the some
initial period and becomes steady later on. The effects of inflation are not
considered in some inventory models. However, from a financial point of
view, an inventory represents a capital investment and must complete
capital funds. Thus, it is necessary to
consider the effects of inflation on the inventory system. Therefore, this
concept is also taken in this model. Shortages are allowed in inventory
with the concept of partial backlogging. Backlogging rate is exponential
decreasing function of time. Holding cost is also taken as time dependent.
A numerical assessment of the theoretical model has been done to
illustrate the theory. The solution obtained has been checked for
sensitivity with the result that the model is found to be quite suitable and
stable.
The proposed model can be extended to stochastic demand pattern.
Also, we could extend the model to incorporate some more realistic
features, such as quantity discount or the unit purchase cost, the inventory
holding cost and others are also fluctuating with time.
Chapter 5 An EPQ Model for Decaying I
-159-
REFERENCES
[1] Deng P.S., Lin R.H.J. and Chu P. (2007). A note on the inventory
models for deteriorating items with ramp type demand, E.J.O.R., 178,
112-120.
[2] Giri B.C., Jalan A.K. and Chaudhuri K.S. (2003). Economic order
quantity model with Weibull distribution deterioration, shortage and
ramp-type demand, I.J.S.S., 34(4), 237-243.
[3] Goswami A. and Chaudhuri K.S. (1991). An EOQ model for
deteriorating items with a linear trend in demand, J.O.R.S., 42(12),
1105-1110.
[4] Goyal S.K. (1986). On improving replenishment policies for linear
trend in demand, Engineering costs and production Economics
(E.C.P.E.), 10, 73-76.
[5] Hill R.M. (1995). Inventory model for increasing demand rate followed
by level demand, J. of Operational Research Society, 46, 1250-1259.
[6] Mandal B. and Pal A.K. (1998). Order level inventory system with
ramp type demand rate for deteriorating items, Journal of
interdisciplinary Mathematics, 1, 49-66.
[7] Manna S.K. and Chaudhuri K.S. (2006). An EOQ model with ramp
type demand rate, time dependent deterioration rate, unit production cost
and shortages, E.J.O.R., 171, 557-566.
[8] Panda S., Senapati S. and Basu M. (2008). Optimal replenishment
policy for perishable seasonal products in a season with ramp-type time
dependent demand, Computers and Industrial Engineering, 54, 2, 301-
314.
Chapter 5 An EPQ Model for Decaying I
-160-
[9] Singh S.R. and Singh T.J. (2007). An EOQ inventory model with
Weibull distribution deterioration, ramp type demand and partial
backlogging, Indian Journal of Mathematics and Mathematical Science,
3, 2, 127-137.
[10] Skouri K. et al. (2009). Inventory models with ramp type demand rate,
partial backlogging and Weibull deterioration rate, European Journal of
Operational Research, 192, 1, 79-92.
[11] Wu J.W., Lin C., Tan B. and Lee W.C. (1999). An EOQ inventory
model with ramp-type demand rate for items with Weibull deterioration,
Information and Management Science, 3, 41-51.
[12] Wu K. S. and Ouyang L.Y. (2000). A replenishment policy for
deteriorating items with ramp type demand rate (Short Communication),
Proceedings of National Science Council ROC (A), 24, 279-286.
[13] Wu K.S. (2001). An EOQ inventory model for items with Weibull
distribution deterioration, ramp type demand rate and partial
backlogging, P.P.C., 12(8), 787-793.