 Open Access
 Total Downloads : 342
 Authors : Ankit Bhojak, U. B. Gothi
 Paper ID : IJERTV4IS090195
 Volume & Issue : Volume 04, Issue 09 (September 2015)
 DOI : http://dx.doi.org/10.17577/IJERTV4IS090195
 Published (First Online): 11092015
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
An EOQ Model with Time Dependent Demand and Weibull Distributed Deterioration
Ankit Bhojak
Department of Statistics
GLS Institute of Commerce, GLS University Ahmedabad, Gujarat, India

B. Gothi
Head, Department of Statistics St. Xaviers College (Autonomous)
Ahmedabad, Gujarat, India
Abstract In this paper, we developed an inventory model for deteriorating items under time dependent demand function. Inventory holding cost is a linear function of time. For deterioration of units we considered two parameters Weibull distribution. In this study, a more realistic scenario is assumed where the part of shortages was backordered and the rest was lost with time. The backordering rate is variable and it is considered exponential function depending on waiting time for the next replenishment. Solution procedure of the developed model is presented with numerical example and its sensitivity analysis.
Keywords EOQ model; deterioration; time dependent demand; weibull distribution; shortages; inventory

INTRODUCTION
Deterioration of produced items is very important factor in inventory management and in any production. In the recent time more research work has been carried out in the inventory management to develop inventory models with deterioration of items and shortages are permitted. Commodities like vegetables, fruits and food items from depletion by direct spoilage while kept in store. It has been observed that the failure of many items may be expressed by Weibull distribution.
Ghare and Scharder [5] first formulated a mathematical model with a constant deterioration rate. Khemlnitsky and Gerchak [7] developed optimal control approach to production systems with inventory level dependent demand. Ruxian , Hongjie Lan and John Mawhinney [12] reviewed on deteriorating inventory study. Vipin Kumar, Singh and Sanjay Sharma [21] developed inventory model for profit maximization production with time dependent demand and partial backlogging. Papachristos and Skouri
[10] gave an inventory model with deteriorating items, quantity discount, pricing and timedependent partial backlogging. Samanta and Ajanta Roy [14] presented a production inventory model with deteriorating items and shortages. Inventory model with price dependent demand and time varying holding cost was given by Ajantha Roy [1]. Vinod Kumar Mishra and Lal Sahab Singh [20] have given deteriorating inventory model for time dependent demand and holding cost with partial backlogging. Kirtan Parmar, Indu Aggarwal and Gothi [9] have formulated an order level inventory model for deteriorating items under varying demand condition.Analysis of inventoryproduction systems with weibull distributed deterioration is given by Azizul Baten and Anton Abdulbasah Kamil [3]. Chen and Lin [4] developed optimal replenishment scheduling for inventory items with weibull distributed deterioration and time varying demand. Sanni [13] studied an economic order quantity inventory model with time dependent weibull deterioration and trended demand.
An EOQ Model for varying items with weibull distribution deterioration and pricedependent demand was given by Begum, Sahoo, Sahu and Mishra [11]. One more condition was imposed in the form of shortage by Wu and Lee [22] in their work, an EOQ inventory model for items with weibull distributed deterioration, shortages and time varying demand. Vikas Sharma and Rekha Rani Chaudhary [19] have also developed an inventory model for deteriorating items with Weibull deterioration with time dependent demand and shortages. Tripathy & Pradhan [17] formulated an integrated partial backlogging inventory model having weibull demand and variable deterioration rate with the effect of trade credit. Amutha and Dr. Chandrasekaran [2] developed an inventory model for deteriorating products with weibull distributed deterioration, time varying demand and partial backlogging. Tripathy and Pradhan [15] have given an EOQ model for weibull deteriorating items with power demand and partial backlogging.
Some more conditions were applied on inventory holding cost by Ghosh and Chaudhuri [6] in their research in an order level inventory model for a deteriorating item with weibull distribution deterioration, timequadratic demand and shortages. Tripathy and Mishra [16] developed an inventory model for Weibull deteriorating items with price dependent demand and time varying holding cost. Kirtan Parmar & U. B. Gothi [8] developed an EOQ model of deteriorating items using three parameter weibull distribution with constant production rate and time varying holding cost.
Vashistha & Gupta [18] has developed an inventory model with weibull distribution deterioration and time dependent demand. We have modified the same inventory model by considering inventory holding cost as a linear function of time and also including production cost to the total cost function to make the model more realistic. The
sensitivity analysis is also carried out by changing the values of all the parameters one by one. We have used different formulae to find costs like IHC, DC, SC and LSC.

NOTATIONS

The following notations are used to develop the mathematical model:

Q t : Inventory level of the product at time t ( 0).

Shortages are allowed and unsatisfied demand is
backlogged at a rate e T t , where the backlogging parameter is a positive constant.

Total inventory cost is a real and continuous function which is convex to the origin.

MATHEMATICAL MODEL AND ANALYSIS
The initial stock is S at time t = 0, then inventory level decreases mainly due to meet up demand with rate


R t : Demand rate varying over time.

a + b t + c t2 and partly from deterioration with rate


t : Deterioration rate

A : Ordering cost per order during the cycle period.

Ch : Inventory holding cost per unit.

Cd : Deterioration cost per unit.
t 1 and reaches to S at t = Âµ. The stock reduces to zero at t = t1 due to demand with rate a + m t where m b + c and the deterioration with rate t 1 . Thereafter, shortages are allowed to occur during the time
1
interval t1 , T and the demand during the period t1 , T

Cs : Shortage cost per unit.

pc : Production cost per unit.
is partially backlogged.

l : Opportunity cost due to lost sale per unit.

S : Initial stock level at the beginning of every
Inventory Level
t t 1
cycle.

S1 : Inventory level at t = Âµ.

S2 : The maximum inventory level during shortage
R t a + b t + c t2
period.

T : Duration of a cycle.

TCt1 , T : Total cost per unit time.
III. ASSUMPTIONS
R t a + m t
S
S1 where m b c
T
Time
The following assumptions are considered to develop this model.

Replenishment rate is infinite.

Leadtime is zero.

A single item is considered over the prescribed period of time.

No repair or replacement of the deteriorated items
O t1 S2
Fig. Graphical representation of inventory system
The rate of change of inventory during the
takes place during a given cycle.

We consider finite time horizon period.

Demand rate R(t) is assumed to be a funcion of time such that
intervals 0 , , , t1 and t1 , T
following differential equations
d Qt
is governed by the
R(t) =a +b t +c{ t t H t }t , where H
t is the Heavisides function defined as
+ t 1 Qt
d t
a + b t + c t2
0 t
(1)
H t 1,
0,
if t
and a is the initial rate of
if t
d Qt + t 1 Qt d t
a + m t where m
b + c
demand, b is the rate with which the demand rate increases. The rate of change in demand rate itself increases at a rate c. a, b and c are positive constants.

Holding cost is linear function of time and it is
d Qt
T t
t t1
(2)
Ch = h + r t (h, r > 0).

t t 1 is the two parameter Weibull distributed deterioration rate, where is scale
a + m t e t1
d t
t T
(3)
Now solving equation (1), (2) and (3) with boundary
parameter and is shape parameter and
0 < < < 1 .
condition Q(0) = S and Qt1 0 we get,
Q t = S 1 t
b t2 c t3 a t + 1
a t + + +
The total cost can be obtained by adding the following costs.
2 3 + 1
b t + 2
+ 2 + 2
c t + 3
+ 3 + 3
1
2
m a
0 t
(4)
Ordering cost OC = A
The deterioration cost during the period
0 , t1
(10)
Q t =
a t t1
t2 t2 +
+ 1
t + 1 t + 1
1
DC C
t 1 Q t dt +
t1
t 1 Q t dt
m
d
1 1
+ + 2
t + 2 t + 2 a t + 1 t t
0
S
a + 1
b + 2
c + 3
m t + 2 t2 t
t t
Cd
+ 1
+ 2 + 2 + 3 + 3
2 1
1
(5)
m t2 t
+ a t + 1 1
t2 t2 1 2
Q t =
a 1 Tt t a + m 1 T 1
1
2
t 1 1 2
2
t3 t3
a 1 m t1
+ 1
+ 2
+ m 1
t t T
3
1
(6)
The shortage cost during the period t1 , T
(11)
In equation (5) , Q S1 and so T
S = a t m t2 2 + a
t 1 1
SC Cs Q t dt
t
2
1 1 1
+ 1 1
1
T t 2
m 2
2
= C a 1 T 1
+ + 2 t1
a t1
s 2
2

m
t2 2
a + m 1 T T t 3
1
1 t T t 2
2 3
1 1
In equation (4), Q S1 which gives
(7)
m T t 4 3 t2
+ 1 t T t 3 1 T t 2
1
b 2
c 3
a + 1
3 4 1 1 2
1
S = 1
S1 a + 2 + 3
+ 1
(12)
b + 2
2 + 2
c + 3
3 + 3
The opportunity cost due to lost sales during the period
t1 , T
In equation (6), QT S2
and hence
(8)
T2 t2
LSC
T
l
t1
a + m t
1 e
T t
dt
S = a 1 T T t a m 1 T 1
m T a 2 2
2 1 2
l
a T T t1 T t1
m T3 t3
2
m 3 3
3 1
(9)
3 T t1
(13)
The holding cost during the period 0 , t1
S 1 a 2
b 3
c 4
t1
A h S
1
2 + 6 +
12
IHC
h r t Q t dt h r t Q t dt
0
a
2
+ b
3
S 1 a 2
b 3
c 4
1
2
2
3
h S
+ +
1
2 6 12
+
c 4
3
4
a 2
b 3
c 4
+ +
m t2
a t + 1
m t + 2
1
2
2
3
3
4
a t
1 1
1 t
1 2 + 1 + 2 1
m t2
a t + 1
m t + 2
a t1
1 1
1 t
1
a 2 t2 m 3 t3 a 2 t 2
2 + 1
+ 2
2 1 6
1
1
2 1
a 2 t2 m 3 t3 a 2 t 2
m 3 t 3
2 1 6
1
1
2 1
2
2
3 1
m
3
3
m t2 1 t 1

t
a t 1 1
2
2
3 1
1
2
+ 1
m t2 1 t 1
S 2 S 2 a 3 b 4 c 5
a t 1 1
1
+ r 2
2
3
+ 8 + 15
2
+ 1
2 2 3 4 5
a 3
b 4
r S
S a + b + c
+
2 2 3 8 15
1 3 2 4
c 5
a 3
+
b 4
+ c
5
+
3
5
1 3 2 4 3 5
2 + 1
+ 2 2 2
m t a t m t t
a t
1
1 1 1 1
m t2
a t + 1
m t + 2 2 t2
1 2
+ 1
+ 2 2
a t1
1 1
1 1
TC t , T
2 + 1
+ 2 2
1 T a
3 3
m 4 4
a 3 3
3
t1
8
t1
1
3
t1
a 3 t3 m 4 t4 a 3 t 3
3 1 8
1
1
3 1
m 4 t 4
2
2
4 1
m
4
4
t1
m t2 2 t 2
2 2 4
a t 1 1
1
2
+ 2
m t2 2 t 2
a t 1 1
1
T t 2
a + m 1 T
2
+ 2
C a 1 T 1
(14)
s
2 2
Production cost
T t 3
1 t T t 2
PC = pc S S2
(15)
3
1 1
4
m T t1 3 3 t2
2
The total cost per unit time is given by
+ t T t 1 T t
1 3 4 1 1 2
1
TCt1 , T
OC IHC SC DC LSC PC
T
S
a + 1
b + 2
c + 3
Cd
+ 1 + 2 + 2 + 3 + 3
m t2 t
+ a t + 1 1
1
2
t 1 1 t 2 2
a 1 m 1
+ 1
+ 2
m T a 2 2 m
3 3
l a T T t1 T t1 T
t1
2
3
p S S
c 2
1
Parameter 
% change 
T 
t1 
TCt1, T 
% changes in TCt1, T 
20 
7.445406199 
5.29016578 
9759.947791 
0.107044875 

10 
7.446835483 
5.290354243 
9765.177951 
0.053514251 

A 
10 
7.449691919 
5.290730829 
9775.633473 
0.053497897 
20 
7.451119074 
5.290918953 
9780.858839 
0.106979459 

20 
7.342798114 
5.300616593 
9325.835862 
4.550175556 

10 
7.394243966 
5.295717103 
9542.005314 
2.337683668 

10 
7.505103902 
5.285035598 
10012.36947 
2.476488128 

20 
7.565040232 
5.279127689 
10269.41498 
5.107346055 

20 
5.048259926 
4.210974646 
4023.445977 
58.82007598 

10 
6.14066536 
4.752816003 
6253.2881 
35.99766711 

10 
9.176093149 
5.811514762 
15991.71499 
63.67502185 

20 
12.12255666 
6.206800713 
31100.4103 
218.3123472 

20 
7.442946559 
5.289841412 
9730.31566 
0.410329403 

10 
7.445608265 
5.290192425 
9750.362325 
0.205152018 

a 
10 
7.450913956 
5.290891916 
9790.44823 
0.205126771 
20 
7.453557987 
5.291240402 
9810.487494 
0.410228412 

20 
7.39193315 
5.287053749 
9554.925649 
2.205444184 

10 
7.42019919 
5.288813518 
9662.84006 
1.100941408 

b 
10 
7.476130652 
5.292241495 
9877.627337 
1.097403945 
20 
7.503801812 
5.29391074 
9984.504834 
2.191293912 

20 
7.046694019 
5.270130313 
8255.748149 
15.50251118 

10 
7.252724663 
5.281124747 
9021.998138 
7.659951222 

c 
10 
7.634206651 
5.298550067 
10501.30021 
7.480688706 
20 
7.811326283 
5.305288724 
11215.01677 
14.78556963 

20 
7.448155361 
5.290556183 
9770.554284 
0.001512462 

10 
7.448209705 
5.290549388 
9770.480401 
0.000756263 

10 
7.448318413 
5.290535795 
9770.332615 
0.000756326 

20 
7.448372776 
5.290528998 
9770.258712 
0.001512716 
Our objective is to determine optimum values
t and T
TABLE – I SENSITIVITY ANALYSIS
such that
TCt1 , T is minimum. Note that
t and T are
1
the solutions of the equations
TCt1 , T 0 &
t1
such that
TCt1 , T 0
T
(17)
2TCt , T 2TC t , T 2TC t , T 2
1
t2
1
T2
1
t T
0
1 1
t t , T = T
1 1
2TCt , T
1
1
t2
0
1 1
t t , T = T
(18)
The optimal solution of the equations in (17) can be obtained by using the software. This has been illustrated by the following numerical example.

NUMERICAL EXAMPLE
Let us consider the following example to illustrate the above developed model. We consider the following values of the parameters A = 300, h = 2, r = 16, = 5, Cd = 18, l
= 10,
= 0.0001, = 5, a = 2, b = 3, c = 5, m = 28, Cs = 17,
= 0.0002 & pc = 15 (with appropriate units of
measurement). We obtain the optimal values
1
T = 7.448264056 units, t = 5.2905425919 units and
optimal total cost
TCt1 , T
= 9770.406510 units by
using appropriate software.

SENSITIVITY ANALYSIS
Sensitivity analysis helps in identifying the effect of optimal solution of the model by the changes in its parameter values. In this section, we study the sensitivity
of total cost per time unit TCt1 , T with respect to the
changes in the values of the parameters A, , Âµ, a, b, c, , h, pc, Cs, Cd, l , m and r.
The sensitivity analysis is performed by considering 10% and 20% increase and decrease in each one of the above parameters keeping all other remaining parameter as fixed. The results are presented in Table 1. The last column of
table shows the % change in TCt1 , T as compared to
the original solution corresponding to the change in parameters values.
td>
0.474813625
Parameter
%
change
T
t1
TCt1, T
% changes in
TCt1, T
20
7.423514172
5.288219977
9677.559835
0.950284678
10
7.435909423
5.289386344
9724.01529
h
10
7.460578348
5.291688823
9816.733642
0.474157667
20
7.472852573
5.292825141
9862.996829
0.947660856
20
7.610933659
5.326173188
9690.57666
0.817057616
10
7.529209709
5.308272694
9731.07186
0.402589707
pc
10
7.368092895
5.272982062
9808.5799
0.390704208
20
7.288692191
5.255590224
9845.591279
0.769515246
20
7.70899727
5.260068445
9378.29687
4.013237731
10
7.56889612
5.276149679
9584.207555
1.905744205
Cs
10
7.343048417
5.303527912
9939.95053
1.735281115
20
7.250290241
5.315322778
10095.23773
3.324643822
20
7.446451526
5.293180369
9756.435631
0.142991795
10
7.447344685
5.291852949
9763.390681
0.071806942
Cd
10
7.449209062
5.289248958
9777.48175
0.072414993
20
7.450179142
5.287971718
9784.615065
0.145424388
20
7.448290504
5.290539379
9770.364749
0.000427436
10
7.44827728
5.290540986
9770.38563
0.000213717
l
10
7.448250832
5.290544198
9770.427392
0.000213715
20
7.448237609
5.290545805
9770.448272
0.000427427
20
7.966953092
5.312540139
9504.831551
2.718156703
10
7.687810269
5.30157837
9652.993137
1.201724553
m
10
7.240214212
5.279778227
9864.625505
0.964330334
20
7.057659597
5.269453198
9940.922426
1.745228462
20
7.065941013
5.277231694
8311.034745
14.9366535
10
7.263975282
5.28540387
9055.049867
7.321667149
r
10
7.62108883
5.293563685
10460.48512
7.062946719
20
7.784171635
5.295077033
11128.07187
13.89568957

CONCLUSION
From the above sensitivity analysis we may conclude that the total cost per time unit TCt1 , T is highly sensitive to
changes in the values of the parameters Âµ, c and r moderately sensitive to changes in the values of the parameters , b, Cs, m and less sensitive to changes in the values of the parameters A, a, , h, pc, Cd, and l .
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