 Open Access
 Total Downloads : 549
 Authors : R. Amutha, Dr. E. Chandrasekaran
 Paper ID : IJERTV2IS50507
 Volume & Issue : Volume 02, Issue 05 (May 2013)
 Published (First Online): 27052013
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
An Inventory Model For Deteriorating Items With Three Parameter Weibull Deterioration And Price Dependent Demand
1R. Amutha
Research Scholar
Department of Mathematics, Presidency College, Chennai05
2Dr. E. Chandrasekaran
Associate Professor
Department of Mathematics, Presidency College, Chennai05
Abstract
The paper presents an inventory model for deteriorating items with price dependent demand. Deterioration rate follows a three parameter Weibull distribution. Shortages are allowed and are completely backlogged. The results are illustrated with the help of numerical example. Sensitivity analyses are carried out to analyze the effect of changes in the optimal solution with respect to change in one parameter at a time.
Keywords: Deterioration items, holding cost, Inventory, Price dependent demand time, Shortages.

Introduction
Inventory models are classified in to three categories (1) Deterioration (2) Obsolescence
(3) no deterioration/no obsolescence. Deterioration makes the product value dull.
Ajanta Roy [1] presented an inventory model for time proportional deterioration rate and demand is function of selling price. The Author discussed the model without shortage and also with shortages in which the shortages are completely backlogged. Mukesh Kumar, Anand Chauhan, Rajat Kumar [8] extended Ajanta Roy models with trade credit. Tripathy C.K and L.M. Pradhan [13] gave a model in which the demand of the product decreases with the increase of time and sale price and deterioration rate follows a three parameter Weibull distribution. Now Tripathy C.K and L.M. Pradhan [14] included salvage value and developed an EOQ model for three parameter Weilbull distribution deterioration rate under permissible delay in payments. Padmanabhan.G, Prem Vrat [9] formulated an EOQ model for perishable items under stock dependent selling rate. Sahoo.N.K., Sahoo .C.K. & Sahoo.S.K described an inventory model for price dependent demand and time varying holding cost. Vikas Sharma and Rekha Rani Chaudhary [16] explained and inventory model for two parameter Weibull deterioration rate.
They found profit for their model. Sanjay JAIN and Mukesh KUMAR [12] explained an inventory model with ramp type demand and three parameter Weibull deterioration rate. The Authors also analyzed and summarized economic order quantity models done by few researchers. There are some products which start deteriorate only after some interval of time. This was explained by taking three parameter Weibull distribution deterioration rate. Anil Kumar Sharma, Manoj Kumar Sharma and Nisha Ramani [2] described an inventory model for two parameter Weibull distribution deterioration rate and demand rate is power pattern. Manoj Kumar Meher, Gobinda Chandra Panda, Sudhir Kumar Sahu [7] adopted a two parameter Weibull distribution deterioration to develop an inventory model under permissible delay in payments. Kun Shan Wu [6] made an attempt in his paper to obtain the optimal ordering quantity of deteriorating items for two parameter Weibull distribution deterioration under shortages and permissible delay in payments. P.K.Tripathy and S.Pradham [15] also define an inventory model with two parameter Weibull distribution as demand rate and deterioration rate increases with time.
In this present paper, we have developed an inventory model for threeparameter Weibull deterioration rate and price dependent demand. Shortages are allowed and are completely backlogged. Holding cost is assumed to be constant. Our aim is to increase the profit.

Assumptions and notations

The demand rate is a function of selling price.

Shortages are allowed and are completely backlogged

Lead time is zero.

Replenishment is instantaneous

A is the set up cost

C is the unit cost of an item

p is the selling price

Demand D (t) = f (p) = ap, where a > p.

C2 is the shortage cost per unit time
(x) (t) = (t)1, 0 < 1 , > 0 and
T 2
p) [ 1 +
2
( ) 2
( 1)( 2)
2 ( )2 2
+
+
]
]
4(2 1)( 1)
– < < is the deterioration rate. At time T1 the Inventory becomes Zero and shortages start occurring.
(6)
Stock loss due Deterioration

h is the constant holding cost.

T is the length of the cycle.
T1
D = (ap) e (t ) 0
T1
dt (ap) dt
0


Mathematical formulation and solution
(T
) 1
2 (T
)2 1
= (ap) [ 1 + 1 –
Let I (t) be the inventory at time T ( 0 t T ) the differential equation for the instantaneous state over
( ) 1
1
–
–
2 ( )2 1
2(2 1)
(0,T) are given by
dI (t)
+ (t )
1
I (t) = – (ap), 0 t T1
]
]
1 2(2 1)
(7)
dt
(1)
Order Quality
dI (t)
dt
(2)
= – (ap), T1 t T
T
Q = [D + (a p)dt ]
0
With boundary condition I (T1) = 0
(T
) 1
2 (T
)2 1
= (ap) [ 1 + 1 + T –
Solving equation (1)
( ) 1
1
–
–
]
]
2 ( )2 1
2(2 1)
1
1
I (t) = – (ap) [(tT1) +
2
1
((t ) 1 – (T – ) 1 ) +
1 1
1 1
(8)
1
2(2 1)
2(2 1)
(3)
((t )2 1 – (T – )2 1 ) ], 0 t T
CQ
Purchase cost =
T
Solving equation (2)
C (T
) 1
2 (T )2 1
= – (ap) (tT ), T t T
= (ap)
[ 1 + 1 + T –1 1
(4)
T 1
2(2 1)
Shortage Cost
( ) 1
2 ( )2 1
–
] (9)
C T 1
2(2 1)
SC = 2 (a p) (tT1) dt
T T1
Total Profit Per unit time is
1
C2
= (ap) (TT1)2
= p (ap) –
T
[Ordering cost + purchase cost +2T
(5)
shortages cost + holding cost]
Holding Cost
1
K(p,T, T1) = p (ap) –
T
{ A +
C2
(ap) (TT1)2 + C (ap)
2
h T1
(T
) 1
( ) 1
–
–
HC =
T
I (t) dt
0
( 1 –
1
1

h
(T ) 2
2 ( )2 1
2 (T ) 1
= (ap) [
T
1
( 1 – T1 (T1 ) ) +
1
1
1
1
+
+
2
2(2 1)
2(2 1)
+ T ) h (ap)
2 (T
)2 2 [
( 1 – T (T – )2 1 )] + h (a
2(2 1) 2( 1) 1 1
1
(T ) 2
2
( ) 2
2 ( )2 2
( 1 T (T – )
1 ) + – ]
2 1 1
1 )
1 )
(T )2 2
2(2 1)
T 2
( 1)( 2)T 2
4T 2 (2 1)( 1)
(10)
2 1 1
( – T1 (T1 )] + h (ap) [ +
K ( p,T )
1 C (T Tv)2
2( 1)
( ) 2
2
+ ]}
+ ]}
2 ( )2 2
= (a2p) –
p
{ 2 – C
T 2
( 2)( 1) 4(2 1)( 1)
<>Let T1= vT, 0 < v < 1
(Tv ) 1
(
(
1
( ) 1
–
–
–
–
1
1
K (p,T) = p (ap) –
T
{ A +
C2
(ap) (TvT)2 + C (ap)
2
2 ( )2 1
2(2 1)
(Tv ) 2
2 (Tv )2 1
+
+
2(2 1)
+ T) + h [ (
1
2
(
(
–
–
(vT ) 1 ( ) 1 – Tv (Tv ) 1 ) + (
1
2 ( )2 1
1
+
+
2 (vT )
–
2 1
2
(Tv )2 2

Tv (Tv – )
2 1
2(2 1)
(Tv)2
)] h [ +
2(2 1)
2(2 1)
+ T ) h (ap) [
2( 1) 2
(
(
1
(Tv ) 2
2

vT (vT – )
1 ) +
( ) 2
+
+
( 2)( 1)
2 ( )2 2
4(2 1)( 1)
]} = 0. (11)
2 For the maximization of profit we set,
2(2 1)
(
(
(vT )2 2
+ ].
+ ].
2( 1)

vT (vT – )
2 1
)] + h(ap) [
K ( p,T )
T = 0 &
Provided
K ( p,T )
p = 0
(vT)2
( ) 2
2 ( )2 2
2 K ( p,T )
( ) (
( ) (
< 0,
2 K ( p,T ) )
2 ( 2)( 1) 4(2 1)( 1)
T 2
p2
< 0 and
K ( p,T )
A C2 2
( 2 K ( p,T ) )( 2 K ( p,T ) ) ( 2 K ( p,T ) )
–
> 0
–
> 0
= –
T T 2 2
(ap) (1 v)
– C(ap) [
T 2
p2
Tp
+
+
T ( 1)v(vT ) (vT ) 1
T 2 ( 1)


Numerical Example
+ +
+ +
( ) 1
2 ( )2 1
Let us consider C = 5, v = 0.2, C2
= 5, = 0.4,
T 2 ( 1) 2T 2 (2 1)
]
]
2T (2 1)v(vT )2 2 (vT )2 1
2T 2 (2 1)
+ h (ap)
[ [–
–
p
T
TP
Q
0.5
32.4215
5.9714
141.5371
108.3591
0.6
32.4534
5.9869
140.8515
109.2179
0.7
32.4866
6.0031
140.1385
110.1126
0.8
32.5210
6.0198
139.3986
111.0397
p
T
TP
Q
0.5
32.4215
5.9714
141.5371
108.3591
0.6
32.4534
5.9869
140.8515
109.2179
0.7
32.4866
6.0031
140.1385
110.1126
0.8
32.5210
6.0198
139.3986
111.0397
T ( 2)v(vT ) 1 (vT ) 2
= 2, = 1, h = 1, a = 50, A = 1000 in proper units we get p = 32.3909, T = 5.9564, TP = 142.1949, Q = 107.5330

Sensitivity Analysis
( 1)( 2)T 2
v2 (vT ) +
v 2T (2 2)(vT )2 1 2 (vT )2 2
4(2 1)( 1)T 2
2 v2 (vT )2
–
2
v 2
] h (ap) [ –
2
a
p
T
TP
Q
60
36.7900
5.2035
346.5236
124.2395
70
41.3870
4.6959
605.7523
138.6358
80
46.0894
4.3198
918.4367
151.5396
H
p
T
TP
Q
2
32.43.03
5.9277
139.9951
106.7867
3
32.4698
5.8998
137.8099
106.0562
4
32.5093
5.8125
135.6390
105.3385
5
32.5488
5.8458
133.4822
104.6336
H
p
T
TP
Q
2
32.43.03
5.9277
139.9951
106.7867
3
32.4698
5.8998
137.8099
106.0562
4
32.5093
5.8125
135.6390
105.3385
5
32.5488
5.8458
133.4822
104.6336
From this table, it can be observed that
C2
p
T
TP
Q
4
31.8084
6.5334
177.8724
121.6404
3
31.1694
7.3674
218.8579
141.8183
2
30.4342
8.6899
267.7428
174.0637
1
29.4950
11.0218
329.7243
234.6714
C2
p
T
TP
Q
4
31.8084
6.5334
177.8724
121.6404
3
31.1694
7.3674
218.8579
141.8183
2
30.4342
8.6899
267.7428
174.0637
1
29.4950
11.0218
329.7243
234.6714

p(t, p) is slightly sensitive to changes in , h and it is highly sensitive to changes in a, C2, C2 and v.

p is slightly sensitive to changes in , h
and moderately sensitivity to changes in a, C2, C2 and v.

Q is slightly sensitive to changes in the values of , h and C and it is highly
sensitive to changes in a, C2 and v.

T is slightly sensitive to changes in the values of , a, h and C and moderately sensitive to changes in C2 and v


Conclusion
C
p
T
TP
Q
6
32.9898
6.0645
124.4512
105.7209
7
33.5930
6.1782
107.3297
103.8447
8
34.2008
6.2979
90.8342
101.8983
9
34.8138
6.4246
74.9689
99.8804
C
p
T
TP
Q
6
32.9898
6.0645
124.4512
105.7209
7
33.5930
6.1782
107.3297
103.8447
8
34.2008
6.2979
90.8342
101.8983
9
34.8138
6.4246
74.9689
99.8804
A deterministic inventory model for deteriorating inventory model with three parameter Weibull distribution deterioration rate has been developed. Demand rate is function of selling price and holding cost is constant occurring shortages and completely backlogged. A numerical example is also given in support of the theory. A future research it may be consider to extend the model under permissible delay in payments.

References
v 
p 
T 
TP 
Q 
0.3 
31.6714 
6.3800 
179.1983 
121.7266 
0.4 
30.8940 
6.1710 
202.9896 
130.9089 
0.5 
30.2681 
5.4801 
206.8682 
129.9765 
0.6 
29.9177 
4.7713 
193.7094 
123.3320 
v 
p 
T 
TP 
Q 
0.3 
31.6714 
6.3800 
179.1983 
121.7266 
0.4 
30.8940 
6.1710 
202.9896 
130.9089 
0.5 
30.2681 
5.4801 
206.8682 
129.9765 
0.6 
29.9177 
4.7713 
193.7094 
123.3320 

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