An Inventory Model for Weibull Deteriorating Items with Linear Demand Pattern
L. K. RAJU 1 , U. K. MISRA 2 , SRICHANDAN MISHRA 3 and G. MISRA 4
1 Department of Mathematics
NIST, Paluru Hills, Berhampur, Odisha, INDIA.
2 Department of Mathematics
Berhampur University, Berhampur, Odisha, INDIA.
3 Department of Mathematics
Govt. Science College, Malkangiri. Odisha, INDIA.
4 Department of Statistics,
Utkala University, Bhubaneswara Odisha, INDIA.
(Received on: June 30, 2012) ABSTRACT
The objective of this model is to investigate the inventory system for perishable items with linear demand pattern where two parameter Weibull distribution for deterioration is considered. The Economic order quantity is determined for minimizing the average total cost per unit time. As the rate of deterioration increases, the optimal time of the inventory decreases and the required number of items for the fixed length of each ordering or the production cycle, the minimum total operational cost of the inventory and the required items for the fulfillment of backorder increases. The application is illustrated with suitable examples.
Keywords: Inventory system, Linear demand, Deterioration, Weibull distribution.
AMS Classification No: 90B 05 .
1. INTRODUCTION
The deterioration of many items during storage period is a real fact. In many inventory models, it is assumed that the items can be stored indefinitely without any
risk of deterioration. However, certain types
of items undergo changes while in storage so
that, with time, they become partially or
entirely unfit for use. Deterioration refers to
damage, spoilage, vaporization, or
obsolescence of the products. There are
441 L. K. Raju, et al., J. Comp. & Math. Sci. Vol.3 (4), 440-445 (2012)
Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)
several types of items that will deteriorate if stored for extended periods of time.
Examples of deteriorating items include metal parts, which are prone to corrosion and rusting, and food items, which are subject to spoilage and decay. Electronic components and fashion clothing also fall into this category, because they can become obsolete over time and their demand will typically decrease drastically.
Inventory control for deteriorating items has been a well-studied problem.
Numerous optimal and heuristic approaches have been developed for modeling and solving different variations of this problem.
C. K. Tripathy, L. M. Pradhan and U.K.
Mishra 8 have developed a model with linear deterioration rate where the demand rate is considered to be constant. Sarbjit and S.S.
Raj 4 have developed a model whose demand as well as perishability rate increases with time. Ghare and Schrader 2 use the concept of deterioration. Covert and Philip 1 also formulated a model with variable rate of deterioration with two parameter Weibull distributions which was further extended by Shah 6 . Sarbjit and Raj 5 also developed the model for items having linear demand and variable Deterioration rate with trade credit.
Tomba et. al. 7 developed a model with linear demand pattern and deterioration with shortages. Sanjay Jain and Kumar 3 established the model having power demand pattern, Weibull distribution deterioration and shortages.
In this paper an attempt has been made to develop an inventory model for perishable items with two-parameter Weibull density function for deterioration and the linear demand pattern is used over a finite planning horizon. Nature of the model
is also discussed for shortage state. Optimal solution for the proposed model is derived and the applications are investigated with the help of some numerical examples.
2. ASSUMPTIONS AND NOTATIONS Following assumptions are made for the proposal model:
i. Single inventory will be used.
ii. Lead time is zero.
iii. The model is studied when shortages are allowed.
iv. Demand follows the linear demand pattern
v. Shortages are allowed and are completely backlogged.
vi. Replenishment rate is infinite but size is finite.
vii. Time horizon is finite.
viii. There is no repair of deteriorated items occurring during the cycle.
ix. The second and higher powers of α
are neglected in this analysis of the model hereafter.
Following notations are made for the given model:
) (t
I = On hand inventory level at any time t , t ≥ 0 .
bt a t
R ( ) = + is the demand rate at time t . : θ = αβ β − 1
θ t , The two-parameter Weibull distribution deterioration rate (unit/unit time). Where 0 < α << 1 is called the scale parameter, β > 0 is the shape parameter.
Q = Total amount of replenishment in the beginning of each cycle.
V = Inventory at time t = 0
T = Duration of a cycle.
c d = The deterioration cost per unit item.
c h = The holding cost per unit item.
c b = The shortage cost per unit.
U =The total average cost of the system.
3. FORMULATION
The objective of the model is to determine the optimal order quantity in order to keep the total relevant cost as low as possible. The optimality is determined for shortage of items. Taking Q
amount of replenishment in the
each cycle, and after fulfilling backorders let
Here we have taken the total duration fixed constant. The objective here is to determine the optimal order quantity in order to keep the total relevant cost as low as possible.
= The deterioration cost per unit item.
= The holding cost per unit item.
= The shortage cost per unit.
=The total average cost of the system.
The objective of the model is to determine the optimal order quantity in order to keep the total relevant cost as low as possible. The optimality is determined for Q be the total amount of replenishment in the beginning of each cycle, and after fulfilling backorders let
V be the level of initial inventory. In the period ( 0 , t 1 ) the inventory level gradually decreases due to market demand and deterioration.
At t 1 , the level of inventory reaches zero and after that the shortages are allowed to occur during the interval [ t , 1 T
fully backlogged. Only the backlogging items are replaced by the next replenishment. The behavior of inventory during the period ( T 0 , ) is depicted in the following inventory-time diagram.
Here we have taken the total duration T as . The objective here is to determine the optimal order quantity in order to keep the total relevant cost as low as
If I (t ) be the on-hand inventory at time t ≥ 0 , then at time t + ∆ t , the on inventory in the interval [ ] 0 t , 1 will be
) ( t t
I + ∆ = I ( t ) − θ ( t ) I ( t ) ∆ t
be the level of initial inventory. In the the inventory level gradually decreases due to market demand and , the level of inventory reaches zero and after that the shortages are allowed
]
T , which are fully backlogged. Only the backlogging items are replaced by the next vior of inventory is depicted in the time diagram.
hand inventory at , the on-hand will be
t t
R ∆
− ( )
443 L. K. Raju, et al., J. Comp. & Math. Sci. Vol.3 (4), 440-445 (2012)
Journal of Computer and Mathematical Sciences Vol. 3, Issue 4, 31 August, 2012 Pages (422-497)
Dividing by ∆ t and then taking as ∆t → 0 we get
=
+ − ( )
)
( 1
t I dt t
t
dI β
αβ − ( a + bt ) ; 0 ≤ t ≤ t 1
(3.1) For the next interval [ ] t , 1 T , where the shortages are allowed we have
t t R t I t t
I ( + ∆ ) = ( ) − ( ) ⋅ ∆ .
Dividing by ∆ t and then taking as ∆t → 0 we get
T t t bt
dt a t
dI ( ) = − ( + ); 1 ≤ ≤ (3.2) The boundary conditions are I ( 0 ) = V and
0 ) ( t 1 =
I .
On solving equation (3.1) with boundary condition we have
; ) 2 ( ) 2 ( ) 1 ( ) ( )
(
1 2 22 2 1 1 1 1
1
−
+ +
− + + −
+
−
=
− β β+ β+ β+ β+β α β
α t a t t a α t t b t t b t t e
t
I
0 ≤ t ≤ t
1(3.3) On solving equation (3.2) with boundary condition we have
) 2 (
) ( )
( 1 b t 1 2 t 2
t t a t
I = − + − ;
T t
t 1 ≤ ≤ (3.4) Using I ( 0 ) = V in equation (3.3) we have
2 1 2
1 1 1
1 1 2 2
+ +
+ + + +
+
= β β
β α β
α b t
b t a t
at V
(3.5) The total amount of deteriorated units in
0 ≤ t ≤ t 1 is given by
dt t I t
t
) (
1
0
∫ α β β − 1
(3.6)
+ + +
+ + +
− +
− +
−
= + + + 1 2 + 2
1 2 1 2
1 1
1 2
1
1 2 1 2 ( 2 ) ( 2 1 )( 1 ) 4 ( 1 )( 2 )
β β
β β
β β
β β
β α β
β β α β
β β
α
αβ β b t
a t b t
a t V t
V t
The total cost function consists of the following elements:
)
(i Holding cost per cycle
∫
10
) (
t
h I t dt
C
(3.7)
+ + +
+ + +
− +
−
−
= + + 1 + 3
2 1 1
1 3
1 2 1
1 2 3 1 ( 1 )( 2 ) 2 ( 2 )( 3 )
β β
β
β β
β α β
β
β α β
α b t
a t V t
t b t t a V C h
)
(ii Deterioration cost per cycle
∫
1−
0
1 ( )
t
d t I t dt
C αβ β
(3.8)
+ + +
+ + +
− +
− +
−
= + + + 1 2 + 2
1 2 1 2
1 1
1 2
1
1 2 1 2 ( 2 ) ( 2 1 )( 1 ) 4 ( 1 )( 2 )
β β
β β
β β
β β
β α β
β β α β
β β
α
αβ C d V β t V t a t b t a t b t
)
(iii Shortage cost per cycle
∫
T
t
b I t dt
C
1
)
(
− + − − −
= 2 2 6 2 3
3 1 2 1 2 3
1 2 1
t b t a T b T t b T T a t a C b
(3.9) Taking the relevant costs mentioned above, the total average cost per unit time of the system is given by
t T V
U 1
) ,
( 1 = {Holding cost + Deterioration cost - Shortage cost} (3.10)
+ + +
+ + +
− +
−
−
= + + 1 + 3
2 1 1
1 3
1 2 1
1 2 3 1 ( 1 )( 2 ) 2 ( 2 )( 3 )
1 β β β
β β
β α β
β
β α β
α b t
a t V t
t b t t a V T C h
+ + +
+ + +
− +
− +
−
+ 1 1 2 1 + 1 1 + 2 1 2 + 1 1 2 + 2
) 2 )(
1 ( 4 )
1 )(
1 2 ( )
2 ( 2 1
2
β β
β β
β β
β β
β α β
β β α β
β β
α
αβ C d V β t V t a t b t a t b t
− + − − −
− 2 2 6 2 3
3 1 2 1 3 2
1 2 1
t b t T a T b t T b T a t a C b
Eliminating V from equation (3.10) we have
= ) ( t 1
U
+ + + +
+
+ + + 2 + 2
1 2
1 1 2 1 1
1 1 2 2
1
1 β β β β
β α β α
β β
α C a t a t b t b t
T d
(3.11)
+ + + +
+
− + + + 1 3 + 2
2 2 1 1 3 1 1
2
1 1 2 2
2
β β
β
β β α
β α β
α b t
b t a t
t a
+
+ + +
− +
− +
− + + + + 1 2 + 2
1 2 1 2
1 1
1 2 ( 2 ) ( 2 1 )( 1 ) 4 ( 1 )( 2 )
1
β β
β β
β β
β α β
β β α β
β t
t b t a
t b a
+ + + +
+ +
−
+ + + +
+
+
+ + + + + 12 +33 1 2 2 1 2
1 3
1 3
1 2 1 2
1
1 2 2 1 1 2 2
β β
β β
β
β β α
β α β α
β α β α
b t b t
a t t
a b t
b t a t
t a C
h
+
+ + +
+ +
−
−
+ 1 +32 1 3
1 2 1
