International Journal of Scientific & Engineering Research Volume 2, Issue 4, April2011 1
ISSN 22295518
An Integrated Partial Backlogging Inventory Model having Weibull Demand and Variable Deterioration rate with the Effect of Trade Credit
P.K. Tripathy, S. Pradhan
Abstract  Demand considered in most of the classical inventory models is constant, while in most of the practical cases the demand changes with time. In this article, an inventory model is developed with time dependent two parameter weibull demand rate whose deterioration rate increases with time. Each cycle has shortages, which have been partially backlogged to suit present day competition in the market. Also the effect of permissible delay is also incorporated in this study. The total cost consists of ordering cost, inventory holding cost, shortage / backordering cost, lost sale cost and deterioration cost are formulated as an optimal control problem using trade credit policy. Optimal solution for the model is derived and the trade credit on the optimal replenishment policy are studied with the help of numerical examples.
Index Terms Inventory, Shortages, partial backlogging, weibull demand, trade credit , variable deterioration, replenishment
—————————— • ——————————
In some real life situation there is a part of demand which
can not be satisfied from the current inventory, leaving the
n the classical inventory economic order quantity (EOQ) model, it was tacitly assumed that the customer must pay for the items as soon as the items are received. However, in practices or when the economy turns sour, the supplier frequently offers its customers a permissible delay in payments to attract new customer. In today’s business transaction, it is frequently observed that a customer is allowed some grace period before settling the accounts with the supplier or the producer. The customer does not have to pay any interest during this fixed period but if the payment gets delayed beyond the period interest will be charged by the supplier. This arrangement comes out to be very advantageous to the customer as he may delay the payment till the end of the permissible delay period. During the period, he may sell the goods, accumulate revenues on the sales and earn interest on that revenue. Thus it makes economic sense for the customer to delay the payment of the replenishment account upto the last day of the settlement period allowed by the supplier on the producer. Similarly for supplier, it helps to attract new customer as it can be considered some sort of loan. Furthermore, it helps in the bulk sale of goods and the existence of credit period serves to reduce the cost of holding stock to the user, because it reduces the amount of capital invested in stock for the duration of the credit period. So, the concept of permissible delay in
payments beneficial both for supplier and customer.

1. P.G. Department of Statistics, Utkal Universitym
Bhubaneswar751004, India, msccompsc@gmail.com
2. Dept. of Mathematics, Orissa Engineering College,
Bhubaneswar751007, India, suchipradhan@yahoo.com
system in stock out. In these systems two situations are mainly considered, all customers wait until the arrival of the next order (Complete back order case) or all customers leave the system (lost sales case). However, in practical, some customers are able to wait for the next order to satisfy their demands during the stock out period, while others do not wish to or can not wait and they have to fill their demands from other sources. This situation is modeled by consideration of partial back ordering in the formulation of the mathematical model. Wee (1999) developes a determinatic inventory model with quantity discount, pricing and partial backlogging when the product in stock deteriorates with time according a weibull distribution. Teng (2002) presents on EOQ model under the conditions of permissible delay. Chen et al. (2003) establish an inventory model having weibull deterioration and time varying demand. Wu et al. (2003) considered an inventory model where deteriorating rate and demand rate are follows the Chen’s model (2003) where shortages are permitted. Papachristos and Skouri (2003) present a production inventory model with production rate, product demand rate and deteriorating rate, all considered as functions of the time. Their model follows shortages and the partial rate is a hyperbolic function of the time up to the order point. They propose an algorithm for finding the solution of the problem. Abad (2003) considers the problem of determining the optimal price and lot size for reseller in which unsatisfied demand is partially backordered. There are several interesting papers related to partial backlogging and trade credits viz. park (1982), Jamal et al. (1997) , Lin et al. (2000), Dye et al. (2007) and their references.
With this motivation, in this paper an attempt is made to formulate an inventory model in time dependence of demand follows a two parameters weibull type with time variable deterioration where unsatisfied
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International Journal of Scientific & Engineering Research Volume 2, Issue 4, April2011 2
ISSN 22295518
demand is partially backlogged and delay payment is allowed. With the help of this model, supplier can easily extract its order quantity, cycle period for shortages as well as payment time to reduce the total inventory cost
To develop the proposed model, the following notations and assumptions are used in this article.
I(t) : Inventory level at time t
Q (t) : Order quantity at time t = 0
(Fig1)
In this model two parameter weibull type of demand is considered with variable rate of deterioration. Depletion of the inventory occurs due to demand (supply) as well as due to inventory i.e. during the period (0,T) and shortages occurs during period (T1, T).
The differential equation of inventory level I (t) w.r.t. time
is given by
dI (t )
vt I t
A, h,C1,
+ =  a t 1
dt
(1)
C2, C3 : Ordering cost, holding cost, deteriorating cost,
shortage cost for backlogged items and the unit
cost of lost sales respectively. All of the cost
With boundary condition I (0) = Q and I (T1) = 0
The solution of equation (1) is
parameters are positive constants. Ie : Interest which can be earned
I (t ) =
a v
+ 2
(t +2  T +2 )+
(2)
Ic : Interest charges which invested in inventory,
Ic Ie
a (T  t ) avt TI
1 2
,0 < t < T
M : Permissible delay in setting the amounts, 0<M< T T : Length of the replenishment cycle
T1 : Time when inventory level comes down to zero, 0
and Q = aT
av T +2

+ 2
(3)
< T1< T
Similarly the differential equation for shortages is given by
TIC1 : Total inventory cost when M < T1
dI (t )
= 
a t
1
(4)
TIC2 : Total inventory cost when T1, < M < T
dt 1 + 8 (T  t )
The solution of equation (4) for shortage is
a 8 t
+1
1. The inventory system involves only one item
2. Replenishment occurs instantaneously on ordering
I(t) = at
(8T  1) 
+ 1
(5)
i.e. lead time is zero and it takes place at an
The deteriorating cost during one cycle is
infinite rate.
T T1
C Q fa t 1dt
 Ca vT +2
(6)
3. The demand rate of any time t is
a t 1
two
DC =
1 
L 0
= + 2
parameter weibull type, where
0 < a << 1, > 0
The holding cost for carrying inventory over the period
are called scale and shape parameter respectively.
4. Q (t) = vt is the variable deterioration rate, where 0
(0,T) is given by
T1 T
+1
+3
< v <<1.
HC = f h I (t )dt = h a T1
avT
 1
( + 9)
(7)
5. Shortages are allowed and unsatisfied demand is O
L + 1
6( + 3) J
backlogged at the rate of 1
1+ 8 (T  t )
The backlogging parameter 8 is a positive constant, where T1 � t < T .
The total amount of shortage cost during the period (T1, T) is given by
T
SC =  C2 f I (t )dt
T1
T=a (8=T  1) (T +1  T +1) +
+ 1
= C2
(8)
a 8
+ 2
+ 2
( + 1)( + 2) (T
 T1 )
J
The amount of lost cost during the period (0,T) is given by
T T a t 1
LC = C3 f 1  1 + 8 (T  t )
T1 L J
T8 a T +1 T
= C3
 8a T1
lT  1 I
(9)
L
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+ 1
+ 1 J
International Journal of Scientific & Engineering Research Volume 2, Issue 4, April2011 3
ISSN 22295518
In this situation since the length of period with
The solutions for the optimal values of T1 and T can be found by solving the following equations simultaneously.
positive stock is larger than the credit period, the buyer can use the sale revenue to earn interest at an annual
a T IC1 = 0 and
aT
aTIC1 = 0
aT1
(13)
rate Ie in (0,T1)
Provided they satisfy the conditions
The interest earned IE1 , is
M
PI a
M +1
a 2TIC
1 > 0
and
a 2TIC
1 > 0
(14)
e 2 2
IE1 = PI e fa
O
t dt =
( + 1)
(10)
a T
aT1
In this situation interest charged IC2=0 and the interest earned per time unit is
T
IE 2
T
= PIe fa
L0
t dt + a T
(M  T )
J
T
= PIe a
MT  a
2T +1
(15)
L + 1 J
(Fig2)
However, beyond the credit period, the unsold stock is supposed to be financed with an annual rate Ic and the interest charged IC1 , is
T1
IC1
= PIc f I (t )dt
M
rT a
I
+1 av (
1 
+ 9)T + 3

= PI I + 1
6( + 3)
+3
I av l M
 M.T
+2 I
I l lL
M +1
+ 3 1 I J
av T M 3
+ a lT1
M  I  1
+ 1 6 J
(11)
(Fig3)
Total inventory cost per unit time is given by
given by
Then the total inventory cost per unit time is
TIC
= 1 [A + HC + SC + LC + DC + IC
 IE ]
1 T 1 1
TIC 2
= 1 [A + HC + SC + LC + DC + IC
 IE 2 ]
1 T Ir=a
T +1
av (
+ 9)T +3
ra (8T  1) T
T I + 1
6( + 3) I
2 l (
+ 1) ( 1
) I T
Ira
T1 a v (
+ 9)T I
= A + hH
L l
1  1
+ c H
J
T +1  T +1
= A + h H
T Il
+1
+ 1  6(
+3
+ 3) I
( +2  +2 ) Ir +1
I L
8 a T T1
8a T
H
8a T
T T1
+ ( + 1)(
+ 2)
+ 
3 I + 1
1 l 
I
+ 1 I
ra (8T  1)
8 a (T
+2  T
+2 )
ra v T
+2
J
Ira
l
T +1
av (
+ 9)T +3
J + C2 H
l
(T +1  T
+ 1
+1 )+
( + 1)(
+ 2)
 C H
1 + PI H
1  1
r 8a +1
T ra vT
+2
1 l + 2 J
C Il + 1
6( + 3)
+ C I T
 8aT
lT  1 I  C
1 I
av M +3 +2
M +1
av M 3T I
3 IH + 1
1 l + 1I
1 H + 2
JIJ
 l  MT1
I  a lT1
M  I + 1
r a 2T +1
+ 2 + 3
PI a M +1
+ 1 6 IJ
 PIe Ha
l
M T 
+ 1 JJ
(16)
 e (12)
+ 1 J
The solutions for the optimal values of T1 and T can be found by solving the following equations simultaneously.
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International Journal of Scientific & Engineering Research Volume 2, Issue 4, April2011 4
ISSN 22295518
a 2TIC
2 = 0
8T 2
and
a 2TI C
2 = 0
aT 2
(17)
of the model analysis by obtaining the optimal inventory length ,optimal economics order quantity, shortage time period and also calculated the total variable cost. From
the numerical examples it is found that for high values of
Provided they satisfy the conditions
the parameter , zero terminal inventory level provides
a 2TIC
aT 2
2 > 0
and
a 2TIC
aT 2
2 > 0 (18)
higher unit of the total variable cost. Thus for simplicity regarding the applicability of the model it is not preferable to adopt zeroterminal inventory level policy.
Case I : A=50, h=2, c1=2, C2=0.8, C3=2.0, a =0.0002, v=0.001, 8 =0.5, p=0.5, Ic= 0.2, Ie=0.1, m =0.05
T1  T  Q  TIC1  
0.8  1.90535191  106.840667  0.000335  0.718680 
1.0  2.2908332  77.122890  0.000459  1.130076 
1.5  2.917128  40.368203  0.001  2.662553 
2.0  3.01722969  24.922125  0.001829  4.927522 
2.5  2.77056753  17.154589  0.002566  7.866730 
3.0  2.4129991  12.736923  0.00282  11.374538 
3.5  2.07280312  9.991002  0.002571  15.332234 
4.0  1.78842979  8.166399  0.00205  19.628761 
4.5  1.56061254  6.889604  0.001485  24.170034 
5.0  1.379736015  5.958621  0.001001  28.880925 
Case II :
The proposed model can be extended in numerous ways. For example we may consider the effect of inflation and time value of money by taking discount rate.
The demand rate function D(t) =a t 1, a, > 0 follows two parameters weibull distribution in time t.
Thus functional form represents increasing, decreasing and constant demand for different values of the parameter
We have
A=50, h2, c1=2, C2=0.8, C3=2.0, a =0.0002, v=0.001, 8
=0.5, p=0.5, Ie=0.1, m =0.5
and
dD(t ) = a (
dt
d 2 D(t )
= a (
dt 2
 1)t 2
 1)(
 2)t 3
(19)
(20)
i) for 0< <1,
D(t ) < 0
and
d D(t ) > 0
the demand rate
dt dt
decreased with time at an increasing rate.
ii) =1, the demand rate becomes steady over time.
iii) 1< � 2,
d D(t )
dt
> 0 ,
d 2 D(t )
dt 2
� 0, then the demand
In this paper, an EOQ inventory model is considered for determining the optimal replenishment schedule under variable deterioration with two parameter weibull type demand where shortages are partially back logged. Also the proposed model in cooperates other
increases with time at decreasing rate.
Case (i) is applicable to the seasonal products towards the end of the seasons.
Case (ii) is applied to the consumer goods long established in the market by their brand name.
Case (iii) is appropriate for some fashionable product lunched in the market
2
realistic phenomenon and practical features such as trade credit period. The credit policy in payment has become a
If >2 then
d D(t ) > 0
dt
and
d D(t ) > 0,
dt 2
which implies
very powerful tool to attract new customers and a good incentive policy for the buyers. In keeping with this reality,
these factors are incorporated into the present model. The model is very useful in retail business. In real market
situations demand rate may be increased, decreased or constant. Keeping these consideration, we have assumed
here that the time dependence of demand follows the two parameters weibull distributions where depending upon the values of the parameter (<1,>1 and >2 ),the degree of nonlinearity varies. (see Appendix I) Numerical examples are presented to justify the claim of each case
that the demand will go on increasing with time at an
increasing rate. Thus there will be an accelerated growth in demand, which is not very realistic.
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