9 different deterioration rates of two warehouse inventory model with time and price dependent demand under inflation and permissible delay in payments 9 raman patel

ISSN No. (Print): 0975-1718
ISSN No. (Online): 2249-3247
Different Deterioration Rates of two Warehouse Inventory Model with
Time and Price Dependent Demand under Inflation and Permissible
Delay in Payments
Raman Patel
Department of Statistics,
Veer Narmad South Gujarat University, Surat, (Gujarat), INDIA
(Corresponding author: Raman Patel)
(Received 03 January, 2018, accepted 27 January, 2018)
(Published by Research Trend, Website: www.researchtrend.net)
ABSTRACT: A two warehouse inventory model with different deterioration rates is developed. Demand is
considered as function of price and time. Holding cost is considered as linear function of time. Inflation factor
is also considered with permissible delay. Shortages are not allowed. Numerical case is given to represent the
model. Affectability investigation is likewise done for parameters.
Keywords: Two warehouse, Different deterioration, Time dependent demand, Price dependent demand, Inflation,
Permissible Delay in Payments
I. INTRODUCTION
Inventory problems for deterioration items have
been studied extensively by many researchers from time
to time. Whitin [25] developed inventory model for
fashion goods deteriorating at the end of prescribed
storage period. Ghar and Schrader [7] developed an
inventory model with a constant rate of deterioration.
Shah and Jaiswal [19] considered an order level
inventory model for items deteriorating at a constant
rate. Alfares [2] developed an inventory model with a
stock level demand rate and a variable holding cost with
the assumption that holding cost increases with time
spent in storage. The related works are found in
(Nahmias [13], Raffat [16], Goyal and Giri [9], Ruxian
et al. [17]).
Buzacott [4] considered inventory model by
considering inflationary impacts into record. Su et al.
[22] developed model under inflation for stock
dependent consumption rate and exponential decay.
Moon et al. [12] developed models for ameliorating /
deteriorating items with time varying demand pattern
over a finite planning horizon taking into account the
effects of inflation and time value of money.
It is generally assumed that a supplier must be paid
for items as and when the customer receives the items.
But many times it happens that the supplier allows
credit for some fixed time period in settling the
payment for the product and is not charged any interest
from the customer for that specified period. However, if
he pays beyond that specified period, then the interest
will be charged. Goyal [8] first considered the
economic order quantity model under the condition of
permissible delay in payments. Goyal’s [8] model was
extended by Aggarwal and Jaggi [1] for deteriorating
items. An inventory model with varying rate of
deterioration and linear trend in demand under trade
credit was considered by Chang and Dye [5]. Teng et
al. [23] developed an optimal pricing lot sizing model
by considering price sensitive demand under
permissible delay in payments. An inventory model for
stock dependent consumption and permissible delay in
payment under inflationary conditions was developed
by Liao et al. [11]. Singh [21] developed an EOQ
model with linear demand and permissible delay in
payments. The effect of inflation and time value of
money were also taken into account. Patel and Patel
[15] developed an eoq model with linear demand under
permissible delay in payments. A literature review on
inventory model under trade credit is given by Chang et
al. [6].
To take advantage of price discounts, many times
retailer decides to buy goods exceeding his Own
Warehouse (OW) capacity. Hence an additional
warehouse is arranged known as Rented Warehouse
(RW) which has better storage facilities with higher
inventory carrying cost and low rate of deterioration. A
two warehouse inventory model for deteriorating items
with linear demand and shortages was developed by
Bhunia [3]. Sana et al. [18] proposed two warehouse
inventory model on pricing decision. Yu et al [26] gave
two warehouse inventory model for deteriorating items
with decreasing rental over time.
International Journal of Theoretical & Applied Sciences, 10(1): 53-65(2018)

Patel 54
Tyagi [24] proposed a two warehouse inventory model
with time dependent demand and variable holding cost.
Sheikh and Patel [20] developed a two warehouse
inventory model under linear demand and time varying
holding cost. Parekh and Patel [14] developed
deteriorating item inventory models for two warehouses
with linear demand under inflation and permissible
delay in payments. Jaggi et al. [10] gave replenishment
policy for non-instantaneous deteriorating items in two
storage facilities under inflation.
Generally the products are such that there is no
deterioration initially. After certain time deterioration
starts and again after certain time the rate of
deterioration increases with time. Here we have used
such a concept and developed the deteriorating items
inventory model.
In this paper we have developed a two warehouse
inventory model with different deterioration rates.
Demand function is price and time dependent. Holding
cost is time varying. Shortages are not allowed.
Numerical case is given to represent the model.
Affectability investigation is likewise done for
parameters.
II. ASSUMPTIONS AND NOTATIONS:
Notations: The following notations are used for the
development of the model:
D(t) : Demand is a function of time and price
(a + bt - ρp, a>0, 0<b<1, ρ>0)
HC(OW) : Holding cost is linear function of time t
(x1+y1t, x1>0, 0<y1<1) in OW.
HC(RW) : Holding cost is linear function of time t
(x2+y2t, x2>0, 0<y2<1) in RW.
A : Ordering cost per order
c : Purchasing cost per unit
p : Selling price per unit
T : Length of inventory cycle
I0(t) : Inventory level in OW at time t
Ir(t) : Inventory level in RW at time t
Ie : Interest earned per year
Ip : Interest paid in stocks per year
R : Inflation rate
Q : Order quantity
tr : Time at which inventory level becomes
zero in RW.
W : Capacity of own warehouse
θ : Deterioration rate in OW during
µ1<t<µ2, 0< θ<1
θt : Deterioration rate in OW during
µ2 ≤ t ≤ T, 0< θ<1
π : Total relevant profit per unit time.
Assumptions: The following assumptions are
considered for the development of model.
• The demand of the product is declining as a function
of time and price.
• Replenishment rate is infinite and instantaneous.
• Lead time is zero.
• Shortages are not allowed.
• OW has fixed capacity W units and RW has
unlimited capacity.
• The goods of OW are consumed only after
consuming the goods kept in RW.
• The unit inventory cost per unit in the RW is higher
than those in the OW.
• Deteriorated units neither be repaired nor replaced
during the cycle time.
• During the time, the account is not settled;
generated sales revenue is deposited in an interest
bearing account. At the end of the credit period, the
account is settled as well as the buyer pays off all
units sold and starts paying for the interest charges
on the items in stocks.
III. THE MATHEMATICAL MODEL AND
ANALYSIS
At time t=0, Q units enters into the system of which
W are stored in OW and rest (Q-W) are stored in RW.
At time tr level of inventory in RW reaches to zero
because of demand and OW inventory remains W.
During the interval (tr,µ1) inventory depletes in OW
due to demand, during interval (µ1, µ2) inventory
depletes in OW due to deterioration at rate θ and
demand. During interval (µ2, T) inventory in OW
depletes due to joint effect of deterioration at rate θt and
demand. By time T both the warehouses are empty.
Let I(t) be the inventory at time t (0 ≤ t ≤ T) as
shown in figure.
Fig. 1.

Patel 55
Hence, the inventory level at time t in RW and OW and
governed by the following differential equations:
rdI (t)
= - (a + bt - ρp),
dt
r0 t t≤ ≤ (1)
0dI (t)
= 0,
dt
r0 t t≤ ≤ (2)
0dI (t)
= - (a + bt - ρp),
dt
r 1t t µ≤ ≤ (3)
0
0
dI (t)
+ θI (t) = - (a+bt - ρp),
dt
1 2µ t µ≤ ≤ (4)
0
0
dI (t)
+ θtI (t) = - (a+bt - ρp),
dt
2µ t T≤ ≤ (5)
with initial conditions I0(0) = W, I0(µ1) = S1, I0(tr)=W,
Ir(0) = Q-W, Ir(tr) = 0 and I0(T) = 0.
Solving equations (1) to (5) we have,
2
r
1
I (t) = Q -W - (a - ρp)t - bt
2
(6)
0I (t) = W (7)
( ) ( )2
0 1 1
1
I (t)=S +(a-ρp) µ t + b µ
2
- -t2
1 (8)
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
2
1 1
2 2
1
1
0
3
2
1 1
1
a µ -t -ρp µ -t + aθ µ -t
2
1 1
- ρpθ µ - t + b µ - t
2 2
1
+ bθ µ - t - aθt µ - t
3
1
+ ρpt µ - t - bθt µ - t
I (t)
2
+ S 1+
=
- t)θ(µ
 
 
 
 
 
 
 
 
 
 
 
2
1
2 2
1 1
3
1
2
1
(9)
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( )
3 3
3 3 2 2
0
4 4 2
2 2 2 2
1
a T-t -ρp T-t + aθ T -t
6
1 1
- ρpθ T -t + b T -t
6 2
I (t) =
1 1
+ bθ T -t - aθt T-t
8 2
1 1
+ ρpθt T-t - bθt T -t
2 4
 
 
 
 
 
 
 
 
 
 
 
(10)
(by neglecting higher powers of θ)
Putting t = tr in equation (6), we get
2
r r
1
Q = W + (a - ρp)t + bt
2
(11)
Putting t = tr in equations (7) and (8), we get
0 rI (t ) = W (12)
( ) ( )2
0 r 1 1 r r
1
I (t )=S +(a-ρp) µ - t + b µ - t
2
2
1 (13)
So from equations (12) and (13), we have
( ) ( )2
1 1 r r
1
S =W (a -ρp) µ -- t - b µ - t
2
2
1 (14)
Putting t = µ2 in equations (9) and (10), we get
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( )
( )
1 2 1 2
1 2
1 2
1 1 2
0
1
a µ -µ -ρp µ -µ + aθ µ -µ
2
1 1
- ρpθ µ - µ + b µ - µ
2 2
1
+ bθ µ - µ - aθt µ - µ
3
I (t) =
1
+ ρpt µ - µ - bθt
-
µ - µ
2
+ S ( )1+ θ µ µ
 
 
 
 
 
 
 
 
 
 
 
2 2
1 2
2 2 2 2
1 2 1 2
3 3
1 2
2 2
1 2
(15)
( ) ( ) ( )
( ) ( )
( ) ( )
( ) ( )
3
3 2
0
4
2 2
1
a T-µ -ρp T- µ + aθ T -µ
6
1 1
- ρpθ T -µ + b T -µ
6 2I (t) =
1 1
+ bθ T -µ - aθµ T-µ
8 2
1 1
+ ρpθµ T-µ - bθt T -µ
2 4
 
 
 
 
 
 
 
 
 
 
 
3
2 2 2
3 2
2 2
4 2
2 2 2
2 2
2 2 2
(16)
So from equations (15) and (16), we have
( )2
2
2 2
2 2
2 2 2 4
2 2 r 2
2 2 2 2 2 2
2 2 1 2 1
2 2 2 2 2 2 4
2 r 1 2 1 2
2 2 2 3
2 r 1 2 r 2
2
1
1
T =
b θµ -2
-aθµ + ρpθµ + 2a - 2ρp
-4bρpθµ + 4bθµ ρpt - 2aθ µ ρp
+ 8aθµ ρp - 2bθ µ ρpµ - 4bθ µ Wµ
+4bθ µ ρpt µ +2abθ µ µ +2bθ µ ρp
- 4abθ µ t µ + 4abθ µ t - 8bWθµ
- 4abθµ +
+
2 2
2 r 1
2 r 1 r 1 r 2
2 3 2 2 2
r 2 2 2 r
2 2 2 2 2 2 4
2 2 1
2 2 3 2 2 2 4
2 2 r 2 r 2
2 2 2 2 6
2 2
4abθµ -8bρpt +4bθρpµ
- 8bθµ ρpt µ + 8abθt µ + 8bθρpt µ
- 8abθt µ + 4bθ µ W - 2b θµ t
- 4ρ p θµ + aρ p θ µ + 8bWθµ
-4bθµ W-4bθ µ ρpt -4abθµ t +a θ µ
- 4a θµ - 8aρp + b θ µ + r
2 4 2 2 2 2 2
2 r
8abt
- 2b θµ + 4a + 4ρ p + 4b t + 8bW
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
(17)

Patel 56
From equation (17), we see that T is a function of W
and tr, so T is not a decision variable.
Based on the assumptions and descriptions of the
model, the total annual relevant profit(π), include the
following elements:
(i) Ordering cost (OC) = A (18)
(ii) ( ) ( )
r
-R
t
1 1 0
0
t
HC OW = x +y t I (t) dte∫
( )
2
T
-
1
R
0
t
1
µ
+ x +y t I (t) dte∫ (19)
(iii) ( ) ( )
r
-Rt
t
2 2 r
0
HC RW = x +y t I (t) dte∫ (20)
(iv)
2
1 2
-Rt
µ T
0 0
µ µ
-Rt
DC = c θ I (t) dt+ θe t I (te ) dt
 
 
 
 
∫ ∫ (21)
(v) ( ) -Rt
T
0
SR = p a + bt - ρp dte
 
 
 
∫ (22)
To determine the interest earned, there will be two cases
i.e. Case I: (0≤M≤ T) and Case II: (M>T).
Case I: (0≤M≤T): In this case the retailer can earn
interest on revenue generated from the sales up to M.
Although, he has to settle the accounts at M, for that he
has to arrange money at some specified rate of interest
in order to get his remaining stocks financed for the
period M to T. So
(vi) Interest earned per cycle:
( )
M
-Rt
1 e
0
IE = pI a + bt - ρp te dt∫ (23)
Case II: (0 ≤T ≤ M):
In this case, the retailer earns interest on the sales
revenue up to the permissible delay period. So
(vii) Interest earned up to the permissible delay period
is:
( ) ( ) ( )
T
-Rt
2 e
0
IE = pI a+bt-ρp t e dt+ a+bT-ρp T M-T
 
 
 
∫
(24)
To determine the interest payable, there will be five
cases i.e.
Interest payable per cycle for the inventory not sold
after the due period M is
Case I: (0≤M≤tr):
(viii) IP1
T
-Rt
p
M
= cI I(t)e dt∫
r r 1
r
t t µ
-Rt -Rt -Rt
p r 0 0
M M t
= cI I (t)e dt + I (t)e dt + I (t)e dt
 
 
 
 
∫ ∫ ∫
2
1 2
µ T
-Rt -Rt
p 0 0
µ µ
+ cI I (t)e dt + I (t)e dt
 
 
 
 
∫ ∫
(25)
Case II: (tr≤M≤ µ1):
(ix) IP2
T
-Rt
p
M
= cI I(t)e dt∫
1 2
1 2
µ µ T
-Rt -Rt -Rt
p 0 0 0
M µ µ
= cI I (t)e dt+ I (t)e dt + I (t)e dt
 
 
 
 
∫ ∫ ∫
(26)
Case III: (µ1≤M≤ µ2):
(x) IP3
T
-Rt
p
M
= cI I(t)e dt∫
2
2
µ T
-Rt -Rt
p 0 0
M µ
= cI I (t)e dt + I (t)e dt
 
 
 
 
∫ ∫ (27)
Case IV: (µ2≤M≤T):
(xi) IP4
T
-Rt
p
M
= cI I(t)e dt∫ (28)
Case V: (M>T):
(xii) IP5 = 0 (29)
(by neglecting higher powers of b and R)
The total profit (πi), i=1,2,3,4 and 5 during a cycle
consisted of the following:
[ ]i i i
1
π = SR-OC-HC(RW)-HC(OW)-DC-IP +IE
T
(30)
Substituting values from equations (18) to (29)
in equation (30), we get total profit per unit. Putting µ1=
v1T, µ2= v2T and value of S1 and T from equation (14)
and (17) in equation (30), we get profit in terms of tr
and p for the five cases as under:
[ ]1 1 1
1
T
(31)
[ ]2 2 1
1
T
(32)
[ ]3 3 1
1
T
(33)
[ ]4 4 1
1
T
(34)
[ ]5 5 2
1
T
(35)
The optimal value of tr* and p* (say), which
maximizes πi can be obtained by solving equation (31),
(32), (33), (34) and (35) by differentiating it with
respect to tr and p and equate it to zero

Patel 57
i.e. i r i r
r
π (t ,p) π (t ,p)
=0, =0, i=1,2,3,4,5
t p
∂ ∂
∂ ∂
(36)
provided it satisfies the condition
2 2
i r i r
2
rr
2 2
i r i r
2
r
π (t ,p) π (t ,p)
t pt
> 0, i=1,2,3,4,5.
π (t ,p) π (t ,p)
p t p
∂ ∂
∂ ∂∂
∂ ∂
∂ ∂ ∂
(37)
IV. NUMERICAL EXAMPLE
Case I: Considering A= Rs.100, W = 65, a = 500,
b=0.05, c=Rs. 25, ρ= 5, θ=0.05, x1 = Rs. 2, y1=0.04, x2 =
Rs. 6, y2=0.08, v1=0.30, v2=0.50, R = 0.06, Ie = 0.12, Ip
= 0.15, M = 0.01 in appropriate units. The optimal
value of tr*=0.0513, p* = Rs. 50.7003,
Profit*=Rs.11775.0326 and Q*=77.6454.
Case II: Considering A= Rs.100, W = 65, a = 500,
b=0.05, c=Rs. 25, ρ= 5, θ=0.05, x1= Rs. 2, y1=0.04, x2 =
Rs. 6, y2=0.08, v1=0.30, v2=0.50, R = 0.06, Ie = 0.12, Ip
value of tr*=0.0497, p* = Rs. 50.5995,
Profit*=Rs.11834.9853 and Q*=77.2761.
Case III: Considering A= Rs.100, W = 65, a = 500,
b=0.05, c=Rs. 25, ρ= 5, θ=0.05, x1=Rs.2, y1=0.04, x2 =
Rs. 6, y2=0.08, v1=0.30, v2=0.50, R = 0.06, Ie = 0.12, Ip
value of tr*=0.0458, p* = Rs. 50.5325,
Profit*=Rs.11889.9319 and Q*=76.3281.
Case IV: Considering A= Rs.100, W = 65, a = 500,
b=0.05, c=Rs. 25, ρ= 5, θ=0.05, x1=Rs.2, y1=0.04, x2 =
Rs. 6, y2=0.08, v1=0.30, v2=0.50, R = 0.06, Ie = 0.12, Ip
value of tr*=0.0345, p* = Rs. 50.4623,
Profit*=Rs.11987.5446 and Q*=73.5453.
Case V: Considering A= Rs.100, W = 65, a = 500,
b=0.05, c=Rs. 25, ρ= 5, θ=0.05, x1=Rs. 2, y1=0.04, x2 =
Rs. 6, y2=0.08, v1=0.30, v2=0.50, R = 0.06, Ie = 0.12, Ip
value of tr*=0.0136, p* = Rs. 50.4505,
Profit*=Rs.12128.5746 and Q*=68.3694.
The second order conditions given in equation (37) are
also satisfied. The graphical representation of the
concavity of the profit function is also given.
Case I
tr and Profit
Graph 1
Case I
pand Profit
Graph 2
Case I
tr, p and Profit
Graph 3

Patel 58
Case II
tr and Profit
Graph 4
Case II
pand Profit
Graph 5
Case II
tr, p and Profit
Graph 6
Case III
tr and Profit
Graph 7
Case III
pand Profit
Graph 8
Case III
tr, p and Profit
Graph 9

Patel 59
Case IV
tr and Profit
Graph 10
Case IV
pand Profit
Graph 11
Case IV
tr, p and Profit
Graph 12
Case V
tr and Profit
.
Graph 13
Case V
pand Profit
Graph 14
Case V
tr, p and Profit
Graph 15

Patel 60
V. SENSITIVITY ANALYSIS
On the basis of the data given in example above we have studied the sensitivity analysis by changing the following
parameters one at a time and keeping the rest fixed.
Table 1.
Case I (0 ≤M≤ tr)
Sensitivity Analysis
Para-meter % tr p Profit Q
a
+20% 0.0725 60.6245 17160.1483 86.5273
+10% 0.0636 55.6581 14342.3121 82.2808
-10% 0.0338 45.7547 9458.6865 72.4775
-20% 0.0086 40.8274 7393.8518 66.6844
θ
+20% 0.0497 50.7061 11772.0687 77.2496
+10% 0.0504 50.7032 11773.5486 77.4228
-10% 0.0521 50.6974 11776.5209 77.8433
-20% 0.0529 50.6945 11778.0134 78.0414
x1
+20% 0.0414 50.7440 11761.9621 75.1960
+10% 0.0463 50.7218 11768.4201 76.4079
-10% 0.0562 50.6795 11781.7975 78.8591
-20% 0.0612 50.6594 11788.7125 80.0983
x2
+20% 0.0603 50.7395 11756.5013 79.8521
+10% 0.0558 50.7201 11765.6983 78.7491
-10% 0.0466 50.6801 11784.5107 76.4916
-20% 0.0419 50.6595 11794.1392 75.3369
A
+20% 0.0815 50.7070 11714.2499 85.0870
+10% 0.0667 50.7030 11743.9460 81.4406
-10% 0.0349 50.6996 11807.7267 73.6029
-20% 0.0176 50.7014 11842.3087 69.3383
M
+20% 0.0513 50.6966 11776.9263 77.6464
+10% 0.0513 50.6984 11775.9785 77.6459
-10% 0.0512 50.7022 11774.0885 77.6203
-20% 0.0512 50.7041 11773.1462 77.6198
R
+20% 0.0539 50.7226 11731.3620 78.2803
+10% 0.0526 50.7116 11753.1905 77.9629
-10% 0.0500 50.6890 11796.8901 77.3278
-20% 0.0482 50.6774 11818.7643 76.8868
ρ
+20% 0.0608 42.3697 9712.0191 79.9436
+10% 0.0564 46.1562 10649.6506 78.8824
-10% 0.0453 56.2548 13150.7991 76.1825
-20% 0.0383 63.1984 14870.9239 74.4680

Patel 61
Table 2.
Case II ( tr ≤M≤ µ1)
a
+20% 0.0696 60.5251 17235.6924 85.6974
+10% 0.0613 55.5580 14409.8691 81.6865
-10% 0.0330 45.6530 9511.3840 72.3173
-20% 0.0086 40.7248 7439.6058 66.6888
θ
+20% 0.0481 50.6052 11832.0740 76.8795
+10% 0.0489 50.6024 11833.5277 77.0778
-10% 0.0505 50.5966 11836.4471 77.4744
-20% 0.0513 50.5937 11837.9130 77.6728
x1
+20% 0.0398 50.6438 11822.0043 74.8219
+10% 0.0447 50.6213 11828.4169 76.0362
-10% 0.0546 50.5784 11841.7071 78.4922
-20% 0.0596 50.5580 11848.5801 79.7338
x2
+20% 0.0588 50.6384 11816.2946 79.5124
+10% 0.0543 50.6191 11825.5697 78.4070
-10% 0.0450 50.5795 11844.5479 76.1196
-20% 0.0403 50.5590 11854.2645 74.9624
A
+20% 0.0801 50.6045 11773.8270 84.7830
+10% 0.0653 50.6012 11803.6985 81.1288
-10% 0.0332 50.5998 11867.9116 73.2004
-20% 0.0158 50.6030 11902.7674 68.9023
M
+20% 0.0488 50.5791 11849.9118 77.0587
+10% 0.0493 50.5892 11842.4040 77.1798
-10% 0.0500 50.6101 11827.6556 77.3475
-20% 0.0503 50.6211 11820.4147 77.4188
R
+20% 0.0525 50.6217 11791.2256 77.9619
+10% 0.0511 50.6107 11813.0978 77.6190
-10% 0.0482 50.5882 11856.8901 76.9083
-20% 0.0464 50.5766 11878.8144 76.4663
ρ
+20% 0.0602 42.2682 9769.8524 79.8328
+10% 0.0553 46.0550 10708.4383 78.6424
-10% 0.0431 56.1544 13212.2091 75.6589
-20% 0.0353 63.0988 14934.2118 73.7405

Patel 62
Table 3.
Case II (µ1 ≤M≤ µ2)
a
+20% 0.0632 60.4610 17308.6533 83.8144
+10% 0.562 55.4925 14473.4174 80.3167
-10% 0.0303 45.5847 9558.4661 71.7289
-20% 0.0072 40.6548 7479.4846 66.4164
θ
+20% 0.0443 50.5383 11887.0900 75.9558
+10% 0.0450 50.5354 11888.5090 76.1296
-10% 0.0466 50.5296 11891.3587 76.5266
-20% 0.0473 50.5267 11892.7895 76.7005
x1
+20% 0.0359 50.5784 11877.1925 73.8712
+10% 0.0408 50.5551 11883.4827 75.0868
-10% 0.0508 50.5107 11896.5375 77.5703
-20% 0.0557 50.4896 11903.2973 78.7887
x2
+20% 0.0550 50.5706 11870.9666 78.5931
+10% 0.0504 50.5518 11880.3761 77.4610
-10% 0.0410 50.5129 11899.6408 75.1449
-20% 0.0362 50.4928 11909.5104 73.9608
A
+20% 0.0766 50.5340 11828.0095 83.9456
+10% 0.0616 50.5323 11858.2365 80.2361
-10% 0.0291 50.5352 11923.3349 72.1971
-20% 0.0114 50.5412 11958.7577 67.8191
M
+20% 0.0431 50.5064 11917.9330 75.6659
+10% 0.0445 50.5189 11903.7985 76.0095
-10% 0.0469 50.5471 11876.3301 76.5967
-20% 0.0479 50.5627 11862.9904 76.8403
R
+20% 0.0489 50.5548 11846.0104 77.0894
+10% 0.0474 50.5437 11867.9616 76.7212
-10% 0.0440 50.5212 11911.9236 75.8854
-20% 0.0421 50.5096 11933.9394 75.4178
ρ
+20% 0.0582 42.1997 9820.7358 79.3639
+10% 0.0525 45.9872 10761.1376 77.9712
-10% 0.0380 56.0886 13270.0054 74.4089
-20% 0.0288 63.0345 14995.7322 72.1384

Patel 63
Table 4.
Case II (µ2 ≤M≤ T)
Para-
meter
% tr p Profit Q
a
+20% 0.0448 60.4034 17445.8747 78.3496
+10% 0.0413 55.4281 14589.5630 76.2691
-10% 0.0223 45.5092 9639.7878 69.9607
-20% 0.0025 40.5746 7546.4780 65.4928
θ
+20% 0.0329 50.4683 11984.8379 73.1480
+10% 0.0336 50.4653 11986.1895 73.3218
-10% 0.0351 50.4592 11988.9032 73.6944
-20% 0.0358 50.4561 11990.2655 73.8684
x1
+20% 0.0243 50.5130 11975.5543 71.0127
+10% 0.0293 50.4872 11981.4643 72.2536
-10% 0.0394 50.4382 11993.7919 74.7637
-20% 0.0445 50.4150 12000.2093 76.0327
x2
+20% 0.0440 50.4980 11967.8234 75.8904
+10% 0.0392 50.4803 11977.6022 74.7059
-10% 0.0294 50.4439 11997.6590 72.3847
-20% 0.0244 50.4251 12007.9545 71.0481
A
+20% 0.0664 50.4531 11923.4101 81.4497
+10% 0.0508 50.4562 11954.6599 77.5842
-10% 0.0169 50.4721 12022.3535 69.1851
-20% 0.0000 50.4790 12059.4626 65.0000
M
+20% 0.0260 50.4474 12041.2131 71.4418
+10% 0.0304 50.4529 12013.9480 72.5312
-10% 0.0379 50.4751 11961.9697 74.3850
-20% 0.0410 50.4912 11937.1966 75.1493
R
+20% 0.0385 50.4872 11943.1291 74.5318
+10% 0.0365 50.4733 11965.3208 74.0386
-10% 0.0321 50.4511 12009.8046 72.9536
-20% 0.0295 50.4397 12032.1058 72.3101
ρ
+20% 0.0520 42.1232 9907.1347 77.8576
+10% 0.0439 45.9132 10852.5082 75.8643
-10% 0.0231 56.0235 13375.6700 70.7264
-20% 0.0097 62.9770 15112.1420 67.4065
From the table we observe that as parameter a
increases/ decreases average total profit and order
quantity increases/ decreases for all five cases.
From the table we observe that as parameter θ
increases/ decreases there is very minor change in
average total profit and order quantity for all five cases.
From the table we observe that as parameter x1
increases/ decreases average total profit and order
quantity decreases/ increases for all five cases.
From the table we observe that as parameters x2, A, R
and ρ increases/ decreases average total profit
decreases/ increases and order quantity increases/
decreases for all five cases.

Patel 64
Table 5.
Case II (M> T)
a
+20% 0.0265 60.4053 17654.7592 70.2463
+10% 0.0212 55.4243 14762.6012 69.7250
-10% 0.0025 45.4872 9752.8237 65.6814
-20% 0.0000 40.4659 7635.0903 65.0000
θ
+20% 0.0126 50.4566 12126.0857 68.1212
+10% 0.0131 50.4536 12127.3290 68.2453
-10% 0.0141 50.4474 12129.8225 68.4934
-20% 0.0146 50.4443 12131.0727 68.6175
x1
+20% 0.0056 50.5073 12117.9762 66.3858
+10% 0.0096 50.4785 12123.2010 67.3770
-10% 0.0176 50.4232 12134.0942 69.3627
-20% 0.0216 50.3966 12139.7572 70.3572
x2
+20% 0.0214 50.4840 12107.3212 70.2982
+10% 0.0175 50.4674 12117.8701 69.3341
-10% 0.0096 50.4332 12139.4417 67.3792
-20% 0.0055 50.4155 12150.4791 66.3637
A
+20% 0.0401 50.4239 12059.2046 74.9400
+10% 0.0271 50.4360 12093.1023 71.7159
-10% 0.0000 50.4636 12165.8654 65.0000
-20% 0.0000 50.3898 12204.0987 65.0000
M
+20% 0.0136 50.4473 12218.5724 68.3695
+10% 0.0136 50.4489 12173.5735 68.3695
-10% 0.0136 50.4521 12083.5758 68.3692
-20% 0.0136 50.4538 12038.5770 68.3691
R
+20% 0.0176 50.4725 12083.0734 69.3584
+10% 0.0156 50.4615 12105.8026 68.8640
-10% 0.0114 50.4394 12151.3944 67.8249
-20% 0.0090 50.4281 12174.2677 67.2307
ρ
+20% 0.0355 42.0950 10024.3757 75.2781
+10% 0.0249 45.8930 10980.3017 71.7363
-10% 0.0014 55.0206 13533.5378 65.3078
-20% 0.0000 62.8968 15290.9818 65.0000
From the table we observe that as parameter M
increases/ decreases average total profit also increases/
decreases for all five cases but for order quantity almost
remains fixed for all five cases.
VI. CONCLUSION
In this paper, we have developed a two warehouse
inventory model for deteriorating items with different
deterioration rates under time and price dependent
demand, and time varying holding cost under
inflationary conditions. Sensitivity with respect to
parameters has been carried out. The results show that
with the increase/ decrease in the parameter values
there is corresponding increase/ decrease in the value of
profit.
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