International Journal of Mathematical Archive-7(4), 2016,
122-132
Available online throug
ISSN 2229 – 5046
International Journal of Mathematical Archive- 7(4), April – 2016 122
A FUZZY TYPE DETERIORATION INVENTORY MODEL
WITH STOCK DEPENDENT SELLING RATE AND PARTIAL BACKLOGGING
SUSHIL KUMAR*
Department of Mathematics & Astronomy,
University of Lucknow, Lucknow- 226007, (U.P.), India.
(Received On: 10-03-16; Revised & Accepted On: 13-04-16)
ABSTRACT
I
n the present paper we considered an inventory model for deteriorating items with stock dependent selling rate and fuzzy type deterioration. Shortages are allowed and completely backlogged. The backlogging rate of unsatisfied demand is assumed to be a decreasing exponential function of waiting time. The purpose of our study is to defuzzify the total profit function by signed distance method and comparing the results of this method with the crisp model. Further a numerical example is also given to demonstrate the developed model and to show the sensitivity analysis of the effect of change of parameters.INTRODUCTION
In some consumer goods it has been observed that the demand of some items is influenced by the amount of on hand inventory stock. The demand of such goods may be increase or decrease according as on hand inventory stock increases or decreases. This situation arise in a super market where is a lot of stock of goods. The effect of deterioration cannot be avoided in any business organization and deterioration is defined by a process in which the stocked items loose part of their value with passage of time.
In past few years some researchers considered the time dependent demand rate because the demand of newly launched products such as fashionable garments, electronic items, mobile phones increases with time and later it becomes constant. In some real life situations the customers suffer the problem of shortage because there are some customers who wait for the next replenishment while the others do not wait for replenishment and go elsewhere as waiting time increases, but in the recent years some researchers gave their attention towards the stock dependent demand rate.
In some consumer goods it has been observed that the demand of some items is influenced by the amount of on hand inventory stock. The demand of such goods may be increase or decrease according as on hand inventory stock increases or decreases. This situation arises in a super market, when there is a lot of stock of goods. In this area the work has been done by the following researchers. Baker and Urban [1] proposed a deterministic inventory model for deteriorating items with stock level dependent demand rate. Mandal and Phaujdar [2] presented an inventory model for deteriorating items with stock level dependent consumption rate. Vrat and Padmanabhan [3] developed an EOQ model for perishable products with stock dependent selling rate. Mandal and Maiti [4] proposed an inventory model for damageable products with stock dependent demand and variable replenishment rate. Dye and Ouyang [5] developed an EOQ model for perishable items with stock dependent selling rate by allowing shortages. Kao and Hsu [6] developed a single period inventory model with fuzzy demand. Hsieh [7] proposed an inventory model with fuzzy production rate. Yao and Chiang [8] developed an inventory model without backorders and defuzzified the fuzzy holding cost by signed distance method and centroid method. Hou [9] presented an inventory model for deteriorating items with stock dependent consumption rate and shortages. He was also used inflation and time discounting in his inventory model. Sujit et al.
[10] presented an economic production quantity model with fuzzy type demand rate and deterioration rate. Syed and Aziz [11] considered a fuzzy inventory model without shortages. Valliaththal and Uthayakumar [12] presented an EOQ model for perishable products with stock dependent selling rate and shortages. Roy et al. [13] developed an inventory model for deteriorating items with stock dependent demand. He was also considered the fuzzy inflation rate and time discounting over a random planning horizon. Sana [14] proposed a lot size inventory model with time varying deterioration rate and stock dependent demand by allowing shortages. Chang et al. [15] determined an optimal replenishment policy for an inventory model of non-instantaneous deteriorating items with stock dependent demand. De and Rawat [16] developed a fuzzy inventory model for deteriorating items without shortages. Nagrare and Dutta [17] developed a continuous review inventory model for perishable products with inventory dependent demand. Sana [18] proposed a control policy for a production system with stock dependent demand.
IJMA- 7(4), April-2016.
The figure 1show the developed model and the figures 2 and 3 show the graphs of total profit function with respect to deterioration parameter and selling rate parameter. The tables 2 and 3 show the variation of total profit function with respect to the parameters
θ
andβ
.In the present paper we consider an inventory model for deteriorating items with stock dependent demand rate. Shortages are allowed and completely backlogged. The backlogging rate of unsatisfied demand is assumed as a decreasing exponential function of waiting time. The purpose of our study is to defuzzify the total profit function by signed distance method and comparing the results of this method with the crisp model. A numerical example is also given to demonstrate the developed model and to shoe the sensitivity analysis of the parameters.
DEFINITIONS AND PRELIMINARIES
When we are considering the fuzzy inventory model then the following definitions are needed.
Definition: A fuzzy set
~
A
on the given universal set X is denoted and defined by
{(
,
Ã(
)
)
:
}
~X
x
x
x
A
=
λ
∈
where,
λ
Ã:
X
→
[
0
,
1
]
, is called the membership function,and,
λ
Ã(
x
)
= degree of x in~
A
Definition: A fuzzy number
~
A
is a fuzzy set on the real line R, if its membership functionλ
Ãhas the following
properties
1.
λ
Ã(
x
)
is upper semi continuous.2.
λ
Ã(
x
)
=0, outside some interval[
a
1,
a
4]
3.
∃
real numbersa
2anda
3,a
1≤
a
2≤
a
3≤
a
4 such thatλ
Ã(
x
)
is increasing on[
a
1,
a
2]
, decreasing on]
,
[
a
3a
4 andλ
Ã(
x
)
=1, for eachx
in[
a
2,
a
3]
Definition: A triangular fuzzy number is specified by the triplet
(
a
1,
a
2,
a
3)
wherea
1
a
2
a
3and defined by itscontinuous membership function
λ
Ã:
X
→
[
0
,
1
]
as follows
≤
≤
−
−
≤
≤
−
−
=
otherwise
a
x
a
a
a
x
a
a
x
a
a
a
a
x
x
A,
0
,
,
)
(
2 32 3
3
2 1
1 2
1
~
λ
Definition:Let
~
A
be a fuzzy set defined on R, then the signed distance of~
A
is defined as[
A
α
A
β
]
d
αA
d
=
∫
L+
R
10 ~
)
(
)
(
2
1
0
,
where,
A
α=
[
A
L( )
α
+
A
R( )]
β
=
[
a
+ −
(
b
a
) ,
α
d
−
(
d
−
c
)
α α
]
,
∈
[0,1]
is an α cut of a fuzzy set~
A
.Definition: If
(
,
,
)
~
c
b
a
A
=
is a triangular fuzzy number then the signed distance of~
A
is defined as)
2
(
4
1
0
,
~
c
b
a
A
d
=
+
+
IJMA- 7(4), April-2016.
© 2016, IJMA. All Rights Reserved 124 Assumptions and Notations- We consider the following assumptions and notations
1. The demand rate is
R
(
t
)
=
α
+
β
I
(
t
),
where α is a positive constant and β is the stock dependent selling rate parameter,0
≤
β
≤
1
2. θ is the deterioration parameter. 3.
δ
is the backlogging parameter. 4. A is the ordering cost per order.5.
h
cis the holding cost per unit per unit time.6.
s
c is the shortages cost per unit per unit time. 7.c
is the purchase cost per unit.8.
p
is the selling price per unit, where.9.
c
3 is the opportunity cost per unit due to lost sales.10.
~
θ
is the fuzzy deterioration parameter. 11.T
is the length of order cycle. 12.T
1 is the time at which shortage starts.13.
TP
(
T
1,
T
)
is the total profit per unit time.14.
(
,
)
~
1
T
T
TP
is the total fuzzy profit per unit time. 15.I
(
t
)
is the inventory level at any time in[
0
,
T
]
. 16. The time horizonT
is infinite.17. The lead time is zero.
18. The replenishment rate is infinite.
Figure-1
CRISP MODEL
Mathematical Formulation-Suppose an inventory system consists S units of the product in the beginning of each cycle. Due to demand and deterioration the inventory level decreases in
[
0
,
T
1]
and becomes zero att
=
T
1. The interval[
T
1,
T
]
is the shortages interval. During the shortages interval[
T
1,
T
]
the unsatisfied demand is backlogged ata rate of
e
−δt, whereδ
is the backlogging parameter and t is the waiting time.The instantaneous inventory level at any time t in [0, T] are governed by the following differential equations
1
0 ),
( I t T
I dt
dI +
θ
=−α
+β
≤ ≤(1) with boundary condition I(0) = S
T
t
T
e
dt
dI
=
−
− t≤
≤
1
,
δ
α
(2)
with boundary condition
I T
( )
1=
0
IJMA- 7(4), April-2016.
t t
Se
e
I
[
( )1
]
( ))
(
β θ β
θ
β
θ
α
+−
+
++
=
2
)}
(
){
(
)}
(
{
2
t
S
t
S
S
I
=
+
α
+
θ
+
β
+
θ
+
β
α
+
θ
+
β
(3)
The solution of equation (2) is
]
[
e
(T T1)e
(T t)I
=
−δ −−
−δ −δ
α
)
2
2
(
2
)
(
T
1t
T
12t
2TT
1Tt
I
=
α
−
+
αδ
−
−
+
(4)The ordering cost per cycle is
T
A
O
C=
(5)The holding cost per cycle is
∫
=
10
)
(
T c
C
I
t
dt
T
h
H
1 2
0
{
(
)}
(
){
(
)}
2
T c C
h
t
H
S
S
t
S
T
α
θ β
θ β α
θ β
=
+
+
+
+
+
+
+
∫
2 3
1 1
1
{
(
)}
(
){
(
)}
2
6
c C
h
T
T
H
ST
S
S
T
α
θ β
θ β α
θ β
=
+
+
+
+
+
+
+
(6)The shortage cost per cycle is
∫
−
=
TT c
C
I
t
dt
T
s
S
1
)
(
3 2
3 2
2 2
1 1
1 1 1
3
3
2
2
c C
s
T
T
T
T
S
TT
T T
TT
T
α
α
αδ
α
= −
−
+
−
+
−
−
(7)Due to lost sales the opportunity cost per cycle in
[
T
1,
T
]
isdt
e
T
c
O
T
T
t T
PC
[
1
]
1
) ( 3
∫
−
− −=
α
δ)]
3
3
(
3
2
[
2
2 1 2
1 3 1 3 1
2 1 2 3
TT
T
T
T
T
TT
T
T
c
O
PC=
+
−
+
−
−
+
δ
αδ
(8)
The purchase cost per cycle in
[
T
1,
T
]
isquantity
ordered
back
c
cS
P
C=
+
×
] 2 2
( 2 ) (
[ 2
1 2
2 1
1 T T T TT T
T S c
PC = +
α
− +αδ
− − + (9)The sales revenue per cycle is
1
1
0
[ (
)
(
) ]
T T
R
T
IJMA- 7(4), April-2016.
© 2016, IJMA. All Rights Reserved 126
2 3 2
2 3
1 1
1
{
(
)}
(
){
(
)}
(
1)
(
1)
2
6
2
6
R
T
T
p
p
S
=
p
β
S
+
T
α
+
S
θ β
+
+
θ β α
+
+
S
θ β
+
+
p T
α
−
αδ
T
−
T
+
αδ
T
−
T
(10) Therefore the total profit per unit time is
]
[
1
)
,
(
1S
RO
CH
CS
CO
PCP
CT
T
T
TP
=
−
−
−
−
−
2
}
{
}
{
)
(
)
(
[
1
)
,
(
2 3 1 1T
c
c
p
c
Sh
S
p
T
c
p
cS
A
T
T
T
TP
=
−
+
+
+
α
+
β
−
c−
α
T
+
α
s
C−
αδ
−
αδ
−
αδ
2
}
))
(
(
))
(
(
{
2 1 3T
c
c
S
h
s
p
S
p
β
α
+
θ
+
β
−
αδ
+
α
c−
cα
+
θ
+
β
−
αδ
−
αδ
+
6
}
2
{
}
{
3 3 2 2 1 3T
c
s
p
TT
c
c
s
p
α
cαδ
αδ
αδ
αδ
cαδ
αδ
−
+
+
+
−
−
+
6
}
))
(
)(
(
2
))
(
)(
(
{
3 1 3 2 2T
c
S
h
s
p
S
p
β
θ
+
β
α
+
θ
+
β
−
αδ
+
αδ
c−
cθ
+
β
α
+
θ
+
β
+
αδ
+
]
2
}
2
{
2
}
2
{
2 1 3 2 2 1 2 3 2 2TT
c
s
p
T
T
c
p
s
cαδ
αδ
αδ
αδ
cαδ
αδ
−
+
+
−
−
+
(11)The necessary condition for
TP
(
T
1,
T
)
to be maximum is that1
1
( , ) 0
TP T T
and T ∂ = ∂
0
)
,
(
1=
∂
∂
T
T
T
TP
, on solving these two equations we find the optimum values of
T and
1T
sayT and
1*T
*for which profit is maximum and the sufficient condition is2
2 2 2 2
1 1 1 1
2 2 2
1 1 1
( , )
( , )
( , )
( , )
0
0
TP T T
TP T T
TP T T
TP T T
and
T
T
T T
T
∂
∂
∂
∂
−
∂
∂
∂ ∂
∂
(12)3 1 1
))
(
(
))
(
(
{
)
(
)
,
(
c
S
h
s
p
S
p
c
Sh
S
p
T
T
T
TP
c cc
α
β
α
θ
β
αδ
α
α
θ
β
αδ
β
−
−
+
+
+
−
+
−
+
+
−
=
∂
∂
−
αδ
c
}
T
1+
{
αδ
p
−
α
s
c+
αδ
c
+
αδ
c
3}
T
+
{
p
β
(
θ
+
β
)(
α
+
S
(
θ
+
β
))
−
αδ
2p
2
2 1
3
2
(
)(
(
))
}
{2
2
c c c
T
s
h
S
c
s
αδ
θ β α
θ β
αδ
αδ
+
−
+
+
+
+
+
2
2 2 2 2
3
}
{
2
3}
1]
2
cT
p
c
p
s
c TT
αδ
αδ
αδ
αδ
αδ
−
+
+
−
−
(13)
))}
(
(
))
(
(
[{
1
)
,
(
3 2 1 1 2β
θ
α
αδ
αδ
α
αδ
β
θ
α
β
+
+
−
+
−
−
−
+
+
=
∂
∂
S
h
c
c
s
p
S
p
T
T
T
T
TP
c c
{
(
)(
(
))
2
(
)(
(
))
2
αδ
θ
β
α
θ
β
αδ
β
θ
α
β
θ
β
+
+
+
−
+
−
+
+
+
+
p
S
p
s
ch
cS
+
αδ
2c
3}
T
1+
{
αδ
2p
−
2
αδ
s
c−
αδ
2c
3}
T
]
(14)1 3
3
}
{
}
{
)
[(
1
T
c
c
s
p
T
c
c
p
s
c
p
T
T
TP
cc
α
δ
αδ
αδ
α
δ
α
αδ
αδ
α
α
+
−
−
−
+
−
+
+
+
=
∂
∂
1 3 2 2 2 3 2 2}
2
{
2
}
2
{
αδ
p
−
αδ
s
c−
αδ
c
T
+
αδ
s
c−
αδ
p
+
αδ
c
TT
+
IJMA- 7(4), April-2016.
1 2
2 1 3 2 2
}
{
)
(
)
(
[
1
]
2
}
2
{
A
CS
p
C
T
p
S
Sh
c
T
T
T
c
s
p
αδ
cαδ
α
β
cα
αδ
−
−
−
−
+
+
+
+
−
−
+
3 2
3
{
(
(
))
2
}
{
α
s
c−
αδ
p
−
αδ
c
−
αδ
c
T
+
p
β
α
+
S
θ
+
β
−
αδ
p
+
α
s
c−
αδ
c
+
3 1
2
1
{
}
2
))}
(
(
S
T
p
s
c
c
TT
h
c
cα
θ
β
αδ
α
cαδ
αδ
αδ
−
+
+
+
−
+
+
−
αδ
p
αδ
s
cαδ
c
T
{
p
β
(
θ
β
)(
α
S
(
θ
β
))
αδ
p
2
αδ
s
c6
}
2
{
23
3 2
2
−
−
+
+
+
+
−
+
+
2
}
2
{
6
}
))
(
)(
(
12
3 2 2
3 1 3
2
T
T
c
p
s
T
c
S
h
cθ
+
β
α
+
θ
+
β
−
αδ
+
αδ
c−
αδ
+
αδ
−
]
2
}
2
{
2 1 3 2
2
TT
c
s
p
αδ
cαδ
αδ
−
−
+
(15)
1
[{
}
{
2
}
{
2
}
]
1 3 2 2
3 2 2
3 2
2
T
c
p
s
T
c
s
p
c
c
p
s
T
T
TP
c c
c
αδ
αδ
αδ
αδ
αδ
αδ
αδ
αδ
αδ
α
−
−
−
+
−
−
+
−
+
=
∂
∂
3 3 1
2
[(
)
{
}
{
}
1
T
c
c
s
p
T
c
c
p
s
C
p
T
+
α
+
α
c−
αδ
−
αδ
−
αδ
+
αδ
−
α
c+
αδ
+
αδ
−
αδ
p
αδ
s
cαδ
c
T
{
2
αδ
s
cαδ
p
αδ
c
}
TT
{
αδ
p
2
αδ
s
c2
}
2
{
2 2 3 1 22
3 2
2
−
−
+
−
+
+
−
+
A
CS
p
C
T
p
S
Sh
c
T
s
p
T
T
c
α
β
cα
α
cαδ
αδ
+
−
+
+
+
+
−
−
+
−
−
]
2
[
(
)
(
)
{
}
{
2
}
3 12 1 3 2
−
αδ
c
−
αδ
c
T
+
p
β
α
+
S
θ
+
β
−
αδ
p
+
αδ
c
−
αδ
c
3−
αδ
c
2
3
{
(
(
))
2
}
c
c
p
S
p
s
T
S
h
α
θ
β
{
β
(
θ
β
)(
α
(
θ
β
))
αδ
2
αδ
2
))}
(
(
22
1
+
+
+
+
−
+
+
+
−
2
}
2
{
6
}
))
(
)(
(
12
3 2 2
3 1 3
2
T
T
c
p
s
T
c
S
h
cθ
+
β
α
+
θ
+
β
−
αδ
+
αδ
c−
αδ
+
αδ
−
]
2
}
2
{
2 1 3 2
2
TT
c
s
p
αδ
cαδ
αδ
−
−
+
(16)c c
c
c
c
s
p
c
T
p
s
s
p
T
T
T
TP
αδ
α
αδ
αδ
αδ
αδ
αδ
αδ
αδ
2
{
)
2
(
)
[(
1
23 2 2
3 1
2
−
+
+
−
+
+
+
−
=
∂
∂
∂
3
2 1 3 2
))
(
(
{
}
[{
1
]
}
p
S
Sh
c
p
S
p
s
c
T
T
c
β
cα
β
α
θ
β
αδ
α
cαδ
αδ
−
−
−
+
+
+
−
+
−
−
−
αδ
c
−
h
c(
α
+
S
(
θ
+
β
))}
T
1+
{
αδ
p
−
α
s
c+
αδ
c
3+
αδ
c
}
T
+
{
p
β
(
θ
+
β
)(
α
+
S
(
θ
+
β
))
−
αδ
2p
+
2
αδ
s
c−
h
c(
θ
+
β
)(
α
+
S
(
θ
+
β
))
{
2
}
]
2
}
2
{
2
}
2 2 3 12
3 2 2
2 1 3 2
TT
c
s
p
T
c
p
s
T
c
αδ
cαδ
αδ
αδ
αδ
cαδ
αδ
+
−
+
+
−
−
−
(17)FUZZY MODEL
IJMA- 7(4), April-2016.
© 2016, IJMA. All Rights Reserved 128 Let
θ
=
(
θ
1,
θ
2,
θ
3)
be a triangular fuzzy numbers then the total profit per unit time in fuzzy sense is2
}
{
}
{
)
(
)
(
[
1
)
,
(
2 3 1 1 ~T
c
c
p
s
T
c
Sh
S
p
T
c
p
cS
A
T
T
T
TP
=
−
+
+
+
α
+
β
−
c−
α
+
α
c−
αδ
−
αδ
−
αδ
2
}
))
(
(
))
(
(
{
2 1 3 ~ ~T
c
c
S
h
s
p
S
p
β
α
+
θ
+
β
−
αδ
+
α
c−
cα
+
θ
+
β
−
αδ
−
αδ
+
6
}
2
{
}
{
3 3 2 2 1 3T
c
s
p
TT
c
c
s
p
α
cαδ
αδ
αδ
αδ
cαδ
αδ
−
+
−
+
−
−
+
6
}
))
(
)(
(
2
))
(
)(
(
{
3 1 3 2 ~ ~ 2 ~ ~T
c
S
h
s
p
S
p
β
θ
+
β
α
+
θ
+
β
−
αδ
+
αδ
c−
cθ
+
β
α
+
θ
+
β
+
αδ
+
]
2
}
2
{
2
}
2
{
2 1 3 2 2 1 2 3 2 2TT
c
s
p
T
T
c
p
s
cαδ
αδ
αδ
αδ
cαδ
αδ
−
+
+
−
−
+
(18)Now we defuzzify the total profit
(
1,
)
~
T
T
TP
by signed distance methodSIGNED DISTANCE METHOD
By signed distance method the total profit per unit time is
)]
,
(
)
,
(
2
)
,
(
[
4
1
)
,
(
1 ~ 3 1 ~ 2 1 ~ 1 ~1
TP
T
T
TP
T
T
TP
T
T
T
T
T
TP
=
+
+
where
2
}
{
}
{
)
(
)
(
[
1
)
,
(
2 3 1 1 ~ 1T
c
c
p
s
T
c
Sh
S
p
T
c
p
cS
A
T
T
T
TP
=
−
+
+
+
α
+
β
−
c−
α
+
α
c−
αδ
−
αδ
−
αδ
2
}
))
(
(
))
(
(
{
2 1 3 ~ 1 ~ 1T
c
c
S
h
s
p
S
p
β
α
+
θ
+
β
−
αδ
+
α
c−
cα
+
θ
+
β
−
αδ
−
αδ
+
6
}
2
{
}
{
3 3 2 2 1 3T
c
s
p
TT
c
c
s
p
α
cαδ
αδ
αδ
αδ
cαδ
αδ
−
+
+
+
−
−
+
6
}
))
(
)(
(
2
))
(
)(
(
{
3 1 3 2 ~ 1 ~ 1 2 ~ 1 ~ 1T
c
S
h
s
p
S
p
β
θ
+
β
α
+
θ
+
β
−
αδ
+
αδ
c−
cθ
+
β
α
+
θ
+
β
+
αδ
+
]
2
}
2
{
2
}
2
{
2 1 3 2 2 1 2 3 2 2TT
c
s
p
T
T
c
p
s
cαδ
αδ
αδ
αδ
cαδ
αδ
−
+
+
−
−
+
2
}
{
}
{
)
(
)
(
[
1
)
,
(
2 3 1 1 ~ 2T
c
c
p
s
T
c
Sh
S
p
T
c
p
cS
A
T
T
T
TP
=
−
+
+
+
α
+
β
−
c−
α
+
α
c−
αδ
−
αδ
−
αδ
2
}
))
(
(
))
(
(
{
2 1 3 ~ 2 ~ 2T
c
c
S
h
s
p
S
p
β
α
+
θ
+
β
−
αδ
+
α
c−
cα
+
θ
+
β
−
αδ
−
αδ
+
6
}
2
{
}
{
3 3 2 2 1 3T
c
s
p
TT
c
c
s
p
α
cαδ
αδ
αδ
αδ
cαδ
αδ
−
+
+
+
−
−
+
6
}
))
(
)(
(
2
))
(
)(
(
{
3 1 3 2 ~ 2 ~ 2 2 ~ 2 ~ 2T
c
S
h
s
p
S
p
β
θ
+
β
α
+
θ
+
β
−
αδ
+
αδ
c−
cθ
+
β
α
+
θ
+
β
+
αδ
+
]
2
}
2
{
2
}
2
{
2 1 3 2 2 1 2 3 2 2TT
c
s
p
T
T
c
p
s
cαδ
αδ
αδ
αδ
cαδ
αδ
−
+
+
−
−
+
2
}
{
}
{
)
(
)
(
[
1
)
,
(
2 3 1 1 ~ 3T
c
c
p
s
T
c
Sh
S
p
T
c
p
cS
A
T
T
T
TP
=
−
+
+
+
α
+
β
−
c−
α
+
α
c−
αδ
−
αδ
−
αδ
2
}
))
(
(
))
(
(
{
2 1 3 ~ 3 ~ 3T
c
c
S
h
s
p
S
p
β
α
+
θ
+
β
−
αδ
+
α
c−
cα
+
θ
+
β
−
αδ
−
αδ
IJMA- 7(4), April-2016.
6
}
2
{
}
{
3 3 2 2 1 3T
c
s
p
TT
c
c
s
p
α
cαδ
αδ
αδ
αδ
cαδ
αδ
−
+
+
+
−
−
+
6
}
))
(
)(
(
2
))
(
)(
(
{
3 1 3 2 ~ 3 ~ 3 2 ~ 3 ~ 3T
c
S
h
s
p
S
p
β
θ
+
β
α
+
θ
+
β
−
αδ
+
αδ
c−
cθ
+
β
α
+
θ
+
β
+
αδ
+
]
2
}
2
{
2
}
2
{
2 1 3 2 2 1 2 3 22
p
c
T
T
p
s
c
TT
s
cαδ
αδ
αδ
αδ
cαδ
αδ
−
+
+
−
−
+
Therefore by the signed distance method the total profit per unit time is
3 1 1 ~
{
2
}
{
4
)
(
4
)
(
4
[
4
1
)
,
(
A
cS
p
c
T
p
S
Sh
c
T
s
p
c
T
T
T
TP
=
−
+
+
α
+
+
β
−
C−
α
+
α
C−
αδ
−
αδ
)
4
2
(
{
}
{
2
}
1 2 32 1 3
2
αβ
αδ
α
α
αδ
αδ
β
θ
θ
θ
β
αδ
+
−
+
−
−
−
+
+
+
+
−
c
T
p
p
s
Ch
Cc
c
T
p
S
C C C
s
p
TT
c
c
s
p
T
Sh
θ
θ
θ
β
4
{
αδ
α
αδ
αδ
}
{
αδ
2
αδ
2
)}
4
2
(
3 1 22 1 3
2
1
+
+
+
+
−
+
+
+
−
−
)
4
2
(
{
3
2
}
2
{
3
2
}
1 2 33 1 2 3 2 3 3
2
αδ
αδ
αδ
αβ
θ
θ
θ
β
αδ
+
+
−
+
+
+
+
−
c
T
s
Cc
p
T
p
)
4
2
(
)
2
4
2
4
2
(
θ
12θ
22θ
32β
2θ
1β
θ
2β
θ
3β
α
θ
1θ
2θ
3β
β
+
+
+
+
+
+
−
+
+
+
+
p
S
h
C
p
s
T
Sh
C C 23 1 3 2 1 2 2 3 2 2 2
1
2
{
2
6
)}
2
4
2
4
2
(
θ
+
θ
+
θ
+
β
+
θ
β
+
θ
β
+
θ
β
+
αδ
−
αδ
−
]
}
2
{
2
}
3 122 2 2 1 3 2
TT
c
s
p
T
T
c
αδ
αδ
Cαδ
αδ
+
−
−
+
(19)
The necessary condition for
(
1,
)
~
T
T
TP
to be maximum is that0 ) , ( 0 ) , ( 1 ~ 1 1 ~ = ∂ ∂ = ∂ ∂ T T T TP and T T T
TP , and solving these equations we find the optimum values of
T and
T1 say
* *
1and T
T for which profit is maximum and the sufficient condition is
2
~ ~ ~ ~
2 2 2 2
1 1 1 1
2 2 2
1 1 1
( , )
( , )
( , )
( , )
0
0
TP T T
TP T T
TP T T
TP T T
and
T
T
T T
T
∂
∂
∂
∂
−
∂
∂
∂ ∂
∂
1 3 1 1 ~)
(
4
)
(
4
[
4
1
)
,
(
T
c
c
h
s
p
p
c
Sh
S
p
T
T
T
T
TP
C CC
α
αβ
αδ
α
α
αδ
αδ
β
−
−
+
−
+
−
−
−
=
∂
∂
C Cs
p
T
Sh
S
p
β
θ
+
θ
+
θ
+
β
−
θ
+
θ
+
θ
+
β
+
αδ
−
α
+
{
(
12
2 34
)
(
12
2 34
)}
14
(
)
4
2
(
{
)
2
(
2
)
1 2 32 1 2 3 2
3
αδ
αδ
αδ
αδ
αβ
θ
θ
θ
β
αδ
+
+
+
−
+
+
+
+
+
c
c
T
s
Cc
p
T
p
)
4
2
(
)
2
4
2
4
2
(
1 2 3 1 2 32 2 3 2 2 2
1
θ
θ
β
θ
β
θ
β
θ
β
α
θ
θ
θ
β
θ
β
+
+
+
+
+
+
−
+
+
+
+
p
S
h
C3 2 2 1 3 2 1 2 2 3 2 2 2
1
2
(
2
2
)}
2
4
2
4
2
(
T
s
c
Sh
Cθ
+
θ
+
θ
+
β
+
θ
β
+
θ
β
+
θ
β
+
αδ
C+
αδ
−
2 2 2 2
3 1
)
4(
2
C)
]
p T
p
s
c TT
αδ
αδ
αδ
αδ
−
+
−
−
(20) 1 3 3 1 ~)
(
4
)
(
4
)
(
4
[
4
1
)
,
(
T
c
c
s
p
T
c
c
p
s
c
p
T
T
T
T
TP
CC
αδ
αδ
αδ
αδ
α
αδ
αδ
α
α
+
+
−
−
−
+
−
+
+
=
∂
∂
p
TT
c
p
s
T
c
s
p
C C2 1 3 2 2 2 3 2 2
(
2
)
2
(
4
)
2
(
2
αδ
−
αδ
−
αδ
+
αδ
−
αδ
+
αδ
+
αδ
+
1 2 2 1 3 2)
(
4
)
(
4
)
(
4
[
4
1
]
)
2
A
cS
p
c
T
p
S
Sh
c
T
T
T
c
s
Cαδ
α
β
Cα
αδ
−
−
−
+
+
+
+
−
−
IJMA- 7(4), April-2016.
© 2016, IJMA. All Rights Reserved 130
2 1 3
2
3
)
2
{
}
(
2
α
s
C−
αδ
p
−
αδ
c
−
αδ
C
T
+
p
αβ
−
αδ
p
+
α
s
C−
α
h
C−
αδ
c
−
αδ
c
T
+
3 2
1 3
2 1 3
2
1
4
(
2
)}
4
2
(
)
4
2
(
{
p
β
S
θ
+
θ
+
θ
+
β
−
Sh
Cθ
+
θ
+
θ
+
β
T
+
αδ
p
−
α
s
C+
αδ
c
+
3
2
)
2
(
3
2
)
2
(
)
3 1 2 3 2 3
3 2 2
1
T
p
c
s
T
c
s
p
TT
c
αδ
αδ
Cαδ
αδ
Cαδ
αδ
αδ
+
−
−
+
+
−
+
)
2
4
2
4
2
(
)
4
2
(
{
αβ
θ
1+
θ
2+
θ
3+
β
+
β
θ
12+
θ
22+
θ
32+
β
2+
θ
1β
+
θ
2β
+
θ
3β
+
p
p
S
6
)}
2
4
2
4
2
(
)
4
2
(
3 1 3 2
1 2 2 3 2 2 2 1 3
2 1
T
Sh
h
Cθ
θ
θ
β
Cθ
θ
θ
β
θ
β
θ
β
θ
β
α
+
+
+
−
+
+
+
+
+
+
−
2 1 3 2 2
2 1 3 2 2
)
2
(
2
)
2
(
2
αδ
s
C−
αδ
p
+
αδ
c
T
T
+
αδ
p
−
αδ
s
C−
αδ
c
TT
+
(21)
Numerical example-Let us consider the following parameters in appropriate units
500
,
10
,
6
,
5
,
4
,
3
,
100
,
2
,
005
.
0
,
5
.
0
,
200
=
=
=
=
=
=
3=
=
=
=
=
β
θ
δ
A
h
cs
cc
c
p
S
α
Table-1: (variation of total profit with respect to the parameter θ)
θ
T
1T
TP
0.005 36.3938 818.2250 7
7.5046 10
×
0.010 36.3942 818.3942
7.5050 10
×
7 0.015 36.3945 818.2710 77.5055 10
×
0.020 36.3949 818.2940 7
7.5059 10
×
As we increase the parameter
θ
then the values of the parametersT
1,
T
andTP
increases.Table-2: (variation of total profit with respect to the parameter β)
β
T
1T
TP
0.5 36.3938 818.2250 7
7.5046 10
×
1.0 36.6280 832.8690 7
8.1140 10
×
2.0 38.1901 938.5680
1.0084 10
×
8 3.0 41.1326 1174.5800 81.6250 10
×
As we increase the parameter
β
then the values of the parameters,T
1,
T
and TP increases.
Figure-2: variation of θ Figure-3: variation of β
