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Posynomial Geometric Programming
Problems with Multiple Parameters
A. K.Ojha and K.K.Biswal
Abstract— Geometric programming problem is a powerful tool for solving some special type non-linear programming problems.
It has a wide range of applications in optimization and engineering for solving some complex optimization problems. Many
applications of geometric programming are on engineering design problems where parameters are estimated using geometric
programming. When the parameters in the problems are imprecise, the calculated objective value should be imprecise as well.
In this paper we have developed a method to solve geometric programming problems where the exponent of the variables in
the objective function, cost coefficients and right hand side are multiple parameters. The equivalent mathematical programming
problems are formulated to find their corresponding value of the objective function based on the duality theorem. By applying a
variable separable technique the multi-choice mathematical programming problem is transformed into multiple one level
geometric programming problem which produces multiple objective values that helps engineers to handle more realistic
engineering design problems.
Index Terms— Duality theorem, Geometric programming, multiple parameters, optimization,. Posynomial.
—————————— ‹ ——————————
1 INTRODUCTION
Various mathematical programming methods have been
formulated to solve many challenging real world problems. Since 1960 some authors [5] have cited that geometric inequality helps to solve special type optimization
problem which is known as geometric programming.
However Duffin, Peterson and Zener[8] laid the foundation stone to solve wide range of engineering problems by
developing basic theories of geometric programming and
its application in their text book. Geometric programming(GP) is a technique for solving polynomial type nonlinear programming problems. One of the remarkable
properties of Geometric programming is that a problem
with highly nonlinear constraints can be stated equivalently with a dual program. If a primal problem is in posynomial form then a global minimizing solution of the
problem can be obtained by solving its corresponding
dual maximization problem because the dual constraints
are linear, and linearly constrained programs are generally easier to solve than ones with nonlinear constraints.
GP problem has a dual impact in the area of integrated
circuit design[4,10,17] manufacturing system design [8,
3],project management[23], maximization of long run and
short term profit[16], generalized geometric programming problem with non positive variables [24] and goal
programming model [1]. Several algorithms due to
•
•
Beighter and Phillips[2], Fang et al.[9], Kortanek[12], Kortanek et al.[13], Peterson[19], Rajgopal and Bricker [22]
and Zhu and Kortanek[25] strengthen the solution of
complicated Geometric programming problem for the
exact known value of cost and constraint coefficients.
Sensitive analysis of various optimal solutions due to
Dembo[6], Dinkle and Tretter [7] and Kyparisis [14] using
Geometric programming technique simplifies certain engineering design problem in which some of the problem
parameters are estimates of the actual value. There are
certain problems in which some of the coefficients may
not be presented in a precise manner. For example, in
project management the time required to complete the
various activities in a research and development project
may be only known approximately. In order to determine
the inventory policy of a novel technology product, the
demand and supply quantities may be uncertain due to
insufficient market information and are specified by
ranges. If some parameters imprecise or uncertain, then
the most liking values are usually adopted to make the
conventional geometric programming workable. This
simplification might result in a derived result which is
misleading. One way to manipulate imprecise parameters
is via probability distributions. However, a probability
distribution requires constructions of prior predictable
regularity or a posterior frequency determination which
may not be possible in certain cases. Uncertain parameters can be considered by applying interval estimates in_____________________________
stead of single values. In the recent papers Liu [16] has
Dr.A. K. Ojha, School of Basic Sciences, IIT Bhubaneswar, Orissa, studied the geometric programming problems considerPin-751013, India.
ing the cost coefficient, constraint coefficients and the
K.K.Biswal Department of Mathematics, CTTC Bhubaneswar, B- right hand sides are interval numbers where the derived
36, Chandaka Industrial Area, Bhubaneswar, Orissa, Pin-751024, objective values also lies in an interval. When the cost coIndia.
efficient, constraint coefficients and exponents of the decision variable in the objective functions of the GP problem
are multiple parameters the problem becoming more
84
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complicated. In this paper we have developed methods
for the solution of GP problem when the cost, constraints,
its right hand side and exponents are multiple parameters. For these multiple parameters we construct multiple
level mathematical programming models to find the
value of the objective function. These results will provide
the decision makers with more information for making
better decisions. The organization of this paper is as follows: Following introduction, mathematical formulations
and methodology for solving geometric programming
problems with multiple parameters have been discussed
to find the respective objective values of the problem in
Section 2. Dual form of GPP has been discussed in Section
3. Some illustrative examples are given in Section 4 for
understanding the problems and finally at Section 5 some
conclusions are drawn from the discussion.
2 Mathematical Formulation
A typical constrained posynomial geometric programming problem is presented as follows:
Z=
T0
n
min
a
g 0 ( x) = ∑ C 0t ∏ x j 0 tj
x
t =1
j =1
middle one is the average of other two.
n
min T0
C 0t ∏ x aj0 tj
∑
x t =1
j =1
such that
Ti
n
∑C ∏ x
it
t =1
Ti
n
t =1
a
≤ bi , i = 1,2,..., m
where
{
= {B
}
{
}
}
{
}
C 0t = C 0Lt , C 0Mt , C 0Ut , C it = C itL , C itM , C itU ,
bi
L
i
, BiM , BiU
}
{
a 0tj = A0Ltj , A0Mtj , A0Utj , a itj = AitjL , AitjM , AitjU
In order to find the objective values of the posynomial
function we will derive its corresponding values with
respect to their counterpart of the parameters. Let us define
For each triplet
(2.1)
x j > 0, j = 1,2,..., n
L
n
min
a
C0t ∏ x j 0 tj
∑
x t =1
j =1
M
Z ,Z ,Z
values of
The posynomial g0(x) is a objective function containing T0
number of terms where as the posynomial gi(x), i = 1,
2,…,m contains Ti terms with m inequality constraints. By
the definition of posynomial all the co-efficients Cit , i =
0,1, 2,…,m and t = 1, 2,…,Tm are positive and the exponents a0tj and aitj are arbitrary constants. Writing the right
hand side of the geometric programming problem given
by (2.1) in more general form, we have
(C , b, a ) ∈ S
ing objective Z of (2.3) by
Let
U
we denote its correspond-
Z (C , b, a )
are the lower, middle and maximum
Z (C , b, a )
defined by
{
}
{
}
Z L = min Z (C , b, a ) : (C , b, a ) ∈ S
Z U = max Z (C , b, a) : (C , b, a ) ∈ S
are the values at the first and third parameter where
{
Z M = Z (C , b, a ) : (C , b, a ) ∈ S
L
n
M
U
Now the above objectives Z , Z , Z can be formulated as geometric programming as:
∑ Cit ∏ x j itj ≤ bi , i = 1,2,..., m
t =1
}
is the value at middle parameter
such that
Ti
(2.3)
j =1
⎧(C , b, a ) : c it ∈ C it , b i ∈ B i , a itj ∈ A itj , ⎫
S=⎨
⎬
⎩1 ≤ t ≤ Ti , i = 1,2,..., m, j = 1,2,..., n
⎭
j =1
T0
a itj
j
x j > 0, j = 1,2,..., n .
subject to
gi ( x) = ∑ Cit ∏ x j itj ≤1, i = 1,2,..., m
85
a
(2.2)
j =1
x j > 0, j = 1,2,..., n
where all bi are positive real numbers. If bi = 1 for all i
then this modified geometric program becomes the original one given by(2.1).
Considering C0t ,Cit , bi , a0tj and aitj are the multiple values
of the corresponding posynomial geometric program given by (2.2)can be reduced in the following form restricting
the number multiple parameter as three number where
n
min
min T0
Z =
C 0t ∏ x aj 0 tj
∑
(C , b, a ) ∈ S x t =1
j =1
L
Ti
subject to
n
∑C ∏ x
it
t =1
a itj
j
≤ b i , i = 1,2,..., m
j =1
x j > 0, j = 1,2,..., n
n
max
min T0
Z =
C 0t ∏ x aj 0 tj
∑
(C , b, a ) ∈ S x t =1
j =1
U
(2.4)
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such that
Ti
n
∑C ∏ x
it
t =1
a itj
j
≤ bi , i = 1,2,..., m
(2.5)
j =1
Now our main objective is to find the minimum value of
ZLand maximum value of ZU against all possible values
on S and such that ZM should give the value of Z against
all possible values of S which will approximately the best
possible values in between ZL and ZU
To derive the minimum value of the model (2.4) against
x j > 0, j = 1,2,..., n
ZM
86
all possible values on S we can set C0t to
T0
n
max
min
C 0t ∏ x aj0 tj
=
∑
(C , b, a ) ∈ S x t =1
j =1
exponent as
C 0t to C 0Lt
and
A0Ltj
Hence the model (2.4) can be transformed to the form
such that
Ti
n
∑C ∏ x
it
t =1
a itj
j
≤ bi , i = 1,2,..., m
min T0 L n A0Ltj
Z =
xj
∑ C0t ∏
x t =1
j =1
L
(2.6)
j =1
such that
x j > 0, j = 1,2,..., n
∑ C (B ) ∏ x
Ti
Since bi in the model (2.4), (2.5), (2.6) may not be equal
to the constant 1 then dividing the constraint coefficients Cit by bi ∀i then it is transformed to the standard form
−1
n
i
t =1
a itj
j
≤ 1, i = 1,2,..., m (2.7)
j =1
Similarly to find the minimum value of the model (2.5)
n
max
min T0
a 0 tj
C
0t ∏ x j
∑
(C , b, a ) ∈ S x t =1
j =1
∑ (C )(b ) ∏ x
−1
n
i
t =1
a itj
j
≤ 1, i = 1,2,..., m (2.8)
j =1
x j > 0, j = 1,2,..., n
∑ (C )(b ) ∏ x
−1
t =1
i
a itj
j
j =1
x j > 0, j = 1,2,..., n
≤ 1, i = 1,2,..., m
n
U
Aitj
j
≤ 1, i = 1,2,..., m
(2.11)
j =1
x j > 0, j = 1,2,..., n
The minimum value of model (2.6) can be calculated at
the corresponding middle counter part of the parameter
by transforming in the form
Ti
min
x
T0
n
t =1
j =1
∑ C0Mt ∏ x j 0tj
( ) ∏x
∑ CitM BiM
t =1
n
−1
AM
such that
Such that
it
max T0 U n A0Utj
xj
∑ C0 t ∏
x t =1
j =1
( ) ∏x
ZM =
n
max
min T0
a 0 tj
C
0t ∏ x j
∑
(C , b, a ) ∈ S x t =1
j =1
Ti
and
for all i.
∑ CitU BiU
t =1
ZM =
B
C it
Under this model (2.5) becomes
Ti
Ti
C it
is
bi
b i are set to C itU
such that
such that
it
U
i
ZU =
x j > 0, j = 1,2,..., n
ZU =
(2.10)
x j > 0, j = 1,2,..., n
and
∑ (C )(b ) ∏ x
≤ 1, i = 1,2,..., m
j =1
minimum when the value of
such that
Ti
AitjL
j
against all such possible values of S, then the ratio
n
min
min T0
Z =
C
x aj 0 tj
0
t
∑
∏
(C , b, a) ∈ S x t =1
j =1
L
it
t =1
n
L −1
i
L
it
(2.9)
−1
n
AitjM
j
j =1
x j > 0, j = 1,2,..., n
≤ 1, i = 1,2,..., m
(2.12)
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3 Dual form of GPP
Since model (2.5) is the conventional geometric programming problem then it can be solved directly by using
primal based algorithm or dual based algorithm[19]. Methods due to Rajgopal and Bricker[22], Beightler and Phillips[2] and Duffin[8] projected in their analysis that the
dual problem has the desirable features of being linearly
constrained and having an objective function with structural properties with more suitable solution. According to
Liu[16] the model (2.5) can be transformed to the corresponding
dual
geometric
problem
as
ZL =
max T0 ⎛ C0Lt
⎞
∏ ⎜ w0t ⎟⎠
w t =1 ⎝
w0 t
∏∏ [(C )(B )]
m
ti
L
it
L
i
−1
t =1 t =1
⎛ wi 0 ⎞
⎜
wit ⎟⎠
⎝
∑ w0t = 1
Ti
∑∑ a
itj
i =1 t =1
subject to
(2,4,6) t1−2 t 2−2 + t 2( −5, −4, −3) t 3−1 ≤ 1
t1 , t2 , t3 > 0
According to the model (3.1),(3.2) and (3.3) the problem can
be transformed to its correspond dual program as
(3.1)
t =1
m
min
: (10,20,30 )t1( −3, −2, −1)t2( 2,3, 4)t3−1
(4.1)
t
+ 40t1t2 + 40t1t2t3
wit
T0
such that
max ⎛ 10 ⎞
⎟
:⎜
Z =
w ⎜⎝ w01 ⎟⎠
wit = 0, j = 1,2,...., n
max T0 ⎛ C0Ut
⎞
⎜
∏
w0t ⎟⎠
w t =1 ⎝
T0
such that
∑w
0t
t =1
m
w0 t
∏∏ [(C )(B )]
m
ti
U
it
U
i
−1
t =1 t =1
=1
⎛ wi 0 ⎞
⎜
wit ⎟⎠
⎝
⎛ 2 w11 ⎞
⎟⎟
⎜⎜
⎝ w11 ⎠
wit
wit ≥ 0, ∀t , i
Z
T0
∑w
such that
t =1
0t
∏∏ [(C )(B )]
m
ti
M
it
t =1 t =1
=1
M
i
−1
⎛ wi 0 ⎞
⎜
wit ⎟⎠
⎝
wit
Ti
∑∑ a
i =1 t =1
itj
w03
(w11 + w12 )(w
11 + w12
)
(4.2)
w01 , w02 , w03 , w11 , w12 ≥ 0
(3.3)
max ⎛ 30 ⎞
⎟
Z =
:⎜
w ⎜⎝ w01 ⎟⎠
w01
U
m
w12
⎛ 40 ⎞
⎟⎟
⎜⎜
⎝ w03 ⎠
− 3w01 + w02 + w03 − 2w11 = 0
− 2 w01 + w02 + w03 − 2 w11 − 5w12 = 0
− w01 + w03 − w12 = 0
(4.3)
i =1 t =1
w0 t
⎛ 1 ⎞
⎟⎟
⎜⎜
⎝ w12 ⎠
w02
w01 + w02 + w03 = 1
(3.2)
Ti
max T0 ⎛ C0Mt
⎞
=
⎜
⎟
∏
w
0t ⎠
w t =1 ⎝
w11
⎛ 40 ⎞
⎟⎟
⎜⎜
⎝ w02 ⎠
subject to
∑∑ aitj wit = 0, j = 1,2,...., n
M
w01
L
wit ≥ 0, ∀t , i
ZU =
87
wit = 0, j = 1,2,...., n
wit ≥ 0, ∀t , i
The model (3.1),(3.2) and (3.3) are the usual dual problem
and it can be solved using the method relating to the dual
theorem.
4 Illustrative Examples
To illustrate the methodology proposed in this paper for
solving a GPP with multiple parameters of cost, constraint coefficients and exponents of the decision variables a few numerical examples are considered.
Example:1
Let us consider the geometric programming problem
which has the following mathematical form:
⎛ 1 ⎞
⎟⎟
⎜⎜
⎝ w12 ⎠
w12
⎛ 40 ⎞
⎟⎟
⎜⎜
w
⎝ 02 ⎠
w2
⎛ 40 ⎞
⎟⎟
⎜⎜
w
⎝ 03 ⎠
(w11 + w12 )(w
11 + w12
subject to
w01 + w02 + w03 = 1
(4.4)
− w01 + w02 + w03 − 2 w11 = 0
4 w01 + w02 + w03 − 2 w11 − 3w12 = 0
− w01 + w03 − w12 = 0
w01 , w02 , w03 , w11 , w12 ≥ 0
w03
)
⎛ 6 w11 ⎞
⎟⎟
⎜⎜
w
⎝ 11 ⎠
w11
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Z
M
max ⎛ 20 ⎞
⎟
:⎜
=
w ⎜⎝ w01 ⎟⎠
w01
⎛ 1 ⎞
⎟⎟
⎜⎜
⎝ w12 ⎠
⎛ 40 ⎞
⎟⎟
⎜⎜
⎝ w02 ⎠
w12
w02
⎛ 40 ⎞
⎟⎟
⎜⎜
⎝ w03 ⎠
w03
(w11 + w12 )(w
11 + w12
⎛ 4 w11 ⎞
⎟⎟
⎜⎜
⎝ w11 ⎠
88
(2, 2.5, 3) x13 x3 + x1−1 x3−1 ≤ (3, 4, 5 )
w11
x2−1 x3( −1, −2, −3) x4−2 + (3,3.5,4) x12 x2 x4 ≤ 1
)
x1 , x2 , x3 , x4 > 0
Now the corresponding dual program is formulated as follows:
subject to
w01 + w02 + w03 = 1
(4.5)
max ⎛ 1 ⎞
⎟
Z =
:⎜
w ⎜⎝ w01 ⎟⎠
w01
L
− 2w01 + w02 + w03 − 2w11 = 0
3w01 + w02 + w03 − 2w11 − 4 w12 = 0
− w01 + w03 − w12 = 0
⎛ 1 ⎞
⎜⎜
⎟⎟
3
w
⎝ 12 ⎠
w01 , w02 , w03 , w11 , w12 ≥ 0
Using LINGO the dual solution of the optimal objective values of
Z can be obtained as: ZL=125.9045, for
0.5767344;
*
03 =
w
0.2821770;
ZM = 194.9390, for
*
w01
*
11 =
w
w11* =0.2815197; w12*
*
= 0.1537485; w02 =
for
*
w01
w11* = 0.3462515; w12*
= 0.1410885;
0.2178230;
= 0.1456535;
0.3277204;
*
w01
*
w02
*
12
w
*
w02
=
=0.1410885,
= 0.5266261;
*
w03
=
= 0.1820669; ZU = 296.2627,
0.4362556;
*
w03
=0.4099959;
= 0.2562475
In the constrained geometric programming problem, the dual optimal solu*
⎛ 3 ⎞
⎜⎜
⎟⎟
⎝ w22 ⎠
w12
w22
⎛ 3 ⎞
⎜⎜
⎟⎟
⎝ w02 ⎠
w02
( w11 + w12 ) ⎛
1 ⎞
⎜⎜
⎟⎟
w
⎝ 21 ⎠
(w11 + w12 )
(w21 + w22 )(w
21 + w22
t
max ⎛ 3 ⎞
⎟
Z =
:⎜
w ⎜⎝ w01 ⎟⎠
t 3*
t 3* =
⎛ 1 ⎞
⎜⎜
⎟⎟
⎝ 5w12 ⎠
t
Z
= 0.6223022 and for
ZU
as
t1* =2.380155; t 2*
t
=1.35740;
0.9398071
The derived optimal solutions for the lower, middle and upper parts of the multiple parameters are the best possible
values as represented by Liu[16].
In the next example we shall set the right hand side of the
constrained as the multiple parameter.
Example:2
Let us consider the geometric programming problem with
multiple parameters in objective function:
min
: (1, 2 , 3) x1( −4, −3, −2 ) x2−1 x3 x4−1
x
+ (3, 5, 7) x1− 2 x2( −3, −2, −1) x3− 2
such that
(4.6)
(4.7)
w01 + w02 = 1
− 4w01 − 2w02 + 3w11 − w12 + 2 w22 = 0
− w01 − 3w02 − w21 + w22 = 0
w01 − 2 w02 + w11 − w12 − w21 = 0
− w01 − 2 w21 + w22 = 0
w01 , w02 , w11 , w12 , w21 , w22 ≥ 0
t
w21
subject to
*
*
as 1 = 1.305470; 2 =
programming problem is obtained for
*
M
*
*
1.390561; 3 =0.4892672, for
as 1 = 1.804540; 2 = 1.422246;
Z
t
w11
)
tions w provide weights of the terms in the constraints of the transformed primal problem. The corresponding primal solution of the geometric
L
⎛ 2 ⎞
⎜⎜
⎟⎟
⎝ 3w11 ⎠
w01
U
w12
⎛ 7 ⎞
⎜⎜
⎟⎟
⎝ w02 ⎠
w02
( w11 + w12 ) ⎛
(w11 + w12 )
⎛ 3 ⎞
⎜⎜
⎟⎟
⎝ 5w11 ⎠
1 ⎞
⎜⎜
⎟⎟
⎝ w21 ⎠
w11
w21
(4.8)
w22
⎛ 4 ⎞
⎜⎜
⎟⎟ (w21 + w22 )( w21+ w22 )
⎝ w22 ⎠
w01 + w02 = 1
− 2w01 − 2w02 + 3w11 − w12 + 2 w22 = 0
− w01 − w02 − w21 + w22 = 0
w01 − 2 w02 + w11 − w12 − 3w21 = 0
− w01 − 2 w21 + w22 = 0
w01 , w02 , w11 , w12 , w21 , w22 ≥ 0
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M
Z
max ⎛ 2 ⎞
⎟
:⎜
=
w ⎜⎝ w01 ⎟⎠
⎛ 1 ⎞
⎜⎜
⎟⎟
⎝ 4 w12 ⎠
⎛ 3.5 ⎞
⎜⎜
⎟⎟
⎝ w22 ⎠
w12
w22
w01
⎛ 5 ⎞
⎜⎜
⎟⎟
w
02
⎝
⎠
w02
⎛ 2 .5 ⎞
⎜⎜
⎟⎟
4
w
11
⎝
⎠
( w11 + w12 ) ⎛
1 ⎞
⎜⎜
⎟⎟
⎝ w21 ⎠
(w11 + w12 )
(w21 + w22 )(w
21 + w22
parameters. In the above discussed two examples it is
understood that the value of the objective remain within
the range for the multiple parameters in exponent, cost
and constrained. In the second example the objective values are in decreasing order due to the exponent of the
decision variable are considered in decreasing order.
Geometric programming has already shown its power in
practice in the past. In many real world geometric programming problem the parameters may not be known
precisely due to insufficient informations and hence this
paper will help the wider applications in the field of engineering problems.
w11
w21
)
(4.9)
subject to
6 Acknowledgments
w01 + w02 = 1
The authors are very much thankful to the anonymous
referees for their valuable suggestion for the improvement of the paper.
− 3w01 − 2w02 + 3w11 − w12 + 2w22 = 0
REFERENCES
− w01 − 2w02 − w21 + w22 = 0
w01 − 2 w02 + w11 − w12 − 2w21 = 0
− w01 − 2 w21 + w22 = 0
w01 , w02 , w11 , w12 , w21 , w22 ≥ 0
The optimal dual and primal solution of the geometric program is
as follows. Dual ZL = 47.47193 for
= 0.9236831E - 07;
Dual ZM = 44.53226 for
*
w01
*
11
*
w02
=
= 0;
= 0.8571429;
*
12
w
w12*
= 0.833333;
*
*
w21
=0.5; w22 =
*
1.83333 the corresponding primal optimal variables are x1 =
*
*
*
0.3205667; x 2 = 1.481980; x3 = 1.064722; x 4 =1.719745 .
0.166666;
w11*
*
w01
*
w02
= 0.1428571;
*
21
*
= 0.1360334E 06; w = 0; w = 0.2857142; w22 =
1.428571 and its corresponding primal optimal variables are
x1* =0.2482863; x 2*
= 3.164843;
Dual ZU = 23.22874 for
x3*
= 1.128281;
x 4* =1.220395.
*
*
=0.8751308; w02 =
w01
0.1248692;
*
w11* =0.5231659E03; w12* =0.2513079; w21
= 0.1248692;
*
w22 =1.124869 and its corresponding primal optimal variables are
x1* = 0.1314416; x 2* = 60.08292; x3* = 1.524756;
x 4* =0.2167734
This example demonstrates that the objective values remains
in the required range when the parameters are multiple in
nature.
5 Conclusions
From 1960 geometric programming problem has undergone several changes. In most of the engineering problems the parameters are considered as deterministic. In
this paper we have discussed the problems with multiple
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Dr.A.K.Ojha: Dr.A.K.Ojha received a Ph.D(mathematics)
from Utkal University in 1997. Currently he is an
Asst.Prof. in Mathematics at I.I.T. Bhubaneswar, India. He
is performing research in Nural Network, Geometric Programming, Genetical Algorithem, and Particle Swarm
Optimization. He has served more than 27 years in different Govt. colleges in the state of Orissa. He has published
22 research papers in different journals and 7 books for
degree students such as: Fortran 77 Programming, A text
book of modern algebra, Fundamentals of Numerical
Analysis etc.
K.K.Biswal:
Mr.K.K.Biswal
received
a
M.Sc.(Mathematics) from Utkal University in 1996. Currently he is a lecturer in Mathematics at CTTC, Bhubaneswar, India. He is performing research works in
Geometric Programming. He is served more than 7 years
in different colleges in the state of Orissa. He has published 2 research papers in different journals.
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