Posynomial Geometric Programming Problems with Multiple

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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

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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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