Structural Design Optimization using Parameterized P-norm Based ...

International Journal of Computer & Organization Trends (IJCOT)–Volume 33 Number1– July 2016

Structural Design Optimization using Parameterized P-norm Based Geometric Programming Technique Mridula Sarkar1, Samir Dey 2, Tapan Kumar Roy3 1

Ph.D Candidate , Department of Mathematics, Indian Institute of Engineering Science and Technology,Shibpur,Howrah-711103,West Bengal,India. 2 Assistant Professor, Department of Mathematics, Asansol Engineering College, Vivekananda Sarani, Asansol-713305, West Bengal,India 3 Professor, Department of Mathematics,Indian Institute of Engineering Science and Technology,Shibpur,Howrah-711103,West Bengal, India. Abstract— In this paper we will make an approach to solve single objective structural model using parameterized p-norm based fuzzy Geometric Programming technique. A structural design model in fuzzy environment has been developed. Here pnorm based generalised triangular fuzzy number (GTFN) is considered as fuzzy parameter so that the decision maker can take advantage of no-exact parameter. Generalised triangular p-norm is discussed with their basic properties and some special cases. In this structural model formulation, the objective function is the weight of the truss; the design variables are the cross-sections of the truss members; the constraints are the stresses in members. A classical truss optimization example is presented here in to demonstrate the efficiency of our proposed optimization approach. The test problem includes a two-bar planar truss subjected to a single load condition. This approximation approach is used to solve this single-objective structural optimization model. The model is illustrated with numerical examples. Keywords— Generalized Triangular Fuzzy Number, p-norm, Geometric Programming, Single Objective Optimization, Structural Optimization. I. INTRODUCTION Optimization seeks to maximize the performance of a system, part or component, while satisfying design constraints. One common form of optimization is trial and error and is used every day. We make decisions, observe the result, and change future actions depending on the success of those decisions. When performing optimization, we wish to minimize (or maximize) the structural design, while considering both design variables and design constraints. Design variables are variables the designer or engineer can freely choose between, for example the thickness of a wall, the material chosen, and the width of a part. The resulting stress, deflection, volume, natural frequency and other

ISSN: 2249-2593

typical performance measures are often considered either as objective functions or as constraints. In practice, the problem of structural design may be formed as a typical non-linear programming problem with non-linear objective function and constraints functions in fuzzy environment. Zadeh [1] first introduced the concept of fuzzy set theory. Then Zimmermann [2] applied the fuzzy set theory concept with some suitable membership functions to solve linear programming problem with several objective functions. Some researchers applied the fuzzy set theory to structural model. For example, Wang et al. [3] first applied -cut method to structural designs where the non-linear problems were solved with various design levels, and then a sequence of solutions were obtained by setting different level-cut value of Rao[4] applied the same α -cut method to design a four–bar mechanism for function generating problem. Structural optimization with fuzzy parameters was developed by Yeh et al. [5].Xu[6] used two-phase method for fuzzy optimization of structures. Shih et al. [7] used level-cut approach of the first and second kind for structural design optimization problems with fuzzy resources. Shih et al.[8] developed an alternative -level-cuts methods for optimum structural design with fuzzy resources. Prabha et al.[18] presents an efficient algorithm to optimize fuzzy transportation problem. Geometric Programming (GP) method is an effective method used to solve a non-linear programming problem like structural problem. It has certain advantages over the other optimization methods. Here, the advantage is that it is usually much simpler to work with the dual than the primal one. Solving a non-linear programming problem by GP method with degree of difficulty (DD) plays essential role. (It is defined as DD = total number of terms in objective function and constraints – total number of decision variables – 1). Since late 1960‟s, GP has been known and used in various fields (like OR, Engineering sciences etc.). Duffin et al. [9] and Zener[10] discussed the basic theories on GP with engineering application in their books. Another

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International Journal of Computer & Organization Trends (IJCOT)–Volume 33 Number1– July 2016

famous book on GP and its application appeared in 1976 (Beightler et al., [11]).The most remarkable property of GP is that a problem with highly nonlinear constraints can be transformed equivalently into a problem with only linear constraints. In real life, there are many diverse situations due to uncertainty in judgments, lack of evidence etc. Sometimes it is not possible to get relevant precise data for the cost parameter. The idea of impreciseness (fuzziness) in GP i.e. fuzzy geometric programming was proposed by Cao [12]. Ojha et al. [14] used binary number for splitting the cost coefficients, constraints coefficient and exponents and then solved it by GP technique. A solution method of posynomial geometric programming with interval exponents and coefficients was developed by Liu [15]. In 2015, Dey and Roy [16]optimized shape design of structural model with imprecise coefficient by parametric geometric programming .Islam and Roy [17] used FGP to solve a fuzzy EOQ modelwith flexibility and reliability consideration and demand dependent unit production cost a space constraint. FGP method is rarely used to solve the structural optimization problem. Dey and Roy [18] solved twobar truss non-linear problem using Intuitionistic fuzzy Optimization Technique. But still there are enormous scopes to develop a fuzzy structural optimization model through fuzzy geometric programming (FGP).The parameter used in the GP problem may not be fixed. It is more fruitful to use fuzzy parameter instead of crisp parameter. In that case we can introduce the concept of fuzzy GP technique in parametric form. In this paper we are making an approach to solve single-objective structural model using parameterized p-norms based fuzzy geometric programming technique. In this structural model formulation, the objective function is to minimize weight of the truss; the design variables are the cross-sections of the truss members; the constraints are the stresses in members. The test problem includes a two-bar planar truss subjected to a single load condition. This approximation approach is used to solve this single-objective structural optimization model. The remainder of this paper is organized in the following way. In section 2, we discuss about single objective structural optimization model. In section 3, we discuss the mathematical Prerequisites. In section 4, we propose the technique to solve single-objective non-linear programming problem using p-norms based fuzzy optimization. In section 5, apply pnorms based fuzzy optimization technique to solve single-objective structural model and numerical illustration is given. Finally we draw conclusions in section 6.

II. MATHEMATICAL FORM OF A SINGLE – OBJECTIVE STRUCTURAL OPTIMIZATION MODEL

In sizing optimization problems the aim is to minimize a single objective function, usually the weight of the structure, under certain behavioural constraints on stress and displacements. The design variables are most frequently chosen to be dimensions of the cross-sectional areas of the members of the structure. Due to fabrication limitations the design variables are not continuous but discrete since cross-sections belong to a certain set. A discrete structural optimization problem can be formulated in the following form (1) Minimize f ( x)

Subject to gi ( x) 0, i 1, 2,......., m

Aj

Rd ,

j 1, 2,......., n

where f ( A) represents objective function, g A is the behavioural constraint, m and n are the number of constraints and design variables, respectively. A given set of discrete values is expressed by R d and design variables A j can take values only from this set.In this paper, objective function is taken as m

f A

i

(2)

Ai li

i 1

and constraints are chosen to be stress of structures i 0 i

gi A

where

i

1 0 i 1,2,..., m

and li are weight of unit volume and

length of i th element, respectively, m is the number of the structural elements, i and i0 are the i th stress and allowable stress, respectively. III. PREREQUISITE MATHEMATICS A. Fuzzy Set Let X is a set (space), with a generic element of X denoted by x , that is X ( x) .Then a Fuzzy set (FS) is defined as  A x, ( x) : x X A

where

 A

FS A .

:X A

[0,1] is the membership function of

( x ) is the degree of membership of the

A. element x to the set  B.

-Level Set or

-cut of a Fuzzy Set

The -level set of the fuzzy set A of X is a crisp set A that contains all the elements of X that have membership values greater than or equal to i.e.  A x : A ( x) , x X, [0,1] . C. P-norm Generalized Triangular Fuzzy Number) a ,a ,a ;w A fuzzy number a p is said to 1

be

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(3)

p-norm

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generalized

2

3

a

p

triangular

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International Journal of Computer & Organization Trends (IJCOT)–Volume 33 Number1– July 2016

number GTFN

4) a p .bp

if its membership function is

p

defined by

a2 x a2 a1

wa 1

a ( x )

p

p

if a1 p

x a3 a3 a2

wa 1

1

1

x

a2

0

x

p

0)

if (a p

0, bp

0)

0)

p

if (a p

(a1b3 , a2 b2 , a3b1; min(wa , wb ))

p

(a3b3 , a2 b2 , a1b1 ; min(wa , wb ))

p

0)

5) a p / bp

p

if a2

if (a p

0, bp 0, b

(a1b1 , a2 b2 , a3b3 ; min(wa , wb ))

a3

otherwise

(a1 / b3 , a2 / b2 , a3 / b1 ;min(wa , wb ))

p

if (a p

(a1 / b1 , a2 / b2 , a3 / b3 ;min(wa , wb ))

p

if (a p

0, bp 0, b

p

0)

if (a p

0, bp

0)

(a3 / b1 , a2 / b2 , a1 / b3 ;min(wa , wb ))

p

IV. MATHEMATICAL ANALYSIS

Where wa represent the maximum degree of

atleast one of the values of a1 , a2 , a3 is not equal to zero. 2) Remark 2: (a1 , a2 , a3 ; wa ) p is said to A (GTFN ) p , a p

1) Geometric Programming Method A geometric program (GP) is a type of mathematical optimization problem characterized by objective and constraint functions that have a special form. GP is a methodology for solving algebraic non-linear optimization problems. Also linear programming is a subset of a geometric programming .The theory of geometric programming was initially developed about three decades ago and culminated in the publication of the seminal text in this area by Duffin, Peterson, and Zener [18]. The general constrained Primal Geometric Programming problem is as follows

be positive (i.e a p

Minimize f 0 ( x)

membership satisfy in 0 wa 1 Also a1 a2 a3 and p is a positive integer. It can be easily observed that when p 1

GTFN

p

reduces to GTFN.

1) Remart 1: A (GTFN ) p , a p

(a1 , a2 , a3 ; wa )

positive (i.e a p

0 ) if and only if

p

is said to be

0 , and

a1

0 ) if and only if a1

0 and

zero. 3) Remark 3: a p  0 if and only if all the values of a1 , a2 , a3 are equal to zero. 4) Remark 4: a p is said to be non- negative if either a p  0 or 0.

D. Arithmatic Operations The arithmetic operations over (GTFN ) p are defined as follows a p (a1 , a2 , a3 ; wa ) p Let bp (b1 , b2 , b3 ; wb ) p be (GTFN ) p , Then 1) a p bp bp

2) a p 3) a p

(a1 b1 , a2

and

(a1 b1 , a2 b2 , a3 b3 ; min( wa , wb )) p p

if if

a

x j 0 tj

(4)

j 1

Subject to Tm

f i ( x)

n

cit t 1

xj

0,

a

x j itj

bi ; i 1, 2,3,......., m

j 1

j 1, 2,.........., n.

Here c0t

0 and a0tj be any real number. The

objective function contains T0 terms and Ti terms in the inequality constraints. Here the coefficient of each term is positive.So it is a constrained posynomial geometric programming problem. Let T T0 T1 ......... Ti be the total number of terms in the primal program. The degree of difficulty (DD) is defined as DD = Total no. of terms – (Total no. of variables -1) = T (n 1) .The dual problem (with the objective function, d (w) where

b2 , a3 b3 ; min( wa , wb ))

( a1 , a2 , a3 ; wa ) ( a3 , a2 , a1 ; wa )

n

c0t t 1

atleast one of the values of a1 , a2 , a3 is not equal to

a p

T0

0 0

p

p

.

w

w(wit ), i 0,1, 2......, m; t 1, 2,.....Ti

.

decision vector) of the geometric programming problem (4) for the general posynomial case is as follows T0

Maximize d ( w) t 1

c0t w0t

w0 t

m

Ti

i 1 t 1

cit

is

wit

bi wit

the

wit

(5)

Subject to T0

w0t

1,

(Normality condition)

t 1

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International Journal of Computer & Organization Trends (IJCOT)–Volume 33 Number1– July 2016

m

Ti

aitj wit

0

for j 1, 2,......, n. (Orthogonality

be p-norm based triangular fuzzy numbers with membership functions

i 0 t 1

Condition) wit 0 i 0,1,........., m; t 1, 2,........Ti . For a primal problem with M variables, T0 T1 ......... Ti terms and n constraints, the dual problem consists of T0

w 1

c0 t

a

c0t

T1 ......... Ti variables

d * ( w* ) w0*t t

x j 0 tj

1, 2,3,..., Ti

* it

w

a

cit

xn itj j 1

cit

T0

 Minimize f 0 ( x)

c0t

x

t 1

a0 tj j

Ti

f i ( x) t 1

xj

n

cit

x

aitj j

j 1

0

for j 1, 2,........, n where c0t , cit , bi are p-norm based generalised

positive triangular fuzzy number. a0tj and aitj are real numbers for all i, t , j . Let c0t (c10t , c20t , c30t ; w) cit b i

(c1it , c2it , c3it ; w)

(b1i , b2i , b3i ; w)

p p

p

1

x c3i c3i c2i

(1 t

(1 i

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Ti ,1 i

x

c2it

x

a2

if c2it

x

c3it

1

p

if c1it

x

c2it

if

x

b2

if b2i

x

p

c3i

the functions 0, w , fcitl : c1it , c2it 0, w ,

fbitl : b1i , b2i

0, w

continuous and

f citr : c20t ( ), c30t

Where are w 0,1 and non-decreasing f c0 tr : c20t ( ), c30t 0, w ,

0, w , f citr : c2it , c3it

0, w ,

Where w 0,1 are continuous and non-increasing function and w is called maximum membership degree. Here cut of c0t , cit , bi are given by

c0t ( )

c0tL ( ), c0tR ( ) 1

1

cit ( )

w

1

p

1

p

(c30t c20t )

w

citL ( ), citR ( ) 1

p

c2it

p

(c20t c10t ) , c30t

1

Find x So as to

p

p

(c2it c1it ) , c3it

w

1

w

1

p

(c3it c2it )

( x1 , x2 , x3 ,......, xn )T

f

R 0

(9) p

c30t t 1

m)

c30t

otherwise

where f c0 tl : c10t ( ), c20t

Minimize

T0 )

x

p

T0

(1 t

if c20t

p

0

j 1

 bi for i 1, 2,........., m 

p

p

b2i x b2i b1i

w 1

c20t

Such that

a2

otherwise

p

n

1

w

( x)

x

if c1it

x c3it c3it c2it

w 1

bi

c20t

p

0

system of linear equations of t j ( j 1, 2,.........., n. ).We can easily find primal

if

if

w 1

Taking logarithms in (6) and (7) and putting t j log x j for j 1, 2,.........., n. we shall get a

x

p

w

( x)

t 1

2) Fuzzy Geometric Programming Problem The formulation of fuzzy geometric programming with fuzzy parameters can be stated as follows Find x ( x1 , x2 , x3 ...........xn )T (8) so as to

1

p

c2it x c2it c1it

w 1

variables from the system of linear equations. Case I: For T n 1 ,the dual program presents a system of linear equations for the dual variables where the number of linear equations is either less than or equal to the number of dual variables. A solution vector exists for the dual variable (Beightler and Philips [20]). Case II: For T n 1 ,the dual program presents a system of linear equations for the dual variables where the number of linear equation is greater than the number of dual variables. In this case, generally, no solution vector exists for the dual variables. However, one can get an approximate solution vector for this system using either the least squares or the linear programming method.

p

if c10t

otherwise

(6)

wit*

1

0

i 1, 2,3,...., m; t 1, 2,3,..., Ti (7)

Ti

p

c10t

x c30t c30t c20t

w 1

j 1

n

c20t

1

p

x

w

( x)

and m 1 constraint. The relation between these problems, the optimality has been shown to satisfy n

c20t

1

w

1

p

n

(c30t c20t ) j 1

such that

m)

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a

x j 0 tj

International Journal of Computer & Organization Trends (IJCOT)–Volume 33 Number1– July 2016

p

c30t

Ti

1

w 1

p

t 1

b2i

1

Now solving above two sub problem by geometric programming technique we can get the upper and lower bound of objective function for each 0,1 .

p

(c30t

f i ( x)

for

1

c20t )

n

a

x j itj

p

1

j 1

(b2i b1i )

w

i 1, 2,................, m

x j 0 for j 1, 2,3....................n, 0,1 Using cut of the p-norm generalised triangular fuzzy number coefficients the above problem reduces to (10) Find x ( x1 , x2 , x3 ,......xn )T so as to T0

Minimize

n

f 0 ( x)

c0tL ( ), c0tR ( ) t 1

IV. NUMERICAL ILLUSTRATION A well-known two-bar [17] planar truss structure is considered. The design objective is to minimize weight of the structural WT A1 , A2 , yB of a statistically loaded two-bar planar truss subjected to stress i A1 , A2 , yB constraints on each of the truss members i 1, 2 .

a

x j 0 tj J 1

Such that Ti

n

f i ( x)

a

citL ( ), citR ( )

x j itj

t 1

biL ( ), biR ( )

j 1

for i 1, 2,.......m a0tj and aitj re real numbers for all i, t , j . The above problem is equivalent to the sub-problem Find x ( x1 , x2 , x3 ,............., xn )T (11) so as to 1

p

T0 L 0

Minimize f ( x)

c20t

1

n

(c20t c10t )

w

t 1

p

a

x j 0 tj j 1

such that p

c2it

Ti

1

1

(c20t

w

f i ( x)

1

p

t 1

b3i

p

1

c10t )

n

a

x j itj

p

Fig. 1 Design of the two-bar planar truss The single-objective structural model can be expressed as

1

j 1

(b3i b2i )

w

for i 1, 2,.........., m

xj

0

for

j 1, 2,...., n,

Find x

( x1 , x2 , x3 ,....., xn )T so as to

Minimize

R 0

p

T0

c30t

1

(12)

p

n

(c30t c20t )

w

t 1

1

l yB

2

A2 xB2 yB 2 (13)

such that

0, w

a0tj and aitj are real numbers for all i, t , j.

f

A1 xB2

Minimize WT A1 , A2 , yB

AB

A1 , A2 , yB

BC

A1 , A2 , yB

a

x j 0tj j 1

P xB2

l

yB

2 T AB

lA1 P xB2

yB 2

C BC

lA2

such that p

c30t

Ti

1

for

xj

(c30t p

t 1

1

0.5

p

w

f i ( x) b2i

1

1

w

c20t )

for

j 1, 2,3.....n,

p

j 1

(b2i b1i )

1

1.5 ; A1

0,1

0, A2

0;

The input data for structural optimization problem (13) is given as follows Nodal load ( P ) (90,100,110;0.8) p KN , Volume Density (  )

a0tj and aitj are real numbers for all i, t , j.

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a

x j itj

i 1, 2,....., m

0

n

yB

(7.6,7.7,7.8;0.8)

p

KN / m3 ,

Length (l ) 2 m; Width ( xB ) (.9,1,1.1, 0.8)

p

Allowable compressive stress  c

(90,100,110;0.8)

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International Journal of Computer & Organization Trends (IJCOT)–Volume 33 Number1– July 2016

Allowable tensile stress T

(145,150,155;0.8)

p

Mpa;

y coordinate of node B ( yB ) (0.5 yB 1.5) Solution: The non-linear structural optimization problem of two bar truss is  Minimize WT ( A1 , A2 , yB ) 2 2 (14) (7.6,7.7,7.8;0.8) A (.9,1,1.1,0.8) (2 y ) 2 A (.9,1,1.1,0.8) y2 p

1

p

B

2

B

Such that (90,100,110;0.8) AB

( A1 , A2 , yB )

,

p

(.9,1,1.1,0.8)

p

100

c

1.5 ,

yB

p

1 1

2

100 1

(2 yB ) 2

p

p

155 1

,

1

p

1

p

(0.1),7.8 1

w

7.7 1

1

BC

100 1

100

p

w

1

1

(10)

w

p

(0.1)

(16)

w

1

1

1

1

1

1

w

1

(10)

1

1

1

1

1

1

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1

1,

p

1

1

p

1 1

p

yB 2

(0.1)

w

p

(10)

w

0; 0.5

1

yB

1,

1.5

0, w

(0.1) 1

150 1

1

w

(2 yB )

2

A2 1.1 1

and 1

p

(0.1)

w p

1

2 p

(0.1)

w

yB 2

p

(10) 1

p

1.1 1 p

w

1

p

(0.1)

(2 yB )2

(5) 1,

2 A1 p

(0.1)

w 1

(17)

( A1 , A2 , yB )

w p

1

2

AB p

7.8

p

(0.1)

w

p

w

w

(2 yB ) 2

(0.1)

(10)

such that

p

110 1

p

1

A1 1.1 1

w

p

(10),110

1

w

(10)

p

(0.1),1.1

p

w

p

Minimize WT R ( A1 , A2 , yB )

(5)

p

w

1

1

p

1 1 (5)

p

p

yB 2

(0.1)

p

w

p

w

1

p

w

2 A2

p

(0.1),7.8

1 1

p

1

p

w

1

A2

( A1 , A2 , yB )

AB

w

(15)

(10)

w

1

1

p

(2 yB )2

(0.1)

2

p

p

w

w

1

(0.1),110

p

1

p

(5),155

p

1

1

p

w

p

(0.1)

p

(10),110

p

100

1

w

p

w

p

p

P

1

p

p

(0.1),1.1 1

p

7.7

p

( A1 , A2 , yB )

A1 , A2

xB

7.7 1

p

(0.1)

p

w p

7.7

1

2 A1

A1

p

c

1

2

p

110

100

(5)

p

(0.1),110

p

w

p

p

Minimize WT L ( A1 , A2 , yB )

2

0, A2 0 cut of  , xB , P , t ,  c are given by

7.7 1

150

1

Using cut above problem (15) is reduced to the two sub problems

2 A1

p

t

1

w

subject to

(.9,1,1.1,0.8)

p

145,150,155;0.8

P

1

p

( A1 , A2 , yB )

xB

w

1

p

(5),155

p

,

p

(90,100,110;0.8)

The

2

2 A2

90,100,110;0.8

yB

1

p

( A1 , A2 , yB )

0.5

t

A1

(90,100,110;0.8)

AB

(2 yB )2

p

150

p

2 A1

145,150,155;0.8

BC

2

(.9,1,1.1,0.8)

p

1

p

p

(10)

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BC

( A1 , A2 , yB ) 1

p

110 1

p

(10)

w 1

p

100 1

1

p

1.1 1

p

(0.1)

w

p

yB 2

(10)

w

1,

2 A2

A1 , A2 0; 0.5 yB 1.5 0, w To apply Geometric Programming Technique we may consider any of the sub problems as Minimize WT ( A1 , A2 , yB ) M A1 L( )

2

2

(2 y 2 B ) A2 L( )

(2 y 2 B )

(18) w31

subject to AB

BC

w01 w11 2w31 2w32 2w33 0, w02 w21 2w41 2w42 0, w32 2w33 2w42 0, The constraints of (20) forms a system of six linear equations with nine unknowns .So the system has infinite number of solutions .However the problem is to select the optimal dual variables w01 , w02 , w11 , w21 , w31 , w32 , w33 , w41 , w42 . We have w01 w11 w31 w32 w33 , w02 w21 1 w31 w32 w33 , w41 1 w31 0.5w32 , w42 0.5w32 w33 Substituting w01 , w02 , w11 , w21 , w41 , w42 in the dual formulation we get Maximize dWT (w31 , w32 , w33 )

N

( A1 , A2 , yB )

2

L( )

1,

2 A1 S

( A1 , A2 , yB )

L( )

2

y

4( w31 w32 w32

2 B

1,

2 A2

L( )

A1

0, A2

0;

0.5

yB

1.5

(19)

subject to ( A1 , A2 , yB )

NA3 2 A1

1,

BC ( A1 , A2 , y B )

SA4 2 A2

1,

(( L( ))2 4) A3 2 4 yB A3 2 2

2 L( ) 2

2 L( )

4 A3

A4

2

2

2 B

4 yB A3

y A4

2

2 B

y A3

1,

M w01

L( )

2

w02

2

1,

M w02

N 2w11

4

w31 w32

w33

w31

(w31 w32 w33 ) w33

w33

L( )

such that w01 w02 1, w01 w11

2

w41 w42

( w31 w32 w33

w33

w33 )

w41 w42 0.5w32 w33

0.5 w32 w33

(21)

To find the optimal w31 , w32 , w33 which maximizes the dual dWT (w31 , w32 , w33 ) we take logarithm of both sides of (21) and get log dWT (w31 , w32 , w33 )

4) w31

w32

4( w31 w32 w32 w41

w33 )

w42

w33 respectively and equating to zero we get

0,

0.5

w42

4

1 w31

w31

log dWT ( w31 , w32 , w33 ) 2 w33 w31

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Sw31 L( ) L

2

0,

Sw32 0,

0

log dWT ( w31 , w32 , w33 ) 2

(20)

2

(0.5w32 w33 )0.5

and N (0.5w32 w33 ) Sw33

2

0, w02 w21

w42 )

(1 w31 0.5w32 )log(1 w31 0.5w32 ) (0.5w32 w33 )log(w41 w42 ) (0.5w32 w33 )log(0.5w32 w33 ) Differentiating partially with respect to w31 , w32 and

2

w41 w42

w31 log w31

w32 log w32 w33 log(w31 w32 w33 )

1 w31 0.5w32 N L( ) w32

w33

(1 w31 0.5w32 ) log Q 2 ( w41

4 N 1 w31 0.5w32

w41

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w32

w21

w31

2

w31

(( L( ))2 4) w31 w32 w33

1 w31 0.5 w32

w41 w42

w33 log w33

S 2w11

w31 w32 w33

w31

w32 log 4 w31 w32 w33

1,

w11

2(w31

(1 w31 w32 w33 )

w33 )

w31 log (Q 2

0.5 yB 1.5 A1 0, A2 0, A3 0, A4 0 This is a signomial Geometric Programming Problem with DD=9-(5+1)=3 The dual formulation is Maximize dWT (w01 , w02 , w11 , w21 , w31 , w32 , w33 , w41 , w41 ) w01

N w32 w33 )

(w31 w32 w33 )log M (w31 w32 w33 )log(w31 w32 w33 ) (1 w31 w32 w33 )log M (1 w31 w32 w33 )log(1 w31 w32 w33 ) (w31 w32 w33 )log N (w31 w32 w33 )log(2(w31 w32 w33 )) (1 w31 w32 w33 )log S (1 w31 w32 w33 )log(2(1 w31 w32 w33 ))

yB2 A3 2 2

2

1 w31 w32 w33

M 1 w31 w32 w33

1 w31 0.5w32

Let ( L( ))2 (2 yB )2 A32 and ( L( )) yB2 A42 Then above problem (18) can be written as Minimize WT ( A1 , A2 , yB ) M ( A1 A3 A2 A4 )

AB

w31 w32 w33

S 2(1 w31 w32 w33 )

2

(2 y B )

M w32 w33

1 1 w31 0.5w32

0

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International Journal of Computer & Organization Trends (IJCOT)–Volume 33 Number1– July 2016

2

log dWT ( w31 , w32 , w33 ) 2 w31 w32

3 w31 w32 w33 2

1 1 w31 w32 w33

log dWT ( w31 , w32 , w33 ) 2 w33 w32 3 w32

w31 2

0.5 1 w31 0.5w32

w33

0

log dWT ( w31 , w32 , w33 ) 1 1 2 w33 0.5w32 w33 w33 It is to be noted that for optimum dual variable * * * the Hassian w31 , w32 , w33 matrix log dWT w31 , w32 , w33 2 2

2

w31

2

2 2

log dWT w31 , w32 , w33 2

2 2

w33 w31

i.e. 2

2 2

2 2

2 2

M

w32 w33

w33 w31

2

0,1

7.7 1

1

p

w

100 1

1

1

log dWT w31 , w32 , w33 2

2

2 2

w32 w33

log dWT w31 , w32 , w33 2

w33

(0.1)

p

(10)

,S p

110 1

1

w

p

w

1

p

(10)

w

0,1

7.8

1

110

1

1

p

p

(0.1) , L

w 1

w

1

0

w

1 1

1

1

p

(0.1)

w

p

p

(10) p

150

p

w

100 1

(5)

N

w31w33

log dWT w31 , w32 , w33

1

p

(10)

p

2

w32

w33 w32

M

1 1

p

w

and for 0

p

(0.1) , L

w32

w31w32

log dWT w31 , w32 , w33 2

0.5w32 w33 1 w32 w31 w33

p

log dWT w31 , w32 , w33

log dWT w31 , w32 , w33 2

* w42 * w w42 * 41

p

w33

w31 w32

log dWT w31 , w32 , w33

w32 w31

log dWT w31 , w32 , w33 2

2

1 w31 0.5w32 1 w32 w31 w33

* w42

p

0,

2

w31

* w41

155 1

w32 w31

log dWT w31 , w32 , w33 2

2

2

2

* w33

* w41

Now for

log dWT w31 , w32 , w33

w31

log dWT w31 , w32 , w33

w

w31w33

2

2

2

* w33 * w32

p

2

log dWT w31 , w32 , w33

* w33

N

log dWT ( w31 , w32 , w33 ) 2 w31

log dWT w31 , w32 , w33

* w32 * w32

we will get optimal solution for A1* , A2* .

must be negative definite . 2

* w31

* 31

1

* w21 * w31 * * w32 w33

log dWT w31 , w32 , w33

w33 w32

* w21

SA4 2 A2

1;

* w31

yB2 A4 2

log dWT w31 , w32 , w33

w32

log dWT w31 , w32 , w33 2

4 A3 2

L2 A4 2

log dWT w31 , w32 , w33

w31w32

w32 w31

log dWT w31 , w32 , w33 2

2

2

log dWT w31 , w32 , w33 2

log dWT w31 , w32 , w33 2

2

L2

yB2 A3 2

2

2

w11* w11*

4 yB x3 2

0.5 0.5w32 w33

log dWT ( w31 , w32 , w33 ) 2 w31 w33

NA3 2 A1

110

1

100

1

(10)

p

p

w

* * * * * * * * * * w01 d * w01 , w02 , w11 , w21 , w31 , w32 , w33 , w41 , w42

MA2 A4

* * * * * * * * * * w02 d * w01 , w02 , w11 , w21 , w31 , w32 , w33 , w41 , w42

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1

p

(10)

the above problem gives the left and right spread of weight interval.

Now from the primal dual relation MA1 A3

p

w

,S (5)

1

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International Journal of Computer & Organization Trends (IJCOT)–Volume 33 Number1– July 2016

VI.TABLE I OPTIMIZED RESULT OF DESIGN VARIABLES OF TRUSS FOR

Level of Possibility or Degree of Uncertainty

0.0

w 0.8 .

Right Left Spread of Design Variables

Norm P=1

Norm P=2

Spread of

Norm

Norm

Design

P=1

P=2

19.272

19.271

0.6975

0.6974

.8708

0.8708

0.8163

0.8163

18.645

19.233

0.6825

0.6965

0.8463

0.8692

0.8202

0.8165

18.038

19.110

0.6678

0.6936

0.8226

0.8645

0.8241

0.8173

17.451

18.903

0.6532

0.6886

0.7995

0.8564

0.8279

0.8186

16.882

18.600

0.6390

0.6814

0.7777

0.8446

0.8316

0.8205

16.331

18.185

0.6250

0.6713

0.7554

0.8283

0.8353

0.8231

15.797

17.620

0.6113

0.6574

0.7343

0.8062

0.8390

0.8268

11.397

16.811

0.5002

0.6372

0.5822

0.7743

0.8436

0.8321

12.974

14.779

Variables WT L A1L L 2

A y

0.1

A

0.2

A

WT A1L 0.3

L 2

A y

WT A1L 0.4

L 2

A y

0.5

A

WT A1L 0.6

L 2

A

y

0.7

A

0.8

L 2

A y

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L B

y

R B

10.115

0.4874

0.4608

WT A1R

0.5265

R 2

0.8552

y

R B

10.283

0.4987

0.4661

WT A1R

0.5338

R 2

0.8537

y

R B

10.522

0.5102

0.4735

WT A1R

0.5443

R 2

y

R B

11.370

0.5220

0.4841

WT A1R

0.5592

R 2

R

A

y

R B

13.548

11.397

0.5339

0.5002

WT A1R

0.5822

R 2

0.8436

R

A

12.113

0.8483

R

A

11.735

0.8514

R

A

11.348

0.8337 L

0.8563

R

A

10.973

0.6316

L B

WT A1L

0.5216

R 2

0.8372 L

L 2

y

0.4573

0.6140

L B

WT A1L

0.4762

WT A1R

0.8406 L

y

R B

10.002

0.5968

L B

0.8569

R

A

10.610

0.8440 L

L 2

y

0.5187

R 2

0.5801

L B

WT A1L

0.4552

0.8473 L

y

R B

0.4653

0.5639

L B

A

WT A1R

0.8506 L

0.8571

R 2

9.938

0.5481

L B

0.5178

WT R A1R

10.258

0.8539 L

L 2

y

0.4546

0.5327

L B

WT A1L

0.4546 0.8571

L

L 2

y

9.917

0.5178

L B

WT A1L

9.9173

R

A y

R B

R

12.974

12.974

0.5460

0.5460

WT A1R

0.5460

0.5845

0.6498

R 2

0.6498

0.6938

R B

0.8301

0.8461

0.6498 0.8301

0.8301

A

y

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International Journal of Computer & Organization Trends (IJCOT)–Volume 33 Number1– July 2016

In general the value of shows that the level of possibility and degree of uncertainty of the obtained information. When the value of increases, the level of possibility becomes greater and the degree of uncertainty become less. From the above result it 0 the widest interval indicates is clear that when that objective value definitely lie into this range. On 0.8 indicates the other hand the possibility level the most possible value of the objective function. In this example the objective value is impossible to fall below 9.917 or exceed 19.272 for p=1,2 and the most possible value lie within 12.974 and 14.779 for p=1,2. VII. CONCLUSIONS In this work, a Geometric Programming for Structural Optimization of two bar truss design problem has been discussed. The considered problem is a highly nonlinear and non-exact in nature. Here the parameter is taken as p norm based generalised triangular fuzzy number and Zadeh‟s extension principle has been used to transform the fuzzy geometric programming problem to a pair of two mathematical programmes. P-norm based fuzzy number can be used in several optimization design problems.

[12] B.Y.Cao‟‟Solution and theory of question for a kind of fuzzy positive genetic programme,Proc,2nd IFSA Congress,Tokyo, pp.205-208, july 1987. [13] Ojha,A.K., Das,A.K.,‟‟Geometric Programming Problem with Coefficients and exponents associated with binary numbers,Int.J.Comput.Sci.,vol.7,pp.49-55,Feb 2010. [14] Liu,S.T., “Posynomial Geometric Programming with interval exponent and coefficients‟‟, Eur.J.Oper.Res.,vol.186,pp.1727,Apr 2008. [15] Dey,S.,Roy,T.K.,‟‟ Truss Design Optimization Using Fuzzy Geometric Programming in Parametric Form‟‟,Journal of computational science,vol.4,400-415,Jan 2014. [16] S Islam,T.K.roy, “ A fuzzy EOQ model with flexibility and reliability consideration and demand dependent unit production cost a space constraint: A fuzzy geometric programming approach”, Applied Mathematics and computation, vol.176, pp.531-544,May2006. [17] Dey,S.,Roy,T.K.,”Optimized Solution of two-bar truss design using intuitionistic optimization technique.‟‟,I.J.Information Engineering and Electronic Bussiness,vol.4,pp.45-51,Aug 2014. [18] Prabha, S, Krishna. Devi,S.,Deepa,S., “An efficient algorithm to obtain the optimal solution for fuzzy transportation problems”, International journal of computer and organization trends,vol.4,pp.32-39,Jan 2014.

ACKNOWLEDGMENT The research work of Mridula Sarkar is financed by Rajiv Gandhi National Fellowship (F117.1/2013-14-sc-wes-42549/(SA-III/Website)),Govt of India. REFERENCES [1] [2]

[3]

[4]

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Zadeh.L.A.. „‟Fuzzy Sets‟‟, Information and Control,vol.8 ,pp. 338-353, 1965. Zimmermann,H.J „‟Fuzzy linear programming with several objective function „‟Fuzzy sets and systems,vol.1,pp.4555,Jan 1978. Wang,G.Y.,Wang,W.Q.,‟‟Fuzzy optimum design of structure‟‟Engineering Optimization,vol. 8,pp. 291300,1985. Rao, S.S., “Description and optimum Design of Fuzzy Mathematical Systems”,Journal of Mechanisms, Transmissions, and Automation in Design, vol. 109,pp.126132, Mar1987. Yeh, Y.C, and Hsu, D.S. “Structural optimization with fuzzy parameters”.Computer and Structure, vol. 37,pp. 917–924, 1990. Xu, C. “Fuzzy optimization of structures by the two-phase method”,Computer and Structure, vol.31, pp.575–580,Dec 1989. Shih,C.J, Lee,H.W. “Level cut Approaches of First and Second Kind for Unique Solution Design in Fuzzy Engineering Optimization Problems ”,Tamkang Journal of Science and Engineering,vol.7,pp. 189-198,Sep 2004. Shih,C.J.,Chi,C.C. and Hsiao,J.H. “Alternative -level-cuts methods for optimum structural design withfuzzy resources”, Computers and Structures ,vol.81, pp.2579–2587,Nov 2003. R.J.Duffin, E.L.Peterson and C.M.zener, Geometric Programming theory and Applications, Wiley, New York, 1967. C.Zener, “Engineering Design by Geometric Programming Wiley, 1971. C.S.Beightler,D.T.Phillips.,Applied Geometric Programming, Wiley, New York 1976.

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