Solution of an Engineering Design Optimization System: A General Fuzzy Programming Technique and Intuitionistic Fuzzy Optimization Technique

Solution of an Engineering Design Optimization System: A General Fuzzy Programming Technique and Intuitionistic Fuzzy Optimization Technique

Samir Dey* and Tapan Kumar Roy

Department of Mathematics, Bengal Engineering and Science University, Shibpur, Howrah-711103, West Bengal, India.

Abstract

The main objective of structural engineers through out design history has been to obtain structure under the prescribed design conditions which can not only withstand external loads safety but also achieve an economic solution. This paper we apply General Fuzzy Non-Linear Programming [GFNLP] technique as well as Intuitionistic Fuzzy optimization[IFO] technique to solve the problem of optimum design of plane truss structures. Our objective is to prove IFO method perform better than GFNLP method. This approach is illustrated on planer truss optimization model and the results are discussed.

Keywords: General Fuzzy non-linear programming, Intuitionistic fuzzy optimization, Engineering Design, Structural Optimization.

Mathematics Subject Classification Code: 90C15, 90C29, 90C70.

1. Introduction

   The science and engineering in real-world problems are often not deterministic or non- crisp as people recognized. Fuzzy set theory [23] was a recent progress of describing certain non-crisp information with fuzzy arising in problems; since then, many fields

   ranging from sciences to industrial, medical and financial application had applied it successfully. From the point of view of engineering, most applications and developments with fuzzy theory belong to the category of measurement, manufacturing and control behavior.However; literatures reported engineering design and their applications with intutionistic fuzzy and fuzzy logic are uncommon in dealing with the fuzziness existing in the real-world problems.

   Zadeh (1965) first gave the concept of fuzzy set theory. Later on Bellman and Zadeh (1970) used the fuzzy set theory to the decision making problem. Tanaka (1974) introduced the objective as fuzzy goal over -cut of a fuzzy constraint set and Zimmerman (1978) gave the concept to an inventory and production problems. Wang et al. [21] first applied -cut method to structural designs where the non-linear problems were solved with various design levels , and then a sequence of solutions are obtained by setting different level-cut value of . Rao [20] applied the same -cut method to design a fourbar mechanism for function generating problem.Yeh and Hsu [22] followed the framework of Wang et al.[21] under different design level of obtaining the optimum design level while the total cost is based on the failure possibility instead of the membership value of satisfaction.

   Intuitionistic Fuzzy Set (IFS) was introduced by K.Atanassov (1986) and seems to be applicable to real world problems. The concept of IFS can be viewed as an alternative approach to define a fuzzy set in case where available information is not sufficient for the definition of an imprecise concept by means of a conventional fuzzy set. Thus it is expected that, IFS can be used to simulate human decision-making process and any activities requiring human expertise and knowledge that are inevitably imprecise or totally reliable. Here the degree of rejection and satisfaction are considered so that the sum of both values is always less than unity (1986). Atanassov also analyzed Intuitionistic fuzzy sets in more explicit way.Atanaossov (1989) discussed open problems in intuitionistic fuzzy sets theory. An interval valued intutionistic fuzzy sets was analyzed by Atanaossov and Gargov (1999).Atanassov and Kreinovich (1999) implemented Intuitionistic fuzzy interpretation of interval data. The temporal intuitionistic fuzzy sets are also discussed by Atanassov (1999).Intuitionistic fuzzy soft sets are also considered by Roy et al. [10].Rough intuitionistic fuzzy sets are analyzed by Rizvi et

   al.[17].Angelov(1997) implemented the optimization in an intuitionistic fuzzy environment. He (1995) also contributed in his another two important papers, based on Intuitionistic fuzzy optimization.Pramnik and Roy (2005) solved a vector optimization problem using Intuitionistic fuzzy goal programming. A transportation model is solved by Jana and Roy (2007) using multi-objective intuitionistic fuzzy linear programming.

   The main objective of a structural engineering is to design structures which withstand external loads safely and at a minimum cost or weight [9,18 and 19].The desire to improve a design without compromising the structural integrity has been a strong driving force behind the development of various optimum design methods.

   In this paper we consider a structural model is subject to geometry area, stress [1].First time we apply General Fuzzy Non-Linear Programming [GFNLP] technique to solve the above mention model. We also apply Intuitionistic Fuzzy Optimization [IFO] technique to solve the above mention model. In this paper our objective is to establish the fact numerically that IFO technique minimizes the weight more than FGNLP technique.

   1.1 Preliminaries.

   Definition 1.1.1. (FS) Let X is a set (space), with a generic element of X denoted by x ,

   that is

   Where

   X (x) .Then a FS is defined as Equation

   A x, A (x) / x X

   A : X [0,1] the membership function of FS A is,

   A (x) [0,1]

   is the degree of

   membership of the element x to the set A.

   Definition 1.1.2. (IFS) For a set X, an IFS A in the sense of Atanassov is given by equation

   A x, A (x), A ( x) / x X

   Where the function

   A : X [0,1]

   and

   A : X [0,1]

   with the

   condition 0 A (x) A (x) 1 , x X .

   The numbers,

   A (x) [0,1]

   and

   A (x) [0,1]

   ,denote the degree of membership and the

   degree of non-membership of the element x to the set A, respectively. For each IFS A in

   X, the amount

   A (x) 1 A (x) A (x)

   is called the degree of indeterminacy

   (hesitation part) ,which may cater to membership value, non-membership value or both.

   Definition 1.1.3: -Level Set or -cut of a Fuzzy Sets: The -level set (or interval of

   confidence at level or -cut) of the fuzzy set A-

   of X is a crisp set A

   that contains all

   the elements of X that have membership values in A-

   A

   A

   A- x, – (x) , x X , [0,1].

   greater than or equal to i.e

2. Mathematical Analysis

   1. General Fuzzy Non-linear Programming (FNLP) Technique to solve Multi- Objective Non-Linear Programming Problem (MONLP):

      A Multi-Objective Non-Linear Programming (MONPL) or Vector Minimization problem (VMP) may be taken in the following form:

      Subject to

      Min f (x) [f1 (x), f2 (x),….., fk (x)]

      T

      T

      j j

      j j

      x X x Rn : g (x) or or b for j 1, 2,…., m

      (2.1.1)

      and

      li xi ui

      (i 1, 2, 3,…., n).

      Zimmermann (1978) showed that fuzzy programming technique can be used to solve the multi-objective programming problem.

      To solve MONLP problem, following steps are used:

      Step 1: Solve the MONLP (2.1.1) as a single objective non-linear programming problem using only one objective at a time and ignoring the others, these solutions are known as ideal solution.

      Step 2: From the result of step 1, determine the corresponding values for every objective at each solution derived. With the values of all objectives at each ideal solution, pay-off matrix canbe formulated as follows:

      f1 (x)

      f * (x1 ) 1	f * (x1 ) 2	….	f * (x1 ) k
      f * (x2 ) 1	f * (x2 ) 2		f * (x2 ) k
      		….	.
      f * (xk ) 1	f * (xk ) 2		f * (xk ) k

      f * (x1 ) 1	f * (x1 ) 2	….	f * (x1 ) k
      f * (x2 ) 1	f * (x2 ) 2		f * (x2 ) k
      		….	.
      f * (xk ) 1	f * (xk ) 2		f * (xk ) k

      x1 x2

      xk

      f2 ( x)

      fk ( x)

      Here

      x1 , x2 ,….., xk

      are the ideal solutions of the objectives

      f1 x,

      f2 x,

      ……,

      fk x respectively.

      r r r r

      r r r r

      So U

      maxf

      x1 , f

      x2 ,…., f

      xk and L

      f * xr

      for r 1, 2,…., k

      r r

      r r

      Where

      Ur and

      L be upper and lower bounds of the rth

      objective function

      fr x

      for

      r

      r

      r 1, 2,…., k .

      Step 3: Using aspiration level of each objective of the MONLP (2.1.1) may be written as follows:

      Find x so as to satisfy

      fr (x) Lr

      x X .

      for r = 1,2,3,,k

      Here objective functions of (2.1.1) are considered as fuzzy constraints. These types of fuzzy constraints can be quantified by eliciting a corresponding membership function:

      r fr (x) 0 if fr (x) Ur

      Ur fr (x) if L

      f (x) U (r 1, 2, 3,…, k)

      (2.1.2)

      Ur Lr

      r r r

      1 if fr (x) Lr

      Having elicited the membership functions (as in (2.1.2)) r=1,2,3,.,k. introduce a general aggregation function

      D 1 1 2 2 k k

      D 1 1 2 2 k k

      (x) G( (f (x)), (f (x)),……, (f (x))) .

      r[fr (x)]

      for

      So a fuzzy multi-objective decision making problem can be defined as

      D

      D

      Max (x)

      subject to x X .

      (2.1.3)

      Here we adopt fuzzy decision based on minimum operator (like Zimmermanns approach (1978).In this case (2.1.3) is known as Fuzzy Non Linear Programming Model (FNLPM).

      Then the problem (2.1.3) using the membership function as in (2.1.2) according to max-min operator is reduces to

      Max (2.1.4)

      Subject to i[fi (x)] for i=1,2,.,k

      x X , [0,1],

      Step 4: Solve (2.1.4) to get optimal solution.

   2. Formulation of Intuitionistic Fuzzy Optimization (IFO)

      When the degree of rejection (non-membership) is defined simultaneously with degree of acceptance (membership) of the objectives and when both of these degrees are not complementary to each other, then IF sets can be used as a more general tool for describing uncertainty.

      To maximize the degree of acceptance of IF objectives and minimize the degree of rejection of IF objectives subject to constraints, we can write:

      max i x, x R, i 1, 2,…….., k n.

      min i x, x R , i 1, 2,…….., k n.

      Subject to

      (2.2.1)

      i x 0

      i x i x

      i x i x 1

      x 0

      th

      th

      Where

      i x

      denotes the degree of membership function of X

      to ith IF sets

      and

      i xdenote the degree of non-membership (rejection) of X from the i IF sets.

      L

      L

      U

      U

   3. An Intuitionistic Fuzzy Approach for Solving Engineering Design Optimization Model (EDOM) with Linear Membership and Non-Membership Functions:

      To define the membership function of EDOM problem, let

      acc k

      and

      acc k

      be the lower

      and upper bounds of the k th objective function. These values are determined as follows: Calculate the individual minimum value of each objective function as a single

      objective IP subject to the given set of constraints. Let

      x* , x* , x* ,, x*

      be the

      1 2 3 k

      L

      L

      respective optimal solution for the k different objective and evaluate each objective function at all these k optimal solution. It is assumed that here at least two of these solutions are different for which the kth objective function has different bounded values.

      For each objective, find lower bound (minimum value)

      acc k

      and the upper bound

      k

      k

      (maximum value) Uacc . But in intuitionistic fuzzy optimization (IFO), the degree of

      L

      L

      rejection (non-membership) and degree of acceptance (membership) are considered so that the sum of both values is less than one. To define membership function of EDOM

      problem, let

      rej k

      and

      Urej be the lower and upper bound of the objective function

      Zk x

      k

      k

      where

      Lacc Lrej Urej Uacc .When the upper and lower bounds for each objective are

      k k k k

      k k k k

      specified then we form IF model and then convert it into a crisp model.

      The linear membership and non-membership function for the objective defined as:

      Zk x is

      1

      U acc Z

      k k

      k k

      x

      if Z

      x Lacc

      Z x k k

      if Lacc Z x U acc

      (2.3.1)

      • L

      • L

      k k Uacc acc k k

      k k k

      0

      0

      if Z

      1 if Z

      x U acc

      k k

      k k

      x U rej

      k k

      Z x Lrej

      Z x k k

      if Lrej Z x U rej

      (2.3.2)

      k k

      k k

      k k U rej Lrej

      k k k

      0

      0

      k k

      k k

      if Z

      x Lrej

      The picture of linear membership and non-membership functions of the objective goal is given below.

      ( k Zk , k Zk )

      k Zk

      1

      k Zk

      L

      L

      acc k

      L

      L

      rej k

      k k

      k k

      Uacc Urej

      Zk x

      Figure-1: Linear membership and non-membership functions for objective goal.

      In case of minimization problem, the lower bound for non-membership function (rejection) is always greater than that of the membership function (acceptance).

      Then the solution of the EDOM problem is summarized in the following steps:

      Step-1. Pick the first objective function and solve it as a single objective IP subject to the constraints .Continue the process K-times for K different objective functions. If all the

      solutions ( i.e

      x* = x* = x* =..= x* . i =1,2,3,.,m. j = 1,2 ,3,..,n) same ,then

      1 2 3 k

      one of them is the optimal compromise solution and go to Step-6.otherwise go to Step-2. However this rarely happens due to the conflicting objective functions. Then the

      Intuitionistic fuzzy goals take the form Z

      x L

      x*

      k 1, 2, 3,……, K .

      k k k

      Step-2. To build membership function, goal and tolerance should be determined at first. Using the ideal solution, obtain in step-1, we find the values of all the objective functions at each ideal solution and construct pay-off matrix as follows:

      Z x* Z x* ….. ….. Z x*

      1 1 2 1 k 1

      Z x* Z x* ….. ….. Z x*

      1 2 2 2 k 2

      …..	…..	…..	…..	…..
      …..	…..	…..		…..	…..

      …..	…..	…..	…..	…..
      …..	…..	…..	…..	…..

      Z x* Z x* ….. ….. Z x*

      1 k 1 2 k 1

      k k 1

      Z x* Z x* ….. ….. Z x*

      1 k 2 k k k

      Step-3. From step-2. we find the upper and the lower bounds of each objectives for the degree of acceptance and rejection corresponding to the set of solutions as follows:

      k k r

      k k r

      U acc maxZ

      x*

      for

      1 r K

      and

      Lacc minZ

      x*

      for

      1 r K

      for

      k k r

      k k r

      degree of acceptance of objectives. We present upper bound and lower bound for the degree of rejection of objectives as follows:

      Lrej Lacc t Uacc Lacc

      with Urej Uacc

      where 0 t 1

      ,t is presented by decision

      k k k k k k

      maker.

      Step-4. The initial intuitionistic fuzzy model becomes (in terms of aspiration levels with each objective)

      Find xij, i 1, 2, 3,…., m. and j 1, 2, 3,……., n

      k k k k

      k k k k

      So as to satisfy

      Z Lacc

      with tolerance Uacc Lacc

      for the degree of acceptance ,for k=1,2,3,.,K.

      k k k k

      k k k k

      and Z

      Urej with toleranceUrej Lrej

      for the degree of rejection, for k=1,2,3,,K.

      Step-5 .Now constructs the membership (acceptance) and non-membership (rejection) functions of objective Zk x by (2.4.1) and (2.4.2).

      Step-6. Then the following Intuitionistic Fuzzy Optimization model can be written as

      max k x, x R, k 1, 2,…….., K.

      mink x, x R , k 1, 2,…….., K.

      Subject to

      k x 0

      k x k x

      k xk x 1

      x 0

      Then the above problem of equation can be reduced following Angelov (1997) to the following form:

      max( );

      subject to

      k k k k

      k k k k

      Z x U acc U acc Lacc ;

      k k k k

      k k k k

      Z x Lrej U rej Lrej ;

      (2.3.3)

      ;

      1;

      0;

      x 0;

      Step-7. Solve equation (2.3.3) by using appropriate mathematical programming algorithm to get an optimal solution and evaluate the K objective function at these optimal compromise solutions.

      Step-8. Stop.

3. APPLICATION

   A two-bar truss shown in Fig.2 is designed to support the loading condition Consider the following data Nodal load ( P ) =100 KN ; Volume density

   B

   B

   ( )= 7.7 KN / m3 ;Length ( l )= 2000 mm ;Width( x )=1000 mm ; Allowable tensile stress( t )=130 MPa with maximum allowable tolerance 20 MPa ;Allowable

   compressive stress( c )= 90 MPa with maximum allowable tolerance 10 MPa ;Cross-

   sectional area of bar 1( A )= (0 mm2 A 1000 mm2 ) ;Cross-sectional area of bar 2

   1 1

   ( A )=(0 mm2 A 1000 mm2 ) ;Y coordinate of node B( y )= (500 mm y

   1500 mm) ;

   2 2 B B

   The structure is subject to constraints in geometry, area, stress [1]. Obviously, this is fuzzy optimization problem.

   Figure-2: Design of the two-bar planar truss

   The Fuzzy Optimization model of the two-bar truss is as follows:

   min W A , A , y

   7.7 A 1 (2 y )2 A 1 y2

   1 2 B

   1 B 2 B

   2

   2

   100 1 (2 yB )

   subject to. G1 A1, A2 , yB

   2A

   130 with tolerance 20;

   1

   100 1 y2

   (3.1.1)

   G A , A , y

   B 90 with tolerance 10;

   2 1 2 B

   2A2

   0.5 yB 1.5 A1 0; A2 0;

   According to Werners approache (1987), we consider two sub-problems. One sub- problem is an optimization problem without tolerance of constraints and other sub- problem is an optimization problem with maximum allowable tolerance of constraints.

   They are

   Sub-Problem-1:

   min W A , A , y

   7.7 A 1 (2 y )2 A 1 y2

   1 2 B

   1 B 2 B

   2

   2

   subject to. G1 A1, A2 , yB

   100 1 (2 yB )

   2A

   130 ;

   1

   100 1 y2

   (3.1.2a)

   G2 A1, A2 , yB

   B 90 ;

   2A2

   0.5 yB 1.5 A1 0; A2 0;

   Sub-Problem-2:

   min W A , A , y

   7.7 A 1 (2 y )2 A 1 y2

   1 2 B

   1 B 2 B

   2

   2

   subject to. G1 A1 , A2 , yB

   100 1 (2 yB )

   2A

   150 ;

   1

   100 1 y2

   (3.1.2b)

   G2 A1, A2 , yB

   B 100 ;

   2A2

   0.5 yB 1.5 A1 0; A2 0;

   Solving these two sub-problems we have lower bound and upper bound of weight W. i.e.125.7 N W 142.3 N .

   So the model with fuzzy constraints (3.1.1) reduces to the following fuzzy optimization problem

   Find

   A1 , A2 and yB

   Such that

   2 2

   2 2

   W A1, A2 , yB 7.7 A1 1 (2 yB ) A2 1 yB 125.7

   with tolerance 16.6 N.

   2

   2

   100 1 (2 yB )

   G1 A1, A2 , yB 130 with tolerance 20; 2A1

   100 1 y2

   G2 A1, A2 , yB

   B 90 with tolerance 10; 2A

   2

   0.5 yB 1.5 A1 0; A2 0;

   So we can apply the max-min operator to obtain the optimal decision.i.e

   Max

   subject to W (A1, A2 , yB ) ;

   1

   1

   G (A1, A2 , yB ) ;

   2

   2

   G (A1, A2 , yB ) ;

   [0,1];

   (3.1.3)

   Where the linear membership function and the corresponding figure for weight

   W ( A1, A2 , yB ) is

   1 if W(A1, A2 , yB ) 125.7

   (A , A , y ) 142.3 W(A1, A2 , yB )

   if 125.7 W(A , A , y ) 142.3

   (3.1.3a)

   W 1 2 B

   16.6

   1 2 B

   0 if W(A1, A2 , yB ) 142.3

   W (A1, A2 , yB )

   1

   125.7 142.3

   W(A1, A2 , yB )

   Figure-3: Linear membership function for W ( A1, A2 , yB )

   And the linear membership function for first constraint corresponding figure is

   1 if G1 (A1, A2 , yB ) 130

   150 G (A , A , y )

   G1 (A1, A2 , yB ) and the

   (A , A , y )

   1 1 2 B

   if 130 G (A , A , y ) 150

   1

   1

   20

   20

   G 1 2 B

   1 1 2 B

   (3.1.3b)

   0 if G1 (A1, A2 , yB ) 150

   1

   1

   G (A1, A2 , yB )

   1

   G1 (A1, A2 , yB )

   130 150

   Figure-4: Linear membership function for

   G1 (A1, A2 , yB )

   And the linear membership function for second constraint corresponding figure is

   1 if G2 (A1, A2 , yB ) 90

   G2 (A1, A2 , yB ) and the

   (A , A , y ) 100 G2 (A1, A2 , yB )

   if 90 G (A , A , y ) 100

   G2 1 2 B

   10

   2 1 2 B

   (3.1.3c)

   0 if G2 (A1 , A2 , yB ) 100

   2

   2

   G (A1, A2 , yB )

   1

   G2 (A1, A2 , yB )

   90 100

   Figure-5: Linear membership function for

   G2 (A1, A2 , yB )

   Then the problem becomes.

   M ax

   subject to

   7.7 A 1 2 y

   7.7 A 1 2 y

   1 B 2 B

   2 A 1 y 2

   2 A 1 y 2

   16.6 142.3;

   16.6 142.3;

   2

   2

   100 1 2 y B

   2 A1

   100 1 y 2

   20 150;

   (3.1.4)

   B 10 100;

   2 A 2

   0.5 y B 1.5;

   [0,1];

   A1 0; A 2 0;

   Solution of the above model by General Fuzzy Non-Linear Programming [GFNLP] Technique in section 2.1 and we get the following results are obtain in Table-1:

   Table-1

   Desgn variable A (mm2 ) 1	Design variable A (mm2 ) 2	Y coordinate of node B YB (m)	Aspiration Level	Weight W (N )
   556.5	677.8	.81	.51331	133.78

   To solve the model (3.1.1) by Intuitionistic Fuzzy Optimization (IFO) Technique,

   Max

   subject to

   W ( A1, A2 , yB ) ; W ( A1 , A2 , yB ) ;

   G ( A1, A2 , yB ) ; G ( A1, A2 , yB ) ;

   (3.1.5)

   2

   2

   1 1

   2

   2

   G ( A1, A2 , yB ) ;

   1

   G ( A1, A2 , yB ) ;

   [0,1]; [0,1]

   Where the membership function

   W ( A1, A2 , yB ) for weight

   W ( A1, A2 , yB )

   is defined in

   (3.1.3a) and the non-membership function

   its figure is defined as

   W ( A1, A2 , yB )

   for weight

   W ( A1, A2 , yB ) and

   0 if W(A1, A2 , yB ) 125.7

   W(A , A , y ) 125.7

   (A , A , y )

   1 2 B

   if 125.7 W(A , A , y ) 130.7

   (3.1.5a)

   5

   5

   W 1 2 B

   1 2 B

   1 if W(A1, A2 , yB ) 130.7

   W (A1, A2 , yB )

   1

   W(A1, A2 , yB )

   125.7 130.7

   Figure-6: Linear non-membership function for W ( A1, A2 , yB )

   1

   1

   And the membership function G ( A1, A2 , yB ) for first constraint G1 (A1, A2 , yB ) is defined

   1

   1

   in (3.1.3b) and the linear non-membership function

   G1 (A1, A2 , yB ) and its figure is defined as

   G ( A1, A2 , yB )

   for first constraint

   0 if G1 (A1, A2 , yB ) 130

   G (A , A , y ) 130

   (A , A , y )

   1 1 2 B

   if 130 G (A , A , y ) 149

   1

   1

   19

   19

   G 1 2 B

   1 1 2 B

   (3.1.5b)

   1 if G1 (A1, A2 , yB ) 149

   1

   1

   G (A1, A2 , yB )

   1

   G1 (A1, A2 , yB )

   130 149

   Figure-7: Linear non-membership function for

   G1 (A1, A2 , yB )

   And the membership function

   G ( A , A , y ) for second constraint

   G2 ( A1, A2 , yB ) is

   2 1 2 B

   2 1 2 B

   2 1 2 B

   2 1 2 B

   defined in (3.1.3c) and the linear non-membership function

   constraint G2 (A1, A2 , yB ) and its figure is defined as

   G ( A , A , y ) for second

   0

   G ( A , A , y ) 90

   if G2 ( A1 , A2 , yB ) 90

   G

   G

   2

   2

   ( A1, A2 , yB

   )

   2 1 2 B

   9

   if 90 G2 ( A1, A2 , yB ) 99

   (3.1.5c)

   1

   if G2 ( A1, A2 , yB ) 99

   2

   2

   G (A1, A2 , yB )

   1

   G2 (A1, A2 , yB )

   90 99

   Figure-8: Linear non-membership function for

   G2 (A1, A2 , yB )

   Then the problem becomes.

   Max

   subject to

   (3.1.6)

   7.7 A 1 2 y

   7.7 A 1 2 y

   2 A 1 y2

   2 A 1 y2

   16.6 142.3;

   16.6 142.3;

   1 B 2 B

   2

   2

   100 1 2 y B

   2A1

   100 1 y 2

   20 150;

   B 10 100; 2A 2

   7.7 A 1 2 y

   7.7 A 1 2 y

   2 A 1 y 2

   2 A 1 y 2

   5 125.7;

   5 125.7;

   1 B 2 B

   2

   2

   100 1 2 y B

   2A1

   100 1 y 2

   19 130;

   B 9 90; 2A 2

   0.5 y B 1.5;

   1;

   [0,1]; [0,1];

   A1 0; A 2 0;

   Solution of the model (3.1.1) by Intuitionistic Fuzzy Optimization (IFO) is obtained in table-2:

   Table-2

   Design variable A* (mm2 ) 1	Design variable A* (mm2 ) 2	Y coordinate of node B Y * (m) B			Weight W (N )
   537.5	659.5	.80	0.2023	0.7977	129.6

4. Analyzing the above tables (Table-1 and table-2) and the following observation can be made:

   We compare TABLE-1 and TABLE-2 and the solution of the Weight of the planer truss bar is minimized in case of Intuitionistic Fuzzy Optimization (IFO) Technique then General Fuzzy Non-linear Programming [GFNLP] .We finally concludes that IFO technique performs better than GFNLP technique.

5. Conclusions and Future Scope of Research

In this paper we consider two bar truss model and is solved by GFNLP method, firstly. Besides this elaborate solution IFO method is also used for improvement of the solutions. We established that, objective of this paper is that, IFO technique usually perform better than GFNLP technique. We conclude that IFO minimizes weight more than GFNLP technique.

This two bar truss model can also be analyzed by Geometric Programming as well as Fuzzy Geometric Programming technique can also be applied to obtain possible improve solutions of this model.

Acknowledgement

The authors are grateful to Prof Samir Kumar Hati, Assistant Professor, Department of Mechanical Engineering, Asansol Engineering College, Asansol, India, for his valuable suggestion.

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