Structural optimization with fuzzy parameters

Structural optimization with fuzzy parameters

conrpurers & SrrucrureJ Vol. 37,No. 6, pp. 917-924,19% Printedin Great Britain. 0045.7949po $3.00 + 0.09 Pergamon Press pk STRUCTURAL O~IMIZATION PA...

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conrpurers & SrrucrureJ Vol. 37,No. 6, pp. 917-924,19% Printedin Great Britain.

0045.7949po $3.00 + 0.09 Pergamon Press pk

STRUCTURAL O~IMIZATION PARAMETERS

WITH FUZZY

YI-CHJZRNGYEH and DIN-SHIU Hsu Department

of Civil Engineering, National Cheng Kung University, Tainan, Taiwan, Republic of China (Receiwd 28 September 1989)

Abtract-In this paper, the process of optimum design of structures is developed in which design parameters with a fuzzy nature are considered. The fuzzy optimum problem is transformed into an ordinary unconstrained optimum problem by expressing the objective function in the form of fuzzy expected cost. The proposed method is much simpler than the conventional fuzzy optimum method and appears to be able to deal with the uncertainty with the minimum of complexity. The proposed method can be shown to possess the following advantages: (1) no p~dete~in~ mapping function is needed; (2) consistency between objective and constrained functions is obtained; (3) the ~rnpu~tion~ process is simplified; and (4) the optimum level design is obtained. Illustrative numerical examples have been presented and the above mentioned advantages have been proved. Moreover, a comparison between various cases for both system failure cost and fuzziness of design parameters is performed.

NOTATION

F x g

Q I. R P( .) I b c D x(.) n( ‘) T E d Ci Cf c, _

*

value of As. It seems more reasonable to assign a transition stage from absolute acceptance to absolute unacceptance. At the same time, the design load P of structures may be constrained by a lower bound value as P 2 P’; in other words, P = P’ is acceptable but P = P’ - AP is unacceptable, even for a very small value of AP. Again, it seems more reasonable to assign a transition stage from absolute acceptance to absolute unacceptance. In this paper the problem of structural optimum design in a fuzzy environment (fuzzy allowable strength, fuzzy loads) is considered. The formulation for structural fuzzy optimum design is derived in the first part of the paper. The attempts of some pioneer researchers (Zimmermann [7], Rao [8], Verdegay [9], Yazenin [lo], Yeh and Hsu [l 11, etc.) are also discussed. Their main idea’ was to replace the measure of objective function and constrained functions on mem~rship functions, and the fuzzy design characterized by its membership function, which can be viewed as intersection of the fuzzy goal and fuzzy constraints, then the optimum design is a design which maximizes the membership function of the design. A new approach based on the concept of fuzzy expected cost for the above mentioned problem is proposed in the second part of the paper. Numerical examples for a three-bar truss and a simple ductile frame are presented to explain the applications of the proposed method in the last part of the paper.

objective function design variable constrained function design parameter load allowable strength probability initial fabricating cost length fuzzy goal fuzzy constraints fuzzy decision possibility mem~rs~p total cost expected cost membership value of satisfaction under one design level initial fabricating materials cost of unit volume component failure cost system failure cost

fuzzy characteristic optimum characteristic 1. INTRODUCTION

A structural system can be described by a set of quantities, some of which are viewed as variables during the optimization process. The optimum design method concerning deterministic parameters has already been very well developed [l, 21. The probability-based structural optimization method can be applied to problems in which design parameters are actually of a random nature 13-61. In fact, in design problems there exists a vast amount of fuzzy information in resistances as well as in applied loads. For example, the stress induced in a structure may be constrained by an upper bound value as s < s”; this implies that s = s” is acceptable but s = s” + As is unac~ptable, even for a very small

2.

FORMULA~ON

OF THE PROBLEM

A classical optimization problem of structures with deterministic parameters can be formulated as follows [l, 21: find X which minimizes F(X) subject tog,(X,Q)
i=l,2

,...,

m

(1)

Y.-C. YEH and D.-S. Hsu

918

where X denotes design variables (for example, the section of components in structural design); Q denotes design parameters (for example, the allowable strength and loads in structural design); F(X) denotes objective function; and g,(X, Q) denotes the jth constrained functions. When structural design is under a fuzzy environment, i.e. design parameters Q are fuzzy numbers [12, 131, fuzzy optimization problems can be formulated as follows [lo, 141: find X which minimizes F(X) subject tog,(X,Q-)
j=l,2

,...,

m

P(X) = mjn(0)) G Gi

(3a)

P(X) = min(N)). c j c,

(3b)

= min(N),

~(0.

G

(4)

C

An optimum solution X* can be selected as the decision for which the membership function is maximum: p(X*) = maxp(X). D

(5)

“D

In fuzzy optimum design, if objective function F(X) and constrained functions g,(X) can be described by membership functions P(X)

and

&

respectively, then the optimum design is X* with P(X*) = maxfi(X) = [email protected](X), P(X)) D

“D = m;xMX), F

“F

g

m;ln(M))l.

There are some disadvantages tioned method.

(7)

where T(a) denotes the total cost; I(a) denotes the initial fabricating cost of the structure; E(a) denotes the expected cost of additional expenses needed that resulted from design constraints is not absolutely satisfied; and a denotes the membership value of satisfication under different design levels. Then the optimum design is a with: find

a E [0, 1] T(a) = I(a) + E(a).

(8)

Obviously, the expected cost E(a) is a function of constrained functions, while constrained functions are functions of design variables. Wang and Wang [16,17] pointed out that it is very important and difficult to obtain proper E(a) values. Wang and Wang’s framework is followed herein while the total cost is measured based on failure possibility instead of measuring the membership value of satisfication under different design levels a. 3. FUZZY OPTIMUM DESIGN OF STRUCTURES BASED ON FUZZY EXPECTED COST

To overcome the above mentioned disadvantages of Bellman’s model and Wang’s model, the concepts of failure possibility and fuzzy expected cost are introduced. 3.1. Failure possibility

P(X),

F

T(a) = I(a) + E(a)

which minimizes

Membership of decision alternative D can be viewed as intersection of the fuzzy goal and fuzzy constraints as follows:

D

An alternative approach has been proposed by Wang and Wang [16,17]. The modified objective function based on expected cost is employed as follows:

(2)

where w denotes fuzzy characteristics. The general framework for fuzzy optimum design follows the fuzzy decision proposed by Bellman and Zadeh [lS]. In fuzzy decision problems the fuzzy goals G and fuzzy constraints C are defined as fuzzy sets in the space of decision alternatives, and are characterized by their membership function as follows:

0)

mapping function in order to transform them into membership values. Strictly speaking, it is difficult to find the proper mapping functions. (2) Membership of design D can be viewed as intersection of membership of the fuzzy objects and membership of the fuzzy constraints. Strictly speaking, consistency of these memberships is doubtful.

(6)

gJ

in the above men-

(1) To have the objective function and the constrained functions fuzzy, we must fix a predetermined

The concept of probability is usually related to the frequency of occurrence of clearly defined events. Usually, the probability values can be captured by repeated experimental data through statistical techniques. In constrast, Zadeh [ 181proposed that a fuzzy set can be regarded as a basis for the theory of possibility. In general, the possibility distribution that is associated with a variable is epistemic rather than physical. In other words, it is nonstatistical in nature; conversely, the probability distribution is statistical in nature. Roughly speaking, possibility is based on a fuzzy definition domain; however, probability is based on a random occurrence space, which is very different.

Structural optimization with fuzzy parameters

919

In a fuzzy environment, the constrained functions with design parameters Q which are fuzzy numbers can be transformed into a failure possibility constrained function as follows:

In the same manner,

where ITis the possibility measurements; gj stands for the jth constrained function. This has been expressed in a form analogous to those in probability-based optimization by Frangopol[3-51 as follows:

c om b’ming this equation with eqn (9), we have (see Fig. 1)

x+j= x(gj(x, Q -) 2 0) =

max 0(x.0-r Eto.4

ltfi = ajcw,pm_a;qlo,~l[n(gj(X, Q -))I

[

P(X) 1 S

Q En* (12b)

It is worthwhile emphasizing that when the function is monotonic and continuous, by means of the vertex method [12, 131,we can use simple and efficient computations of extended algebraic operations on fuzzy numbers, i.e.

where P is the probability measurement and gj stands for the jth constrained function. Recalling possibility theory [19,20], let n(x) be a possibility distribution induced by a fuzzy set S in X. Let A be a nonfuzzy set defined in X; the possibility that x belongs to A is n(A), where n(A) is the possibility measurement induced by n(x), and we have

1

max Vi i where

(13)

[c, d] = interval of y

WI = y:;

Ax)

= y,‘,“,” [4x)1.

(11)

[ S 1 possibil I

itr

Vi = the coordinates of the ith vertex in an mdimensional rectangle X, x X, x - . . x X,,,, possibility

rr

n R

L

f.o____A

f.o.-_A

tb)

L

(a) possibility

R

II 8, (X,cl-)

I

Fig. 1. (a) Possibility distribution of design parameter: load L. (b) Possibility distribution of design parameter: allowable strength R. (c) Possibility distribution of fuzzy constrained function.

920

Y.-C. YEHand D.-S. Hsu

The distinct cut levels of y by means of the distinct cut levels of X can then be obtained. Most of the structural systems that are composed of multiple components can be classified as seriesconnected, parallel-connected or combination systems. Generally, in reliability analysis, the structural systems can be regarded as series-connected systems leading to small errors even when the system possesses indeterminacy [21]. Structural systems which are modeled in series systems mean that the failure of any one or more of these components implies the failure of the entire system. If J!?,denotes the failure of component i, then the event of failure of series systems E, can be formulated as [22,23]: E,=(E,)U(E*)U...

U (E,)U . . U(E,).

(14a)

In order to make the optimization design more rational and overcome the aforementioned disadvantages, the fuzzy expected cost model is used for solving the problem of fuzzy optimum design as follows. (i) By means of transformation of failure possibility constrained functions, the fuzzy optimum design model is formulated in equation (18): find X which minimizes F(X)

(14b)

3.2. Fuzzy expected cost Expected cost in a fuzzy environment can be analogous to that in a random environment. The expected cost in a random environment can be written as:

(18)

where 7c/I= s,(x,z:,

Recalling possibility theory [19,20], the possibility of failure of the system z,~ can be formulated as: 7cfi= 7c(E,) = m,ax(n(E,)) = m,ax7rfi.

j=l,2,...,m

subjecttorc/,,<7[fi

ml[n(gi(X e *

))I

and Q stands for the upper limit of n,,. (ii) To express the modified objective function in terms of fuzzy expected cost, fuzzy optimum design can be formulated and written as: find X which minimizes T(X) = I(X) + E(X)

(19)

where Z(X) = c,.~:x;r,,

where C, is the measured cost of event i and Pi is the probability of occurrence of event i. The expected cost in a fuzzy environment can be written as: E = max(C;xj).

where C,j(X) and C,,(X) denote the failure cost of component j and the failure cost of a system with design variables A’,respectively. P,, and P/3denote the failure probability of component j and the failure probability of a system with design variables X, respectively. The expected cost in a fuzzy environment can be written as:

x

where T(X) denotes the total cost; I(X) denotes the initial fabricating cost of the structure; E(X) denotes the expected cost of additional expenses needed that resulted from design constraints’ unsatisfied state; Ci denotes the unit cost of initial fabricating materials; and 1 denotes the dimension of the structure. If C,(X) and C,s(X) are independent of design variable A’,then E(X) in eqn (19) can be simplified as follows:

Hence, the fuzzy optimum problem can be dealt with and formulated in the form expressed in eqn (20):

E(X) = fmaxK,j(x)~y,W)l *

+ max[CP)~~&Vl I

I

(15b)

Consequently, with design variables X, the expected cost of additional expenses in a random environment can be written as follows:

I

E(X) = f max[C,,(X).rr/,(X)]

(17)

where 7fr, and nfi denote the failure possibility of component j and the failure possibility of a system with design variables X, respectively.

find X which minimizes T(X)=C,*CX,I,+f

i_i

C.n +c,,.?I,, f) fi

Structural optimization with fuzzy parameters

1

1

921

n

=

el/YR

25/2$

-

(23)

R

The unit cost of initial fabricating materials and the cost for the jth component failure are given as C, = 1.O and C,-,= 0, respectively. Firstly, in order to survey the effect of the system failure cost C,,, there are four different cases (numbered l-4 in Table 1) with C/ equal to 10, 100, 1000 and 10,000. In addition, in order to survey the effect of the fuzziness of design parameters, the system failure cost is taken as C, = 100, and the possibility distribution of the fuzzy allowable strength R is taken as follows:

1

Fig. 2. Three-bar truss.

I[ = eW7 - GV* where

(24)

R

In eqn (24) four cases for different (a, b) pairs which are numbered 5-8 in Table 2 are calculated with (a, 6) = (252.9, (25, 5), (25,7.5) and (25, 10). Obviously, the fuzzy nature has been revealed in these cases by parameter b. The source fuzzy optimum design formulation is displayed as follows:

(20)

7rfi =

m,ax 7c,] .

In the present procedure, the fuzzy optimum problem has been transformed into an ordinary unconstrained optimum problem. It can be solved easily by conventional unconstrained optimization methods [ 1,2].

find X which minimizes F = 2,/? X, + X2 subject to

4. NUMERICAL

Example

EXAMPLES

1 t?l =

The three-bar truss shown in Fig. 2 has frequently been used as an example in structural optimization literature [l, 21. In the case of possibility distribution, fuzzy loads L,, L, and fuzzy allowable strength R are given as shown in eqns (21)-(23): 71=

el/2(LI - 1om

XL,-R
g2 =

2x1

J-2x:+2x,x2

(21)

Li

g3=

n = el12(L2- W)

(22)

4 +$x2

J 2x:+2x,x,

L,-R
L

4

_



fix:+2x,x2

XL,-R
L2

Table 1. Fuzzy optimum design for different system failure cost C, Case 1 2 3 4

Cfi

x:

10 100 1000 10,000

1.05 1.31

1.60 1.90

x:

I(X)

E(X)

T*

0.553 0.758 0.860 0.968

3.513 4.474 5.395 6.347

0.430 0.401 0.412 0.437

3.943 4.875 5.807 6.783

5 0.0430 0.00401 0.000412 0.0000437

Table 2. Fuzzy optimum design for different fuzziness of allowable strength R Case

C,?

a

b

x:

x:

W)

E(X)

T*

%r

5 6 7 8

100 100 100 100

25 25 25 25

2.5 5 7.5 10

1.31 1.96 2.83 3.60

0.758 1.138 1.58 1.91

4.474 6.667 9.606 12.08

0.401 1.475 5.139 13.39

4.875 8.142 14.74 25.47

0.00401 0.0148 0.0514 0.134

922

Y.-C. YEH and D.-S. Hsu

PPW

c-

1

T

ZP

H

Z,

ZI

i

Example 2

A simple ductile frame, shown in Fig. 3, is subjected to a total gravity load P, and an equivalent static earthquake load P2. Assume that the frame is built of ductile material with elastic-perfectly plastic behavior. Assume also that the failure of the frame is through the formulation of fully plastic hinge mechanisms, as shown in Fig. 4. By the principle of virtual work, the constrained functions of the respective mechanisms can be shown to be as follows:

rLI

g, = P,H -4S,Z,

Fig. 3. A simple ductile frame. H = 15, L = 20.

g,=P,H+PL/2-4S,Z,-2S,Z, g, = P, L/2 - 2s,z,

We can transform the problem into the fuzzy expected cost model of eqn (20) as follows:

- 2syzz

g,=P,H+PL/2-2S,Z,-4S,,Z,

find X which minimizes where Z, and Z, are plastic modulus of column and beam, respectively and Hand L are length of column and beam, respectively. Assume the initial fabricating cost I as shown below:

T(X) = 2Jj: X, + X, + c,s. 5 where

I = 22, H + Z,L.

The conventional unconstrained optimization method [ 1,2] can then IX used for solving the formula above; the solutions are listed in Tables 1 and 2.

Mechanism

In this problem, Z, and Z, are design variables, and loads P,, P2 and yield stress S, are design parameters. Assume that the loads P,, P2 and the yield stress S, are all fuzzy numbers. In the case of a

1

Mechanism

2

I I

Mechanism

3

(25)

Mechanism 4 Fig. 4. Major hinge mechanisms.

Structural optimization with fuzzy parameters

923

Table 3. Fuzzy optimum design for different system failure cost C, Case 1 2 3 4 5

C/II 1 10 100 1000 10.000

G 0.0715 0.0838 0.0930 0.102 0.110

z: 0.0715 0.0838 0.0930 0.102 0.110

I

E

T*

3.57 4.12 4.65 5.09 5.49

0.278 0.204 0.193 0.168 0.155

3.85 4.39 4.84 5.25 5.64

7% 5.64.E -4 1.09.E-6 4.9.E-8 3.0.E-9 0

3-2 7.77.E-2 4.85.E-3 4.14.E-4 3.31.E-5 2.81.E-6

5-J 1.92.E-3 2.93*E-6 1.4.E-8 l.O.E-9 0

=f4

7.77.E-2 4.85.E-3 4.14.E-4 3.31.E-5 2.81.E-6

Table 4. Fuzzy optimum design for different fuzziness of yield stress S, Case

b

6 7 8 9

125 250 500 1000

possibility

z: 0.0715 0.0930 0.119 0.232

z: 0.0715 0.0930 0.119 0.232

I

E

T*

4.19 4.65 5.94 11.6

0.118 0.193 0.437 2.93

4.31 4.84 6.38 14.5

the fuzzy numbers are given

distribution,

as follows: n = e1/2VI- Io%V

(26)

pi

41 l.O.E-9 4.9.E-8 7.3.E -4 5.2.E -4

*/z 2.8.E-9 4.14.E-4 7.36.E-4 2.53.E-4

5-J 0 1.4.E-8 l.O.E-5 9.8.E-4

We can transform the problem expected cost model of eqn (20):

94

2.8.E-9 4.14.E -4 7.36.E -4 2.53.E-4

into the fuzzy

find Z, and Z, which minimizes T = I + E where

II = el/2(P2 - 30/1OP

I=2Z,H+Z2L

(27)

PZ H=

E=C,,.nfi el’2’s’.- 5oooI25w

(28)

where

s,

Firstly, in order to survey the effect of the system failure cost C,S, there are five different cases (numbered l-5 in Table 3), with C,, equal to 1, 10, 100, 1000 and 10,000 times the initial fabricating cost. In addition, in order to survey the effect of the fuzziness of design parameters, the system failure cost is taken as C,$equal to 100 times the initial fabricating cost, and the possibility distribution of the yield stress S, is taken as shown below: n =

e1/2(s,-u/w

(29)

?rfi =

g,(x, r_:z,

ml In (gj (x3

Q N))I.

The conventional unconstrained optimization method [ 1,2] can then be used for solving the formula above; the solutions are listed in Tables 3 and 4. 5. CONCLUSION

SY

In eqn (29) four cases for different (a, b) pairs which are numbered 6-9 in Table 4 are calculated with (a, b) = (5000, 125), (5000,250), (5000,500) and (5000, 1000). Obviously, the fuzzy nature has been revealed in these cases by parameter b. The source fuzzy optimum design formulation is displayed as follows: find Z, and Z2 which minimizes F = 22, H + Z,,!, subject to g, = P,H - 4S,Z, < 0 gz=PzH+PL/2-4~,,Z,-2SyZ2<0 g, = P, L/2 - 2&Z, - 2&Z, Q 0 g, = P2H + PL/2 - 2S,Z, - 4sYzr 6 0.

The optimization of structures containing fuzzy information has been considered. Illustrative examples have been presented and it has been observed that the calculation task involved is much simpler than that of the deterministic structural optimization. Moreover, the proposed method is superior to the classical structural fuzzy optimum design method in the following ways. (1) The proposed method needs no predetermined mapping function in order to transform them into membership values. (2) The proposed method considers consistency between objective function and constrained functions by the concept of fuzzy expected cost. (3) The proposed method transforms the fuzzy optimum formula into an unconstrained optimum formula, so the computational step is easy and simplified.

924

Y.-C. YEH and D.-S. Hsu

(4) The proposed method can be used to find the optimum level design by an unconstrained optimization procedure. Two more conclusions can be drawn. (1) From numerical examples, one can find that the optimum initial fabricating cost is proportional to its system failure cost, and for the optimum failure possibility the converse is true. The optimum fuzzy expected cost is almost independent of its system failure cost. (2) From numerical examples, one can find that the optimum initial fabricating cost, the optimum fuzzy expected cost and the optimum failure possibility are all proportional to the fuzziness of the design parameter. The fuzzier the design parameter, the larger the mentioned values obtained.

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14. J. Munro and P.-H. Chang, Optimal plastic design with imprecise data. J. Engng Mech. 112, 888-903 (1986).

15. R. E. Bellman and L. A. Zadeh, Decision-making in a fuzzy environment. Management Sci. 17(4), 141-164 REFERENCES 1. U. Kirsch, Optimum Structural Design. McGraw-Hill, New York (1981). 2. G. N. Vanplatts, Numerical Optimization Techniquesfor Engineering Design with Application. McGraw-Hill, New York (1984). 3. D. M. Frangopol, Interactive reliability-based structural optimization. Comput. Struct. 19, 559-563 (1984). 4. D. M. Frangopol, Towards reliability-based computer aided optimization of reinforced concrete structures. Engng Opt. 8, 301-313 (1985). 5. D. M. Frangopol, Multicriteria reliability-based structural optimization. Struct. Safety 3, 154-159 (1985). 6. D. S. Hsu, Reliability constraint on optimum design. In Recent Developments in Structural Optimization (Edited

by Y. Chang), pp. 72-85. ASCE, New York (1986). 7. H.-J. Zimmermann, Fuzzy programming and linear programming with several objective functions. Fuzzy sets syst. 1, 45-55 (1977). 8. S. S. Rao, Optimum design of structures in a fuzzy environment. AIAA Jnl25, 1633-1636 (1987).

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17. G.-Y. Wang and W.-Q. Wang, Fuzzy optimum design of structures. Engng Opt. 8, 291-300 (1985). 18. L. A. Zadeh, Fuzzy sets as a basis for a theory of possibility. Fuzzy Sets Syst. 1, 3-28 (1978). 19. D. Dubois and H. Prade, Fuzzy Sets and SystemsTheory and Application. Academic Press, New York (1980). 20. G. J. Klir and T. A. Folger, Fuzzy Sets, Uncertainty and Information. Prentice-Hall, Englewood Cliffs, NJ (1988). 21. T. J. P. Yao and H. Y. Yeh, Formulation of structural reliability. J. struct. Div., ASCE 95, No. ST12, 2611-2619 (1969). 22. A. H-S. Ang and W. H. Tang, Probability Concepts in Engineering Planning and Design. John Wiley, New York (1984). 23. P. Thoft-Christensen and M. J. Baker, Structural Reliability Theory and its Applications. Berlin (1982).