Game theory approach for multiobjective structural optimization

E

DMJ-7949 87 f3 00 l 0.w 1937 Pergamon Journal: Ltd.

GAME THEORY APPROACH FOR MULT~OBJECTIVE STRUCTURAL OPTIMIZATION s. s. RAO School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907, U.S.A. (Receked

21 April t986)

Abstract-The concept of Pareto-optimal solution is discussed in the context of a multiobjective structural optimization problem and the commonly used methods of generating Pareto-optimal solutions are

indicated. The graphical interpretations of the non-cooperative and cooperative game theory approaches are presented for a two-criteria optimization problem. The relationship between Pareto-optimal solutions and game theory is described. A computational procedure is developed for solving a general multiobjective optimization problem using cooperative game theory. The procedure is illustrated with two numerical examples.

INTRODUCTION

Recent advances in structural optimization resulted in the development of techniques, mainly based on nonlinear programming, for handling problems with a constantly increasing number of design variables and constraints. Usually a scalar-valued objective function is optimized over a feasible design space and the result is then used as a guiding device in striving for the best practicable structure. However, there often exist several structural design problems, which involve several, usually conflicting, objectives to be considered by the designer. A promising approach for solving this type of problem Seems to be multiobjective nonlinear programming where a vectorvalued objective function is examined. The problem is stated as: Find the vector of design or control variables X which minimizes the vector of criterion or objective functions

subject to gj(X) IO,

i =: 1,2,. . . , m,

and the objective functions A(X) may be- noncommensurable. The multiobjective optimization arose in a natural fashion in mathematical economics; its use in engineering and structural design is relatively recent. A variety of techniques and applications of multiobjective optimization have been developed in the past several years. The earliest work reporting the 119

consideration of multiple objectives in mathematical programming appears to be that of Kuhn and Tucker [I]. The progress in the field of mutticriteria optimization was summarized by Hwang and Masud [Z] and Stadler [3]. The conversion of a single objective structural optimization problem into a multiobjective problem by treating the constraint functions as additional objectives was discussed in 141.The weighting method, the constraint method, and the minimax approaches of generating Pareto optima were given by Koski along with structural design examples [5]. A method for generating Pareto-optimal set, for multicriteria problems capable of being formulated as serial stage-state discrete dynamic programs, was presented in [6]. The development of a computer program called MOEDM, which provides the designer with useful trade-off information from which sound design decisions can be made for linear multiobjective problems was described in Dlesk and Liebman [7]. The e-constraint method was used by Carmichael [S] for generating the Pareto-optimal solutions of a five bar planar truss. The role of game theory in the engineering design process was discussed by Vincent (91.The basic definitions were given without any computational algorithms or numerical examples. The multiobjective optimum design of a clamped-hinged beam was determined by Adali [IO] by approximating the area by linear splines. The numerical results for a number of design examples were presented in the form of graphs of optimal trade-off curves. A comparison of the computational efficiencies of the weighting, noninferior set estimation and constraint methods was presented along with a study of the abifities of the methods to produce an approximation of the Pareto-optimal set [ll]. The multicriterion optimization of elastic stress limited isostatic trusses was considered and a numerical method for determining the Pareto-optimal set of the problem was deveioped [ 121.The application of several multiobjective optim~ation techniques was indicated in

s. s. RAO

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the context of the design of vibration isolation systems by Rao [ 131. Usually there exists no unique solution which would give an optimum for all the objective functions simultaneously. Thus a new optimality concept, different from that used in scalar optimization, is to be used in finding the solution of the vector optimization problem. The concept of Pareto optimality has been found to be quite useful in this context. Several methods were suggested for generating the Paretooptimal solutions. The game theory approach for generating the best compromise (Pareto-optimal) solution is presented in this work afong with numerical examples, PARETO-OPTIMAL

SOLUTION

A vector X*o S is called Pareto-optimal for the probiem of eqn (If if and oniy if X f S and imply that i = 1,2,. . . , k _KW SWI*) for f;(X)=f;(X*) for i = 1.2,. ..,k (171. Verbally, the vector X* is Pareto-optimal if there exists no feasible vector X which would decrease some objective function without causing a simultaneous increase in at Ieast one objective function. Usually several Pareto optima exist for a vector optimi~tion problem and additional information is needed to order the Paretooptimal set. This clearly makes it possible to bring in additional considerations not included in the optimization model, thus making the multiobjective approach a flexible technique for most design problems. Several numerical methods have been suggested for solving a vector optimization problem. Each method, in general, generates a different Paretooptimal solution. In the most commonly used approach, known as the weighting method, a scalar objective function is fo~ulated as a weighted sum of the individual objective functions. If a E Rk denotes the vector of weighting coefficients, the problem takes the form mi$ize

where f? is the ideal feasible solution of f;. ej =f: + A,, and A, are positive increments. This means that one objective function is taken as a scalar objective function and the others are constrained by suitably chosen constants E,. By systematically varying 4, the entire set of Pareto-optimal solutions can be generated even in nonconvex situations. In this work, a game theory approach is presented for finding the best compromise (Pareto-optimal) solution of the multiobjective optimization problem. GAME THEORY APPROACH The game theory approach can be seen with reference to a two objective, two design variable optimization problem whose graphical representation is shown in Fig. 1. Letf, (xi, x2) andfr(x, , x2) represent two scalar objectives and xi and x2 two scalar design variables. It is assumed that one player is associated with each objective. The first player wants to select a design variable x, which will minimize his pay-off& and similarly the second player seeks a variable x, which will minimize his own pay-offf,. Iff, andf, are continuous, then the contours of constant values of h and 52 appear as shown in Fig. I. The dotted lines passing through 0, and OL represent the loci of rational (minimizing) choices for the first and second player for a fixed value of x2 and x,, respectively. The intersection of these two lines, if it exists, is a candidate for the two objective minimization problem assuming that the players do not cooperate with each other (non-coo~rative game). In Fig. 1, the point N (x:,x:) represents such a point. This point, known as a Nash equilibrium solution, represents a stable equilibrium condition in the sense that no player can deviate unilaterally from this point for further improvement of his own criterion [15]. This point has the characteristic that

fl(.C* x?) $4 (XIYx:1 and

a*f(X),

(3) fi(X?% x:1 S-iW

where s=(xER”}gj(X)So,

j=1,2

,-..,

m}.

(4)

The vector a can be normalized so that the sum of its components which are non-negative and not ail zero, is equal to one. Now Pareto-optimal solutions can be generated by parametrically varying weights a, in the objective function. One disadvantage of this method is that it is incapable of producing the whole Paretooptimal set for certain nonconvex problems. In the <-constraint method 131,the e-constraint problem can be formulated as minimizefl(X) subject to&(X) < 4; and X o S,

(6)

I x*)3

(7)

where x1 may be to the left or right of x? in eqn (6) and x2 may lie above or below x: in eqn (7). Extension of the idea to a k-player non-cooperative game gives the mathematical definition of a Nash equilibrium solution as fi(x:,x:,...,

Xk*)Ifi(X,,X2*,...~X~)

x:)sx?(x:,~2,...,x:) ficq,xr,..., . (8) x~)sf&:rX:,...,-d A&:,x:,..., ~

i = 1,2, . . . , k and i + i

(5)

Sofar it has been assumed that there exists only one

Multiobjective structural optimization

Fig. I. Cooperative and non-cooperative game solutions. Nash equilibrium point, i.e. the dotted lines in Fig. 1 intersect only at one point. An interesting situation occurs when the two lines intersect at more than one point. In this case, since the values offi andf2 are different Nash equilibrium points, any player can have the advantage of declaring his/her move first thereby forcing the other players to play at the equilibrium point of his/her own choice. In a cooperative game, the two players agree to cooperate with each other and hence any point in the shaded region S of Fig. 1 will provide both of them with a better solution than their respective Nash equilibrium solutions [16]. Since the region S does not provide a unique solution, the concept of Paretooptimal (non-infe~or) solutions can be introduced to eliminate many solutions from the region S. It can be seen that all points in the region S can be eliminated except those on the continuous line O,ACQDBO, which represents the loci of tangent points between the contours off, and f2. Every point on this fine has the property that it is not dominated by any other point in its neighborhood, i.e. f,(e)

Ifi

and

where Q is a point lying on the line 0,02 and P is a neighboring point. Thus all points of S that do not lie on the fine O,O, need not be considered during cooperative play. The set of all points lying on AB is known as a Pareto-optimal set and is denoted by S,.

Since S, represents the solution szt to be considered in a cooperative game, the main task in a multicriteria optimization problem will be to determine the solution set S,. After determining the Pareto-optimal set, one has to pick up a particular element from the set by adopting a systematic procedure. If it is possibIe to convert all the criteria involved in the problem to some common units, then the problem will be greatly simplified. If this is not possible, further rules of negotiation in the form of a supercriterion or bargaining model should be specified before selecting a particular element from the set .S,. A procedure for finding the set S, and an element of S, based on a su~rc~te~on is presented in the following section. COMPUTATIONAL PROCEDURE

The cooperative game theory approach of solving the multicriteria optimization problem can be stated as follows. k players are assumed to correspond to the k objectives; one for each objective. While playing the game (i.e. while designing the structure), each player will try to improve his/her own conditions (i.e. to decrease the value of his/her own objective function). The players will start bargaining from their respective reference (starting) values and put a joint effort in maximizing a subjective criterion (su~r~te~on) formed by themselves. It is assumed that each player has analyzed his/her own criterion before starting the game to find the maximum possible benefit he/she can obtain. This will also help him,her in guaranteeing against the worst value. This analysis is necessary since each player should know the extreme conditions

s. s. Rio

122

of his/her own and others so that none of them begins bargaining from a reference value which is unrealistic (i.e. unacceptable to the other players). The extreme values for each player are determined as follows. At any starting feasible design vector X,, the objective function values are found and positive constant multipliers M,, mz, , m, are chosen such that

i th player, indicates that during the cooperative play.

he/she should not expect a value for his objective better than 4(X:) but is guaranteed that his/her objective will never be worse than F,,. Assuming that all the players start negotiation by taking their worst values as reference values, a supercriterion S can be constructed as S = fi

{Fiu- MX:)},

(14)

i-l

mifi(X,)

= mzf2(X,) = ... = m&(X,)

where M is a constant. functions as

= IV, (9)

By redefining the objective

&(X)=m&X);

i=1,2

,...,

k,

F,(X?) FZCG) f-,(x:) F*(x:)

Fk(X?)

[PI=

F,(c, X) = i ciFi(X) 1-I

(IO)

one can notice that any design vector which minimizes the functionf;(X) will also minimize the function FJX). This scaling is done to make all the objective functions numerically equal at a particular design X,. Hereafter, it will be assumed that the k players correspond to the k scaled objective functions given by eqn (10). Starting from the design vector X,, each objective function h(X); i = 1,2, . . . , k is minimized subject to the constraints of eqn (2). Then a matrix [P] is constructed as

F,(F)

where Xy, a Pareto-optimal solution, minimizes a combined objective function F,(c, X) defined by

.

subject to the constraints conditions

(15)

of eqn (2) and cis satisfy the

i = 1,2,...,k

ci 2 0, and

,$, ci = 1.

(16)

From eqn (14) it can be seen that all the players will be interested in maximizing the criterion S. As X: is implicitly indepenent on the c,s, the problem now is to determine the optimum values of c,s for which S of eqn (14) attains a maximum value. The procedure of obtaining the optimum vector

(11)

c:

c*=

H .

C;:

It can be seen that the diagonal elements in the matrix [P] are the minima in their respective columns. Defining Fiu as FiU=maxFJXT);

i=l,2,

begins by assuming any vector c and improving it in the subsequent iterations by moving along the steepest ascent directions of S through appropriate step lengths.

:;2, a rectangular

as

matrix [R] is

ILLUSTRATIVE

EXAMPLES

Example 1

[R] =

Fi(X:)

Flu

(13)

This matrix gives the extreme values of all the players. For example, the ith row, which corresponds to the

The two bar truss shown in Fig. 2 is considered for illustrating the game theory approach. The area of cross section of the members (A ) and the position of the joints 1 and 2 (x) are treated as design variables. The truss is assumed to be symmetric about the y axis. The coordinates of joint 3 are held constant. The weight of the truss and the displacement of the joint 3 are considered as the objective functions fi and h. The stresses induced in the members are constrained to be smaller than the permissible stress, u,,. In addition, lower bounds are placed on the design

Multiobjective structural optimization

b?(X)=

123

P(x, - I)(1 +x:)o.5

_

c

-a,50

(20)

g,(X) = xI” - x, IO

(21)

g4(X) = ,y!” -x2 IO,

(22)

where 5, = x/h, x2 = A/A,,, E = Young’s modulus, p = density of the material, and xi’) = lower bound on ?c,; i = 1,2. The supercriterion (S) is chosen as the maximization of the product of deviations of the individual objective functions from their respective worst possible values: S(X) = M(X) -f;““IMV)

Fig. 2. Two-bar truss (example I). variables. Thus the problem becomes: minimize f, (X) = 2$1x, Jix Ph(l +x;)‘.S(l L(X) =

2 fi

(17)

+x;)o.s (18)

Ex;x,

subject to g,(x)=

P(1 +x,)(1

+x:)”

d I 0

2&,x,

‘k3 r

-

Ii1 i

!

(1%

O

\ X

-fT”l,

wheref;““” =fi(Xf) andfy =f;,(X:). The problem data is taken as p = 0.283 lb/in’, h = 100 in. A,, = 1 in2, P = 10,000 lb, E = 30 x IO6Ib/in2, u. = 20,000 lb/in2, .x{‘)= 0.1 and x ‘2)= 1.0. The contours of the two objective functions in the feasible design space are shown in Fig. 3 and the ideal feasible solutions fr and ff are also identified. The contours of the objective function in the weighting method (with equal weights forf, andf,) are shown in Fig. 4. It is to be noted that the objective functions are normalized so that they have the same numerical value at the initial design point during numerical optimization. The constraints g, and g, are normalized by dividing throughout by cro. The constraints g, and g4 are

W=80.00

A//

1

0.25 0.00

0.25

0.50

0.75

(23)

1.00

1.25 1.50 1.75 XI Fig. 3. Contours of the objective functions.

2.00

2.25

124

2.25

1.50 X2

1.25

1.00

i\

0.75 7

0.50

0.25 __ 0.00

0.25

0.50

0.75

I.00

1.25

1.50

1.75

2.00

2.25

XI

Fig. 4. Contours of the objective function in the weighting method. normalized by dividing throughout by their respective lower bound values. The results of numerical optimization are shown in Table 1. It can be observed that the optimum design vectors obtained bear mixed characteristics of XT and Xz. Example

2

The twenty-five bar space truss shown in Fig. 5 is considered with the objectives of minimizing the weight, minimizing the deflection of nodes 1 and 2, and maximizing the fundamental frequency of vibration. The truss is required to support the two load conditions given in Table 2 and is designed subject to constraints on the member stresses as well as Euler

Table I. Results of m~tiobj~tive

Quantity X=G’

Minimization offi 0.5295 0.6743

buckling. The minimum and maximum allowable areas of the members are specified as 0.1 in’ and S.Oin’, respectively. The aiiowabie stresses for all members are specified as 40,000 psi in both tension and compression. The Young’s modulus is taken as E = 10’ psi and the material density as p = 0.1 Ib/in3. The members are assumed to be tubular with a nominal diameter to thickness ratio of D/t = 100 SO that the buckling stress becomes

bbi =

- 100.01nm, , i=l,2 SL:

f J,;!bI iz(m.1 * = ul 9: (psi)

optimi~tjo~

Minimization offi 0.8612 2.5OOoi

of example 1

Weighting method (c, = c* = l/2) 0.7635 1.0540

Game theory approach 0.768 1.1408 1

200.1214 45.1615

233.2904 38.5847

93.1723 93.7941

101.7106 86.4693

36.1493 0.0943

186.7361 0.0182

75.0595 0.0442

81.4137 0.0408

19,994.9t 3889.7

4033.6 300.9

25.

(24)

The member areas are linked in the following groups:

* F+

,...,

9747.6 1306.9

8996.0 1180.1

t Active constant: F, = nt,A, F2= m&; m,, m, are constants; ci, = stress in member 1. o2 = stress in member 2.

Multiobjective structural optimization

125

Fig. 5. Twenty-five bar truss (example 2). A,; A, = A, = A, = A,; A, = A, = A, A,2=A,3; A,,=A,,=A,,=A,,; = A,, ; 4, = 4, = A,, = A,,. Thus independent areas are selected as the

= A,; A,0 = A,,; A,s=A,9=Azo a total of eight design variables.

The values of the objective functions at the starting design vector are taken as F,, = F?, = 500 and F,, = - 500. The results of minimization of the three individual objective functions are given in Table 3. The single criterion optimization problems are solved by using the method of feasible directions [14]. The matrix [P] is obtained as

Table 2. Loads acting on twenty-five bar truss Joint

1

2

3

6

0

0

0

0

Load condition 1, loadr in pouna!s

FY FZ

20,ooa -5,000

F,

-5,000 10,000

-

- 20,000 -5,000

FX 0 0 Load condition 2, loa& in pout& FX 1000 0 - 10,000 5,000

0

0

500

500

0

0

[PI=

2448.1753

100.0015

-509.7537

L 908.4379 352.3705

439.4703 624.3236

- 53 788.6644 1.8644

Design vector, X(in’)

Weight (lb) Deflection (in.) Fundamental frequency (Hz) Active behavior constraints Active side constraints Supercriterion, S

Starting voint

Minimization of weight

1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 330.7208 1.5417

lo. 1 0.8023 0.7479 ‘0.1 0.1245 0.5712 0.9785 0.8025 233.0726

68.8648

73.2535

0 0 -0.2362 x 10’

(25)

As stated earlier, the diagonal elements of [P] can be seen to be the smallest elements in their respective columns.

Table 3. Results of minimization of individual objective functions Ouantitv

1 .

1.9250

Minimization of deflection 3.793 1

Maximization of freauencv *o. 1

l5.0 +5.0 3.3183 *5.0 ‘5.0 *5.0 *5.0 1619.3258 0.3083

0.7977 0.7461 0.7282 0.8484 1.9944 1.9176 4.1119 600.8789 1.3550

70.2082

108.6224

9t

0

43

2 0

8

1 0

t Buckling of members 2, 5,7,8, 19 and 20 in load condition 1 and members 13, 16 and 24 in load condition 2. $ Buckling of members 2, 5, 7 and 8 in load condition I.

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126

Table 4. Results of maximization of the supercriterion Initial c, c, = c* = C]= l/3

Quantity

Design vector X(in’)

Weight (lb) Deflection (in.) Fundamental frequency (Hz) Active behavior constraints Active side constraints Supercriterion, .S

lo.I 1.1464 1.3156 0.4838 40.1 1.3315

*o. 1 0.8718 0.9841 0.2602 0.1003 0.5447 0.9669 2.3169 326.94828 1.4152 90.8159

1.8558 4.4960 596.5181 0.9401 100.2154

4t

0

I

It can be seen from these results that the minimization of weight resulted in 624.3% increase in deflection and 32.56% decrease in fundamental frequency compared to their respective optimum values. Similarly the optimization of deflection led to 694.8% increase in weight and 35.37% reduction in fundamental frequency. The maximization of the fundamental frequency yielded a design with 257.8% increase in weight and 439.5% increase in deflection. The active constraints at each of these optimum points are also indicated in Table 4. The [RI-matrix is constructed as

VI =

[ -788.6644

2448.1753 624.3236 .

2 1.0761 x IOn

0.4833 x lo*

t Buckling of members 19 and 20 in load condition condition 2.

352.3705 I~.~15

Optimal c. c, = 0.1433, c? = 0.3628. cj = 0.4939

(26)

- 509.7537 I

The first player, corresponding to the first objective, cannot claim a value lower than 352.3705 for his/her objective while he/she is guaranteed that it will never exceed the value of 2448.1753. Similarly the second and third rows of [R J indicate the two extreme values of the second and third players respectively. Now the supercriterion S is constructed and maximized with respect to the parameters cis without vioIating eqn (16). For this, initially c, , c, and c, are taken as l/3 each, and their values are changed in the subsequent iterations depending on the gradient of the supercriterion, and by taking a suitable step length along that direction. The procedure is terminated when the function S attains a local maximum. The initial and the optimum values of c and S obtained are given in Table 4 along with the corresponding design vectors. It can be seen that although initially equal weights are given to all the objectives, at the optimum solution (where S is maximum) the weightage of the first objective has decreased and that of the third has increased substantially. The increase in the value of the su~rc~terion obtained is about

1 and members 12 and 16 in load

122.65%. The optimum design vector given in Table 4 is the required Pareto-optimal solution (which maximizes S) of the design problem. From the matrix [R] and Table 4, it is found that at the end of the play, although no player could drive his/her objective near to his/her own minimum, all of them have reduced their values significantly in comparison to their respective reference values. CONCLUSIONS

The concepts of Pareto-optimal solution and game theory are presented. Methods of generating the Pareto-optimal solution set are described. The theoretical basis of game theory is explained in graphical relationship between Pareto-optimal terms. The solutions and game theory is described. The generation of Pareto-optimal solutions and the selection of a particular Pareto-optimal solution according to game theory approach are illustrated with simple truss design examples. Although only two and three objective optimization problems are considered in this work, the procedure is quite general and is applicable to any multiobjective optimization problem. Acknowiedgemenrs-The

research was partly supported by the Flight Dynamics Laboratory, WPAFB through Universal Energy Systems, Task # 84-25, contract no. F33615-83-c-3000. The useful suggestions made by Dr V. B. Venkayya during the progress of this work are gratefully acknowledged. REFERENCES

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