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Multiob jective Optimization of Trusses using Genetic Algorithms

Carlos A. Co ello Co ello

Alan D. Christiansen

Engineering Design Centre, University of Plymouth, Plymouth, PL4 8AA, UK



Department of Computer Science, Tulane University, New Orleans, LA 70118, USA

Abstract:

In this paper we propose the use of the genetic algorithm (GA) as a tool to solve multiobjective

optimization problems in structures. Using the concept of min-max optimum, a new GA-based multiobjective

optimization technique is proposed and two truss design problems are solved using it. The results produced by

this new approach are compared to those produced by other mathematical programming techniques and GA-based

approaches, proving that this technique generates better trade-os and that the genetic algorithm can be used as a

reliable numerical optimization tool.

Keywords

genetic algorithms, multiob jective optimization, vector optimization, multicriteria optimization,

structural optimization, truss optimization

1 Intro duction

In most real-world problems, several goals must b e satised simultaneously in order to obtain an optimal

solution. The multiple ob jectives are typically conicting and non-commensurable, and must b e satised simul-

taneously. For example, we might want to b e able to minimize the total weight of a truss while minimizing

its maximum deection and maximizing its maximum allowable stress. The common approach in this sort of

problem is to cho ose one ob jective (for example, the weight of the structure) and incorp orate the other ob jectives

as constraints. This approach has the disadvantage of limiting the choices available to the designer, making the

optimization pro cess a rather dicult task.

Another common approach is the combination of all the ob jectives into a single ob jective function. This

technique has the drawback of mo delling the original problem in an inadequate manner, generating solutions that

will require a further sensitivity analysis to b ecome reasonably useful to the designer.

A more appropriate approach to deal with multiple ob jectives is to use techniques that were originally designed

for that purpose in the eld of Op erations Research. Work in that area started a century ago, and many approaches

have b een rened and commonly applied in economics and control theory.

This paper addresses the imp ortance of multiob jective structural optimization and reviews some of the basic

concepts and part of the most relevant work in this area. Also, we discuss the suitability of a heuristic technique

inspired by the mechanics of natural selection (the genetic algorithm, or GA) to solve multiob jective optimization

problems. We also intro duce a new metho d, based on the concept of min-max optimum. The new metho d

is compared with other GA-based multiob jective optimization metho ds and some mathematical programming

techniques. We show that the new metho d is capable of nding b etter trade-os among the comp eting ob jectives.

Our approach is tested on two well-known truss optimization problems. We p erform these tests with a computer

program called MOSES, which was develop ed by the authors to exp eriment with new and existing multiob jective

optimization algorithms.



This work was done while the author was at Tulane University.

2 Previous Work on Multiob jective Structural Optimization

The rst application of multiob jective optimization concepts in structural mechanics app eared in a 1968 pap er

by Krokosky [35]. In this early pap er, Krokosky devised a method that accommo dates the designer's preference

of the design requirements. In his approach, Krokosky required the most desirable and least desirable values

of each of the decision variables and the dierent levels of desirability of various combinations of such decision

variables provided through a ranking matrix. Krokosky adopted a random search technique to nd the b est

trade-o correlating the dierent ob jectives (i.e., desired values of the decision variables) in terms of the a priori

chosen design parameters. This technique happ ens to b e computationally inecient and impractical b ecause it is

sometimes very dicult, or even imp ossible, to get all the requirements in terms of one quantity [45]. Krokosky's

approach was later applied to the optimal design of sandwich panels [55]. Rao et al. [45] used utility theory to

overcome some of the drawbacks of Krokosky's approach when dealing with optimal material choice problems.

Stadler [58] noted the scientic application of the concept of Pareto optimality to problems of natural structural

shap es. He used this concept to compute optimal initial shap es of uniform shallow arches. Rao et al. [47, 48]

showed signicant work in multiob jective structural optimization with uncertain parameters. Rao was one of the

rst to p oint out the imp ortance of incorp orating concepts from game theory into structural optimization, and used

several mathematical programming techniques such as global criterion, utility function, goal programming, goal

attainment, b ounded ob jective function and lexicographic metho ds, to solve multiob jective structural optimization

problems. A more extended analysis of the use of game theory as a design to ol may b e found in [61], although

no applications are included.

Carmichael [7] prop osed the use of the

-constraint metho d to the multiob jective optimum design of trusses.

A more formal treatment of the sub ject wass given by Koski and Silvennoinen [33], who prop osed a numerical

metho d that generates the Pareto optimal set of an isostatic truss, based on the exact solution of bicriterion

subproblems. An extension to this work was published later [34] by the same authors. In this latter pap er the

authors prop osed the scalarization of the original multiob jective optimization problem by using norm metho ds

and the reduction of the dimension of the problem by a partial weighting metho d. They used trusses (b oth

isostatic and hyp erstatic) to illustrate these metho ds.

Fu and Frangopol [19] formulated a multiob jective structural optimization technique based on structural reli-

ability theory. This approach was illustrated solving a hyperstatic truss and a frame system. El-Sayed et al. [12]

used linear goal-programming techniques with successive linearizations to solve nonlinear structural optimization

problems. Their application was a three bar truss with uncertainties in b oth load magnitude and direction.

Ha jela and Shih [29] presented a slight variation of the global criterion approach, used in conjunction with

a branch and b ound algorithm, to solve multiob jective optimization problems that involve a mix of continuous,

discrete and integer design variables. A simply supp orted I-b eam and a comp osite laminated b eam were included

to exemplify their approach. Another variant of the global criterion approach was suggested by Saravanos and

Chamis [52] to design lightweight, low-cost comp osite structures. Tseng and Lu [60] applied goal programming,

compromise programming and the surrogate trade-o metho d to the selection of system parameters and large

scale structural design optimization problems.

Grandhi et al. [25] presented a reliability-based decision criterion approach for multiob jective optimization

of structures with a large numb er of design variables and constraints. Lounis and Cohn [38] used a pro jected

Lagrangian algorithm to transform the multiob jective optimization of prestressed concrete structures into a single-

ob jective optimization problem.

Finally, the b o ok by Eschenauer et al. [13] is a very valuable guide to some of the most relevant work in

multiob jective design optimization in the last few years. Go o d surveys on multiob jective structural optimization

may b e found in Stadler [59], Duckstein [11] and Co ello [10].

3 Basic Concepts

Multiob jective optimization (also called multicriteria optimization, multiperformance or vector optimization)

can b e dened as the problem of nding [41]:

a vector of decision variables which satises constraints and optimizes a vector function whose elements

If this was the case, then 



would b e a desirable solution, but we normally never have a situation like this, in

which all the

( 

) have a minimum in

at a common p oint



. An example of this ideal situation is shown in

Figure 1. However, since this situation rarely happ ens in real-world problems, then we have to establish a certain

criteria to determine what would b e considered as an \optimal" solution.

3.1 Pareto Optimum

The concept of

Pareto optimum

was formulated by Vilfredo Pareto in 1896 [42], and constitutes by itself the

origin of research in multiob jective optimization. We say that a p oint 



2 F

Pareto optimal

if for every 

2 F

either,

(

( 

) =

( 



)) (5)

or, there is at least one

such that

( 

)

> f

( 



) (6)

In words, this denition says that 



is Pareto optimal if there exists no feasible vector 

which would decrease

some criterion without causing a simultaneous increase in at least one criterion. Unfortunately, the Pareto

optimum almost always gives not a single solution, but rather a set of solutions called

non-inferior

non-

dominated

solutions.

3.2 Pareto Front

The minima in the Pareto sense are going to b e in the b oundary of the design region, or in the lo cus of the

tangent p oints of the ob jective functions. This region is called the

Pareto Front

. In general, it is not easy to nd

an analytical expression of the line or surface that contains these p oints, and the normal pro cedure is to compute

the points

and their corresponding

(

). When we have a sucient amount of these, we may proceed to

take the nal decision.

3.3 Min-Max Optimum

The idea of stating the

min-max optimum

and applying it to multiob jective optimization problems, was taken

from game theory, which deals with solving conicting situations. The min-max approach to a linear mo del was

prop osed by Jutler [40] and Solich [40]. It has been further develop ed by Osyczka [39], Rao [46] and Tseng and

Lu [60].

The min-max optimum compares relative deviations from the separately attainable minima. Consider the

ob jective function for which the relative deviation can b e calculated from

( 

) =

( 

)

(7)

or from

( 

) =

( 

)

( 

)

(8)

It should b e clear that for (7) and (8) we have to assume that for every

and for every 

2 F

( 

)

= 0.

If all the ob jective functions are going to b e minimized, then equation (7) denes function relative increments,

whereas if all of them are going to b e maximized, it denes relative decrements. Equation (8) works conversely.

Let 

( 

) = [

( 

)

; : : : ; z

( 

)

; : : : ; z

( 

)]

b e a vector of the relative increments which are dened in

. The

comp onents of the vector

( 

) will b e evaluated from the formula

(

( 

)) =

max

( 

)

; z

( 

)

(9)

Now we dene the min-max optimum as follows [40]:

A p oint 



2 F

is min-max optimal, if for every 

2 F

the following recurrence formula is satised:

Step 1:

( 



) =

min

2 F

max

( 

)

(10)

and then

, where

is the index for which the value of

( 

) is maximal.

If there is a set of solutions

 F

which satises Step 1, then

Step 2:

( 



) =

min

max

I ; i

( 

)

(11)

and then

; i

, where

is the index for which the value of

(

) in this step is maximal.

If there is a set of solutions

 F

which satises step

1 then

Step r:

( 



) =

min

max

I ; i

( 

)

(12)

and then

; i

, where

is the index for which the value of

( 

) in the

th step is maximal.

If there is a set of solutions

 F

which satises Step

1, then

Step k:

( 



) =

min



( 

)

max

I ; i

for i

I and i

(13)

where

( 



)

; : : : ; v

( 

) is the set of optimal values of fractional deviations ordered non-increasingly.

This optimum can b e describ ed in words as follows. Knowing the extremes of the ob jective functions which

can b e obtained by solving the optimization problems for each criterion separately, the desirable solution is the

one which gives the smallest values of the relative increments of all the ob jective functions.

The point 



2 F

which satises the equations of Steps 1 and 2 may b e called the b est compromise solution

considering all the criteria simultaneously and on equal terms of imp ortance.

4 Mathematical Programming Techniques

Multiob jective optimization is a very well known topic in Op erations Research, and many techniques have

b een develop ed to deal with such problems. The rst step in developing a mathematical programming technique

to deal with multiob jective optimization problems is to b e able to identify, given a set of feasible solutions to the

problem, which of them are Pareto optimal. After that, it will b e desirable to agree up on a further concept of

absolute optimality in this context, in case the designer desires a single nal solution. For the scop e of this pap er,

we will adopt the concept of min-max optimum for that purp ose.

Osyczka provides in his bo ok [40] an algorithm based on the contact theorem [37], which can identify Pareto

optimal solutions from a given set of feasible solutions. This algorithm was implemented by the authors of this

pap er and incorp orated in MOSES together with another one that nds the optimum in the min-max sense [10].

4.1 Monte Carlo Metho ds

We also implemented the two Monte Carlo metho ds used by Osyczka [40] to nd the min-max optimum.

These metho ds are called

exploratory

b ecause a p oint is generated by means of a rule which disregards the results

previously obtained. In particular, the Monte Carlo metho d picks up a certain numb er of p oints at random over

the estimated range of all the variables of the problem. This is done formally by obtaining the randomly selected

value for

from the following formula



(

) for

= 1

;

; : : : ; n

(14)

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