Optimization

203

7

Multiobjective Optimization

7.1 Introduction

In previous chapters, optimization problems with a single objective function

were discussed and these problems were with or without constraints. Typical

single-variable objective functions are cost minimization, efciency maximiza-

tion, weight minimization, and so on. The solution to single-variable optimiza-

tion problems results in a single point in the design space and the corresponding

objective function value at that point gives the minimum value of the function.

In the multiobjective optimization problem, two or more objective func-

tions are to be simultaneously optimized. For example, the criteria in manu-

facturing a product could be cost minimization and efciency maximization.

The general form of a multiobjective optimization problem can be mathe-

matically stated as

Minimize

f

k

(x) k = 1, 2,…, K (7.1)

subject to

g

i

(x) ≤ 0 i = 1, 2,…, m < n (7.2)

h

j

(x) = 0 j = 1, 2,…, r < n (7.3)

x

l

≤ x ≤ x

u

(7.4)

where x is a vector of n design variables given by

x =





























x

x

x

n

1

2



204 Optimization: Algorithms and Applications

The solution to a multiobjective problem results in a number of points

in the objective function space referred to as Pareto optimal solutions. For a

multiobjective problem with two objective functions (the rst function is

efciency maximization and the second function is cost minimization), a

typical Pareto optimal front is shown in Figure 7.1. The rst objective ( f

1

)

function “efciency” is along the x-axis of this gure and the y-axis contains

the second objective ( f

2

) function “cost.” The Pareto optimal front is obtained

using the principle of domination. In this concept, each solution is compared

to check whether it dominates another solution or not.

A solution x

1

is said to dominate another solution x

2

if the following condi-

tions are satised

• The solution x

1

is no worse than x

2

in all objectives.

• The solution x

1

is better than x

2

in at least one objective.

Consider points A and C for domination. Clearly, point C dominates point

A in both the objective functions. However, point C is itself dominated by

at least one of the points in the Pareto optimal front. The points along the

Pareto optimal front are referred to as nondominated solutions. In Figure 7.1,

the Pareto optimal front is convex. However, this front can be concave, par-

tially convex/concave or discontinuous. The trade-off between the objective

functions denes the shape of the Pareto front.

In this chapter, we discuss the methods for obtaining the nondominated

solutions for a multiobjective optimization problem. These methods will be

applied on some well-known test functions. The road map of this chapter

is shown in Figure 7.2. The weighted sum approach, ε-constraint method,

goal programming, and utility function method are explained as the tech-

niques for solving multiobjective problems. In the weighted sum approach,

C

B

A

Pareto optimal front

f

2

f

1

Nondominated solutions

FIGURE 7.1

Pareto optimal front.

205Multiobjective Optimization

different objectives are combined into a single objective function using dif-

ferent weights. This method is simple and easy to implement. However, it

can locate one Pareto point in one optimization run using the gradient-based

method. The particle swarm optimization (PSO) technique, which works

with a number of solution points, can locate the Pareto front on one single

run. In the ε-constraint method, one objective function is minimized and

remaining objective functions are transformed into constraints which are

to be specied by the user. The transformed problem is then solved using

the gradient-based method. The method can locate the Pareto fronts of the

nonconvex problems. In goal programming, a target is set for each of the

objective functions and the optimizer aims to minimize the deviations from

the set goals. In the utility function method, all the objectives are combined

into a single function which is then solved along with the constraints. In

the last section, shape optimization of a reentry body is carried out that has

two conicting objectives: weight minimization and stability maximization,

along with constraints.

7.2 Weighted Sum Approach

The simplest approach to solve a multiobjective optimization problem is to

combine all the objective functions into a single objective function, which

then can be solved using any of the methods described in previous chap-

ters. Different objective functions can be combined into a single objective

Multiobjective optimization

Weighted sum approach

ε-Constraints method

Goal programming

Utility function method

Application

FIGURE 7.2

Road map of Chapter 7.