Multiobjective Optimization

Prepared By Mohammed Amer Kamil

The engineer is often confronted with “simultaneously”

minimizing (or maximizing) different criteria. The

structural engineer would like to minimize weight and

also maximize stiffness; in manufacturing, one would

like to maximize production output and quality while

minimizing cost and production time.

These practical applications are we refer to them as

multiobjective optimization problems.

Introduction

Mathematically, the problem with multiple objectives may be stated as

where the weights wi are ≥ 0, ∑wi = 1. The weights are chosen from experience.

Example:Each unit of product Y requires 2 hours of machining in the first cell and 1 hour in the second cell. Each unit of product Z requires 3 hours of machining in the first cell and 4 hours in the second cell. Available machining hours in each cell = 12 hours. Each unit of Y yields a profit of $0.80, and each unit of Z yields $2. It is desired todetermine the number of units of Y and Z to be manufactured to maximize:(i) total profit(ii) consumer satisfaction, by producing as many units of the superior quality product Y. If x1 and x2 denote the number of units of Y and Z, respectively, then the problem is:maximize f1 = 0.8 x1 + 2 x2 and f2 = x1subject to 2x1 + 3 x2 ≤ 12x1 + 4 x2 ≤ 12x1, x2 ≥ 0

Concept of Pareto Optimality

The preceding problem is shown graphically in Fig.1. Point A is the solution if only f1 is to be maximized, while point B is the solution if only f2 is to be maximized. For every point in x1 − x2 space, there is a point ( f ((x1), f (x2)) in criterion space.

Referring to Fig. 2, we observe the following :•There is interesting aspect of points lying on the line A − B: no point on the line is “better” than any other point on the line with respect to both objectives. •A point closer to A will have a higher value of f1 than a point closer to B but at the cost of having a lower value of f2. •In other words, no point on the line “dominates” the other.•Furthermore, a point P in the interior is dominated by all points within the triangle as shown in Fig. 2. •the line segment A − B represents the set of “nondominated” points or Pareto points in Ω.•We refer to the line A − B as the Pareto curve in criterion space.•This curve is also referred to as the Pareto efficient frontier or the nondominated frontier.

• general, no solution vector X exists that maximize all the objective

functions simultaneously..

• A feasible solution X is called Pareto optimal if there exists no

other feasible solution Y such that

fj(Y) ≤ fi(X) for i = 1, 2, . . . , k

with fj(Y) < fi(X) for at least one j.

• In other words, a feasible vector X is called Pareto optimal if there

is no other feasible solution Y that would maximize some objective

function without causing a simultaneous decrease in at least one

other objective function.

Definition of Pareto Optimality

Example 2Referring to Fig. 3-a, consider the feasible region and the problem of maximizing f1 and f2. The (disjointe) Pareto curve is identified and shown as dotted lines in the figure. If f1 and f2 were to be minimized, then the Pareto curve is as shown in Fig. 3-b

• Several methods have been developed for solving a

multiobjective optimization problem.

• Some of these methods will be briefly described in the

following slides.

• Most of these methods basically generate a set of Pareto

optimal solutions and use some additional criterion or rule

to select one particular Pareto optimal solution as the

solution of the multiobjective optimization problem.

Solving a multiobjective optimization problem

Weighted Sum Method

• Weight of an objective is chosen in proportion to the

relative importance of the objective.

• Scalarize a set of objectives into a single objective by

adding each objective pre-multiplied by a user supplied

weight .

Advantage

•Simple

Disadvantage

•It is difficult to set the weight vectors to obtain a

Pareto-optimal solution in a desired region in the

objective space

•It cannot find certain Pareto-optimal solutions in the

case of a nonconvex objective space

• Keep just one of the objective and restricting the

rest of the objectives within user-specific values .

ε-Constraint Method

• Keep f2 as an objective Minimize f2(x)• Treat f1 as a constraint f1(x) ≤ ε1

Advantage•Applicable to either convex or non-convex problemsDisadvantage•The ε vector has to be chosen carefully so that it is within the minimum or maximum values of the individual objective function

Lexicographic MethodWith the lexicographic method, preferences are imposed by ordering the objectives according to their importance or significance, rather than by assigning weights. The objective functions are arranged in the order of their importance. Then, the following optimization problems are solved one at a time:

• Here, i represents a function’s position in the preferred sequence, and fj(x*j ) represents the minimum value for the jth objective function, found in the jth optimization problem.

Advantages :

•it is a unique approach to specifying preferences.

•it does not require that the objective functions be

normalized.

•it always provides a Pareto optimal solution.

Disadvantages :

•it can require the solution of many single objective problems

to obtain just one solution point.

•it needs additional constraints to be imposed.