Theory-Lesson 11(final solution)

Question 1

What is pareto optimality index and what its drawback?

Answer 1

Pareto optimality index is a measure of how “good” a pareto solution is for every OF.

It’s measure is called ineffectiveness index
q=1-(1-(1/p^(1/n))^n)

where n is the dimensionality of the problem

DRAWBACK:this technique becomes ineffective for high dimensionality because q tends to 1 meaning that the
number of the pareto points are almost as many as the total sample points.

Question 2

What is k-optimality ?

Answer 2

A point is k-optimal if and only if it is pareto optimal for all subsets of n-k objectives, where n is the number of OFs.

for k=0 -> 0-optimality which is the basic pareto solution

for k=n ->n-optimality which corresponds to the utopia point.

for k=n-1 we have the next best compromise except from the utopia point.

Of course, the higher the k the better.

Question 3

What is the corollary property of k-optimality?

Answer 3

A k-optimal point has at least k+1 objectives better than any other point.

pareto-optimal point-> 1 improved objective

k-optimal point->k+1 improved objectives

Question 4

What is the drawback of k-optimality?

Answer 4

Crisp selection of k, k can only be an integer.

Also, k does not give any further information, just provides the number k.

This approach can fail if the maximum attainable optimality is 0-optimality which gives back the pareto set.

Question 5

What is k-ε optimality and its advantage over k-optimality?

Answer 5

kε-optimality is used to overcome the problem of k-optimality.

ε is a measure of the difference between the OF’s actual values.

If the difference between the OFs values is smaller than the indifference threshold εi, the OF will be considered as not so bad and evaluated with a value between 0 and 1.

So each OF have its own indifference threshold εi, which represents the minimum meaningful difference in the OF values.

k-ε allows to have a continous degree of optimality and therefore it can have intermediate values when compared to k-optimality which deals with integer numbers.

ε’s are chosen by the decision maker.

k-ε can obtain a single solution, the one with the highest