Scalarization-based methods

How can we convert an objective function to be minimized into an objective function to be maximized?

• Add -1

• Multiply it by -1

• You cannot

Answer explanation

The objective function must be multiplied by -1. This also works for converting an objective function to be minimized into one to be maximized.

• It is undefined

• It can be specified by a domain expert (a decision maker)

Answer explanation

The whole domain of x is the feasible set in the case of no constraints. In practice, one could also ask the decision maker in the case of absence of constraints.

Which of the given statements concerning the three sets of objective vectors in the figure are true?

• All the sets are Pareto optimal

• Set A and B are Pareto optimal

• Set C is weakly Pareto optimal

• All the sets are weakly Pareto optimal

Answer explanation

Pareto optimal solutions are a subset of weakly Pareto optimal solutions.

When computing a trade-off as given in (6), in which given case should we be extra careful?

• There is no need to be careful

• The denominator is zero

Answer explanation

We do not wish to divide by zero...

Which of the following statements is true concerning the solutions shown in the figure below?

• The circles are properly Pareto optimal, but the squares are not

• The squares are properly Pareto optimal, but the circles are not

• The circles and squares and properly Pareto optimal, but the stars are not

• The stars are properly Pareto optimal

Answer explanation

With proper Pareto optimality, there should be a meaningful trade-off between solutions in the objective function values when switching from one solution to another.

What is the ideal point, and what is the nadir point according to the payoff-table?

• Impossible to say

Answer explanation

The ideal is taken from the diagonal of the table and the nadir by finding the maximum value of each column.

• Because we go always one step beyond

• There is no special reason, just for fun

• To avoid dividing by zero

• As a counterpoint to the dystopian point

Answer explanation

Once again, to not divide by zero. Fun fact, the dystopian point is a real concept in multiobjective optimization, but seldom used.

• The goals are unachievable

• We divide by zero

• The constraint is not feasible

• The solution found might not be Pareto optimal

Answer explanation

One of the main weaknesses of goal programming is that it stops once it finds a feasible point near the reference point.