A two-slope achievement scalarizing function for interactive multiobjective optimization

The use of achievement (scalarizing) functions in interactive multiobjective optimization methods is very popular, as indicated by the large number of algorithmic and applied scientific papers that use this approach. Key parameters in this approach are the reference point, which expresses desirable objective function values for the decision maker, and weights. The role of the weights can range from purely normalizing to fully preferential parameters that indicate the relative importance given by the decision maker to the achievement of each reference value. Technically, the influence of the weights in the solution generated by the achievement scalarizing function is different, depending on whether the reference point is achievable or not. Besides, from a psychological point of view, decision makers also react in a different way, depending on the achievability of the reference point. For this reason, in this work, we introduce the formulation of a new achievement scalarizing function with two different weight vectors, one for achievable reference points, and the other one for unachievable reference points. The new achievement scalarizing function is designed so that an appropriate weight vector is used in each case, without having to carry out any a priori achievability test. It allows us to reflect the decision maker's preferences in a better way as a part of an interactive solution method, and this can cause a quicker convergence of the method. The computational efficiency of this new formulation is shown in several test examples using different reference points.