Reference point method with importance weighted ordered partial achievements

Abstract

The Reference Point Method (RPM) is a very convenient technique for interactive analysis of the multiple criteria optimization problems. The interactive analysis is navigated with the commonly accepted control parameters expressing reference levels for the individual objective functions. The partial achievement functions quantify the DM satisfaction from the individual outcomes with respect to the given reference levels, while the final scalarizing achievement function is built as the augmented max–min aggregation of the partial achievements. In order to avoid inconsistencies caused by the regularization, the max–min solution may be regularized by the Ordered Weighted Averages (OWA) with monotonic weights which combines all the partial achievements allocating the largest weight to the worst achievement, the second largest weight to the second worst achievement, and so on. Further, following the concept of the Weighted OWA (WOWA), the importance weighting of several achievements may be incorporated into the RPM. Such a WOWA RPM approach uses importance weights to affect achievement importance by rescaling accordingly its measure within the distribution of achievements rather than by straightforward rescaling of achievement values. The recent progress in optimization methods for ordered averages allows one to implement the WOWA RPM quite effectively as extension of the original constraints and criteria with simple linear inequalities. There is shown that the OWA and WOWA RPM models meet the crucial requirements with respect to the efficiency of generated solutions as well as the controllability of interactive analysis by the reference levels.