Multi-Criteria Decision Analysis (MCDA) is extensively used across diverse industries to assess and rank alternatives. Among numerous MCDA methods developed to solve real-world ranking problems, TOPSIS remains one of the most popular choices in many application areas. TOPSIS calculates distances between the considered alternatives and two predefined ones, namely the ideal and the anti-ideal, and creates a ranking of the alternatives according to a chosen aggregation of these distances. However, the interpretation of the inner workings of TOPSIS is difficult, especially when the number of criteria is large. To this end, recent research has shown that TOPSIS aggregations can be expressed using the means (M) and standard deviations (SD) of alternatives, creating MSD-space, a tool for visualizing and explaining aggregations. Even though MSD-space is highly useful, it assumes equally important criteria, making it less applicable to real-world ranking problems. In this paper, we generalize the concept of MSD-space to weighted criteria by introducing the concept of WMSD-space defined by what is referred to as weight-scaled means and standard deviations. We demonstrate that TOPSIS and similar distance-based aggregation methods can be successfully illustrated in a plane and interpreted even when the criteria are weighted, regardless of their number. The proposed WMSD-space offers a practical method for explaining TOPSIS rankings in real-world decision problems.

There has been a great deal of interest about negotiations having interdependent issues and nonlinear utility spaces as they arise in many realistic situations. In this case, reaching a consensus among agents becomes more difficult as the search space and the complexity of the problem grow. Nevertheless, none of the proposed approaches tries to quantitatively assess the complexity of the scenarios in hand, or to exploit the topology of the utility space necessary to concretely tackle the complexity and the scaling issues. We address these points by adopting a representation that allows a modular decomposition of the issues and constraints by mapping the utility space into an issue-constraint hypergraph. Exploring the utility space reduces then to a message passing mechanism along the hyperedges by means of utility propagation. Adopting such representation paradigm will allow us to rigorously show how complexity arises in nonlinear scenarios. To this end, we use the concept of information entropy in order to measure the complexity of the hypergraph. Being able to assess complexity allows us to improve the message passing algorithm by adopting a low-complexity propagation scheme. We evaluated our model using parametrized random hyper- graphs, showing that it can optimally handle complex utility spaces while outperforming previous sampling approaches.

Experiments show that these approaches achieve high effectiveness Negotiation scenarios involving nonlinear utility (measured as high optimality rates and low failure rates functions are specially challenging, because traditional for the negotiations) in the evaluation scenario they describe negotiation mechanisms cannot be applied. (Section 2). However, as we will show empirically in Section Even mechanisms designed and proven useful for 5.2, these approaches perform worse as the circumstances of nonlinear utility spaces may fail if the utility space the scenario turn harder (that is, when the utility functions is highly nonlinear. For example, although both are highly nonlinear, like in B2B interactions or distributed contract sampling and constraint sampling have automated control systems). Under these circumstances, the been successfully used in auction based negotiations failure rate increases drastically, raising the need for an alternative with constraint-based utility spaces, they tend approach.