This paper presents NexClass, a Decision Support System (DSS) which supports classification of alternatives into predefined non-ordered categories according to their performance on evaluation criteria and implements a novel classification algorithm based on multicriteria analysis and outranking relations. More detailed, assignment to classes is based on the concept of non-exclusivity, which defines at what degree an alternative can be included in a specific category. For each category, a threshold is defined by the decision maker, which indicates its limit with respect to the evaluation criteria. Alternatives are next evaluated according to the criteria and non-excluding degrees are calculated for each category. Finally, an alternative is assigned to the category for which non-excluding degree gets the lowest value. NexClass DSS implements the above classification algorithm, providing a user-friendly interface, which supports decision makers to formulate and solve classification problems. In addition to the methodology and the DSS, we present a real world application at a classification problem in banking environment. Our findings derived from evaluation experiments in the banking environment provide valid evidence that the proposed methodology and the DSS effectively support decision makers in classification decisions.