Gandibleux, 2002, for a review). Representations and definitions The MCDM problem can be represented in the criterion space or the decision space. Alternatively, if different criteria are combined by a weighted linear function, it is also possible to represent the problem in the weight space. Below are the demonstrations of the criterion and weight spaces as well as some formal definitions. Criterion space representation Let us assume that we evaluate solutions in a specific problem situation using several criteria. Let us further assume that more is better in each criterion. Then, among all possible solutions, we are ideally interested in those solutions that perform well in all considered criteria. However, it is unlikely to have a single solution that performs well in all considered criteria. Typically, some solutions perform well in some criteria and some perform well in others. Finding a way of trading off between criteria is one of the main endeavors in the MCDM literature. Mathematically, the MCDM problem corresponding to the above arguments can be represented as subject to where is the vector of k criterion functions (objective functions) and is the feasible set, . If is defined explicitly (by a set of alternatives), the resulting problem is called a multiple-criteria evaluation problem. If is defined implicitly (by a set of constraints), the resulting problem is called a multiple-criteria design problem. The quotation marks are used to indicate that the maximization of a vector is not a well-defined mathematical operation. This corresponds to the argument that we will have to find a way to resolve the trade-off between criteria (typically based on the preferences of a decision maker) when a solution that performs well in all criteria does not exist. Decision space representation The decision space corresponds to the set of possible decisions that are available to us. The criteria values will be consequences of the decisions we make. Hence, we can define a corresponding problem in the decision space. For example, in designing a product, we decide on the design parameters (decision variables) each of which affects the performance measures (criteria) with which we evaluate our product. Mathematically, a multiple-criteria design problem can be represented in the decision space as follows: where is the feasible set and is the decision variable vector of size n. A well-developed special case is obtained when is a polyhedron defined by linear inequalities and equalities. If all the objective functions