on the efficient frontier. Any point (supported or not) can be reached. The second term in the objective function is required to avoid generating inefficient solutions. Figure 3 demonstrates how a feasible point, , and an infeasible point, , are projected onto the nondominated points, and , respectively, along the direction using an achievement scalarizing function. The dashed and solid contours correspond to the objective function contours with and without the second term of the objective function, respectively. Solving MCDM problems Different schools of thought have developed for solving MCDM problems (both of the design and evaluation type). For a bibliometric study showing their development over time, see Bragge, Korhonen, H. Wallenius and J. Wallenius [2010]. Multiple objective mathematical programming school (1) Vector maximization: The purpose of vector maximization is to approximate the nondominated set; originally developed for Multiple Objective Linear Programming problems (Evans and Steuer, 1973; Yu and Zeleny, 1975). (2) Interactive programming: Phases of computation alternate with phases of decision-making (Benayoun et al., 1971; Geoffrion, Dyer and Feinberg, 1972; Zionts and Wallenius, 1976; Korhonen and Wallenius, 1988). No explicit knowledge of the DM's value function is assumed. Goal programming school The purpose is to set apriori target values for goals, and to minimize weighted deviations from these goals. Both importance weights as well as lexicographic pre-emptive weights have been used (Charnes and Cooper, 1961). Fuzzy-set theorists Fuzzy sets were introduced by Zadeh (1965) as an extension of the classical notion of sets. This idea is used in many MCDM algorithms to model and solve fuzzy problems. Ordinal data based methods Ordinal data has a wide application in real-world situations. In this regard, some MCDM methods were designed to handle ordinal data as input data. For example, Ordinal Priority Approach and Qualiflex method. Multi-attribute utility theorists Multi-attribute utility or value functions are elicited and used to identify the most preferred alternative or to rank order the alternatives. Elaborate interview techniques, which exist for eliciting linear additive utility functions and multiplicative nonlinear utility functions, may be used (Keeney and Raiffa, 1976). Another approach is to elicit value functions indirectly by asking the decision-maker a series of pairwise ranking questions involving choosing between hypothetical alternatives (PAPRIKA method; Hansen and Ombler, 2008). French school The French school focuses on decision aiding, in particular the ELECTRE family of outranking methods that originated in France during the mid-1960s. The method was first proposed by