the first and second objectives, respectively. The red boundary between those two extreme points represents the efficient set. It can be seen from the figure that, for any feasible solution outside the efficient set, it is possible to improve both objectives by some points on the efficient set. Conversely, for any point on the efficient set, it is not possible to improve both objectives by moving to any other feasible solution. At these solutions, one has to sacrifice from one of the objectives in order to improve the other objective. Due to its simplicity, the above problem can be represented in criterion space by replacing the with the as follows: subject to We present the criterion space graphically in Figure 2. It is easier to detect the nondominated points (corresponding to efficient solutions in the decision space) in the criterion space. The north-east region of the feasible space constitutes the set of nondominated points (for maximization problems). Generating nondominated solutions There are several ways to generate nondominated solutions. We will discuss two of these. The first approach can generate a special class of nondominated solutions whereas the second approach can generate any nondominated solution. Weighted sums (Gass & Saaty, 1955) If we combine the multiple criteria into a single criterion by multiplying each criterion with a positive weight and summing up the weighted criteria, then the solution to the resulting single criterion problem is a special efficient solution. These special efficient solutions appear at corner points of the set of available solutions. Efficient solutions that are not at corner points have special characteristics and this method is not capable of finding such points. Mathematically, we can represent this situation as = subject to By varying the weights, weighted sums can be used for generating efficient extreme point solutions for design problems, and supported (convex nondominated) points for evaluation problems. Achievement scalarizing function (Wierzbicki, 1980) Achievement scalarizing functions also combine multiple criteria into a single criterion by weighting them in a very special way. They create rectangular contours going away from a reference point towards the available efficient solutions. This special structure empower achievement scalarizing functions to reach any efficient solution. This is a powerful property that makes these functions very useful for MCDM problems. Mathematically, we can represent the corresponding problem as = }, subject to The achievement scalarizing function can be used to project any point (feasible or infeasible)