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[ edit ] There are several ways to generate nondominated solutions. We will discuss two of these.
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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[ edit ] 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.
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 max wT. Achievement scalarizing function Wierzbicki,  Figure 3. Projecting points onto the nondominated set with an Achievement Scalarizing Function 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. 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, g1, and an infeasible point, g2, are projected onto the nondominated points, q1 and q2, respectively, along the direction w 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. For a bibliometric study showing their development over time, see Bragge, Korhonen, H.
Wallenius and J. Wallenius . 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, . Fuzzy-set theorists Fuzzy sets were introduced by Zadeh  as an extension of the classical notion of sets. This idea is used in many MCDM algorithms to model and solve fuzzy problems. 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, are used Keeney and Raiffa, . The method was first proposed by Bernard Roy Roy, . Evolutionary multiobjective optimization school EMO EMO algorithms start with an initial population, and update it by using processes designed to mimic natural survival-of-the-fittest principles and genetic variation operators to improve the average population from one generation to the next.
The goal is to converge to a population of solutions which represent the nondominated set Schaffer, ;  Srinivas and Deb, . Then the decision-maker evaluates the relative importance of its various elements by pairwise comparisons. The AHP converts these evaluations to numerical values weights or priorities , which are used to calculate a score for each alternative Saaty, .
A consistency index measures the extent to which the decision-maker has been consistent in her responses. AHP is one of the more controversial techniques listed here, with some researchers in the MCDA community believing it to be flawed[ citation needed ]. The underlying mathematics is also more complicated[ vague ], though it has gained some popularity as a result of commercially available software.
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Multiple-criteria decision analysis