An analysis of the final solution

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An analysis of the final solution

An analysis of the final solution

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.

Problem Tree / Solution Tree Analysis

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, [17] 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.

Multiple-criteria decision analysis - Wikipedia

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. The purpose of vector maximization is to approximate the nondominated set; originally developed for Multiple Objective Linear Programming problems Evans and Steuer, ; [19] Yu and Zeleny, [20].

Phases of computation alternate with phases of decision-making Benayoun et al. No explicit knowledge of the DM's value function is assumed. 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, [25]. Fuzzy-set theorists Fuzzy sets were introduced by Zadeh [26] 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, [27].

The method was first proposed by Bernard Roy Roy, [28].The Origins of the Final Solution: The Evolution of Nazi Jewish Policy, September March (Comprehensive History of the Holocaust) [Christopher R.

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