Lecture
In any decision-making problem, it is necessary to construct the Pareto — Edgeworth set in order to exclude clearly unacceptable alternatives and narrow down the range of options for the DM's choice. A distinctive feature of the structure of the Pareto — Edgeworth set and its image in criterion space - the Pareto front — is that they are unstable with respect to small transformations (including linear ones) of the criterion space, can be disconnected (even for connected original sets of alternatives and their vector estimates), and cannot be found by linear methods. In particular, the widely used simplex methods construct only the effective vertices, i.e., the vertices of a convex polyhedron, whereas the Pareto — Edgeworth set, even in the linear case, may turn out to be non-convex, which accounts for its ability to describe reality well, since reality can in practice be more complex than the linear models underlying simplex methods. For the practical construction of the Pareto — Edgeworth set in a decision-making problem with two efficiency criteria, it is useful to employ a simplified geometric interpretation, which makes it possible to visually represent both the process of finding this set and the resulting outcomes. When there are three or more criteria, it is advisable to apply computer modeling and one of the many available numerical algorithms. Moreover, even the theoretically best possible approximation of the Pareto front has a substantial error, whose asymptotic estimate increases sharply as the number of criteria grows. Therefore, for approximation, it is advisable to use the most accurate and efficient numerical algorithms.
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