By Xavier Gandibleux

The generalized zone of a number of standards choice making (MCDM) will be outlined because the physique of equipment and methods wherein the worry for a number of conflicting standards may be officially integrated into the analytical technique. MCDM is composed in general of 2 branches, a number of standards optimization and multi-criteria selection research (MCDA). whereas MCDA is usually keen on a number of standards difficulties that experience a small variety of choices frequently in an atmosphere of uncertainty (location of an airport, form of drug rehabilitation program), a number of standards optimization is usually directed at difficulties formulated inside of a mathematical programming framework, yet with a stack of goals rather than only one (river basin administration, engineering part layout, product distribution). it truly is concerning the most recent therapy of a number of standards optimization that this e-book is worried. i glance at this e-book as a well equipped and well-rounded presentation of what I view as ”new wave” themes in a number of standards optimization. on reflection to the origins of MCDM, most folks agree that it used to be no longer till concerning the early Nineteen Seventies that a number of standards optimization c- gealed as a box. at the present, and for approximately the subsequent fifteen years, the point of interest was once on theories of a number of aim linear programming that subsume traditional (single criterion) linear programming, algorithms for characterizing the effective set, theoretical vector-maximum dev- opments, and interactive tactics

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Extra resources for Multiple criteria optimization : state of the art annotated bibliographic surveys

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It is an old idea to try to complement a given optimization problem with minimal value I) by a dual problem with supremal value S, remember the dual variational principles of Dirichlet and Thompson (cf. g. O. Friedrichs [98] or simply the pair of dual programs in linear optimization. The reasons for the introduction of a useful dual problem are the following: The dual problem has (under additional conditions) the same optimal value as the given “primal” optimization problem, but solving the dual problem could be done with other methods of analysis or numerical mathematics.

So strong duality means that and touch each other or with other words, and don’t overlap. Otherwise we speak of a pair of dual problems with a duality gap (in the scalar case at the beginning of this chapter that would mean I > S). Having a pair of strongly dual multicriteria programs their feasible elements respectively give estimations of the set of efficient elements of (P) “from above and below” (with respect to C). Lemma 3 Assume (P), (D) to be weakly dual. If there are such that then is minimal for (P), maximal for (D) and (P), (D) are strongly dual.

S. , and Cho [174]; Lee and Kum [178]; Li, Yang and Chen [182]; Lin [182]; Lin, Yang, and Yao [184]; Noor [207]; Oettli [209]; Siddiqi, Ansari, and Ahmad [240]; Rapcsak [228]; Yang [285, 286, 287, 288, 289]; Yang and Chen [290]; Yang and Goh [291], Yao [293, 294]; Yu and Yao [298]). 36 MULTIPLE CRITERIA OPTIMIZATION Existence of solutions for generalized vector variational inequalities and complementarity problems (Chang, Thompson, and Yuan [54]; Chen [55]; Chen and Hou [60]; Chen and Yang [64, 65]; Danilidis and Hadjisavvas [76]; Isac and Yuan [140]; Kazmi [158]; Lai and Yao [170]; Yin and Xu [297]; Qun [226]).

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