By Josef Gruber (auth.), Dr. Dr. habil. Andranik Tangian, Prof. Dr. Josef Gruber (eds.)
For numerous many years, students have constructed equipment for fixing optimization difficulties which emerge in economics, econometrics, operations study, and different disciplines. a substantial attempt has been made to build equations from which constraints will be derived, yet strangely little has been performed to build the opposite a part of optimization types: the scalar-valued aim functionality, the limited greatest or minimal of which supplies the optimum answer. The given quantity is meant to draw cognizance to the matter, to offer the key achievements within the box and to stimulate additional learn and teaching.
Read or Download Constructing Scalar-Valued Objective Functions: Proceedings of the Third International Conference on Econometric Decision Models: Constructing Scalar-Valued Objective Functions University of Hagen Held in Katholische Akademie Schwerte September 5–8, 1995 PDF
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Extra info for Constructing Scalar-Valued Objective Functions: Proceedings of the Third International Conference on Econometric Decision Models: Constructing Scalar-Valued Objective Functions University of Hagen Held in Katholische Akademie Schwerte September 5–8, 1995
Considering zg, and sO as fixed, and zl, sl as variables, F is indeed the sum of a function of Z3, and of a function of s. On the Preferences Characterization of Additively Separable Utility 35 3 2 Figure 11. Extension of additively decomposed utility from 2 to 3 commodities JRt, the preference relation ~ can be represented by Having shown that, in an additively decomposed utility function, we observe that any additively decomposed utility function representing the same preference relation is obtained from u by an increasing linear transformation.
It is easy to verify that the condition CM holds for F, as it should be. c) = 2, IF(d) = 3, IF(e) = 4 . The condition RV is obviously satisfied as weH. Therefore, the function IF so obtained can be taken as an objective function in the family of optimization problems (2) which generates the indirect value function F initially given. Let us emphasize that 1F differs from the original objective function 1 we started with. Such nonuniquiness of objective functions underlying the same family of scalar optimization problems is a typical case for rather complicated families U of admissible sets.
By the independence assumption, er' '" w' implies 1 '" 6 (by which notation we mean that er' is indifferent to w', and 1 is indifferent to 6 ). By the same assumption, ß'" ß' implies 6 '" 6'. Similarly, w' '" er implies 6' '" 1'. Now, by transitivity of indifference, h "" 6, 6 "" 6', 6' "" 'Y'l implies h "" iL which was to be established. In the horizontal plane at elevation zg, the preference preorder induced by ~ can, therefore, be represented by U1(Zt} + U2(Z2). At the same time, on lR~, ~ can be represented by a continuous utility function u.