Abstract
A decision-maker who aims to select the ”best” collection of alternatives from the finite set of available ones might be severely restricted in the design of the selection method. If the representation of valuations of available alternatives is subject to invariance under linear scaling, such as the choice of the unit of measurement, a sensible way to compare choices is to compare weighted sums of individual valuations corresponding to these choices. This scaling invariance, in conjunction with additional reasonable axioms, provides a characterization of linear 0-1 programming objective functions.
The problem of finding an optimal subset of available data to be aggregated, allowing for use of different aggregation methods for different subsets of data, is also addressed. If the input data in the optimal aggregation problem are measured on a ratio scale and if the aggregation must be unanimous and symmetric, the arithmetic mean is the only sensible aggregation method.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Clemen, R.T.: Combining Forecasts: A Review and Annotated Bibliography. Intl. J. Forecasting 5, 559–583 (1989)
Dyer, J.S., Fishburn, P.C., Steuer, R.E., Wallenius, J., Zionts, S.: Multiple Criteria Decision Making, Multiattribute Utility Theory: The Next Ten Years. Management Science 38(5), 645–654 (1992)
Eichhorn, W.: Functional Equations in Economics. Addison-Wesley, Reading (1978)
Krantz, D.H., Luce, R.D., Suppes, P., Tversky, A.: Foundations of Measurement, vol. I. Academic Press, New York (1971)
Labreuche, C., Grabisch, M.: The Choquet Integral for the Aggregation of Interval Scales in Multicriteria Decision Making. Fuzzy Sets and Systems 137(1), 11–26 (2003)
Luce, R.D., Krantz, D.H., Suppes, P., Tversky, A.: Foundations of Measurement, vol. III. Academic Press, New York (1990)
Marichal, J.-L.: On Choquet and Sugeno Integrals as Aggregation Functions. In: Grabisch, M., Murofushi, T., Sugeno, M. (eds.) Fuzzy Measures and Integrals, pp. 247–272. Physica Verlag, Heidelberg (2000)
Pekeč, A.: Limitations on Conclusions from Combinatorial Optimization Models. Ph.D. Dissertation, Rutgers University (1996)
Roberts, F.S.: Measurement Theory. Addison-Wesley, Reading (1979)
Roberts, F.S.: Limitations of Conclusions Using Scales of Measurement. In: Pollock, S.M., Rothkopf, M.H., Barnett, A. (eds.) Handbooks in OR & MS, vol. 6, pp. 621–671. North-Holland, Amsterdam (1994)
Sen, A.K.: Collective Choice and Social Welfare. North-Holland, Amsterdam (1984)
Sen, A.K.: Choice, Welfare and Measurement. Harvard University Press, Cambridge (1987)
Suppes, P., Krantz, D.H., Luce, R.D., Tversky, A.: Foundations of Measurement, vol. II. Academic Press, New York (1989)
Yager, R.R.: On Ordered Weighted Averaging Aggregation Operators in Multicriteria Decision Making. IEEE Trans. Systems Man Cybernet. 18, 183–190 (1988)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Pekeč, S. (2011). Scaling Invariance and a Characterization of Linear Objective Functions. In: Brafman, R.I., Roberts, F.S., Tsoukiàs, A. (eds) Algorithmic Decision Theory. ADT 2011. Lecture Notes in Computer Science(), vol 6992. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24873-3_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-24873-3_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-24872-6
Online ISBN: 978-3-642-24873-3
eBook Packages: Computer ScienceComputer Science (R0)