Aggregation of interacting criteria

Aggregation (from Latin aggregare, to be accepted) stands for summary . It is about the appropriate combination of various criterion values ​​to form a global value, for example with the aim of obtaining the most objective possible ranking between competing individuals. The individual criteria can influence ( interact ) with one another . This aggregation takes place using so-called aggregation functions , which are often suitable mean values ​​of the individual criteria, see. See also the articles Aggregate Function and Aggregation (OLAP) , which were written from the point of view of data aggregation .

Simple example

The following table shows in the first three columns the performance (in points) of 4 students in the subjects mathematics, physics and German. The numerical values ​​in the other columns are explained in the course of the article.

Surname	mathematics	physics	German	AT THE	GM	OWA (1)	OWA (2)	CI
Peter	19th	16	10	15th	16.0	13.6	17.2	13.40
Paul	13	15th	17th	15th	14.6	14.4	16.0	15.07
Petra	19th	18th	8th	15th	16.4	13.2	17.6	12.33
Paula	8th	19th	18th	15th	14.4	13.2	17.6	15.33

Aggregation functions

Let there be a set of criteria with the characteristics that are to be combined into a global value . ${\displaystyle U=\{1,\dots ,n\}}$ ${\displaystyle n}$ ${\displaystyle x_{1},\dots ,x_{n}}$ ${\displaystyle y}$

Arithmetic mean (AM)

This is one of the simplest aggregation functions:

${\displaystyle y=1/n\sum _{i=1}^{n}x_{i}}$.

AM completely dispenses with the setting of priorities and interaction among the criteria; in the example, the AM column shows no difference between the students.

Weighted Average (GM)

The different weights allow you to set priorities: ${\displaystyle p_{i}}$

${\displaystyle y=\sum _{i=1}^{n}p_{i}x_{i};\quad p_{i}\geq 0;\quad \sum _{i=1}^{n}p_{i}=1}$.

If you want to rate the natural sciences higher for certain objectives and B. selects the column GM in the table. With language preference, of course, the picture is different (not included in the table). However, an interaction between the criteria is not yet taken into account here. ${\displaystyle p_{\text{Math}}=p_{\text{Ph}}=0,4;\quad p_{\text{De}}=0,2}$

Ordered Weighted Average (OWA)

Ordered weighted averages (Engl. Ordered Weighted Average (OWA) ) have been considered for the first time in 1988 by Ronald Robert Yager, see also. If the criterion values ​​are ordered according to size, then OWA is defined by ${\displaystyle x_{(1)}\leq x_{(2)}\leq \dots \leq x_{(n)}}$

${\displaystyle y=\sum _{i=1}^{n}p_{i}x_{(i)};\quad p_{i}\geq 0;\quad \sum _{i=1}^{n}p_{i}=1}$.

Extreme OWA's are and that you get for or . ${\displaystyle y=\min(x_{1},\dots ,x_{n})}$ ${\displaystyle y=\max(x_{1},\dots ,x_{n})}$ ${\displaystyle p_{1}=1}$ ${\displaystyle p_{n}=1}$

If you want to reward the best possible general education in the example, you have to give the smallest criterion value a high weight, e.g. B. . Then the column OWA (1) results in the table. If, on the other hand, you are rewarded for having top values ​​in at least one subject, then you give the highest criterion value a high weight, e.g. B. . Then the column OWA (2) results in the table. At OWA the criteria interact. B. only set the smallest criterion value if all criterion values ​​are known. The disadvantage of the previous aggregation functions, however, is that they cannot take redundancies or synergies between criteria into account. In the example there is no difference between Petra and Paula. However, one could argue that a student who is good at math will almost automatically be good at physics too; H. the performances in these two subjects show a certain redundancy. ${\displaystyle p_{1}=0,5\quad p_{2}=0,3\quad p_{3}=0,2}$ ${\displaystyle p_{1}=0,1\quad p_{2}=0,3\quad p_{3}=0,6}$

Discrete Choquet Integral (CI)

The discrete CI is the most flexible aggregation function. It is defined by

${\displaystyle y=\sum _{i=1}^{n}p_{(i)}x_{(i)};\quad p_{(i)}=\mu (A_{(i)})-\mu (A_{(i+1)});\quad A_{(i)}=\{(i),(i+1),\dots ,(n)\};\quad A_{(1)}=U;\quad A_{(n+1)}=\emptyset }$.

Where is a normalized capacitance (i.e. ). is the set of criteria whose values ​​are at least as large as the i-th value in the ranking . For three criteria , the discrete CI is detailed as follows: ${\displaystyle \mu }$ ${\displaystyle \mu (U)=1}$ ${\displaystyle A_{(i)}}$ ${\displaystyle x_{(i)}}$ ${\displaystyle x_{(1)}\leq x_{(2)}\leq \dots \leq x_{(n)}}$ ${\displaystyle U=\{1,2,3\}}$

${\displaystyle y=[1-\mu ((2),(3))]x_{(1)}+[\mu ((2),(3))-\mu ((3))]x_{(2)}+\mu ((3))x_{(3)}={\begin{cases}\ [1-\mu (2,3)]x_{1}+[\mu (2,3)-\mu (3)]x_{2}+\mu (3)x_{3},&{\text{wenn}}\quad x_{1}\leq x_{2}\leq x_{3}\\\ [1-\mu (2,3)]x_{1}+[\mu (2,3)-\mu (2)]x_{3}+\mu (2)x_{2},&{\text{wenn}}\quad x_{1}\leq x_{3}\leq x_{2}\\\ [1-\mu (1,3)]x_{2}+[\mu (1,3)-\mu (3)]x_{1}+\mu (3)x_{3},&{\text{wenn}}\quad x_{2}\leq x_{1}\leq x_{3}\\\ [1-\mu (1,3)]x_{2}+[\mu (1,3)-\mu (1)]x_{3}+\mu (1)x_{1},&{\text{wenn}}\quad x_{2}\leq x_{3}\leq x_{1}\\\ [1-\mu (1,2)]x_{3}+[\mu (1,2)-\mu (2)]x_{1}+\mu (2)x_{2},&{\text{wenn}}\quad x_{3}\leq x_{1}\leq x_{2}\\\ [1-\mu (1,2)]x_{3}+[\mu (1,2)-\mu (1)]x_{2}+\mu (1)x_{1},&{\text{wenn}}\quad x_{3}\leq x_{2}\leq x_{1}\end{cases}}}$.

If, for example, with mathematics, physics and German and more ${\displaystyle U=\{M,P,D\}}$ ${\displaystyle M}$ ${\displaystyle P}$ ${\displaystyle D}$

${\displaystyle \mu (M)=\mu (P)=\mu (D)=1/3\quad }$ (all subjects are equivalent)

${\displaystyle \mu (M,P)=0,4<2/3\quad }$ (Redundancy between math and physics)

${\displaystyle \mu (M,D)=\mu (P,D)=0,7>2/3\quad }$ (slight synergy),

then for example for Peter ( ) ${\displaystyle x_{D}<x_{P}<x_{M}}$

${\displaystyle y=[1-0,4]\cdot 10+[0,4-0,33]\cdot 16+0,33\cdot 19=13,4}$.

The other results can be found in the CI column of the table. For more complicated examples, especially practical applications, software is required.

Web links

• M. Grabisch: An introduction to the Choquet integral

Individual evidence

1. ↑ ^{a b} Grabisch, M., Marichal, J.-L., Mesiar, R. and E. Pap (2009): Aggregation Functions . Cambridge University Press
2. ↑ Yager, RR (1988): On ordered weighted averaging aggregation operators in multi-criteria decision making , IEEE Transactions on Systems, Man and Cybernetics 18, 183-190
3. ^ Yager, RR and J. Kacprzyk (1997): The Ordered Weighted Averaging Operators: Theory and Application , Kluwer
4. ↑ Simon James (2016): An Introduction to Data Analysis using Aggregation Functions in R , Springer