(Direct links to the operators: AM, WM, OWA, WOWA)
AM(a_{1},...,a_{N}) = Σ_{i=1}^{N} a_{i} / N
WM(a_{1},...,a_{N}) = Σ_{i=1}^{N} p_{i} a_{i}
In this definition, p_{i} corresponds to the weight or relevance of the i-th information source. Weights should be positive and add to one.OWA(a_{1},...,a_{N}) = Σ_{i=1}^{N} w_{i} a_{σ(i)}
where σ corresponds to a permutation of the a_{i} so that they are ordered from the largest one to the lowest one. So, a_{1} will be the largest of the a_{i} and a_{N} will be the lowest of the a_{i}.In this definition, p_{i} correspond to the weight of the ith data after ordering them. In this way, weights permits to express us whether we want to give importance to low, high or central data. Weights should be positive and add to one.Yager, R.R., "On Ordered Weighted Averaging Aggregation Operators in Multi-Criteria Decision Making," IEEE Transactions on Systems, Man and Cybernetics 18, 183-190, 1988.
WOWA(a_{1},...,a_{N}) = Σ_{i=1}^{N} ω_{i} a_{σ(i)}
where σ corresponds to a permutation of the a_{i} so that they are ordered from the largest one to the lowest one,and where &omega is defined by:ω_{i} = w^{*}(Σ_{j≤i} p_{σ(j)}) - w^{*}(Σ_{j<i} p_{σ(j)})
where, as in the case of the OWA operator, w_{i} correspond to the weight of the ith data after ordering them (w is a weighting vector as in the weighted mean), and where w^{*} is a function that interpolates the points (i/n, Σ_{j≤i}p_{j}) and the point (0,0), and that should be a straight line if the points can be interpolated in this way.V. Torra, The Weighted OWA operator, Int. J. of Intel. Systems, 12 (1997) 153-166. Paper @ interscience.wiley
The interpolation method used to build w^{*} is described in:V. Torra, The WOWA operator and the interpolation function W*: Chen and Otto's interpolation method revisited, Fuzzy Sets and Systems, 113:3 (2000) 389-396. Paper @ Sciencedirect
We present a discussion of different alternative interpolation methods in:V. Torra, Z. Lv, On the WOWA operator and its interpolation function, Int. J. of Intelligent Systems, 24:10 (2009) 1039-1056. Paper @ Wiley online
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