[ Online version - http://nearfar.org/x/regime/ ]
1. There are N items whose ordering is given by the ordinal weights
vector(OWV). For example, if there are 4 items, the OWV
[1 4 2 3] gives their ordering.
2. A regime vector (RV) is an ordered set of N numbers from the set {-1, 0,
1}. For example, [+1 -1 -1 +1] is a RV of size 4.
3. Each OWV can have multiple cardinal weights vector (CWV) corresponding to
the same ordering. For example, if OWV is [1 4 2 3] then [0.40 0.35 0.20
0.05], [0.45 0.30 0.15 0.10], etc.. can be the corresponding CWV. It is to
be noted that the sum of numbers in CWV must be equal to 1. Formally, CWV
(cardinal weights vector) is a set of N numbers when they add up to 1 and
they are ordered in the ordering specified in OWV.
4. The 'score' of a RV for the given OWV is equal to the probability that *RV
x CWV > 0*. For example: OWV = [1 4 2 3] and RV = [+1 -1 -1 +1]. If CWV = [
0.40 0.35 0.20 0.05], then RV x CWV > 0. But if CWV = [0.45 0.30 0.15 0.10],
then RV x CWV < 0. For all possible values of CWV (for the given OWV), the
probability that RV x CWV > 0 gives the 'score' of RV.
Write a computer program that takes the OWV and a RV and outputs the score
of that RV.
Note
This challenge is part of the Regime algorithm and involves Linear
multiobjective programming. If you have questions on clarification of the
above description, send me an email <[EMAIL PROTECTED]>. If
you need to the study bigger context, I can send you the paper that
describes the regime algorithm in brief.
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