sec-fixed-total-validity.tex

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\begin{document}
\label{sec:fixed-total-validity}
Fixed Total Validity (FTV) is a procedure to score pilots on their best task
performances, rather than all their tasks. Fixed Total Validity means the sum
(total) of available points (validity) is set (fixed) to the same value for
each competitor.
\begin{align*}
    FTVfactor &= 0.2 \lor 0.25 \\
    CalculatedFTV &= (1 - FTVfactor) * \sum_{t : Task}\frac{AvailablePoints_t}{1000}
\end{align*}
To calculate a pilot’s FTV score, for all his flights:
\begin{enumerate}
    \item
        Calculate a performance percentage for each day by dividing the pilot's
        day score by the day’s available points
    \item
        Arrange all flights in descending order of performance percentage
    \item
        Total up the flights' raw day scores (not performance percentages) in
        order of performance percentage until the sum of validities for those
        scores reaches the pre-decided Fixed Total Validity value.
\end{enumerate}
If the last score added takes that pilot's total validity above the Fixed Total Validity, then only a fraction
of that score is used so that the pilot's total validity is equal to the Fixed Total Validity.
\begin{align*}
    \forall t : t \in ScoredTasks : Performance_{p, t} &= \frac{Score_{p, t}}{AvailablePoints_t} \\
    \\
    SortedPerformance_p &= sortDescending(\forall t : t \in ScoredTasks : Performance_{p, t}) \\
    \\
    OrderedValidities_p &= orderByPerformance(SortedPerformance_p, xs) \\
    xs &= \forall t : t \in ScoredTasks : \frac{AvailablePoints_t}{1000} \\
    \\
    OrderedScores_p &= orderByPerformance(SortedPerformance_p, ys) \\
    ys &= \forall t : t \in ScoredTasks : Score_{p, t} \\
    \\
    FTVScore_p &= \sum_{u = 0}^{numberOfTasks} OrderedScores_{p, u} * \max(1, \frac{\min(0, a)}{OrderValidites_{p, u}}) \\
    a &= CalculatedFTV - (u > 0 : \sum_{v = 0}^{\min(0, u - 1)} OrderedValidites_{p, v})
\end{align*}
\end{document}