I'm facing a mathematical challenge related to selectivity freedom under overlapping conditions. To make it more relatable, let's consider an analogy.
Imagine a farmer who has just completed his apple harvest. The harvested apples can be either green or red. Additionally, some apples, irrespective of their color, may have worms. Now, the farmer wants to select a group of apples from his harvest that satisfies certain conditions related to their color (green or red) and whether or not they have worms.
Until now, I've been able to calculate the exclusion of apples considering only one characteristic at a time - color or worm-status. For instance, if the farmer has a basket comprising $50$ green apples and $50$ red apples, and he needs a minimum of $75$% green apples in his selection, the exclusion is calculated as follows:
$$\begin{align}\text{Selectivity freedom} = \text{Green Min Ratio}\cdot\text{Green Ratio in Basket} + \text{Residual Ratio}\cdot\text{Any Apple on Basket Ratio} = 0.75 * 0.5 + 0.25 * 1 = 0.625\end{align}$$
So, the exclusion is $1 - 0.625 = 0.375\text{ or }37.5$%
However, now we have a more intricate situation. The farmer's basket has $40$ green and worm-free apples, $10$ apples green nor worm-free, $20$ red and worm-free apples, and $30$ apples that are red with worm. The challenge is to determine the exclusion when the conditions overlap, like needing a minimum of $75$% for green apples and $50$% for worm-free apples.
Currently, I'm trying to address this by transforming the problem into a mathematical optimization scenario. I'm using linear programming to maximize the number of apples selected while satisfying all conditions to minimize the exclusion.
Here's a brief overview of my approach: $$x_1 =\text{ Green and worm minimum}$$ $$x_2 =\text{Green and free-worm minimum}$$ $$x_3 =\text{Red and worm minimum}$$ $$x_4 =\text{Red and free-worm minimum}$$
Green_worm_pct $= 0.1$ (for this example) Green_free_worm_pct $= 0.4$ (for this example) Red_worm_pct $= 0.3$ (for this example) Red_free_worm_pct $= 0.2$ (for this example)
Objective function: Green_worm_pct $x_1 + $ Green_free_worm_pct $x_2 +$ Red_worm_pct $x_3 + $ Red_free_worm_pct $x_4$
Constraints: $x_1 + x_2 \ge $Green_min
$x_3 + x_4 \ge$ Red_min
$x_1 + x_3 \ge $free_worm_min
$x_2 + x_4 \ge$ worm_min
$x_1 + x_2 + x_3 + x_4 = 1$
Bounds: All the variables are in $(0,1)$
The problem is then to find the $X[i]$s that maximize the sum of $X[i]$ for all apples '$i$', while satisfying the constraints. This corresponds to maximizing the number of apples selected while satisfying all conditions.
I would appreciate any thoughts, suggestions or alternative methods to address this issue. Thanks in advance!