Problem: Optimally distribute a fixed amount of work among a set of workers of different performance Consider there is amount of work $W$ to be done and set of $N$ workers who has different performance: $p=\{p_1,p_2,\dots,p_N\}$. Consider that the work is divisable and can be distributed among the workers: $W=w_1+w_2+\dots+w_N$. Let $w=\{w_1,w_2,\dots,w_N\}$ be the distribution of work. 
How can one choose the optimal work distribution $w_{opt}$ so that the work is completed in least time: $max_i (\frac{w_{{opt}_i}}{p_i}) \rightarrow min$ ?
 A: This is a minimax optimization problem: 
$$
\min_{p_i} \max_{i}\frac{w_i}{p_i}
$$
subject to 
$$
\sum_{i=1}^{n}{w_i}=W.
$$
First consider the maximum between $\frac{w_i}{p_i}$ and $\frac{w_j}{p_j}$ and suppose that the sum of $w_i$ and $w_j$ is fixed. Minimum of the maximum of  $\frac{w_i}{p_i}$ and $\frac{w_j}{p_j}$ is obtained for $ \frac{w_i}{p_i}=\frac{w_j}{p_j}$. To see this, suppose that the minimum of maximum is obtained for  $ \frac{w_i}{p_i} > \frac{w_j}{p_j}$. Then you can always decrease  $ \frac{w_i}{p_i}$ by $\delta>0$ and yet keeping it as maximum while the minimum of maximum is decreased. 
Now suppose that the minimum of maximum is obtained by choosing $(w_1,...,w_n)$. If for $w_i$ and $w_j$, $\frac{w_i}{p_i}$ is not equal to $ \frac{w_j}{p_j}$. Suppose w.l.g. $\frac{w_i}{p_i} > \frac{w_j}{p_j}$. Then using same procedure as before, you can find a new sequence $(w_1,..,w_i-\delta,...,w_j+\delta,w_n)$ such that the minimum of maximum decreases or does not change. 
Therefore the minimum of maximum is obtained for $\frac{w_i}{p_i} = \frac{w_j}{p_j}$ for all $i,j$ which gives the following solution:
$$
\displaystyle w_i=W\frac{p_i}{\displaystyle\sum_{j=1}^n p_j}
$$
