The Metalco Company desires to blend a new alloy of $40\%$ tin, $35\%zmc,al$ seteral available alloys $($labeled $1,2,$dots,$5)$ having the composition listed in the table below.

$\backslash$table$[[$Composition$,$Alloy$],[1,2,3,4,5],[$Percentage of zinc,$10,25,45,20,50],[$Percentage of lead,$30,60,10,30,10],[$Cost $(\frac{\text{\$}}{l}b),22,20,25,24,27]]$

The problem is to determine the proportions of these alloys that should be blended to produce the new alloy at a minimum cost. Formulate an LP for this problem. $($Formulation only.$)$

$3.$ Consider the following variation of round$-$the$-$clock worker scheduling problem discussed in class. Suppose some workers may work in shifts that consist of a single $4hr$ time$-$slot, while others may work $8hr$ shifts consisting of two contiguous time$-$slots. Formulate this variation as a linear optimization model. $($Formulation only.$)$

Hint: Consider variables $x_i$ for $8hr$ shifts and $y_i$ for $4hr$ shifts.