linear programming
Let $x_i$ be the number of workers to be hired that starts its shift on day $i$. So we want to minimize the function $f( {\bf } ) = \sum_{i=1}^7 x_i$. The constraints:
$$ x_1 \leq 17 $$
$$ x_1 + x_2 \leq 13 $$
$$ x_1 + x_2 + x_3 \leq 15 $$
$$ x_1+x_2+x_3+x_4 \leq 19 $$
$$x_1 + x_2 + x_3 + x_4 + x_5 \leq 14 $$
$$ x_2 +x_3 + x_4 + x_5 + x_6 \leq 16 \; \; \; \text{day 1 needs 2 days off} $$
$$ x_3 + x_4 + x_5 + x_6 + x_7 \leq 11 $$
$$ x_i \geq 0 $$
Is this a correct formulation?
Do it like this: introduce a new binary decision variable $z \in \{0,1\}$ and write the objective as: $$ \min -c_1x_1 - c_2z$$ subject to: $$ z\ge x_2$$ $$ z \ge x_3 $$ $$ z \ge x_4 $$
plus all remaining original constraints.
In short, we are simply modeling $ z = \max\{x_2,x_3,x_4\}$ and changing the problem type to min-max.
The requirement space is the set of all nonnegative vectors ${\bf x} = (x_1,x_2,x_3,x_4,x_5,x_6)$ that satisfy the linear combination $\sum_{i=1}^6 {\bf a}_i x_i$ where ${\bf a}_i$ is the ith column of matrix $A$. Graphically, you can depict a combination of vectors as a cone. For example, the cone generated by $a_1$ and $a_2$ is:
Since the cone contains $b$, this combination of vectors has a feasible $x$ ($x_1=8/3$, $x_2=2/3$). The other combinations of vectors that contain $b$ are $(a_1,a_4)$, $(a_1,a_5)$, $(a_1,a_6)$, $(a_2,a_3)$, $(a_3,a_4)$ and $(a_3,a_6)$. If you find the $x$ for each of these combinations and compute the objective value for each of those $x$, the one with the largest objective value is optimal.
However, the question is flawed. There is no optimal solution, since $x=[0; 0; c; 0]$ is feasible to the original problem for any $c\geq 6$. The objective value is $2c$, which can grow arbitrarily large.
Best Answer
Not really.
Guide:
The nurse who are present on day $1$ are the ones who start from day $4,1,5,6,7$. Hence,
$$x_1 + x_4 + x_5 + x_6 + x_7 \ge 17.$$
Use the same strategies on various days.
Also, we have to impose nonnegative integer constraints and remember to write down the objective function.