A simple examination in a hospital shows that the hospital needs the following number of nurses at different times of the day:
\begin{array}{|c|} \hline \text{course} & \text{time} & \text{least number of needed nurses} \\ \hline 1& 6 \;\text{am}\; \text{to}\; 10\; \text{am} & 60&\\ \hline 2& 10\; \text{am}\; \text{to}\; 2\; \text{pm} & 70&\\ \hline 3&2\;\text{pm}\; \text{to}\; 6\; \text{pm} & 60&\\ \hline 4&6\; \text{pm}\; \text{to}\; 10\; \text{pm} & 50&\\ \hline 5& 10\;\text{pm}\; \text{to}\; 2\; \text{am} & 20&\\ \hline 6& 2\;\text{am}\; \text{to}\; 6\; \text{am} & 30&\\ \hline \end{array}
As soon as the nurses arrive at the hospital, they introduce themselves at the beginning of each course and start their work and can work for $8$ hours. The hospital wants to know how many nurses it needs to hire so that it has the necessary manpower for different periods of work. Formulate the problem in mathematical programming.
Define $ x_i $ as the number of nurses starting their work at the beginning of the $ i $ th course, then we are going to find the $6$-tuple $(x_1,x_2,x_3,x_4,x_5,x_6)$ to minimize $z=\sum_{i=1}^{6} x_i$ assuming the following relations hold: $$x_1 \ge 60$$ $$x_1+x_2 \ge 70$$ $$x_2+x_3 \ge 60$$ $$x_3+x_4 \ge 50$$ $$x_4+x_5 \ge 20$$ $$x_5 +x_6\ge 30$$ $$x_6 +x_1\ge 6 0$$ $$x_i \in \mathbb Z_{ \ge 0},\; i \in \{1,...,6\}$$
However my source answer removes the first constraint $x_1 \ge 60$, but I think that's necessary.(Assuming the hospital starts its work at $6$ am.)
I'm not looking for the answer itself, instead I want to know whether the inequality $x_1 \ge 60$ should be mentioned or not.
The $x_1 \ge 60$ constraint should be omitted. Every nurse covers two consecutive courses, and every course is covered by two consecutive sets of nurses. So every variable should appear twice, and every constraint should contain two variables.