Part a$)$ How would you modify the model to imply that at most $10$ part$-$time nurses of type $1$ can be hired through contract type $4$ in period $1?(3$ points$)$

Part b$)$ How would you modify the model to imply that, in each period, at most $15$ part$-$time nurses $($regardless of their types$)$ can be hired through contract type $3?(3$ points$)$

Part c$)$ How would you modify the model to imply that the number of hired permanent nurses of type $1$ cannot be $10$ units more than the total number of part$-$time nurses of type $1$ hired during the planning horizon? $(3$ points$)$

Part d$)$ How would you modify the model to imply that the absolute difference between the number of hired permanent nurses of type $4$ and the number of part$-$time nurses of the same type hired in period $6$ is less than $7$ units? $(3$ points$)$

Part e$)$ How would you modify the model to imply that the absolute difference between the number of hired permanent nurses of type $3$ and the number of part$-$time nurses of the same type hired in period $7$ is more than $10$ units? $(3$ points$)$

Part f$)$ How would you modify the model to imply that, if less than $10$ permanent nurses are hired, at least $10$ part$-$time nurses $($regardless of their types$)$ must be hired in period $1?(3$ points$)$

Part g$)$ How would you modify the model to imply that, either more than $19$ permanent nurses are hired, or least $16$ part$-$time nurses of type $3$ must be hired in period $5?(3$ points$)$

Part h$)$ How would you modify the model to imply that, the total amount of overtime of different type of nurses working in period $1$ cannot be more than $400$ hours? $(3$ points$)$

In Lecture $2,$ the following model was presented to formulate a nurse budgeting problem with different features including different units, different time periods, and different types of contracts for hiring part$-$time nurses:

min${\displaystyle \underset{\text{r}\text{i}\text{n}\text{R}}{\overset{?}{}}}{c}_r{x}_r+{\displaystyle \underset{\text{r}\text{i}\text{n}\text{R}}{\overset{?}{}}}{\displaystyle \underset{\text{t}\text{i}\text{n}\text{T}}{\overset{?}{}}}{c}_{rt}^{\text{'}}{o}_{rt}+{\displaystyle \underset{\text{r}\text{i}\text{n}\text{R}}{\overset{?}{}}}{\displaystyle \underset{\text{t}\text{i}\text{n}\text{T}}{\overset{?}{}}}{\displaystyle \underset{jin{J}_r}{\overset{?}{}}}{c}_{rtj}^{\text{'}\text{'}}{z}_{rtj}$

Subject $to$:

$x_r={\displaystyle \underset{iin{I}_r}{\overset{?}{}}}{x}_{\text{r}\text{i}\text{t}}^{\text{'}},$rinR,tinT

$o_{rt}={\displaystyle \underset{iin{I}_r}{\overset{?}{}}}{o}_{\text{r}\text{i}\text{t}}^{\text{'}},$rinR,tinT

$y_{rt}={\displaystyle \underset{iin{I}_r}{\overset{?}{}}}{y}_{\text{r}\text{i}\text{t}}^{\text{'}},$rinR,tinT

$y_{rt}={\displaystyle \underset{jin{J}_r}{\overset{?}{}}}{\displaystyle \underset{?}{\overset{?}{}}}$