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

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

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

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

$y_{rt}={\displaystyle \underset{jin{J}_r}{\overset{?}{}}}{\displaystyle \underset{?}{\overset{?}{}}}]t$ and $[t$

Subject to:

Decision variables

$x_r=$ The number of permanent nurses of type $r$ to hire for the planning horizon.

$y_{rt}=$ The number of part$-$time nurses of type $r$ to hire from the agency in period $t.$

$o_{rt}=$ The amount of overtime for all nurses of type $r$ in period $t.$

$x_{\text{r}\text{i}\text{t}}^{\text{'}}=$ The number of permanent nurses of type $r$ that work in unit $i$ in period $t.$

$y_{\text{r}\text{i}\text{t}}^{\text{'}}=$ The number of part$-$time nurses of type $r$ that work in unit $i$ in period $t.$

$o_{\text{r}\text{i}\text{t}}^{\text{'}}=$ The amount of overtime for all nurses of type $r$ that work in unit $i$ in period $t.$

$z_{rtj}=$ The number of part$-$time nurses of type $r$ to hire from the agency in period $t$ through contract type,

Part a$)$ How would you modify the model to imply that at least $10$ part$-$time nurses of any type must be hired through contract type $3$ in period $2?(3$ points$)$

$\frac{7}{10}$

Part b$)$ How would you modify the model to imply that the number of hired permanent nurses of type $3$ cannot be more than $6$ units less than the total number of part$-$time nurses of type $3$ hired during the planning horizon? $(3$ points$)$

Part c$)$ How would you modify the model to imply that the absolute difference between the number of hired permanent nurses of type $2$ and the number of part$-$time nurses of the same type hired in period $5$ is more than $15$ units? $(3$ points$)$

Part d$)$ How would you modify the model to imply that the absolute difference between the number of hired permanent nurses of type $5$ and the number of part$-$time nurses of the same type hired in any period is less than $6$ units? $(3$ points$)$

Part e$)$ How would you modify the model to imply that, if more than $12$ permanent nurses are hired, at most $15$ part$-$time nurses $($regardless of their types$)$ must be hired in period $5?(3$ points$)$

Part f$)$ How would you modify the model to imply that, either more than $20$ permanent nurses are hired, or at most $15$ part$-$time nurses of type $2$ must be hired in period $4?(3$ points$)$

Part g$)$ How would you modify the model to imply that, the total amount of overtime of different types of nurses working in period $4$ cannot be more than $400$ hours, unless the total number of hired permanent nurses is more than $12?(3$ points$)$

Part h$)$ How would you modify the model to imply that, in at most two of the periods, more than $12$ parttime nurses $($regardless of their types$)$ can be hired through contract type $2?(3$ points$)$