Problem $4.(10$ pts$)$ Radiation Therapy & Elastic Goals

Consider the radiation therapy problem where a doctor wants to minimize the total dosage of

radiation delivered to the cells $($where cells are represented by pixel $($i$,$ j$)).$ The constraints are

such that the dosage over the tumor area will be at least a target level $(R_L)$ and that the dosage

over the critical area $($normal cells$)$ will be at most a target level $(R_H).$

$($a$)$ Complete the following LP$.$

Let $D^{P_{i,j}}=$ unit dose delivered to pixel $(i,j)$ by beamlet $p,$

$R_L=$ the lower bound of dosage exposed to tumor cells,

and $R_U=$ the upper bound of the dosage exposed to normal cells.

The decision variables are $w=(w_1,$dots,$W_n),$ where $w_p$ is the intensity weight assigned to beamlet

$p$ for $p=1$ to $n$;

$($b$)$ It is far better to set the parameters $(R_L$ and $\{$:$R_H)$ conservatively while allowing some

violation. If there is no feasible solution to the original problem, there will definitely be a

feasible solution to the following nonlinear program since one can choose $Y_{ij}$ to guarantee

feasibility. In the following NLP model, the objective function would try to set all of the $y^{\text{'}}$ s to

$0.$ Failing that, it would set all of the $y^{\text{'}}$ s as small as possible.

Complete the following NLP model.

${?}_{\text{M}\text{i}\text{n}\text{i}\text{m}\text{i}\text{z}\text{e}}{\displaystyle \underset{(i)}{\overset{?}{}}},j(Y_{ij}{)}^2$

$D_{i,j}$

$R_L$ for $\frac{i,j}{i}nT(tumo\frac{r}{c}$ancer cells$)$

$R_H$ for $(i,j)inC(critica\frac{l}{n}$ormal cells$)$

$W_{p,},Y_{ij}0$ for all $p,$ for all $(i,j)$