# Formulation of assignment problem as Linear Optimization

$$\begin{array}{cccccc} \hline & 1& 2&3&4&5 \\ \hline \text{Tumor} & 0.4 & 0.3 & 0.25 & 0.7 & 0.5 \\ \text{Organ 1} & 0.1 & 0 & 0 & 0.1 & 0.2 \\ \text{Organ 2} & 0.1 & 0 & 0.15 & 0 & 0.1 \\ \text{Organ 3} & 0 & 0.1 & 0 & 0 & 0 \\ \text{Organ 4} & 0 & 0.2 & 0.1 & 0.1 & 0 \\ \hline \end{array}$$

The intensity of radiation which can be used in the treatment is limited to 60 Gray and there are upper bounds on the quantity of radiations coming from each position as follows

$$\begin{array}{cc} \hline \text{Position} & \text{Upper limit} \\ \hline 1 & 12 \\ 2 & 13\\ 3 & 10\\ 4 & 15\\ 5 & 15\\ \hline \end{array}$$

You want to maximize the quantity of radiations affecting the tumor but also respect the following tolerance levels for each adjacent organ

$$\begin{array}{cc} \hline \text{Organ #} & \text{Max Gray per organ} \\ \hline 1 & 5.5 \\ 2 & 9.0\\ 3 & 6.0\\ 4 & 2.4\\ \hline \end{array}$$

How to formulate the above problem as a linear optimization problem to find the quantity or radiation which has to be supplied from each position in order to maximize the quantity received by the target tumor, but respecting the limit of radiation per organ?

Thanks.

It is not an assignment problem.

You get to decide the intensity of each position. Suppose $$y_i$$ is the intensity from position $$i$$ that you decide to apply, then the quantity of treatment of tumor that we want to maximize is

$$\max_y0.4y_1+0.3y_2+0.25y_3+0.7y_4+0.5y_5$$

Also you need to limit the total intensity, hence $$\sum_i y_i \le 60$$

Exercises left for you:

• Also you have to limit the intensity from each angle. I will leave this task to you. For the first treatment. $$y_1 \le 12$$.

• Also, note that we have to consider the tolerance level of each organ. I will also leave this task for you. For the first organ, the constraint is $$0.1y_1+0.1y_4+0.2y_5 \le 5.5.$$

Don't forget the nonnegative constraint too.