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In the anti-tumor treatment with radiotherapy, it is possible to irradiate the tumoral mass from different positions with different intensities. For each of these possibilities, however, one has to take into account the harmful side effects that the treatment causes to the adjacent organs. Suppose you have a discrete set of positions from which the tumor can be irradiated and want to decide with what level of intensity to irradiate the tumor from each position. You must consider a number of adjacent organs to preserve and for every possible position you have the data expressing the percentage of radiation that would affect the tumor and the percentage of radiation that would strike each of the adjacent organs. In fact, the following table shows the percentage of radiation going from each position (on the columns) to the tumor and to the adjacent organs (on the rows).

\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.

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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.

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