33 In treating a brain tumor with radiation, physicians want the maximum amount of radiation possible to bombard the tissue containing the tumors. The constraint is, however, that there is a maximum amount of radiation that normal tissue can handle without suffering tissue damage. Physicians must therefore decide how to aim the radiation so as to maximize the radiation that hits the tumor tissue subject to the constraint of not damaging the normal tissue. As a simple example of this situation, suppose six types of radiation beams (beams differ in where they are aimed and their intensity) can be aimed at a tumor. The region containing the tumor has been divided into six regions:

three regions contain tumors and three contain normal tissue. The amount of radiation delivered to each region by each type of beam is shown in Table 117.

If each region of normal tissue can handle at most 40 units of radiation, then which beams should be used to maximize the total amount of radiation received by the tumors?

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TABLE 117 Normal Tumor 1 2 3 1 2 3 Beam 12 16 12 20 12 8 10 6 9 8 13 4 12 12 9 4 11 2363 20 12 6 1 18 15 8 2 13 10 17 3 18 13 5 8 7 7 10 10 10 640 16 4 14 5 10 6