Optimization Models: Exercise 1
Name: A0
August 15, 2019
•Read carefully every statement and answer only what is asked. Show your work.
1. Write an optimization model that would solve the following problems. Define clearly all the variables
and parameters used. Find the optimal solution using the Graphic Method
(a) Radiation therapy involves using an external beam treatment machine to pass ionizing radiation
through the patients body, damaging both cancerous and healthy tissues: the radiation dosage
throughout the tumor region must be large enough to kill the malignant cells, which are slightly
more radiosensitive, yet small enough to spare the healthy cells. A typical therapy involves using
two beams from different directions. The impact of the beams on the different types of tissue is
shown in the table below.
Fraction absorbed
Area Beam 1 Beam 2 Goal (Kilorads)
Healthy anatomy 0.4 0.5 as little as possible
Critical tissues 0.3 0.1 no more than 2.7
Tumor region 0.5 0.5 exactly 6
Center of tumor 0.6 0.4 at least 6
At what dosage do the combination of these 2 beams accomplish all the goals?
(b) Sunco processes oil into aviation fuel and heating oil. It costs $40 to purchase each 1,000 barrels
of oil, which is then distilled and yields 500 barrels of aviation fuel and 500 barrels of heating
oil. Output from the distillation may be sold directly or processed in the catalytic cracker. If
sold after distillation without further processing, aviation fuel sells for $60 per 1,000 barrels, and
heating oil sells for $40 per 1,000 barrels. It takes 1 hour to process 1,000 barrels of aviation
fuel in the catalytic cracker, and these 1,000 barrels can be sold for $130. It takes 45 minutes
to process 1,000 barrels of heating oil in the cracker, and these 1,000 barrels can be sold for
$90. Each day, at most 20,000 barrels of oil can be purchased, and 8 hours of cracker time are
available. Formulate an LP to maximize Suncos profits.
