Lab 1
Wyndor problem
Wyndor is a company specialized in the the production of 2 products:
• Product 1: An 8-foot glass door with aluminum framing
• Product 2: A 4 x 6 foot double-hung wood-framed window
Wyndor has 3 plants to produce these products. Product 1 requires some of the
production capacity in Plants 1 and 3, but none in Plant 2. Product 2 needs only
Plants 2 and 3. More information can be found in the table below
Solution
𝑥1= number of batches produced of product 1
𝑥2= number of batches produced of product 2
max 𝑧 = 3000𝑥1 + 5000𝑥2
s.t.
𝑥1 ≤ 4 (time available plant 1)
2𝑥2 ≤ 12 (time available plant 2)
3𝑥1 + 2𝑥2 ≤ 18 (time available plant 3)
𝑥1, 𝑥2 ≥ 0 (non-negativity constraints)
Radiation therapy problem
, Mary has been diagnosed with cancer and is to receive radiation therapy. This
therapy involves using two external beam treatment machines. Because these
beams can also cause damage to healthy parts of the body, the design of
radiation therapy is a very delicate process. The goal of the design is to select the
dosage of radiation (measured in kilorads) of the two beams that minimize the
total dosage reaching the healthy anatomy.
The following table indicates how many kilorads each part of the body absorbs
per dosage of radiation.
To prevent exposing Mary with an overdose of radiation, there are restrictions on
the total dosage from both beams. These restrictions are:
• The absorption over the critical tissues must not exceed 2.7 kilorads
• The average absorption over the entire tumor must equal 6 kilorads
• The center of the tumor must absorb at least 6 kilorads
Solution
𝑥1= dose (in kilorads) of beam 1
𝑥2= dose (in kilorads) of beam 2
m𝑖𝑛 𝑧 = 0.4𝑥1 + 0.5𝑥2
s.t.
0.3𝑥1 + 0.1𝑥2 ≤ 2.7 (critical tissues)
0.5x1 + 0.5𝑥2 = 6 (entire tumor)
0.6𝑥1 + 0.4𝑥2 ≥ 6 (center tumor)
𝑥1, 𝑥2 ≥ 0 (non-negativity constraints)
Wyndor problem
Wyndor is a company specialized in the the production of 2 products:
• Product 1: An 8-foot glass door with aluminum framing
• Product 2: A 4 x 6 foot double-hung wood-framed window
Wyndor has 3 plants to produce these products. Product 1 requires some of the
production capacity in Plants 1 and 3, but none in Plant 2. Product 2 needs only
Plants 2 and 3. More information can be found in the table below
Solution
𝑥1= number of batches produced of product 1
𝑥2= number of batches produced of product 2
max 𝑧 = 3000𝑥1 + 5000𝑥2
s.t.
𝑥1 ≤ 4 (time available plant 1)
2𝑥2 ≤ 12 (time available plant 2)
3𝑥1 + 2𝑥2 ≤ 18 (time available plant 3)
𝑥1, 𝑥2 ≥ 0 (non-negativity constraints)
Radiation therapy problem
, Mary has been diagnosed with cancer and is to receive radiation therapy. This
therapy involves using two external beam treatment machines. Because these
beams can also cause damage to healthy parts of the body, the design of
radiation therapy is a very delicate process. The goal of the design is to select the
dosage of radiation (measured in kilorads) of the two beams that minimize the
total dosage reaching the healthy anatomy.
The following table indicates how many kilorads each part of the body absorbs
per dosage of radiation.
To prevent exposing Mary with an overdose of radiation, there are restrictions on
the total dosage from both beams. These restrictions are:
• The absorption over the critical tissues must not exceed 2.7 kilorads
• The average absorption over the entire tumor must equal 6 kilorads
• The center of the tumor must absorb at least 6 kilorads
Solution
𝑥1= dose (in kilorads) of beam 1
𝑥2= dose (in kilorads) of beam 2
m𝑖𝑛 𝑧 = 0.4𝑥1 + 0.5𝑥2
s.t.
0.3𝑥1 + 0.1𝑥2 ≤ 2.7 (critical tissues)
0.5x1 + 0.5𝑥2 = 6 (entire tumor)
0.6𝑥1 + 0.4𝑥2 ≥ 6 (center tumor)
𝑥1, 𝑥2 ≥ 0 (non-negativity constraints)
