QBUS2310 Practice Midsemester Exam
Semester 1, 2022
ˆ Please carefully read the instructions on the exam site for time limits and upload instructions.
ˆ This is an open book exam: you may look at and refer to the lecture slides and the book, but you
cannot refer to materials outside these.
ˆ You may also use Python/Gurobi to assist you with computations if needed.
Question: 1 2 3 4 Total
Points: 10 10 10 20 50
i
QBUS2310 Practice Midsemester Exam Semester 1, 2022
1. (10 points) Formulate an equivalent linear program for the following model:
max
x
min{x1, x2, x3}
s.t. max{|x1 − x2|, |x2 + x3|, |x3 − x1| −min{x2, x3}} ≤ min{x1 + x2, x3 − |x2|}
x3 ≥ 1 + x2
x1, x2, x3 ≥ 0.
2. (10 points) Formulate the dual linear program of the following model:
min
x
6x1 − 6x2 − 3x4
s.t. −5x1 + 7x2 − 7x3 − 2x4 = 1
2x1 − x2 − 4x3 + 2x4 = −4
7x1 − 5x2 + 4x3 + 4x4 = 4
−6x1 − 4x2 − 2x3 + 6x4 ≤ −3
x1 ≥ 0, x2 free, x3 ≤ 0, x4 free.
3. (10 points) Consider the following LP:
min
x
x1 + 4x2
s.t. 2x1 + 3x2 ≥ −1
9x1 + 5x2 ≤ 3
x1 ≤ 0, x2 ≤ 0.
Using complementary slackness, verify whether x =
(
0,− 13
)
is optimal or not.
4. (20 points) Top Brass Trophy Company makes large championship trophies for youth sports leagues. At
the moment, they are planning production for semester 1 sports: basketball and rugby. Each type of trophy
has a wood base and an engraved plaque. The amount of wood require, profit per trophy and total demand
for semester 1 differs by the type of trophy. These numbers are given in the following table:
basketball rugby
plaques required per trophy 1 1
wood required per trophy 40cm 20cm
profit per trophy $12 $9
demand 600 1000.
Furthermore, we know that 1450 plaques and 450m of wood are available in total. Every trophy of unmet
demand requires you to pay a penalty of $2. Unsold trophies must go into storage at a cost of $1 per trophy.
Formulate an LP to find the production plan that maximizes the amount you earn.
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