UNIVERSITY OF TORONTO
Rotman School of Management
RSM 270
Students found committing any dishonest practices will be immediately
dismissed from the examination and subject to such discipline as the Faculty or
the University may impose. Examples of dishonest practices include (but are
not limited to):
• Using books or other media except those authorized by the examiners,
• Speaking or communicating with other students,
• Purposely displaying notes or memoranda to other students
Question 1 (14 points) [15 minutes]
Consider the following linear programming problem:
Max 3A + 3B
Subject to:
2A + 4B ≤ 12
6A + 4B ≤ 24
2A +3B ≤ 12
A,B ≥ 0
a) (4 points) Plot the feasible region
b) (3 points) Is there any redundant constraints? Give your reason.
c) (4 points) Find the optimal solution and value
d) (4 points) If the objective function is changed to 2A + 6B, what will the optimal solution be?
Solution.
a)
RSM 270H1S – L0101 Final Exam Page 1 of 12
, B
6
5
4
Optimal Solution
A = 3, B = 1.5
3 Value of Objective Function = 13.5
(3,1.5)
2
1
(0,0) A
1 2 3 4 5 6
(4,0)
b) 2A +3B ≤ 12 is redundant
c) Draw objective function line , Final Optimal value and the optimal solution.
d)
B Optimal Solution
A = 0, B = 3
3 Value of Objective Function = 18
2
1
(0,0) A
1 2 3 4 5 6 7 8 9 10
RSM 270H1S – L0101 Final Exam Page 2 of 12
, Question 3 (14 points) [20 minutes]
The management of Hart-House Company is trying to determine the amount of each of two
products to produce over the coming planning period. The following information concerns labor
availability, labor utilization, and product profitability:
Product (hours/unit) Labor-Hours
Department 1 2 Available
A 1.00 0.35 100
B 0.30 0.20 36
C 0.20 0.50 50
Profit contribution/unit $30.00 $15.00
a) (6 points) Develop a linear programming model of the Hart-House Company problem.
b) (8 points) Suppose that a maximum of 10, 6, and 8 hours of overtime may be scheduled in
departments A, B, and C, respectively. The cost per hour of overtime is $18 in department
A, $22.50 in department B, and $12 in department C. Formulate a linear programming
model that can be used to determine the optimal production quantities if overtime is made
available.
Hint: You do not need to find an optimum solution (neither in part (a) nor in part (b))!
Solution a)
a. Let x1 = units of product 1 produced
x2 = units of product 2 produced
Max 30x1 + 15x2
s.t.
x1 + 0.35x2 ≤ 100 Dept. A
0.30x1 + 0.20x2 ≤ 36 Dept. B
0.20x1 + 0.50x2 ≤ 50 Dept. C
RSM 270H1S – L0101 Final Exam Page 3 of 12
Rotman School of Management
RSM 270
Students found committing any dishonest practices will be immediately
dismissed from the examination and subject to such discipline as the Faculty or
the University may impose. Examples of dishonest practices include (but are
not limited to):
• Using books or other media except those authorized by the examiners,
• Speaking or communicating with other students,
• Purposely displaying notes or memoranda to other students
Question 1 (14 points) [15 minutes]
Consider the following linear programming problem:
Max 3A + 3B
Subject to:
2A + 4B ≤ 12
6A + 4B ≤ 24
2A +3B ≤ 12
A,B ≥ 0
a) (4 points) Plot the feasible region
b) (3 points) Is there any redundant constraints? Give your reason.
c) (4 points) Find the optimal solution and value
d) (4 points) If the objective function is changed to 2A + 6B, what will the optimal solution be?
Solution.
a)
RSM 270H1S – L0101 Final Exam Page 1 of 12
, B
6
5
4
Optimal Solution
A = 3, B = 1.5
3 Value of Objective Function = 13.5
(3,1.5)
2
1
(0,0) A
1 2 3 4 5 6
(4,0)
b) 2A +3B ≤ 12 is redundant
c) Draw objective function line , Final Optimal value and the optimal solution.
d)
B Optimal Solution
A = 0, B = 3
3 Value of Objective Function = 18
2
1
(0,0) A
1 2 3 4 5 6 7 8 9 10
RSM 270H1S – L0101 Final Exam Page 2 of 12
, Question 3 (14 points) [20 minutes]
The management of Hart-House Company is trying to determine the amount of each of two
products to produce over the coming planning period. The following information concerns labor
availability, labor utilization, and product profitability:
Product (hours/unit) Labor-Hours
Department 1 2 Available
A 1.00 0.35 100
B 0.30 0.20 36
C 0.20 0.50 50
Profit contribution/unit $30.00 $15.00
a) (6 points) Develop a linear programming model of the Hart-House Company problem.
b) (8 points) Suppose that a maximum of 10, 6, and 8 hours of overtime may be scheduled in
departments A, B, and C, respectively. The cost per hour of overtime is $18 in department
A, $22.50 in department B, and $12 in department C. Formulate a linear programming
model that can be used to determine the optimal production quantities if overtime is made
available.
Hint: You do not need to find an optimum solution (neither in part (a) nor in part (b))!
Solution a)
a. Let x1 = units of product 1 produced
x2 = units of product 2 produced
Max 30x1 + 15x2
s.t.
x1 + 0.35x2 ≤ 100 Dept. A
0.30x1 + 0.20x2 ≤ 36 Dept. B
0.20x1 + 0.50x2 ≤ 50 Dept. C
RSM 270H1S – L0101 Final Exam Page 3 of 12
