Final 2019
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School
University of Toronto *
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Course
1005
Subject
Industrial Engineering
Date
Dec 6, 2023
Type
Pages
2
Uploaded by CorporalMoosePerson987
APS1005:Operations Research for Engineering Management
Instructor: Daniel Frances
August 20, 2019
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Q1:
A generator manufacturing company has signed a contract with an energy company. As to contract,
the manufacturer has to produce generators and ship them within four months. The number of generators
that have to be shipped is as follows:
Month
# of generators to be shipped
May
2500
June
3500
July
4300
August
5000
The manufacturer’s production line generally works in two shifts: shift 1 and 2. The maximum production
during the first shift is 2600 generators. and during the second shift is half of the first shift, i.e. 1300
generators. The production cost is 1600$ per generator for the first shift, and due to the extra energy
costs during second shift, this cost increases by 60%. The holding cost of each produced motor is 50$
monthly. The manufacturer wants to minimize their costs.
a:
For 20 marks, formulate the problem as an LP problem. Clearly state and explain all the components
of your model (i.e. assumptions, variables, constraints and objective function)
b:
For 10 marks, solve the LP, and report the optimal solution and objective.
c:
For 4 marks, report and interpet the shadow prices and reduced costs.
1
Q2:
A company is organizing a holiday dinner for their employees and needs to buy 60 dinners to
provide for their 60 employees. The company works with a caterer that offers a set of 5 possible dinner
menus: A,B,C,D and E. Each dinner of a given menu has a specific price
p
j
, nutrition factor
f
j
and a
label indicating whether it is vegan or not
v
j
∈ {
0
,
1
}
, with values as shown below.
Menu
Price (
p
j
)
Nutrition Factor (
f
j
)
Vegan or not (
v
j
)
A
15
3
0
B
18
5
0
C
20
3.5
1
D
17
4
0
E
19
2.5
1
The company needs a minimum of 15 vegan dinners, and the total nutrition factor has to be at least
190. The company wants to minimize the total cost of the dinners. The caterer’s policies impose the
following constraints on the company’s menu selection:
•
A minimum buying policy of 6 for menu purchase, e.g., if the company wants to buy dinners of a
given menu, they need to buy at least 6 of them.
•
Menus A and E cannot be selected together.
•
If the company orders menu D, the number of dinners ordered from menu D has to be at least
twice the number of dinners ordered from menu C.
a:
For 23 marks, model this problem as an IP problem. Clearly state and explain all the components
of your model (i.e. assumptions, variables, constraints and objective function).
b:
For 10 marks, solve the IP.
Q3:
A robot
R
is located at the bottom left corner in a 2-dimensional maze that is made up of 4-by-4
(i.e.
total of 16) grid locations as visualized by Table 1.
The robot
R
can move in 4 directions (i.e.,
up, down, left, right) within the grid, and must end up in the goal location
G
in at most 9 moves. A
block
B
represents a type of location that the robot
R
cannot move to or from. The numbers on the
grid locations represent the one-time reward
Q
that can be collected by the robot
R
by moving to the
respective location.
1
1
G
2
2
B
B
1
R
2
Table 1: Visualization of the maze for Q1.
a:
For 23 marks, formulate a DP model such that the robot
R
navigates the maze and reaches the goal
location
G
with maximum total reward
Q
.
Clearly present your assumptions, stages, states, decision
variables, transition relation, immediate return and cumulative return.
b:
For 10 marks, solve the DP.
2
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