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### Question 1: Plants A, B, and C produce products I, II, and III. The rate of production (units per hour) of each product in the different plants is given in the table below: PRODUCT PLANT A B C I 4 6 5

Question Preview:

Question 1:
Plants A, B, and C produce products I, II, and III. The rate of production (units per
hour) of each product in the different plants is given in the table below:
PRODUCT
PLANT
A B C
I 4 6 5
II 3 4 3
III 2 3 2
The three plants may purchase a limited number of production hours at different
prices according to the following table:
PLANT MAXIMUM NUMBER OF HOURS PRICE PER HOUR
A 250 \$ 10
B 300 \$ 8
C 200 \$ 12
The demand and the unit sale price for the 3 products are given in the fol...

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Question Preview:

Question 1:
Plants A, B, and C produce products I, II, and III. The rate of production (units per
hour) of each product in the different plants is given in the table below:
PRODUCT
PLANT
A B C
I 4 6 5
II 3 4 3
III 2 3 2
The three plants may purchase a limited number of production hours at different
prices according to the following table:
PLANT MAXIMUM NUMBER OF HOURS PRICE PER HOUR
A 250 \$ 10
B 300 \$ 8
C 200 \$ 12
The demand and the unit sale price for the 3 products are given in the following table:
PRODUCT DEMAND PRICE PER UNIT
I 500 units \$ 50
II 800 units \$ 40
III 700 units \$ 30
The demand for each product must be met. Any additional unit produced is sold at the
same price.
The goal is to maximize the total profit.
1. Formulate as an LP model.
2. Implement your model in Python and solve it using Gurobi Solver. What is the
optimal production policy and what is the profit under this policy? (Provide a
[20 Marks for Question 1; 10 for a correct formulation and 10 for the Python
model]
Question 2:
You have solved a minimization problem with 3 variables and 2 constraints and have
printed out the following sensitivity report:
Microsoft Excel Sensitivity Report
Final Reduced Objective Allowable Allowable
Cell Name Value Cost Coefficient Increase Decrease
\$C\$4 Value: X1 0 3 5 1E+30 3
\$D\$4 Value: X2 0 1 3 1E+30 1
\$E\$4 Value: X3 1 0 4 2 4
Constraints
Cell Name Value Price R.H. Side Increase Decrease
\$F\$8 Constraint1: 2 2 2 1E+30 1
\$F\$9 Constraint2: 2 0 1 1 1E+30
a. Does the solution have multiple optimal solutions? Provide a short
explanation.
b. What is the optimal objective function value if the RHS value of the first
constraint increases to 7? Provide a short explanation.
c. What is the optimal objective function value if the RHS value of the first
constraint decreases to 1? Provide a short explanation.
d. What is the optimal objective function value if the RHS value of the second
constraint increases to 3? Provide a short explanation.
e. Will the current solution remain optimal if the objective function coefficients
for X1 and X3 both decrease by 1? Provide a short explanation.
[25 Marks for Question 2; 5 for each section]
Question 3:
A company produces and sells three products (A, B and C). the operations manager is
tasked with planning the level of production for the next 4 months. It is assumed that
any number of products produced can be sold, generating the following per unit profit
(differences are due to seasonality of the products):
PRODUCT A PRODUCT B PRODUCT C
Month 1 \$ 140 \$ 155 \$ 210
Month 2 \$ 140 \$ 140 \$ 220
Month 3 \$ 150 \$ 150 \$ 230
Month 4 \$ 150 \$ 155 \$ 230
Two resources are required to produce each product, labour hours and raw materials.
The following table provide the number of units of each resource required to produce
each product.
PRODUCT A PRODUCT B PRODUCT C
LABOUR (HOURS) 2 3 4
RAW MATERIAL (KG) 4 3 5
The company has already placed future orders for raw material for the next 4 months.
These orders cannot be updated. You can assume that any amount of raw material that
is not used in a given month can be carried to and used in the next month(s) without
incurring any holding cost. For example, if you receive 100 kilograms of raw material
in Month 1 and choose to only use 90 kilograms for production, 10 kilograms are
The company has planned its workforce and has a certain number of available labour
hours for each of the next 4 months. The differences between months are due to
planned annual leave of its employees. In order to increase productivity, a survey
among workers has been undertaken and the company has determined the number of
overtime it will be able to use each month. Clearly, the company can decide not to
have the employees work over time even if they are willing to. Furthermore, unused
labour cannot be used in the future (for example, if you have 100 hours of labour in
Month 1, but only choose to use 90 you cannot “store” the 10 unused hours for future
use.) Due to the fact that the company’s employees earn different salaries, the hourly
cost of overtime will vary over time.
The relevant information is provided in the table below:
The goal of the company is clearly to maximize the total 4-month profit.
1. Formulate as an LP model.
2. Implement your model in Excel and solve it using Solver. Provide a
3. Provide a complete policy suggestion based on your results. Include any
information that may be relevant to your CEO.
[55 Marks for Question 3; 35 for the correct LP formulation; 10 for an
appropriate Excel Model and solution; 10 for Section 3]
RAW
MATERIAL
ORDERED
(KG)
AMOUNT OF
REGULAR
LABOUR
AVAILABLE
(HOURS
AMOUNT OF
OVERTIME
AVAILABLE
(HOURS)
OVERTIME
COST PER
HOUR
Month 1 120 100 30 \$ 25.0
Month 2 140 80 20 \$ 25.0
Month 3 150 90 30 \$ 28.0
Month 4 110 100 10 \$ 30.0

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### Solution preview

QUESTION 1
1. Formulate as an LP model.
Total products produced per hour;
X1: 4+6+5=15 units/hr.
X2: 3+4+3=10 units/hr.
X3: 2+3+2=7 units/hr.
Profit;
(15*50)-\$10= \$740
(10*40)-\$8= \$392
(7*30)-\$12= \$198
Hence Objective function

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