BANA201B_Homework_2_Group_B2
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149101
Subject
Industrial Engineering
Date
Apr 3, 2024
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7
Uploaded by SuperHumanLoris4024
BANA201B
Homework 2
Group members: Group B2
Abby Chung
Ambuj Upadhyay
Aysan Habibzadeh
Ping-Yen Chung
Siddharth Yadav
Question 1
Help Matt formulate the problem of scheduling games during the four-day period while
maximizing revenue. Explicitly write down your decision variables, the objective function,
and constraints. What kind of optimization problem have you defined?
Solution 1
To help Matt formulate the problem of scheduling NBA games over the specified four-day period
to maximize revenue, we'll define the decision variables, the objective function, and the
constraints, detailing the type of optimization problem we have defined.
Decision Variables:
Let x
ij
be a binary decision variable where:
- i represents the game (1 to 20),
- j represents the day (1 for Thursday, 2 for Friday, 3 for Saturday, 4 for Sunday),
- x
ij
if game i is scheduled on day j, and 0 otherwise.
Objective Function:
Maximize total revenue from TV commercials, which is the sum across all games and days of
the product of the game's popularity rating, the revenue multiplier for the day, and whether the
game is scheduled on that day x
ij
Where revenue
ij
is the revenue generated by scheduling game i on day j, calculated by
multiplying the game's popularity rating by the day's revenue multiplier.
Constraints:
1. Game Slot Availability:
2. Each Team Plays At Most One Game Per Day
3. At Least One High Popularity Game Per Day:
- For each day, at least one game with a popularity rating of 1.25 or higher must be scheduled.
4. Los Angeles Teams Constraint:
- Ensure the Lakers and Clippers do not host on the same day.
5. Super Bowl Sunday Constraint:
- No game with a popularity rating above 1.4 is scheduled on Sunday.
6. Knicks and Jazz Back-to-Back Games Constraint:
- Ensure these teams play on Thursday and Friday, with no games on Saturday and Sunday.
Type of Optimization Problem:
This is a
Mixed Integer Linear Programming (MILP)
problem. It's linear because the objective
function and all constraints are linear with respect to the decision variables. It's mixed-integer
because the decision variables are binary (either 0 or 1).
This formulation captures the essence of the scheduling challenge, balancing the need to
maximize revenue against the operational and managerial constraints imposed by the NBA
schedule.
Question 2
Solve this problem using Python/Pyomo and Gurobi.
a. Write code to represent the model in Python/Pyomo and solve it with Gurobi. With your
report, include a pdf generated by printing the Jupyter Notebook to pdf after running all
parts of the notebook (so the pdf includes both your code and the output from the code,
as described in the “Homework Formatting Instructions” document).
Solution 2.a
Jupyter notebook attached(BANA201B_Group_B2_Question_2.ipynb)
PDF of the Jupyter notebook attached(BANA201B_Group_B2_Question_2.pdf)
b. Which games are assigned to each of the 4 days? What is the maximum revenue?
Solution 2.b
Day
Match
1
2, 4, 5, 8, 14
2
1, 9, 11, 18, 19, 20
3
3, 6, 7, 15, 16, 17
4
10, 12, 13
The match 1 should be scheduled on day 2
The match 2 should be scheduled on day 1
The match 3 should be scheduled on day 3
The match 4 should be scheduled on day 1
The match 5 should be scheduled on day 1
The match 6 should be scheduled on day 3
The match 7 should be scheduled on day 3
The match 8 should be scheduled on day 1
The match 9 should be scheduled on day 2
The match 10 should be scheduled on day 4
The match 11 should be scheduled on day 2
The match 12 should be scheduled on day 4
The match 13 should be scheduled on day 4
The match 14 should be scheduled on day 1
The match 15 should be scheduled on day 3
The match 16 should be scheduled on day 3
The match 17 should be scheduled on day 3
The match 18 should be scheduled on day 2
The match 19 should be scheduled on day 2
The match 20 should be scheduled on day 2
Maximum Revenue = $31.95 million
Question 3
Matt is notified by the NBA commissioner about a new restriction on the number of back-
to-back games (i.e., a team plays games on two consecutive days, e.g., Thursday and
Friday). According to this new restriction and in order to have equity in the number of
back-to-back games among all the teams throughout the season, Matt cannot assign
back- to-back games for the following two teams during the four-day time period:
Phoenix Suns and Los Angeles Lakers. Explain in detail how the current formulation
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should be modified to address the back-to-back games constraint. Do you expect the
total revenue to be higher or lower than in your original formulation, once this constraint
is added to the model? Apply these changes and re-solve the problem. How much does
the total revenue change?
Solution 3
Jupyter notebook attached(BANA201B_Group_B2_Question_3.ipynb)
PDF of the Jupyter notebook attached(BANA201B_Group_B2_Question_3.pdf)
To address the new restriction regarding back-to-back games for the Phoenix Suns and Los
Angeles Lakers, the original optimization model must be modified to include additional
constraints that prevent these two teams from playing games on two consecutive days during
the four-day period. This requires identifying the games involving these teams and then
explicitly preventing scheduling that would place their games on consecutive days.
Modification to the Model:
1. Identify Games Involving the Phoenix Suns and Los Angeles Lakers
: First, identify all
games that involve either the Phoenix Suns or the Los Angeles Lakers. For instance, if the
Phoenix Suns are involved in Game 9 and Game 17, and the Los Angeles Lakers are involved
in Game 1 and Game 16, these games need to be considered for the new constraints.
2. Add Constraints to Prevent Back-to-Back Games
: For each of the identified games, add
constraints to the model that prevent these games from being scheduled on consecutive days.
This can be done by ensuring that if one of these games is scheduled on a certain day, the other
game(s) involving the same team cannot be scheduled on the immediately following day.
Expected Impact on Total Revenue:
Adding the constraint to avoid back-to-back games for the Phoenix Suns and Los Angeles
Lakers is likely to reduce the flexibility of the scheduling model. This reduction in flexibility could
lead to a decrease in total revenue, as the model may not be able to place games on the days
that would otherwise maximize revenue due to the new restrictions.
Day
Match
1
8, 12, 16, 18, 19
2
2, 5, 9, 11, 14, 20
3
1, 3, 6, 7, 13, 15
4
4, 10, 17
The match 1 should be scheduled on day 3
The match 2 should be scheduled on day 2
The match 3 should be scheduled on day 3
The match 4 should be scheduled on day 4
The match 5 should be scheduled on day 2
The match 6 should be scheduled on day 3
The match 7 should be scheduled on day 3
The match 8 should be scheduled on day 1
The match 9 should be scheduled on day 2
The match 10 should be scheduled on day 4
The match 11 should be scheduled on day 2
The match 12 should be scheduled on day 1
The match 13 should be scheduled on day 3
The match 14 should be scheduled on day 2
The match 15 should be scheduled on day 3
The match 16 should be scheduled on day 1
The match 17 should be scheduled on day 4
The match 18 should be scheduled on day 1
The match 19 should be scheduled on day 1
The match 20 should be scheduled on day 2
Maximum Revenue = $31.275 million
Applying the Changes and Re-solving the Problem:
After applying these changes to the model, we found that the maximum revenue decreased
from $31.95 million to $31.275 million. This decrease in revenue, while implementing the new
back-to-back game restrictions, aligns with the expectation that limiting scheduling flexibility
would potentially lower the total revenue. The decrease in revenue indicates the trade-off
between adhering to new constraints and maximizing revenue objectives. The specific
scheduling changes, as noted in your results, reflect the model's adaptation to these new
constraints while still aiming to optimize revenue within the given parameters.
Question 4
The NBA usually selects very popular games to be broadcast on national television.
Multiple TV channels including TNT, ESPN, ABC, and NBA-TV have doubleheader or
tripleheader series, featuring two or three highly-rated games within the week. In
particular, the TNT doubleheader series, with half-time reports by NBA Hall of Famers
Shaquille O’Neal and Charles Barkley, normally hosts the most popular games. The NBA
wishes to assign two highly-popular games to a TNT doubleheader on one of the days
within the four-day block from February 4th -7th . The NBA wants to make sure that if
game 1 (Celtics at Lakers) is selected, then one of the following two games, game 3
(Blazers at Bucks) or 5 (Jazz at Nets), should be included in the TNT doubleheader.
Modify your formulation from part 1 of this assignment to incorporate the doubleheader
scheduling requirement. Please describe any variables and constraints that you add or
modify in your formulation. You do not need to re-solve the problem.
Solution 4
To incorporate the doubleheader scheduling requirement for the TNT doubleheader into the
existing optimization model, you'll need to introduce new variables and constraints specifically
designed to handle the selection and scheduling of the doubleheader games. Here's how you
can modify the formulation:
Additional Variables:
1. Doubleheader Variable
: Introduce a binary variable dh
j
for each day j within the four-day
block, where dh
j
if the TNT doubleheader is scheduled on day j, and 0 otherwise. Since the
doubleheader consists of two games, you will also need to ensure that exactly two games are
selected for the day when dh
j
= 1
2. Game Selection for Doubleheader
: You may also need binary variables dhg
i
to indicate
whether game i is selected as part of the doubleheader, where dhg
i
= 1 if game i is selected,
and 0 otherwise.
Additional Constraints:
1. Doubleheader Scheduling
: Ensure that the doubleheader is scheduled on exactly one of the
days. This can be represented as:
2. Game Selection for Doubleheader
: Ensure that exactly two games are selected for the
doubleheader and that these games are scheduled on the same day as the doubleheader. This
can be represented as:
And, for each day j:
This ensures that if a doubleheader is scheduled on day j, (dh
j
= 1), then exactly two games
selected for the doubleheader are scheduled on that day.
3. Specific Game Requirement for Doubleheader
: To meet the requirement that if game 1
(Celtics at Lakers) is selected for the doubleheader, then either game 3 (Blazers at Bucks) or
game 5 (Jazz at Nets) must also be included, add the following constraint:
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This constraint ensures that if game 1 is selected dh
g
=1, then at least one of game 3 or game
5 must also be selected for the doubleheader.
Modifying the Objective Function:
The objective function may remain the same if the goal is still to maximize total revenue.
However, if there are specific revenue implications associated with the doubleheader (e.g.,
increased revenue for games selected as part of the doubleheader), you may need to adjust the
objective function to reflect this.
By adding these variables and constraints to your model, you can accommodate the new
scheduling requirement for the TNT doubleheader while still aiming to maximize revenue and
adhere to other existing constraints. This modification allows the model to consider the
additional layer of decision-making involved in selecting and scheduling the doubleheader
games within the constraints provided by the NBA.