Management Science I: Homework 5
Fall 2023 | Due Wednesday 2/28/24 (@ 11:59 PM)
There are three problems in this homework assignment, totaling 60 points. You may submit
either typed or handwritten solutions (in addition to screenshots where requested). You will
submit the homework using Gradescope (“Homework 5”), and you will submit the
accompanying spreadsheet on Canvas.
General reminders for assignments:
● Different questions build off each other in the homework. You will receive partial credit
if you do the right thing using an upstream incorrect answer (e.g., if you use the wrong
formula in an Excel cell in one part, and then rely on that cell later). We can only know
whether you were doing the right thing if you include an explanation of how you got
your numbers – so include brief (1 sentence or phrase) explanations of your logic if you
want to be eligible for partial credit.
● As stated in the syllabus, you may work in groups of up to 4 people. A couple of
reminders: do not collaborate outside of these groups, and do not use Generative AI
tools to assist with any of the problem solutions.
Homework overview
● Problem 1: IP constraints
● Problem 2: Goal programming pt. 1
● Problem 3: Goal programming pt. 2
, Problem 1: Logical Constraints [20 pts, 5 pts each]
Suppose you are deciding how much to fund five candidate projects ( F 1 , … F5 . Fi = $ funded to
project i). For each of the conditions below, write out how the constraint(s) required to reflect
the desired requirements. If you introduce any auxiliary variables, you must define them (and
whether they binary, continuous, or integer?).
1. Each funded project (i.e., if the funding amount is greater than 0) must receive at least
$300 in funding.
Y i is a binary variable that equals 1 if project i is funded and 0
otherwise.
F i ≥ 300∗Y i
2. At most one project can receive more than $500 in funding.
5
∑ F i >500- M∗Y i
i=1
3. Project 5 must receive the most funding out of all projects (it can be tied with others,
but there cannot be a project with more funding than 5 receives).
F 5 ≥ Fi , i∈ { 1,2,3 , 4 }
4. The total funding amount must be either $500, $1000, or $1500. Hint: you want the RHS
of a constraint on total funding to equal exactly one of three values.
This requirement can be modeled by introducing three binary variables
Y 500, Y 1000, Y 1500. Where Y 500 = 1 if the total funding is $500; Y 1000 =1 if the
total funding is $1000; Y 1500 =1 if the total funding is $1500
5
∑ F i=500∗¿ Y 500+1000∗Y 1000+1500∗Y 1500 ¿
i=1
When to use big m is very based on intuition
M is arbitrary larger number
Fall 2023 | Due Wednesday 2/28/24 (@ 11:59 PM)
There are three problems in this homework assignment, totaling 60 points. You may submit
either typed or handwritten solutions (in addition to screenshots where requested). You will
submit the homework using Gradescope (“Homework 5”), and you will submit the
accompanying spreadsheet on Canvas.
General reminders for assignments:
● Different questions build off each other in the homework. You will receive partial credit
if you do the right thing using an upstream incorrect answer (e.g., if you use the wrong
formula in an Excel cell in one part, and then rely on that cell later). We can only know
whether you were doing the right thing if you include an explanation of how you got
your numbers – so include brief (1 sentence or phrase) explanations of your logic if you
want to be eligible for partial credit.
● As stated in the syllabus, you may work in groups of up to 4 people. A couple of
reminders: do not collaborate outside of these groups, and do not use Generative AI
tools to assist with any of the problem solutions.
Homework overview
● Problem 1: IP constraints
● Problem 2: Goal programming pt. 1
● Problem 3: Goal programming pt. 2
, Problem 1: Logical Constraints [20 pts, 5 pts each]
Suppose you are deciding how much to fund five candidate projects ( F 1 , … F5 . Fi = $ funded to
project i). For each of the conditions below, write out how the constraint(s) required to reflect
the desired requirements. If you introduce any auxiliary variables, you must define them (and
whether they binary, continuous, or integer?).
1. Each funded project (i.e., if the funding amount is greater than 0) must receive at least
$300 in funding.
Y i is a binary variable that equals 1 if project i is funded and 0
otherwise.
F i ≥ 300∗Y i
2. At most one project can receive more than $500 in funding.
5
∑ F i >500- M∗Y i
i=1
3. Project 5 must receive the most funding out of all projects (it can be tied with others,
but there cannot be a project with more funding than 5 receives).
F 5 ≥ Fi , i∈ { 1,2,3 , 4 }
4. The total funding amount must be either $500, $1000, or $1500. Hint: you want the RHS
of a constraint on total funding to equal exactly one of three values.
This requirement can be modeled by introducing three binary variables
Y 500, Y 1000, Y 1500. Where Y 500 = 1 if the total funding is $500; Y 1000 =1 if the
total funding is $1000; Y 1500 =1 if the total funding is $1500
5
∑ F i=500∗¿ Y 500+1000∗Y 1000+1500∗Y 1500 ¿
i=1
When to use big m is very based on intuition
M is arbitrary larger number
