## Math3272 Linear Programming课程简介

The book is a comprehensive text on linear programming, covering both the theoretical foundations and practical applications of the subject. It is intended for graduate-level students and researchers in mathematics, engineering, and related fields.

Some of the topics covered in the book include linear algebra, convex analysis, duality theory, sensitivity analysis, and algorithms for solving linear programming problems. The book also includes numerous examples and exercises to help readers develop their understanding of the material.

If you have access to SpringerLink through your institution, you should be able to access the book online. You can search for the book on the SpringerLink website and click on the “Access” button to log in and view the book.

## PREREQUISITES

Instructor: Mikhail Lavrov
Location: Mathematics 112
Lecture times: 5:00pm to 6:15pm on Tuesday and Thursday
Textbook: Linear Programming: Foundations and Extensions by Robert
Vanderbei. You can access it online via SpringerLink (you will have to click Log in, in the top right, and access it via KSU’s subscription).
Office hours: Tuesday 12:00pm to 1:00pm, Wednesday 4:00pm to 5:00pm in Mathematics 245.
D2L page: https://kennesaw.view.usg.edu/d2l/home/2672229
During the scheduled office hours, you are welcome to stop by my office without an appointment: answering your questions is the reason I’m there! Outside that time, please email me at [email protected] and I will either answer your question by email, or we will find a different time to meet.
(However, it is difficult for me to adjust my schedule on very short notice, so please email me the day before if you want to set up a meeting.)

D2L will be used to submit assignments (these will be posted both here and on D2L, for convenience) and to view grades. The syllabus is also posted on D2L.

## Math3272 Linear Programming HELP（EXAM HELP， ONLINE TUTOR）

(a) Write an integer linear program that finds maximum the number of (prof, student,time) meetings that can happen.
(b) Explain how to get the actual meeting schedule from your integer linear program.
(c) Suppose you are now also given, for each professor, a “bigshot score” $b_{p_t}$ that is a positive number that says how important it is to satisfy that professor’s demands. For every meeting that professor $p_i$ takes with a student in her preference list $M_{p_i}$, we get $b_{p_i}$ points. Describe how to modify your integer linear program to find a schedule that maximizes the number of bigshot points your schedule gets.
(d) One more extension: visit days are successful if students meet with a lot of people, even if neither the student or the professor put one another in their preference lists. Describe how to modify your integer linear program so that you get $b_{p_i}$ points if you schedule one of professor $p_i$ ‘s meetings with a student on her list $M_{p_i}$ but only 1 point if you schedule her with someone who is not on her preference list. Assume now that students will meet with anyone they are assigned to, but the time constraints must still be satisfied.

(a) Integer linear program that finds the maximum number of (prof, student, time) meetings:

Let $M$ be the set of students, $P$ be the set of professors, and $T$ be the set of timeslots. Let $x_{i,j,k}$ be a binary variable that takes the value 1 if and only if student $i\in M$, professor $j\in P$, and timeslot $k\in T$ meet. Then the integer linear program can be written as:

Maximize $\sum_{i\in M}\sum_{j\in P}\sum_{k\in T} x_{i,j,k}$

Subject to: $\sum_{j\in P}\sum_{k\in T} x_{i,j,k} = 1 \hspace{10mm} \forall i \in M$ (Each student meets exactly one professor) $\sum_{i\in M}\sum_{k\in T} x_{i,j,k} \leq 1 \hspace{10mm} \forall j \in P$ (Each professor meets at most one student) $\sum_{i\in M}\sum_{j\in P} x_{i,j,k} \leq 1 \hspace{10mm} \forall k \in T$ (Each timeslot is occupied by at most one meeting) $x_{i,j,k} \in {0,1} \hspace{10mm} \forall i\in M, j\in P, k\in T$ (Binary variables)

(b) To get the actual meeting schedule, we can simply look at the values of the $x_{i,j,k}$ variables that are equal to 1. For each such variable, we can say that student $i$, professor $j$, and timeslot $k$ meet.

(c) Modified integer linear program that maximizes the bigshot points:

We can modify the objective function of the previous integer linear program to maximize the sum of bigshot points earned by scheduling meetings with professors in their preference lists. Let $b_{j}$ be the bigshot score of professor $j\in P$. Then the modified integer linear program can be written as:

Maximize $\sum_{i\in M}\sum_{j\in P}\sum_{k\in T} b_{j}x_{i,j,k}$

Subject to: $\sum_{j\in P}\sum_{k\in T} x_{i,j,k} = 1 \hspace{10mm} \forall i \in M$ (Each student meets exactly one professor) $\sum_{i\in M}\sum_{k\in T} x_{i,j,k} \leq 1 \hspace{10mm} \forall j \in P$ (Each professor meets at most one student) $\sum_{i\in M}\sum_{j\in P} x_{i,j,k} \leq 1 \hspace{10mm} \forall k \in T$ (Each timeslot is occupied by at most one meeting) $x_{i,j,k} \in {0,1} \hspace{10mm} \forall i\in M, j\in P, k\in T$ (Binary variables)

(d) Modified integer linear program that considers visit days:

We can modify the previous integer linear program to also consider visit days. We can introduce an additional set $S$ of possible students that professors may meet on the visit day. Let $M_{j}$ be the set of students in professor $j$’s preference list, and let $b_{j}$ be

1. “Greg’s problem (sort of).” You have two images $A$ and $B$. Within each image are sets of points $P_A=\left{p_1^A, \ldots, p_m^A\right}$ and $P_B=\left{p_1^B, \ldots, p_n^B\right}$. Each point has a label $\ell\left(p_i\right)$, and several points will have the same labels. You are given a set $E \subseteq P_A \times P_B$ of pairs of points that are allowed to be assigned to each other.
You want to assign points in $P_A$ to points in $P_B$ subject to the following conditions:
(1) all points in $A$ must be assigned to some point in $B$ (some points in $B$ might go assigned though).
(2) two points $p_i^A, p_j^B$ can be assigned to each other only if $\left(p_i^A, p_j^B\right) \in E$.
(3) two points $p_i^A, p_j^B$ can be assigned to one another only if they have the same label: $\ell\left(p_i^A\right)=\ell\left(p_j^B\right)$.
(4) several points from $A$ can be be assigned to a single point in $B$, but we require that the number of points assigned to points with the same label in $B$ to differ by at most 1 (including those to which we assign 0 points). That is, if 2 points from $A$ are assigned to point $p_1^B$ then if $\ell\left(p_3^B\right)=\ell\left(p_1^B\right)$, we can’t assign 4 points from $A$ to $p_3^B$. Another example: if we assign 0 points to $p_7^B$ then no other point with the same label can be assigned more than 1 point from $A$.

This problem can be modeled as a bipartite graph where the nodes in the left partition represent points in $A$ and the nodes in the right partition represent points in $B$. An edge is present between a node $p_i^A$ and $p_j^B$ if $(p_i^A, p_j^B) \in E$.

Let $x_{i,j}$ be a binary variable that takes value $1$ if point $p_i^A$ is assigned to point $p_j^B$ and $0$ otherwise. We can formulate the problem as an integer linear program as follows:

\begin{align*} \text{maximize} \quad & 0 \ \text{subject to} \quad & \sum_{j=1}^{n} x_{i,j} = 1 && \forall i \in {1, \ldots, m} \ & \sum_{i=1}^{m} x_{i,j} \leq \ell(p_j^B) && \forall j \in {1, \ldots, n} \ & \sum_{i=1}^{m} x_{i,j} \leq 1 && \forall j \in {1, \ldots, n} \ & x_{i,j} = 0 && \text{if } (p_i^A, p_j^B) \notin E \ & x_{i,j} = 0 && \text{if } \ell(p_i^A) \neq \ell(p_j^B) \ & x_{i,j} \in {0, 1} && \forall i \in {1, \ldots, m}, j \in {1, \ldots, n} \end{align*}

The objective function is not important in this case, since we don’t want to optimize any quantity. The first constraint ensures that each point in $A$ is assigned to exactly one point in $B$. The second constraint enforces the limit on the number of points assigned to a point with the same label in $B$. The third constraint ensures that no more than one point in $A$ is assigned to a point in $B$. The fourth and fifth constraints handle the conditions (2) and (3) in the problem statement.

To get the actual assignment, we can inspect the values of the variables $x_{i,j}$ that are set to $1$. Each such variable indicates that point $p_i^A$ is assigned to point $p_j^B$.

To modify the objective function to take into account the bigshot score $b_{p_i}$ of each professor $p_i$, we can change the objective function to maximize the sum of bigshot scores of professors whose demands are satisfied:

\begin{align*} \text{maximize} \quad & \sum_{i=1}^{m} b_{p_i} \sum_{j=1}^{n} x_{i,j} \ \text{subject to} \quad & \sum_{j=1}^{n} x_{i,j} = 1 && \forall i \in {1, \ldots, m} \ & \sum_{i=1}^{m} x_{i,j} \leq \ell(p_j^B) && \forall j \in {1, \ldots, n} \ & \sum_{i=1}^{m} x_{i,j} \leq 1 && \forall j \in {1, \ldots, n\

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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