Math 381 - Discrete Mathematical Modeling

Lecture Summaries

10M - 11/26/11

• Comments on the 1st drafts.
• Grammar: repetition, verb-noun agreement, consistent tense.
• Project: clear description of model
• References: at least 4.
• Presentations start Wednesday.

9W - 11/23/11

• Markov Chains.
• Ergodic Markov chains: the Cola example.
• The steady state of an ergodic Markov chain.
• Left and right eigenvectors.
• Applications of the steady state.

9M - 11/21/11

• The Poisson distribution in R.
• Sampling from a Poisson distribution: the Fishtank example.
• Interarrival times in a Poisson distribution: the Exponential distribution.
• Continuous random variables, Probability density function.
• R code for the Fishtank example using a Poisson distribution.
• A nice page on the Exponential distribution.
• A brief introduction to Continuous random variables.

8F - 11/18/11

• Sampling from a discrete distribution: the Stock example.
• Analyzing data using R.
• The Bernoulli distribution, the Binomial distribution.
• The Poisson distribution.
• Wolfram's page on the Poisson distribution goes over the derivation I did in class.

8W - 11/16/11

• Comments on the format of your paper.
• More on the Dice example.
• More on the Fishtank example.
• Using R to create, and sample from, a discrete distribution.
• The Stock example: R code and data file.

8M - 11/14/11

• Monte Carlo simulations: the Fishtank example.
• Sampling from a distribution.
• The Uniform distribution, the Normal distribution.
• Sampling from a Bernoulli distribution.
• Discrete distributions: the Dice example.
• R code for the Fishtank example.
• R code for the Dice example.
• I gave an A+ to this paper the last time I taught the class.
• Here is another good paper.

7W - 11/09/11

• Worker scheduling: heuristics.
• The transport matrix.
• More complex if/then constraints.
• R code for solving the Traveling Salesman Problem.
• I can easily solve very large IPs with GUSEK.
• The homework due Monday will not be collected.

7M - 11/07/11

• Integer programming: the Automobile Manufacturer example.
• Either/Or constraints in an Integer Program.
• If/Then constraints.
• Worker scheduling: an introduction.
• lpSolve works well to solve IPs in R.
• You can also use GLPK to solve larger IPs in R.
• The NEOS Server can solve your big IP.

6F - 11/04/11

• Comments on the midterm.
• The first draft of your paper is due 11/21/11.
• The Traveling Salesman Problem.
• Subtour elimination constraints.
• Introducing new variables to formulate complex constraints in an IP.
• Read Sections 9-1 and 9-2.
• Homework due 11/14/11: Section 9-2 #1, 6, 12.

6W - 11/02/11

• Midterm exam.
• Here are Solutions to the Midterm.

6M - 10/31/11

• Midterm review.
• Graphs: Shortest path, minimal spanning tree, maximum flow.
• Linear programming.
• Markov chains and the Traveling salesman problem.
• Here are solutions to Homework 4 and Homework 5.

5F - 10/28/11

• Comments on the homework: 8-3 #16.
• Complete graphs, Bipartite graphs, Complete bipartite graphs.
• The Traveling Salesman Problem.
• A greedy algorithm and an upper bound.
• Formulating the TSP as an integer program, Subtours.
• Branch and bound techniques.
• The Georgia Tech TSP webpage: Take a look at the games.
• Read Section 9-6.

5W - 10/26/11

• Comments on the homework: 3-3 #10.
• Projects: Group meetings, data collection, methods.
• Read this sample paper: The Missouri Lottery.
• Multiperiod linear programming problems.
• Linear Programming: the Multiperiod Work Scheduling example.
• Still more on structured matrices and R.
• R code for solving the Multiperiod Work Scheduling example.
• Integer Programs: some difficulties.
• Read Sections 3-10, 3-11, 3-12.

5M - 10/24/11

• Revised proposals due Friday.
• Midterm exam is on Nov. 2, a week from Wednesday.
• Linear Programming: the Work Scheduling example.
• Decision variables: these can take some thought.
• More on structured matrices and R.
• R code for solving the Work Scheduling example.
• Linear Programming: the Capital Budgeting example.
• Multi-year decision making and Net Present Value.
• Read Sections 3-5 and 3-6.

4F - 10/21/11

• Please pick up your old homework.
• Revised proposals due next Friday.
• Midterm exam is on Nov. 2, a week from Wednesday.
• Linear Programming: the Farm example.
• Decision variables, the Objective function.
• Constraints and Linear inequalities.
• The Graphical method. Tight constraints at the solution.
• Read Sections 3-2 and 3-3.
• Homework due 10/28/11: Section 8-3 #5,8,16; Section 3-3 #10; Section 3-5 #5.
• Do not worry about the "cut" in 8-3 #5.

4W - 10/19/11

• Project proposals: Data - gathering, processing, data entry.
• Project proposals: Methods - Integer Programming, Monte Carlo simulation, Approximate methods.
• Project proposals: Scope - Restricting the problem, Expanding the problem.
• Optimization: Decision Variables, Objective and Constraints.
• Maximum Flow: the Job Assignment example.
• Using R to create a large structured matrix.
• R code for solving the Job Assignment example.
• Go to the MAC talk on Friday at 2:30 in Kane 210.
• Read Section 8-3. Skip LINGO and Ford-Fulkerson.

4M - 10/17/11

• Project proposals back Wednesday. Start thinking about gathering data.
• More graph theory: Maximum Flow problems.
• Maximum Flow: the Pipeline example.
• Modeling max flow as a linear program.
• Totally unimodular matrices and integer solutions.
• R code for solving the Pipeline Example.

3F - 10/14/11

• Applications of Shortest Path Problems: the library example.
• Prim's algorithm and Kruskal's algorithm.
• Minimal Spanning Trees: an example of Prim's algorithm.
• You can do many graphical algorithms in R using the RBGL package, but it's a little tricky to install. Use the method shown in the link from last class. Talk to me if you have problems.
• Here is some documentation for RBGL.
• Homework due 10/21/11: Section 8-2 #2,7,9 and Section 8-6 #2,4
• Note: Do Dijkstra's algorithm by hand on 8-2 #2,7.

3W - 10/12/11

• Dijkstra's Algorithm: an example.
• Comments on Computational Complexity.
• Applications of Shortest Path Problems: the car example.
• This is a helpful URL for getting graph theory working in R.
• Read section 8-6 in the text.

3M - 10/10/11

• Groups: Please email me the names in your group. (No need if you are in a Carson Center group.)
• Written project proposals are due Friday.
• Graph Theory: Basic Definitions.
• The Shortest Path Problem and Dijkstra's Algorithm.
• Read section 8-1 and 8-2 in the text. Skip the section on the "Transshipment Problem".
• The Wikipedia article on Graph Theory is quite good.
• Here is a link to the LyX typesetting program that was mentioned in class.
• Here's a solution to Homework 2.

2F - 10/07/11

• Groups: Please email me the names in your group. (No need if you are in a Carson Center group.)
• Carlson Center groups: Please contact your client promptly.
• I suggest you set up a weekly meeting time for your group.
• Monte Carlo Simulation example: The law firm. Here's some R code for this problem: law_firm_monte_carlo.R
• Homework due 10/14/11: Section 3-2 #5; 3-4 # 2; 17-6 #4,7 (Do NOT solve the LPs graphically.)
• Solve both LP problems using R (or the software of your choice). See the class log for 10/03/11 for an example.
• The section numbers should work for OR by Winston (4th edition). Please check if you are using that book.

2W - 10/05/11

• Comments on forming groups. Email me if you want to make a presentation to the class on Friday.
• Project proposals are due on 10/14/2011, a week from Friday.
• Absorbing Markov Chains: A dice game. Here's some R code to solve this problem: dice_game.R
• Applications of Markov Chains to Work Force Planning: The law firm. Here's some R code for this problem: law_firm.R
• Comment on Ergodic Markov Chains.

2M - 10/03/11

• More linear programming: The diet problem. Here's some R code to solve this problem: diet.R
• Remember to sign up for the Carlson Center projects by Wednesday.
• Markov Chains: A dice game.
• Homework due Friday. These problems are in the textbook p. 17 #1,2,3; p.441 #2.
• These problems are also in the book Operation Research by Wayne Winston. It is on reserve in Odegaard, but the page numbers are different.
• The same homework in Winston (2nd Edition): Section 3-1 #1,2,3; Section 19-6 #2

1F - 09/30/11

• Presentation by Michaelann Jundt, Director of the Carlson Center.
• Please consider doing one of their projects. Visit their website and click "Online Registration" to get started.
• More on linear and integer programming.
• The lifeboat example.
• Using R to solve LPs and IPs. You can download R for free here.
• This is a nice quick R reference.
• I've created a class bulletin board to help facilitate the formation of groups. You can access it here.

1W - 09/28/11

• Overview of course
• Introduction to the modeling process.
• Linear Programming
• The statistical programming language R can be used to solve Linear and Integer Programs. Here is a comprehensive introduction to R.
• Here's the R code I tried to run today: life-boats-1.R and life-boats-2.R.
• Syllabus
• Homework #1.
• The next MAC talk is on Linear Programming. These talks are for a general audience.