Video 1: Introduction
The slides from all videos in this lecture can be downloaded here: An Introduction to Integer Optimization (PDF - 1.1MB).
9 Integer Optimization
9.2 Sports Scheduling: An Introduction to Integer Optimization
Quick Question
Suppose that you are trying to schedule 3 games between 6 teams (A, B, C, D, E, and F) that will occur simultaneously. Which of the following are feasible schedules? Select all that apply.
Explanation Each of the teams has to play exactly one of the other teams for the games to occur simultaneously. In the second option, C is playing twice, which is impossible. In the fourth option, B and C are both playing twice.
How many different feasible schedules are there?
Explanation There are 15 different feasible schedules. We can count them by observing that A can play any of the 5 teams. Once this is fixed, we have 4 teams left. There are 3 ways to make two pairs out of 4 teams. So in total, there are 5\*3 = 15 different schedules. Here is a list of all of them: A plays B, C plays D, E plays F A plays B, C plays E, D plays F A plays B, C plays F, D plays E A plays C, B plays D, E plays F A plays C, B plays E, D plays F A plays C, B plays F, D plays E A plays D, B plays C, E plays F A plays D, B plays E, C plays F A plays D, B plays F, C plays E A plays E, B plays C, D plays F A plays E, B plays D, C plays F A plays E, B plays F, C plays D A plays F, B plays C, D plays E A plays F, B plays D, C plays E A plays F, B plays E, C plays D
Quick Question
For each of the decisions below, indicate if the decision variables would be binary, integer, or neither.
- We have 20 students, and we want to assign them to one of two groups.
Explanation The first and third decisions require binary decision variables, since they are both assignment problems. In the first case, we'll have a binary decision variable for each student (20 decision variables). In the third case, we'll have a binary decision variable for each person (15 decision variables). The second decision requires integer decision variables, since the owner needs to decide how many of each item to send to each store (15 decision variables). Since fractional items would not make sense, the decisions are integer. The fourth decision does not need binary nor integer decision variables, because the amount in grams can be fractional.
- The owner of 5 clothing stores needs to decide how many shirts, pants, and hats to send to each store, given historical sales data.
Explanation The first and third decisions require binary decision variables, since they are both assignment problems. In the first case, we'll have a binary decision variable for each student (20 decision variables). In the third case, we'll have a binary decision variable for each person (15 decision variables). The second decision requires integer decision variables, since the owner needs to decide how many of each item to send to each store (15 decision variables). Since fractional items would not make sense, the decisions are integer. The fourth decision does not need binary nor integer decision variables, because the amount in grams can be fractional.
- After try-outs, the coach of a basketball team needs to decide which people should make the team (15 people tried out).
Explanation The first and third decisions require binary decision variables, since they are both assignment problems. In the first case, we'll have a binary decision variable for each student (20 decision variables). In the third case, we'll have a binary decision variable for each person (15 decision variables). The second decision requires integer decision variables, since the owner needs to decide how many of each item to send to each store (15 decision variables). Since fractional items would not make sense, the decisions are integer. The fourth decision does not need binary nor integer decision variables, because the amount in grams can be fractional.
- A fertilizer company is trying to decide how much (in grams) of three different compounds to add to each bag of fertilizer.
Explanation The first and third decisions require binary decision variables, since they are both assignment problems. In the first case, we'll have a binary decision variable for each student (20 decision variables). In the third case, we'll have a binary decision variable for each person (15 decision variables). The second decision requires integer decision variables, since the owner needs to decide how many of each item to send to each store (15 decision variables). Since fractional items would not make sense, the decisions are integer. The fourth decision does not need binary nor integer decision variables, because the amount in grams can be fractional.
Note that you will need to answer all of the questions above before checking your answers.
Quick Question
Suppose we had two more teams in our tournament (for a total of 6 teams). Each division would have 3 teams. So each team plays two teams twice (the teams in their division), and each team plays three teams once (the teams in the other division). This means that the tournament will last for 7 weeks. How many decision variables would we have?
Exercise 1
Numerical Response
Explanation
We would have 105 decision variables because we have 7 weeks, and 15 different pairs of teams.
How many constraints would we have? Don’t include the constraints that force the variables to be binary when counting the constraints here. (HINT: We would have 6 division constraints, since each pair in each division needs to play twice.)
Exercise 2
Numerical Response
Explanation
We would have 6 division constraints, 9 non-division constraints (each of the three teams in one division has to play each of the three teams in the other division), and 42 constraints to make sure each team only plays one team each week (6 teams times 7 weeks).
CheckShow Answer
Quick Question
Suppose we want to add a constraint that teams A and B must play in week 4 (we want the last game to be a divisional one). Given the current model, which of the following constraints would model this correctly? Select all that apply.
Explanation The second and third constraints would both model this correctly. We can either force the decision variable for week 4 to 1, or we can make sure that only one game is played in the earlier weeks.
Video 2: The Optimization Problem
Video 3: Solving the Problem
In this video, we’ll be using the spreadsheet SportsScheduling. If you are using LibreOffice or OpenOffice, please download and open the spreadsheet SportsScheduling (ODS). If you are using Microsoft Excel, please download and open the spreadsheet SportsScheduling (XLSX). The following spreadsheets have the completed model as it is at the end of the video: SportsScheduling_Complete (ODS) and SportsScheduling_Complete (XLSX).
Video 4: Logical Constraints
Video 5: The Edge
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