Previously in Chapter 6

• Using binary variables• Unintended options• Lack of sensitivity analysis

Agenda

• Quiz logistics

• Finish binary variables– Traveling salesman problem

• Nonlinear functions– Finishes optimization– (Will come back to chapter 5 later.)

Quiz Survey

Should I move it? 11 Yes, 3 No

Most preferred Acceptable

Mon-Tue 4 6

Tue-Wed 4 9

Wed-Thur 3 9

Thur-Fri 3 9

Fri-Sat 1 6

Quiz

• Tue 11am to Wed 11am

• office hours Mon 4-5

• moving Wed office hours to Tue 4-5

Agenda

• Quiz logistics

• Finish binary variables– Traveling salesman problem

• Nonlinear functions– Finishes optimization– (Will come back to chapter 5 later.)

Traveling Salesman Problem

Chicago

Detroit

Atlanta

Traveling Salesman Problem

• Data: Cities and Distances

• Goal: minimize travel time to tour all cities(visit each city exactly once)

Traveling Salesman Problem (TSP)

• Famous problem

• Vehicle Routing

• Job Sequencing (p207)– 5 tasks that need to be done on a machine– Setup time for a task depends on the task

that just completed

Job Sequencing

task 1 task 2

task 3

task 4task 5

start/finish

Solving TSP

1. Setup an assignment problem

Solving TSP

1. Setup an assignment problem

2. Any subtours (loops of length k<n) ?

Solving TSP

1. Setup an assignment problem

2. Any subtours (loops of length k<n) ?

3. Add subtour constraints• sum of variables ≤ k-1• now binary variables needed

Solving TSP

1. Setup an assignment problem2. Any subtours (loops of length k<n) ?3. Add subtour constraints

• sum of variables ≤ k-1• now binary variables needed

4. Repeat (go to stop 2)• often you finish quickly

(though no guarantees)

Nonlinear functions

• product terms– chemical processes– area, volume– variance

• log terms– utility functions– entropy

Nonlinear Functions

• Sensitivity Analysis– Like a derivative

• Sqrt() function– Help the solver by defining it for x<0

• Solution may depend on starting point– Solvers use hill-climbing methods

Nonlinear Example

• Given a set of rectangular boxes, how many can you fit into a container?