Optimization begins with a question that sounds simple but usually is not: what does best mean?
Shortest route, lowest cost, fewest mistakes, maximum fairness, fastest schedule, least wasted time - each phrase creates a different mathematical problem. The numbers may stay the same, but the objective changes the answer.
Three parts of the model
Most optimization problems have:
- choices - what you are allowed to decide,
- constraints - what your decision must obey,
- an objective - what you are trying to minimize or maximize.
Students should learn to name these before calculating. Otherwise the computation can look precise while the model is vague.
A COSMO activity
Give students a small map with distances between locations. Ask them to find a route that visits every point.
Then change the objective:
- minimize total distance,
- minimize the longest single trip,
- minimize the number of turns,
- make the workload as balanced as possible between two drivers.
The same map becomes several different problems. That is the lesson.
Where applied math gets interesting
Optimization is powerful because it forces assumptions into the open. But it is also dangerous if students forget that the objective was chosen by a person.
A model can optimize what it measures and ignore what it leaves out. That is why COSMO teaches optimization as a conversation between math and context, not as a magic answer machine.