circuit design and computer hardware
Example: a robotic arm has to create a complicated electronic circuit by putting a conducting material on a non-conducting base. The robotic arm has to traverse the whole graph that makes up the circuit at least once. In order to reduce the time the roboter needs to create one electronic circuit, the way it has to traverse needs to be minimized, that is one needs a good approximate solution to the traveling salesman problem. I know of examples where improved heuristic approximation algorithms have increased the output by several percents (the student doing the math thesis on this was rewarded by the company producing these circuits with several hundred thousands of dollars. No, it wasn't me.).