Solve task in the Applied combinatorial (discrete )optimization

The solution should be in English and i need it to be written in Word for any changes i might add ,it's home take task in the Applied combinatorial for master student in Advanced combinatorics program, the solution should not be copy from the internet or Chat AI, and if it needs to draw a graph or a table please , and please the rest details we can talk about them i will also provide many files including description from lectures . Also, I will provide my last work and the way maybe it can inspire you with solving.