How to model and solve the machine maker problem, manwork and machinework at the same time?

Question

Eddy is an mechanic and he can create machine to make specific tool. In 8 hour he can make a machine that make the same tool as he can make in 1 hour for 4 hour, that mean 1 tool. More clearly: 1 hour= 1 tool; 8 hour= 1 machine that make 1 tool in 4 hour; 1 machine 4 hour= 1 tool. How to maximine number of tool he can make in one week, one month or one year, ten years, assuming he has 10 hours of work a day. And how to generalise this math problem (books or articles about it is welcome)

Answer 1

Think about the problem in terms of costs of making the machine. A machine costs $8$ tools to make and return $1$ tool for every $4$ hours is works. Hence a machine breaks even (repays itself in terms of tools) if it runs for at least $8\cdot4$ hours.

Assuming the $10$ hours a day limit applies to human work as well as to machines, at $4$ days, that is, at $40$ hours, it is irrelevant (in terms of the number of tools produced) whether only tools are made or whether one machine is first made and then tools are produced.

With this logic, you can derive what is optimal. Namely, denote by $r$ the number of hours still available for work. Then if $r<40$ it is optimal to produce tools only. If $r\geq40$ it is optimal to produce exactly one machine now and decide what to do after that using the same rule. (That is, after the machine is produced, the remaining time will be $r-8$.)