You are the project manager for a construction company. A project has exceeded its budget and is still incomplete. You estimate that the project requires $25$ additional person$-$days of effort or equivalent in order to be completed. You have been granted permission to use additional resources $($labour and machinery$)$ to help deliver the project. However, you must keep the cost to a minimum in doing this.

You may hire additional labour for $200$ per person, per day. You may hire additional machines for $2,000$ per machine, per day. Each machine$-$day employed reduces the number of required person$-$days by $5.$ You can only secure $15$ days$$ worth of work from suitably trained workers, and there is only one suitable machine available on the market to hire, for up to four days.

Formulate the problem as a linear program including all constraint equations and the cost function and sketch it graphically, labelling the feasible region. $(15$ marks$)$

Solve the linear program by any method of your choice and estimate the optimal number of workers and machines to deliver the project at the lowest cost.