$2.$ Aircraft Engine Maintenance $(15$ points$)-$ Periodic preventive maintenance is carried out on aircraft engines, where an important component must be replaced. The number of aircraft scheduled for such maintenance over the next $6$ months is estimated at $200,180,300,198,230,$ and $290,$ respectively. All maintenance work is done in the month that parts are available, where a used component may be replaced with a new or an overhauled component. The overhauling of used components may be done in a local repair facility, where they will be ready for use at the beginning of next month, or they may be sent to a central repair shop where a delay of $2$ months $($as such maintenance can occur in the third month$)$ is expected. The repair cost in the local shop is $$120$ per component. At the central facility, the cost is only $$35$ per component. An overhauled component used in a later month will incur an additional storage cost of $$1\mathrm{.}50$ per unit per month. New components may be purchased at $$200$ each in month $1,$ with a $5\%$ price increase every $2$ months. A new component is available immediately $($no delay$).$ Formulate a model to the problem and solve to determine the optimal schedule for satisfying the demand for the component over the next $6$ months. Also consider that the planning occurs within the first month. So all parts are ordered in the first month to satisfy maintenance requirements