Problem $3$:

An end product is being produced in a large manufacturing plant. The bill of material diagram is as follows:

The lead time is given in weeks.

The MPS schedule for this end product for the following four weeks, ten weeks from now is as follows:

$\backslash$table$[[$Week $10,$Week $11,$Week $12,$Week $13],[10,14,19,9]]$

The fixed order cost is $$100$ for all the units, and the holding cost is $$0\mathrm{.}6$unit$/$week for all the units.

a$)$ Assuming the silver meal lot production for all the components, determine the planned order releases for component $P.$

b$)$ Suppose the production capacity for item $C$ is $55$ units per week. If we start producing at week $8,$ is it possible to find a planned order release without backordering? If backordering is allowed at a cost of $$5$ unit$/$item$,$ formulate a mathematical program that minimize the total cost from item C$.$