ask an expert $4.$ A maintenance planner would like to plan the mechanical maintenance activities over the following planning horizon. Assume that we have three time periods with maintenance loads of $300,250,$ and $350$ hours of grade $1$ work, and $100,50,$ and $150$ hours of grade $2$ work, respectively. The in$-$house regular capacity is $200,350,$ and $300$ hours for periods $1,2,$ and $3$ respectively. The overtime capacity is at most $\backslash (25\backslash \%\backslash )$ of the in$-$house capacity.

Subcontracting is abundant, and technically, there is no limit on this source. The cost of performing one in$-$house man$-$hour is taken to be $1$ unit cost, an overtime man$-$hour costs $\backslash (50\backslash \%\backslash )$ more than this regular time, i$.$e$.,1\mathrm{.}5$ cost units, and the sub$-$contracting costs $2$ cost units. Backlogging of one man$-$hour costs $0\mathrm{.}3$ unit cost.

a$)$ Develop the maintenance plan and calculate its corresponding cost.

b$)$ Formulate the maintenance plan that satisfies the following constraints, and find its minimum$-$cost solution:

$-$ All grade $2$ work should be met by regular in$-$house crafts as much as possible.

$-$ If it is not possible to satisfy grade $2$ work by regular in$-$house crafts, use overtime.

$-$ No backlog is allowed for grade $2$ work.

$-$ The priority $1$ work could be met with overtime or sub$-$contract maintenance.