Merton's production manager suggests purchasing Model $101$ or Model $102$ engines from an outside supplier in order to relieve the capacity problem in the engine assembly department. If Merton decides to pursue this alternative, it will be effectively "renting" capacity: furnishing the necessary materials and engine components, and reimbursing the outside supplier for labor and overhead. Should the company adopt this alternative? If so$,$ what is the maximum rent it should be willing to pay for a machine$-$hour of engine assembly capacity? What is the maximum number of machine$-$hours it should rent?

Merton is considering the introduction of a new truck, to be called Model $103.$ Each Model $103$ truck would give a contribution of $$2,000.$ The total engine assembly capacity would be sufficient to produce $5,000$Model$103s$ per month, and the total metal stamping capacity would be sufficient to produce $4,000$Model$103s.$ The new truck would be assembled in the Model $101$ assembly department, each Model $103$ truck requiring only half as much time as a Model $101$ truck.

$($a$)$ Should Merton produce Model $103$ trucks?

$($b$)$ How high would the contribution on each Model $103$ truck have to be before it became worthwhile to produce the new model?

Engines can be assembled on overtime in the engine assembly department. Suppose production efficiencies do not change and $2,000$ machine$-$hours of engine assembly overtime capacity are available. Direct labor costs are higher by $50\%$ for overtime production. While variable overhead would remain the same, monthly fixed overhead in the engine assembly department would increase by $$0\mathrm{.}75$ million. Should Merton assemble engines on overtime?

Merton's president, in arguing that maximizing short$-$run contribution was not necessarily good for the company in the long run, wanted to produce as many Model $101s$ as possible. After some discussion, it was agreed to maximize the monthly contribution as long as the number of Model $101$ trucks produced was at least three times the number of Model $102$s$.$ What is the resulting "optimal" product mix?