$6\mathrm{.}2$ Make or Buy: A sudden increase in the demand for smoke detectors has left Acme Alarms with insufficient capacity to meet demand. The company has seen monthly demand from its retailers for its electronic and battery$-$operated detectors rise to $20,000$ and $10,000,$ respectively, and Acme wishes to continue meeting demand. Acme$$s production process involves three departments: Fabrication, Assembly, and Shipping. The relevant quantitative data on production and prices are summarized below.

DepartmentMonthly Hours AvailableHours$/$Unit $($Electronic$)$Hours$/$Unit $($Battery$)$Fabrication$20000\mathrm{.}150\mathrm{.}10$Assembly$42000\mathrm{.}200\mathrm{.}20$Shipping$25000\mathrm{.}100\mathrm{.}15$Variable cost$/$unit$$18\mathrm{.}80$$$16\mathrm{.}00$Retail price$$29\mathrm{.}50$$$28\mathrm{.}00$

The company also has the option to obtain additional units from a subcontractor, who has offered to supply up to $20,000$ units per month in any combination of electronic and battery$-$operated models, at a charge of $$21\mathrm{.}50$ per unit. For this price, the subcontractor will test and ship its models directly to the retailers without using Acme$$s production process.

a$.$ Acme wants an implementable schedule, so all quantities must be integers. What are the maximum profit and the corresponding make$/$buy levels?

b$.$ Compare the maximum profit in $($a$)$ to the maximum profit achievable without integer constraints. Does the integer solution correspond to the rounded$-$off values of the noninteger solution? By how much $($in percentage terms$)$ do the integer restrictions alter the value of the optimal objective function?