A niche bicycle manufacturer manufactures bicycles in two different sizes. David, the company's owner$-$manager, has just received the following forecasts for the next six months.

$\backslash$table$[[,,],[,,$N$],[1,0,,$o$],[00,50,$v$.],[$uni$,0,$D$],[$ts$,$un$,$e$],[90,$its,c$.],[0,50,],[,0,]]$

Jan. $600300$

Feb. $700500$

Mar. $1,100400$

Apr. $1,100600$

a$.$ Under what circumstances is it appropriate to develop just one plan rather than two $($one for each size$)?$

b$.$ Suppose forecasts for the two sizes are summed to obtain one forecast for each month. Currently David employs $27$ full$-$time, highly skilled employees, each of whom can produce $50$ bikes per month. Because skilled labour is in short supply in the area, David would like to keep them permanently. There is no inventory on hand at present, but David would like to have $300$ units on hand at the end of April. A maximum of $200$ bikes can be produced during overtime each month. Determine the minimum cost production plan using these unit costs: Regular $$50$ Inventory $$2\mathrm{.}00$ per month Overtime $$75$ Back order $$10\mathrm{.}00$ per month $($Hint: Use overtime in November, December, and April.$)$