For the following problem, $($i$)$ formulate, $($ii$)$ solve graphically by sketching the

constraints and a few isoprofit lines, $($iii$)$ solve using the simplex tableau.

A small manufacturer produces two products: plastic mugs which sell at a profit of $5$ per mug,

and plastic boxes which sell at a profit of $4$ per box. Each mug requires $1$ kg of plastic to

manufacture, and each box requires $2$ kg of plastic. The supply of plastic is limited to $20$ kg per

day. The manufacturer has a total of $16$ hours of manufacturing labour available per day. Each

$2$

mug requires $2$ hours of labour and each box requires $1$ hour of labour. Market limits dictate that

mugs should be manufactured at a rate of no more than $6$ per day. At what rate should the

manufacturer make mugs and boxes to maximize the profit rate?