$2.$

Example

A small firm builds two types of garden shed.

Type A requires $2$ hours of machine time and $5$ hours of craftsman time.

Type B requires $3$ hours of machine time and $5$ hours of craftsman time.

Each day there are $30$ hours of machine time available and $60$ hours of craftsman time. The profit on each type A shed is $60$ and on each type B shed is $84.$

Formulate the appropriate linear programming problem.

Solution

$($a$)$ Unknowns

Define

x $=$ number of Type A sheds produced each day,

y$=$ number of Type B sheds produced each day.

$($b$)$ Constraints

Machine time:

$2$x$+3$y

Craftsman time:

$5$x$+5$y

and

x$20,$y$20$

$($c$)$

Profit

P$=60$x$+84$y

So$,$ in summary, the linear programming problem is

maximise P$=60$x$+84$y

subject to $2$x $+3$y

x$+$ y

x $>=0$

y $>=$Example

A small firm builds two types of garden shed.

Type A requires $2$ hours of machine time and $5$ hours of

craftsman time.

Type B requires $3$ hours of machine time and $5$ hours of

craftsman time.

Each day there are $30$ hours of machine time available and $60$

hours of craftsman time. The profit on each type A shed is $60$

and on each type B shed is $84.$

Formulate the appropriate linear programming problem.

Solution

$($a$)$ Unknowns

Define

$x=$ number of Type A sheds produced each day,

$y=$ number of Type B sheds produced each day.

$($b$)$ Constraints

Machine time: $,2x+3y30$

Craftsman time: $,5x+5y60$

and $,x0,y0$

$($c$)$ Profit

$P=60x+84y$

So$,$ in summary, the linear programming problem is

maximise $P=60x+84y$

subject $to2x+3y30$

$x+y12$

$x0$

$y00$