Problem $2(60$ marks$)$

Homemade fruit pie

A retired couple supplements their income by making fruit pies, which they sell to a local grocery store. During the month of September, they produce apple and grape pies. The apple pies are sold for $$5\mathrm{.}20$ and the grape pies are sold for $$4\mathrm{.}50$ to the grocer. The couple is able to sell all of the pies they produce owing to their high quality. They use fresh ingredients. Flour and sugar are purchased once each month. For the month of November, they have $1,200$ cups of sugar and $2,500$ cups of flour. Each apple pie requires $2$ cups of sugar and $3$ cups of flour, and each grape pie requires $1\mathrm{.}5$ cups of sugar and $3$ cups of flour. Each cup of sugar costs $20$ cents and each cup of flour costs $10$ cents. If the couple working together can make an apple pie in $4\mathrm{.}5$ minutes and a grape pie in $4$ minutes. They plan to work no more than total $50$ hours.

The couple is interested in maximizing profit and hired you to advise them about how many apple pies and grape pies should be produced to obtain the largest possible profit. Please provide them with a linear programming model in graphical form to accomplish their objective.

$($b$)$ If the couple requests you to develop the algebraic model, what equations you will provide him?

Let X $=$ number of apple pies to produce

Y $=$ number of grape pies to produce

i$.$ What would be the profit per unit of apple pie and grape pie? $(2$ marks$)$

Profit Calculation:

Apple pie:

Grape pie:

ii$.$ What would be the objective function? $(2$ marks$)$

iii. What would be the constraints equations? $(4$ marks$)$

iv$.$ Identify number of apple pies and grape pies will be produced and the total profit at the corner point a$,$ b$,$ c$,$ d$,--?(4$ marks$)$

$$

$($c$)$ With respect to the primary decisions facing the couple, what recommendation would you provide? $(1$ mark$)$

$($d$)$ How much profit will the couple realize if they operate at the optimal solution? $(1$ mark$)$

Your Answer:

$($e$)$ How much sugar, flour and labour hours they can use according to this solution? $(9$ marks$)$

i$)$ Sugar for producing apple pies $(1$ mark$)$

Your Answer:

ii$)$ Sugar for producing grape pie $(1$ mark$)$

Your Answer:

iii$)$ Flour for producing apple pies $(1$ mark$)$

Your Answer:

iv$)$ Flour for producing grape pie $(1$ mark$)$

Your Answer:

v$)$ Labour minutes for producing apple pie $(1$ mark$)$

Your Answer:

vi$)$ Labour minutes for producing grape pie $(1$ mark$)$

Your Answer:

vii$)$ Which constraint$($s$)$ is$/$are binding and which is$/$are non$-$binding? $(3$ marks$)$

$($f$)$ Determine the allowable increase and allowable decrease of the profit from apple pie. $(4$ marks$)$

$($g$)$ How much profit would increase if the couple could increase his labour hour by $1$ extra hour? $(4$ marks$)$

Your Answer:

$($h$)$ Determine the allowable increase for the Labour hour constraint. $(2$ marks$)$

Your Answer:

$($i$)$ Determine the allowable decrease for the labour hour constraint? $(2$ marks$)$

Your Answer:

$($j$)$ In an effort to increase profit, should the couple try and get more labour hours to produce more apple pies and grape pies? Circle the correct response and explain. $(2$ marks$)$

Yes No Cannot tell

Explain:

$($k$)$ How much profit would increase if the couple could increase sugar by $1$ extra unit? $(3$ marks$)$

Your Answer:

$($l$)$ Determine the allowable increase for the sugar constraint. $(2$ marks$)$

Your Answer:

$($m$)$ Determine the allowable decrease for the sugar constraint? $(2$ marks$)$

Your Answer:

$($n$)$ In an effort to increase profit, should the couple try and get more sugar to produce more apple pies and grape pies? Circle the correct response and explain. $(2$ marks$)$

Yes No Cannot tell

Explain: $($o$)$ If the couple could add $5$ extra hours $(300$minutes$)$ of labour or $100$ cups of sugar, which option should he choose? How much will profit increase? $(2$ marks$)$