QUESTION $3$: LINEAR PROGRAMMING

$30$ MARKS

Jack Turnbull is a young pig farmer who has modernised his farm and is exploring all areas to cut cost. One of his major cost is that of pig feed. He has just acquired a lucrative contract from a major slaughtering house, which states some minimum standard for pig feed which must be maintained. Jack currently buys two brands of feed, Lean Growth and Low Fat Growth. A pound of Lean Growth cost $$\mathrm{.}02$ and a pound of Low Fat Growth cost $$0\mathrm{.}03.$

Jack would like to determine the lowest cost diet that meets the minimum monthly feeding requirement per pig.

The table below contains relevant information about the composition of Lean Growth and Low FatGrowth, as well as the minimum monthly requirement for each nutritional ingredient for pigs as stipulated by the slaughter house.

Composition of Each Pound of Feed

$\backslash$table$[[$INGREDIENT$,$LEAN GROWTH,$\backslash$table$[[$LOW FAT$],[$GROWTH$]],\backslash$table$[[$MINIMUM$],[$MONTHLY$],[$REQUIREMENT$]]],[$A$,5oz.,10oz.,90oz.],[$B$,4oz.,3oz.,48oz.],[$C$,0\mathrm{.}5oz.,0,1\mathrm{.}5oz.],[$D$,0,0\mathrm{.}75oz.,1\mathrm{.}5oz.]]$

a$)$ Express the decision variables in terms of $x_1$ and $x_2$

b$)$ Derive Jack's objective function in terms of $x_i$ and $x_2$

c$)$ What are the constraints under which Jack has to operate

d$)$ Graphically illustrate the Feasible Region

e$)$ Label the corner points using coordinates e$.$g B$(2,6)$

f$)$ What is the combination that optimises the objective function

g$)$ Determine what it will cost Jack based on his objective

$2$ marks

$3$marks

$7$ marks

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$6$ marks

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$3$ marks