Tina\'s Material Company hauls gravel to a construction site, using a small truc

Question

Tina's Material Company hauls gravel to a construction site, using a small truck and a large truck. The carrying capacity and operating cost per load are given in the accompanying table. Tina must deliver a minimum of 160 cubic yards per day to satisfy her contract with the builder. The union contract with her drivers requires that the total number of loads per day is a minimum of 9. How many loads should be made in each truck per day to minimize the total cost?

Capacity in cubic yards for a small truck are 20 and for a large truck are 80.

Cost per load for a small truck are $82 and for a large truck are $47.

Explanation / Answer

The mathematcal model of problem is

Let small truck carries x cubic loads and large truck carries y cubic loads .

Thus

Cost [C=82x+47y]

sub to.

capacity

[20x+80y>=160]

and

union

[x+y>=9] , [x,y>=0]

Thus

Min [C=82x+47y]

s.t.

[x+4y>=8]

[x+y>=9]

[x,y>=0]

Now draw graphs of constraints

The solution of inequality is in first quadrant right of green line.

Thus pssible solutionare P(9,0) and Q(0,9)

C at P(9,0)=82 x 9+47 x 0=738

C at (0,9)=82 x 0+47 x 9= 423

Thus minimum of C= 423 and it will at Q(0,9)

i.e x=0 ,y=9 and cost =423.

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