A manufacturing company receives orders for engines from two assembly plants.

Question

question continued:  Plant I needs at least 45 ​engines, and plant II needs at least 32 engines. The company can send at most 120 engines to these assembly plants. It costs ​$30 per engine to ship to plant I and ​$50 per engine to ship to plant II. Plant I gives the manufacturing company ​$20 in rebates toward its products for each engine they​ buy, while plant II gives similar ​$10 rebates. The manufacturer estimates that they need at least ​$1500 in rebates to cover products they plan to buy from the two plants. How many engines should be shipped to each plant to minimize shipping​ costs? What is the minimum​ cost?
How many engines should be shipped to each plant to minimize shipping​ costs, subject to the given​ constraints?

The number of engines to send to plant I is?
The number of engines to send to plant II is?
 

Answer 1

Let x=number of engines for Plant I and y=number of engines for Plant II.

The constraints are represented by border lines on the diagram.

The red lines indicate the first constraint concerning the minimum number of engines for each plant. The area to the right of the vertical red line signifies x≥45, while the area above the horizontal red line signifies y≥32.

The manufacturer’s production capacity is that x+y≤120, represented by the area under the green capacity line. Combining these first two constraints, we have the area ABCD. The shipping costs in total are 30x+50y.

The rebate constraint is 20x+10y≥1500, which reduces to 2x+y≥150, represented by the area above the blue line.

Combining all the constraints we have the area of ABCDE. The values of x and y must reside in this area. 

If we calculate the shipping costs 30x+50y for each vertex of the figure we get:

A=$5100, B=$4500, C=$3850, D=$3370 and E=$4350.

So D has the minimum total shipping costs, when Plant I has 59 engines ($1770 shipping costs) and Plant II has 32 engines ($1600 shipping costs). The manufacturer receives exactly $1500 in rebates.