How many engines should be shipped to each plant to minimize shipping? costs, subject to the given? constraints?

Question

A manufacturing company receives orders for engines from two assembly plants. Plant I needs at least 4545 ?engines, and plant II needs at least 3232 engines. The company can send at most 140140 engines to these assembly plants. It costs ?$3030 per engine to ship to plant I and ?$4040 per engine to ship to plant II. Plant I gives the manufacturing company ?$2020 in rebates toward its products for each engine they? buy, while plant II gives similar ?$1515 rebates. The manufacturer estimates that they need at least ?$15001500 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?

Answer 1

Betty,

The special symbol changes made by the editor have made this a little confusing, so I hope I'm understanding the question correctly.

 

This type of question generally requires that you write some equations based on the constraints, find their intersections (usually by graphing) and use the coordinates of those intersections to plug into the cost equation for comparison. Graphing is not practical in this editor, so the best I can do is show you the equations and discuss.

 

If x = engines for Plant I  and y = engines for Plant II, then:

 

x ≥ 4545       y ≥ 3232       x + y ≤ 140140

 

rebates = 2020x + 1515y ≥ 15001500

 

shipping cost = 3030x + 4040y

 

If we graph the first three equations we get a large triangle that the values must fall within. When we plot the rebates equation, another "at least" equation, we fin that it is completely outside and below the triangle of the first three. That means that it has no effect on the outcome. The only points we need to check are the vertices of the triangle. A(4545, 135595), B(136908, 3232), C(4545, 3232)

 

Plug each of these into the cost equation and we get:

 

A = $561,575,150

 

B = $427,888,520

 

C = $ 26,828,630

 

So point C gives you the number of engines to ship and the least cost. I can't help but think something is missing from this question but there's nothing else I can offer unless you have more information. I hope this at least helps your understanding.