Question

A manufacturing company receives orders for engines from two assembly plants. Plant I needs at least 45 engines, and plant II needs at least 32 engines. The company can send at most 140 engines to these assembly plants. It costs $35 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 $15 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

Solution: First we will create the linear equation required to solve the problem , and then for solving the linear variable equation we can use excel sover (or online linear equation solving calculator).

Now, Let the number of engines supplied to Plant 1 be x

and the number of engines supplied to Plant 2 be y.

Given, Plant 1 needs atleast 45 engines and plant 2 needs atleast 32 engines.

Therefore, x >= 45

y >= 32

Again, the company can send atmost 140 engines , so x + y <= 140.

Now, plant 1 gives 20$ rebates for receiving 1 engine , so total rebate given by plant 1 = 20x

Now, plant 2 gives 15$ rebates for receiving 1 engine , so total rebate given by plant 2 = 15x

Manufacturer needs atleast 1500$ rebates , hence we have

20x + 15y >= 1500

Also, it cost 35$ for shipping to plant 1 and 50$ for shipping to plant 2 , hence total shipping cost is 35x + 50y

We have to minimize the shipping cost, considering the constraint. So we have the following scenario

Minimize, 35x + 50y

Constraint : x >= 45

  y >= 32

  x + y <= 140

20x + 15y >= 1500

Putting these values in the excel solver (or in online equation solver) , we will get

x= 51, y = 32.

So, number of engines send to plant 1 = 51

number of engines send to plant 1 = 32

Also, minimum cost = 35x + 50y  = (35*51 +50*32 ) = 3385