# A manufacturing company receives orders for engines from two aseembly 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...

A manufacturing company receives orders for engines from two aseembly 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 $30 per engine to ship to plant I and $40 per engine to ship to plant II. Plants I gives the manufacturing company $20 in rebates toward its products for each engine they buy, while plant II gives similar $15 rebates. The mnufacturer 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, subject to the given constraints?

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Let x be the number of engines that will be shipped to Plant I. And, let y be the number of engines that will be shipped to Plant II.

Then, express the given conditions in math form.

One of the given conditions is Plant I needs at least 45 engines. Its equivalent equation is:

`xgt=45`

Also, Plant II needs at least 32 engines. Its math form is:

`ygt=32`

Next, the company can send at most 140 engines to these assembly plants. So, its equivalent equation is:

`x + y lt= 140`

And, the last condition is that the manufacturer needs at least $1500 rebates. Since Plant I gives $20 in rebates toward its products for each engine they bought, while plant II gives similar $15 rebates, then the equation for this is:

`20x + 15ygt=1500`

And this simplifies to:

`4x+3ygt=300`

Hence, there are four constraints. These are:

EQ1: `xgt=45`

EQ2: `ygt=32`

EQ3: `x + ylt=140`

EQ4: `4x+3ygt=300`

Next, determine the objective function. Since the goal is to minimize the shipping cost, then consider the cost of delivering the engines to each plant.

Since it costs $30 per engine to ship to Plant I, then the shipping costs of delivering x number of engines to this plant is:

`C_I= 30x`

For plant II, its cost is $40 per engine. So, the shipping costs of delivering y number of engines to this plant is:

`C_(II)=40y`

And, the total shipping costs for the two plants is:

`C_I + C_(II) = 30x + 40y`

Thus, the objective function is:

`z=30x + 40y`

To determine the values of x and y that will minimize z, graph the four inequalities. Then, consider only the region that satisfies them all.

Hence, the region that satisfy the four inequalities is:

*(Note: Green - Boundary line of EQ1. Red - Boundary line of EQ2.*

*Yellow - Boundary line of EQ3. Purple - Boundary line of EQ4. )*

Then, consider the corner points of the bounded region. These are

(45,95)

(45,40)

(51,32) and

(108,32)

Plug-in these points to the objective function z=30x+40y.

For (45,95), the resulting value of z is:

`z=30(45) + 40(95)=5150`

For (45,40), the resulting value of z is:

`z=30(45)+40(40)=2950`

For (51,32), the resulting value of z is:

`z=30(51)+40(32)=2810`

And for (108,32), the resulting value of z is:

`z=30(108)+40(32)=4520`

So among the four corner points, it is point (51,32) that yields a minimum value of z which is 2810.

T**herefore, to minimize the shipping costs, 51 engines should be sent to Plant I and 32 engines to Plant II.**

The 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 $30 per engine to ship to plant I and $40 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 rebates of $15. The manufacturer estimates that they need at least $1500 in rebates to cover products they plan to buy from the two plants. The shipping cost of the company has to be minimized.

From the information given, the rebate given by Plant I is $20 and the shipping cost to send an item to this plant is $30. If the company supplies to Plant II, it only gets a rebate of $15 while the shipping cost is higher at $40.

To minimize shipping cost the company should send as many engines as it can to Plant I. If it is obliged to send at least 32 engines to Plant II, that should be done. The rebate received for this is 32*15 = 480. As the company requires a rebate of $1500, the remaining amount can be got if 51 engines are sent to Plant II. The total shipping costs incurred are 51*30 + 32*40 = $2810.

The company should send 32 engines to Plant II and 51 engines to Plant I.