minimization

[meminusgarfield]

Not sure which section I need to post this in, but I am having trouble with a linear programming problem. I am supposed to use graphical methods to minimize:

z=4x+7y

subject to:
x-y >= 1
3x+2y>= 18
x>= 0
y>=0

The answer is in the back of the book, but I don't understand how they got it. My textbook skips a lot of the steps. I used the echelon method to find x and y, but I don't know where to go from there. Any help would be much appreciated.

 

[mmm4444bot]

I am supposed to use graphical methods

I used the echelon method to find x and y Do you mean the lines' intersection point ?

:?: Have you graphed these two inequalities in Quadrant I, yet ?

x - y ? 1

3x + 2y ? 18

The area where they overlap forms a shape, for x and y non-negative (i.e., the constrained solution area on the graph).

You need to evaluate z at the two vertex points because, in linear programming, the minimum (or maximum) always occurs at one of the vertex points of the solution area.

The intersection point is one of the vertices; if you graph the inequalities together, you'll see the other.

If you would like more help, please be more specific about any parts where you're stuck and what you've already tried.

PS: If your book "skipped" graphing, too, you can see plenty of lessons and examples HERE.

 

[meminusgarfield]

Thank you for getting back so quickly - I sure appreciate it. I was actually able to figure out the minimization after staring at it for a couple of hours. Our book took us on through maximization, and the corner point theorum. No problem there, but when they asked us to formulate linear inequalities from story problems I got lost again. Here is an example I have been working on (to no avail).

The Meyers Company produces small engines for several manufacturers. The company receives orders from two assembly plants for their Top-flight engine. Plant 1 needs at least 45 engines, and plant 2 needs at least 32 engines. The company can send at most 90 engines to these two assembly plants. It costs $30 per engine to ship to plant 1 and $40 per engine to ship to plant 2. Plant 1 gives Miers $20 in rebates toward its products for each engine they buy, while plant 2 gives similar
$15 rebates. Miers estimates that they need $1200 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?

I know how to graph and solve the problems once they are set up, but I don't know how to set up the problem from the example above. The book shows z as a varable, with x and y being constraints. I would show my work to you, but I am stuck at the beginning. If I could figure out how to set up the problem, I am sure I could solve it.

 

[mmm4444bot]

As with most word problems, start by reading the entire exercise; then, assign symbols to represent any specific unknown values for which the exercise asks.

:?: What does this exercise ask for ?

How many engines should be shipped to each plant

Pick variable names for these two unknowns. Write them down, in case you forget what the symbols mean.

Let x = the number of engines shipped to plant 1

Let y = the number of engines shipped to plant 2

Now go through the given information again, line by line, and look for relationships involving the numbers x and y.

Plant 1 needs at least 45 engines, and plant 2 needs at least 32 engines.

x ? 45

y ? 32

The company can send at most 90 engines to these two assembly plants.

x + y ? 90

It costs $30 per engine to ship to plant 1 and $40 per engine to ship to plant 2.

30x represents dollars to ship x engines to plant 1

40y represents dollars to ship y engines to plant 2

The sum of these two expressions represents the total shipping cost.

Plant 1 gives Miers $20 in rebates toward its products for each engine they buy, while plant 2 gives similar $15 rebates.

20x represents dollars rebated from plant 1

15y represents dollars rebated from plant 2

Miers estimates that they need $1200 in rebates to cover products they plan to buy from the two plants.

20x + 15y ? 1200

What is the minimum [total shipping] cost ?

Let z = total shipping cost

z = 30x + 40y

Graph the four inequalities above, and then evaluate z at the vertex points of the solution area.

I welcome specific questions.



PS: With future exercises, please start a new thread for each new exercise.

 

[meminusgarfield]

thank you.