# Linear Programming 1: Maximization -Extreme/Corner Points

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Welcome to the first video in this Linear

Programming series. In this video, we will graphically solve a

basic maximization linear programming problem using the extreme or corner point approach.

What we have here is a linear programming or LP model.

The X and Y in this model are referred to as Decision Variables.

They tell us what quantity to buy, produce, sell, or transport, and so on.

The “2X + 5Y” here is referred to as the objective function which we want to maximize.

In linear programming, we either maximize or minimize the objective.

The next 2 lines are called constraints. They are restrictions that shape how you attain

the objective. Let’s call them C1 and C2 for reference

purposes. Since we’re dealing with real life objects,

we do not expect them to have negative values. So the last line here tells us that both X

and Y have to be greater or equal to 0. We call them the non-negativity constraints.

To solve this model graphically, we begin by finding points that satisfy the constraint

lines. For the first constraint, when x = 0, y equals

8. And when y = 0, x = 16.

For the second constraint, when x = 0, y equals 15.

And when y = 0, x = 9. Now when drawing the graph, we usually just

stay in the first quadrant here where both X and Y are positive, because of the non-negativity

constraints. So for the first constraint, we have the points

(0, 8) and (16, 0). We join those two points for the constraint

line. We do the same for constraint 2: (0, 15) and

(9, 0) and then draw the constraint line. Since these constraints are less than or equal

to constraints, they will be satisfied in the region below the lines towards the origin.

Therefore, the region satisfying both constraints simultaneously is this one here.

It is called the feasible region. That is, any point in this region is a feasible solution.

In particular, the optimal or the best solution will occur at an extreme point or corner point

of the feasible region. These are the corner points for this feasible

region. Let’s label them 1 to 4. The optimal solution to this Linear Programming

problem will occur in at least one of them. To decide which one is optimal, we will find

the coordinates of the points, plug them into the objective function and then choose the

best. At corner point 1, the coordinates are clearly

(0, 0). At point 2, (0, 8).

At point 4, (9, 0). Now those are easy to see.

For corner point 3, we can see that the coordinates are (6, 5) by eyeballing. That is, looking

at it very closely. But eyeballing is not usually the best way to go when finding the

intersection of 2 lines, especially when manually drawing the graph.

So let’s see a way to solve the 2 equations simultaneously to determine the actual coordinates.

Here are the lines for the 2 constraints. Now suppose I choose to eliminate X, note

that the coefficient of X here in C2 is 5, then I can simply multiply the first equation

by 5 to give 5X + 10 Y = 80. And then subtract C2 from the new equation.

So 5X cancels 5X. 10Y minus 3Y is 7Y.

And 80 minus 45 gives 35. And on dividing both sides by 7, we have Y

= 5. To now find X, we substitute Y = 5 into any

of these 3 equations. Suppose we choose C1. Then we have X + 2(5)

= 16. That is, X + 10 = 16.

X = 16 -10 And X = 6.

So the X, Y coordinates for extreme point 3 are indeed (6 and 5).

Next we determine the optimal solution point by finding the corner point that gives the

best value of the objective function. The objective function was to maximize 2X + 5Y. As you case see here the objective function

or its value is sometimes represented with Z.

So now, the x-y coordinates at point (1) are (0, 0).

And substituting that in the objective function gives 2(0) + 5(0) which equals 0.

At point (2) we have (0, 8), so the objective function value is 2(0) + 5(8) which equals

40. The coordinates at point 3 are (6, 5), so

Z equals 2(6) + 5(5) which equals 37. And finally at Point (4) with (9, 0), Z = 2(9)+ 5(0) which gives 18. So here we have it. Point (2) provides the

highest value of the objective function. So the optimal solution occurs at point (2) and

it is X = 0, and Y = 8. And The corresponding objective function value

is 40. And that concludes the solution to this LP problem. Thanks for watching.

Adesoji AluThanks so much

Munisa RaimovaGreaaat thanks

Jazzmyne PipkinsYou are my lifesaver!! Thank you!!

Salma Abdallahthank you

Payal PhalkeThanks Sir , you saved me !

Shakhzoda SabirovaThank you a lot

jackie reyesThank you so much for this video my professor doesn't explain linear programming well. I was so stressed since midterms are coming up but I feel more confident about doing the exam now! 🙂

Solomon MollaWow this is best explanation video thank you a lot

Alialdin MohamadThat was very helpful man,thanks

Mritunjay SinghYour way of explanation is awesome.

Sonu MU have given very clear explanation Sir. Tnq u so much

undermygarageexcellent, and well explained. thank you!!

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The SourceWonderful, sir. I loved it. Respects and thanks!

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awesomeI watch it first time and understand

Thank you so much

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Athank you so much!!

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Goran PavlovicBRAVO. This is best explained video on YT.

Tencha RuizThank you so so much. Your summarization was spot on and the detail between the corner points and Zeda was the missing link (6,5), this was the learning I needed to help with understanding the basics. I was worried about missing this part and building a bad foundation for the next few weeks in class.

malayali cornerThnk u sir

sharu lathaclear and easy to understand. ……tq a lot!

anna babuwhat if more than one corner point is unknown…pls reply fasttttttttttt

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i a m a b d o r a h m a nThanks man

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Haniz LandGreat way of explanation – thank u !

Timothy John CorpuzThe best and easily digestible for LP so far. This is better than our An hour and a half session explaining LP by our professor

oyshee OYSHEEThanks for video….Its vrry helpful….

Alka TiwariYou explained us very well. Thank Sir for helping other students.

Brian GlusovichI just understood 3 days of lecture in 5 minutes, thank you!