Find the maximum value of the objective function f(x, y)=8 x+5 y given the constraints shown. {

Find the maximum value of the objective function f(x, y)=8...

Key Concepts

Feasible Region

The feasible region is the set of all points that satisfy all the constraints in a linear programming problem. It is typically a convex polygon or polyhedron, and any optimal solution must lie within this region.

Constraints

Constraints in linear programming are the set of linear inequalities or equations that limit the values of the decision variables. They represent the conditions or restrictions that must be satisfied for the solution to be considered feasible.

Corner Point Theorem

The corner point theorem, also known as the fundamental theorem of linear programming, states that if an optimal solution exists for a linear programming problem, then at least one of the vertices, or corner points, of the feasible region will be an optimal solution. This theorem forms the basis for methods that solve linear programming problems by evaluating the objective function at these extreme points.

Linear Programming

Linear programming is an optimization technique where the goal is to maximize or minimize a linear objective function subject to a set of linear constraints. It is widely used in various fields to allocate resources efficiently and to solve decision-making problems under given conditions.

Objective Function

The objective function is the expression that needs to be optimized in a linear programming problem. It defines the criterion for success—whether to maximize profit, minimize cost, or achieve any other specific goal—using a linear combination of decision variables.

Transcript

00:01 So we're going to be given an objective function f of x and y, and that's going to be equal to 8x plus 5y.

00:14 So we want to optimize or maximize this function, and it's going to be really easy to do it without any constraints.

00:21 So we're going to add some constraints.

00:23 These constraints are going to come in the form of inequalities.

00:26 The first one is going to be 2x plus y is less than or less than or less.

00:32 Equal to 7, x plus 2y is less than or equal to 5, then we have the two inequalities that constrain us to the first quadrant, which are x is greater than equal to 0, y is greater than or equal to 0.

00:55 These values here are just constrain us to real numbers or positive numbers, including 0, and these two set the other boundaries.

01:03 So let's go through and let's graph these two inequalities out on our graphing calculators, but first we got to get into y equals mxxb format.

01:13 So here we just subtract both sides by two.

01:15 We get y is less than or equal to negative two x plus seven.

01:22 And here we have to subtract by x and divide both sides by two.

01:26 So we get y is less than or equal to negative one half x plus 5 halves.

01:38 Now let's go through and let's graph these out.

01:42 So you can grab out the first one, which is y is less than or equal to negative 2x plus 7.

01:54 That gives us our first region, right? then we have x, oops, sorry, y minus 1 half x plus 5 plus 5 half x plus 5 halves.

02:13 Oh, sorry, that should be a less than or equal to.

02:18 So that's what we want.

02:20 Then we have, of course, x is greater than equal to zero, and y is greater than or equal to zero.

02:29 So i'm going to unmark these because we know we're working in the first quadrant.

02:33 It just makes it a little easier to see.

02:35 Therefore, we know that we have to have our solution is within this region, within the solution region right here, or the feasible region.

02:46 Okay, i'll mark those just so you don't forget them.

02:51 Within this feasible region.

02:53 And we also know that it's going to be one of the, one of these points here could be 0 .0.

03:02 It could be 3 .50.

03:05 It could be 1 .8 comma 3 .4 or it could be 0 .2 .5...

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