Solve the optimization problem. Maximize P=12 x+15 y subject to the following constraints. {

Solve the optimization problem. Maximize P=12 x+15 y subject...

Key Concepts

Corner Point Method

The corner point method is a technique used to solve linear programming problems by evaluating the objective function at each of the vertices (corner points) of the feasible region. Because the optimal solution in a linear programming problem occurs at one of these vertices, this method efficiently identifies the best outcome.

Feasible Region

The feasible region is the set of all possible points that satisfy all the constraints of the linear programming problem. It often forms a convex polygon or polyhedron in the decision variable space, and the optimal solution is typically found at one of the vertices of this region.

Objective Function

The objective function is the function that needs to be maximized or minimized in an optimization problem. It represents the goal of the problem, such as maximizing profit or minimizing cost, and is typically expressed as a linear combination of the decision variables.

Linear Programming

Linear programming is a method used in optimization where an objective function is maximized or minimized subject to a set of linear constraints. This technique is widely applied in various fields such as operations research, economics, and engineering to find the best outcome in a mathematical model whose requirements are represented by linear relationships.

Constraints

Constraints in a linear programming problem are the linear equations or inequalities that define the limitations within which the objective must be optimized. They restrict the values that the decision variables can take and shape the feasible region of the problem.

Transcript

00:01 Here in this problem we have to maximize the function p is equals to 12 times x plus 15 y under the given constraints that is x is greater than or equals to 3 y is greater than or equals to 1 x is less than or equals to 10 and y is less than or equals to 14 now first of all we have to graph this inequality on a graph paper so let's take a graph paper now so this is a graph paper now first of all we'll find the solution set for the inequality x greater than or equals to 3 for that i'll be graphing the equation x equal to 3 so this line would represents x equal to 3 so this line will represents x equal to 3 now the solution set for this inequality would lie on either side of the line either this side or this side we have to check and for that we have to take a point on either side of the line suppose i take 4 comma 0 now we'll check the inequality at this point whether it's satisfy or not so when 4 comma 0 here x coordinate is 4 so 4 is greater than or equals to 3 if i plug the value of x here we get 4 is greater than or equals to 3 which is true so the solution set for this inequality would lie towards this side now again we'll do the same thing for the second inequality that is y is greater than or equals to 1 for that i'll be graphing the equation y equals to 1 so this line will represents y equals to 1 now again we have to check whether the solution set for this inequality would lie this side or this side of the line let's draw this line with a different color so that we can differentiate the two inequality easily it should be a straight line so this line will represent y is greater than or equals to one now again we have to check whether the solution set lie this side or this side for that we have to take a point on either side of the line so i'm taking the point on this side that is 2 .0 now we'll check at this point whether the inequality is true or false so here the x coordinate is 0 and the y coordinate is 2 so we're going to plug the value of y coordinate here so we get 2 is greater than or equals to 1 which is true 2 is greater than 1 that is true so the solution set for this inequality would lie this side now we'll be graphing the same third inequality that is x is less than or equals to 10 for that i'll be graphing the equation x equal to 10 so this line will represent x is equals to 10 now again we have to check that the solution set would lie this side or this side of the line of the green line so for that we will be again taking a point on either side of the line so i'm taking the point 8 comma 0 this point and we'll go into check whether at this point the inequality is satisfied or not so the value of x coordinate here is 8 so we're going to plug the value of x coordinate as 8 here so we get 8 is less than or equals to 10 which is true it is less than 10 that is true so the solution set for this inequality would lie this side now the final inequalities left that is y is less than or equals to 14 so for that i'll be graphing the equation y equals to 14 so this line should be a straight line you can use your ruler also to make this line so this line will represent y is equals to 14 now the solution set for the inequality would lie on either side of this orange line so i'm taking a point on this side to check whether the inequality is true or false at this point here x coordinate is 0 and the y coordinate is 15 so we're going to plug the value of y coordinate here we'll get 15 is less than or equals to 14 which is four false 15 is not less than 14 15 is greater than 14 the solution set would not like this side it would like this side because at this side the inequality fails so this side would be the solution set now we can conclude that this region would be the solution set for the given set of inequality this is a feasible region in which our function will be maximized and we know that the function attains maximum value at one or more corner points and there are four corner points.

06:28 This is first corner point, second corner point, third corner point and the fourth corner point.

06:34 Now we'll be writing the coordinate of these corner points...

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