The Vertices Of A Feasible Region Are \[$(14,2)\$\], \[$(0,9)\$\], \[$(6,8)\$\], And \[$(10,3)\$\]. What Is The Maximum Value Of The Objective Function \[$P\$\] If \[$P = 180x + 250y\$\]?A. 2,940 B.

by ADMIN 200 views

The Maximum Value of the Objective Function P

In linear programming, the objective function is a mathematical expression that represents the goal of the problem. The maximum value of the objective function is the highest value that the function can attain, given the constraints of the problem. In this article, we will discuss how to find the maximum value of the objective function P, given the vertices of a feasible region and the equation of the objective function.

The feasible region is the set of all possible solutions to the problem. It is the region in the coordinate plane that satisfies all the constraints of the problem. In this case, the vertices of the feasible region are given as (14,2), (0,9), (6,8), and (10,3). These points represent the maximum and minimum values of the variables x and y, subject to the constraints of the problem.

The objective function P is given by the equation P = 180x + 250y. This equation represents the goal of the problem, which is to maximize the value of P. The coefficients of x and y in the equation represent the weights or priorities assigned to each variable. In this case, the weight of x is 180 and the weight of y is 250.

To find the maximum value of P, we need to evaluate the objective function at each of the vertices of the feasible region. This is because the maximum value of P must occur at one of the vertices, since the objective function is linear and the feasible region is a polygon.

Let's evaluate the objective function at each of the vertices:

  • At (14,2), P = 180(14) + 250(2) = 2520 + 500 = 3020
  • At (0,9), P = 180(0) + 250(9) = 0 + 2250 = 2250
  • At (6,8), P = 180(6) + 250(8) = 1080 + 2000 = 3080
  • At (10,3), P = 180(10) + 250(3) = 1800 + 750 = 2550

The maximum value of the objective function P occurs at the vertex (6,8), where P = 3080. This is the highest value that the objective function can attain, given the constraints of the problem.

The maximum value of the objective function P is 3080.

The problem of finding the maximum value of the objective function P is a classic example of a linear programming problem. The solution involves evaluating the objective function at each of the vertices of the feasible region and selecting the vertex that yields the highest value. This approach is known as the "vertex method" or "corner point method" in linear programming.

In this article, we have discussed how to find the maximum value of the objective function P, given the vertices of a feasible region and the equation of the objective function. We have also provided a step-by-step solution to the problem, using the vertex method. This approach is useful for solving linear programming problems, where the objective function is linear and the feasible region is a polygon.

  • [1] Chvatal, V. (1983). Linear Programming. W.H. Freeman and Company.
  • [2] Dantzig, G.B. (1963). Linear Programming and Extensions. Princeton University Press.
  • [3] Hillier, F.S., & Lieberman, G.J. (2015). Introduction to Operations Research. McGraw-Hill Education.
  • Linear programming
  • Objective function
  • Feasible region
  • Vertex method
  • Corner point method
  • Maximum value
  • Optimization
  • Operations research