Question

A linear programming model is given as follows:

Minimize Z = 8x₁ + 6x₂
Subject to 4x₁ + 2x₂ 20
-6x₁ + 4x₂ 12
x₁ + x₂ ≥6
X₁, X₂ ≥0
(Solve by geometric method, do not use Excel Solver)
Define the feasible solution area and obtain the optimal solution graphically

Answer 1

The question involves using a geometric method to solve a linear programming problem by graphing the constraints to find the feasible solution area, then identifying the optimal solution by applying the objective function Z.

Step-by-step explanation:

The student's question involves finding the optimal solution of a linear programming problem through the geometric method. The constraints given need to be translated into linear equations and graphed to define the feasible solution area. Constraints will appear as straight lines on a graph, and the feasible area will be the region where all the inequalities are satisfied simultaneously.

The objective function, Z = 8x₁ + 6x₂, will then be used to find the minimum value within this feasible region. Graphically, one would draw lines representing different values of Z and shift these lines until the last point within the feasible region is reached, which will be the solution that minimizes Z.

To graph the inequalities, we first convert each constraint into equation form (e.g., y = b + mx) and plot them, taking note of where the lines intersect as well as the axes intercepts. Once graphed, the area that satisfies all constraints represents the feasible region.

Vertices of this region are potential candidates for the optimal solution. By assessing the objective function Z at these vertices, we can determine which one gives the smallest value for Z, thereby identifying the optimal solution.