Question

Find the minimum value of Z = 3x + 7y by the graphical method under the following constraints:
\(x + y \le 8, 3x + 5y \ge 0, x \ge 0, y \ge 0\)

Hint:

In a graphical LPP, the optimal solution (maximum or minimum) for a bounded feasible region is always found at one of its corner points. Always identify all constraints; sometimes a constraint might be redundant (as \(3x+5y \ge 0\) was here, given \(x \ge 0, y \ge 0\)), which can simplify the problem.

Answer 1

Step 1: Understanding the Concept:
This is a Linear Programming Problem (LPP). The goal is to find the minimum value of a linear objective function \(Z\) subject to a set of linear inequalities called constraints. The graphical method involves plotting these constraints to identify a feasible region and then testing the corner points of this region in the objective function.
Step 2: Key Formula or Approach:
1. Treat the inequalities as equations to plot the boundary lines.
2. Determine the region that satisfies each inequality.
3. Identify the common region satisfying all constraints, which is the feasible region.
4. Find the coordinates of the vertices (corner points) of the feasible region.
5. Evaluate the objective function \(Z = 3x + 7y\) at each vertex.
6. The smallest value of Z among these will be the minimum value.
Step 3: Detailed Explanation or Calculation:
The constraints are:
1. \(x + y \le 8\)
2. \(3x + 5y \ge 0\)
3. \(x \ge 0\)
4. \(y \ge 0\)
Graphing the Constraints:
- The constraints \(x \ge 0\) and \(y \ge 0\) restrict the feasible region to the first quadrant of the Cartesian plane.
- The constraint \(3x + 5y \ge 0\) is satisfied by all points in the first quadrant (since x and y are non-negative). Therefore, this constraint is redundant and does not further restrict the feasible region defined by the other constraints.
- To plot \(x + y \le 8\), we first draw the line \(x + y = 8\). This line passes through the points (8, 0) and (0, 8). The inequality \(\le\) means the feasible region lies on and below this line.
Identifying the Feasible Region and Corner Points:
The feasible region is the area in the first quadrant bounded by the lines \(x=0\), \(y=0\), and \(x+y=8\). This forms a triangle.
The vertices (corner points) of this triangular feasible region are:
- O(0, 0): The origin.
- A(8, 0): The x-intercept of the line \(x+y=8\).
- B(0, 8): The y-intercept of the line \(x+y=8\).
Evaluating the Objective Function Z at Corner Points:
We now evaluate \(Z = 3x + 7y\) at each vertex:
- At O(0, 0): \(Z = 3(0) + 7(0) = 0\)
- At A(8, 0): \(Z = 3(8) + 7(0) = 24\)
- At B(0, 8): \(Z = 3(0) + 7(8) = 56\)
Step 4: Final Answer
Comparing the values of Z (0, 24, 56), the minimum value is 0, which occurs at the point (0, 0).