Solve the following Linear Programming Problem graphically :
Maximize \( Z = \frac{2x}{5} + \frac{3y}{10} \)
subject to constraints
\( 2x + y \leq 1000 \)
\( x + y \leq 800 \)
\( x, y \geq 0 \).
Show Hint
To plot lines quickly, find the x-intercept and y-intercept by setting \( y=0 \) and \( x=0 \) respectively.
In "less than or equal to" (\( \leq \)) constraints with positive coefficients, the feasible region is usually toward the origin.
Concept: • Identify the feasible region by plotting the linear inequalities on a coordinate plane.
• Find the coordinates of the corner points (vertices) of the feasible region.
• Evaluate the objective function at each corner point.
• The point that yields the highest value for \( Z \) is the optimal solution.
Step 1: Plot the boundary lines and find the feasible region
Convert inequalities to equations to find boundary lines:
1. \( L_1: 2x + y = 1000 \). At \( x = 0, y = 1000 \); at \( y = 0, x = 500 \). Points are \( (0, 1000) \) and \( (500, 0) \).
2. \( L_2: x + y = 800 \). At \( x = 0, y = 800 \); at \( y = 0, x = 800 \). Points are \( (0, 800) \) and \( (800, 0) \).
3. \( x, y \geq 0 \) represents the first quadrant.
The feasible region is the area bounded by these lines and the axes toward the origin.
Step 2: Find the point of intersection of the boundary lines
Solve the system:
\( 2x + y = 1000 \) ...(i)
\( x + y = 800 \) ...(ii)
Subtract (ii) from (i):
\( (2x - x) + (y - y) = 1000 - 800 \implies x = 200 \).
Substitute \( x = 200 \) in (ii):
\( 200 + y = 800 \implies y = 600 \).
The intersection point is \( (200, 600) \).
Step 3: Evaluate the objective function at corner points
\includegraphics[width=1\linewidth]{Q31_sol.png}
The corner points of the feasible region are \( O(0,0), A(500,0), B(200,600), \) and \( C(0,800) \).
Objective Function: \( Z = 0.4x + 0.3y \)
1. At \( O(0,0): Z = 0.4(0) + 0.3(0) = 0 \)
2. At \( A(500,0): Z = 0.4(500) + 0.3(0) = 200 \)
3. At \( B(200,600): Z = 0.4(200) + 0.3(600) = 80 + 180 = 260 \)
4. At \( C(0,800): Z = 0.4(0) + 0.3(800) = 240 \)
The maximum value of \( Z \) is 260 at the point \( (200, 600) \).
