Step-by-Step Solution
Topic: Linear Programming Problems (LPP)
(I) IDENTIFYING CONSTRAINTS
The feasible region is bounded by three line segments forming the lower boundary. We determine the inequalities by looking at the lines and the region (which is above/away from origin).
1. Line connecting D(0, 8) and C(1, 5):
This corresponds to the equation 3x + y = 8 (Check: 3(0)+8=8 and 3(1)+5=8).
Since the region is “above” this line, the constraint is: 3x + y ≥ 8.
2. Line connecting C(1, 5) and B(2, 4):
This corresponds to the equation x + y = 6 (Check: 1+5=6 and 2+4=6).
Constraint: x + y ≥ 6.
3. Line connecting B(2, 4) and A(10, 0):
This corresponds to the equation x + 2y = 10 (Check: 2+2(4)=10 and 10+0=10).
Constraint: x + 2y ≥ 10.
4. Non-negative constraints: x ≥ 0, y ≥ 0.
3x + y ≥ 8
x + y ≥ 6
x + 2y ≥ 10
x, y ≥ 0
(II) MINIMIZE COST
Objective Function: Z = 16x + 20y
We use the Corner Point Method. The corner points of the unbounded region are D, C, B, and A.
| Corner Point | (x, y) | Z = 16x + 20y |
|---|---|---|
| D | (0, 8) | 16(0) + 20(8) = 160 |
| C | (1, 5) | 16(1) + 20(5) = 116 |
| B | (2, 4) | 16(2) + 20(4) = 32 + 80 = 112 |
| A | (10, 0) | 16(10) + 20(0) = 160 |
The minimum value appears to be 112 at point B(2, 4).
Verification for Unbounded Region:
Since the region is unbounded, we must check if the inequality 16x + 20y < 112 (or 4x + 5y < 28) has any common points with the feasible region.
The line 4x + 5y = 28 passes through (7, 0) and (0, 5.6). Point B(2,4) lies on this line. Since the feasible region is “above” the boundary lines and 4x + 5y < 28 represents the region “below” this line, there are no common points within the feasible region.
⚠️ Silly Mistake Audit: Unbounded Regions
Don’t Stop at the Table!
For unbounded feasible regions, the minimum value found in the corner point table is not automatically the final answer. You must verify that no other point in the region gives a smaller value by plotting the inequality Objective Function < Min Value. If there’s an overlap, no minimum exists. If no overlap, your answer is correct.