Exercise 12.1
Solve the following Linear Programming Problem graphically: Maximise Z = x + y, subject to x – y ≤ –1, –x + y ≤ 0, x, y ≥ 0.
Step 1: Analyse the constraints.
Constraint 1: x – y ≤ –1, which means y ≥ x + 1 (the feasible region is above the line y = x + 1).
Constraint 2: –x + y ≤ 0, which means y ≤ x (the feasible region is below or on the line y = x).
x, y ≥ 0.
Step 2: Determine if a feasible region exists.
Constraint 1 requires y ≥ x + 1, so y > x.
Constraint 2 requires y ≤ x.
It is impossible to have simultaneously y > x and y ≤ x.
Therefore, there is no point (x, y) that satisfies all the constraints at the same time.
Step 3: Conclusion.
The feasible region is empty — no feasible solution exists. Hence the linear programming problem has no solution, and Z = x + y has no maximum value.
Answer: The problem has no feasible solution (the constraints are inconsistent), so Z = x + y has no maximum value.