Question

Under what condition will a linear programming system containing objective function variables have no common feasible region?

• A. When the objective function coefficients are set strictly to zero.
• B. When the system of linear constraints is mutually inconsistent, meaning there is no overlapping coordinate space that satisfies all conditions at once.
• C. When all constraints are written using strictly non-negative bounding limits.
• D. When the optimal solution point matches one of the outer corner coordinates.

Hint:

When an LPP system produces no overlapping feasible region, it is classified as an infeasible solution space, meaning it is impossible to find a maximum or minimum value for the objective function.

Answer 1

The Correct Option is B

Concept: In Linear Programming Problems (LPP), the feasible region is the shared collection of all coordinate points on a graph that satisfy every constraint inequality simultaneously.

Step 1: Analyze the geometric structure of constraint half-planes.
Each individual constraint inequality forms a shaded half-plane boundary line across a coordinate grid. For a feasible region to exist, these shaded areas must overlap, creating a shared boundary space.

Step 2: Evaluate the cause of an infeasible state.
If the constraint requirements conflict with each other (for example, requiring \( x \le 2 \) and \( x \ge 5 \) at the same time), their shaded regions will point in opposite directions and never overlap. This mutual inconsistency means there are zero coordinate points that satisfy all conditions at once, resulting in no feasible region.