Step 1: Understanding the Concept:
In Linear Programming (LP), the "Feasible Region" is the set of all possible points \((x, y)\) that satisfy all the constraint inequalities simultaneously. Geometrically, each constraint inequality represents a "half-plane." The feasible region is the intersection of all these half-planes. If a point lies in the feasible region, it is a valid candidate for the optimal solution.
The objective function is then evaluated within this region to find the maximum or minimum value. If there are no points that satisfy every single constraint at the same time, the feasible region is the empty set (\(\emptyset\)). This state is known as "Infeasibility."
Step 2: Detailed Explanation:
Linear programming models are built on several constraints, including non-negativity constraints like \(x \geq 0\) and \(y \geq 0\).
If a system of constraints is mutually inconsistent, it means that the geometric regions defined by individual constraints do not overlap.
For example:
Constraint 1: \(x + y \geq 10\) (points far from origin)
Constraint 2: \(x + y \leq 5\) (points near origin)
There is no coordinate point \((x, y)\) that can simultaneously be greater than \(10\) and less than \(5\). Consequently, there is no shared space, and no feasible region exists.
Without a feasible region, the objective function has no "territory" to explore, making it impossible to find a maximum or minimum value. This situation often arises due to logical errors in model formulation or contradictory requirements in real-world scenarios.
Option (A) is wrong because zero coefficients just make the function value constant (zero), but a region could still exist. Option (C) defines a standard feasible region in the first quadrant. Option (D) describes a valid bounded solution.
Step 3: Final Answer:
An LP system has no feasible region when the constraints are inconsistent and possess no common intersection in the coordinate system.