Operations Research — Linear Programming (graphical method): In a two-variable graphical model of linear programming, the region defined collectively by all inequality constraints together with the non-negativity restrictions (x ≥ 0, y ≥ 0) is called what?

Introduction / Context:
In linear programming (LP), especially when teaching with two decision variables, we often use a graphical solution method. The first task is to translate verbal limits (such as resource capacities and minimum requirements) into linear inequalities. Those inequalities, together with the non-negativity requirements (e.g., x ≥ 0 and y ≥ 0), carve out a portion of the plane that contains every combination of decision variables that is allowed by the model. Knowing the correct name for this region is fundamental before applying the objective function to find an optimal point.

Given Data / Assumptions:

• An LP has linear constraints (equalities or inequalities) and non-negativity restrictions.
• We are using the 2-variable graphical viewpoint (coordinates on a plane).
• The objective function is a separate expression to be optimized over the allowed points.

Concept / Approach:
The set of all points (x, y) that satisfy every constraint simultaneously is the feasible solution region (often shortened to “feasible region”). Only points in this region are candidates for the optimal solution. The objective function does not define the region; rather, it is evaluated over the region to locate a best point at a vertex or along an edge, depending on the model’s structure.

Step-by-Step Solution:

Verification / Alternative check:
You can randomly test any point within the shaded intersection by substituting it into each constraint and the non-negativity conditions. If all are satisfied, the point is feasible, confirming the meaning of the region.

Why Other Options Are Wrong:

• Non-negativity restrictions: These are only part of the defining conditions.
• Objective function: Guides optimization but does not define the allowed region.
• Constraints: Individual rules; the region is the intersection of all such rules.
• None of the above: Incorrect because “feasible solution region” is the standard term.

Common Pitfalls:
Confusing a single constraint’s half-plane with the overall feasible region; forgetting to include x ≥ 0 and y ≥ 0 so the region incorrectly extends into negative axes; believing the objective function line determines feasibility.

Final Answer:
feasible solution region