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1.4.3 Vertices and Boundaries of Feasible Regions

Identify and analyze the corner points and boundary lines that define the feasible region.

定义

A feasible region is the set of all points that satisfy all constraints in a system of linear inequalities. The vertices (or corner points) of a feasible region are the points where two or more boundary lines intersect, and they are located at the corners of the region. The boundary lines are the lines formed by the equations corresponding to each inequality constraint. For a bounded feasible region, the vertices are the extreme points where the optimal value of a linear objective function is typically achieved. Each boundary line is determined by replacing the inequality sign with an equality sign in the original constraint. The feasible region can be bounded (enclosed) or unbounded (extending infinitely in one or more directions), and vertices only exist at the intersections of boundary lines that satisfy all original inequality constraints.

核心公式

  • \(\text{Boundary line: } ax + by = c \text{ (from constraint } ax + by \leq c \text{ or } ax + by \geq c\text{)}\)
  • \(\text{Vertex: intersection point of two boundary lines, found by solving } \begin{cases} a_1x + b_1y = c_1 \ a_2x + b_2y = c_2 \end{cases}\)
  • \(\text{Objective function at vertex: } Z = mx + ny \text{ (evaluated at each vertex)}\)
  • \(\text{Feasible region: } \{(x,y) : a_1x + b_1y \leq c_1, a_2x + b_2y \leq c_2, \ldots, a_nx + b_ny \leq c_n\}\)
  • \(\text{Optimal solution: } \max/\min(Z) \text{ occurs at a vertex of the feasible region (Vertex Theorem)}\)

易错点

  • ⚠️ Forgetting to check that a vertex candidate actually satisfies ALL original inequality constraints—a point may be at the intersection of two boundary lines but still be outside the feasible region if it violates another constraint
  • ⚠️ Confusing boundary lines with the feasible region itself; students often shade the wrong side of a line or fail to recognize that the feasible region is the intersection of all half-planes, not just one
  • ⚠️ Failing to identify all vertices, especially in unbounded regions, or missing vertices that occur at the intersection of a boundary line with an axis (where x=0 or y=0)
  • ⚠️ Assuming the optimal value always occurs at a vertex without checking; while this is true for linear programming, students must verify they have correctly identified all vertices and evaluated the objective function at each one