Business Mathematics and Statistics · Ch 10 — Operations Research (Linear Programming Problem, Network Analysis)
Graphing Linear Inequalities and the Feasible Region
Graphing Linear Inequalities and the Feasible Region
Solving an LPP graphically first requires plotting each constraint as a region on a graph, exactly as a linear inequality in two variables is graphed.
Graphing a Linear Inequality — Three Steps
- Boundary line: replace the inequality sign with and draw the resulting line, most easily using its two intercepts (put for the x-intercept, for the y-intercept).
- Test point: substitute a convenient point not on the line (the origin, if the line doesn't pass through it) into the original inequality — if it holds, shade that side; if not, shade the other side.
- Solid or dashed: a non-strict inequality () gets a solid boundary (included); a strict inequality () gets a dashed boundary (excluded).
| Constraint | Boundary line (x-intercept, y-intercept) | Test point (origin) | Shaded side | Boundary style |
|---|---|---|---|---|
| and | — true | Origin's side | Solid () | |
| and | — false | Away from the origin | Dashed () |
A real LPP has several constraints together, considered as a system. The feasible region is the set of points satisfying every constraint simultaneously — found by shading each inequality's half-plane on the same axes and keeping only their common overlap (never their combined union). Because every business LPP also carries the non-negativity restrictions , the feasible region is always confined to the first quadrant. …
The set of points satisfying every constraint of an LPP (including non-negativity) at once — the intersection, never the union, of every individua …
A point where two boundary lines of the feasible region (or a boundary line and an axis) meet; the objective function is evaluated only at these points, ne …