ETF2480 Chap.2 Feasible Regions and Optimal Solutions
Feasible Regions and Optimal Solutions
A feasible region contains every decision that satisfies all constraints. In two dimensions, each inequality keeps one side of a line and their intersection forms a polygon or another convex set. Parallel objective contours reveal how improvement reaches an extreme point, a whole optimal edge, or continues without bound. This geometry gives meaning to solver statuses.
Optimal means the algorithm established the best feasible value under the model; infeasible means no point meets all rows; unbounded means the objective can improve indefinitely; and a limit status may return a candidate without a proof. Reading status before reading solution values is therefore a core modelling habit.
What this chapter covers
- 01
Half-planes and feasible intersections
- 02
Extreme points and objective contours
- 03
Unique and alternate optima
- 04
Infeasible constraint systems
- 05
Unbounded improving directions
- 06
Status-first solver interpretation
Worked example · free
Compare every vertex of a service plan
- +1The axis vertices are (0,0), (36,0) and (0,27).
- +2The two capacity lines intersect at x = 18 and y = 18, which satisfies both rows.
- +2Objective values are 0, 1,080, 1,350 and 1,440 respectively.
- +1Choose (18,18) and report that both limits bind in the continuous model.
Key terms
- Feasible region
- The intersection of all constraint sets and variable domains.
- Extreme point
- A corner of a polyhedral feasible region that can support a linear optimum.
- Alternate optima
- Multiple feasible decisions sharing the same best objective value, often along an edge.
- Infeasible model
- A model whose constraints and domains have no common solution.
- Unbounded model
- A feasible model whose objective can improve indefinitely in the chosen direction.
Feasible Regions and Optimal Solutions FAQ
Why do linear optima occur at corners?
A linear contour moves across a convex polygon without bending. Its final contact in the improving direction includes an extreme point, although an entire edge can tie when the contour is parallel to that edge. This result supports vertex checking only for small graphical models.
What should I do when a model is infeasible?
Check units, inequality directions, duplicated requirements and the intended period first. Then identify the conflicting business commitments. A solver conflict report can isolate rows, but management must decide whether data, capacity or a service requirement can legitimately change.
Does unbounded mean infinite real profit?
Usually it signals that the mathematical model omitted a relevant limit or has an incorrect sign. Inspect demand, capacity, budget, linking constraints and variable domains. The real organisation is bounded even when the supplied equations fail to express that boundary.
How should I verify a plotted optimum?
Solve its binding equations exactly, substitute the coordinates into every restriction and compare the objective with all other feasible vertices. Report the status before interpreting any displayed decision values.
Exam move
Draw small models until status becomes visual. Shade each half-plane, mark the common region and draw two parallel objective contours to identify the improving direction. Calculate vertex coordinates algebraically rather than estimating them from the picture. Create one example for each status: a single best corner, a tied edge, incompatible minimum and maximum requirements, and an omitted upper bound.
Then connect each picture to solver language. When software reports a status, write the modelling question it triggers before inspecting values. For infeasibility, search for conflict; for unboundedness, search for a missing limiter; for alternate solutions, decide whether the business values flexibility; for an optimal result, verify feasibility and objective value.
This habit turns solver output into an explanation instead of a screenshot. Use a repeatable vertex ledger with columns for coordinates, row checks and objective value. Generate candidates from axis intercepts and pairwise intersections, then discard them only after substituting into every restriction. To study alternate optima, change the objective slope until it matches one boundary and identify both endpoint solutions.
For an infeasible example, locate the smallest conflicting pair of requirements and translate that conflict back into business language. For an unbounded example, draw an improving ray and name the absent capacity or policy rule that would normally stop it. Compare a bounded feasible region with an unbounded region that still has a finite optimum, because geometry alone does not determine objective status.
When using software, save the status message before reading decision values and never interpret placeholder values from a failed solve. End each drill by sketching how one right-hand-side change moves a boundary and predicting whether the old optimum remains feasible.
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