Graphical method of solving LPP
Once a linear programming problem has been formulated, someone still has to find the actual best values of and . For two-variable problems, CBSE's chosen tool is the graphical method: sketch every constraint as a straight line, shade the region where all of them hold at once, and read the answer off the corners of that region. The idea rests on one clean fact — a linear objective function never peaks in the middle of a region, only at its edges, and in fact only at its vertices. That single observation, the Corner Point Theorem, turns an infinite search over every point in the feasible region into a short, exact search over a handful of corner points. This article builds the theorem into a repeatable procedure, the Corner Point Method, extends it to the trickier case of an unbounded region, and works through two fully solved examples — one bounded maximisation and one unbounded minimisation — of exactly the kind the syllabus expects.
Because the objective function is linear, its optimal value over a feasible region always occurs at a corner point, so evaluating at every vertex (with an extra half-plane check if the region is unbounded) finds the optimum.
The core idea
Why corners, not the interior. A linear function has no curvature — it rises (or falls) steadily in one direction and has no interior hill or valley to peak at. Over a convex region bounded by straight edges, that steady rise must therefore be carried furthest at a vertex, where two edges meet and the region can extend no further in the favourable direction. This is stated formally as the theorem below.
where: are the decision variables, are the fixed cost/profit coefficients from the problem statement, and is the value of the objective function being maximised or minimised.
Corner Point Theorem. If a linear programming problem has an optimal solution, it occurs at a corner point (vertex) of the feasible region. If two adjacent corners both give the same optimal -value, then every point on the segment joining them is also optimal, and the LPP has infinitely many optimal solutions.
Corner Point Method — the standard procedure. Graph every constraint as a boundary line, shade the half-plane each inequality permits (test a convenient point such as the origin), and mark the region common to all of them, noting whether it is bounded or unbounded. List every corner point, finding coordinates either by inspection of intercepts or by solving pairs of boundary equations simultaneously. Evaluate at each corner. If the region is bounded, the largest and smallest of these values are the maximum and minimum of — full stop. If the region is unbounded, a candidate optimum needs one further check, given in the derivation below, before it can be trusted.
Derivation
The goal is to show, using only the geometry of half-planes, why the largest corner-point value found on an unbounded feasible region is merely a candidate maximum, and to state the exact test that confirms or rejects it — this is the single most commonly mis-applied step in Class 12 LPP answers, so it deserves its own justification rather than being taken on faith.
Step 1. Suppose the corner points give a largest value at some vertex . On a bounded region this settles the matter immediately, because the whole region lies within a finite closed shape and , being linear, cannot exceed the largest value found on the vertices of that closed shape — no interior or edge point can beat every vertex.
Step 2. An unbounded region has no such finite boundary in every direction; it stretches out indefinitely along at least one direction. So the argument in Step 1 no longer closes off the possibility that keeps climbing as or grows without limit somewhere out along that unbounded stretch.
Step 3. The fix is to test the open half-plane directly against the feasible region. If this half-plane shares no point at all with the feasible region, then no feasible point can reach a -value beyond , and is confirmed as the true maximum. If the half-plane does intersect the feasible region, can be pushed past somewhere out along the unbounded stretch, so was never the maximum — indeed the region may have no maximum at all.
Step 4. The same logic runs in reverse for a minimum on an unbounded region: test whether shares any point with the feasible region for the smallest corner value . Both checks reduce to one geometric question — does the region reach past the candidate value in the relevant direction — answered by inspection of the sketch, not by further algebra.
Worked example
Solve graphically: minimise subject to , , , .
Step 1. Sketch the boundary lines. For : intercepts and ; testing the origin gives , which is false, so the feasible side lies away from the origin. For : intercepts and ; testing the origin gives , true, so the feasible side includes the origin.
Step 2. Find where the two lines cross. Doubling gives ; subtracting this from gives , and back-substituting gives . The lines meet at .
Step 3. List the corner points. Satisfying both inequalities together with gives corners , and ; the region extends outward above the line , so it is unbounded.
Step 4. Evaluate at each corner: at , ; at , ; at , . The smallest of these is at .
Step 5. Since the region is unbounded, apply the open half-plane test from the derivation: check whether shares any point with the feasible region. It does not, because the entire feasible region lies on the far side of from that half-plane, so no feasible point can score below .
Answer: the minimum value is , attained at .
Common traps
- Testing only the "obvious" corners. Every vertex of the feasible region must be evaluated — skipping one, especially an intersection found by solving equations rather than reading an intercept, can silently produce the wrong optimum.
- Skipping the half-plane check on an unbounded region. The largest or smallest corner value there is only a candidate; without the open half-plane test, a maximisation problem in particular can report a false optimum.
- Misreading intercepts while plotting. A single arithmetic slip in finding where a line crosses the axes shifts the whole feasible region and every corner point that follows.
- Confusing which corner gives the max versus the min. Tabulate every corner point with its -value systematically rather than judging the answer visually from the sketch.
Ask the tutor: Want to work through your own LPP constraints step by step, or check whether your feasible region is bounded before you commit to an answer?
Why does the optimal value always occur at a corner point and never somewhere along an edge or inside the region?
The objective function is a plane tilted at a constant slope — it has no interior peak or dip of its own, so it can only rise steadily in one direction across the feasible region. Since the region is bounded by straight edges, that steady rise is carried furthest at a vertex, where the boundary can extend no further in the favourable direction; an edge or interior point always has a neighbouring vertex that scores at least as well. The one exception is when two adjacent corners tie for the optimum, in which case the whole edge joining them is optimal too.
How do I decide whether a feasible region is bounded or unbounded just from the constraints?
Sketch the region and look at whether it can be enclosed inside some finite circle. Constraints of the form (something) (a number), such as , tend to cap the region on that side, while constraints of the form (something) (a number), such as , push the boundary outward and, if not capped by another constraint, let the region stretch away indefinitely. In the worked example, the constraint leaves the region open above the line , which is why it is unbounded even though a constraint is also present.
If I forget the half-plane check on an unbounded region, could my "maximum" actually be wrong?
Yes, and this is the single most common mark-losing mistake in this topic. On an unbounded region, may keep climbing without limit as you move outward along the unbounded stretch, in which case the LPP has no maximum at all, even though every corner point gives some finite value. The corner values only tell you about the vertices you can see; the open half-plane test is what checks the parts of the region you cannot read off a table. Always run it before declaring a maximum on an unbounded region — minimisation problems are safer, since is far less often unbounded below when .