Linear Programming • Topic 2 of 2

Graphical Solution & the Corner-Point Method

For two variables, an LPP is solved graphically.

The feasible region

Plot each constraint line and shade the side satisfying its inequality. The overlap of all shaded regions (including $x,y\ge0$) is the feasible region. It is a convex polygon; its vertices are the corner points.

Corner-point theorem

If an optimal value of the objective function exists, it occurs at a corner point of the feasible region. Method:

  1. Find all corner points (intersections of boundary lines).
  2. Evaluate $Z=ax+by$ at each corner.
  3. The largest value is the maximum; the smallest is the minimum.

Bounded vs unbounded

If the feasible region is bounded, both a maximum and a minimum exist. If it is unbounded, an optimum may not exist — you must check whether $Z$ can grow without limit.

Solved Examples
1
Worked Example
Maximise $Z=3x+2y$ over corner points $(0,0),(4,0),(0,5),(2,3)$.
Solution

$Z(0,0)=0,\ Z(4,0)=12,\ Z(0,5)=10,\ Z(2,3)=6+6=12$. Maximum $Z=12$ (attained at $(4,0)$ and $(2,3)$).

2
Worked Example
Minimise $Z=x+y$ over corner points $(2,0),(0,3),(1,1)$.
Solution

$Z(2,0)=2,\ Z(0,3)=3,\ Z(1,1)=2$. Minimum $Z=2$.

3
Worked Example
Where does the optimum of a (bounded) LPP occur?
Solution

At a corner point of the feasible region, by the corner-point theorem.

4
Worked Example
Find the corner point of $x+y=4$ and $x=y$.
Solution

Solving $x=y$ and $x+y=4$: $2x=4\Rightarrow x=2,\ y=2$. Corner point $(2,2)$.

Key Points

  • Feasible region = overlap of all constraint half-planes (a convex polygon).
  • The optimum, if it exists, is at a corner point.
  • Evaluate $Z$ at every corner; largest = max, smallest = min.
  • Bounded region $\Rightarrow$ both max and min exist; unbounded needs a check.
Quick Check — 4 MCQs
Tap an option to check your answer0 / 4
Q1.By the corner-point theorem, the optimum occurs at:
Explanation: Optima are attained at vertices.
Q2.Maximum of $Z=3x+2y$ over $\{(0,0),(4,0),(0,5),(2,3)\}$ is:
Explanation: Both $(4,0)$ and $(2,3)$ give $12$.
Q3.The feasible region of an LPP is always:
Explanation: Intersection of half-planes is convex.
Q4.If the feasible region is bounded, then:
Explanation: Bounded regions guarantee both.
Practice more MCQs → 📝 Assignment · 35 marks