The objective function and testing the vertices

The objective function

• The objective function is the quantity you want to maximise (profit, output) or minimise (cost, waste).
• It is written in terms of the decision variables, e.g. profit:

$P = 40x + 30y$

• $P$ — the value being optimised (here, profit in dollars).
• $40$ — the profit per table (the amount each unit of $x$ contributes).
• $30$ — the profit per chair (the amount each unit of $y$ contributes).
• The objective function is not a constraint — it has no inequality sign.

Why the optimum is at a vertex

• The objective function has a constant slope, so lines of equal value $P = c$ are all parallel.
• As $c$ grows, the line $P = c$ sweeps across the feasible region without changing direction.
• The last point it touches before leaving the region is a corner — so the maximum (and the minimum) must occur at a vertex.
• This is the key idea that makes the method work: you only ever need to check the corners.

The corner-point (vertex) method

• List every vertex of the feasible region (found on the previous page).
• Substitute each vertex into the objective function.
• Compare the values:
  • The largest value is the maximum.
  • The smallest value is the minimum.
• State which vertex wins and what the optimal value is.

The sliding objective-line method

• An alternative to testing every corner:
  • Draw one objective line for a convenient value, e.g. $40x + 30y = 120$.
  • Slide it parallel in the direction that increases $P$ (away from the origin for a positive objective).
  • The last vertex the line meets inside the region is the optimum.
• Both methods give the same answer; the vertex method is safer under exam pressure, the sliding line is a good check.