Interpreting the optimum in context

Interpreting the solution in context

• The maths gives a vertex; the answer is a sentence about the situation.
• State the optimal values of the decision variables with their meaning and units ("make 3 tables and 4 chairs").
• Give the optimal value of the objective too ("for a maximum profit of $240").
• Check it is sensible — non-negative amounts, within every constraint, and reasonable for the context.

Integer-only solutions

• Many contexts only allow whole numbers — you cannot make $2.5$ tables or pack $\tfrac{1}{3}$ of a tray.
• The graphical optimum may land on a vertex with fractional coordinates.
• When that happens:
  • Do not simply round — a rounded point can fall outside the feasible region.
  • Test the whole-number points near the optimal vertex that are still feasible.
  • Choose the feasible integer point that gives the best objective value.

Solutions on an edge (multiple optima)

• Usually one vertex wins. But if the objective line is parallel to a binding constraint edge, then every point along that edge gives the same optimal value.
• This is a multiple-optima (or "infinitely many solutions") case.
• How to spot it: two adjacent vertices give the equal best objective value.
• How to report it: state that any point on that edge is optimal — often giving the business useful flexibility in what to make.

Checking your answer makes sense

• Substitute the optimal point back into every constraint — it must satisfy them all.
• Confirm the objective value matches your vertex test.
• Re-read the question — answer exactly what was asked (a plan, a profit, or both), with units.