Optimization and Mathematical Modelling
Optimization questions ask you to translate a written situation into constraints and an objective, then find the best feasible choice. The exam tests the whole modelling cycle, not just the final number.
The modelling cycle
Define the variables with their units; write the objective function (the quantity to maximize or minimize); translate every condition into an inequality or equation; add the non-negativity constraints x ≥ 0 and y ≥ 0; graph the boundary lines; shade the feasible region, testing a point such as (0, 0) to choose the correct side; list the vertices; evaluate the objective at each vertex.
The corner-point principle
For a linear objective over a bounded polygonal feasible region, the optimum occurs at a vertex. If the variables must be whole numbers, check integer points close to the best vertex, because the true optimum may lie just inside an edge rather than exactly on a corner.
Quadratic optimization
When the model is quadratic, the optimum is at the vertex: complete the square or use x = −b/(2a). Example of a standard result: for a fixed perimeter, the rectangle of maximum area is a square.
Worked example
A club sells x standard and y premium T-shirts. Constraints: x + y ≤ 60 (silk-screening hours), 2x + y ≤ 100 (budget), x ≥ 0, y ≥ 0. Profit is P = 8x + 12y. Vertices and values: (0, 0) → 0; (50, 0) → 400; (0, 60) → 720; the intersection of x + y = 60 and 2x + y = 100 is x = 40, y = 20 → 320 + 240 = 560. The maximum is 720 at (0, 60): 60 premium shirts and no standard shirts. Feasibility check: 0 + 60 = 60 ≤ 60 ✓ and 2(0) + 60 = 60 ≤ 100 ✓. State the answer in context: 60 premium T-shirts give the greatest profit, $720.
Traps that cost marks
- Shading the wrong side of a constraint line.
- Evaluating the objective at an intersection that is not a vertex of the feasible region.
- Forgetting non-negativity, then reporting a negative production level.
- Rounding a fractional optimum when items must be whole units.
- Giving only the optimal value and omitting the values of the variables and the contextual conclusion.