Optimisation models

Objective. Formulate and critique operational decisions.

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Learn: the key idea

Optimisation models translate real decisions into mathematical objectives and constraints, then interrogate whether the chosen formulation truly reflects the operational problem. Building one requires identifying decision variables, writing the objective (cost, profit, time) as a function of them, listing binding constraints (resources, demand, capacity), then solving and checking the solution's practical sense. Example: a firm making x and y units of two products with profit 5x+4y, limited by 2x+y≤100 labour hours, builds an LP; solving gives the production plan maximising profit while respecting labour. Critiquing a model means asking whether omitted constraints (like storage) would change the answer.

Worked example

An optimisation model for a workshop maximises profit P = 4x + 3y subject to 2x + y ≤ 10 and x, y ≥ 0. Find the maximum profit.

1 $List vertices of the feasible region: (0,0), (5,0), (0,10)$
2 $Evaluate P at (5,0): P = 20$
3 $Evaluate P at (0,10): P = 30$
4 $Compare: the maximum profit is 30, found at (0,10)$

Answer: 30

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Common mistake. Students treat the first mathematical formulation as final without checking units, sign conventions, or omitted real-world constraints; a profit model ignoring capacity can recommend infeasible production levels.

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