MGMT90280 · Managerial Decision Analytics
Nonlinear Programming (NLP)
Nonlinear Programming (NLP) is the optimisation topic in MGMT90280 Managerial Decision Analytics at the University of Melbourne, where a decision model keeps variables, an objective and constraints but the objective and/or a constraint is nonlinear — most often a quadratic profit model, Profit = Revenue − Cost. You maximise it in Excel with the GRG Nonlinear Solver engine, and by hand by setting the derivatives to zero and, when a constraint bites, re-solving on the binding boundary. It typically appears inside the 50% final exam's optimisation question, paired with linear programming.
What this chapter covers
- 01What makes a program nonlinear: powers (Q²), products (P·Q) and ratios in the objective or a constraint
- 02The profit model: Profit = Revenue − Cost, with quadratic cost aQ + bQ² + f or quadratic revenue when price falls with quantity
- 03GRG Nonlinear — the Excel Solver engine for smooth nonlinear models (replaces Simplex LP)
- 04Local vs global optima: GRG climbs to the nearest peak; use multiple starting points (multi-start)
- 05When a local optimum is global: a concave objective (single downward parabola) has one peak
- 06Unconstrained optimum: set every partial derivative ∂Z/∂Qᵢ = 0
- 07Second-order (max) test: confirm concavity, second derivative < 0
- 08Binding-boundary substitution: if the free optimum is infeasible, set the constraint to an equality, substitute and re-optimise
- 09Reading the result: state the reject/accept decision, the optimal decision variables and the dollar profit
Profit maximisation with a binding capacity
- +2Model. Revenue = p·q = (100 − 2q)q = 100q − 2q². Profit Z = Revenue − Cost = (100q − 2q²) − (20q + 80) = −2q² + 80q − 80. The q² term makes it nonlinear.
- +2Unconstrained optimum. dZ/dq = −4q + 80 = 0 ⇒ q = 20. Second derivative d²Z/dq² = −4 < 0 ⇒ Z is concave, so this stationary point is a maximum (and, being concave, the global one).
- +2Feasibility check. q = 20 exceeds the capacity 15 ⇒ the free optimum is infeasible, so reject it. Because Z is still rising up to q = 20, on 0 ≤ q ≤ 15 the best point is the right-hand boundary ⇒ the capacity constraint is binding at q = 15.
- +2Profit. At q = 15: price = 100 − 2(15) = $70, Revenue = 70×15 = $1,050, Cost = 20×15 + 80 = $380. Maximum profit = 1,050 − 380 = $670.
Key terms
- Nonlinear program (NLP)
- An optimisation model whose objective and/or at least one constraint contains a nonlinear term — a power, a product of variables, or a ratio (e.g. a quadratic Q²).
- Profit = Revenue − Cost
- The standard NLP objective in this subject: Z = P·Q − (aQ + bQ² + f), where a = variable cost/unit, b = cost curvature, f = fixed cost, P = price, Q = output.
- GRG Nonlinear
- The Generalised Reduced Gradient engine in Excel Solver used for smooth nonlinear models; an iterative hill-climber that steps uphill until the slope flattens.
- Local vs global optimum
- A local optimum is best only in its neighbourhood; the global optimum is best overall. GRG guarantees only a local optimum unless the objective is concave (for a max) over a convex region.
- Concave objective
- A downward-curving objective (second derivative < 0, a single peak). For a maximisation problem, a concave objective means any local optimum is also the global optimum.
- Binding constraint
- A constraint that holds with equality at the optimum (its slack is zero); relaxing it would change the optimal objective value.
- Binding-boundary substitution
- When the unconstrained optimum violates a constraint, set that constraint to an equality, use it to eliminate a variable, substitute into the objective, and re-optimise on the boundary.
- Second-order (max) test
- Confirming a stationary point is a maximum by checking curvature: the second derivative is negative (one variable) or the Hessian is negative definite (several variables).
Nonlinear Programming (NLP) FAQ
Do I need calculus, or is Excel Solver enough?
Both. The exam rewards the by-hand method — write Profit = Revenue − Cost, set the derivative(s) to zero, run the feasibility check, and substitute on the binding boundary if needed — and you should be able to reproduce or check the answer with Excel Solver's GRG Nonlinear engine. A final number with no working scores little.
Why can Excel Solver give different answers on the same model?
GRG Nonlinear is a hill-climber that stops at the nearest peak, so on a bumpy (non-concave) objective the reported optimum can depend on the starting values. Try several starting points (multi-start) and keep the best. For a concave profit parabola there is only one peak, so this problem does not arise.
Can AI help me with Nonlinear Programming in MGMT90280?
Yes — Sia can explain the method step by step: how to set up Profit = Revenue − Cost, take the derivatives, run the feasibility check and do binding-boundary substitution, using practice-style numbers. It supports your understanding but will not sit your assessment or generate answers to submit, and it cannot promise any particular grade — always follow the LMS rules on permitted help.
Exam move
Drill the five-step loop until it is automatic: (1) write the model as Profit = Revenue − Cost and simplify; (2) set each partial derivative to zero for the unconstrained optimum; (3) confirm a maximum with the second-derivative sign; (4) check feasibility against every constraint; (5) if a constraint is violated, substitute it as an equality and re-optimise on the binding boundary. Practise stating the reject/accept decision in words and quoting both the decision variables and the dollar profit, and be ready to explain the local-vs-global caveat of GRG Nonlinear. In the final exam, nonlinear programming usually shares the optimisation question with linear programming, so budget your time in proportion to the marks and confirm the exam date, duration and permitted materials on the LMS.
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