MGMT90280 · Managerial Decision Analytics
Introduction to MDA & Linear Programming (LP) + Sensitivity Analysis
This opening chapter of University of Melbourne MGMT90280 Managerial Decision Analytics introduces prescriptive analytics: turning a business decision into a linear program (LP) with decision variables, a linear objective (Max or Min) and linear constraints, then solving it in Excel Solver. It shows how to linearise percentage rules and how to read the Solver Sensitivity Report — reduced cost, shadow price, allowable increase/decrease, and binding vs non-binding constraints (slack). These skills anchor Question 1 of the 50% final exam, where LP formulation and sensitivity interpretation are paired with a nonlinear-programming model.
What this chapter covers
- 01The four ingredients of an LP: decision variables, linear objective, linear constraints, non-negativity
- 02The general LP form Max/Min Z = Σ cⱼxⱼ subject to Σ aᵢⱼxⱼ {≤,=,≥} bᵢ, xⱼ ≥ 0
- 03Linearising percentage / ratio rules (e.g. A ≥ 40% of A+B becomes 0.60A − 0.40B ≥ 0)
- 04The feasible region as a convex polygon; why the optimum sits at a corner (vertex)
- 05Reduced cost: how much a zero-valued variable's coefficient must improve to enter the solution
- 06Shadow price (dual value): change in optimal Z per one-unit increase in a constraint's RHS
- 07Binding vs non-binding constraints and slack/surplus; non-binding ⇒ shadow price = 0
- 08Allowable increase/decrease as the validity range for shadow prices and objective coefficients (1E+30 = infinity)
- 09Solving LPs in Excel Solver with Simplex LP and reading the Sensitivity Report
Graphical product-mix LP + reading the sensitivity report
- +1Define variables: x₁ = Standard desks made, x₂ = Premium desks made, both ≥ 0.
- +1Objective: Max Z = 30x₁ + 50x₂ (total profit in dollars).
- +2Constraints: carpentry 1x₁ + 2x₂ ≤ 40; finishing 1x₁ + 1x₂ ≤ 30; x₁, x₂ ≥ 0.
- +2The two resource lines cross where x₁ + 2x₂ = 40 and x₁ + x₂ = 30; subtracting gives x₂ = 10, so x₁ = 20. The other feasible corners are (0,20), (30,0) and (0,0).
- +1Evaluate Z at each corner: (20,10) → 30·20 + 50·10 = 1100; (0,20) → 1000; (30,0) → 900; (0,0) → 0. The best corner is (20,10).
- +1Decision: make 20 Standard and 10 Premium desks for maximum profit $1,100. Both constraints are fully used (carpentry 20 + 2·10 = 40; finishing 20 + 10 = 30) so both are binding, slack = 0. The report's shadow prices — carpentry $20/hour, finishing $10/hour — say one extra carpentry-hour adds $20 and one extra finishing-hour adds $10, so carpentry is the tighter bottleneck to relieve first.
Key terms
- Decision variable
- An unknown quantity the model is free to choose (e.g. units to make, hours to assign). Always defined with its unit; every variable is constrained to be non-negative (xⱼ ≥ 0).
- Objective function
- The single linear expression to be maximised (profit, output) or minimised (cost, time), written Z = c₁x₁ + c₂x₂ + … + cₙxₙ, where cⱼ is the profit or cost per unit of variable xⱼ.
- Binding constraint
- A constraint satisfied as an equality at the optimum — the resource is fully used, so slack = 0. A binding constraint generally has a non-zero shadow price.
- Slack (or surplus)
- The gap between a constraint's left- and right-hand sides at the optimum. Slack > 0 means spare resource (non-binding); slack = 0 means the constraint is binding.
- Reduced cost
- For a decision variable that is zero in the optimal solution, the amount its objective coefficient must improve before it becomes worth using. A variable already greater than zero has reduced cost 0.
- Shadow price (dual value)
- The change in the optimal objective value Z per one-unit increase in a constraint's right-hand side, valid only within that constraint's allowable increase/decrease range. A non-binding constraint has shadow price 0.
- Allowable increase / decrease
- How far a right-hand side (for shadow prices) or an objective coefficient (for the current solution) can change before the reported value stops being valid. Solver prints 1E+30 to mean infinity — no limit.
- Feasible region
- The set of all points satisfying every constraint — a convex polygon in two variables. Because the objective is linear, the optimum always lies at a corner (vertex) of this region.
Introduction to MDA & Linear Programming (LP) + Sensitivity Analysis FAQ
Why must percentage rules be linearised before Solver can use them?
A rule like 'x₁ must be at least 40% of x₁+x₂' written as x₁/(x₁+x₂) ≥ 0.40 is a ratio, which is non-linear, so Simplex LP cannot handle it. Multiply both sides by the (non-negative) denominator and move every variable to one side: x₁ − 0.40(x₁+x₂) ≥ 0, i.e. 0.60x₁ − 0.40x₂ ≥ 0. The inequality direction never flips because you multiply by a non-negative quantity, and the result is a clean linear constraint with a constant right-hand side.
What is the difference between a shadow price and a reduced cost?
A shadow price belongs to a constraint: it is the change in the optimal objective value per one extra unit of that constraint's right-hand side (only valid inside its allowable range). A reduced cost belongs to a variable: for a variable that is zero in the optimum, it is how much that variable's objective coefficient must improve before it enters the solution. Confusing the two is a common marked error — one prices a resource, the other prices bringing an unused variable into play.
Can AI help me with linear programming and sensitivity analysis in MGMT90280?
Yes — Sia can explain the method step by step: how to define decision variables, write and linearise constraints, set up the model in Excel Solver, and interpret each column of the Sensitivity Report (reduced cost, shadow price, slack, allowable increase/decrease). It works through practice problems with you and checks your reasoning, but it does not sit assessments for you, hand over exam or assignment answers, or guarantee any grade — and Assignment 1 is a Respondus quiz that prohibits generative AI, so use Sia only for study and revision.
Exam move
Master the exam Question 1 workflow until it is automatic: define variables (with units), write the objective (Max or Min), list each linear constraint, linearise every percentage rule (show the multiply-out step), and add non-negativity. Then practise reading a Solver Sensitivity Report as one-line answers — binding constraints have slack 0 and non-zero shadow prices, non-binding ones have shadow price 0, and a shadow price is valid only inside its allowable increase/decrease. Budget your time by marks: with 100 marks over 120 writing minutes, spend about 1.2 minutes per mark (a 20-mark question ≈ 24 minutes) and use the 30-minute reading time to plan. The final exam is open-book with a Casio FX-82; confirm the exact date and permitted materials on the LMS for the end-of-semester (~November 2026) sitting.
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