QBUS2310 Management Science
The Complete Exam Bible · Semester 2 2026QBUS2310 Overview
- The University of Sydney
- Semester 2 2026
- Level 2
- Mathematical optimisation
Management Science studies how to turn complex operational choices into mathematical models that can be solved and checked. The unit begins with prescriptive analytics: define the decision variables, separate parameters from choices, state an objective, and describe every operational limit as an equation or inequality.
- Core method Define choices, goal, limits and domains
- Verification Check feasibility and an independent certificate
- Assessment mix Assignments, written test and final exam
- Tools Python, Gurobi and RSOME support modelling
How QBUS2310 is assessed
| Component | Weight | Format |
|---|---|---|
| Final exam Written exam | 40% | Written exam in the formal exam period, two hours |
| Assignment 0 A written assignment submitted through Canvas. | 8% | Practical skill Early Feedback Task in Week 03 |
| Mid-Semester Test A mid-semester test | 20% | Written test in Week 07, one and a half hours |
| Assignment 1 A written assignment submitted through Canvas. | 12% | Practical skill task in Week 07 |
| Assignment 2 A written assignment submitted through Canvas. | 20% | Practical skill task in Week 13 |
What QBUS2310 covers
The unit moves from mathematical decision models and linear optimisation through geometry, modelling, duality, integer decisions, networks, and optimisation under uncertainty. Each chapter pairs formulation with a verification method so the result remains connected to the management decision.
Optimisation Decisions and Linear Models
Decision variables, objectives, constraints, feasibility and linear model conventionsManagement science turns a decision into a mathematical model. The essential move is to separate choices from given information, state one measurable objective, and translate every operational limit into a constraint.
- Prescriptive analytics and decision quality
- Decision variables and parameters
- Objectives and constraint systems
- Feasible and optimal solutions
- Infeasibility and unboundedness
- Linear objective and constraint tests
Linear Algebra, Convexity and Model Structure
Vectors, matrices, linear independence, convex combinations and convex functionsOptimisation models compress many variables and constraints into vector and matrix notation.
- Sets, membership and quantifiers
- Vector and matrix dimensions
- Transpose and matrix products
- Linear independence and rank
- Convex combinations
- Convex sets and line segments
Feasible Geometry and Basic Solutions
Halfspaces, active constraints, vertices, basic feasible solutions and standard formEach linear inequality cuts space into a halfspace, and their intersection forms a polyhedron. Geometry makes feasibility, binding constraints, and candidate optima visible.
- Halfspaces and polyhedra
- Active and inactive constraints
- Slack and surplus
- Vertices and extreme points
- Basic solutions and rank
- Basic feasible solutions
Building Linear Optimisation Models
Resource, transport, staffing, inventory and cash-flow formulationsThe hardest modelling step is not solving an optimisation model but deciding what every symbol means. Different applications share conservation, capacity, coverage, and balance patterns.
- Manufacturing resource models
- Diet lower-bound models
- Transportation balances
- Cyclic workforce coverage
- Inventory flow conservation
- Regular and overtime capacity
Linearisation and Piecewise Objectives
Epigraph variables, maximum functions, absolute values and concave maximisationSome nonlinear-looking objectives can be represented exactly by a linear model when they are finite piecewise-linear envelopes. The direction of each auxiliary inequality matters.
- Epigraph reformulation
- Maximum of affine functions
- Sum of maxima
- Absolute-value objectives
- Piecewise-linear convexity
- Concave maximisation
Duality, Certificates and Shadow Prices
Weak duality, complementary slackness, infeasibility certificates and sensitivityDuality gives a second optimisation model whose feasible values bound the primal objective. Matching primal and dual values can certify optimality without searching every feasible plan.
- Primal and dual viewpoints
- Weak duality bounds
- Strong duality conditions
- Complementary slackness
- Dual feasibility
- Farkas certificates
Zero-Sum Games as Linear Programs
Mixed strategies, guaranteed payoff, equalisation and maximin formulationsA finite two-player zero-sum game can be written as a linear program because one player chooses probabilities to maximise a guaranteed payoff against every opposing pure action. The opponent solves the dual minimisation problem.
- Payoff matrices
- Pure and mixed strategies
- Probability simplex
- Guaranteed payoff
- Maximin formulation
- Opponent dual problem
Integer Models and Branch-and-Bound
Binary decisions, logical links, LP relaxations, integrality gaps and branchingInteger restrictions represent indivisible quantities, yes-or-no choices, and logical structure. Removing integrality creates an LP relaxation that supplies a bound, not necessarily an implementable plan.
- Integer and binary variables
- Mixed-integer models
- Fixed-charge logic
- Set covering
- LP relaxation
- Integrality gap
Network Flow and Assignment Models
Conservation, capacities, shortest paths, assignment and max-flow dualityNetwork structure turns many large linear models into node and arc balances. A flow entering a transshipment node must leave it, while source and sink balances encode supply and demand.
- Nodes, arcs and direction
- Flow conservation
- Arc costs and capacities
- Minimum-cost flow
- Shortest-path models
- Assignment structure
Optimisation Under Uncertainty and Risk
Worst-case models, expected value, recourse framing, VaR and CVaRDeterministic models treat uncertain quantities as known, which can produce fragile decisions. Uncertainty can be handled through expected-value models, worst-case protection, staged recourse, or explicit risk measures.
- Sources of model uncertainty
- Expected-value decisions
- Worst-case protection
- Scenario and recourse framing
- Loss distributions
- Value at Risk
Linear optimisation then provides the core language for resource allocation, capacity planning, production, transportation, workforce coverage, inventory, and short-term financing. Geometry links halfspaces, active constraints, vertices, and basic feasible solutions. Algebra and convexity explain why those pictures generalise beyond two variables.
Later topics use duality to produce bounds and certificates, shadow prices to interpret local resource value, and linearisation to represent finite piecewise-linear objectives. Discrete decisions require integer and binary variables, LP relaxations, integrality-gap reasoning, and branch-and-bound. Network structure brings conservation, capacity, shortest-path, assignment, and max-flow ideas into one family.
A zero-sum game becomes a maximin linear program over probabilities. Uncertain decisions introduce expected value, worst-case protection, recourse framing, Value at Risk, and Conditional Value at Risk. Throughout, the important habit is verification: a solver output matters only when the formulation is correct, the candidate is feasible, and an independent bound or structural check supports the conclusion.
The recurring discipline is to connect each model family to its own proof obligation. A linear candidate needs feasibility and a bound; a network needs conservation at every node; a mixed strategy needs probabilities that sum to one and a guarantee against every response; an integer incumbent needs comparison with a valid relaxation bound; and a tail-risk calculation needs an explicit loss and probability convention.
Inequality direction carries meaning: capacity commonly uses total use ≤ availability, while service or nutrition requirements commonly use delivery ≥ need. A good written solution also distinguishes an exact reformulation from an approximation, states the conditions behind an extreme-point or integrality claim, and limits sensitivity conclusions to the range where the active structure remains unchanged.
These checks make the result reproducible even when software is used for the final solve.
Worked example · free
Resource allocation with a dual certificate
- +1Enumerate the feasible vertices: (0,0), (4,0), (3,2), and (0,5).
- +1Evaluate the objective at each vertex to obtain 0, 12, 13, and 10.
- +1Choose (3,2), which is feasible and has the largest vertex value, 13.
- +1Use dual multipliers u=v=1. They are feasible and give 5u+8v=13, matching the primal value.
Key terms
- Optimisation Model
- A mathematical representation of choices, goals, and operational limits.
- Feasible Solution
- A choice that satisfies every constraint and variable-domain rule.
- Linear Program
- An optimisation model with a linear objective, linear constraints, and continuous variables.
- Basic Feasible Solution
- A feasible point determined by enough independent active constraints.
- Duality
- A relationship between a primal model and a companion bounding model.
- Shadow Price
- A local marginal value associated with changing a constraint bound.
- Integer Program
- A model requiring selected decision variables to take integer values.
- Flow Conservation
- A balance requiring inflow and outflow to match at a transshipment node.
- Mixed Strategy
- A probability distribution over actions in a game.
- Conditional Value at Risk
- The average loss in a specified worst tail.
QBUS2310 FAQ
What does Management Science teach?
It teaches how to formulate decisions as optimisation models, solve or bound those models, and interpret the result. The course connects linear and integer programming with geometry, duality, network flows, strategic games, and uncertainty, while emphasising assumptions and critical evaluation.
How is the unit assessed?
The official Semester 2 offering uses three written practical assignments, a mid-semester written test, and a final written exam. The published weights are 8%, 12%, 20%, 20%, and 40%, which total 100%. Check the current unit page for operational dates and instructions.
Do I need to know programming?
The unit uses optimisation software and Python-based tools, but mathematical formulation comes first. The published mid-semester format is written and has no programming component. Software should be used to solve a model you can define, audit, and explain.
What should I check after a solver returns an answer?
Check variable units, constraint directions, feasibility, objective value, and the solver status. Then find an independent certificate where possible, such as vertex comparison, a matching dual value, flow conservation, or a relaxation bound.
How should I prepare for the written tests?
Practise translating new stories into variables, objectives, and constraints without looking at a solution. Follow each formulation with a verification pass and a short managerial interpretation. Keep handwritten revision notes aligned with the current assessment instructions.
Questions students ask in each QBUS2310 chapter
- Why does prescriptive analytics and decision quality matter?
- Decision Variable: A controllable quantity whose value the optimisation model chooses. Parameter: Given information that describes costs, capacities, requirements, or other fixed inputs. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation. Chapter 01 · Optimisation Decisions and Linear Models
- Why does sets, membership and quantifiers matter?
- Matrix Product: An array operation defined only when the inner dimensions of the factors agree. Linear Independence: A property meaning no vector in a set can be formed as a nontrivial combination of the others. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation. Chapter 02 · Linear Algebra, Convexity and Model Structure
- Why does halfspaces and polyhedra matter?
- Halfspace: One side of a hyperplane, including its boundary, described by a linear inequality. Active Constraint: A constraint that holds with equality at the point being examined. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation. Chapter 03 · Feasible Geometry and Basic Solutions
- Why does manufacturing resource models matter?
- Flow Balance: An equation matching what enters, leaves, is produced, or is consumed by an entity. Coverage Constraint: A requirement that enough scheduled resources serve a time period or demand group. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation. Chapter 04 · Building Linear Optimisation Models
- Why does epigraph reformulation matter?
- Epigraph Variable: An auxiliary variable used as an upper bound on objective components. Affine Function: A linear expression plus a constant. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation. Chapter 05 · Linearisation and Piecewise Objectives
- Why does primal and dual viewpoints matter?
- Dual Problem: A companion optimisation model that prices or combines the primal constraints. Weak Duality: The rule that every feasible dual value bounds every feasible primal value in the appropriate direction. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation. Chapter 06 · Duality, Certificates and Shadow Prices
- Why does payoff matrices matter?
- Zero-Sum Game: A game in which one player's gain is the other player's equal loss. Mixed Strategy: A probability distribution over pure actions. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation. Chapter 07 · Zero-Sum Games as Linear Programs
- Why does integer and binary variables matter?
- Integer Program: An optimisation model requiring some or all variables to take integer values. Binary Variable: A zero-or-one variable representing a discrete choice. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation. Chapter 08 · Integer Models and Branch-and-Bound
- Why does nodes, arcs and direction matter?
- Flow Conservation: The requirement that inflow and outflow balance at a transshipment node. Arc Capacity: An upper bound on flow along a directed connection. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation. Chapter 09 · Network Flow and Assignment Models
- Why does sources of model uncertainty matter?
- Expected Value: A probability-weighted average outcome. Worst-Case Model: A model that protects against the most adverse outcome in a stated uncertainty description. Keep these two roles separate when checking the model, because confusing them changes the feasible set, bound, or operational interpretation. Chapter 10 · Optimisation Under Uncertainty and Risk
How to study for the exam
Learn by alternating modelling and auditing. On one day, formulate a new production, transport, inventory, or risk problem from prose. On the next, inspect a finished formulation and locate a deliberate defect in its units, indices, timing, or inequality direction. Keep a compact glossary of theorem conditions and a separate error log.
Recompute small examples by hand before using software, because geometry, conservation, and primal-dual bounds reveal mistakes that a status message cannot explain. For assessment preparation, practise complete written solutions: define variables, state the objective, list constraints and domains, solve or reason to a candidate, verify it independently, and translate the result back into the decision context.
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