AAE5301 Chap.1 Optimisation Foundations for Low-altitude Services
Optimisation Foundations for Low-altitude Services
From service question to mathematical decision
Optimisation starts by separating what an operator can choose from what the scenario supplies. Decision variables describe assignments, opened sites, vehicle counts or selected arcs. Parameters supply demand, time, cost, capacity and service targets. The objective ranks feasible choices, while constraints preserve physical limits and service promises.
This chapter also distinguishes LP, integer and mixed-integer models and shows why a solution must be translated back into operational language.
Build the model as a chain of claims
A defensible formulation states the decision, unit and domain of every variable before writing expressions. Each objective term then has a common interpretation and compatible unit.
Every constraint should be readable as a resource, logical or service statement, with its index range declared. This order makes missing decisions visible: if a proposed policy cannot be represented by a variable, it cannot be selected by the solver.
It also prevents fixed demand or travel time from being mistaken for a choice.
Validate before interpreting
Check dimensions, limiting cases and feasibility before celebrating an objective value. Removing a capacity limit should not make the best result worse, while raising required service should not make a minimisation problem cheaper without another change.
After solving, translate nonzero variables into actions, test the plan against the original scenario and name any omitted uncertainty. Optimisation supports a decision only when the mathematical result and the operational story agree.
What this chapter covers
- 01
Decision variables and parameters
- 02
Objectives with explicit units
- 03
Constraints and feasible sets
- 04
Binary, integer and continuous domains
- 05
LP, IP, MIP and MILP classification
- 06
Feasibility and sensitivity checks
Worked example · free
Allocate missions under endurance limits
- 2Define one binary variable for each drone-mission pair.
- 5Write value maximisation, mission uniqueness and per-drone endurance constraints.
- 3Select missions and verify the objective and constraints.
Key terms
- Objective Function
- The measurable benefit to maximise or cost to minimise over feasible decisions.
- Constraint
- A mathematical rule representing a physical, resource or service limit.
- Binary Variable
- A decision variable restricted to zero or one for a yes-or-no choice.
- Feasible Point
- A complete decision satisfying every constraint and domain restriction.
- Binding Constraint
- A constraint met at equality by the selected solution and often responsible for the trade-off.
Optimisation Foundations for Low-altitude Services FAQ
Why should variables come before the objective?
A symbol is meaningful only after it states what can be chosen, its unit and its domain. Defining variables first prevents fixed data from being treated as controllable and reveals whether a binary, integer or continuous decision is required.
Does a higher objective always mean a better service?
Only among decisions that satisfy the model and only for the service promise encoded by the objective. A high profit result can still be unacceptable if urgency, fairness or reliability was omitted from the constraints.
How do I classify a mixed-integer model?
Inspect expressions and domains. If objective and ordinary constraints are linear and at least one decision is integer while another is continuous, the model is a mixed-integer linear program.
How can I test a formulation without solving a large instance?
Create a tiny instance whose answer is obvious by inspection, such as two demand points and one candidate vehicle. Substitute one feasible and one deliberately infeasible decision into every constraint. This catches reversed inequalities, missing indices and incorrect units before solver output makes the model look authoritative. Then vary one parameter and check whether the response direction matches the operational meaning.
What should an optimisation conclusion report?
Report the selected actions, the objective value with units, the binding limits and at least one sensitivity or caveat. A list of variable values is not yet a recommendation. Explain what the operator should open, assign or schedule, why the choice beats the relevant alternative, and which assumption could change it.
How do binding constraints help explain a solution?
A binding constraint has no remaining slack at the reported solution and often identifies the resource or service promise limiting improvement. Interpret it in operational units, then test whether relaxing it changes the decision or only the objective. A nonbinding constraint can still matter in another scenario, so binding status explains the current instance rather than proving permanent irrelevance.
Pair this check with reduced-cost or sensitivity information only when the model type and solver output support that interpretation.
Exam move
Take one short scenario and rewrite it three times with a different objective: cost, served demand and waiting. Keep the physical constraints unchanged, then compare which decision changes. This exposes the value judgement carried by the objective and makes model classification easier to recall. Build a symbol table with decision, unit, index and domain columns, then annotate each constraint with its operational sentence.
Finish by testing a zero-demand case, a tight-capacity case and one deliberately infeasible case. This routine makes sign, unit and domain errors easier to find than rereading algebra alone. Summarise the final solution in one implementable sentence.
