Use simulations (disaster planning, pilot training)

9 Modelling – Using Simulations

Learning Objectives (AO1‑AO3)

• AO1 – Knowledge: Define simulations, what‑if analysis and goal‑seek; recognise where they are used (disaster planning, pilot training, finance, population, climate, queues, traffic, construction).
• AO2 – Application: Apply what‑if analysis and goal‑seek in spreadsheets; construct simple simulation models for the contexts above.
• AO3 – Analysis: Evaluate the strengths and limitations of a simulation, interpret results and make justified recommendations.

What Is a Simulation?

A simulation is a computer‑based model that imitates the behaviour of a real‑world system over time. It allows users to explore “what would happen if…?” without affecting the actual system.

Key Features of Effective Simulations

• Realism – accurate representation of physical, logical or procedural aspects.
• Interactivity – users can change variables and instantly see the effect.
• Scalability – can model anything from a single workstation to a city‑wide network.
• Feedback – provides data for analysis, decision‑making and improvement.

Simulation Development Process

1. Define objectives and scope.
2. Collect reliable data and identify the variables.
3. Develop the mathematical/algorithmic model (e.g., algebraic formula, differential equation, stochastic process).
4. Select and implement suitable software.
5. Validate and calibrate the model against real‑world data.
6. Run a range of scenarios and record outcomes.
7. Analyse results, draw conclusions and refine the model.

Modelling Techniques

What‑If Analysis

• Definition: Changing one or more input variables to see how the outputs are affected.
• Spreadsheet‑only example (Finance):
  1. Cell B2 – loan amount (£10 000)
  2. Cell B3 – annual interest rate (%). Create a data table that varies this rate from 3 % to 7 % in 0.5 % steps.
  3. Cell B4 – monthly repayment calculated with =PMT(B3/12, 5*12, -B2).
  4. Result: students see how a 1 % rise in interest increases the monthly payment by ≈£15.
• Disaster‑evacuation example (already in notes) – varies walking speed v_avg to observe total evacuation time T_evac.

Goal‑Seek

• Definition: Finding the input value that produces a desired output.
• Spreadsheet‑only example (Business):
  1. Cell B2 – unit selling price (unknown).
  2. Cell B3 – fixed costs (£12 000).
  3. Cell B4 – variable cost per unit (£8).
  4. Cell B5 – target profit (£5 000).
  5. Cell B6 – profit formula =B2*1000 – B3 – B4*1000.
  6. Use “Goal Seek” → set B6 to 5000 by changing B2. Result: required selling price ≈ £25.
• Pilot‑training example (already in notes) – finds thrust percentage that yields a climb rate of 1500 ft min⁻¹.

Characteristics of Modelling Software

Characteristic	Spreadsheet (Excel, Google Sheets)	Specialised Modelling Tools (AnyLogic, MATLAB, R, Simulink)
Ease of use	Very high – familiar UI, built‑in formulas	Variable – requires programming or modelling‑language knowledge
Validation & testing	Cell‑audit, formula‑trace	Unit‑test frameworks, debugging tools
Scalability	Limited by row/column count and calculation speed	Designed for large‑scale, multi‑agent or continuous‑time models
What‑if & goal‑seek support	Native data‑tables, Scenario Manager, Goal Seek	Usually via scripting or dedicated experiment modules
Visualization	Charts, conditional formatting	3‑D graphics, animation, GIS integration
Cost & accessibility	Often already available in schools	May require licences; many academic licences are free

Common Modelling Contexts (Brief Case‑Studies)

1. Queue Management – School Cafeteria

• Goal: Minimise average waiting time for students.
• Variables: Arrival rate (students min⁻¹), service time per student (seconds), number of service points.
• Model: M/M/c queue formula \[ W_q = \frac{L_q}{\lambda},\qquad L_q = \frac{\rho^c}{c!(1-\rho)}\frac{P_0}{(1-\rho)^2} \] where \(\lambda\) = arrival rate, \(\mu\) = service rate, \(\rho = \lambda/(c\mu)\).
• What‑if analysis: Vary number of service points (c = 2, 3, 4) and record resulting \(W_q\). Students see a sharp drop when moving from 2 to 3 counters.
• Goal‑seek: Set a target waiting time of 2 min and use Goal Seek to find the required service rate \(\mu\) (i.e., how fast each cashier must serve).

2. Traffic‑Flow – Urban Intersection

• Goal: Reduce average vehicle delay at a signalised junction.
• Variables: Arrival flow (veh h⁻¹), green‑time allocation (seconds), cycle length (seconds).
• Model: Webster’s delay formula \[ D = \frac{C(1-g/C)^2}{2(1-\lambda)} + \frac{\lambda^2}{2\mu(1-\lambda)}, \] where \(C\) = cycle length, \(g\) = effective green time, \(\lambda\) = degree of saturation, \(\mu\) = saturation flow rate.
• What‑if analysis: Change green‑time from 20 s to 30 s while keeping cycle length at 60 s; observe the impact on delay.
• Goal‑seek: Set a maximum acceptable delay of 30 s and determine the minimum green‑time required.

Other Contexts Mentioned in the Syllabus

• Financial forecasting – interest‑rate or sales‑growth scenarios.
• Population growth – birth‑rate, death‑rate, migration.
• Climate change – greenhouse‑gas emission levels.
• Construction scheduling – task‑duration variations.

Case Study 1: Disaster‑Planning Simulation

Simulations help emergency services prepare for floods, earthquakes, industrial accidents, etc.

Stage	Key Activities	Typical Tools
Data Collection	Geographic information, population density, infrastructure maps	GIS, census databases
Model Development	Define hazard zones, propagation rules, resource‑allocation algorithms	MATLAB, Python (NumPy, Pandas), AnyLogic
Scenario Creation	Vary severity, time of day, response times (what‑if)	Simulation‑engine UI, spreadsheet data tables
Execution	Run multiple iterations, capture evacuation times, casualties	Monte‑Carlo engine, batch processing
Analysis	Statistical comparison, identify bottlenecks	R, Excel (pivot tables, Goal Seek)

Evacuation‑time formula (used for what‑if):

\[ T_{\text{evac}} = \frac{D}{v_{\text{avg}}} + t_{\text{prep}} \]

• D – average distance to safety (km)
• v_avg – average walking speed (km h⁻¹)
• t_prep – preparation time per person (min)

Using a spreadsheet data table, set D = 3 km, t_prep = 5 min and vary v_avg from 1.0 to 1.4 km h⁻¹. T_evac falls from 185 min to 131 min, illustrating the impact of crowd‑management measures.

Case Study 2: Pilot‑Training Simulation

Flight simulators replicate aircraft behaviour, allowing pilots to practise normal and emergency procedures safely.

Component	Function	Real‑world Equivalent
Flight‑Dynamics Model	Calculates aircraft motion using aerodynamic equations	Physical aircraft response
Avionics Suite	Displays instruments, navigation data	Cockpit panels
Environmental Model	Generates weather, turbulence, terrain	Atmospheric conditions
Control Interface	Joystick, throttle, pedals for user input	Pilot controls

Fundamental lift equation used in the dynamics model:

\[ L = \tfrac{1}{2}\,\rho\,V^{2}\,S\,C_{L} \]

• L – lift (N)
• \rho – air density (kg m⁻³)
• V – true airspeed (m s⁻¹)
• S – wing area (m²)
• C_{L} – lift coefficient (dimensionless)

Goal‑Seek example: Set a target climb rate of 1500 ft min⁻¹, keep weight and air density constant, and use Goal Seek to find the thrust percentage that satisfies the lift equation.

Comparing Disaster Planning and Pilot‑Training Simulations

Aspect	Disaster Planning	Pilot Training
Primary Goal	Optimise emergency response	Develop pilot proficiency
Time Scale	Hours to days	Seconds to minutes per scenario
Stakeholders	Government agencies, NGOs, public	Aviation authorities, airlines, pilots
Validation Method	Historical incident comparison, drills	Flight‑test data, certification standards
Typical Software	GIS + AnyLogic / Python	Dedicated flight‑sim platforms, MATLAB/Simulink

Assessment Checklist for Simulation Projects (AO1‑AO3)

• Clear definition of objectives and success criteria (AO1).
• Comprehensive data collection with source citations (AO1).
• Appropriate mathematical model (e.g., algebraic, differential, stochastic) (AO2).
• Verification that the model behaves correctly in test cases (AO2).
• Documentation of assumptions, limitations and validation results (AO3).
• Use of what‑if analysis and, where relevant, goal‑seek to explore alternatives (AO2).
• Interpretation of results with reference to real‑world implications and recommendations (AO3).

Summary

Simulations are powerful tools for modelling complex systems where real‑world experimentation is impractical or risky. By following a structured development process, selecting suitable software, and applying techniques such as what‑if analysis and goal‑seek, students can create realistic models for disaster planning, pilot training, queue management, traffic flow and many other contexts. The ability to analyse outcomes, evaluate model reliability and communicate recommendations fulfills the Cambridge IGCSE/A‑Level modelling objectives.