🧠 Problem Solving: Consider Wider Problems (9694)
Hello future Problem Solvers! Welcome to the final section of the Problem Solving strand. If the earlier chapters focused on getting the 'right answer' based on specific information, this chapter teaches you how to ensure that answer actually works in the messy, ever-changing real world.
This section is all about testing the robustness of your solutions and learning how to build better, more realistic representations of situations—we call these models.
4.1 Identify the Impact of a Change to a Problem
You’ve just worked hard to solve a complex problem—maybe scheduling an event or optimising a shipping route. But what happens if the rules change *after* you’ve found the perfect solution? The skill here is anticipating external factors and knowing when your solution breaks.
Why do we need to consider changes?
In real life, circumstances are rarely static. A solution that works on Tuesday might fail spectacularly on Friday if a new factor intervenes. Problems often ask you to consider the implications of a change to the scenario. This means you must assess how a new piece of information affects the validity and feasibility of your original answer.
Key Idea: Scenario Robustness
A robust solution is one that can handle small changes without completely collapsing. When a scenario changes, you must logically trace the knock-on effects.
Step-by-Step: Assessing the Impact of Change
1. Understand the Original Solution: Identify the key parameters and constraints used to achieve the initial answer (e.g., travel time was 30 minutes, maximum capacity was 50).
2. Identify the Change: Clearly state what new factor has been introduced (e.g., a road closure, a price hike, an increase in demand).
3. Assess the Implication (The Link): How does this change interact with the original parameters? (e.g., Road closure means travel time increases to 50 minutes).
4. Adjust the Solution: Determine what adjustment is needed to the original solution to accommodate the new reality.
Example Scenario (from the syllabus):
You solved a scheduling problem for a series of concerts, calculating performance timings. A news report comes out stating that a major road leading to the venue is closed, increasing travel time for 80% of attendees by 30 minutes.
Implication: This change affects the attendance. People might arrive late or not come at all, making the original timings unrealistic.
Adjustment: You might need to delay the start time of the first performance, or introduce longer breaks between sets, or even reschedule the concerts entirely to a day without the road closure.
💡 Common Mistake to Avoid:
Don’t just state the change; you must explain the consequences of the change on the outcome. (e.g., Don't just say 'road closed,' say 'road closed, meaning attendees will be delayed, impacting ticket sales and forcing a timing adjustment').
4.2 Develop a Model
A model in Problem Solving is a simplified representation of a real-world system used to find a solution. Think of it like a simplified map: it shows the essential elements needed for the task (roads, landmarks) but leaves out unnecessary details (every single tree or lamp post).
The Limitation of Simple Models
Often, the model you start with is too simple. It might work for basic calculations, but it fails when faced with the complexity of the real world.
For instance, a simple cost model for a taxi journey might be: Cost = Fixed Fee + (Distance × Price per km).
This model doesn't account for waiting time, peak-hour surcharges, or tolls. If you try to apply this simple model to a complicated journey, the predicted cost will be wrong.
Skill 1: Identifying Missing Features to Include
The first step in developing a better model is recognizing what has been omitted from the current representation. You need to identify features of the situation being modelled which need to be included.
How do you spot a missing feature?
- Check for anomalies: Does the model consistently fail to predict the correct outcome? (e.g., The calculated taxi fare is always too low).
- Look at constraints: Are there factors limiting the outcome that haven't been accounted for? (e.g., Limited working hours, weather conditions).
- Identify hidden costs or rules: Are there exceptions or conditional charges? (e.g., "If the journey is over 10 km, a £5 cleaning fee applies").
Example: A model predicts the time needed to produce 100 shirts. It accounts for sewing time but not machine maintenance.
Missing Feature: Time lost due to maintenance/breakdowns.
Skill 2: Adjusting the Model
Once you have identified the missing features, the final, crucial step is to adjust a model to incorporate additional features. This means formally introducing the new element into your system of calculation or rules.
How to Incorporate New Features:
1. Adding a New Parameter/Variable: This is necessary for quantifiable elements.
Example: If you missed the waiting time in the taxi model, you add a new variable: + (Waiting Time × Price per minute).
2. Adding Conditional Rules (Thresholds): This is necessary when the calculation changes based on whether a limit is passed.
Example: If the cost changes after a certain distance, you add a threshold rule: "If Distance > 20 km, increase Price per km by 10%". (Remember the syllabus mentions calculations that vary past a threshold value.)
3. Establishing Relationships: This links the new feature to existing features.
Example: Machine maintenance time is not fixed, but is related to how many shirts are produced (after every 500 shirts, 1 hour of maintenance is required).
Did you know?
Developing sophisticated models is what statisticians, engineers, and financial analysts do every day! You are learning how to bridge the gap between pure mathematics and practical application.
Quick Review: The Two Main Skills
1. Impact of Change (4.1): A sudden external factor happens. How does it make the old solution invalid? What adjustment is required?
2. Develop a Model (4.2): The initial model is flawed/incomplete. What feature is missing? How do you formally build that feature into the model (via variables or rules)?