Introduction to Finding Optimal Solutions

Welcome! In this chapter, we are diving into one of the most practical parts of Thinking Skills: Finding Optimal Solutions. If you have ever tried to figure out the cheapest way to travel to a concert, or how to fit the most activities into a single weekend, you have already practiced this skill!

In the context of your exam (especially Paper 3: Problem Analysis and Solution), an "optimal solution" isn't just any answer; it is the best possible answer based on the rules provided. Whether you are trying to maximize profit or minimize time, this chapter will show you how to hunt down that perfect result systematically.

What is an "Optimal" Solution?

In Thinking Skills, "optimal" usually means one of two things:

  1. Maximizing: Finding the highest possible value (e.g., most points, most profit, most people helped).
  2. Minimizing: Finding the lowest possible value (e.g., least cost, shortest time, minimum waste).

Note: Before you can find an optimal solution, you often need to know how to search for solutions in general. If you haven't yet, you might want to briefly review the chapters on Searching for solutions in simple and complicated situations.

Step 1: Identify Your Goal and Constraints

Before you start calculating, you must be clear on what the problem is asking for and what the "rules" are. These rules are called constraints.

The Goal: Are you looking for the maximum or the minimum? Always underline this in your question paper!

The Constraints: These are the limits. Examples include:
- Budget: You cannot spend more than \( \$100 \).
- Time: The task must be finished by \( 09:00 \).
- Capacity: The van can only hold \( 500 \text{ kg} \).
- Logic: You cannot be in two places at once.

Quick Tip: The "Boundary" Rule

Often, the optimal solution lies right at the edge of a constraint. For example, if you want to maximize profit and can sell up to \( 50 \) items, the optimal solution is likely found by looking at what happens when you sell exactly \( 50 \) (or as close to it as the other rules allow).

Don't just guess! To find the best solution, you need a method. Here are three common ways to approach it:

1. The "Extreme Cases" Method

Try the most extreme options first. If you want to minimize cost, try the cheapest individual items first. If that doesn't satisfy the constraints, move to the next cheapest option. This narrows down the "search space" quickly.

2. The Systematic Trial

If there are only a few possibilities, list them out.
Example: If you have three different routes (A, B, and C), calculate the time for each:
- Route A: \( 15 \text{ mins} \)
- Route B: \( 12 \text{ mins} \)
- Route C: \( 18 \text{ mins} \)
The optimal (minimum) is clearly Route B.

3. Working Backwards

Sometimes, the "end state" is fixed. If you know you must arrive at \( 18:00 \), work backwards through the travel times to find the latest possible (optimal) departure time.

Real-World Example: The Delivery Van

The Scenario: You need to deliver \( 10 \) boxes.
- Big boxes weigh \( 20 \text{ kg} \) and earn you \( \$10 \).
- Small boxes weigh \( 10 \text{ kg} \) and earn you \( \$6 \).
- Your van has a limit of \( 60 \text{ kg} \).
Goal: Find the optimal (maximum) profit.

Step-by-Step Solution:
1. Test only big boxes: \( 3 \times 20 \text{ kg} = 60 \text{ kg} \). Profit: \( 3 \times \$10 = \$30 \).
2. Test only small boxes: \( 6 \times 10 \text{ kg} = 60 \text{ kg} \). Profit: \( 6 \times \$6 = \$36 \).
3. Test a mix: \( 2 \text{ big} (40 \text{ kg}) + 2 \text{ small} (20 \text{ kg}) = 60 \text{ kg} \). Profit: \( (2 \times \$10) + (2 \times \$6) = \$20 + \$12 = \$32 \).

Comparing the results: \( \$30 \), \( \$36 \), and \( \$32 \). The optimal solution is \( \$36 \).

Common Pitfalls to Avoid

1. Forgetting one constraint: It is easy to find a "great" solution that actually breaks a small rule hidden in the text. Always double-check your final answer against every bullet point in the question.

2. Stopping too early: Just because you found a "good" answer doesn't mean it's the "optimal" one. Always ask yourself: "Could I do better if I changed one variable?"

3. Misreading "Minimum" vs "Maximum": This sounds simple, but in the heat of an exam, students often find the highest cost when they were asked for the lowest.

Communicating Your Reasoning

In the 2028 syllabus, there is an increased focus on communicating reasoning. For Paper 3, you may be asked to justify why your solution is optimal. This means you shouldn't just provide the number; you should explain your process.

How to justify:
- Show your calculations clearly.
- Use units (e.g., \( \$ \), \( \text{kg} \), \( \text{hours} \)).
- Explain why other options were rejected (e.g., "Any further increase in weight would exceed the \( 60 \text{ kg} \) limit").

Quick Review Box

Goal: Find the absolute best outcome (Max or Min).
Constraints: The rules/limits you must stay within.
Method: Use systematic searching or test extreme boundaries.
Verification: Check your final answer against all rules.
Justification: Label your numbers and explain your steps.

Key Takeaways

1. Optimization is about balance: You are trying to get the best result while juggling various restrictions.

2. Calculations must be organized: As per the 2028 conventions, label your values (e.g., write "Total Cost = ..." rather than just a string of numbers).

3. Complexity grows: In Paper 3, scenarios will have more information. Take it slow, identify the constraints one by one, and then start your search for the optimal solution.

Don't worry if this seems tricky at first! Optimization is a skill that improves with practice. Start with simple "maximization" puzzles and gradually work your way up to complex Paper 3 scenarios.