Unit 1.3: Optimal Problem Formulation

Formulating a problem is often more difficult than solving it. It involves translating a descriptive physical problem into a purely mathematical model.

Step-by-Step Procedure

1. Identify Design Variables: What can you change? (e.g., Length, Width).
2. Identify the Objective: What is the goal? (e.g., Minimize Cost). Express it in terms of variables.
3. Identify Constraints: What are the rules? (e.g., Max Stress, Min Volume).
4. Set Bounds: What are the limits? (e.g., Dimensions > 0).

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Detailed Example 1: Design of a Soft Drink Can

Problem: Design a closed cylindrical soft drink can to hold a volume of V = 330 ml, such that the material used (Total Surface Area) is minimized.

1. Design Variables:

• Radius: r
• Height: h
• Vector: X = [r, h]ᵀ

2. Objective Function:

• Minimizing material means minimizing Surface Area (A).
• Top/Bottom Area: 2 * πr²
• Side Area: 2πrh
• Minimize f(r, h) = 2πr² + 2πrh

3. Constraints:

• Volume must be exactly 330 ml.
• Volume formula: V = πr²h
• Eq Constraint: πr²h = 330 => πr²h - 330 = 0

4. Bounds:

• Radius and height cannot be negative.
• r > 0, h > 0

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Detailed Example 2: Rectangular Garden Fencing

Problem: You have 100 meters of fencing wire. You want to make a rectangular garden that has the maximum possible area.

1. Design Variables:

• Length: x
• Width: y

2. Objective Function:

• Maximize Area A = x * y
• (Or Minimize f(x, y) = -xy)

3. Constraints:

• Total perimeter must be 100m.
• Perimeter P = 2x + 2y
• Eq Constraint: 2x + 2y = 100 => x + y - 50 = 0

4. Bounds:

• x > 0, y > 0