1. In the design space, plot the objective function contours for the following unconstrained problem and identify the Pareto optimal set:

    Minimize: f₁(x) = (x₁ − 2)² + (x₂ − 3)²

    f₂(x) = (x₁ − 5)² + (x₂ − 8)²

    Draw the gradients of each function at any point on the Pareto optimal set in the design space. Comment on the relationship between the two gradients.

2. Sketch the Pareto optimal set for Problem 1 in the criterion space. Find the utopia point. Is the utopia point attainable?

3. A constrained optimization of two variables and two objectives is given below:

    Minimize:  f₁(x) = (x₁ − 2)² + (x₂ − 3)²

    f₂(x) = (x₁ − 5)² + (x₂ − 8)²

    Subject to: g₁ = −x₁ − x₂ + 10 ≤ 0

    g₂ = −10 − 2x₁ + 3x₂ ≤ 0

4. A right circular cone shown below is considered for a redesign. The objective is to use a minimum material (surface area) to achieve a maximum volume. More specifically, we want to minimize both the lateral area S and total surface area T of the cone for a volume no less than 250 cm³ by varying the base radius r and height h of the cone. Note that T = S + B, in which B is the based area B = πr². The size of the cone is constrained by r ∈ (1, 10) cm and h ∈ (1, 20) cm.

5. Solve Problem 4 using the weighted-sum method. Find at least 10 solutions by varying the weights (including a case for w₁ = 0.1 and w₂ = 0.9). Identify those solutions in the Pareto front in the criterion space.

6. Solve Problem 4 using the weighted min–max method, assuming w₁ = 0.1 and w₂ = 0.9. Compare results with those of Problem 5. Comment on the pros and cons of the weighted min–max and the weighted-sum method based on the observation made in solving the cone problem.

7. Solve Problem 4 using the NBI methods by creating at least five points on the CHIM.

8. Solve Problem 3 using the min–max method.

9. Derive Eq. 5.21c of Example 5.7: β=12[(α1+α2)+2−(α12−α22)]. Hint: (α₁ − β)² + (α₂ − β)² = 1.