Discrete Optimisation

- Solve the following two problems with both exhaustive enumeration and branch and bound
- Problem 1 is a mixed integer linear optimisation problem (the problem has both discrete and continuous variables). Do not use intlinprog (from MATLAB) to solve this problem, for exhaustive enumeration solve it by enumerating through the discrete variables and then use linprog to find the continuous variables. For branch and bound use linprog or Excel Solver to find the partial solutions.
- Problem 2 is a discrete nonlinear optimisation problem. For branch and bound use
fmincon or Excel Solver to find the partial solutions.
- Write a report:
• Describe the process of finding the solution: how many evaluations were needed for exhaustive enumeration? What path did the search take for branch and bound? How many partial and full evaluations were needed for branch and bound?
• Include your MATLAB code for exhaustive enumeration
• Include any code or an image of any spreadsheets used for branch and bound
• Draw the trees for branch and bound. For each node state:
- Which variables are constrained
- The partial or full solution
- Whether or not the solution is feasible
- Whether or not the node has been pruned

Problem 1

Minimise:

f = 4x₁ + 5x₂ + 3x₃ + 6x₄ + 4x₅ + 5x₆ + 7x₇

Subject to:

g₁ = 4x₁ + 3x₂ + 6x₃ + 5x₄ + x₅ + x₆ + 3x₇ ≥ 50
g₂ = 7x₁ + 2x₂ + 2x₃ + 6x₄ + 3x₇ ≤ 70
g₃ = 6x₁ + 5x₂ + 3x₃ + 3x₄ + x₅ + 8x₆ + x7 ≥ 40
x₁, x2, x3, x4 ∈ {1, 2, 3, 4}
x₅, x₆, x₇ ≥ 0

Problem 2

An I-beam is shown in the figure to the right. Given the following equations and constraints, develop a mathematical model and find the dimensions of a beam with a minimal cross sectional area.

Cross sectional area: A = x₁x₂ + 2x₃x₄ - 2x₂x₄ cm²
Section modulus: S = x₁(x₃x₄ + x₁x₂ ) cm³
Bending moment: M = 400 kNm
Axial force: P = 130 kN

Bending stress: σ_B = 1000M/S    MPa
Axial stress: σ_P = 10P/A    MPa

stress constraint:  σ_B + σ_P - 250 ≤ 0     MPa

Buckling constraint: x₁/x₂ -145 ⁴√(1 + σ_P/σ_B)²/(1 + 173(σ_P/σ_B)² ≤ 0)

And subject to the following constraints on plate thickness and width:

x₁ : 37, 39, 41
x₂ : 1.1, 1.2, 1.3
x₃ : 30, 32, 34
x₄ : 0.8, 1.0, 1.2