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# 49928: Design Optimisation for Manufacturing

Assignment 2

Due: 9:00 am 18/06/2018

Solve the following problems with both exhaustive enumeration and branch and bound.

Write a short report:

Describe the process of finding the solution for both methods.

For branch and bound show the branch evaluated at each iteration (both problems should need 6 branches each).

Describe the differences in the way the different methods find the solution (think about the number of full and partial evaluations, remember that you are trying to demonstrate that you understand exhaustive enumeration and branch and bound).

Include any code you wrote.

Implementing a full branch and bound algorithm may be too difficult, instead you can manually perform branch and bound, solving each iteration with MATLAB or Excel.

Problem 1 (8 marks)

Minimise:

f  = 3x1 + 3x2 + 5x3 + 4x4 + 3x5 + 5x6 + 8x7

Subject to:

g1 = 4x1 + 3x2 + 6x3 + 5x4 + x5 + x6 + 3x7 ≥ 40

g2 = 7x1 + 2x2 + 2x3 + 6x4 + 3x7 ≤ 90

g3 = 6x1 + 5x2 + 3x3 + 3x4 + x5 + 8x6 + x7 ≥ 50

x1, x2, x3, x4 {1, 2, 3, 4}

x5, x6, x7 ≥ 0

Problem 2 (7 marks)

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:

Section modulus:

Bending moment:

Axial force:

Bending stress:

Axial stress:

Stress constr

A = x1x2 + 2x3x4 − 2x2x4

S   = x1(x3x4 + x16x2 )

M  = 400

P    = 150

σ        = 1000M

BS

σ        = 10P

PA

σB + σP − 200 ≤ 0

cm2

cm3

kNm

kN

M P a

M P a

M P a

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 x √ (1+ σP )2