1 Assignment 2 (Due date: 9am June 23, 2016) Solve the following two problems by using (1) exhaustive enumeration and (2) Branch and Bound algorithm. (Hint: For Branch and Bound, you can use MATLAB Toolbox or Excel Solver to find the optimal solutions to the continuous relaxations of the problems). Question I: (8 marks) Minimise x 1 + 3x2 + 2x3 +8x4 + 9 x5 + 4x6 + 7x7 Subject to: 5x 1 + 2x2 + 3x3 + 6x4 + x5 + 5x6 + x7 ≥120 3x 1 + 4x2 + x3 + 2x4 + 3x6 ≤ 70 3x 1 + 5x2 + x3 + 7x4 + 2 x5 +8x6 + 6x7 ≥ 30 x 1 ∈ {1, 2,3, 4}, x2 ∈ {1, 2,3, 4}, x3 ∈ {1, 2,3, 4}, x4 ∈ {1, 2,3, 4}, x5, x6, x7 ≥ 0 Question II: (7 marks) An I-beam of the type shown in the Figure below is to be chosen in structural application. The data and the relevant relationships are given: Cross-sectional area: 2 A = x1x2 + 2x3x4 − 2 x2x4 cm Section modulus: 3 S = x1(x3x4 + x1x2 / 6) cm Bending moment: M = 400 kNm Axial force: P = 150 kN Bending stress: MPa S M B 1000 σ = 2 Axial stress: MPa PA P 10 σ = Stress constraint: MPa σ B +σ P − 200 ≤ 0 Buckling constraint: 0 1 173( ) (1 ) 145 4 2 2 1 2 ≤ + + − P B P B xx σ σ σ σ The plate widths and thickness are available in following sizes: x1: 60, 61, 62 x2: 0.4, 0.5, 0.6 x3: 32, 36, 40 x4: 0.7, 0.8, 0.9 Choose the beam of minimum cross section.