49928: Design Optimisation for Manufacturing Assignment 2: Discrete Optimisation Due: 9:00 am Monday 15/10/2018 ● Solve the following two problems with both exhaustive enumeration and branch and bound...

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49928: Design Optimisation for Manufacturing Assignment 2: Discrete Optimisation Due: 9:00 am Monday 15/10/2018 ● Solve the following two problems with both exhaustive enumeration and branch and bound ● The assignment is worth 15 marks in total (15% of your final mark for the subject) ● Exhaustive enumeration is worth 2.5 marks for each problem, branch and bound is worth 5 marks for each problem. ● 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 (8 marks) Minimise: x x x x x xf = 4 1 + 5 2 + 3x3 + 6 4 + 4 5 + 5 6 + 7 7 Subject to: x x x x x 0g1 = 4 1 + 3 2 + 6 3 + 5 4 + x5 + x6 + 3 7 ≥ 5 x x x x 0g2 = 7 1 + 2x2 + 2 3 + 6 4 + 3 7 ≤ 7 x x x x x 0g3 = 6 1 + 5 2 + 3 3 + 3 4 + x5 + 8 6 + x7 ≥ 4 , x , x , x 1, 2, 3, 4}x1 2 3 4 ∈ { , x , xx5 6 7 ≥ 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: x x x x xA = x1 2 + 2 3 4 − 2 2 4 mc 2 Section modulus: (x x )S = x1 3 4 + 6 x x1 2 mc 3 Bending moment: 00M = 4 Nmk Axial force: 30P = 1 Nk Bending stress: σB = S 1000M PaM Axial stress: σP = A 10P PaM Stress constraint: 50 σB + σP − 2 ≤ 0 PaM Buckling constraint: 45 x2 x1 − 1 √4 (1+ )σB σP 2 1+173( )σB σP 2 ≤ 0 And subject to the following constraints on plate thickness and width: 37, 39, 41x1 : 1.1, 1.2, 1.3x2 : 30, 32, 34x3 : 0.8, 1.0, 1.2x4 :