1 / 129

Instructor: Assist. Prof. Dr.- Ing . Mostafa Ranjbar

Multidisciplinary Engineering Design Optimization (MCE 540 Graduate Course – Mechanical Engineering Department). Instructor: Assist. Prof. Dr.- Ing . Mostafa Ranjbar

charlenew
Download Presentation

Instructor: Assist. Prof. Dr.- Ing . Mostafa Ranjbar

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Multidisciplinary Engineering Design Optimization (MCE 540 Graduate Course – Mechanical Engineering Department) • Instructor: • Assist. Prof. Dr.-Ing. Mostafa Ranjbar • Ph.D. (Dr-Ing.), Multidisciplinary Engineering Design Optimization of Structures,TechnischeUniversität Dresden, Germany, 2011 • M.Sc., Vibration Monitoring and Fault Diagnosis of Structures, TarbiatModares University, Tehran, Iran, 2000 • B.Sc., Mechanical Engineering, Shiraz university, Iran, 1998

  2. MULTIDISCIPLINARY SYSTEM DESIGN Optimization ______________________________________AN INTRODUCTION LECTURE #1

  3. Today’s Lecture • Introduction to Optimization • Modern Meta-Heuristic Optimization • Design of Experiments, RSMs & Meta-Modeling • Robust Design • Tools • Applications • Recent Developments & Challenges • Future • Q&A

  4. MULTIDISCIPLINARY SYSTEM DESIGN Optimization ______________________________________INTRODUCTION TO SYSTEM LECTURE #1

  5. The World Around Us “All modern products are designed as a SYSTEM”

  6. The World Around Us

  7. The World Around Us

  8. The World Within Us SYSTEMS?

  9. More Examples of Systems

  10. More Examples of Systems

  11. More Examples of Systems

  12. More Examples of Systems SYSTEMS? Level Specific Name • System Launch vehicle • Subsystem Propulsion • Element SRM • Component Ignition Device • Part Igniter

  13. More Examples of Systems COMPARTMENTALIZATION • Helicopter as an example of a Multidisciplinary Complex System “Helicopters don’t fly. They beat the air into submission.” Dr. Ed Smith

  14. f(t) input + x(t) output - - c k RS Model MODELING Meta Model The Modeling Space Model Physical system World

  15. MULTIDISCIPLINARY SYSTEM DESIGN Optimization ______________________________________OPTIMIZATION LECTURE #1

  16. WHAT IS OPTIMIZATION? • “Making things better” • “Generating more profit” • “Determining the best” • “Do more with less”

  17. WHAT IS OPTIMIZATION? “The determination of values for design variableswhich minimize (maximize) the objective, while satisfying all constraints” Principles of Optimal Design: Modeling and Computation 2d Ed. by Panos Y. Papalambros and Douglass J. Wilde, Cambridge University Press, New York, 1988, 2000.

  18. OPTIMIZATION Design Space: The space of working (Hill in this case) Objective: Find the Highest Point. Design Variables: Longitude and latitude. Constraints: Stay inside the fences.

  19. OPTIMIZATION

  20. OPTIMIZATION Objective Function Constraints Bounds Design Variables

  21. Responses Derivatives ofresponses (design sensitivities) SOLVING OPTIMIZATION PROBLEMS • Optimization problems are typically solved using an iterative algorithm: Model Constants Designvariables Optimizer

  22. LOCAL AND GLOBAL OPTIMA LOCAL OPTIMA maxima Local maxima Local minima minima GLOBAL MINIMA

  23. Optimization Problems

  24. Optimization Problems

  25. HISTORICAL PERSPECTIVE • Lagrange (1750): constrained minimization • Cauchy (1847): steepest descent • Dantzig (1947): Simplex method (LP) • Kuhn, Tucker (1951): optimality conditions • Karmakar (1984): interior point method (LP) • Bendsoe, Kikuchi (1988): topology optimization

  26. Lagrange • Joseph-Louis Lagrange (25 January 1736 – 10 April 1813), born Giuseppe Lodovico (Luigi) Lagrangia, was a mathematician and astronomer • Significant contributions to all fields of analysis, to number theory, and to classical and celestial mechanics. • On the recommendation of Euler and d'Alembert, in 1766 Lagrange succeeded Euler as the director of mathematics at the Prussian Academy of Sciences in Berlin, • Lagrange's treatise on analytical mechanics (MécaniqueAnalytique, 4. ed., 2 vols. Paris: Gauthier-Villars et fils, 1888-89), written in Berlin and first published in 1788, offered the most comprehensive treatment of classical mechanics since Newton and formed a basis for the development of mathematical physics in the nineteenth century. • The method of Lagrange multipliers gives a set of necessary conditions to identify optimal points of equality constrained optimization problems. • This is done by converting a constrained problem to an equivalent unconstrained problem with the help of certain unspecified parameters known as Lagrange multipliers.

  27. Lagrange Multipliers (  ) • The classical problem formulation minimize f(x1, x2, ..., xn) Subject to h1(x1, x2, ..., xn) = 0 can be converted to minimize L(x, l) = f(x) -  h1(x) where L(x, ) is the Lagrangian function is an unspecified positive or negative constant called the Lagrangian Multiplier • New problem is: minimizeL(x, l) =f(x)-h1(x) • Suppose that we fix  =  *and the unconstrained minimum of L(x; l) occurs at x = x* and x* satisfies h1(x*) = 0, then x* minimizes f(x) subject to h1(x) = 0. • Trick is to find appropriate value for Lagrangian multiplier l. • This can be done by treating l as a variable, finding the unconstrained minimum of L(x, l) and adjustinglso that h1(x) = 0 is satisfied.

  28. Method • Original problem is rewritten as: minimize L(x, l) =f(x)-lh1(x) • Take derivatives of L(x, l) with respect to xi and set them equal to zero. • If there are n variables (i.e., x1, ..., xn) then you will get n equations with n + 1 unknowns (i.e., n variables xi and one Lagrangian multiplier l) • Express all xi in terms of Langrangian multiplier l • Plug x in terms of l in constraint h1(x) = 0 and solve l. • Calculate x by using the just found value for l. • Note that the n derivatives and one constraint equation result in n+1 equations for n+1 variables!

  29. Multiple constraints • The Lagrangian multiplier method can be used for any number of equality constraints. • Suppose we have a classical problem formulation with k equality constraints minimize f(x1, x2, ..., xn) Subject to h1(x1, x2, ..., xn) = 0 ...... hk(x1, x2, ..., xn) = 0 This can be converted in minimizeL(x, l) = f(x) - lT h(x) Where lT is the transpose vector of Lagrangian multpliers and has length k

  30. EXAMPLE • HONDA CARS Factory manufactures HONDA CITY and HONDA CIVIC • MANUFACTURING CONSTRAINT; PLANT CAPACITY = 90 cars per day

  31. EXAMPLE I can’t manage cost or manufacturability We cannot manage the assembly line with both cars The engineer at his management desk The manufacturing engineer in the machine shop

  32. EXAMPLE • COST of Manufacturing; C (x, y)= 6x2 + 12y2

  33. EXAMPLE • VARIABLES x = No. of HONDA CITY cars produced y = No. of HONDA CIVIC cars produced • COST of Manufacturing; C (x, y)= 6x2 + 12y2 • OBJECTIVE: MINIMIZE COST • CONSTRAINT: 90 cars per day x + y = 90 • Original problem is rewritten as: minimize L(x, l) =f(x)-lh1(x)

  34. EXAMPLE OR x + y - 90= 0 • VARIABLES x = No. of HONDA CITY cars produced y = No. of HONDA CIVIC cars produced • COST of Manufacturing; C (x, y)= 6x2 + 12y2 • OBJECTIVE: MINIMIZE COST • CONSTRAINT: 90 cars per day x + y = 90 • Original problem is rewritten as: minimize L(x, l) =f(x)-lh1(x)

  35. EXAMPLE C (x, y)= 6x2 + 12y2 - l (x + y - 90) Fx = 12x – l Fy = 24y– l F l= – x – y + 90 • VARIABLES x = No. of HONDA CITY cars produced y = No. of HONDA CIVIC cars produced • COST of Manufacturing; C (x, y)= 6x2 + 12y2 • OBJECTIVE: MINIMIZE COST • CONSTRAINT: 90 cars per day x + y = 90 • Original problem is rewritten as: minimize L(x, l) =f(x)-lh1(x)

  36. EXAMPLE C (x, y)= 6x2 + 12y2 - l (x + y - 90) Fx = 12x – l = 0 Fy = 24y– l = 0 F l= – x – y + 90 = 0 • VARIABLES x = No. of HONDA CITY cars produced y = No. of HONDA CIVIC cars produced • COST of Manufacturing; C (x, y)= 6x2 + 12y2 • OBJECTIVE: MINIMIZE COST • CONSTRAINT: 90 cars per day x + y = 90 • Original problem is rewritten as: minimize L(x, l) =f(x)-lh1(x)

  37. EXAMPLE C (x, y)= 6x2 + 12y2 - l (x + y - 90) Fx = 12x – l = 0 ; x = l / 12 Fy = 24y– l = 0 ; y= l / 24 F l= – x – y + 90= 0; l = 720 x = 60 ; y= 30 C (x, y)= 6(60)2 + 12(30)2 ? • VARIABLES x = No. of HONDA CITY cars produced y = No. of HONDA CIVIC cars produced • COST of Manufacturing; C (x, y)= 6x2 + 12y2 • OBJECTIVE: MINIMIZE COST • CONSTRAINT: 90 cars per day x + y = 90 • Original problem is rewritten as: minimize L(x, l) =f(x)-lh1(x)

  38. LINEAR PROGRAMMINGCHARACTERISTICS of LP MODELS

  39. LINEAR PROGRAMMING • CHARACTERISTICS

  40. General LP format

  41. LINEAR PROGRAMMINGSTEPS

  42. Steps for LP formulation • Step 1: define decision variables • Step 2: define the objective function • Step 3: state all the resource constraints • Step 4: define non-negativity constraints

  43. LINEAR PROGRAMMINGMAXIMIZATION PROBLEM

  44. A Maximization Model Example The Beaver Creek Pottery Company produces bowls and mugs. The two primary resources used are special pottery clay and skilled labour. The two products have the following resource requirements for production and profit per item produced (that is, the model parameters). Resource available: 40 hours of labour per day and 120 pounds of clay per day. How many bowls and mugs should be produced to maximizing profits give these labour resources? LP formulation Example 1: Max Problem

  45. LP Model Formulation A Maximization Example • Product mix problem - Beaver Creek Pottery Company • How many bowls and mugs should be produced to maximize profits given labor and materials constraints? • Product resource requirements and unit profit:

  46. Sample of LP Let xi be denoted as xi product to be produced, and i = 1, 2 or Let x1 be numbers of product 1 to be produced and x2 be numbers of product 2 to be produced Maximize Z=$40x1 + 50x2 subject to 1x1 + 2x2  40 hours of labor 4x2 + 3x2 120 pounds of clay x1, x2 0 Decision variables Objective function Constraints

  47. Max LP problem Step 1: define decision variables Let x1=number of bowls to produce/day x2= number of mugs to produce/day Step 2: define the objective function maximize Z = $40x1 + 50x2 where Z= profit per day Step 3: state all the resource constraints 1x1 + 2x2 40 hours of labor ( resource constraint 1) 4x1 + 3x2 120 pounds of clay (resource constraint 2) Step 4: define non-negativity constraints x10; x2  0 Complete Linear Programming Model: maximize Z=$40x1 + 50x2 subject to 1x1 + 2x2  40 4x2 + 3x2  120 x1, x2  0

  48. LP Model Formulation A Maximization Example Resource40 hrs of labor per day Availability: 120 lbs of clay Decision x1 = number of bowls to produce per day Variables: x2 = number of mugs to produce per day Objective Maximize Z = $40x1 + $50x2 Function:Where Z = profit per day Resource1x1 + 2x2 40 hours of labor Constraints: 4x1 + 3x2 120 pounds of clay Non-Negativityx1  0; x2  0 Constraints:

  49. LP Model Formulation A Maximization Example Complete Linear Programming Model: Maximize Z = $40x1 + $50x2 subject to: 1x1 + 2x2  40 4x1 + 3x2  120 x1, x2  0

More Related