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EML4552 - Engineering Design Systems II (Senior Design Project)

EML4552 - Engineering Design Systems II (Senior Design Project). Optimization Theory and Optimum Design Dynamic Programming. Hyman: Chapter 10. Basic Concepts. Optimization in Design From Concept Selection to Optimum Design Optimization Theory and Methods

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EML4552 - Engineering Design Systems II (Senior Design Project)

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  1. EML4552 - Engineering Design Systems II(Senior Design Project) Optimization Theory and Optimum Design Dynamic Programming Hyman: Chapter 10

  2. Basic Concepts • Optimization in Design • From Concept Selection to Optimum Design • Optimization Theory and Methods • Large number of design choices: Dynamic Programming • Optimization with continuous variables • Linear programming • Non-linear programming and search methods • Lagrange multipliers

  3. Why Optimum Design? • Find system with minimum ‘cost’-’weight’-’fuel usage’-…etc. that will fulfill the functional specification • Find system with maximum ‘capability’ within certain constraints (cost, weight, etc.) • Competitive pressure drives towards optimum design

  4. Optimization • Minimize (Maximize) an Objective Function of certain Variables subject to Constraints

  5. Design Optimization • Concept Generation • Concept Selection • System Architecture • Detailed Design • Manufacturing • Operational Experience Design Optimization starts with System Architecture and becomes an integral part of the design process through the lifetime of the product OPTIMIZATION

  6. Dynamic Programming • Optimization of systems that feature ‘stages’ • Large number of stages • Large number of choices per stage • Apparently very large number of choices (yet finite) can be efficiently explored and an optimum found with dynamic programming • Dynamic programming allows for a consistent search of the optimum in multi-stage problems • “Efficiency” of dynamic programming increases with the problem size

  7. B2 C2 D2 18 20 15 14 10 A 13 18 12 E 16 10 17 20 B1 C1 D1 Dynamic Programming - Example:Optimum Routing of a Transmission Line • Find least cost to build transmission between A and E and going through (B1 or B2), (C1 or C2), and (D1 or D2)

  8. Dynamic Programming - Example

  9. Dynamic Programming - Example • In this case the combination set of paths is very small, optimum can be found by exhaustive search and inspection • We needed to compute the ‘objective function’ 8 times to determine the minimum • What happens if the number of choices is so large that it becomes impractical to conduct an exhaustive search? • We need a structured approach to find the optimum

  10. Dynamic Programming - Example • Most D.P. problems can be solved by moving forward or backwards through the stages analyzing one stage at a time • Consider working backwards from point E • There are only two paths leading to point E • Tabulate costs for all the paths leading to the last stage

  11. B2 C2 D2 18 20 15 14 10 A 13 18 12 E 16 10 17 20 B1 C1 D1 Dynamic Programming - Identify “Stages” Stage 1 Stage 4 Stage 3 Stage 2

  12. Stage 1

  13. Stage 2 • There are four possible paths to consider in this stage, paths that begin in C1 or C2, and end on D1 or D2 • Tabulate all the costs for the paths in this stage • Combine with costs from previous stage to compute total cost for Stage 1 + Stage 2 • For each beginning point of Stage 2, pick an optimum to arrive at the end point and eliminate those paths that cannot be optimum (basic principle of D.P.)

  14. Stage 2

  15. Stage 3 • Repeat previous approach and prepare a table with the four possible paths for this stage • Only consider the optimum possibilities for the paths from the end of Stage 3 (beginning of Stage 2) to the end point E • identify the optimum paths that go from the beginning of Stage 3 to the end point E (basic principle of D.P.)

  16. Stage 3

  17. Stage 4 • Repeat procedure for the last stage, now there are only 2 paths to consider in in this stage • Apply basic principle of D.P. to determine the optimum path that covers all four stages

  18. Stage 4

  19. Reconstruct the “Optimum Path”

  20. B2 C2 D2 18 20 15 14 10 A 13 18 12 E 16 10 17 20 B1 C1 D1 Dynamic Programming - Example:Optimum Routing of a Transmission Line • In this example the optimum could be determined by inspection, but as system complexity increases, dynamic programming is needed

  21. Stage n Stage n-1 Stage 1 Dynamic Programming

  22. Example: Gas Pipeline Operation Minimize Fuel Consumption through Compressor Pressure Settings

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