Eml4552 engineering design systems ii senior design project
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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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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

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


Why optimum design

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


Optimization

Optimization

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


Design optimization

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


Dynamic programming

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


Dynamic programming example optimum routing of a transmission line

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)


Dynamic programming example

Dynamic Programming - Example


Dynamic programming example1

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


Dynamic programming example2

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


Dynamic programming identify stages

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


Stage 1

Stage 1


Stage 2

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.)


Stage 21

Stage 2


Stage 3

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.)


Stage 31

Stage 3


Stage 4

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


Stage 41

Stage 4


Reconstruct the optimum path

Reconstruct the “Optimum Path”


Dynamic programming example optimum routing of a transmission line1

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


Dynamic programming1

Stage n

Stage n-1

Stage 1

Dynamic Programming


Example gas pipeline operation

Example: Gas Pipeline Operation

Minimize Fuel Consumption through

Compressor Pressure Settings


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