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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

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 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 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 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 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




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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