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# EML4552 - Engineering Design Systems II (Senior Design Project) - PowerPoint PPT Presentation

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)

Optimization Theory and

Optimum Design

Dynamic Programming

Hyman: Chapter 10

• 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

• 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

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

• 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

• 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

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)

• 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

• 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

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

• 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.)

• 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.)

• 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

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

Stage n-1

Stage 1

Dynamic Programming

Minimize Fuel Consumption through

Compressor Pressure Settings