1. EML4552 - Engineering Design Systems II (Senior Design Project)
Optimization Theory and
Optimum Design
Dynamic Programming
Hyman: Chapter 10
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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
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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
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4. Optimization
Minimize (Maximize) an Objective Function of certain Variables subject to Constraints
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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
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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
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7. 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) A B1 B2 C1 C2 D1 D2 E 14 16 20 18 15 17 10 13 12
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8. Dynamic Programming - Example
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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
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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
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11. Dynamic Programming - Identify “Stages”B1 B2 C1 C2 D1 D2 E 14 16 20 18 15 17 10 13 12
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12. Stage 1
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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.)
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14. Stage 2
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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.)
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16. Stage 3
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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
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18. Stage 4
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19. Reconstruct the “Optimum Path”
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20. 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 A B1 B2 C1 C2 D1 D2 E 14 16 20 18 15 17 10 13 12
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21. Dynamic Programming
Stage n
Stage n-1
Stage 1
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22. Example: Gas Pipeline Operation
Minimize Fuel Consumption through
Compressor Pressure Settings
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