EML4552 - Engineering Design Systems II (Senior Design Project) Optimization Theory and Optimum Design Dynamic Programming Hyman: Chapter 10 EML 4550 - Spring’08

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 EML 4550 - Spring’08

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 EML 4550 - Spring’08

Optimization Minimize (Maximize) an Objective Function of certain Variables subject to Constraints EML 4550 - Spring’08

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 EML 4550 - Spring’08

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 EML 4550 - Spring’08

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 EML 4550 - Spring’08

Dynamic Programming - Example EML 4550 - Spring’08

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 EML 4550 - Spring’08

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 EML 4550 - Spring’08

Dynamic Programming - Identify “Stages” B1 B2 C1 C2 D1 D2 E 14 16 20 18 15 17 10 13 12 EML 4550 - Spring’08

Stage 1 EML 4550 - Spring’08

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.) EML 4550 - Spring’08

Stage 2 EML 4550 - Spring’08

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.) EML 4550 - Spring’08

Stage 3 EML 4550 - Spring’08

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 EML 4550 - Spring’08

Stage 4 EML 4550 - Spring’08

Reconstruct the “Optimum Path” EML 4550 - Spring’08

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 EML 4550 - Spring’08

Dynamic Programming Stage n Stage n-1 Stage 1 EML 4550 - Spring’08

Example: Gas Pipeline Operation Minimize Fuel Consumption through Compressor Pressure Settings EML 4550 - Spring’08