Determining the optimal solution to a real-world engineering problem IEEE TEP Activity 3: It’s all about Optimization 2010 Spring Technical English Program Санкт-Петербург, Рассия May 15, 2010 Determining the optimal solution to a real-world engineering problem

IEEE TEP: All About Optimization OPTIMIZE a quantity of interest MINIMIZE MAXIMIZE Time Risk Cost Effort Energy Efficiency Performance Return Revenue Profit

IEEE TEP: All About Optimization Example problem You need to buy some filing cabinets. Two choices Cabinet X and Cabinet Y

IEEE TEP: All About Optimization Example problem Cabinet X Cabinet Y Costs $10 per unit Costs $20 per unit Requires six (6) square feet of floor space Requires eight (8) square feet of floor space Holds eight (8) cubic feet of files Holds twelve (12) cubic feet of files

IEEE TEP: All About Optimization Your role -- is to optimize!!! You have been given $140 for this purchase, though you don't have to spend that much. The office has room for no more than 72 square feet of cabinets. How many of which model should you buy, in order to maximize storage volume?

IEEE TEP: All About Optimization Your approach?

IEEE TEP: All About Optimization Variables of interest?

IEEE TEP: All About Optimization Mathematical Model – Define Variables What are my variables of interest? x: number of model X cabinets purchased y: number of model Y cabinets purchased Also called DECISION VARIABLES.

IEEE TEP: All About Optimization Mathematical Model – Define Variables What are my variables of interest? x: number of model X cabinets purchased y: number of model Y cabinets purchased Also called DECISION VARIABLES. Other quantities of interest based on the information provided?

IEEE TEP: All About Optimization Mathematical Model – Define Variables What are my variables of interest? x: number of model X cabinets purchased y: number of model Y cabinets purchased Also called DECISION VARIABLES. Other quantities of interest based on the information provided: Cost: C = 10x+20y

IEEE TEP: All About Optimization Mathematical Model – Define Variables What are my variables of interest? x: number of model X cabinets purchased y: number of model Y cabinets purchased Also called DECISION VARIABLES. Other quantities of interest based on the information provided: Cost: C = 10x+20y Space: S = 6x + 8y

IEEE TEP: All About Optimization Mathematical Model – Define Variables What are my variables of interest? x: number of model X cabinets purchased y: number of model Y cabinets purchased Also called DECISION VARIABLES. Other quantities of interest based on the information provided: Cost: C = 10x+20y Space: S = 6x + 8y Volume: V = 8x + 12y

IEEE TEP: All About Optimization Any Constraints?

IEEE TEP: All About Optimization Mathematical Model – Any constraints? Are there any constraints on cost (money)?

IEEE TEP: All About Optimization Mathematical Model – Any constraints? Are there any constraints on cost (money)? Yes – I only have $140 to purchase

IEEE TEP: All About Optimization Mathematical Model – Any constraints? Are there any constraints on cost (money)? Yes – I only have $140 to purchase i.e. Cost: C < 140 i.e. 10x + 20y < 140 or y < – ( 1/x) + 7

IEEE TEP: All About Optimization Mathematical Model – Any constraints? Are there any constraints on space?

IEEE TEP: All About Optimization Mathematical Model – Any constraints? Are there any constraints on space? Yes - no more than 72 square feet of cabinets

IEEE TEP: All About Optimization Mathematical Model – Any constraints? Are there any constraints on space? Yes - no more than 72 square feet of cabinets i.e. Space: S < 72 i.e. 6x + 8y < 72 or y < – ( 3/ 4)x + 9

IEEE TEP: All About Optimization All Constraints together

IEEE TEP: All About Optimization All Constraints together Are there any constraints on cost (money)? Yes – I only have $140 to purchase i.e. Cost: C < 140 i.e. 10x + 20y < 140 or y < – ( 1/x) + 7 Are there any constraints on space? Yes - no more than 72 square feet of cabinets i.e. Space: S < 72 i.e. 6x + 8y < 72 or y < – ( 3/ 4)x + 9

IEEE TEP: All About Optimization What about volume?

IEEE TEP: All About Optimization Mathematical Model – Any constraints? Are there any constraints on volume? Not really We are trying to MAXIMIZE VOLUME i.e. MAXIMIZE: Volume: V = 8x + 12y

IEEE TEP: All About Optimization Mathematical Model – FINAL MODEL MAXIMIZE: VOLUME: V = 8x + 12y subject to: COST: y < – ( 1/x) + 7 and SPACE: y < – ( 3/ 4)x + 9

IEEE TEP: All About Optimization Let’s look at the Feasible Region y 10 8 6 x = 0 4 2 x 2 4 8 10 12 6 14 y = 0

IEEE TEP: All About Optimization Let’s look at the Feasible Region y 10 8 SPACE: y < – ( 3/ 4)x + 9 6 x = 0 4 2 x 2 4 8 10 12 6 14 y = 0

IEEE TEP: All About Optimization Let’s look at the Feasible Region y 10 8 6 x = 0 4 COST: y < – ( 1/x) + 7 2 x 2 4 8 10 12 6 14 y = 0

IEEE TEP: All About Optimization Let’s look at the Feasible Region y 10 8 y < – ( 3/ 4)x + 9 6 x = 0 4 2 y < – ( 1/x) + 7 x 2 4 8 10 12 6 14 y = 0

IEEE TEP: All About Optimization Let’s Look at the Feasible Region y 10 8 y < – ( 3/ 4)x + 9 6 x = 0 4 2 y < – ( 1/x) + 7 x 2 4 8 10 12 6 14 y = 0

IEEE TEP: All About Optimization Let’s Look at the Feasible Region y 10 Corner Points 8 y < – ( 3/ 4)x + 9 6 x = 0 4 2 y < – ( 1/x) + 7 x 2 4 8 10 12 6 14 y = 0

IEEE TEP: All About Optimization Finally the solution Solutions are available on the corner of the feasible region

IEEE TEP: All About Optimization Let’s look at the Feasible Region y 10 Corner Points 8 y < – ( 3/ 4)x + 9 6 x = 0 4 2 y < – ( 1/x) + 7 x 2 4 8 10 12 6 14 y = 0

IEEE TEP: All About Optimization Evaluate the volume at corner points V = 8x + 12y At (0, 0) --> V = 0 At (8, 3) --> V = 100 At (0, 7) --> V = 84 At (12, 0) --> V = 84 Optimal solution is (8,3)

IEEE TEP: All About Optimization Your approach Mathematical Model – Write the equations that represent the system/scenario Decision Variables Objective function Constraints 2) Feasible Region – Draw the feasible region on an x-y plane that will show the constraints 3) Solution evaluation – pick points to evaluate the objective function

IEEE TEP: All About Optimization Today’s tasks Form pairs Read the problem provided to you Optimize!