1. 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

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

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

4. 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

5. 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?

6. IEEE TEP: All About Optimization
Your approach?

7. IEEE TEP: All About Optimization
Variables of interest?

8. 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.

9. 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?

10. 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

11. 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

12. 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

13. IEEE TEP: All About Optimization
Any Constraints?

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

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

16. 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

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

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

19. 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

20. IEEE TEP: All About Optimization
All Constraints together

21. 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

22. IEEE TEP: All About Optimization
What about volume?

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

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

32. 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

33. 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)

34. 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

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