1. Page 1 Page 1 ENGINEERING OPTIMIZATION Methods and Applications A. Ravindran, K. M. Ragsdell, G. V. Reklaitis Book Review

2. Page 2 Page 2 Chapter 4: Linear Programming Part 1: Abu (Sayeem) Reaz Part 2: Rui (Richard) Wang Review Session June 25, 2010

3. Page 3 Page 3 Finding the optimum of any given world – how cool is that?!

4. Page 4 Page 4 Outline of Part 1 Formulations Graphical Solutions Standard Form Computer Solutions Sensitivity Analysis Applications Duality Theory

5. Page 5 Page 5 Outline of Part 1 Formulations Graphical Solutions Standard Form Computer Solutions Sensitivity Analysis Applications Duality Theory

6. Page 6 Page 6 What is an LP? An LP has An objective to find the best value for a system A set of design variables that represents the system A list of requirements that draws constraints the design variables The constraints of the system can be expressed as linear equations or inequalities and the objective function is a linear function of the design variables

7. Page 7 Page 7 Types Linear Program (LP): all variables are real Integer Linear Program (ILP): all variables are integer Mixed Integer Linear Program (MILP): variables are a mix of integer and real number Binary Linear Program (BLP): all variables are binary

8. Page 8 Page 8 Formulation Formulation is the construction of LP models of real problems: To identify the design/decision variables Express the constraints of the problem as linear equations or inequalities Write the objective function to be maximized or minimized as a linear function

9. Page 9 Page 9 The Wisdom of Linear Programming “Model building is not a science; it is primarily an art that is developed mainly by experience”

10. Page 10 Page 10 Example 4.1 Two grades of inspectors for a quality control inspection At least 1800 pieces to be inspected per 8-hr day Grade 1 inspectors: 25 inspections/hour, accuracy = 98%, wage=$4/hour Grade 2 inspectors: 15 inspections/hour, accuracy= 95%, wage=$3/hour Penalty=$2/error Position for 8 “Grade 1” and 10 “Grade 2” inspectors Let’s get experienced!!

11. Page 11 Page 11 Final Formulation for Example 4.1

12. Page 12 Page 12 Example 4.2

13. Page 13 Page 13 Nonlinearity “During each period, up to 50,000 MWh of electricity can be sold at $20.00/MWh, and excess power above 50,000 MWh can only be sold for $14.00/MW” Piecewise  Linear in the regions (0, 50000) and (50000, ∞)

14. Page 14 Page 14 Let’s Formulate Plant/Reservoir APlant/Reservoir B Conversion Rate per kilo-acre-foot (KAF)400 MWh200 MWh Capacity of Power Plants60,000 MWh/Period35,000 MWh/Period Capacity of Reservoir20001500 Predicted Flow Period 120040 Period 213015 Minimum Allowable Level1200800 Level at the beginning of period 11900850 PH1Power sold at $20/MWhMWh PL1Power sold at $14/MWhMWh XA1Water supplied to power plant AKAF XB1Water supplied to power plant BKAF SA1Spill water drained from reservoir AKAF SB1Spill water drained from reservoir BKAF EA1Reservoir A level at the end of period 1KAF EB1Reservoir B level at the end of period 1KAF

15. Page 15 Page 15 Final Formulation for Example 4.2

16. Page 16 Page 16 Outline of Part 1 Formulations Graphical Solutions Standard Form Computer Solutions Sensitivity Analysis Applications Duality Theory

17. Page 17 Page 17 Definitions Feasible Solution: all possible values of decision variables that satisfy the constraints Feasible Region: the set of all feasible solutions Optimal Solution: The best feasible solution Optimal Value: The value of the objective function corresponding to an optimal solution

18. Page 18 Page 18 Graphical Solution: Example 4.3 A straight line if the value of Z is fixed a priori Changing the value of Z  another straight line parallel to itself Search optimal solution  value of Z such that the line passes though one or more points in the feasible region

19. Page 19 Page 19 Graphical Solution: Example 4.4 All points on line BC are optimal solutions

20. Page 20 Page 20 Realizations Unique Optimal Solution: only one optimal value (Example 4.1) Alternative/Multiple Optimal Solution: more than one feasible solution (Example 4.2) Unbounded Optimum: it is possible to find better feasible solutions improving the objective values continuously (e.g., Example 2 without ) Property: If there exists an optimum solution to a linear programming problem, then at least one of the corner points of the feasible region will always qualify to be an optimal solution!

21. Page 21 Page 21 Outline of Part 1 Formulations Graphical Solutions Standard Form Computer Solutions Sensitivity Analysis Applications Duality Theory

22. Page 22 Page 22 Standard Form (Equation Form)

23. Page 23 Page 23 Standard Form (Matrix Form) (A is the coefficient matrix, x is the decision vector, b is the requirement vector, and c is the profit (cost) vector)

24. Page 24 Page 24 Handling Inequalities Using Bounds Slack Using Equalities Surplus

25. Page 25 Page 25 Unrestricted Variables In some situations, it may become necessary to introduce a variable that can assume both positive and negative values!

26. Page 26 Page 26 Conversion: Example 4.5

27. Page 27 Page 27 Conversion: Example 4.5

28. Page 28 Page 28 Recap

29. Page 29 Page 29 Outline of Part 1 Formulations Graphical Solutions Standard Form Computer Solutions Sensitivity Analysis Applications Duality Theory

30. Page 30 Page 30 Computer Codes For small/simple LPs: Microsoft Excel For High-End LP: OSL from IBM ILOG CPLEX OB1 in XMP Software Modeling Language: GAMS (General Algebraic Modeling System) AMPL (A Mathematical Programming Language) Internet http: / /www.ece.northwestern.edu/otc

31. Page 31 Page 31 Outline of Part 1 Formulations Graphical Solutions Standard Form Computer Solutions Sensitivity Analysis Applications Duality Theory

32. Page 32 Page 32 Sensitivity Analysis Variation in the values of the data coefficients changes the LP problem, which may in turn affect the optimal solution. The study of how the optimal solution will change with changes in the input (data) coefficients is known as sensitivity analysis or post-optimality analysis. Why? Some parameters may be controllable  better optimal value Data coefficients from statistical estimation  identify the one that effects the objective value most  obtain better estimates

33. Page 33 Page 33 Example 4.9 100 hr of labor, 600 lb of material, and 300hr of administration per day Product 1Product 2Product 3 Unit profit1064 Material Needed10 lb4 lb5 lb Admin Hr2 hr 6 hr

34. Page 34 Page 34 Solution A. Felt, ‘‘LINDO: API: Software Review,’’ OR/MS Today, vol. 29, pp. 58–60, Dec. 2002.

35. Page 35 Page 35 Outline of Part 1 Formulations Graphical Solutions Standard Form Computer Solutions Sensitivity Analysis Applications Duality Theory

36. Page 36 Page 36 Applications of LP For any optimization problem in linear form with feasible solution time!

37. Page 37 Page 37 Outline of Part 1 Formulations Graphical Solutions Standard Form Computer Solutions Sensitivity Analysis Applications Duality Theory (Additional Topic)

38. Page 38 Page 38 Duality of LP Every linear programming problem has an associated linear program called its dual such that a solution to the original linear program also gives a solution to its dual Solve one, get one free!!

39. Page 39 Page 39 Find a Dual: Example 4.10 Objective coefficients Constraint constants Reversed Columns into constraints and constraints into columns

40. Page 40 Page 40 Find a Dual: Example 4.10

41. Page 41 Page 41 Some Tricks “Binarization” If OR AND Finding Range Finding the value of a variable http://networks.cs.ucdavis.edu/ppt/group_meeting_22may2009.pdf

42. Page 42 Page 42 Binarization x is positive real, z is binary, M is a large number For a single variable For a set of variable

43. Page 43 Page 43 If Both x and y are binary If two variables share the same value If y = 0, then x = 0 If y = 1, then x = 1 If they may have different values If y = 1, then x = 1 Otherwise x can take either 1 or 0

44. Page 44 Page 44 OR A, x, y, and z are binary M is a large number If any of x,y,z are 1 then A is 1 If all of x,y,z are 0 then A is 0

45. Page 45 Page 45 AND x, y, and z are binary If any of x,y are 0 then z is 0 If all of x,y are 1 then z is 1

46. Page 46 Page 46 Range x and y are integers, z is binary We want to find out if x falls within a range defined by y If x >= y, z is true If x <= y, z is true

47. Page 47 Page 47 Finding a Value A,B,C are binary If x = y, C y is true x takes the value of y if both the ranges are true

48. Page 48 Page 48 Thank You! Now Part 2 begins….