ENGINEERING OPTIMIZATION Methods and Applications

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# ENGINEERING OPTIMIZATION Methods and Applications - PowerPoint PPT Presentation

ENGINEERING OPTIMIZATION Methods and Applications. A. Ravindran, K. M. Ragsdell, G. V. Reklaitis. Book Review. Chapter 4: Linear Programming. Part 1: Abu (Sayeem) Reaz Part 2: Rui (Richard) Wang. Review Session June 25, 2010. Finding the optimum of any given world – how cool is that?!.

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ENGINEERING OPTIMIZATION

Methods and Applications

A. Ravindran, K. M. Ragsdell, G. V. Reklaitis

Book Review

Chapter 4: Linear Programming

Part 1: Abu (Sayeem) Reaz

Part 2: Rui (Richard) Wang

Review Session

June 25, 2010

Outline of Part 1

• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory

Outline of Part 1

• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory

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

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

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

The Wisdom of Linear Programming

“Model building is not a science; it is primarily an art that is developed mainly by experience”

Example 4.1

• Two grades of inspectors for a quality control inspection
• At least 1800 pieces to be inspected per 8-hr day

25 inspections/hour, accuracy = 98%, wage=\$4/hour

15 inspections/hour, accuracy= 95%, wage=\$3/hour

• Penalty=\$2/error

Let’s get experienced!!

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, ∞)

Outline of Part 1

• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory

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

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

Graphical Solution: Example 4.4

• All points on line BC are optimal solutions

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!

Outline of Part 1

• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory

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)

Handling Inequalities

Slack

Using Equalities

Surplus

Using Bounds

Unrestricted Variables

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

Outline of Part 1

• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory

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

Outline of Part 1

• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory

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

Example 4.9

100 hr of labor, 600 lb of material, and 300hr of administration per day

Solution

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

Outline of Part 1

• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory

Applications of LP

For any optimization problem in linear form with feasible solution time!

Outline of Part 1

• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications

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!!

Reversed

Constraint constants

Objective coefficients

Columns into constraints and constraints into columns

Find a Dual: Example 4.10

Some Tricks

• • “Binarization”
• If
• • OR
• • AND
• • Finding Range
• • Finding the value of a variable

http://networks.cs.ucdavis.edu/ppt/group_meeting_22may2009.pdf

Binarization

• x is positive real, z is binary, M is a large number
• For a single variable
• • For a set of variable

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

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

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

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

Finding a Value

• A,B,C are binary
• • If x = y, Cy is true

x takes the value of y if both the ranges are true

Thank You!

Now Part 2 begins….