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

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Engineering optimization methods and applications

ENGINEERING OPTIMIZATION

Methods and Applications

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

Book Review


Engineering optimization methods and applications

Chapter 4: Linear Programming

Part 1: Abu (Sayeem) Reaz

Part 2: Rui (Richard) Wang

Review Session

June 25, 2010


Engineering optimization methods and applications

Finding the optimum of any given world

– how cool is that?!


Engineering optimization methods and applications

Outline of Part 1

  • Formulations

  • Graphical Solutions

  • Standard Form

  • Computer Solutions

  • Sensitivity Analysis

  • Applications

  • Duality Theory


Engineering optimization methods and applications

Outline of Part 1

  • Formulations

  • Graphical Solutions

  • Standard Form

  • Computer Solutions

  • Sensitivity Analysis

  • Applications

  • Duality Theory


Engineering optimization methods and applications

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


Engineering optimization methods and applications

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


Engineering optimization methods and applications

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


Engineering optimization methods and applications

The Wisdom of Linear Programming

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


Engineering optimization methods and applications

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!!


Engineering optimization methods and applications

Final Formulation for Example 4.1


Engineering optimization methods and applications

Example 4.2


Engineering optimization methods and applications

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, ∞)


Engineering optimization methods and applications

Let’s Formulate


Engineering optimization methods and applications

Final Formulation for Example 4.2


Engineering optimization methods and applications

Outline of Part 1

  • Formulations

  • Graphical Solutions

  • Standard Form

  • Computer Solutions

  • Sensitivity Analysis

  • Applications

  • Duality Theory


Engineering optimization methods and applications

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


Engineering optimization methods and applications

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


Engineering optimization methods and applications

Graphical Solution: Example 4.4

  • All points on line BC are optimal solutions


Engineering optimization methods and applications

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!


Engineering optimization methods and applications

Outline of Part 1

  • Formulations

  • Graphical Solutions

  • Standard Form

  • Computer Solutions

  • Sensitivity Analysis

  • Applications

  • Duality Theory


Engineering optimization methods and applications

Standard Form (Equation Form)


Engineering optimization methods and applications

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)


Engineering optimization methods and applications

Handling Inequalities

Slack

Using Equalities

Surplus

Using Bounds


Engineering optimization methods and applications

Unrestricted Variables

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


Engineering optimization methods and applications

Conversion: Example 4.5


Engineering optimization methods and applications

Conversion: Example 4.5


Engineering optimization methods and applications

Recap


Engineering optimization methods and applications

Outline of Part 1

  • Formulations

  • Graphical Solutions

  • Standard Form

  • Computer Solutions

  • Sensitivity Analysis

  • Applications

  • Duality Theory


Engineering optimization methods and applications

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


Engineering optimization methods and applications

Outline of Part 1

  • Formulations

  • Graphical Solutions

  • Standard Form

  • Computer Solutions

  • Sensitivity Analysis

  • Applications

  • Duality Theory


Engineering optimization methods and applications

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


Engineering optimization methods and applications

Example 4.9

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


Engineering optimization methods and applications

Solution

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


Engineering optimization methods and applications

Outline of Part 1

  • Formulations

  • Graphical Solutions

  • Standard Form

  • Computer Solutions

  • Sensitivity Analysis

  • Applications

  • Duality Theory


Engineering optimization methods and applications

Applications of LP

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


Engineering optimization methods and applications

Outline of Part 1

  • Formulations

  • Graphical Solutions

  • Standard Form

  • Computer Solutions

  • Sensitivity Analysis

  • Applications

  • Duality Theory (Additional Topic)


Engineering optimization methods and 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!!


Engineering optimization methods and applications

Reversed

Constraint constants

Objective coefficients

Columns into constraints and constraints into columns

Find a Dual: Example 4.10


Engineering optimization methods and applications

Find a Dual: Example 4.10


Engineering optimization methods and applications

Some Tricks

  • • “Binarization”

  • If

  • • OR

  • • AND

  • • Finding Range

  • • Finding the value of a variable

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


Engineering optimization methods and applications

Binarization

  • x is positive real, z is binary, M is a large number

  • For a single variable

  • • For a set of variable


Engineering optimization methods and applications

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


Engineering optimization methods and applications

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


Engineering optimization methods and applications

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


Engineering optimization methods and applications

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


Engineering optimization methods and applications

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


Engineering optimization methods and applications

Thank You!

Now Part 2 begins….


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