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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

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?!


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

    • 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!!




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 Theory (Additional Topic)


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….


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