Chapter 4 (Linear Programming)

ENGINEERING OPTIMIZATIONMethods and Applications

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

Book Review

Chapter 4: Linear Programming

Part 1: Abu (Sayeem) ReazPart 2: Rui (Richard) Wang

Review SessionJune 25, 2010

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

Outline of Part 1Outline of Part 1

• Formulations

• Graphical Solutions

• Standard Form

• Computer Solutions

• Sensitivity Analysis

• Applications

• Duality Theory

What is an LP?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

TypesTypes

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

FormulationFormulation

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 ProgrammingThe Wisdom of Linear Programming

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

by experience”

Example 4.1Example 4.1Two 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!!

Final Formulation for Example 4.1Final Formulation for Example 4.1

Example 4.2Example 4.2

NonlinearityNonlinearity“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, ∞)

Let’s FormulateLet’s Formulate

Plant/Reservoir A Plant/Reservoir B

Conversion Rate per kilo-acre-foot (KAF) 400 MWh 200 MWh

Capacity of Power Plants 60,000 MWh/Period 35,000 MWh/Period

Capacity of Reservoir 2000 1500

Predicted Flow

Period 1 200 40

Period 2 130 15

Minimum Allowable Level 1200 800

Level at the beginning of period 1 1900 850

PH1 Power sold at $20/MWh MWh

PL1 Power sold at $14/MWh MWh

XA1 Water supplied to power plant A KAF

XB1 Water supplied to power plant B KAF

SA1 Spill water drained from reservoir A KAF

SB1 Spill water drained from reservoir B KAF

EA1 Reservoir A level at the end of period 1 KAF

EB1 Reservoir B level at the end of period 1 KAF

Final Formulation for Example 4.2Final Formulation for Example 4.2


• Formulations


• Standard Form



• Applications

• Duality Theory

DefinitionsDefinitions

• 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 4.3Graphical 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 4.4Graphical Solution: Example 4.4

• All points on line BC are optimal solutions

RealizationsRealizations

• 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!


• Formulations


• Standard Form



• Applications

• Duality Theory

Standard Form (Equation Form)Standard Form (Equation Form)

Standard Form (Matrix Form)Standard Form (Matrix Form)

(A is the coefficient matrix, x is the decision vector, b isthe requirement vector, and c is the profit (cost) vector)

Handling InequalitiesHandling Inequalities

Using Bounds

Slack Using Equalities

Surplus

Unrestricted VariablesUnrestricted Variables

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

Conversion: Example 4.5Conversion: Example 4.5

RecapRecap


• Formulations


• Standard Form



• Applications

• Duality Theory

Computer CodesComputer 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


• Formulations


• Standard Form



• Applications

• Duality Theory

Sensitivity AnalysisSensitivity 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 4.9Example 4.9

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

Product 1 Product 2 Product 3

Unit profit 10 6 4

Material Needed 10 lb 4 lb 5 lb

Admin Hr 2 hr 2 hr 6 hr

SolutionSolution

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


• Formulations


• Standard Form



• Applications

• Duality Theory

Applications of LPApplications of LP

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


• Formulations


• Standard Form



• Applications

• Duality Theory (Additional Topic)

Duality of LPDuality 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!!

Find a Dual: Example 4.10Find a Dual: Example 4.10

Objective coefficients

Constraint constants

Reversed

Columns into constraints and constraints into columns

Find a Dual: Example 4.10Find a Dual: Example 4.10

Some TricksSome Tricks• “Binarization”

• If

• OR

• AND

• Finding Range

• Finding the value of a variable

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

BinarizationBinarization

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

• For a single variable

• For a set of variable

xzM

ii

xz

M

*z x M

*ii

z x M

IfIf

• 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

x y

x y

OROR

• 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

x y zAM

A x y z

ANDAND

• 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

1

z xz yz x y

RangeRange

• 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

1x yzM

1y xzM

Finding a ValueFinding 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

1

1

y

x yAM

y xBM

C A B

Thank You!Thank You!

Now Part 2 begins….Now Part 2 begins….

Chapter 4 (Linear Programming)

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Transcript of Chapter 4 (Linear Programming)