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ENGINEERING OPTIMIZATIONMethods and Applications
A. Ravindran, K. M. Ragsdell, G. V. Reklaitis
Book Review
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Chapter 4: Linear Programming
Part 1: Abu (Sayeem) ReazPart 2: Rui (Richard) Wang
Review SessionJune 25, 2010
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Finding the optimum of any given world – how cool is that?!
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Outline of Part 1Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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Outline of Part 1Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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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
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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
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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
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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”
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Example 4.1Example 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!!
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Final Formulation for Example 4.1Final Formulation for Example 4.1
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Example 4.2Example 4.2
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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, ∞)
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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
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Final Formulation for Example 4.2Final Formulation for Example 4.2
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Outline of Part 1Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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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
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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
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Graphical Solution: Example 4.4Graphical Solution: Example 4.4
• All points on line BC are optimal solutions
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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!
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Outline of Part 1Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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Standard Form (Equation Form)Standard Form (Equation Form)
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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)
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Handling InequalitiesHandling Inequalities
Using Bounds
SlackUsing Equalities
Surplus
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Unrestricted VariablesUnrestricted Variables
In some situations, it may become necessary to introduce a variable that can assume both positive and negative values!
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Conversion: Example 4.5Conversion: Example 4.5
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Conversion: Example 4.5Conversion: Example 4.5
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RecapRecap
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Outline of Part 1Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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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
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Outline of Part 1Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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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
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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
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SolutionSolution
A. Felt, ‘‘LINDO: API: Software Review,’’ OR/MS Today, vol. 29, pp. 58–60, Dec. 2002.
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Outline of Part 1Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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Applications of LPApplications of LP
For any optimization problem in linear form with feasible solution time!
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Outline of Part 1Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory (Additional Topic)
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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!!
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Find a Dual: Example 4.10Find a Dual: Example 4.10
Objective coefficients
Constraint constants
Reversed
Columns into constraints and constraints into columns
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Find a Dual: Example 4.10Find a Dual: Example 4.10
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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
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BinarizationBinarization
• x is positive real, z is binary, M is a large number
• For a single variable
• For a set of variable
xz
M
ii
xz
M
*z x M
*ii
z x M
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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
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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 zA
MA x y z
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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 x
z y
z x y
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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 yz
M
1y xz
M
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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 yA
My x
BM
C A B
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Thank You!Thank You!
Now Part 2 begins….Now Part 2 begins….