Transcript Chapter 4 (Linear Programming)

ENGINEERING OPTIMIZATION
Methods and Applications
A. Ravindran, K. M. Ragsdell, G. V. Reklaitis
Book Review
Page 1
Chapter 4: Linear Programming
Part 1: Abu (Sayeem) Reaz
Part 2: Rui (Richard) Wang
Review Session
June 25, 2010
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Finding the optimum of any given world
– how cool is that?!
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Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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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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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
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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
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The 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 4.1
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 4.1
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Example 4.2
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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, ∞)
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Let’s Formulate
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
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
Period 1
200
40
Period 2
130
15
Minimum Allowable Level
1200
800
Level at the beginning of period 1
1900
850
Predicted Flow
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Final Formulation for Example 4.2
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Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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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
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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
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Graphical Solution: Example 4.4
• All points on line BC are
optimal solutions
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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!
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Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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Standard Form (Equation Form)
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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)
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Handling Inequalities
Slack
Using Equalities
Surplus
Using Bounds
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Unrestricted 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 4.5
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Conversion: Example 4.5
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Recap
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Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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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
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Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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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
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Example 4.9
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
100 hr of labor, 600 lb of material, and 300hr of administration per day
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Solution
A. Felt, ‘‘LINDO: API: Software Review,’’ OR/MS Today, vol. 29, pp. 58–60, Dec. 2002.
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Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
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Applications of LP
For any optimization problem in linear form with feasible solution time!
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Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory (Additional Topic)
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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!!
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Find a Dual: Example 4.10
Reversed
Constraint constants
Objective coefficients
Columns into constraints and
constraints into columns
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Find a Dual: Example 4.10
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Some 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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Binarization
• x is positive real, z is binary, M is a large number
• For a single variable
x
z
M
z  x*M
• For a set of variable
z
x
i
i
M
z   xi * M
i
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If
• Both x and y are binary
• If two variables share the same value
x y
• If y = 0, then x = 0
• If y = 1, then x = 1
• If they may have different values
x y
• If y = 1, then x = 1
• Otherwise x can take either 1 or 0
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OR
• A, x, y, and z are binary
x yz
A
M
A x yz
• 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
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AND
• x, y, and z are binary
zx
zy
z  x  y 1
• If any of x,y are 0 then z is 0
• If all of x,y are 1 then z is 1
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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
x  y 1
z
M
• If x <= y, z is true
y  x 1
z
M
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Finding a Value
• A,B,C are binary
• If x = y, Cy is true
x  y 1
A
M
y  x 1
B
M
Cy  A  B
x takes the value of y if both the ranges are true
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Thank You!
Now Part 2 begins….
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