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
Page 2
Finding the optimum of any given world
– how cool is that?!
Page 3
Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
Page 4
Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
Page 5
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
Page 8
The Wisdom of Linear Programming
“Model building is not a science; it is
primarily an art that is developed mainly
by experience”
Page 9
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!!
Page 10
Final Formulation for Example 4.1
Page 11
Example 4.2
Page 12
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
Page 14
Final Formulation for Example 4.2
Page 15
Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
Page 16
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
Page 17
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
Page 18
Graphical Solution: Example 4.4
• All points on line BC are
optimal solutions
Page 19
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!
Page 20
Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
Page 21
Standard Form (Equation Form)
Page 22
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)
Page 23
Handling Inequalities
Slack
Using Equalities
Surplus
Using Bounds
Page 24
Unrestricted Variables
In some situations, it may become necessary to introduce a
variable that can assume both positive and negative values!
Page 25
Conversion: Example 4.5
Page 26
Conversion: Example 4.5
Page 27
Recap
Page 28
Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
Page 29
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
Page 30
Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
Page 31
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
Page 32
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
Page 33
Solution
A. Felt, ‘‘LINDO: API: Software Review,’’ OR/MS Today, vol. 29, pp. 58–60, Dec. 2002.
Page 34
Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory
Page 35
Applications of LP
For any optimization problem in linear form with feasible solution time!
Page 36
Outline of Part 1
• Formulations
• Graphical Solutions
• Standard Form
• Computer Solutions
• Sensitivity Analysis
• Applications
• Duality Theory (Additional Topic)
Page 37
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!!
Page 38
Find a Dual: Example 4.10
Reversed
Constraint constants
Objective coefficients
Columns into constraints and
constraints into columns
Page 39
Find a Dual: Example 4.10
Page 40
Some Tricks
• “Binarization”
• If
• OR
• AND
• Finding Range
• Finding the value of a variable
http://networks.cs.ucdavis.edu/ppt/group_meeting_22may2009.pdf
Page 41
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
Page 42
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
Page 45
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
Page 46
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
Page 47
Thank You!
Now Part 2 begins….
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