ENGINEERING OPTIMIZATION

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ENGINEERING OPTIMIZATION Methods and Applications
A. Ravindran, K. M. Ragsdell, G. V. Reklaitis
Book Review
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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,
  wage4/hour
• Grade 2 inspectors
• 15 inspections/hour, accuracy 95, wage3/hour
• Penalty2/error
• Position for 8 Grade 1 and 10 Grade 2
  inspectors

Lets 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, 8)
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Lets 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) Conversion Rate per kilo-acre-foot (KAF) 400 MWh 200 MWh
Capacity of Power Plants Capacity of Power Plants 60,000 MWh/Period 35,000 MWh/Period
Capacity of Reservoir Capacity of Reservoir 2000 1500
Predicted Flow Predicted Flow Predicted Flow Predicted Flow
Period 1 200 40
Period 2 130 15
Minimum Allowable Level Minimum Allowable Level 1200 800
Level at the beginning of period 1 Level at the beginning of period 1 1900 850
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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. 5860, 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
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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_2
2may2009.pdf
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Binarization

• x is positive real, z is binary, M is a large
  number
• For a single variable
• For a set of variable

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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

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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

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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

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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 gt y, z is true
• If x lt y, z is true

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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
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Thank You! Now Part 2 begins.