Deterministic Optimization Models

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Deterministic Optimization Models

• Understanding the iSIGHT Abstraction 6/16/05

Material taken from Chapter 2 of Optimization in
OperationsResearch by Rardin.
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Topics

• Nomenclature
• Sample Problem Formulation and Intuitive Feel.
• Standard Problem Formulation
• Linear versus Nonlinear Optimization
• Continuous vs. Discrete Optimization
• Multiobjective Optimization

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Nomenclature
Simulations C, Fortran, Java, Lisp Some take
hours or days
INPUTS Design Variables Independent
Variables Starting Point lower bound upper
bound Parameters Fixed values
OUTPUTS Dependent Parameters Objective max or
min Output Main Constraints gt or lt Output
Output
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Two Crude Petroleum Example
Two Crude Petroleum runs a small refinery on the
Texas coast. The refinery distills crude
petroleum from two sources, Saudi Arabia and
Venezuela, into the three main products
gasoline, jet fuel and lubricants. The two
crudes differ in chemical composition and yield
different product mixes. Each barrel of Saudi
crude yields 0.3 barrel of gasoline, 0.4 barrel
of jet fuel, and 0.2 barrel of lubricants. Each
barrel of Venezuelan crude yields 0.4 barrel of
gasoline, 0.2 barrel of jet fuel and 0.3 barrel
of lubricants. The remaining 10 is lost to
refining. The crudes differ in cost and
availability. Two Crude can purchase up to 9000
barrels per day from Saudia Arabia at 20 per
barrel. Up to 6000 barrels per day of Venezuelan
petroleum are available at the lower cost of 15
per barrel. Two Crudes contracts require it to
produce 2000 barrels per day of gasoline,1500
barrels per day of jet fuel and 500 barrels per
day of lubricants.Develop an optimization model
to satisfy these requirements at minimum
cost. We will use this model throughout this
lecture.
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Two Crude Problem Description
Petroleum refinery distills crude from Saudi and
Venezuela
Customer Demand(barrels) 2000 gas, 1500 jet,
500 lub
Saudi Number of Barrels Purchased ????
Cost Per Barrel 20 Max Production Available
9000 Refined barrel .3 gas, .4 jet, .2
lub. Venezuela Number of Barrels Purchased
???? Cost Per Barrel 16 Max Production
Available 6000 Refined barrel .4 gas, .2
jet, .3 lub.
How can Two Crude Most Efficiently Meet Demands?
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Visual Tie In

INPUTS Design Variables x1, x2 0 ltx1lt9 0
ltx2lt6 Parameters cost S 20, V 16 refined
S .3, .4, .2 V .4, .2, .3 customer demand
G 2, JF 1.5, L .5
OUTPUTS Cost Barrels Gas Barrels Jet Fuel Barrels
Lubricants Objective min Cost Main
Constraints Barrels Gas gt 2
Barrels JetFuel gt 1.5 Barrels Lub. gt
.5
C ODE
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General Mathematical Format
The general form of a mathematical program or
(single objective) optimization model is min or
max f(x1,,xn)
Subject to
lt gt
g(x1,,xn)
bi i 1,,m
Where f, g1,,gm are given functions of decision
variables x1,,xn, and b1,,bm are specified
constant parameters.
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Two Crude Petroleum Model
min 20x1 16x2 s.t. 0.3x1 0.4x2 gt
2 0.4x1 0.2x2 gt 1.5 0.2x1 0.3x2 gt
.500 x1 lt 9 x2 lt 6 x1,x2 gt 0
Design Variables? Variable Type Constraints? Main
Constraints? Objective?
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Constraint Satisfaction for feasible design
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Class Exercise (2-1 Rardin)

• Formulate the standard formulation
• Show 2D Feasible Region and Objective Func.
  Contours

The Joel Table Company sells two models of five
leg tables. The basic version uses a wood top,
requires 0.6 hours to assemble, and sells for a
profit of 200. The deluxe version takes 1.5
hours to assemble because of glass top, and sells
for a profit of 350. The company has 300 legs,
50 wood tops, 35 glass tops and 63 hours of
assembly time. Joel wants to maximize profit and
he knows that he can sell all you can make.
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Solution

• max 200x1350x2 (maximize total profit)
• Subject to
• 5x15x2 lt 300 (legs)
• 0.6x11.5x2 lt 63 (assembly hours)
• x1lt50 (wood tops)
• x2lt35(glass tops)
• x1gt0
• x2gt0

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Linear Functions
A function is linear if it is a constant weighted
sum of decision variables. Otherwise it is
nonlinear.
f(x1,x2) 20x1 15x2 f(w1,w2,w3) w12 8w2
w32
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Recognizing Linear Functions
Assuming xs are decision variables, determine
whethereach of the following functions are
linear or nonlinear.
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Linear Programs Defined
An optimization model in functional form is a
linear program (LP) if the (single) objective
function f and all the constraint functions g1,
, gm are linear in the decision variables.
Also, decision variables should be able to take
on whole-number or fractional values.
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Nonlinear Programs Defined
An optimization model in functional form is a
nonlinear program (NLP) if the (single) objective
function f or any of the constraint functions
g1,,gm is nonlinear in the decision variables.
Also, decision variables should be able to take
on whole-number or fractional values.
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Recognizing Linear and Nonlinear Programs
Assuming ys are decision variables and all other
symbols are constant, determine if followingare
linear programs or nonlinear programs.
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Class Exercise (Rardin 2-18)
Assuming that wj are decision variables and all
other symbols Are constant, determine if each of
the following is a linear programor a non linear
program.
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Discrete or Integer Programs
Aka integer (linear or nonlinear)
programs mixed integer (linear or nonlinear)
programs combinatorial optimization problems
A variable is discrete if it is limited to a
fixed or countable set of values. Often the
choices are only 0 or 1 (binary variable).
A variable is continuous if it can take on any
value in a specifiedinterval.
When there is an option, such as when optimal
variable magnitudes are likely to be large
enough that fractions haveno practical
importance, modeling with continuous variablesis
prefered. (Does your program support option?)
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Choosing Discrete Versus Continuous Variables
Decide whether a discrete or continuous variable
would bebest employed to model each of the
following quantities.
The operating temperature of a chemical process.
The warehouse slot assigned a particular product
The amount of money converted from yen to dollars
The number of corvettes produced annually.
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Constraints with Discrete Variables
In choosing among a collection of 16 investment
projects,variables
1 if project is selected
wj
0 otherwise
Express each of the following constraints in
terms of thesevariables.
At least one of the first eight projects must be
selected.
At most three of the last eight projects must be
selected.
Either project 4 or project 9 must be selected,
but not both
Project 11 can be selected only if project 2 is
also
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Integer and Mixed Integer Programs
An optimization model is an integer program (IP)
if any ofits decision variables is discrete. If
all variables are discrete,the model is a pure
integer program otherwise, it is a mixedinteger
program.
Case 1 wj gt 0, j1,,q Case 2 wj 0 or
1, j1,,p wp1 gt 0 and integerCase 3 wj gt
0 j1,,p wp1 gt 0 and integer
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Integer Linear and Integer Nonlinear Programs
A discrete or integer programming model is an
integer linearprogram (ILP) if its (single)
objective function and all mainconstraints are
linear.A discrete or integer programming model
is an integer nonlinearprogram (INLP) if its
(single) objective function or any of itsmain
constraints is nonlinear.
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Recognizing ILPs and INLPs
Determine if each of the following is a LP, NLP,
ILP, MINLP
Max 3w1 14w2 w3 s.t. w1 lt w2 w1 w2 w3
10 wj 0 or 1 j1,,3
Min 3w1 9w2/ w3 s.t. w1 - w2lt 0 w1 w2 w3
10 w1 gt 0 w2 w3 gt 1
Min 3w1 14w2 w3 s.t. w1w2lt1 w1 w2 w3
10 wj gt 0 j1,,3 w1 integer
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Multiple Objectives
All models considered so far have a single
objective.
A multiobjective optimization model maximizes or
minimizesmore than one objective function at the
same time.
Multiple objectives for a automobile may be all
of the followingmax reliability f1(x1,,xn)
max mileage f2(x1,,xn) min cost to
produce f3(x1,,xn)
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NEOS Optimization Treehttp//www-fp.mcs.anl.gov/o
tc/Guide/OptWeb/
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Lab Objectives

• Define a iSIGHT calculation with inputs and
  outputs
• Define a standard iSIGHT optimization problem
  formulation
• Execute an optimization using default
  optimization plan

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Lab Exercise
GM Communications is choosing a cable for a new
16,000 meter telephone line. The following table
shows the diameters available, along with the
associated cost, resistance, and attenuation of
each per meter.
Diameter Cost Resistance Attenuation 4 0.092 0.279
0.00175 5 0.112 0.160 0.00130 6 0.141 0.120 0.0
0161 9 0.420 0.065 0.00095 12 0.719 0.039 0.0004
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• The company wishes to choose the least cost
  combination of wires that will provide a new line
  with at most1600 ohms resistance and 8.5
  decibels attenuation.
• Formulate a mathematical programming model with
  three main constraints to choose an optimal
  combination of wires using decision variables
  x4, x5, x6, x9 and x12
• Put your formulation into iSIGHT. Create a task
  called Communication and create a Calculation to
  implement your equations. Formulate the iSIGHT
  Optimization Model in the Parameters window. If
  you have time then execute the model with the
  default task plan and write down the answer.

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Lab Exercise
The National Science Foundation (NSF) has
received 4 proposals from professors to undertake
new research in OR methods. Each proposal can be
accepted for funding next year at the level (in
thousands of dollars) shown in the following
table or rejected. A total of 1 million is
available for the year.
Proposal 1 2 3 4 Funding 700 400 300 600 Score
85 70 62 93
Scores represent the estimated value of doing
each body of research that was assigned by
NSFsadvisory panel. Formulate a discrete
optimization model to decide what projects to
accept to maximize total score within the
available budget using decision variables pj 1
if proposal j is selected and
0 otherwise. Enter and set up the model in
iSIGHT.
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Lab 1 Follow up

• Optional slides

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What does the iSIGHT client model support?
Which of the following are supported by
iSIGHT? Continuous Nonlinear Program Continuous
Linear Program Integer Linear Program Integer
Nonlinear Program Mixed Integer Linear
Program Mixed Integer Nonlinear
Program Multiple Objective Continuous Nonlinear
Program Multiple Objective Mixed Integer
Nonlinear Program
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iSIGHT Parameter Table
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iSIGHT Abstraction

• Single objective (combines multiple objectives
  into a single weighted objective called
  Objective)
• Objective is considered a nonlinear function
• Every constraint is considered nonlinear function
• One constraint is always added by iSIGHT.
  TaskProcessStatus. You cannot remove it.
• Design variables can be integer, discrete or
  continuous.

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Supposed Difficulties of Classical Numerical
Optimization

• Computational time increases as the number of
  design variables increases. In addition, these
  methods tend to become numerically
  ill-conditioned.
• Optimization techniques have no stored experience
  or intuition on which to draw.
• Most algorithms will get stuck at a suboptimal
  design.
• If the analysis program is not theoretically
  precise, the results may be misleading.
  Optimization will take advantage of analysis
  errors to provide mathematical design
  improvements.
• Most optimization algorithms have difficulty
  dealing with discontinuous functions.
• Convergence to optimal solution depends on chosen
  initial solution.
• Algorithm efficient in solving one problem may
  not be efficient in solving a different
  optimization problem.
• Many analysis/simulation programs were not
  written with automated design in mind.