Deterministic Operations Research Models

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Deterministic Operations Research Models

• J. Paul Brooks
• Jill R. Hardin
• Department of Statistical Sciences and Operations
  Research
• November 28, 2006

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Food for Thought

• Daily snackpeanuts and popcorn
• Need at least 12 grams of protein and at least 24
  grams of carbs
• Peanuts (serving size 1 oz)
• 6 grams protein
• 6 grams carbs
• Popcorn (serving size 1 cup)
• 2 grams protein
• 6 grams carbs

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Food for Thought

• How many servings of each is most cost effective,
  
• while still meeting your carb/protein
  requirements?
• Costco
• Peanuts 0.25 per oz
• Popcorn 0.15 per cup
• Sams
• Peanuts 0.25 per oz
• Popcorn 0.30 per cup
• BJs
• Peanuts 0.35 per oz
• Popcorn 0.10 per cup

1 oz peanuts 3 cups popcorn
4 oz peanuts
6 cups popcorn
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Food for Thought

• Possible solutions defined by
• Nutritional content of each food
• Nutritional requirements
• Solution quality determined by
• Cost of each food
• How did you find a solution?
• What would you do if the problem involved many
  foods and many nutritional requirements?

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Mathematical Programming

• Represents decisions to be made with decision
  variables
• Optimizes the objective functiona function of
  the decision variables
• Respects constraints or restrictions on the
  values that can be assigned to the variables.

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Back to Peanuts and Popcorn

• What decisions must be made?
• Number of oz of peanuts
• Number of cups of popcorn
• What is the objective?
• Minimize total cost
• Costco
• Sams
• BJs

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Back to Peanuts and Popcorn

• What are the constraints?
• Minimum level of protein intakeat least 12 grams
• Minimum level of carb intakeat least 24 grams
• Nonnegative number of servings

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The Mathematical Program
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A Graphical Representation
Popcorn
8
4
Peanuts
2
6
10
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A Graphical Representation
Popcorn
8
Feasible Region
4
Peanuts
2
6
10
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Facts about solutions to Math Programs

• Fact 1 A solution might not exist. Why?
• Infeasibilitythere might be no solution that
  satisfies every constraint. May have to be
  flexible on one or more constraint.
• Unboundednesswe can make the objective value as
  large (or small) as we wish. Typically indicates
  a missing constraint.

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General Classes of Math Programs

• Linear Programs (LP)
• Integer Programs (IP)
• Nonlinear Programs (NLP)

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General Classes of Math Programs

• Linear Programs (LP)
• Objective is a linear function of the decision
  variables
• Constraints can be expressed as linear functions
  of the decision variables
• All variables can take fractional values
• Relatively easy to solve

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Facts about solutions to Math Programs

• Fact 2
• For a linear program, if a solution does exist,
  one will be at a corner point (also called an
  extreme point).
• This allows us to find solutions very quickly,
  because it limits the search space.

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Corner Points for Snack Problem
Popcorn
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Feasible Region
(0,6)
4
(1,3)
Peanuts
2
6
10
(4,0)
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General Classes of Math Programs

• Integer Programs (IP)
• Linear objective, linear constraintsjust like an
  LP.
• One or more variables are limited to integer
  values
• Allows binary (0/1, yes/no) variablesdramatically
  increases modeling power!
• Harder to solve, but for most problems we can do
  it with enough time.
• Many advanced techniques have been developed to
  decrease solution time. Software handles most
  general cases fairly easily, but if not, consult
  an expert (e.g. Jill or Paul!)

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IP Feasible Regions
Feasible region
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General Classes of Math Programs

• Nonlinear
• Objective or some constraint(s) cannot be
  expressed as linear function of the decision
  variables.
• Some special cases are easy (or easier) to
  handle
• Quadratic objective/linear constraints
• Convex objective and feasible region
• In general, very difficult to solve. Hard to
  tell when we have local versus global optimum.
  Often tackled with metaheuristics (genetic
  algorithms, simulated annealing, etc.)

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Local versus Global
Local Maxima
Global Maximum
Local Minima
Global Minimum
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Modeling with Binary Variables

• In treating a brain tumor with radiation, we want
  to bombard the tissue containing the tumors with
  the maximum possible amount of radiation. The
  constraint is, of course, that there is a maximum
  amount of radiation that normal tissue can handle
  without suffering tissue damage. Physicians must
  therefore decide how to aim the radiation to
  accomplish these aims.

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Modeling with Binary Variables

• As a simple example of this situation, suppose
  there are six types of radiation beams (beams
  differ in where they are aimed and their
  intensity) that can be aimed at a tumor. The
  region containing the tumor has been divided into
  six regions three regions contain tumors and
  three contain normal tissue. The amount of
  radiation delivered to each region by each type
  of beam is given in the table. If each region of
  normal tissue can handle at most 60 units of
  radiation, which beams should be used to maximize
  the total amount of radiation received by the
  tumors?

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Modeling with Binary Variables
Beam Normal 1 Normal 2 Normal 3 Tumor 1 Tumor 2 Tumor 3
1 24 18 12 30 18 9
2 18 15 9 27 23 12
3 14 12 20 20 15 26
4 6 18 18 9 27 24
5 14 6 17 20 8 21
6 12 11 11 15 15 15
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Modeling with Binary Variables

• What are the decisions to be made?
• Which beams to use
• More specifically, for each beam, should we use
  it? A yes/no decision.
• Binary variables are ideal here.

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Modeling with Binary Variables

• What is the objective?
• Maximize total radiation delivered to tumors
• Each beam used delivers radiation to each tumor
• Six possible beams
• When variable is zero (i.e. beam not used) no
  radiation delivered when variable is 1 (i.e.
  beam used) full amount of radiation delivered.

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Modeling with Binary Variables

• What are the constraints?
• Maintain acceptable radiation levels in normal
  tissue
• Specifically, each normal region should receive
  no more than 60 total units of radiation from all
  beams

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AMPL and the NEOS Server

• Solving mathematical programs typically requires
• two things
• Model file
• Reflects structure of the problem
• Data-independent
• Data file for a specific instance

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AMPL and the NEOS Server

• Many languages available for writing models.
  Well use AMPL (www.ampl.com).
• The (free)NEOS Server for Optimization allows us
  to
• submit model and data files
• choose solver
• obtain a solution
• www-neos.mcs.anl.gov

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Applications of Math Programming

• Nurse staffing/scheduling
• Haplotype inference
• Protein Threading
• Sequence Alignment
• Therapy design (radiotherapy, brachytherapy, HIV
  treatment)
• Vaccine selection
• Design of organ allocation regions
• Flux Balance Analysis