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. Food for Thought. Daily snack—peanuts and popcorn Need at least 12 grams of protein and at least 24 grams of carbs

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Deterministic operations research models

Deterministic Operations Research Models

J. Paul Brooks

Jill R. Hardin

Department of Statistical Sciences and Operations Research

November 28, 2006


Food for thought

Food for Thought

  • Daily snack—peanuts 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


Food for thought1

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

  • Sam’s

    • Peanuts $0.25 per oz

    • Popcorn $0.30 per cup

  • BJ’s

    • Peanuts: $0.35 per oz

    • Popcorn: $0.10 per cup

1 oz peanuts

3 cups popcorn

4 oz peanuts

6 cups popcorn


Food for thought2

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?


Mathematical programming

Mathematical Programming

  • Represents decisions to be made with decisionvariables

  • Optimizes theobjective function—a function of the decision variables

  • Respectsconstraintsor restrictions on the values that can be assigned to the variables.


Back to peanuts and popcorn

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:

      • Sam’s:

      • BJ’s:


Back to peanuts and popcorn1

Back to Peanuts and Popcorn

  • What are the constraints?

    • Minimum level of protein intake—at least 12 grams

    • Minimum level of carb intake—at least 24 grams

    • Nonnegative number of servings


The mathematical program

The Mathematical Program


A graphical representation

A Graphical Representation

Popcorn

8

4

Peanuts

2

6

10


A graphical representation1

A Graphical Representation

Popcorn

8

Feasible Region

4

Peanuts

2

6

10


Facts about solutions to math programs

Facts about solutions to Math Programs

Fact 1: A solution might not exist. Why?

  • Infeasibility—there might be no solution that satisfies every constraint. May have to be flexible on one or more constraint.

  • Unboundedness—we can make the objective value as large (or small) as we wish. Typically indicates a missing constraint.


General classes of math programs

General Classes of Math Programs

  • Linear Programs (LP)

  • Integer Programs (IP)

  • Nonlinear Programs (NLP)


General classes of math programs1

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


Facts about solutions to math programs1

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.


Corner points for snack problem

Corner Points for Snack Problem

Popcorn

8

Feasible Region

(0,6)

4

(1,3)

Peanuts

2

6

10

(4,0)


General classes of math programs2

General Classes of Math Programs

Integer Programs (IP)

  • Linear objective, linear constraints—just like an LP.

  • One or more variables are limited to integer values

  • Allows binary (0/1, yes/no) variables—dramatically 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!)


Ip feasible regions

6

4

2

1

3

5

IP Feasible Regions

Feasible region


General classes of math programs3

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.)


Local versus global

Local versus Global

Local Maxima

Global Maximum

Local Minima

Global Minimum


Modeling with binary variables

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.


Modeling with binary variables1

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?


Modeling with binary variables2

Modeling with Binary Variables


Modeling with binary variables3

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.


Modeling with binary variables4

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.


Modeling with binary variables5

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


Ampl and the neos server

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


Ampl and the neos server1

AMPL and the NEOS Server

  • Many languages available for writing models. We’ll 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


Applications of math programming

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


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