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

Peanuts (serving size = 1 oz) 6 grams protein

6 grams carbs

Popcorn (serving size = 1 cup) 2 grams protein 6 grams carbs

Food for ThoughtHow 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

6 cups popcorn

4 oz peanuts

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 Represents decisions to be made with

decision variables

Optimizes the objective function—a function of the decision variables

Respects constraints or restrictions on the values that can be assigned to the variables.

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:

1x

2x

1 20.25 0.15x x1 20.25 0.30x x1 20.35 0.10x x

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

1 26 2 12x x

1 26 6 24x x

1 20, 0x x

The Mathematical Program

1 2

1 2

1 2

1 2

min 0.25 0.15

subject to 6 2 12

6 6 24

0, 0

x x

x x

x x

x x

2 6 10

4

8

1 26 2 12x x

A Graphical Representation

1 26 6 24x x

Peanuts

Popcorn

2 6 10

4

8

1 26 2 12x x

A Graphical Representation

1 26 6 24x x

Feasible Region

Peanuts

Popcorn

Facts about solutions to Math ProgramsFact 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

Linear Programs (LP)

Integer Programs (IP)

Nonlinear Programs (NLP)

General Classes of Math ProgramsLinear 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 ProgramsFact 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

2 6 10

4

8

Feasible Region(0,6)

(1,3)

(4,0)Peanuts

Popcorn

General Classes of Math ProgramsInteger 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

1 3 5

2

4

6

Feasible region

General Classes of Math ProgramsNonlinear 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 GlobalLocal Maxima

Local Minima

Global Minimum

Global Maximum

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 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 VariablesBeam Normal

1Normal 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

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.

1 if beam is selected

0 otherwisei

ix

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.

1 2

6

(30 18 9) (27 23 12)min

(15 15 15)

x x

x

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 beams1 2 3 4 5 6

1 2 3 4 5 6

1 2 3 4 5 6

24 18 14 6 14 12 60

18 15 12 18 6 11 60

12 9 20 18 17 11 60

x x x x x x

x x x x x x

x x x x x x

AMPL and the NEOS ServerSolving 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 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 solutionwww-neos.mcs.anl.gov

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