Deterministic Operations Research Models J Paul Brooks Jill

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

Food for Thought p Daily snack—peanuts and popcorn p Need at least 12 grams of protein and at least 24 grams of carbs p Peanuts (serving size = 1 oz) p n 6 grams protein n 6 grams carbs Popcorn (serving size = 1 cup) n n 2 grams protein 6 grams carbs

Food for Thought How many servings of each is most cost effective, while still meeting your carb/protein requirements? p p p Costco: n Peanuts: $0. 25 per oz 1 oz peanuts n Popcorn: $0. 15 per cup 3 cups popcorn Sam’s n Peanuts $0. 25 per oz n Popcorn $0. 30 per cup 4 oz peanuts BJ’s n Peanuts: $0. 35 per oz n Popcorn: $0. 10 per cup 6 cups popcorn

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

Mathematical Programming p Represents decisions to be made with decision variables p Optimizes the objective function—a function of the decision variables p Respects constraints or restrictions on the values that can be assigned to the variables.

Back to Peanuts and Popcorn p What decisions must be made? n n p Number of oz of peanuts Number of cups of popcorn What is the objective? n Minimize total cost Costco: p Sam’s: p BJ’s: p

Back to Peanuts and Popcorn p What are the constraints? n Minimum level of protein intake—at least 12 grams n Minimum level of carb intake—at least 24 grams n Nonnegative number of servings

The Mathematical Program

A Graphical Representation Popcorn 8 4 2 6 10 Peanuts

A Graphical Representation Popcorn 8 Feasible Region 4 2 6 10 Peanuts

Facts about solutions to Math Programs Fact 1: A solution might not exist. Why? p Infeasibility—there might be no solution that satisfies every constraint. May have to be flexible on one or more constraint. p Unboundedness—we can make the objective value as large (or small) as we wish. Typically indicates a missing constraint.

General Classes of Math Programs p Linear Programs (LP) p Integer Programs (IP) p Nonlinear Programs (NLP)

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

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

Corner Points for Snack Problem Popcorn 8 Feasible Region (0, 6) 4 (1, 3) 2 (4, 0) 6 10 Peanuts

General Classes of Math Programs Integer Programs (IP) p Linear objective, linear constraints—just like an LP. p One or more variables are limited to integer values p Allows binary (0/1, yes/no) variables—dramatically increases modeling power! p Harder to solve, but for most problems we can do it with enough time. p 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 Feasible region 4 2 1 3 5

General Classes of Math Programs Nonlinear p Objective or some constraint(s) cannot be expressed as linear function of the decision variables. p Some special cases are easy (or easier) to handle: n n p 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 Maximum Local Maxima Local Minima Global Minimum

Modeling with Binary Variables p 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 p 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 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

Modeling with Binary Variables p What are the decisions to be made? n n n 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 Variables p What is the objective? n n 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 Variables p What are the constraints? n n 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 Solving mathematical programs typically requires two things: p Model file n n p Reflects structure of the problem Data-independent Data file for a specific instance

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

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