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Optimización Combinatoria: Una Introducción

Dr. Juan Frausto Solís

Tecnológico de Monterrey

Campus Cuernavaca

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CONTENTS

Course Data Subjects The rules of the game Introduction to LP:

Linear Programming to Optimization Theory

Modeling Problems

Some Complexity Remarks

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Course´s Data

Dr. Juan Frausto Solís juan.frausto@itesm.mxa n.frausto@itesm.mx http://campus.cva.itesm.mx/jfrausto/ Address:

• Reforma 182-A Col Lomas de Cuernavaca

• Temixco Morelos, 62589 Mexico

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Course´s Data

Dairigido a:• Estudiantes de doctorado• Estudiantes de Maestrìa con

posibilidad de hacer investigaciòn independiente

Propósito del curso:• Una introducción a los

métodos de Optimización Combinatoria. Habilidades para investigación original en el área.

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Temas

1. Introducción 2. Linear Programming (LP). Método simplex General 3. Variantes del Método Simplex. RSM; Dual 4. Degeneración 5. Interior-Point Methods (IPM) 6. Evolutive Programming. Simplex Genetic, Simplex

Annealing 7. Simplex Cosine 8. Optimización Combinatoria en Algoritmos de Redes:

SVM; NN 9. Problemas duros Clásicos: SAT, Graph Coloring,.. 10. Problemas duros no clásicos: GRASPING, Futbol

Robótico, Inversión Bursatil,..

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Bibliography

G.B. Dantzig, Linear Programming, Springer David G: Luemberger, Programación Lineal y No

Lineal, Addison Wesley Iberoamerica Jorge Nocedal, Operations Research, Springer A. D. Belegundu, Optimization Concepts and

Applications in Engineering, Prentice Hall Hamdy Taha, Operations Research, An

Introduction. Prentice Hall Autores: Golberg, Gary Johnson, Papadimitrou,..

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Rules of the GameConcept

Work Hard

Work Hard

Work Hard

Work Hard

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Optimización Combinatoria

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Many Extensions collectively knownAs Mathematical Programming: Non LP,Int Prog, Stocastic Prog, Comb Opt,Network Flow Maximization

1947: G.W. Dantzig propose LP to setgeneral objectives and arrive at a detailedschedule to meet these goals.

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OPTIMIZATION AREAOPTIMIZATION AREA

optimize z = f(x)Subject to x

Constraint

Unconstraint

LinearNo Linear

ContinuosInteger

One variableMulti-variable

MultiobjectiveOne-objective

Tipo deProblemaType ofProblem

x

xz

z y

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What Mathematical Programming is?

Math Prog = Optimization Theory

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Math Prog Vs LP

Math Prog: Branch of Math dealing withtechniques for Maximizing or Minimizing anObjective function, subject to linear, no linear,and integer constraints.

LP: Spetial case of MP concerned withMaximizing or Minimizing a linear Objectivefunction in many variables and subject toLinear equality and inequality constraints

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Why the Programming word is into LP?

For many applications, the solutionCan be interpreted as a program,Namely, a statement of the timeAnd quantity of actions toPerformed by the system so that itMay move from its given statusTowards some defined objective

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Size of LP Problems

Small Problems: constraints 1000

Medium Problems: 1000 constraints 2000

Large Problems: 2000 constraints 10000

Very Large Problems: constraints 10,000

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Some Simple Examples

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A product Mix Problem

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The case of DES-K, a founiture company

DES-k manufactures four models of desks. Each one is first constructed in the carpentry shop and is next sent to the finishing Shop, where it is varnished, waxed, and polished. The number of man-hours of labor required in each shop and the number of hours available in each shop are known.The raw materials and supplies are available in adequate supply and all desks produced can be sold.The DES-k wants to know the optimal product mix,that is, the quantities to make of each type of deskthat will maximize profit.

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Resource Allocation Model in a Manufacturing Company

Many activities compete for resources, such as machine capacity at a plant. The available quantities of some resources may be insufficient to accomodate all the demands placed on them.Moreover, some activities may consume severalresources in producing desired outputs. LP models allow resources to be allocated across the entire sytem being analized to determine how scarce resources can be optimal used.

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Ajax Computer Co. Has a Resource Allocation Problem

Ajax sells three types of computers with the followingNet Profit for each sold computer:

Personal Computer Alpha: $350NoteBook Computer Beta: $470Workstation Gamma : $610

Net profit equals the sales price of each computer minusthe direct costs of´purchasing components, producingComputer cases, and assembling and testing theComputer. We assume that all production during the week will be sold immediatly.

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Ajax test capacity

This week, 120 hours are available onThe A-Line Test equipment Where assembled´s and ´s are tested, and 48 hours are availableOn the C-Line test equipment where ´s assembledAre tested. The testing of each computer takes1 hour.

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Ajax Production constraint for this week: Labor availability

Production is constrained on the availabilityof 2000 labor hours for product assembly. Each requires 10 labor hours,each requires 15 labor hours andEach requires 20 labor hours.Other activities are involved, but are not veryimportant.

We want to allocate these resourcesto maximize net profits for the week.

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Building an LP Model

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Before you can put a problem into a computer and efficiently find a solution, you must first abstract it, which means you have to build a mathematical model.

Formulating Linear Programs

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Formulating Linear Problem

The process of building a math model is so important than solving it, because this process provides insight about how the system works.

The model also helps organize essential information about it.

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It is easy to formulate a Problem?

To formulate a problem of the real word could be most difficult than to solve it!!

This is because of the richness, variety, and ambiguity than exists in the real world or

Because of our ambiguos understanding of the real world.

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How to get a Model: The row (material balance) Approach

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Step 0

Understanding the Problem

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Step 1: Define the Decision Variables

X= Number of ´s to be assembled tested and sold during the week

X = Number of ´s to be assembled tested and sold during the week

X= Number of ´s to be assembled tested and sold during the week

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Step 2.What is the Objective Defining the Objective Function

Objective: To Maximize the Net Profit

Net Profit= 350 X+ 470X + 610X

Max Z = 350 X+ 470X + 610X

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Step 3: Constraints

A Line Test Capacity:

X+ X 120

C line Test Capacity

X 48

Labor availability

10 X+ 15X + 20 X 2000

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Step 4: No Negativity variables

X 0 X 0 X 0

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Final Step: Joining Everything

Max Z = 350 X+ 470X + 610X

Subjecto to: X+ X 120 A-Line test Capacity X 48 C-Line Test Capacity10 X+ 15X + 20 X 2000 Labor availability

X 0; X 0 ; X 0

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Model Structure

Max Z = 350 X+ 470X + 610X

Subjecto to: X+ X 120 X 48 10 X+ 15X + 20 X 2000

X 0; X 0 ; X 0

Right Hand Sides

Non NegativeVariables

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Structure Model: Activities

Max Z = 350 X+ 470X + 610X

Subjecto to: X+ X 120 X 4810 X+ 15X + 20 X 2000

X 0; X 0 ; X 0

Asociated with eachDecision variableIs an activityDescribing the rateAt which theDecision variableConsumesResources.

1 010

X Activity

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How to get a Model?

Row approach

Material Balance Approach

Column Approach

Recipe/Activity Approach

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DES-K With Data

DES-K manufactures four models of desks. Each desk is first constructed in the carpentry shop and then is sent to the finiship shop, where it is varnished, washed and polished. The number of man hours of labor requerid on each shop is:

Desk1 (hrs)

Desk2 (hrs)

Desk3 (hrs)

Desk4 (hrs)

Available

Carpentry Shop

Finishing Shop

4

1

9

1

7

3

10

40

6,000

4,000

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DES-K With Data

Because of limitations in capacity of the plant,no more than 6,000 man hours can be expected in the carpentry shop and 4000 in the finishing shop in the next six months. The profit (revenue minus costs) from the sale of each item is:

Desk1 Desk2 Desk3 Desk4

Profit $12 $20 $18 $40

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DES-K With Data

Determine the optimal production mix assuming that the raw material and supplies are available in adequate supply amd all desks produced will be sold.

That is, determine the quantities to make each type of product which maximize profit

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A System: It is decomposable into a number of elementary functions: the activities.

Column (Recipe/Activity)Approach

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Column (Recipe/Activity)Approach

Activitymen,material,equipment

Final orIntermediateproductsitems

Activity: Like a recipe in a cookbook. Activity level: Quantity of each activity

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Column (recipe/Activity)Approach

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STEPs on Column (recipe/Activity)Approach

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Step 1: Define the activity Set

Decompose the entire system into all of its elementary functions, the activities or processes and

Choose a unit for each type of activity or process in terms of which its quantity or level can be measured.

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STEP 1: Define The activity Set.Exemple: DES-K

Activity: • Manufacturing a desk

Activity Level: Number of desks manufactured = Decision Variable.

Manufacturing Desk1: X1 Manufacturing Desk2: X2 Manufacturing Desk3: X3 Manufacturing Desk4: X4

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Step 2: Define the set itemsExemple: DES-K

1. Object Classes: Desks 2. Items:

• Capacity carpentry shop (in man hours)• Capacity finishing shop (in man hours)• Costs (measured in dollars)

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Step 3: Define The Input-Output Coefficients aij

Determine the quantity of each item i consumed or produced by the operation of each activity j at its unit level.

Ex: aij= amount of time in shop i required to manufacture one desk j

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STEP 3: Input-Output Coefficient

Manufacturing1 unit of Desk 1

4 hours of carpentrycapacity

1 hour of finishingcapacity

$12

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STEP 3: Input-Output Coefficients

Activities

Items Desk1 desk2 Desk3 Desk 4

1. Carpentry Capacity (hrs)

4 9 7 10

2. Finishing capacity (hrs)

1 1 3 40

3. Costs (profit)

$

12 20 18 40

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Step 4. Specify the Exogenous Flows bi

Specify:• Exogenous amounts of each item being supplied (required) to the system

Capacity in the carpentry and finishing are inputs to each activities:

6000 maximun (carpentry) 4000 maximun(finishing shop)

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Step 5. Balance Equations1. Assign unknown activity levels x1,x2,x3,...,

2. For each item write the material balance

Activities

Items Desk1 desk2 Desk3 Desk 4

1. Carpentry Capacity (hrs) 4 9 7 10

2. Finishing capacity (hrs) 1 1 3 40

3. Costs (profit) $ 12 20 18 40

4x1+9x2+7x3+10x4 6000x1+ x2 +3x3+40x4 4000

f.O: Z= 12x1+20x2+18x3+40x4

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JOINING EVERYTHING

Max Z= 12x1+20x2+18x3+40x4

4x1+9x2+7x3+10x4 6000x1+ x2 +3x3+40x4 4000

s.t.

X1,X2,X3,X40

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Some Complexity Remarques

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Complexity of Problems/Algorithms

Resources invested to Solve a Problem

Computational View:Time, Space

Temporal, Spatial Complexity

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Problem:

A set of questions about a subject Instance (of a Problem). A particular

question about a problem.

Subjectx s.t

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Exemple

Problem: Min Z= F(X) s.t X

Instance of a Problem:Min Z = 3 X1+ 4 X2s.t 3 X1 +4 X2 12X1,X2 0

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Complexity of ALgorithms

Time/Memory invested for an algorithm for solving a problem

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Temporal Complexity of Algorithms

Input size

Execution time

exponential

polinomial

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Deterministic (Non Deterministic) Algorithm

(Not) All its instructons are deterministic

Ex. Classicals Simplex, Montecarlo Gauss Jordan

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Polinomial/No-Polinomial Problem

Polinominal They are at least one deterministic algorithm able to solve all their instances in a polinomial time

Non Polinomial Problem.• Thery are not any deterministic

algorithm able to solve all their instances in a polinomial time.

• They are at least one non-deterministic algorithm able to solve all their instances in a polinomial time.

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Good News,l Bad News

Linear Programming is a Polinomial Problem

Linear Programming is used in many other Optimization Problems

Simplex is the most effective Algorithm for many practical problems

Simplex is exponential

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NPH = optimization + X

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Linear Programming AlgorithmsMenu

DETERMINÍSTICOS

NO DETERMINÍSTICOS

• Simplex Genetic•Simplex Annealing

• Simplex, •Revised•LU SImplex• Interior Point •Simplex Cosine

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Computational Methods for Optimization: An Introduction

Dr. Juan Frausto Solís

Tecnológico de Monterrey

Campus Cuernavaca