introduction to (fundamentals of) operations research · introduction to (fundamentals of)...

Introduction to (Fundamentals of)Operations Research

(Week 1: Introduction)

Jose Rui Figueira

Universidade de LisboaInstituto Superior Tecnico

figueira@tecnico.ulisboa.pt

February 15-16, 2016

General Contents

1. Syllabus

2. Operations Research

3. The Science of “Better”

This slides are currently available for personal use of IST LEGI FIOstudents in an unpublished draft form only. The slides cannot becopied, reproduced, or distributed in any form.

Syllabus

Contents

1. Instructors and Course Home

2. Course Meeting Times

3. Prerequisites

4. Description and calendar

5. Course Objectives

6. Format

7. Readings

8. Lecture Notes and Assignments

9. Interesting Links

Instructor and Course Home (1)

I Instructors.

name: Jose Rui Figueira (JRF).phone: 21 423 35 81e-mail: figueira@tecnico.ulisboa.pt

webpage: fenix.tecnico.ulisboa.pt/homepage/ist14525office: 2-N10.24 (TagusPark).

name: Joao Lourenco (JL).e-mail: joao.lourenco@tecnico.ulisboa.pt

webpage: fenix.tecnico.ulisboa.pt/homepage/ist14341

J.R. Figueira (IST) FIO February 15-16, 2016 6 / 51

Instructor and Course Home (2)

I Course.

taught in: Spring 2016 (second semester 2015-2016).

IST course label: FIO (Fundamentos de Investigacao Operacional).

level: Undergraduate

course home: fenix.tecnico.ulisboa.pt/disciplinas/FIO51113/2015-2016/2-semestre

campus: TagusPark

Course Meeting Times

I Weakly Workload.

Lectures (T): 2 sessions of 1.5 hours.(Monday afternoon and Tuesday morning)

Problems (PB): 1 session of 1.5 hours.(Monday and Tuesday afternoon)

I Office Meeting Times (JRF).

Monday (morning): 11:00 - 12:30.

Attention (T classes)! In-class material

Electronic devices should be switch off:

1. (Standard) cell (mobile) phones.

2. Smartphones.

3. Tablets.

4. Laptops.

5. . . .

3. Prerequisites

Mathematics, Economics, Computer Science, SystemsTheory

Main Topics (only basic notions are needed):

1. Systems Theory.

2. Linear Algebra.

3. Computational Mathematics.

4. Euclidean Geometry.

5. Combinatorics.

6. The Design and Analysis of Algorithms.

7. Microeconomics.

8. Statistics.

Brief Description

The course will present a thorough introduction to the fundamentalmodeling aspects, algorithmic techniques, and application fields oflinear programming, project management, inventory management,and forecasting.

Particular attention will be devoted to the possible relations withother disciplines within the Industrial Engineering and Managementundergraduate and graduate courses as well as with disciplines taughtin other courses at IST.

As it will be shown in this course the use of operations researchmodels is frequent when dealing with real-world decision making prob-lems where the need for determining an optimal solution accordingto some measure is present.

The course is particularly devoted to the main facets of linearprogramming: modeling, solving, analyzing, and interpreting the re-sults.

All of these aspects will be covered by the items presented innext slide.

Introduction to Operations Research

Main Topics:

1. Introduction (JRF - 1 Week).

2. Modeling (JRF - 1 Week).

3. Linear Programming (JRF - 6 Weeks).

4. Project Management (JL - 1 Week).

5. Inventory Management (JL - 2 Weeks).

6. Forecasting (JL - 2 Weeks).

5. Course Objectives

Objectives or Goals

I Provide the students with enough experience in modeling, solving, analyzing,interpreting, and making recommendations using OR models within the scopeof the ones taught in this course.

I Improve the students skills for being able to recognize when OR modelsframeworks are the most adequate for real-world problems they are dealingwith.

I Develop multi and interdisciplinary student skills.

I Promote team work.

6. Format

Format of the Course

1. 4.5 hours of lectures and problems classes each week.

2. Regular homework (assignments).

3. Two in-class exams or a final exam.

6. Format

Examination Procedures

I Students should arrive 15 minutes before the start of each exam.

I Students should carry ONLY the essential equipment they need for the exam(no other material is allowed; bags and coats should be left outside theroom).

I The answers should be written in the same booklet containing the questions.Answers provided in supplementary pages will not be assessed.

I During the exam, students should not communicate (it includes talking, eyecontact, ...) with or try to disturb other students.

6. Format

Essential Equipment

I Graffiti pencil, sharpener, and rubber.

I Pens (different colors are allowed, except red) and pen erasers.

I Geometry set (ruler, triangle, and protractor).

I Calculators:ONLY basic (4-function operation primary school calculator) andelementary (with square root and percentage) calculators are allowed. Basicscientific are also allowed.

I Scientific, financial (with graphical and memory functions) and moresophisticated calculators should be left outside the examination rooms.

I Cell Phones should be switch off and left in your back pack,....

6. Format

Grading: Activities and Percentages

I First in-class Exam: 50% (minimum 7.5).

I Second in-class Exam: 50% (minimum 7.5).

I Final Exam: 100% (the same as the two Re-Dos with a minimum of 7.5 ineach part).

7. Readings

Readings

I Main reference:

Hillier, F. and Lieberman, G. (2009). Introduction to Operations Research.The McGraw-Hill Companies, Inc., New York, USA, Ninth edition.

I Portuguese version:

Hillier, F. and Lieberman, G. (2013). Introducao a Pesquisa Operacional.McGraw-Hill, Rio de Janeiro, Brasil, 9a Edicao.

I Software (Install IORTutorial): Book by H&L

8. Lecture Notes and Assignments

Lecture Notes and Assignments

I Slides will be available for students at a webpage.

I Additional Lectures Notes will be provided when necessary.

I Student’s representative:

Fernando Goncalves (fernando.a.goncalves@tecnico.ulisboa.pt)

9. Interesting Links

Some Links

I Michael Trick’s OR Page: (mat.gsia.cmu.edu).

I Some OR Societies:

Portugal: APDIO (apdio.pt).Europe: EURO (www.euro-online.org).Globe: INFORMS (www.informs.org).

I Some OR Consultancy Companies:

Portugal: Bana Consulting (www.bana-consulting.pt).France: Euro Decision (www.eurodecision.fr).USA: Innovative Scheduling (www.innovativescheduling.com).

Operations Research (OR)

Contents

1. The Origins of OR

2. The Nature of OR

3. The Impact of OR

4. Recent Application Fields

5. OR in LEGI and MEGI

6. OR at IST (Undergraduate Courses)

7. Careers in OR (OR Analysts)

History and Pre-History of OR

I Some Words on the Pre-History of OR (from Euclid to Farkas and muchmore).

I Military Operations during World War II.

I Linear Programming: A Tremendous Growth.

I G. Dantzig’s Simplex Method (1947).

I L. Khachiyan’s Ellipsoid Method (1979).

I N. Karmarkar’s interior point method (1984).

I 80s: Strong Impact of Complexity Analysis in OR.

History and Pre-History of OR (cont.)

I 80s: Efficiency of Computers for Solving Large Scale Problems.

I 80s: Heuristics and Nature Inspired Algorithms for OR.

I 80s and 90s: Increasing of Polyhedral Studies for CombinatorialOptimization.

I 90s: Development of Semidefinite and Conic Programming.

I 21st Century: Hybrid, Cooperative, and Partial Optimization for SolvingReal-World Applications.

2. The Nature of OR

The Nature of OR

I Interdisciplinary: OR puts together several academic fields to form a singlebody of knowledge or discipline.

I Multidisciplinary: In the sense that an OR project is in general formed ofpeople from different disciplines.

I Transdisciplinary: New knowledge, produced within the OR community, mayin general be claimed by other disciplines or even to lead to the birth of anew field.

I Cross-disciplinary: Whenever OR makes an explanation of a certain aspectthrough the use of a language of a different discipline.

2. The Nature of OR

Anatomy of the OR Body of knowledge(Mind Maps Templates at:www.statistiker-wg.de/pgf/tutorials/mindmap.htm)

OperationsResearch

Mathematics

Algebra

Geometry

Analysis

GraphTheory

ComputerScience

ComputationalTheory

Algorithmsand DataStructures

ArtificialIntelligence

SoftwareEngineering

Social andBehavioralSciences

Economics

Sociology

Psychology

CognitiveSciences

SystemsTheory

Cybernetics

ComplexSystems

OrganizationalSystems

LivingSystems

Putting alltogether

3. The Impact of OR

A. Ravi Ravidran Personal view: Impact Areas

Impact of OR in Systems Theory:

I Interactions: “Communication among the different functions of the systemcan be improved.”

I Correctives: “Once the map of how the organization works is laid out, it ismuch easier to see where gridlock or backlogs occur and then take correctivemeasures.”

I Efficiency: “Seeks to eliminate all forms of waste and increase the effectiveuse of tools.”‘

Some OR Emerging Application Fields

I Rating or Ranking Alternatives (Decision Analysis).

I Psychology.

I Dynamic Pricing (in airlines companies).

I Supply Chain Management.

I Puzzles, Mazes, and Games.

I HIV/AIDS Intervention Studies.

I Nanotechnology.

I Medicine, Bio-medicine, Genetics, and Biology.

Some OR Emerging Application Fields (cont.)

I Removal of space debris (orbital debris, space junk, or space waste).

I Social Networks (as for example facebook).

I Sports.

I Forensic Science.

I Geographical Information Systems (GIS) for several purposes.

I Neuroscience.

“Countries” Raking by The “Notsobadmoon Agency”

Rank “Country” Score

1 Olele 90.4 N

2 Corkland 84.8 H

3 Atlantic Woodland 67.9 –

4 Great Iberia 67.0 N

5 Cape Braz Mocagola 50.2 H

6 S. Guinetom 45.5 N

7 Benspopo Country 30.0 N

8 Hill South Kingdom 30.0 N

9 Quasiland 27.5 N

10 Pasdechanceland 13.0 H

(Template table by Stefan Kottwitz)

Design of Puzzles, Mazes, and Games

A Maze

InaccessibleLoop

(Maze by Ki Sang Lee)

Space Junk Removal: Assignment Problem

(Graph by the authors of the tkz-berge.sty package)J.R. Figueira (IST) FIO February 15-16, 2016 32 / 51

Nano-products Risk Assessment

(Picture by Tom Bombadil)

Evaluation of a Crime Scene by two Experts

Path ofEntry-Exit

Safety

SceneBoundaries

Security Area

Communications

(Template picture by Alain Matthes)J.R. Figueira (IST) FIO February 15-16, 2016 34 / 51

5. OR in LEGI and MEGI

Special Attention is Devoted to the Following ORDisciplines

1. Introduction to Operations Research.

2. Intermediate Operations Research.

3. Decision Analysis.

4. Risk Evaluation and Management.

5. Project Management.

6. Operations Management.

7. Supply Chain Management.

8. Operations Planning and Control.

9. Logistics and Distribution.

10. Simulation of Processes and Operations.

Courses and Cross Fertilization with OR

I Aerospace Engineering.(Aerospace Vehicle Design Optimization.)

I Applied Mathematics and Computation.(Network Optimization for Rounding Matrix Entries.)

I Architecture.(Site Location for New Green Architecture Buildings.)

I Biological Engineering.(Nature Inspired Algorithms for Population Evolution.)

I Biomedical Engineering.(Decision Aiding Models for ART.)

I Chemical Engineering.(Optimization for Chemical Processes Design.)

I Civil Engineering.(Planning and Design Models for Construction Industry.)

I Communication Networks Engineering.(Path Selection for a Communication Network.)

I Electrical and Computer Engineering.

(Optimal Compensation in Power Systems.)

I Electronics Engineering.

(Consistency Circuits Checking.)

I Environmental Engineering.

(Multi-Criteria Environmental Impact Assessment.)

I Geological and Mining Engineering.

(Estimation of the Available Geological Resources.)

I Information Systems and Computer Engineering.

(Multi-Criteria Evaluation of Information Systems.)

I Materials Engineering.

(Cutting Models for Steel Rods.)

I Mechanical Engineering.

(Optimization for the Design of a Robot Control System.)

I Naval Architecture and Marine Engineering.

(Design of Marine Structures.)

I Physics Engineering.

(Optimization Techniques for the Nonlinear Pendulum Problem.)

A Glance at Possible Job Areas

I HiTech Project Selection and Management.

I Telecommunication Systems.

I Health and Human Services.

I Military Planning and Military Operations Product Development.

I Energy Planning.

I Governance and Decision Support.

I Supply-Chain and Logistics Engineering.

I Railroads, Airlines, Trucking, Maritime,. . ., Transportation Systems.

I Environmental Risk Analysis.

I Network Design and Management.

I Decision Analysis Consultancy Services.

I Resource Allocation.

I Academia.

OR Analyst Main Skills: A Personal View (for an ORAnalyst with Background in Engineering andManagement from IST)

I Open Mind Person.

I Team Worker (International and Multi-cultural teams).

I General Background Favoring a Multi and Interdisciplinary Approach.

I Excellent OR Modeler (Soft OR is important).

I Excellent OR Software user.

OR: The Science of “Better”

Contents

1. OR to Help Making Decisions

2. Decision Making Process

3. Modeling in OR

4. Some Types of Standard Models

5. A Very Simple Linear Programming Model

6. Linear Programming Axioms

1. OR to Help Making Decisions

What is OR about?

INFORMS:

“ OR The Science of Better”

Short (Personal) Definition:

OR is a field of knowledge that makes use of analytical toolsfor helping decision makers to build a decision making process.

The Main Components

1. Decision Making Context:

I Decision Maker (DM) and other Stakeholders.

I Their main concerns.

I Resources available.

2. Problem Description:

I Major points of view.

I Potential options or alternatives (solutions).

I Expected results (best solutions, ranking of solutions, . . .).

3. Formal Model:

I Feasible set of options (solutions).

I Criteria (objective functions).

I Uncertainty.

I Aggregation Operator.

I Results and (sensitivity) robustness studies.

4. Recommendations:

The Main Components: Some Comments

1. As a process it comprises several steps or stages.

2. Presenting it in a linear way is only for a sake of simplicity.

3. It is a co-construction process between DMs and Analysts.

4. Each step will be detailed in class.

3. Modeling in OR

OR Models

Constraints

Structuring Problems

Decision Variables Objective Function(s)

(xj, j = 1, . . . , n)

- Problem unknowns- Nonnegativity of variables

(gi(x) 6 bi, i = 1, . . . , m)

- Restricts variables ranges- Feasible region

(fq(x), q = 1, . . . , p)

- One or more objectives- Measuring solutions

(Soft OR Methods)- Built a DM situation

4. Some Types of Standard Models

Looking for Standard Models (A Rough Taxonomy)

Problem Type Question (Ex) Problem Name (Ex)

Location Where to locate antennas? Set Cover, Facility Location

Covering and Partitioning How to divide a set into subsets? Set Cover, Cluster Sum, Job Assignment

Network (Design) What is the network that minimizes costs? Spanning Tree

Routing What is the optimal tour? Traveling Salesperson

Assignment How to assign workers to machines? Job Assignment

Allocation How to allocate resources to projects? Knapsack, Job Assignment

Sequencing What is the optimal order to perform tasks? Traveling Salesperson, Job Sequencing

Scheduling What is the best task arrangement over time? Job Sequencing

. . . . . . . . .

Problem Statement (Decision Making Situation)

The Wyndor Glass Co. (see Hillier and Lieberman, 2005) produces high-quality windows and glass doors andhas three production plants: Plant 1, Plant 2, and Plant 3. The company wants to introduce two new products:

Product 1 (P1): A new type of glass door.

Product 2 (P2): A new type of window.

Product 1 requires to be processed in Plants 1 and 3, but not in Plant 2. Product 2 needs only Plants 2 and3. We know the following data shown in the table:

(1) Number of hours of production time needed in each plant to produce one batch of each new product.

(2) Number of hours of production time available per week in each plant for these new products.

(3) Profit per batch of each new product.

Plant P1 P2 Available Time1 1 0 42 0 2 123 3 2 18

Price 3 5

The company needs to decide how many batches of each new product to be produced per week so as tomaximize its profit.

Linear Programming Model

maximize 3x1 + 5x2subject to: x1 6 4 Plant 1

2x2 6 12 Plant 23x1 + 2x2 6 18 Plant 3x1, x2 > 0 Nonnegativity

Decision variables:

x1: Number of batches of product 1 to be produced per week.

x2: Number of batches of product 2 to be produced per week.

What kind of model? (Resource) Allocation!

Objective function: Its domain, codomain, and image

1. 〈X, R, F〉 is a triplet that allows the definition of a function f : X→ R.

2. X ⊂ Rn is the function domain, where elements of X are the input of theform x = (x1, x2, . . . , xn).

3. R is the function codomain or target set: the set of all output of the formf (x).

4. F is a subset of the Cartesian product X×R.

5. Z ⊂ R is the image of the function f : the set of all output of the form f (x),where x ranges over all the input of X.

6. A function is surjective (or onto or a surjection) if and only if its codomainequals its image.

7. Our function f : X→ R, defined by f : x 7→ ∑nj=1 cjxj + c0, or equivalently

f (x) = ∑nj=1 cjxj + c0, is an (affine) linear function and not surjective.

6. Linear Programming Axioms

Basic Axioms required for Linear Programming

1. Proportionality: This axiom requires the value of each term in each function(objective function and left-hand side of each constraint) is strictlyproportional to the value of the decision variable in the term.

2. Additivity: This axiom requires that a contribution of each decision variableto each function is independent of the values of this decision variable.

3. Divisibility: This axiom requires that the value of all decision variables shouldbe freely taken from the values defined within their ranges (each decisionvariable may assume, for example, fractional values).

4. Certainty: This axiom requires all the data model (objective functioncoefficients, right-hand side values, and technological coefficients) be knownwith certainty (constant).

introduction to (fundamentals of) operations research · introduction to (fundamentals of)...

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