1 od introduction s

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OPTIMIZAO E DECISOOPTIMIZATION AND DECISION

Joo Miguel da Costa Sousa

Alexandra Moutinho

Instituto Superior Tcnico, Dep. Engenharia MecnicaSeco de Sistemas, Grupo de Controlo Automao e Robtica

Pav. Eng. Mecnica III, 1049001 Lisboa, Portugal

Tel.: (+351)218417471/7, e-mail:{j.sousa,moutinho}@dem.ist.utl.pt


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Program

1. Introduction to optimization. Introduction to OperationsResearch.

2. Linear Programming: Simplex. Duality Theory andSensitivity Analysis.

3. Transportation and Assignment Problems4. Network Optimization Models

Joo Miguel da Costa Sousa / Alexandra Moutinho 2

5. ynam c rogramm ng6. Integer Programming7. Nonlinear Programming:Quadratic Programming. Convex

Programming.

8. Metaheuristics: Tabu search, Simulated annealing, GeneticAlgorithms, Ant Colony Optimization.9. Game Theory NEW!

10. Decision Analysis: Decision Trees. Utility Theory.


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Bibliography

F. Hillier and G. Lieberman. Introduction to OperationsResearch, 8th Edition. McGrawHill, 2005.

http://highered.mcgraw-hill.com/sites/0073017795/information_center_view0/ J. Kennedy, R. C. Eberhart and Y. Shi. Swarm Intelligence.

Morgan Kaufmann Publishers, 2002.


. .

The MIT Press. July 2004. R. Fletcher. Practical Methods of Optimization, 2nd Edition,

John Wiley, 2000.

J. Nocedal and S.Wright. Numerical Optimization. Springer,1999.

Michael Pinedo. Scheduling. Theory, Algorithms and Systems,2nd Edition, Prentice Hall, 2002.


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Assessment Process

Exam (minimum grade: 9,5 / 20);

Project (minimum grade: 9,5 / 20): project assignment: 5 November;

project deadline: 11 December;


ora presen a on: e ween 14 an 1 ecem er.

Final Grade = 0,7 * Exam + 0,3 * Project

Requested effort (see Planning).


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INTRODUCTION


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Roots of operations research

Military services early in World War II. Urgent need to allocate scarce resources to operations and

activities in an effective manner. Scientists were asked to do research on (military) operations.

Examples:


Effective methods of using radars to win the Air Battle ofBritain

Better management of convoy and antisubmarineoperations to win the Battle of North Atlantic.

After the WW II, it became apparent that problemscaused by increasing complexity and specialization inorganizations required the same tools.


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Roots of operations research

Two main factors for rapid growth of OR:

1. Large progress in improving the OR techniquesduring the war.

An example is the development of the simplex method(G.


Dantzig, 1947) for solving linear programming problems. Many standard OR tools were developed before the 50s.

2. The computer revolution: a large amount of

computation is required to deal with OR problems. During the 80s, the PC and related OR software brought

the use of OR to a much larger number of people. Today


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Nature of Operations Research

OR is applied to conduct and coordinate operations

(i.e., the activities) within an organization. Applied to many areas: manufacturing, transportation,

construction, telecommunications, financial planning,


ea care, m ary, pu c serv ces, e c, e c.OR uses techniques resembling the way research is

conducted in many scientific fields.

Formulate the problem, including gathering data; constructa model; conduct experiments; validate the model.

OR is also concerned with management and decisionmaking.


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Nature of Operations Research

OR attempts to find a best (optimal) solution;

search for optimality is an important theme in OR.As OR requires many and broad aspects, it is usually

necessary to use a team approach, including areas


such as: mathematics, statistics and probability theory, economics,

business administration, computer science, engineering

and physical sciences, behavioral sciences and the specialtechniques of OR.


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Impact of Operations Research

Improvement of efficiency in numerous organizations

around the world, and improving economy.IFORS (International Federation of Operations

Research Societies) and INFORMS (Institute for


Operations Research and the Management Sciences).INFORMS has many journals, including Interfaces.

Next table presents some examples of award-winning

applications reported in Interfaces (to see moredetails see page 4 of Hilliers book).


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Impact of Operations Research

Organization Application Year of pub. Annual savings

The Netherlands

Rijkwaterstaat

Develop national water

management policy, including mix

facilities, operating procedures andpricing.

1985 $15 million

Citgo Petroleum

Corporation

Optimize refinery oper., supply,

distribution and marketing of

1987 $70 million


pro uc s.

San Francisco Police

Dept.

Optimally schedule and deploy

police patrol officers.

1989 $11 million

China Optimally select and schedule

massive projects for meeting the

countrys future energy needs.

1995 $425 million

Samsung

Electronics

Develop methods of reducing manu-

facturing times and inventory levels.

2001 $200 million more

revenue

Continental Airlines Optimize reassignment of crews to

flights when a disruption occurs.

2003 $40 million


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Algorithms and Courseware

Algorithm a systematic solution procedure forsolving a particular type of problem.

OR Courseware of Hilliers book and CD-ROM. OR Tutor teach the algorithms


Excel Solver or Premium Solver for Education

LINDO and modeling language LINGO

CPLEX and modeling system MPL elite state-of-the-art

software package for large and challenging OR problems.

We will mostly use Excel and MATLAB (optimizationtoolbox) for solving optimization problems.


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OPERATIONS RESEARCH

MODELING APPROACH


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Phases of an OR study

1. Define the problem and gather relevant data.

2. Formulate a mathematical model for the problem.3. Develop a computer algorithm for deriving solutions

to the roblem from the model.


4. Test the model and refine it as needed.

5. Prepare the ongoing application of the model asprescribed by management.

6. Implement.

Usually some cycles are necessary.


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1. Defining the problem

Practical problems are initially described in a vague,

imprecise way.

OR teams work in an advisory capacity: they dontonly solve the problem, they also advise

mana ement.


To be completely sure about the appropriateobjectives (together with the management) is animportant aspect.

Objectives should be as specific as possible, butconsistent with high-level objectives of the

organization.


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1. Defining the problem

For profit making organizations, objective can be the

long-run profit maximization (including R&D).In practice, this is not enough, and must be combined

with other objectives, such as: improve worker


morale or increase company prestige.Five parties affected by a firm: owners, employees,

customers, suppliers andgovernment(nation).

Besides making profit, a company has broader socialresponsibilities that must also be recognized.


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1. Gathering relevant data

Data is needed to understand the problem and as

input for the mathematical model.Often it is necessary to install a management

information system to deal with the necessary data.


Much of the data is quite soft (rough estimates).Biggest data problem: too many data is available

(gigabytes or terabytes).

Data mining is often required to deal with the data.


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1. Example: Police Department

Recall the San Francisco PD problem.

New system provided annual savings of $11 million,annual increase of $3 million in traffic citationrevenues, and 20% improvement of response times.


Appropr ate o ect ves oun or t is stu y:1. Maintain a high level of citizen safety (establish desiredlevel of protection).

2. Maintain a high level of officer morale (balance workloadequitable amongst officers).

3. Maintain the cost of operation (minimizing number of

officers to satisfy objectives 1 and 2).


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2. Formulating a model

Mathematical models are idealized representations.

Decision variables:x1,x2,,xn.Objective (cost) function:J =f(x1,x2,,xn).


1 2 1 5Constants in the objective function and constraints

are called parameters.

Determining values for the parameters is crucial.These values are based on data and can be uncertain.

Thus, a sensitivity analysis is necessary.


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2. Formulating a model

Linear programming model is often used. It can be

applied to very different problems.Models are an abstract idealization of the problem.

Models must be tractable ca able of bein solved .


To assure high correlation between predictions of themodel and real world data, testing and model

validation must be performed.

Measure of performance combining the multipleobjectives is needed.


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3. Deriving solutions from the model

Develop a (computer-based) procedure for deriving

solutions to the problem from the model.Sometimes, one of the standard algorithms is

applied using readily available software packages.


Search for an optimal (best) solutionfor the model.Herbert Simon (Nobel Laureate) points out that

satisficing (= satisfactoryand optimizing) is much

more prevalent than optimizing in practice.


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3. Deriving solutions

OR seeks for optimal solutions, but time or cost

restrictions may demand for heuristic procedures to

find good suboptimal solutions.Recently, efficient and effective meta-heuristics

have been develo ed for desi nin heuristics for


particular types of problems.One solution is commonly not enough, sopost-

optimal analysis is needed to find alternative

solutions.Post-optimal analysis demands for sensitivity

analysis.


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3. Sensitivity analysis

Sensitive parameter:

For a mathematical model with specified values for all itsparameters, the models sensitive parameters are theparameters whose value cannot be changed without


.

Post-optimality analysis involves obtaining several

solutions that contain improved approximations.

This cycle is repeated until the improvements in thesucceeding solutions become too small to warrantcontinuation.


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4. Testing the model

Developing a large mathematical model is analogousto developing a large computer program:

First version of computer program contain many bugs thatare corrected by thoroughly testing the program.

First version of mathematical ro ram contain man flaws


and some parameters have not been estimated correctly. Small bugs can remain in the program or model.

This process of testing and improving a model is

known as model validation.Revision of a complete model must include an

outsider.


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5. Preparing to apply the model

When model is ready, install a well documented

system for applying it as prescribed by management.

Inputs for the model can be obtained from databases

or information systems.


Ifinteractivityis needed, a decision support systemis installed to help managers in their decision making.DSS can take months (or longer) to be implemented.

Example: Continental Airlines developed the decisionsupport system CrewSolver(it was running onSeptember 11, 2001).


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6. Implementation

Phases:

OR team gives management an explanation of the system. These two parties share the responsibility for developing

procedures to put the system in operation.


Personal involved is indoctrinated, and system is initiated.

Feedback when system is in use is essential toevaluate model.

Documentation is crucial to ensure reproducibility.Crucial for studies of controversial public policies.

Example: studies for localization of future Lisbon airport.


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Discussion

This discipline focuses on constructing and solving

mathematical models, but these are only part of theoverall process of an optimization study.

Optimization is deeply intertwined with the use of


computers.There are many exceptions to the rules prescribed:

OR requires considerable ingenuity and innovation.

1 od introduction s

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