optimizaÇÃo e decisÃo optimization and decision od... · optimizaÇÃo e decisÃo optimization...

OPTIMIZAÇÃO E DECISÃOOPTIMIZATION AND DECISION

João Miguel da Costa Sousa

Alexandra Moutinho

Instituto Superior Técnico, Dep. Engenharia Mecânica Secção de Sistemas, Grupo de Controlo Automação e Robótica

Pav. Eng. Mecânica III, 1049—001 Lisboa, Portugal

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

Program

1. Introduction to optimization. Introduction to Operations Research.

2. Linear Programming: Simplex. Duality Theory and Sensitivity Analysis.

3. Transportation and Assignment Problems4. Network Optimization Models5. Dynamic Programming

João Miguel da Costa Sousa / Alexandra Moutinho 2

5. Dynamic Programming6. Integer Programming7. Nonlinear Programming:Quadratic Programming. Convex

Programming.

8. Metaheuristics: Tabu search, Simulated annealing, Genetic Algorithms, Ant Colony Optimization.

9. Game Theory NEW!

10. Decision Analysis: Decision Trees. Utility Theory.

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.

� Marco Dorigo and Thomas Stützle. Ant Colony Optimization.


� Marco Dorigo and Thomas Stützle. Ant Colony Optimization. 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.

Assessment Process

�Exam (minimum grade: 9,5 / 20);

�Project (minimum grade: 9,5 / 20):

� project assignment: 5 November;

� project deadline: 11 December;

� oral presentation: between 14 and 18 December.


� oral presentation: between 14 and 18 December.

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

�Requested effort (see Planning).

INTRODUCTION

Origins of Operations Research

�As the complexity and specialization in an

organization increase, it becomes more and more

difficult to allocate available resources to the various

activities in a way that is most effective for the

organization as a whole.


organization as a whole.

�This kind of problems and the need to find a way to

solve them provided the environment for the

emergence of operations research (OR).

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 of


�Effective methods of using radars to win the Air Battle of Britain

�Better management of convoy and antisubmarine operations to win the Battle of North Atlantic.

�After the WW II, it became apparent that problems caused by increasing complexity and specialization in organizations required the same tools.

Roots of operations research

� Two main factors for rapid growth of OR:

1. Large progress in improving the OR techniques

during the war.

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


� 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 50’s.

2. The computer revolution: a large amount of

computation is required to deal with OR problems.

� During the 80’s, the PC and related OR software brought

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

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,

health care, military, public services, etc, etc.


health care, military, public services, etc, etc.

�OR uses techniques resembling the way research is

conducted in many scientific fields.

� Formulate the problem, including gathering data; construct

a model; conduct experiments; validate the model.

� OR is also concerned with management and decision

making.

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, engineeringand physical sciences, behavioral sciences and the special

techniques of OR.

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 more

details see page 4 of Hillier’s book).

Impact of Operations Research

Organization Application Year of pub. Annual savings

The Netherlands

Rijkwaterstaat

Develop national water

management policy, including mix

facilities, operating procedures and

pricing.

1985 $15 million

Citgo Petroleum

Corporation

Optimize refinery oper., supply,

distribution and marketing of

products.

1987 $70 million


products.

San Francisco Police

Dept.

Optimally schedule and deploy

police patrol officers.

1989 $11 million

China Optimally select and schedule

massive projects for meeting the

country’s 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

Algorithms and Courseware

�Algorithm – a systematic solution procedure for

solving a particular type of problem.

�OR Courseware of Hillier’s book and CD-ROM.

� OR Tutor – teach the algorithms

� IOR Tutorial (implemented in Java)


� IOR Tutorial (implemented in Java)

� 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 (optimization

toolbox) for solving optimization problems.

OPERATIONS RESEARCH MODELING APPROACH

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 problem from the model.


to the problem from the model.

4. Test the model and refine it as needed.

5. Prepare the ongoing application of the model as

prescribed by management.

6. Implement.

� Usually some cycles are necessary.

1. Defining the problem

�Practical problems are initially described in a vague,

imprecise way.

�OR teams work in an advisory capacity: they don’t

only solve the problem, they also advise

management.


management.

�To be completely sure about the appropriate

objectives (together with the management) is an

important aspect.

�Objectives should be as specific as possible, but

consistent with high-level objectives of the

organization.

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 and government (nation).

�Besides making profit, a company has broader social

responsibilities that must also be recognized.

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.

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 citation

revenues, and 20% improvement of response times.

� Appropriate objectives found for this study:


� Appropriate objectives found for this study:

1. Maintain a high level of citizen safety (establish desired

level of protection).

2. Maintain a high level of officer morale (balance workload

equitable amongst officers).

3. Maintain the cost of operation (minimizing number of

officers to satisfy objectives 1 and 2).

2. Formulating a model

�Mathematical models are idealized representations.

�Decision variables: x1, x2,…, xn.

�Objective (cost) function: J = f (x1, x2,…, xn).

�Constraints: example; x1 + 3x2 x1 – x5 ≤ 20


�Constraints: example; x1 + 3x2 x1 – x5 ≤ 20

�Constants 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.

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 (capable of being solved).


�Models must be tractable (capable of being solved).

�To assure high correlation between predictions of the

model and real world data, testing and model

validation must be performed.

�Measure of performance combining the multiple

objectives is needed.

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) solution for the model.

�Herbert Simon (Nobel Laureate) points out that

satisficing (= satisfactory and optimizing) is much

more prevalent than optimizing in practice.

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-heuristicshave been developed for designing heuristics for


have been developed for designing heuristics for

particular types of problems.

�One solution is commonly not enough, so post-

optimal analysis is needed to find alternative

solutions.

�Post-optimal analysis demands for sensitivity analysis.

3. Sensitivity analysis

�Sensitive parameter:

� “For a mathematical model with specified values for all its

parameters, the model’s sensitive parameters are the

parameters whose value cannot be changed without

changing the optimal solution.”


changing the optimal solution.”

�Post-optimality analysis involves obtaining several

solutions that contain improved approximations.

�This cycle is repeated until the improvements in the

succeeding solutions become too small to warrant

continuation.

4. Testing the model

�Developing a large mathematical model is analogous

to developing a large computer program:

� First version of computer program contain many bugs that

are corrected by thoroughly testing the program.

� First version of mathematical program contain many flaws


� First version of mathematical program contain many 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.

4. Examples

� In the Netherlands Rijkwaterstaat study, model

validation had three main parts:

� Checking results of the 50 models for changes in

parameters.

� Retrospective tests (use of historical data to reconstruct


� Retrospective tests (use of historical data to reconstruct

the past) were done.

� Careful technical review of model, methodology and

results by experts unaffiliated with the project.

� In the Citgo Petroleum Corp. study, model of refinery

operations was tested using input and output data for

a series of months to fix the model inputs.

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.

� If interactivity is needed, a decision support system


� If interactivity is 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 decision

support system CrewSolver (it was running on

September 11, 2001).

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.


� Personal involved is indoctrinated, and system is initiated.

�Feedback when system is in use is essential to

evaluate model.

�Documentation is crucial to ensure reproducibility.

Crucial for studies of controversial public policies.

�Example: studies for localization of future Lisbon airport.

Discussion

�This discipline focuses on constructing and solving

mathematical models, but these are only part of the

overall 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.

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