1OPTIMIZAO E DECISO

Joo Miguel da Costa Sousa

Technical University of Lisbon, Instituto Superior TcnicoDep. of Mechanical Engineering, Center of Intelligent Systems/IDMEC

1049-001 Lisboa, Portugal, E- mail: jmsousa@ist.utl.pt,http://www.dem.ist.utl.pt/~jsousa

Program

1. Introduction to optimization. Operations Research2. Linear Programming: Simplex. Sensitivity Analysis.3. Transportation and Assignment Problems4. Dynamic Programming5. Integer Programming6. Nonlinear Programming. Quadratic Programming.

Convex Programming.7. Metaheuristics. Tabu search, Simulated Annealing,

Genetic Algorithms, Ant Colony Optimization.8. Game Theory9. Decision Analysis. Decision Trees. Utility Theory.

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2Bibliography

? 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 Sttzle. Ant ColonyOptimization. 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 andSystems, 2nd Edition, Prentice Hall, 2002.

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Evaluation method

? Exam (minimum grade: 9.5/20)

? Project (minimum grade: 9.5/20)? project assignment: November

? project deadline: December

? oral presentation: between last week of lectures

? Final Grade = 0.7*Exam + 0.3*Project

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3INTRODUCTION

Origins of Operations Research

? As the complexity and specialization in an organizationincrease, it becomes more and more difficult toallocate available resources to the various activities ina way that is most effective for the organization as awhole.

? This kind of problems and the need to find a why tosolve them provided the environment for theemergence of operations research (OR).

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4Roots 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 WW II, it became apparent that problems causedby 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 techniques duringthe 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 ofcomputation 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 ofpeople. Today

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5Nature 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.

? OR uses techniques resembling the way research isconducted in many scientific fields.? Formulate the problem, including gathering data;

construct model; conduct experiments; validate 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; searchfor optimality is an important theme in OR.

? As OR requires many and broad aspects, it is usuallynecessary to use a team approach, including areassuch as:? mathematics, statistics and probability theory,

economics, business administration, computer science,engineering and physical sciences, behavioral sciencesand the special techniques of OR.

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6Impact of operations research

? Improvement of efficiency in numerous organizationsaround the world, and improving economy.

? IFORS (International Federation of OperationsResearch Societies) and INFORMS (Institute forOperations Research and the Management Sciences).

? INFORMS has many journals, including Interfaces.

? Next table present some examples of award-winningapplications reported in Interfaces (to see more detailssee page 4 of Hilliers book).

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Impact of operations research

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Organization Application Year of pub. Annual savings

The NetherlandsRijkwaterstaat

Develop national water managementpolicy, including mix facilities,operating procedures and pricing.

1985 $15 million

Citgo PetroleumCorporation

Optimize refinery operations, supply,distribution and marketing ofproducts.

1987 $70 million

San FranciscoPolice Dept.

Optimally schedule and deploy policepatrol officers.

1989 $11 million

China Optimally select and schedulemassive projects for meeting thecountrys future energy needs.

1995 $425 million

SamsungElectronics

Develop methods of reducing manu-facturing times and inventory levels.

2001 $200 millionmore revenue

ContinentalAirlines

Optimize reassignment of crews toflights when a disruption occurs.

2003 $40 million

7Algorithms 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

? 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-artsoftware package for large and challenging ORproblems.

? Later, we will use MATLAB with GA, ACO, etc.13

OPERATIONS RESEARCHMODELING APPROACH

8Phases 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 solutionsto the problem 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 management.

? Be completely sure about the appropriate objectives(together with the management) is an importantaspect.

? Objectives should be as specific as possible, butconsistent with high-level objectives of theorganization.

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9Defining the problem

? For profit making organizations, objective can be thelong-run profit maximization (including R&D).

? In practice, this is not enough, and must be combinedwith other objectives, such as: improve worker moraleor increase company prestige.

? Five parties affected by a firm: owners, employees,customers, suppliers and government (nation).

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

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Gathering relevant data

? Data is needed to understand the problem and asinput for the mathematical model.

? Often it is necessary to install a managementinformation 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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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.

? 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 (balanceworkload equitable amongst officers).

3. Maintain the cost of operation (minimizing number ofofficers 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).? Constraints: example; x1 + 3x2 x1 x5 ? 20

? Constants in the objective function and constraints arecalled parameters.

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

? Thus, a sensitivity analysis is necessary.

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Formulating a model

? Linear programming model is often used. It can beapplied to very different problems.

? Models are an abstract idealization of the problem.

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

? To assure high correlation between predictions of themodel and real world data, testing and modelvalidation 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 derivingsolutions to the problem from the model.

? Sometimes, one of the standard algorithms is appliedusing readily available software packages.

? Search for an optimal (best) solution for the model.

? Herbert Simon (Nobel Laureate) points out thatsatisficing (= satisfactory and optimizing) is muchmore prevalent than optimizing in practice.

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Deriving solutions

? OR seeks for optimal solutions, but time or costrestrictions may demand for heuristic procedures tofind good suboptimal solutions.

? Recently, efficient and effective metaheuristics havebeen developed for designing heuristics for particulartypes of problems.

? One solution is commonly not enough, so postoptimalanalysis is needed to find alternative solutions.

? Postoptimal analysis demands for sensitivity analysis.

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Sensitivity analysis

? Sensitive parameter:? For a mathematical model with specified values for all

its parameters, the models sensitive parameters arethe parameters whose value cannot be changedwithout changing the optimal solution.

? Postoptimality analysis involves obtaining severalsolutions 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 analogous todeveloping 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 manyflaws and some parameters have not been estimatedcorrectly.

? Small bugs can remain in the program or model.

? This process of testing and improving a model is knownas model validation.

? Revision of a complete model must include an outsider.

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Examples

? In the Netherlands Rijkwaterstaat study modelvalidation had three main parts:? Checking results of the 50 models for changes in

parameters.

? Retrospective tests (use of historical data toreconstruct the past) were done.

? Careful technical review of model, methodology andresults by experts unaffiliated with the project.

? In the Citgo Petroleum Corporation study, model ofrefinery operations was tested using input and outputdata for a series of months to fix the model inputs.

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5. Preparing to apply the model

? When model is ready, install a well documentedsystem for applying it as prescribed by management.

? Inputs for the model can be obtained from databasesor information systems.

? If interactivity is needed, a decision support system isinstalled 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 developingprocedures to put the system in operation.

? Personal involved is indoctrinated, and system is initiated.

? Feedback when system is in use is essential to evaluatemodel.

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

? Optimization is deeply intertwined with the use ofcomputers.

? There are many exceptions to the rules prescribed:OR requires considerable ingenuity and innovation.

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