Fondamenti di Ricerca Operativa

Federico MalucelliStefano Gualandi (esercitazioni)

Fabio Panozzo (laboratorio)

1Wednesday, March 7, 2012

What is Operations Research (OR)?Support in Decision Making

in any complex systemHOW?

Mathematical modelsQuantitative methodsEfficient algorithms

Use the science to help in taking decisions

Maths

Computer Science

2Wednesday, March 7, 2012

What kind of decisions?

Strategic level:e.g. design a railroadTactical level:e.g. how many trains to useOperational level:e.g. schedule the trains

At any level from top management to everyday problems

3Wednesday, March 7, 2012

Examples

One colleague of yours has been hired by a bank in LondonHe proposed an OR model to allocate bonds to portfoliosAnother student had to organize the tutoring in a climbing centerHe proposed an OR model to maximize the opening hours

OR is really helpful!4Wednesday, March 7, 2012

Practical cases of OR application

Why did you enroll in this course?What route did you follow to reach Politecnico?Where do you eat and what do you eat?

Probably you do not apply consciously OR methodsbut you solve OR problems

5Wednesday, March 7, 2012

One important adviceOR is easier to learn if you put your hands on the problems

componentsPhone AScore A

Phone BScore BObjective:

Maximize the overall score6Wednesday, March 7, 2012

Solution

Are you sure that the solution that you found is the best one?How can you certify it?How much are you willing to spend to have one more display, or keyboard, or memory board?

7Wednesday, March 7, 2012

How OR approaches the problem

DecisionsWhat decisions did you take?How do you characterize your solution?RulesHow do you combine components to obtain a product?ObjectiveWhat do you have to optimize?

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Decisions = variables

xA = how many phones of type A

xB = how many phones of type B

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Rules = ConstraintsThe number of used components cannot exceed the available ones

1xA + 2xB ! 10 display2xA + 2xB ! 18 memory1xA + 3xB ! 12 transmission2xA + 3xB ! 21 keyboard1xA ! 9 navigation1xA ! 10 cameraxA, xB ! 0 non negativity

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Objective function

Maximize the overall score

max 3xA + 8xB

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One simple solution methodExploit the fact that we have only 2 variables

Give a geometric representation

xA

xB

5

1xA + 2xB ! 10

10

9

9

2xA + 2xB ! 18

4

12

xA + 3xB ! 12

7

2xA + 3xB ! 21 xA ! 9

xA ! 10

Feasible regioninfinite number of solutions

“Interesting” solutionsxA = 0, xB = 4 v = 32

xA = 6, xB = 2 v = 34

xA = 8, xB = 1 v = 32

xA = 9, xB = 0 v = 27

12Wednesday, March 7, 2012

Success applications of ORYear Company Application Result

90 Taco Bell personnel scheduling 7.6 M$ saved per year

92 Harris Semiconductors production planning from 50% to 95%jobs on time

95 GM - car rental division car fleet distribution avoided the business failure

96 HP - printers re-planned the production plant production doubled

99 IBM reorganized the logistic chain

750M$ saved per year

2003 GMproduction scheduling

(30 plants in 10 countries)

2G$ saved

2007 Dutch railroad timetable designIncreased punctuality

and customer satisfaction

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Interesting web sites

http://www.scienceofbetter.org/classification of success applicationshttp://www.informs.org/American OR societyFranz Edelman Award recordhttp://www.airo.org/Italian OR society

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Optimization in Public Transport

ATM surface vehicle control room

15Wednesday, March 7, 2012

Timetable design

A C B

bus line

relief point

C

B

A

8:00 10:00 14:0012:00 18:0016:00 22:0020:00

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Vehicle Scheduling

C

B

A

8:00 10:00 14:0012:00 18:0016:00 22:0020:00

17Wednesday, March 7, 2012

Crew Scheduling

C

B

A

8:00 10:00 14:0012:00 18:0016:00 22:0020:00

18Wednesday, March 7, 2012

Games

or any other fantasy sport19Wednesday, March 7, 2012

Fantasy sportset up a team buying real players following some rules (number of backwards, forwards, etc.)

limited budget of fantasy money

each week depending on the players stats of the last game played a score is attributed to each team

Where does OR comes into play?

pointer to an article about OR and fantasy baseballhttp://lionhrtpub.com/orms/orms-8-10/frfantasy.html

We will study models and algorithms for approaching the problem

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Skills of an OR expert

Analyze the problemKnowledge of the application fieldCommunication skills to understand and explain

Knowledge of the OR methods

Ability in building mathematical models

Interpret the results

21Wednesday, March 7, 2012

Program of the courseIntroduction to OROptimization problems and their formulations; decision variables, objective function, constraints. Modeling techniques

Graphs and network flows problemsTrees and graphs. Shortest paths, maximum flow, minimum cost flow, assignments. Some data structures and solution algorithms. Complexity analysis

Linear ProgrammingDuality theory, complementary slackness, the simplex method; geometrical and economical interpretation. Basic solutions and optimality conditions

Integer Linear Programming and combinatorial optimizationRelaxations and branch and bound algorithm. Heuristic algorithms: greedy and local search.

Basics of Non Linear ProgrammingUnconstrained optimization. Convex programming. Newton method, gradient method, conjugate gradient method

22Wednesday, March 7, 2012

Organizzazione del corsoAttività di laboratorio (10 ore): modellazioneutilizzo di un pacchetto software (scaricabile)

Esame scritto: 28 points

2 Prove in itinereSe entrambe sufficienti esonerano dallo scrittoUna insufficienza/assenza è recuperabile (1 sola volta) a qualsiasi esame scritto (fa fede la consegna)

Esame di laboratorio opzionale (3-4 punti) on line

Punti bonus (max 2 a testa): domande durante le lezioni

Appunti ed esercizi svolti disponibili sul sito del corsohttp://home.dei.polimi.it/malucell

Ricevimento: Mercoledì dalle 16 alle 1823Wednesday, March 7, 2012

Forward pointers

Non linear optimization

Ottimizzazione(Corso di Laurea Magistrale in Ingegneria Matematica)

24Wednesday, March 7, 2012

Master thesis in optimizationOptimize the home healthcare service(in collaboration with a health service provider)

Design of broadband access networks

Network design problems with Shared Risk Group protection

Special design and routing problems in transportation

Timetable design problems

Optimization in food production

Classical combinatorial optimization problems

25Wednesday, March 7, 2012

Puzzle

Select one item per category

In a long and dangerous hiking you have to select items to bring with you in the backpack

Backpack capacity: 25 Kg.

do not overcome the capacity

maximize the survival points

The first team providing the optimal solution (and proving it) gains 1pt bonus

26Wednesday, March 7, 2012