Fondamenti di Ricerca Operativa D

Federico MalucelliBernardetta AddisStefano Gualandi

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

in any complex system

HOW?

Mathematical modelsQuantitative methods

Efficient algorithms

Use the science to help in taking decisions

Maths

Computer Science

What kind of decisions?

Strategic level:e.g. design a railroad

Tactical level:e.g. how many trains to use

Operational level:e.g. schedule the trains

At any level from top management to everyday problems

Examples

One colleague of yours has been hired by a bank in LondonHe proposed an OR model to allocate bonds to portfolios

Another student had to organize the tutoring in a climbing centerHe proposed an OR model to maximize the opening hours

OR is really helpful!

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

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

componentsPhone A

Score A

Phone BScore BObjective:

Maximize the overall score

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?

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?

Decisions = variables

xA = how many phones of type A

xB = how many phones of type B

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

1xA + 2xB ! 10 display

2xA + 2xB ! 18 memory

1xA + 3xB ! 12 transmission

2xA + 3xB ! 21 keyboard

1xA ! 9 navigation

1xA ! 10 camera

xA, xB ! 0 non negativity

Objective function

Maximize the overall score

max 3xA + 8xB

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

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

03 GMproduction

scheduling (30 plants in 10 countries)

2G$ saved

05 Canadian Pacific Railroad train scheduling 170M$ per year

Interesting web sites

http://www.scienceofbetter.org/classification of success applications

http://www.informs.org/American OR societyFranz Edelman Award record

http://www.airo.org/Italian OR society

Emergency service in Milan (118)Guarantee an intervention time of less than 8 min.

Statically place ambulances in the area (Piazzole)

Assign ambulances to calls

Assign patients to hospitals

Route ambulances in the city

Assign personnel to the call center so that dropped calls are minimized

Extract data from a database of 10 years

Visualization on a GIS

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 and mathematical models

Interpret the results

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. Basics of computational complexity. Heuristic algorithms: greedy and local search.

Applications of OR in computer science and telecommunications: Routing, scheduling, location.

Course organization

Lab activity: modelingwe will use a modeling and solution package (downloadable)

Mid term examsthe sum of mid term exams = 28 pointsone insufficient grade can be recovered (at most once)

Optional lab exam (4 additional points)

Lecture notes and (solved) exercises available on the course web sitehttp://home.dei.polimi.it/malucell

Consulting hours: Tuesday from 15 to 17.