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Fondamenti di Ricerca Operativa D -...
Transcript of Fondamenti di Ricerca Operativa D -...
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?
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
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
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.