Introduzione al corso RO2 LM Ingegneria Gestionale Laura Galli Dipartimento di Informatica Largo B. Pontecorvo 3, 56127 Pisa [email protected] http://www.di.unipi.it/~galli 29 Settembre 2014 Ricerca Operativa 2 Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa A.A. 2014/15 L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 1 / 25 Conoscenze richieste Corso di base di Ricerca Operativa • Programmazione Lineare • Algoritmo del Simplesso • Dualit` a • Programmazione Lineare su grafi • Cammino minimo • Flusso massimo • Flusso di costo minimo L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 2 / 25 Obiettivo e argomenti del corso Obiettivo Lo scopo del corso `e quello di fornire gli strumenti teorico-algoritmici necessari per modellare e risolvere problemi di Programmazione Intera (IP), che sorgono nella gestione di sistemi complessi, quali la produzione industriale o la gestione di servizi. Il tema principale del corso `e il concetto di (ri)-formulazione. Argomenti • Cenni sulla complessit` a computazionale • Teoria e metodi di Programmazione Lineare Intera • Modellazione e risoluzione di alcuni classici problemi di ottimizzazione L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 3 / 25 Il processo decisionale complessit` a propriet` a modello (analisi poliedrale) Teoria var. continue/intere funz. lineari/non lineari 1 o pi` u obiettivi Algoritmo Modello esatto/euristico/approx. general purpose/ad hoc Problema L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 4 / 25 Riferimenti Laura Galli Dipartimento di Informatica Largo B. Pontecorvo 3 - 56127 Pisa Edificio C - 2◦ piano - stanza 345 DN tel. 050 2212715 e-mail: [email protected] ricevimento: gioved`ı 17.00-19.00 Orario del corso • luned`ı 13.30-15.30 aula F4 • marted`ı 10.30-13.30 aula F4 L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 5 / 25 Materiale per il corso Pagina web del corso http://www.di.unipi.it/optimize/Courses/ Ricerca Operativa II (LM Ingegneria Gestionale) Testi e appunti • lezioni alla lavagna... • ...materiale fornito durante il corso • Appunti RO • M. Pappalardo, M. Passacantando, Ricerca Operativa, Edizioni Plus, 2010. L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 6 / 25 Dettagli del corso Learning is better if you participate • Math on the blackboard...take notes! • Italian & English • I may call on you during the class (GASP !!!) • Readings... • Homeworks... • Plz always check out the AVVISI section of the course web page before the class! L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 7 / 25 Grading Esame • Prova scritta sugli argomenti trattati. • I contenuti dell’esame sono quelli del corso dell’anno accademico a cui si riferisce l’appello, anche per gli studenti che avessero seguito il corso in anni precedenti. • Non ` e permesso l’uso di appunti, libri etc. • La consegna di una prova scritta annulla le prove scritte consegnate in precedenza. • Academic Integrity: if you cheat, you make me sad :-( and sick :-S L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 8 / 25 Let’s start Today’s outline • About this class: done X • IP: Why would you possibly care? • IP warm up: First definitions, Modeling with IP Homework (due next time we meet!) • Be nice: send me an email with your name and surname. Thx! L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 9 / 25 Programming ??? I hate Programming !!! Question What does “programming” mean? L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 10 / 25 Programming ??? I hate Programming !!! Question What does “programming” mean? Answer Planning. L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 11 / 25 Programming ??? I hate Programming !!! Question What does “programming” mean? Answer Planning. MP Mathematical Programming (optimization) is the selection of a best element (with regard to some criteria) from some set of available alternatives. L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 12 / 25 Mathematical Optimization MP Literally, these are “mathematical models for planning”. Also called optimization models. MP The essential element is the existence of an objective. MP Some classifications of mathematical programs: • Linear/nonlinear • Convex/nonconvex • Discrete/continuous • Stochastic/deterministic L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 13 / 25 Programming with Integers ?!?! IP Integer Programming is about decision making with integers. More precisely, decision making in which some of the decisions may take only certain integer values. Question Why would you ever want to program with integers? L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 14 / 25 Programming with Integers ?!?! Question Why would you ever want to program with integers? Answer If the variable is associated with a physical entity that is indivisible, then it must be integer. L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 15 / 25 Programming with Integers ?!?! Question Why would you ever want to program with integers? Answer If the variable is associated with a physical entity that is indivisible, then it must be integer. Important • Sometimes a continuous approximation to the discrete (integer) decision is accurate enough for practical purposes... • but in many cases it is not! / L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 16 / 25 Binary 0-1 Variables We can use 0-1 (binary) variables for a variety of purposes. • Modeling yes/no decisions. • Enforcing disjunctions. • Enforcing logical conditions. • Modeling fixed costs. • Modeling piecewise linear functions. In most of these cases the continuous approximation to the discrete decision is not accurate enough for practical purposes. / L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 17 / 25 MIP Mixed-Integer Linear Programming Problem: (MILP) max {c T x + hT y | A x + G y ≤ b, x ∈ Zn+ , y ∈ Rp+ } • Instance: c ∈ Rn , h ∈ Rp , A ∈ Rm×n , G ∈ Rm×p • MATH NOTATION QUIZ • What the heck is Rp+ ? • What the heck is Zn+ ? • What is the objective function? • What are the decision variables? How many? • What are the constraints? How many? • What is the feasibile region? Can you guess what it is like? L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 18 / 25 Integer Programming Classes • We call the problem mixed integer programming due to the presence of continuous variables • In some problems x are allowed to take on values only 0 or 1 • Such variables are called binary. • Integer Programs involving only binary variables are called binary integer programs (BIPs). (x ∈ Bn ) • (Pure) Integer Programming • G , h, y not present: max {c T x | A x ≤ b, x ∈ Zn+ } • Mixed 0 − 1 Programming • x ∈ Bn . G , h, y present: max {c T x + hT y | A x + G y ≤ b, x ∈ Bn+ , y ∈ Rp+ } • Linear Programming (!) • c, A, x not present: max {hT y | G y ≤ b, y ∈ Rp+ } L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 19 / 25 Solutions • A solution is an assignment of values to variables. • A solution can hence be thought of as an n-dimensional vector. • A feasible solution is an assignment of values to variables such that all the constraints are satisfied. • The objective function value of a solution is obtained by evaluating the objective function at the given point. • An optimal solution (assuming maximisation) is one whose objective function value is greater than or equal to that of all other feasible solutions. • Note that a math program may not have a feasible solution. • Question: What are the different ways in which this can happen? L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 20 / 25 General Comments • Note that many textbooks consider maximization problems by default. • Many people (myself included) often consider minimization as the default, so be ready to be confused. • One further assumption we will make, and never mention again, is that the input data is rational. e.g. (c ∈ Qn ) • This is an important assumption since with irrational data, certain “intuitive” results no longer hold. Such as what? • A computer can only understand rational data anyway, so this is not an unreasonable assumption. • Note that you implicitly assumed this in your linear programming course anyway! L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 21 / 25 How Hard is Integer Programming? • Solving general integer programs can be much more difficult than solving linear programs. • This is more than just an empirical statement • There is a whole theory surrounding it • You will learn some of the gory details • There is no polynomial-time algorithm for solving general MIPs. • Solving the associated linear programming relaxation results in an upper bound on the optimal solution to the MIP (assuming maximization). • In general, an optimal solution to the LP relaxation does not tell us much about an optimal solution to the MIP. • Why does solving the LP relaxation not necessarily yield a good solution? • Rounding to a feasible integer solution may be difficult or impossible. • Rounding may result in a solution far from optimal. • The optimal solution to the LP relaxation can be arbitrarily far away from the optimal solution to the MIP. • We can bound the difference between the optimal solution to the LP and the optimal solution to the MIP. How? L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 22 / 25 Integer Programming and Convexity • The feasible region of an integer program is nonconvex. • The nonconvexity is of a rather special form, though all forms of nonconvexity are in some sense equivalent. • Although the feasible set is nonconvex, there is a convex set over which we can optimize in order to get a solution. Why? • The challenge is that we do not know how to describe that set. • Even if we knew the description, it would in general be way too large to write down explicitly. • Integer variables can be used to express other forms of nonconvexity, as we will see later on. L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 23 / 25 Basic Themes Our goal will be to expose the geometrical structure of the feasible region (at least near the optimal solution). We can do this by: • Outer approximation • Inner approximation • Division An important component of the algorithms we consider will be mechanisms for computing bounds by either: • Relaxation • Duality When all that fails, we will employ a basic principle: divide large, difficult problems into smaller ones: • Logic (conjunction/disjuntion) • Implicit enumeration • Decomposition L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 24 / 25 Combinatorial Optimization • A combinatorial optimization problem: COP = (N, F) consists of • A finite ground set N • A finite set F ⊆ 2N of feasible solutions • Costs cj ∀ j ∈ N P • The cost of F ∈ F is c(F ) = j∈F cj • The combinatorial optimization problem is then max {c(F ) | F ∈ F} • Many COPs can be written as IPs... • We’ll see some examples soon...stay tuned! L. Galli Corso di Ricerca Operativa 2 - Laurea Magistrale in Ingegneria Gestionale Universit` a di Pisa 25 / 25
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