INDR 501 OPTIMIZATION MODELS AND ALGORITHMS. Metin Türkay Department of Industrial Engineering, Koç University, Istanbul

Transcription

1 INDR 501 OPTIMIZATION MODELS AND ALGORITHMS Metin Türkay Department of Industrial Engineering, Koç University, Istanbul Fall 2014

2 COURSE DESCRIPTION This course covers the models and algorithms for optimization problems. The theory and properties of solution methods for linear programming problems will be covered. INDR 501 Metin Türkay 2

3 TEXTBOOK Bazaraa, M.S., J.J. Jarvis and H.D. Sherali, Linear Programming and Network Flows, 4th edition, Wiley, 2010, New Jersey. INDR 501 Metin Türkay 3

4 GRADING Midterm I 20% Midterm II 20% Homework 20% Final Exam 40% A+ A A B B B C C INDR 501 Metin Türkay 4

5 COURSE WEB SITE INDR 501 Metin Türkay 5

6 DECISION MAKING Structuring the problem Analyzing the problem Define the problem Quantitative Analysis Qualitative Analysis Identify the alternatives Summary and Evaluation Determine the criteria Make the decision INDR 501 Metin Türkay 6

7 QUANTITATIVE ANALYSIS Quantitative Analysis Process 1) Model development 2) Data preparation 3) Model solution 4) Analysis of the solution and report generation Potential reasons for a quantitative analysis approach to decision making The problem is complex, has significant impact, is large-scale or repetitive. INDR 501 Metin Türkay 7

8 TWO PRIMARY STAGES Modeling Models are representations of real objects or systems Building a model helps understanding a system Generally, experimenting with models (compared to experimenting with the real system) requires less time, is less expensive, involves less risk Solution & Analysis Determining the best solutions by applying an algorithm and interpreting the results. INDR 501 Metin Türkay 8

9 MATHEMATICAL MODELS Mathematical models represent real world problems through a system of mathematical relationships (formulas and expressions) based on key assumptions, estimates, or statistical analyses Examples of mathematical models Simulation models, econometric models, time series models, mathematical programming models INDR 501 Metin Türkay 9

10 MATH.PROGR. MODELS Relate decision variables with input parameters. Maximize or minimize some objective function subject to constraints. Objec&ves (minimize risk, maximize profit, etc. ) Constraints (capaci5es, budget limits, etc.) INDR 501 Metin Türkay 10

11 DEFINING THE PROBLEM Study the relevant system and develop a welldefined statement of the problem Objectives Constraints Interrelationships Alternatives Time Limits INDR 501 Metin Türkay 11

12 DEFINING THE PROBLEM Ø Decision Variables: variable values to be determined. Ø Objective Function: measure of performance Ø Constraints: any restrictions on the values that can be assigned to decision variables. Ø Parameters: the constants in the constraints and the objective function. INDR 501 Metin Türkay 12

13 SOLUTION APPROACHES Ø HEURISTIC ü traditional choice for many problems ü expert knowledge from experience is used for making decisions ü a feasible solution can be found ü no guarantee on the quality of solution Ø SIMULATION ü the choice for the 1980 s and 1990 s ü a feasible solution is not guaranteed ü quality of the solution is not a concern ü incremental improvement by trial and error Ø OPTIMIZATION ü newly emerging choice ü feasible solution is always found if there is one ü optimal solution is guaranteed for a large class of problems ü theory is not fully understood INDR 501 Metin Türkay 13

14 COMPARISON OF APPROACHES OPTIMIZATION SIMULATION HEURISTIC INDR 501 Metin Türkay 14

15 DEGREES OF FREEDOM A mathematical model for any problem consists of: n variables m equations The degrees of freedom: n-m i (where m i is the number of independent equations) The problem is optimization if n > m i the number of decision variables: n-m i The problem is simulation if n=m i there are no decision variables The heuristic system has no clear relationship between n and m i INDR 501 Metin Türkay 15

16 OPTIMIZATION maximize z=f(x) subject to g(x) 0 x L x x U Optimization Problems are categorized into: LP: Linear Programming Problems NLP: Nonlinear Programming Problems MILP: Mixed-Integer Linear Programming Problems MINLP: Mixed-Integer Nonlinear Programming Problems INDR 501 Metin Türkay 16

17 LINEAR PROGRAMMING feasible region Assumptions: 1. Additivity: contribution of all variables to the objective function and constraints are additive 2. Proportionality: contribution of all variables to the objective function and constraints are proportional to their levels 3. Divisibility: variables can have any real value 4. Certainty: values of c, a ij, b, x L and x U are known and fixed, variables do not have a probability distribution maximize z=c T x subject to Ax = b x L x x U Z objective function x n-vector of variables A mxn matrix (m<n) c n-vector b m-vector Solution Methods: 1. Simplex method (Dantzig, 1949) 2. Interior point method (Karmarkar, 1984) INDR 501 Metin Türkay 17

18 LINEAR PROGRAMMING George Dantzig Founder of the simplex method Father of Linear Programming INDR 501 Metin Türkay 18

19 NOTED CHARACTERS Vassily Leontieff Leonid Kantorovich & Nobel Prize in Economics, 1973 Tjalling C. Koopmans Nobel Prize in Economics, 1975 Herbert A. Simon Nobel Prize in Economics, 1978 Carlos Slim Net worth:$73 bil Taught LP INDR 501 Metin Türkay 19

20 NONLINEAR PROGRAMMING feasible region maximize z=f(x) subject to g(x) 0 x L x x U Assumptions: 1. Divisibility: variables can have any real value 2. Certainty: values of c, a ij, b, x L and x U are known and fixed, variables do not have a probability distribution Solution Methods: 1. Newton type (Karush, 1939, Kuhn&Tucker, 1951) 2. Reduced gradient (Fletcher&Powell, 1963) INDR 501 Metin Türkay 20

21 MIXED-INTEGER LINEAR PR. integer solutions ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú convex hull Assumptions: 1. Additivity: contribution of all variables to the objective function and constraints are additive 2. Proportionality: contribution of all variables to the objective function and constraints are proportional to their levels 3. Certainty: values of c, a ij, b, x L and x U are known and fixed, variables do not have a probability distribution maximize z=c T x+dy subject to Ax+By = e x L x x U y {0,1} Solution Methods: 1. Cutting Plane (Gomory, 1958) 2. Branch and Bound (Land&Doig, 1960) 3. Branch and Cut (Johnson, 2000) INDR 501 Metin Türkay 21

22 MIXED-INTEGER NONLINEAR ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú ú integer solutions outer-approximation Benders decomposition extended cutting plane logic-based methods Assumptions: 1. Certainty: values of c, a ij, b, x L and x U are known and fixed, variables do not have a probability distribution maximize z=c T x+dy subject to Ax+By = e x L x x U y {0,1} Solution Methods: 1. Benders Decomposition (Geoffrion, 1972) 2. Branch&Bound (Gupta&Ravindran, 1985) 3. Outer Approximation (Duran&Grossmann, 1986) 4. Extended Cutting Plane (Westerlund&Pettersson, 1995) 5. Logic Based Methods (Türkay&Grossmann, 1996) INDR 501 Metin Türkay 22

23 METİN TÜRKAY Education: BS, MS: METU, Ankara PhD: Carnegie Mellon Univ, PA Experience: Koç University (2000- ) Lecturer, Rutgers, NJ (1997) Industrial Experience Project Manager, Ceceli Industries, Ankara ( ) Principal Consultant, Mitsubishi Corpora5on, Japan ( ) Consultant, İstanbul Metropolitan Planning Center ( ) Principal Consultant, ZER A.Ş. (SCM&Logis5cs in Koç Holding; ) INDR 501 Metin Türkay 23