indr 501 optimization models and …home.ku.edu.tr/~mturkay/indr501/indr501_intro_2014_web.pdfthis...
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INDR 501 OPTIMIZATION MODELS AND
ALGORITHMS
Metin Türkay Department of Industrial Engineering, Koç University, Istanbul
Fall 2014
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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.
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TEXTBOOK
Bazaraa, M.S., J.J. Jarvis and H.D. Sherali, “Linear Programming and Network Flows”, 4th edition, Wiley, 2010, New Jersey.
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GRADING
Midterm I 20% Midterm II 20% Homework 20% Final Exam 40%
A+ A
98-100 90-97
A- 85-89 B+ 80-84 B 75-79 B- 70-74 C+ 65-69 C 60-64
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http://home.ku.edu.tr/~mturkay/indr501/
COURSE WEB SITE
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DECISION MAKING
Analyzing the problem
Structuring the problem
Define the problem
Determine the criteria
Identify the alternatives
Quantitative Analysis
Summary and Evaluation
Qualitative Analysis
Make the decision
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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.
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§ 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.
TWO PRIMARY STAGES
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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
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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.)
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DEFINING THE PROBLEM
Study the relevant system and develop a well-defined statement of the problem
§ Objectives § Constraints § Interrelationships § Alternatives § Time Limits
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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.
DEFINING THE PROBLEM
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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
SOLUTION APPROACHES
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COMPARISON OF APPROACHES
OPTIMIZATION
SIMULATION
HEURISTIC
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A mathematical model for any problem consists of: n variables m equations
The degrees of freedom: n-mi (where mi is the number of independent equations)
The problem is optimization if n > mi the number of decision variables: n-mi
The problem is simulation if n=mi there are no decision variables
The heuristic system has no clear relationship between n and mi
DEGREES OF FREEDOM
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OPTIMIZATION
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
maximize z=f(x) subject to g(x) ≤ 0
xL≤x≤xU
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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, aij, b, xL and xU are known and fixed, variables do not have a probability distribution
Solution Methods: 1. Simplex method (Dantzig, 1949) 2. Interior point method (Karmarkar, 1984)
maximize z=cTx subject to Ax = b
xL≤x≤xU Z objective function x n-vector of variables A mxn matrix (m<n) c n-vector b m-vector
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LINEAR PROGRAMMING
George Dantzig § Founder of the simplex method § “Father” of Linear Programming
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NOTED CHARACTERS
Vassily Leontieff Leonid Kantorovich & Nobel Prize in Economics, 1973 Tjalling C. Koopmans
Nobel Prize in Economics, 1975
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Herbert A. Simon Nobel Prize in Economics, 1978
Carlos Slim Net worth:$73 bil Taught LP
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NONLINEAR PROGRAMMING
Assumptions: 1. Divisibility: variables can have any real
value 2. Certainty: values of c, aij, b, xL and xU 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)
maximize z=f(x) subject to g(x) ≤ 0
xL≤x≤xU
feasible region
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MIXED-INTEGER LINEAR PR.
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, aij, b, xL and xU are known and fixed, variables do not have a probability distribution
Solution Methods: 1. Cutting Plane (Gomory, 1958) 2. Branch and Bound (Land&Doig, 1960) 3. Branch and Cut (Johnson, 2000)
maximize z=cTx+dy subject to Ax+By = e
xL≤x≤xU
y∈{0,1}
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
integer solutions
convex hull
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MIXED-INTEGER NONLINEAR
Assumptions: 1. Certainty: values of c, aij, b, xL and xU are
known and fixed, variables do not have a probability distribution
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)
maximize z=cTx+dy subject to Ax+By = e
xL≤x≤xU
y∈{0,1}
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
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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 (1990-‐1992) • Principal Consultant, Mitsubishi Corpora5on, Japan (1997-‐2000) • Consultant, İstanbul Metropolitan Planning Center (2002-‐2005) • Principal Consultant, ZER A.Ş. (SCM&Logis5cs in Koç Holding; 2008-‐2012)
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