INDR 501 OPTIMIZATION MODELS AND

ALGORITHMS

Metin Türkay Department of Industrial Engineering, Koç University, Istanbul

Fall 2014

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

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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