IEOR 4004: Introduction to Operations Research Deterministic Models January 22, 2014

Syllabus 20% homework assignments 40% midterm 40% final exam Lectures Monday, Wednesday 7:10pm-8:25pm Recitations: Friday 12:30pm-2pm Instructor: Juraj Stacho (myself) – office hours: Tuesday 1pm-2pm Teaching assistant (TA): Itai Feigenbaum – office hours: Friday, after recitations 2:15pm-3:15pm 1 st homework is already available on CourseworksCourseworks 1 st homework is already available on CourseworksCourseworks

Summary Goal of the course: Learn foundations of mathematical modeling of (deterministic) optimization problems Linear programming – problem formulation (2 weeks) Solving LPs – Simplex method (4 weeks) Network problems (2 weeks) Integer Programming (1.5 weeks) Dynamic Programming (1.5 weeks) Non-linear Programming (1 week time permitting) 2 weeks reserved for review (1 before midterm) all parameters known goal is to minimize or maximize

What this course is/is not about Is about: mathematical modeling (problem abstraction, simplification, model selection, solution) algorithms foundations of optimization deterministic models example real-world models typical model: medium- term production/financial planning/scheduling Not about: coding (computer programming) engineering (heuristics, trade-offs, best practice) stochastic problems (uncertainty, chance) solving problems on real-world data modeling risk, financial models, stock markets, strategic planning

Mathematical modeling Simplified (idealized) formulation Limitations – Only as good as our assumptions/input data – Cannot make predictions beyond the assumptions We need more maps

Mathematical modeling Problem Model Problem simplification Model formulation Algorithm selection Numerical calculation Interpretation Sensitivity analysis

Model selection trade-off between and Simple Complex Model Predictive power (quality of prediction) High Low Easy to compute a solution (seconds) Hard (if not impossible) to compute a solution (lifetime of the universe) Solving Linear programming Network algorithms Integer programming Dynamic programming accuracy (predictive power) model simplicity (being able to solve it)

Modeling optimization problems Optimization problem – decisions – goal (objective) – constraints Mathematical model – decision variables – objective function – constraint equations

Formulating an optimization problem Linear program Decision variables Objective Constraints Domains x 1, x 2, x 3, x 4 2x 2 + 3x 3 x 1 − 2x 2 + x 3 − 3x 4 ≤ 1 − x 1 + 3x 2 + 2x 3 + x 4 ≥ − 2 x 1 ≥ 0 x 3 in {0,1} x 4 in [0,1] Linear constraints Linear constraints Sign restriction + x 1 (1 − 2x 2 ) 2 (x 1 ) 2 + (x 2 ) 2 + (x 3 ) 2 + (x 4 ) 2 ≤ 2 − 2x 1 x 3 = 1 Linear objective Linear objective x 2 ≠ 0.5 Minimize

Mathematical modeling Deterministic = values known with certainty Stochastic = involves chance, uncertainty Linear, non-linear, convex, semi-definite