IEOR 4004 Introduction to Operations Research Deterministic Models

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

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

all parameters known Summary goal is to minimize or maximize 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)

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: mediumterm 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 simplification Model formulation Model Algorithm selection Numerical calculation Problem Interpretation Sensitivity analysis Solution

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

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

Formulating an optimization Linear program problem Linear • Decision variables objective • Objective • Constraints • Domains Linear constraints x 1, x 2, x 3, x 4 x 1 Minimize 2 x 2 + 3 x 3 + (1 − 2 x 2)2 (x 1)2 + (x 2)2 + (x 3)2 + (x 4)2 ≤ 2 x 1 − 2 x 2 + x 3 − 3 x 4 ≤ 1 − x 1 + 3 x 2 + 2 x 3 + x 4 ≥ − 2 x 1 x 3 = 1 x 1 ≥ 0 x 2 ≠ 0. 5 x 3 in {0, 1} x 4 in [0, 1] Sign restriction

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