Operations Research 1-Optimization Spring 2018

Office hour: Tue, Thr, 13:00 – 14:30 or by appointment Instructor: Sungsoo Park (sspark@kaist.ac.kr), Building E2-2, room 4112, Tel: 3121 Office hour: Mon, Wed 14:30 – 16:30 or by appointment (telephone or email) TA: Jaehee Jeong (wogmlalszk@kaist.ac.kr ), Sunhyeon Kwon (khyun34@kaist.ac.kr ) Building E2-2, room 4113, Tel: 3161 Office hour: Tue, Thr, 13:00 – 14:30 or by appointment Homepage: http://solab.kaist.ac.kr/, download class notes, homeworks. Text: “Linear Programming”, Vasek Chvatal, Freeman, 1983 and class handouts   OR-1 Opt. 2018

Grading: Midterm 30 - 40%, Final 40 - 50%, (closed book/notes) Homework 10 - 20% (Including computer assignments, CPLEX/Xpress-MP) Characteristics of the course More emphasis on the theory and algorithm for linear programming Mathematical ideas. Understand the logic and convince yourself. Ideas interrelated. We will build up ideas upon some previous results. Not understanding some logic will cause trouble later. Be steady in studying. Prerequisite: Basic Linear Algebra or consent of instructor. OR-1 Opt. 2018

Class codes Read the code of academic integrity. No copying of homework. You may consult with others on homework problems, but writing should be your own. Each Copying (for all involved students) will be penalized by dropping your final grade by one level (e.g. 𝐵 + → 𝐵 0 ). One grade down for 4 or more classes missed. One additional level of grade down for each two additional classes missed (e.g., If you deserve 𝐴 − but missed 4 classes, you will receive 𝐵 + . And, if you missed 6 classes, you will receive 𝐵 0 , and so on.) Random check of attendance. Give notice for unavoidable situations. For a unavoidable situation, e.g. illness, death of a relative, it will not be counted as a missed class. Scheduled midterm, final exam time will be adhered to. No makeup exam. If you missed an exam, your final grade will be determined upon the rank of your score on the exam you have taken with some disadvantages. OR-1 Opt. 2018

Origins of OR Contributions of scientists and engineers during world war II. Battle of Britain (radar sites selection and control of anti-aircraft systems), anti-submarine warfare in Atlantic (using bombers), Design of B-29, .. See the handout. Also https://www.informs.org/About-INFORMS/History-and-Traditions/Bibliographies/The-Origins-of-OR for more material. Societies INFORMS (the Institute for Operations Research and Management Sciences) : https://www.informs.org Korean Operations Research Society (한국경영과학회) : http://www.korms.or.kr Korean Institute of Industrial Engineers (대한산업공학회): http://kiie.org (MOS (Mathematical Optimization Society) : http://www.mathopt.org/ ) OR-1 Opt. 2018

After the war, methodologies used by the scientists adopted by government, industry, and academic societies were formed. Called Operations Research (US), Operational Research(UK, Europe) (운용과학), Management Science (경영과학) Characteristics: Use of mathematical models to solve the design, evaluation, decision making problems arising in industry, government, military, …. E=mC2, F=ma, … OR-1 Opt. 2018

Nature of OR “research on operations” Applied mathematics + computer science + management Models : Deterministic models, Optimization (확정적 모형, OR-I) Stochastic models (확률적 모형, OR-II) Needed background: Algebra, calculus, discrete mathematics, probability, statistics, data structures, algorithms, data base, programming skills, …) Important thrusts in early stages   1. Technical progress (Simplex method for linear programming, Dantzig, 1947) 2.  Invention of computer and PC OR-1 Opt. 2018

Study areas Deterministic models Linear programming(선형계획법, linear optimization):1975, Nobel prize, Kantorovich, Koopmans (efficient allocation of resources) Nonlinear programming(비선형계획법):1990 Nobel prize, Markowitz (portfolio selection problem) OR-1 Opt. 2018

Integer Programming(정수계획법), Combinatorial optimization (조합최적화) Knapsack problem Traveling Salesman Problem (외판원문제) Given n cities, and distances cij between city i and j. What is the shortest sequence to visit each city exactly once and return to the starting city? ( Applications: PCB assembly, Off-shore drilling, vehicle routing (delivery/pick-up problem), bio, …) web site: http://www.math.uwaterloo.ca/tsp/ OR-1 Opt. 2018

Networks and graphs: 𝐺=(𝑉,𝐸) Shortest path to move from Inchon to Kangnung? (Shortest path problem) Logistics, Telecommunication routing, … Connect the cities with roads (or communication lines) in a cheapest way. (Minimum spanning tree problem) How much commodities (or packets) can we send from Kwangju to Daegu if edges have limited capacities? (maximum flow problem) Seoul Inchon Kangnung Daejeon Kwangju Pusan Daegu : node, vertex : edge, link (undirected) arc (directed) OR-1 Opt. 2018

Dynamic programming (multi-stage decision making) If a system changes over time and the status of the system in the next period depends on the current status and decisions made, what is the best decision in each stage to optimize our goal in the end? Not the formalized problems, but refer the structured steps (methods) used to solve problems involving many stages. Game theory Investigate the best strategy when the outcome of cooperation and/or competition between people or groups depends on the collective decisions made by individual person/group. Economics, Marketing (Nobel prizes: 1994, Nash, Harsanyi, Selten, ; 2012, Roth, Shapley) OR-1 Opt. 2018

Problems for which polynomial running time algorithms exist. Computational complexity: Theory that investigates the inherent difficulty of problems. (Polynomial running time of algorithms vs. Exponential time) Turing machine model of computation. NP-completeness. NP-complete (NP-hard) problems: Knapsack problem, Traveling salesman problem, node packing (independent set, stable set) problem in a graph … Easily solvable problems: shortest path problem with nonnegative arc costs, minimum spanning tree problem, maximum flow problem, … Problems for which polynomial running time algorithms exist. A little bit of changes in the problem structure may make the problem hard. Shortest path problem with nonnegative arc costs vs. shortest path problem with negative arc costs allowed Minimum spanning tree problem vs. Steiner tree problem (𝑆⊆𝑉, 𝑆: Steiner nodes) Useful tool when we try to solve some new problems. Note that the basic models may appear as subproblems in a big problem. Also the models may be hidden in the real problem in some unexpected way. Identifying the hidden model may be crucial. OR-1 Opt. 2018

Stochastic models (OR-II) Markov chain Queueing theory Decision analysis Simulation Reliability OR-1 Opt. 2018

Recently, uncertainties of data are reflected in the optimization models Stochastic Programming Many scenarios with probability of occurring. Minimize the expected objective value. (e.g. two stage model (stochastic programming with recourse)) Robust Optimization Uncertainty set defined for data. Minimize the objective value for all possible realization of data in uncertainty set. e.g. constraints are given as 𝑗 𝑎 𝑖𝑗 𝑥 𝑗 ≤ 𝑏 𝑖 , 𝑖=1,…𝑚. 𝑎 𝑖𝑗 ∈[ 𝑎 𝑖𝑗 − 𝑎 𝑖𝑗 , 𝑎 𝑖𝑗 + 𝑎 𝑖𝑗 ] and at most Γ 𝑖 (≥0, integer) coefficients in 𝑖th constraint can vary simultaneously within the interval. Chance-constrained problem Each constraint: 𝑃 𝑗 𝑎 𝑗 𝑥 𝑗 ≤𝑏 ≥1−𝜀 (e.g. 𝜀=0.01), 𝑎 𝑗 are random variables. OR-1 Opt. 2018