Ordinal Optimization: Soft Optimization for Hard Problems Wisdom consists of knowing when to avoid perfection. — Horowitz’s Rule* * The Complete Murphy’s Law by Arthur Bloch, Price Stern Sloan Publisher, Los Angeles, 1991. ORDINAL OPTIMIZATION SOFT OPTIMIZATION FOR HARD PROBLEMS Yu-Chi Ho Harvard University Massachusetts, USA Tsinghua University Beijing, China Qian-Chuan Zhao Tsinghua University Beijing, China Qing-Shan Jia Tsinghua University Beijing, China Yu-Chi Ho, PhD, Professor Qian-Chuan Zhao, Ph.D. Harvard University Professor of Automation Engineering Cambridge, MA, USA Center for Intelligent & Networked Tsinghua University Systems Beijing, China Tsinghua University e-mail: [email protected] Beijing, China e-mail: [email protected] Qing-Shan Jia, PhD, Lecturer Center for Intelligent & Networked Systems Tsinghua University Beijing, China e-mail: [email protected] Library of Congress Control Number: 2007927989 Ordinal Optimization: Soft Optimization for Hard Problems by Yu-Chi Ho, Qian-Chuan Zhao and Qing-Shan Jia ISBN-13: 978-0-387-37232-7 e-ISBN-13: 978-0-387-68692-9 Printed on acid-free paper. © 2007 Springer Science+Business Media, LLC. All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now know or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks and similar terms, even if the are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. 9 8 7 6 5 4 3 2 1 springer.com To Sophia, You have made life worth living and ideas possible. ——Larry To Betty, You have made life simple and beautiful. ——Qian-Chuan To Huai-Jin Jia and Guo-Hua Zhou, You have made my life possible. ——Qing-Shan Table of Contents Preface------------------------------------------------------------xiii Acknowledgements------------------------------------------xv I Introduction--------------------------------------------------------------------1 II Ordinal Optimization Fundamentals------------------------------7 1 Two basic ideas of Ordinal Optimization (OO) -----------------------------7 2 Definitions, terminologies, and concepts for OO----------------------------9 3 A simple demonstration of OO------------------------------------------------13 4 The exponential convergence of order and goal softening----------------15 4.1 Large deviation theory----------------------------------------------------16 4.2 Exponential convergence w.r.t. order-----------------------------------21 4.3 Proof of goal softening----------------------------------------------------26 4.3.1 Blind pick------------------------------------------------------------26 4.3.2 Horse race -----------------------------------------------------------28 5 Universal alignment probabilities---------------------------------------------37 5.1 Blind pick selection rule ------------------------------------------------- 38 5.2 Horse race selection rule------------------------------------------------- 39 6 Deterministic complex optimization problem and Kolmogorov equivalence----------------------------------------------------------------------48 7 Example applications-----------------------------------------------------------51 7.1 Stochastic simulation models-------------------------------------------- 51 7.2 Deterministic complex models-------------------------------------------53 8 Preview of remaining chapters------------------------------------------------54 III Comparison of Selection Rules---------------------------------------57 1 Classification of selection rules ----------------------------------------------60 2 Quantify the efficiency of selection rules-----------------------------------69 viii Table of Contents 2.1 Parameter settings in experiments for regression functions---------73 2.2 Comparison of selection rules-------------------------------------------77 3 Examples of search reduction-------------------------------------------------80 3.1 Example: Picking with an approximate model--------------------80 3.2 Example: A buffer resource allocation problem------------------84 4 Some properties of good selection rules-------------------------------------88 5 Conclusion-----------------------------------------------------------------------90 IV Vector Ordinal Optimization------------------------------------------93 1 Definitions, terminologies, and concepts for VOO-----------------------94 2 Universal alignment probability---------------------------------------------99 3 Exponential convergence w.r.t. order---------------------------------------104 4 Examples of search reduction------------------------------------------------106 4.1 Example: When the observation noise contains normal distribution ----------------------------------------------------106 4.2 Example: The buffer allocation problem -----------------------------108 V Constrained Ordinal Optimization------------------------------113 1 Determination of selected set in COO--------------------------------------115 1.1 Blind pick with an imperfect feasibility model----------------------115 1.2 Impact of the quality of the feasibility model on BPFM------------119 2 Example: Optimization with an imperfect feasibility model------------122 3 Conclusion--------------------------------------------------------------------124 VI Memory Limited Strategy Optimization----------------------125 1 Motivation (the need to find good enough and simple strategies) -----126 2 Good enough simple strategy search based on OO-----------------------128 2.1 Building crude model----------------------------------------------------128 2.2 Random sampling in the design space of simple strategies--------133 3 Conclusion----------------------------------------------------------------------135 VII Additional Extensions of the OO Methodology---------137 1 Extremely large design space------------------------------------------------138 2 Parallel implementation of OO----------------------------------------------143 2.1 The concept of the standard clock-------------------------------------144 Table of Contents ix 2.2 Extension to non-Markov cases using second order approximations------------------------------------------------------------147 2.2.1 Second order approximation--------------------------------------148 2.2.2 Numerical testing--------------------------------------------------152 3 Effect of correlated observation noises-------------------------------------154 4 Optimal Computing Budget Allocation and Nested Partition-----------159 4.1 OCBA----------------------------------------------------------------------160 4.2 NP--------------------------------------------------------------------------164 5 Performance order vs. performance value----------------------------------168 6 Combination with other optimization algorithms-------------------------175 6.1 Using other algorithms as selection rules in OO---------------------177 6.1.1 GA+OO-------------------------------------------------------------177 6.1.2 SA+OO--------------------------------------------------------------183 6.2 Simulation-based parameter optimization for algorithms----------186 6.3 Conclusion----------------------------------------------------------------188 VIII Real World Application Examples----------------------------189 1 Scheduling problem for apparel manufacturing---------------------------190 1.1 Motivation-----------------------------------------------------------------191 1.2 Problem formulation-----------------------------------------------------192 1.2.1 Demand models----------------------------------------------------193 1.2.2 Production facilities------------------------------------------------195 1.2.3 Inventory dynamic-------------------------------------------------196 1.2.4 Summary -----------------------------------------------------------197 1.3 Application of ordinal optimization-----------------------------------198 1.3.1 Random sampling of designs-------------------------------------199 1.3.2 Crude model--------------------------------------------------------200 1.4 Experimental results-----------------------------------------------------202 1.4.1 Experiment 1: 100 SKUs------------------------------------------202 1.4.2 Experiment 2: 100 SKUs with consideration on satisfaction rate -------------------------------------------------204 1.5 Conclusion----------------------------------------------------------------206 2 The turbine blade manufacturing process optimization problem-------207 2.1 Problem formulation-----------------------------------------------------208 2.2 Application of OO-----------------------------------------------------213 2.3 Conclusion--------------------------------------------------------------219 3 Performance optimization for a remanufacturing system--------------220 3.1 Problem formulation of constrained optimization-----------------220 3.2 Application of COO---------------------------------------------------224 x Table of Contents 3.2.1 Feasibility model for the constraint----------------------------224 3.2.2 Crude model for the performance------------------------------224 3.2.3 Numerical results-------------------------------------------------225