Proportional optimization and fairness

International Series in Operations Research & Management Science Volume 127 Series Editor Frederick S Hillier Stanford University, CA, USA INT SERIES IN OPERATIONS RESEARCH & MANAGEMENT SCIENCE Series Editor: Frederick S Hillier, Stanford University Special Editorial Consultant: Camille C Price, Stephen F Austin State University Titles with an asterisk (*) were recommended by Dr Price Axsäter/ INVENTORY CONTROL, 2nd Ed Hall/ PATIENT FLOW: Reducing Delay in Healthcare Delivery Józefowska & W˛eglarz/ PERSPECTIVES IN MODERN PROJECT SCHEDULING Tian & Zhang/ VACATION QUEUEING MODELS: Theory and Applications Yan, Yin & Zhang/ STOCHASTIC PROCESSES, OPTIMIZATION, AND CONTROL THEORY APPLICATIONS IN FINANCIAL ENGINEERING, QUEUEING NETWORKS, AND MANUFACTURING SYSTEMS Saaty & Vargas/ DECISION MAKING WITH THE ANALYTIC NETWORK PROCESS: Economic, Political, Social & Technological Applications w Benefits, Opportunities, Costs & Risks Yu/ TECHNOLOGY PORTFOLIO PLANNING AND MANAGEMENT: Practical Concepts and Tools Kandiller/ PRINCIPLES OF MATHEMATICS IN OPERATIONS RESEARCH Lee & Lee/ BUILDING SUPPLY CHAIN EXCELLENCE IN EMERGING ECONOMIES Weintraub/ MANAGEMENT OF NATURAL RESOURCES: A Handbook of Operations Research Models, Algorithms, and Implementations Hooker/ INTEGRATED METHODS FOR OPTIMIZATION Dawande et al/ THROUGHPUT OPTIMIZATION IN ROBOTIC CELLS Friesz/ NETWORK SCIENCE, NONLINEAR SCIENCE and INFRASTRUCTURE SYSTEMS Cai, Sha & Wong/ TIME-VARYING NETWORK OPTIMIZATION Mamon & Elliott/ HIDDEN MARKOV MODELS IN FINANCE del Castillo/ PROCESS OPTIMIZATION: A Statistical Approach Józefowska/JUST-IN-TIME SCHEDULING: Models & Algorithms for Computer & Manufacturing Systems Yu, Wang & Lai/ FOREIGN-EXCHANGE-RATE FORECASTING WITH ARTIFICIAL NEURAL NETWORKS Beyer et al/ MARKOVIAN DEMAND INVENTORY MODELS Shi & Olafsson/ NESTED PARTITIONS OPTIMIZATION: Methodology and Applications Samaniego/ SYSTEM SIGNATURES AND THEIR APPLICATIONS IN ENGINEERING RELIABILITY Kleijnen/DESIGN AND ANALYSIS OF SIMULATION EXPERIMENTS Førsund/ HYDROPOWER ECONOMICS Kogan & Tapiero/ SUPPLY CHAIN GAMES: Operations Management and Risk Valuation Vanderbei/ LINEAR PROGRAMMING: Foundations & Extensions, 3rd Edition Chhajed & Lowe/BUILDING INTUITION: Insights from Basic Operations Mgmt Models and Principles Luenberger & Ye/LINEAR AND NONLINEAR PROGRAMMING, 3rd Edition Drew et al/ COMPUTATIONAL PROBABILITY: Algorithms and Applications in the Mathematical Sciences* Chinneck/ FEASIBILITY AND INFEASIBILITY IN OPTIMIZATION: Algorithms and Computation Methods Tang, Teo & Wei/ SUPPLY CHAIN ANALYSIS: A Handbook on the Interaction of Information, System and Optimization Ozcan/ HEALTH CARE BENCHMARKING AND PERFORMANCE EVALUATION: An Assessment using Data Envelopment Analysis (DEA) Wierenga/ HANDBOOK OF MARKETING DECISION MODELS Agrawal & Smith/ RETAIL SUPPLY CHAIN MANAGEMENT: Quantitative Models and Empirical Studies Brill/ LEVEL CROSSING METHODS IN STOCHASTIC MODELS Zsidisin & Ritchie/ SUPPLY CHAIN RISK: A Handbook of Assessment, Management & Performance Matsui/ MANUFACTURING AND SERVICE ENTERPRISE WITH RISKS: A Stochastic Management Approach Zhu/ QUANTITATIVE MODELS FOR PERFORMANCE EVALUATION AND BENCHMARKING: Data Envelopment Analysis with Spreadsheets ∼A list of the early publications in the series is found at the end of the book∼ Wieslaw Kubiak Proportional Optimization and Fairness 123 Wieslaw Kubiak Memorial University Faculty of Administration John’s NL Canada A1B 3X5 wkubiak@mun.ca ISBN: 978-0-387-87718-1 e-ISBN: 978-0-387-87719-8 DOI: 10.1007/978-0-387-87719-8 Library of Congress Control Number: 2008934787 c Springer Science+Business Media, LLC 2009 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 known or hereafter developed is forbidden The use in this publication of trade names, trademarks, service marks, and similar terms, even if they 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 Printed on acid-free paper springer.com To My Inka i Michał Preface If the beginning provides countless possibilities, then why not to start with few questions? Why are cars of different colors spread along an assembly line rather then batched together in a single long sequence of the same color? How to make equal priority jobs progress at the rates proportional to their lengths so that a job twice the length of another one gets a shared resource allocated twice the time of the other job up to any point in time? Or a client who pays three times more for its computations than another client gets its computations to progress three times faster than the other client’s by getting more processor and bandwidth allocations? How to make sure that the Internet gateway bandwidth is shared fairly so that the community sharing the network is not reduced to few getting all and most nothing? All these questions deal with proportional representation either according to the demand for particular car color, or according to the job length or its right to resources, or according to the reciprocal of the packet size to name just few They are fundamental even more so today when we are surrounded by systems enabled by technology to work in a justin-time mode since this mode very principle requires a steady, smooth, and evenly spread progress of tasks in time The progress is proportional to the demand for the tasks’s outcomes As a thinker and futurist Alvin Toffler [1] in his Financial Times interview points out “Global positioning satellites are key to synchronising precision time and data streams for everything from mobile phone calls to ATM withdrawals They allow just-in-time productivity because of precise tracking.” What is somewhat surprising is that all these questions that seem so far apart have similar underlying framework, which is simply speaking to build a finite or infinite often cyclic sequence; we shall refer to it as a just-in-time sequence, on a finite n letter alphabet where each letter is spread “as evenly as possible” and occurs with a given rate or a given number of times The problem of finding such a sequence is not only a mathematical one since there is no mathematical definition of “as evenly as possible” that would satisfactorily capture the challenge behind this phrase The problem can find many mathematical formulations, but none will probably satisfy all Thus, one way of approaching the problem is to use the wellknown apportionment theory and especially its house monotone methods to build the desired just-in-time sequence vii viii Preface The apportionment problem has its roots in the proportional representation system designed for the House of Representatives of the United States where each state receives seats in the House proportionally to its population The theory has been in the making for more than 200 years now and its exciting story as well as main results can be found in an excellent book by Balinski and Young [2], see also more recent book by Young [3], and Balinski’s popular introduction in [4] The title of Balinski and Young’s book speaks for itself: “Fair Representation: Meeting the Ideal of One Man, One Vote.” Its main underlying message is that the ideal is not one but many and that we can only hope to agree on one by stating some “obvious” axioms that it must meet and then find a method that would deliver a solution meeting these axioms, or to prove that one does not exist This process may, however, not save us from falling into various anomalies that not contradict the axioms yet may be at odds with the commonly accepted sense of fair representation This book argues that the apportionment methods, in particular the John Quincy Adams’s and the Thomas Jefferson’s, have been widely, yet unknowingly, rediscovered and used in resource allocation and sequencing computer, manufacturing, and other real-life technical systems Sometimes without a clear understanding of what solutions they lead to in terms of their properties The properties which have been well researched and known from the apportionment literature but missing in the technical one, either computer science or operations research This lack of proper context may have resulted, as we argue in some parts of this book, in overlooking other apportionment methods, in particular the Daniel Webster’s method, that may offer a number of additional attractive properties, like being better balanced than either the Adams’s or the Jefferson’s The axiomatic approach favored by the apportionment theory for the proportional representation systems is preferred over an optimization approach championed by operations research scientists since the problem with the latter approach is in the words of Balinski and Young from [2] as follows: “The moral of this tale is that one cannot choose objective functions with impunity, despite current practices in applied mathematics The choice of an objective is, by and large an ad hoc affair Of much deeper significance than the formulas that are used are the properties they enjoy.” We think, however, that in order to adequately address the proportional representation problems listed at the beginning of this preface and others we need to study them not only through the apportionment theory but through optimization as well After all the questions of quantifying excess inventory and shortage in just-in-time manufacturing, the throughput error in stride scheduling, or the relative and absolute bounds in fair queueing are clearly important By doing so, we also realize that the optimization reveals a new role of the well-known apportionment methods, the Webster’s method in particular The optimization moreover reveals connections with the well-known and still open mathematical conjectures as the Fraenkel’s Conjecture, see Tijdeman [5] for a brief account and Chap 6, finally it relates to the multimodular functions minimization, introduced by Hajek [6] and later developed by Altman et al [7], which aims at evenly spreading the demand and workload in computer and supply chains Preface ix The question of which objective function to choose we settle by choosing either total deviation or maximum deviation objective functions Our solution method is general enough to include a large class of point deviation functions The choice of objective functions follows sometime the choice made by Monden who, in his seminal book [8], described the Goal Chasing Method of Toyota by using the square point deviation function which apparently follows the minimization of square error in the least squares method of Carl Friedrich Gauss The attractive feature of this optimization is that it can be done efficiently, though certain intriguing computational complexity issues remain open, and produce solutions which have many though not all, by the Impossibility Theorem of Balinski and Young [2], desirable properties identified by the theory and practice of apportionment The book intends to chart a solid common ground for discussing and solving problems ranging from sequencing mixed-model just-in-time assembly lines, through just-in-time batch production, balancing workloads in event graphs to bandwidth allocation in the Internet gateways and resource allocation in operating systems From problems in mathematics of social sciences through operations research and computer science problems, it argues that the apportionment theory and the optimization based on deviation functions provide natural benchmarks in this process However, the process has just started and this book is to provide just a small stepping stone on the way to this common ground Needless to say it will be a great pleasure for the author if the book’s topic finds its followers The book includes mostly very recent results – some of them published recently, some of them new and yet unpublished It includes ten main chapters Chapter briefly reviews main results of the apportionment theory used in the remainder of the book It emphasizes the axiomatic approach to the apportionment problem and to the construction of the just-in-time sequences The approach relies on the divisor methods, in particular parametric methods advocated by Balinski and Young [2], and their desirable properties embedded in the resulting just-in-time sequences Chapter considers the problems of deviation minimization, the total and the maximum deviation, as tools for obtaining just-in-time sequences It formulates these problems as nonlinear integer optimization and presents efficient algorithms for their solution The algorithms are based on the concept of ideal positions, closely related to the Webster’s apportionment method They transform the deviation minimization problems to either the assignment or the bottleneck assignment problem, respectively, and then solve the latter The algorithms run in time which is polynomial in the length of the outcome just-in-time output sequence Chapter proves that there exist cyclic solutions that minimize the total deviation for symmetric point deviation functions, the same is shown for the maximum deviation It also proves that limiting optimization to the sequences with the bottleneck deviation not exceeding renders some functions of point deviation equivalent The oneness property claims that limiting search for optimal just-in-time sequences to those with bottleneck not exceeding will be optimal in general However, the chapter shows that all optimal just-in-time sequences for some instances may have the bottleneck deviation higher than – thus showing that the oneness does not hold generally Chapter gives a more efficient algorithm for the maximum absolute deviation (referred to x Preface as bottleneck) deviation The absolute value function of deviation results in optimal bottleneck being always less than 1, and allows to develop strong upper and lower bounds on the optimal bottleneck These bounds and other properties of the bottleneck optimal just-in-time sequences are used in the application to the Liu–Layland problem, stride scheduling, fair queueing, and others in the subsequent chapters Chapter also shows that the optimal bottleneck just-in-time sequences for n = are in fact Webster’s sequences of apportionment and the most regular words at the same time; thus, they optimize the throughput of any two cyclic process sharing a common resource This new observation underlines again the advantages of the Webster’s sequences for other than apportionment problems Chapter further exploits the properties of just-in-time sequences with small bottleneck deviations, which are understood as those less than 12 The question is what are the instances that admit this small bottleneck deviation? The answer given in the chapter is that there is only one, called the power-of-two instance that results in this small bottleneck deviation for n ≥ The chapter also shows the connection between the small bottleneck deviation problem and the famous Fraenkel’s Conjecture, which states that the only distinct rates for which it is possible to build a balanced word on three or more letters come essentially from the power-of-two instances Finally, the chapter presents the small bottleneck problem in the broader context of regular sequences and multimodular functions they minimize The applications of multimodular functions to workload balancing in event graphs (for instance the queues and supply chains) are also discussed in the chapter Chapter addresses the response time variability minimization problem, where the average response time for a client is a reciprocal of its desirable rate Thus, being as close as possible to the average response time aims at achieving the “as evenly as possible” goal The response time variability is one of the main objectives in stride scheduling as well The chapter shows that the problem is NP-hard, proposes exact and heuristic solutions, and reports computational experiments with the latter Chapter proves that the optimal bottleneck sequences make tasks progress at the rates close enough to the tasks’ processing time to request interval ratios so that they solve the Liu–Layland problem – likely the best known scheduling problem in the hard real-time systems It also gives necessary conditions for the apportionment divisor methods to solve the Liu–Layland problem, and proves that the quota-divisor methods solve the Liu– Layland problem as well Finally, the chapter presents solutions to some special cases of the pinwheel scheduling problem given by the bottleneck optimal justin-time sequences Chapter focuses on the problem of constructing just-in-time sequences for supply chains so that the temporal capacity constraints imposed by suppliers are respected The constraints are modeled by giving the limiting, supplydependent proportions p: q that stipulate that at most p out of any q models delivered by the supply chain must be supplied by a particular supplier Though the problem of finding such a sequence is NP-hard in the strong sense the chapter discusses a number of approaches: synchronized delivery and periodic synchronized delivery for better balancing workloads in supply chains Finally, the chapter points out a potential for using tools developed by the combinatorics on words to design the justin-time sequences having desirable properties, and discusses the class of balanced Preface xi words in this role in more detail Chapter 10 looks into the problem of fairness in fair queueing and stride scheduling It shows that both use the Jefferson’s and Adams’s method of apportionment, and both are peer-to-peer fair However, the chapter also argues that the Webster’s method could prove a better yet untested choice for fair queueing and stride scheduling The chapter gives also a closer look at the measures and criteria typically used in the fair queueing and stride scheduling and analyzes them using the apportionment theory and just-in-time optimization tools developed in Chaps 2, 5, and Finally, Chap 11 extends the models developed in Chaps 2, 3, and to manufacturing environments with variable processing and set-up times This is a departure from the usual assumption of negligible variability resulting in an simplification, often criticized, of unit times and synchronized lines assumed in the applications of just-in-time sequences The chapter’s approach is based on batching to smooth out the variability of processing and set-up times, and then on sequencing the batches to minimize the total deviation or alternatively to gain the advantages of the Webster’s method The approach is applied to a real-life problem arising in an automotive pressure hose manufacturer The computational experiments with both algorithms are also presented in the chapter Special thanks go to my friends and colleagues, listed here in a random order, for ´ their encouragement and support: Prof Dominique de Werra (Ecole Politechnique F´ed´erale de Lausanne), Profs Jan We¸glarz and Jacek Bła˙zewicz (Pozna´n University of Technology), Prof Albert Corominas (Universitat Polit`ecnica de Catalunya), Prof Jacques Cariler (Universit´e de Technologie de Compi`egne), Prof Erwin Pesch (University of Siegen), Prof Moshe Dror (University of Arizona), Prof Gerd Finke (Universit´e Joseph Fourier), and Prof Marek Kubale (Gda´nsk University of Technology) I am indebted in particular to Dr Cynthia Philips (Sandia National Laboratories) and Dr Bruno Gaujal (INRIA-Grenoble) for pointing me to a number of important references Finally, I wish to acknowledge the research support of the Natural Sciences and Engineering Research Council of Canada without which many of my research projects on just-in-time would simply not happen St John’s, Canada Wieslaw Kubiak ... FEASIBILITY AND INFEASIBILITY IN OPTIMIZATION: Algorithms and Computation Methods Tang, Teo & Wei/ SUPPLY CHAIN ANALYSIS: A Handbook on the Interaction of Information, System and Optimization. .. measures and criteria typically used in the fair queueing and stride scheduling and analyzes them using the apportionment theory and just-in-time optimization tools developed in Chaps 2, 5, and Finally,... EVALUATION AND BENCHMARKING: Data Envelopment Analysis with Spreadsheets ∼A list of the early publications in the series is found at the end of the book∼ Wieslaw Kubiak Proportional Optimization and Fairness