Winter 2008 CS567 Stochastic Linear/Integer Programming Guest Lecturer: Xu, Huan
|
|
- Jasmine Cain
- 5 years ago
- Views:
Transcription
1 Wnter 2008 CS567 Stochastc Lnear/Integer Programmng Guest Lecturer: Xu, Huan Class 2: More Modelng Examples 1 Capacty Expanson Capacty expanson models optmal choces of the tmng and levels of nvestments to meet future demands of gven product. Here we consder the case of power expanson for electrcty generaton: to fnd optmal levels of nvestments n varous types of power plants to meet future electrcty demand. We dscuss three modelng here: statc determnstc model, dynamc model and stochastc model. 1.1 Statc Determnstc Model Three propertes of a gven power plant can be sngled out: the nvestment cost r, the operatng cost q and the avalablty factor a whch ndcates the percent of tme the power plant can effectvely be operated. Demands for electrcty can be consdered a sngle product, but the level of demand vares over tme. Analysts usually use load duraton curve to descrbe the demand over tme n decreasng order of demand level. The load duraton curve can be approxmated by a pecewse constant curve wth m segments. Let d 1 = D 1, d j = D j D j 1, j = 2,, m represent the addtonal power demand n mode j wth a tme τ j. In the statc stuaton, the problem conssts of fndng the optmal nvestment foe each mode,.e.., to fnd a partcular plant such that the total cost of effectvely producng electrcty durng the tme τ j : { } r + q τ j (j) = arg mn.,,n The soluton of the statc model captures one essental feature of the problem: base load demand should be covered by equpment wth low operatng costs and peak load demand should be covered by equpment wth low nvestment costs. a 1.2 Dynamc Model At least four elements justfy usng a dynamc or multstage model: 1. the long-term evoluton of equpment costs; 2. the long-term evoluton of the load curve; 3. the appearance of new technologes; 4. the obsolescence of currently avalable equpment. Of sgnfcant mportance s the evoluton of demand n both the total energy demanded, and the peak level whch determnes the total capacty that must be avalable. Hence, the followng multstage model s proposed to handle ths: 1
2 t = 1,, H ndex the perods or stages; = 1,, n ndex the avalable technologes; j = 1,, m ndex the operatng modes n the load duraton curve; a = avalablty factor of ; L = lfe tme of ; g t = exstng capacty of at tme t, decded before t = 1; r t = unt nvestment cost for at tme t; q t = unt producton cost for at tme t; d t j = maxmal power demanded n mode j at tme t; τ t j = duraton of mode j at tme t; x t = new capacty made avalable for technology at tme t; w t = total capacty of avalable at tme t; yj t = capacty of effectvely used at tme t n mode j. The electrcty generaton H-stage problem can be defned as H rw t t + qτ t jy t t j s.t.: mn x,y,w w t = w t 1 + x t x t L, = 1,, n, t = 1,, H, yj t = dt j, j = 1,, m, t = 1,, H, yj t a (g t + w), t = 1,, m, t = 1,, H, x, y, w 0. The objectve functon s the sum of nvestment plus mantenance costs and operatng costs. Compared to the statc model, the factor a goes to the constrant. 1.3 Stochastc Model In contrast to the dynamc model whch assumes that the evolutons are all determnstc, we can also consder them as truly random. Ths leads to the stochastc model. The major dfference here s x t now represent the new capacty of decded at tme t, whch becomes avalable at x t+ ( s the constructon delay). We use boldface to represent random varables: x t = new capacty decded at tme t for equpment ; w t = total capacty of avalable at tme t; 2
3 ξ = the vector of random parameters at tme t; The stochastc model s then: s.t.: mn x,y,w E ξ H r t w t + q t τ t jy t j w t = w t 1 + x t x t L, = 1,, n, t = 1,, H, yj t = dt j, j = 1,, m, t = 1,, H, y t j a (g t + w t ), = 1,, m, t = 1,, H, x, y, w 0. Notce here, the decson w t, y t only depend on the realzaton of the random vector up to tme t, but can not depend on future realzatons of the random vector. That s, t s non-antcpatve. If the decson varable (w t, y t ) were not dependent on ξ t, the objectve functon can be wrtten as E ξ r t w t + E ξ q t τ t yj t = r w t t + (q τ j )y t j. t j t j That s, t becomes a determnstc formulaton. Problem 1 s a multstage stochastc lner program wth a specal property: block separable recourse. Ths property stems from a separaton that can be made between the aggregate level decson (x t, w t ) and the detaled-level decsons y t. Suppose future demands are always ndependent of the past. In ths case, the decson on capacty to nstall n the future at some t only depends on avalable capacty and does not depend on the outcomes up to tme t. The same x t must then be optmal for any realzaton of ξ. The only stochastc decson s n the operaton-level: y t, whch now depends separately on each perod s capacty. Thus, ths mult-perod problem becoms a less complex two-perod problem. Consder the followng example: we have a two-perod three-operatng-mode problem, wth n = 4, = 1, a = 1; g = 0. The only random varable s d 1 = ξ. The other demands are d 2 = 3 and d 3 = 2. The nvestment costs are r 1 = (10, 7, 16, 6) wth producton costs q 2 = (4, 4.5, 3.2, 5.5) and load duraton τ 2 = (10, 6, 1). We also add a budget constrant to keep all nvestment below 120. The resultng stochastc program s: 3 mn : 10x x x E ξ [ τj 2 (4y1j y2j y3j y4j)] 2 s.t.: 10x x x , 3 x 1 + yj 2 0, = 1, 4, y y1 2 = ξ, y yj 2 = d 2 j, j = 2, 3 x, y 0. 3 (1)
4 Assumng that ξ takes on the values 3, 5 and 7 wth probabltes 0.3, 0.4 and 0.3 respectvely, an optmal soluton ncludes x 1 = (2.67, 4.00, 3.33, 2.00) wth optmal value of If we consder the expected parameter formulaton, the optmal soluton s (0.00, ) wth an objectve value 365. However, f we use ths soluton n the stochastc problem, then wth probablty 0.3 the demands are not satsfed. Formulaton of 1 requres that the demand s always satsfed, n practce t s often relaxed to a probablty constrant: n 1 P [ a (g t + w) t D t j] α, t, for some α (0, 1).Ths s often called a chance constrant or probablty constrant n stochastc programmng. Notce ths constrant s equvalent to a determnstc constrant: n 1 a (g t + w) t (F t ) 1 (α), t, where F t s the dstrbuted functon of m dt j. 2 Desgn for Manufacturng Qualty Consder a desgner decdng varous product specfcaton to acheve some measure of product cost and performance. However, the specfcaton may not completely determne the characterstcs of each manufactured product. Key characterstcs of the product are often random. For example, every tem ncludes varatons due to machnng or other processng, and each customer also does not use the product n the same way. Thus, cost and performance become random varables, and stochastc programmng can help n ths case, because determnstc methods can yeld costly results that are only dscovered after producton has begun. In ths secton, we consder the desgn of a smple axle assembly for a bcycle cart. The desgner must determne the specfed length w and the dameter ξ of the axle. Together, these quanttes determne the performance characterstc of the product. The goal s to determne a combnaton to acheve the greatest expected proft. In the producton process, the actual dmensons are not exactly those that are specfed. For ths example, we suppose that the length w can be produced exactly but the dameter ξ s a random varable ξ(x) that depends on a specfed value x that represents, for example the settng of a machne. We assume the followng trangular dstrbuton of ξ(x): (100/x 2 )(ξ 0.9x) f 0.9x ξ x, f x (ξ) = (100/x 2 )(1.1x ξ) f x ξ 1.1x, 0 otherwse. The decson s to determne w and x subject to w w max and x x max, n order to maxmze the expected utlty. Further assume that we sell as many as we can produce under the maxmum sellng prce, and the maxmum sellng prce depends on the length and s expressed as r(1 e 0.1w ), 4
5 where r s the maxmum possble for any such products. Our producton costs for labor and equpment are assumed to be fxed, so only materal cost s varable. Ths cost s proportonal to the mean value of the specfed dmenson because materal s acqured before the producton process. That s, c(wπx 2 )/4, for some unt cost c. We have certan qualty constrant: w ξ , w 3 ξ When ether of these constrants s volated, the axle deforms, whch ncurs a qualty cost proportonal to the square of the volaton. That s: Q(w, x, ξ) = mn y {qy 2 w/ξ 3 y 39.27; w 3 /ξ 4 300y 63169}, and the expected qualty cost gven w and x s Q(w, x) = Q(w, x, ξ)f x (ξ)dξ =q 1.1x 0.9x The overall problem s to fnd ξ (100/x 2 ) mn{ξ 0.9x, 1.1x ξ}[max{0, w/ξ , w 3 /300ξ }] 2 dξ. max (total revenue er tem - manufacturng cost per tem -expected qualty cost). Mathematcally, we wrte ths as max z(w, x) = r(1 e 0.1w ) c(wπx 2 )/4 Q(w, x) s.t. 0 w w max, 0 x x max. Let w max = 36, x max = 1.25, r = 10, c = 0.025, q = 1. The optmal soluton s w = 33.6 and x = where the objectve functon s If we consder the expected parameter formulaton, we get a soluton w = and x = 0.963, wth the optmal value It seems that the expected proft s even better than the stochastc soluton. That s because, the optmal value usng the expected parameter overestmate the true expected proft. Substtute ths soluton nto the stochastc formulaton, the expected proft s Hence we see that the VSS s
Amiri s Supply Chain Model. System Engineering b Department of Mathematics and Statistics c Odette School of Business
Amr s Supply Chan Model by S. Ashtab a,, R.J. Caron b E. Selvarajah c a Department of Industral Manufacturng System Engneerng b Department of Mathematcs Statstcs c Odette School of Busness Unversty of
More informationOnline Appendix. t=1 (p t w)q t. Then the first order condition shows that
Artcle forthcomng to ; manuscrpt no (Please, provde the manuscrpt number!) 1 Onlne Appendx Appendx E: Proofs Proof of Proposton 1 Frst we derve the equlbrum when the manufacturer does not vertcally ntegrate
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationk t+1 + c t A t k t, t=0
Macro II (UC3M, MA/PhD Econ) Professor: Matthas Kredler Fnal Exam 6 May 208 You have 50 mnutes to complete the exam There are 80 ponts n total The exam has 4 pages If somethng n the queston s unclear,
More informationPortfolios with Trading Constraints and Payout Restrictions
Portfolos wth Tradng Constrants and Payout Restrctons John R. Brge Northwestern Unversty (ont wor wth Chrs Donohue Xaodong Xu and Gongyun Zhao) 1 General Problem (Very) long-term nvestor (eample: unversty
More informationAssortment Optimization under MNL
Assortment Optmzaton under MNL Haotan Song Aprl 30, 2017 1 Introducton The assortment optmzaton problem ams to fnd the revenue-maxmzng assortment of products to offer when the prces of products are fxed.
More informationEEL 6266 Power System Operation and Control. Chapter 3 Economic Dispatch Using Dynamic Programming
EEL 6266 Power System Operaton and Control Chapter 3 Economc Dspatch Usng Dynamc Programmng Pecewse Lnear Cost Functons Common practce many utltes prefer to represent ther generator cost functons as sngle-
More informationAbsorbing Markov Chain Models to Determine Optimum Process Target Levels in Production Systems with Rework and Scrapping
Archve o SID Journal o Industral Engneerng 6(00) -6 Absorbng Markov Chan Models to Determne Optmum Process Target evels n Producton Systems wth Rework and Scrappng Mohammad Saber Fallah Nezhad a, Seyed
More informationComputing Correlated Equilibria in Multi-Player Games
Computng Correlated Equlbra n Mult-Player Games Chrstos H. Papadmtrou Presented by Zhanxang Huang December 7th, 2005 1 The Author Dr. Chrstos H. Papadmtrou CS professor at UC Berkley (taught at Harvard,
More informationLecture 14: Bandits with Budget Constraints
IEOR 8100-001: Learnng and Optmzaton for Sequental Decson Makng 03/07/16 Lecture 14: andts wth udget Constrants Instructor: Shpra Agrawal Scrbed by: Zhpeng Lu 1 Problem defnton In the regular Mult-armed
More informationSimultaneous Optimization of Berth Allocation, Quay Crane Assignment and Quay Crane Scheduling Problems in Container Terminals
Smultaneous Optmzaton of Berth Allocaton, Quay Crane Assgnment and Quay Crane Schedulng Problems n Contaner Termnals Necat Aras, Yavuz Türkoğulları, Z. Caner Taşkın, Kuban Altınel Abstract In ths work,
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationThe Second Anti-Mathima on Game Theory
The Second Ant-Mathma on Game Theory Ath. Kehagas December 1 2006 1 Introducton In ths note we wll examne the noton of game equlbrum for three types of games 1. 2-player 2-acton zero-sum games 2. 2-player
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationEconomics 101. Lecture 4 - Equilibrium and Efficiency
Economcs 0 Lecture 4 - Equlbrum and Effcency Intro As dscussed n the prevous lecture, we wll now move from an envronment where we looed at consumers mang decsons n solaton to analyzng economes full of
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationThe Value of Demand Postponement under Demand Uncertainty
Recent Researches n Appled Mathematcs, Smulaton and Modellng The Value of emand Postponement under emand Uncertanty Rawee Suwandechocha Abstract Resource or capacty nvestment has a hgh mpact on the frm
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationECE559VV Project Report
ECE559VV Project Report (Supplementary Notes Loc Xuan Bu I. MAX SUM-RATE SCHEDULING: THE UPLINK CASE We have seen (n the presentaton that, for downlnk (broadcast channels, the strategy maxmzng the sum-rate
More informationLecture 10 Support Vector Machines. Oct
Lecture 10 Support Vector Machnes Oct - 20-2008 Lnear Separators Whch of the lnear separators s optmal? Concept of Margn Recall that n Perceptron, we learned that the convergence rate of the Perceptron
More informationOn the Multicriteria Integer Network Flow Problem
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 5, No 2 Sofa 2005 On the Multcrtera Integer Network Flow Problem Vassl Vasslev, Marana Nkolova, Maryana Vassleva Insttute of
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationBasic Statistical Analysis and Yield Calculations
October 17, 007 Basc Statstcal Analyss and Yeld Calculatons Dr. José Ernesto Rayas Sánchez 1 Outlne Sources of desgn-performance uncertanty Desgn and development processes Desgn for manufacturablty A general
More informationFUZZY FINITE ELEMENT METHOD
FUZZY FINITE ELEMENT METHOD RELIABILITY TRUCTURE ANALYI UING PROBABILITY 3.. Maxmum Normal tress Internal force s the shear force, V has a magntude equal to the load P and bendng moment, M. Bendng moments
More informationA NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegian Business School 2011
A NOTE ON CES FUNCTIONS Drago Bergholt, BI Norwegan Busness School 2011 Functons featurng constant elastcty of substtuton CES are wdely used n appled economcs and fnance. In ths note, I do two thngs. Frst,
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationA SEPARABLE APPROXIMATION DYNAMIC PROGRAMMING ALGORITHM FOR ECONOMIC DISPATCH WITH TRANSMISSION LOSSES. Pierre HANSEN, Nenad MLADENOVI]
Yugoslav Journal of Operatons Research (00) umber 57-66 A SEPARABLE APPROXIMATIO DYAMIC PROGRAMMIG ALGORITHM FOR ECOOMIC DISPATCH WITH TRASMISSIO LOSSES Perre HASE enad MLADEOVI] GERAD and Ecole des Hautes
More informationCHAPTER 7 STOCHASTIC ECONOMIC EMISSION DISPATCH-MODELED USING WEIGHTING METHOD
90 CHAPTER 7 STOCHASTIC ECOOMIC EMISSIO DISPATCH-MODELED USIG WEIGHTIG METHOD 7.1 ITRODUCTIO early 70% of electrc power produced n the world s by means of thermal plants. Thermal power statons are the
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationAnnexes. EC.1. Cycle-base move illustration. EC.2. Problem Instances
ec Annexes Ths Annex frst llustrates a cycle-based move n the dynamc-block generaton tabu search. It then dsplays the characterstcs of the nstance sets, followed by detaled results of the parametercalbraton
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationAn Admission Control Algorithm in Cloud Computing Systems
An Admsson Control Algorthm n Cloud Computng Systems Authors: Frank Yeong-Sung Ln Department of Informaton Management Natonal Tawan Unversty Tape, Tawan, R.O.C. ysln@m.ntu.edu.tw Yngje Lan Management Scence
More informationCournot Equilibrium with N firms
Recap Last class (September 8, Thursday) Examples of games wth contnuous acton sets Tragedy of the commons Duopoly models: ournot o class on Sept. 13 due to areer Far Today (September 15, Thursday) Duopoly
More informationCOS 521: Advanced Algorithms Game Theory and Linear Programming
COS 521: Advanced Algorthms Game Theory and Lnear Programmng Moses Charkar February 27, 2013 In these notes, we ntroduce some basc concepts n game theory and lnear programmng (LP). We show a connecton
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationUncertainty in measurements of power and energy on power networks
Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationDynamic Programming. Lecture 13 (5/31/2017)
Dynamc Programmng Lecture 13 (5/31/2017) - A Forest Thnnng Example - Projected yeld (m3/ha) at age 20 as functon of acton taken at age 10 Age 10 Begnnng Volume Resdual Ten-year Volume volume thnned volume
More informationANSWERS. Problem 1. and the moment generating function (mgf) by. defined for any real t. Use this to show that E( U) var( U)
Econ 413 Exam 13 H ANSWERS Settet er nndelt 9 deloppgaver, A,B,C, som alle anbefales å telle lkt for å gøre det ltt lettere å stå. Svar er gtt . Unfortunately, there s a prntng error n the hnt of
More informationxp(x µ) = 0 p(x = 0 µ) + 1 p(x = 1 µ) = µ
CSE 455/555 Sprng 2013 Homework 7: Parametrc Technques Jason J. Corso Computer Scence and Engneerng SUY at Buffalo jcorso@buffalo.edu Solutons by Yngbo Zhou Ths assgnment does not need to be submtted and
More informationSingle-Facility Scheduling over Long Time Horizons by Logic-based Benders Decomposition
Sngle-Faclty Schedulng over Long Tme Horzons by Logc-based Benders Decomposton Elvn Coban and J. N. Hooker Tepper School of Busness, Carnege Mellon Unversty ecoban@andrew.cmu.edu, john@hooker.tepper.cmu.edu
More informationInteractive Bi-Level Multi-Objective Integer. Non-linear Programming Problem
Appled Mathematcal Scences Vol 5 0 no 65 3 33 Interactve B-Level Mult-Objectve Integer Non-lnear Programmng Problem O E Emam Department of Informaton Systems aculty of Computer Scence and nformaton Helwan
More informationCS 2750 Machine Learning. Lecture 5. Density estimation. CS 2750 Machine Learning. Announcements
CS 750 Machne Learnng Lecture 5 Densty estmaton Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square CS 750 Machne Learnng Announcements Homework Due on Wednesday before the class Reports: hand n before
More informationCOMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
Avalable onlne at http://sck.org J. Math. Comput. Sc. 3 (3), No., 6-3 ISSN: 97-537 COMPARISON OF SOME RELIABILITY CHARACTERISTICS BETWEEN REDUNDANT SYSTEMS REQUIRING SUPPORTING UNITS FOR THEIR OPERATIONS
More informationA Simple Inventory System
A Smple Inventory System Lawrence M. Leems and Stephen K. Park, Dscrete-Event Smulaton: A Frst Course, Prentce Hall, 2006 Hu Chen Computer Scence Vrgna State Unversty Petersburg, Vrgna February 8, 2017
More informationCapacity Constraints Across Nests in Assortment Optimization Under the Nested Logit Model
Capacty Constrants Across Nests n Assortment Optmzaton Under the Nested Logt Model Jacob B. Feldman School of Operatons Research and Informaton Engneerng, Cornell Unversty, Ithaca, New York 14853, USA
More informationPrice competition with capacity constraints. Consumers are rationed at the low-price firm. But who are the rationed ones?
Prce competton wth capacty constrants Consumers are ratoned at the low-prce frm. But who are the ratoned ones? As before: two frms; homogeneous goods. Effcent ratonng If p < p and q < D(p ), then the resdual
More informationWhich Separator? Spring 1
Whch Separator? 6.034 - Sprng 1 Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng Whch Separator? Mamze the margn to closest ponts 6.034 - Sprng 3 Margn of a pont " # y (w $ + b) proportonal
More informationf(x,y) = (4(x 2 4)x,2y) = 0 H(x,y) =
Problem Set 3: Unconstraned mzaton n R N. () Fnd all crtcal ponts of f(x,y) (x 4) +y and show whch are ma and whch are mnma. () Fnd all crtcal ponts of f(x,y) (y x ) x and show whch are ma and whch are
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationMarkov Chain Monte Carlo Lecture 6
where (x 1,..., x N ) X N, N s called the populaton sze, f(x) f (x) for at least one {1, 2,..., N}, and those dfferent from f(x) are called the tral dstrbutons n terms of mportance samplng. Dfferent ways
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationCIE4801 Transportation and spatial modelling Trip distribution
CIE4801 ransportaton and spatal modellng rp dstrbuton Rob van Nes, ransport & Plannng 17/4/13 Delft Unversty of echnology Challenge the future Content What s t about hree methods Wth specal attenton for
More informationNewton s Method for One - Dimensional Optimization - Theory
Numercal Methods Newton s Method for One - Dmensonal Optmzaton - Theory For more detals on ths topc Go to Clck on Keyword Clck on Newton s Method for One- Dmensonal Optmzaton You are free to Share to copy,
More informationReal-Time Systems. Multiprocessor scheduling. Multiprocessor scheduling. Multiprocessor scheduling
Real-Tme Systems Multprocessor schedulng Specfcaton Implementaton Verfcaton Multprocessor schedulng -- -- Global schedulng How are tasks assgned to processors? Statc assgnment The processor(s) used for
More informationForesighted Demand Side Management
Foresghted Demand Sde Management 1 Yuanzhang Xao and Mhaela van der Schaar, Fellow, IEEE Department of Electrcal Engneerng, UCLA. {yxao,mhaela}@ee.ucla.edu. Abstract arxv:1401.2185v1 [cs.ma] 9 Jan 2014
More informationTechnical Note: Capacity Constraints Across Nests in Assortment Optimization Under the Nested Logit Model
Techncal Note: Capacty Constrants Across Nests n Assortment Optmzaton Under the Nested Logt Model Jacob B. Feldman, Huseyn Topaloglu School of Operatons Research and Informaton Engneerng, Cornell Unversty,
More informationP R. Lecture 4. Theory and Applications of Pattern Recognition. Dept. of Electrical and Computer Engineering /
Theory and Applcatons of Pattern Recognton 003, Rob Polkar, Rowan Unversty, Glassboro, NJ Lecture 4 Bayes Classfcaton Rule Dept. of Electrcal and Computer Engneerng 0909.40.0 / 0909.504.04 Theory & Applcatons
More information8. Modelling Uncertainty
8. Modellng Uncertanty. Introducton. Generatng Values From Known Probablty Dstrbutons. Monte Carlo Smulaton 4. Chance Constraned Models 5 5. Markov Processes and Transton Probabltes 6 6. Stochastc Optmzaton
More informationPricing and Resource Allocation Game Theoretic Models
Prcng and Resource Allocaton Game Theoretc Models Zhy Huang Changbn Lu Q Zhang Computer and Informaton Scence December 8, 2009 Z. Huang, C. Lu, and Q. Zhang (CIS) Game Theoretc Models December 8, 2009
More informationSampling Theory MODULE VII LECTURE - 23 VARYING PROBABILITY SAMPLING
Samplng heory MODULE VII LECURE - 3 VARYIG PROBABILIY SAMPLIG DR. SHALABH DEPARME OF MAHEMAICS AD SAISICS IDIA ISIUE OF ECHOLOGY KAPUR he smple random samplng scheme provdes a random sample where every
More informationMin Cut, Fast Cut, Polynomial Identities
Randomzed Algorthms, Summer 016 Mn Cut, Fast Cut, Polynomal Identtes Instructor: Thomas Kesselhem and Kurt Mehlhorn 1 Mn Cuts n Graphs Lecture (5 pages) Throughout ths secton, G = (V, E) s a mult-graph.
More information8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars
More informationPh 219a/CS 219a. Exercises Due: Wednesday 23 October 2013
1 Ph 219a/CS 219a Exercses Due: Wednesday 23 October 2013 1.1 How far apart are two quantum states? Consder two quantum states descrbed by densty operators ρ and ρ n an N-dmensonal Hlbert space, and consder
More informationLecture 7: Boltzmann distribution & Thermodynamics of mixing
Prof. Tbbtt Lecture 7 etworks & Gels Lecture 7: Boltzmann dstrbuton & Thermodynamcs of mxng 1 Suggested readng Prof. Mark W. Tbbtt ETH Zürch 13 März 018 Molecular Drvng Forces Dll and Bromberg: Chapters
More informationChapter Newton s Method
Chapter 9. Newton s Method After readng ths chapter, you should be able to:. Understand how Newton s method s dfferent from the Golden Secton Search method. Understand how Newton s method works 3. Solve
More informationRyan (2009)- regulating a concentrated industry (cement) Firms play Cournot in the stage. Make lumpy investment decisions
1 Motvaton Next we consder dynamc games where the choce varables are contnuous and/or dscrete. Example 1: Ryan (2009)- regulatng a concentrated ndustry (cement) Frms play Cournot n the stage Make lumpy
More informationResource Allocation with a Budget Constraint for Computing Independent Tasks in the Cloud
Resource Allocaton wth a Budget Constrant for Computng Independent Tasks n the Cloud Wemng Sh and Bo Hong School of Electrcal and Computer Engneerng Georga Insttute of Technology, USA 2nd IEEE Internatonal
More informationCSC 411 / CSC D11 / CSC C11
18 Boostng s a general strategy for learnng classfers by combnng smpler ones. The dea of boostng s to take a weak classfer that s, any classfer that wll do at least slghtly better than chance and use t
More informationNorm Bounds for a Transformed Activity Level. Vector in Sraffian Systems: A Dual Exercise
ppled Mathematcal Scences, Vol. 4, 200, no. 60, 2955-296 Norm Bounds for a ransformed ctvty Level Vector n Sraffan Systems: Dual Exercse Nkolaos Rodousaks Department of Publc dmnstraton, Panteon Unversty
More informationDO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR. Introductory Econometrics 1 hour 30 minutes
25/6 Canddates Only January Examnatons 26 Student Number: Desk Number:...... DO NOT OPEN THE QUESTION PAPER UNTIL INSTRUCTED TO DO SO BY THE CHIEF INVIGILATOR Department Module Code Module Ttle Exam Duraton
More information1 Derivation of Rate Equations from Single-Cell Conductance (Hodgkin-Huxley-like) Equations
Physcs 171/271 -Davd Klenfeld - Fall 2005 (revsed Wnter 2011) 1 Dervaton of Rate Equatons from Sngle-Cell Conductance (Hodgkn-Huxley-lke) Equatons We consder a network of many neurons, each of whch obeys
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More informationDUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More informationResearch Article Green s Theorem for Sign Data
Internatonal Scholarly Research Network ISRN Appled Mathematcs Volume 2012, Artcle ID 539359, 10 pages do:10.5402/2012/539359 Research Artcle Green s Theorem for Sgn Data Lous M. Houston The Unversty of
More informationECE 534: Elements of Information Theory. Solutions to Midterm Exam (Spring 2006)
ECE 534: Elements of Informaton Theory Solutons to Mdterm Eam (Sprng 6) Problem [ pts.] A dscrete memoryless source has an alphabet of three letters,, =,, 3, wth probabltes.4,.4, and., respectvely. (a)
More informationHila Etzion. Min-Seok Pang
RESERCH RTICLE COPLEENTRY ONLINE SERVICES IN COPETITIVE RKETS: INTINING PROFITILITY IN THE PRESENCE OF NETWORK EFFECTS Hla Etzon Department of Technology and Operatons, Stephen. Ross School of usness,
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationUniversity of California, Davis Date: June 22, 2009 Department of Agricultural and Resource Economics. PRELIMINARY EXAMINATION FOR THE Ph.D.
Unversty of Calforna, Davs Date: June 22, 29 Department of Agrcultural and Resource Economcs Department of Economcs Tme: 5 hours Mcroeconomcs Readng Tme: 2 mnutes PRELIMIARY EXAMIATIO FOR THE Ph.D. DEGREE
More informationThe optimal delay of the second test is therefore approximately 210 hours earlier than =2.
THE IEC 61508 FORMULAS 223 The optmal delay of the second test s therefore approxmately 210 hours earler than =2. 8.4 The IEC 61508 Formulas IEC 61508-6 provdes approxmaton formulas for the PF for smple
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More informationCredit Card Pricing and Impact of Adverse Selection
Credt Card Prcng and Impact of Adverse Selecton Bo Huang and Lyn C. Thomas Unversty of Southampton Contents Background Aucton model of credt card solctaton - Errors n probablty of beng Good - Errors n
More informationThermal-Fluids I. Chapter 18 Transient heat conduction. Dr. Primal Fernando Ph: (850)
hermal-fluds I Chapter 18 ransent heat conducton Dr. Prmal Fernando prmal@eng.fsu.edu Ph: (850) 410-6323 1 ransent heat conducton In general, he temperature of a body vares wth tme as well as poston. In
More informationIntroduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:
CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and
More informationAn Interactive Optimisation Tool for Allocation Problems
An Interactve Optmsaton ool for Allocaton Problems Fredr Bonäs, Joam Westerlund and apo Westerlund Process Desgn Laboratory, Faculty of echnology, Åbo Aadem Unversty, uru 20500, Fnland hs paper presents
More informationx = , so that calculated
Stat 4, secton Sngle Factor ANOVA notes by Tm Plachowsk n chapter 8 we conducted hypothess tests n whch we compared a sngle sample s mean or proporton to some hypotheszed value Chapter 9 expanded ths to
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More informationLecture Randomized Load Balancing strategies and their analysis. Probability concepts include, counting, the union bound, and Chernoff bounds.
U.C. Berkeley CS273: Parallel and Dstrbuted Theory Lecture 1 Professor Satsh Rao August 26, 2010 Lecturer: Satsh Rao Last revsed September 2, 2010 Lecture 1 1 Course Outlne We wll cover a samplng of the
More information1. Inference on Regression Parameters a. Finding Mean, s.d and covariance amongst estimates. 2. Confidence Intervals and Working Hotelling Bands
Content. Inference on Regresson Parameters a. Fndng Mean, s.d and covarance amongst estmates.. Confdence Intervals and Workng Hotellng Bands 3. Cochran s Theorem 4. General Lnear Testng 5. Measures of
More informationSolution (1) Formulate the problem as a LP model.
Benha Unversty Department: Mechancal Engneerng Benha Hgh Insttute of Technology Tme: 3 hr. January 0 -Fall semester 4 th year Eam(Regular) Soluton Subject: Industral Engneerng M4 ------------------------------------------------------------------------------------------------------.
More informationLOW BIAS INTEGRATED PATH ESTIMATORS. James M. Calvin
Proceedngs of the 007 Wnter Smulaton Conference S G Henderson, B Bller, M-H Hseh, J Shortle, J D Tew, and R R Barton, eds LOW BIAS INTEGRATED PATH ESTIMATORS James M Calvn Department of Computer Scence
More informationThe Feynman path integral
The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space
More informationFinite Element Modelling of truss/cable structures
Pet Schreurs Endhoven Unversty of echnology Department of Mechancal Engneerng Materals echnology November 3, 214 Fnte Element Modellng of truss/cable structures 1 Fnte Element Analyss of prestressed structures
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationThe Minimum Universal Cost Flow in an Infeasible Flow Network
Journal of Scences, Islamc Republc of Iran 17(2): 175-180 (2006) Unversty of Tehran, ISSN 1016-1104 http://jscencesutacr The Mnmum Unversal Cost Flow n an Infeasble Flow Network H Saleh Fathabad * M Bagheran
More informationAssortment Optimization under the Paired Combinatorial Logit Model
Assortment Optmzaton under the Pared Combnatoral Logt Model Heng Zhang, Paat Rusmevchentong Marshall School of Busness, Unversty of Southern Calforna, Los Angeles, CA 90089 hengz@usc.edu, rusmevc@marshall.usc.edu
More information