Dynamic Cost Allocation for Economic Lot Sizing Games
|
|
- Sherilyn Holt
- 6 years ago
- Views:
Transcription
1 Dynamc Cost Allocaton for Economc Lot Szng Games Alejandro Torello H. Mlton Stewart School of Industral and Systems Engneerng Georga Insttute of Technology Atlanta, GA elson A. Uhan Mathematcs Department Unted States aval Academy Annapols, MD ovember 6, 2013 Abstract We consder a cooperatve game defned by an economc lot szng problem wth concave orderng costs over a fnte tme horzon, n whch each player faces demand for a sngle product n each perod and coaltons can pool orders. We show how to compute a dynamc cost allocaton n the strong sequental core of ths game,.e. an allocaton over tme that exactly dstrbutes costs and s stable aganst coaltonal defectons at every perod of the tme horzon. 1 Introducton and Motvaton Producton and nventory management are areas n whch cooperaton among ndependent agents has an ntutve appeal. For example, ndependent retalers sharng a warehouse may want to combne orders from a common suppler to enjoy economes of scale derved from larger order quanttes. Vewed from a system-wde perspectve, combnng orders lowers the total cost, but the order consoldaton cannot be acheved unless the ndependent retalers can agree on a far way to splt ths cost. Over the last decade or more, cooperatve game theory research n producton, dstrbuton and nventory models has endeavored to determne far cost allocatons under a varety of settngs. The economc lot szng problem s one of the canoncal producton and nventory models studed n operatons research and management scence, almost snce the feld s ncepton Wagner and Whtn In a cooperatve settng, ths model would capture the stuaton descrbed above, wth each retaler facng ts own demand over the plannng horzon that must be satsfed wth product orders or nventory, but wth coaltons of retalers beng able to order product together. The cooperatve game derved from ths model was proposed n van den Heuvel et al and subsequently studed n Chen and Zhang 2006, Gopaladeskan et al However, all of these papers approach the problem from a statc perspectve; that s, the authors assume an optmal soluton s mplemented over the entre horzon, and seek an up-front statc cost allocaton n the core of ths cooperatve game. 1
2 Unfortunately, a statc cost allocaton even one n the core suffers from a number of sgnfcant drawbacks. Frst, t assumes each retaler would be able to cover ts porton of the entre plannng horzon s cost up front, a sgnfcant fnancal burden that s unrealstc n many settngs. Second, once the allocated costs are collected from the partcpatng retalers, there may be ncentves for ndvdual retalers or coaltons to defect later on n the plannng horzon f they fnd themselves n an advantageous poston. Fnally, a statc allocaton gnores the typcal rollng horzon approach to these models, n whch one or a few perods solutons are mplemented, and then a new model s formulated wtpdated parameters. Wthn a rollng horzon approach, t s unclear how a statc allocaton could be mplemented. Our goal n ths paper s to develop dynamc cost allocatons for the economc lot szng problem. Unlke ther statc counterparts, dynamc allocatons can be constraned to allocate costs as they are ncurred. They can also be desgned so that no coalton ever has an ncentve to defect throughout the entre plannng horzon. Furthermore, because the costs are allocated as they are ncurred, dynamc allocatons can also be ncorporated nto a rollng horzon framework. Our allocaton depends conceptually on extendng the core concept to dynamc settngs e.g. see Kranch et al. 2005, and contnues our work n Torello and Uhan 2013 for mult-perod models wth lnear costs. Cooperatve game theory has been successfully appled n many related models, and we cannot hope to adequately revew them here. Instead, we refer the nterested reader to Cachon and etessne 2006, agarajan and Sošć Two n-depth references on lot szng and more general producton plannng and nventory management are Pochet and Wolsey 2006, Zpkn Ths paper has two remanng sectons. In Secton 2, we ntroduce the model and revew some relevant results. Then, n Secton 3, we defne the strong sequental core, the dynamc verson of the core, and show how to compute such a dynamc allocaton. 2 The Economc Lot Szng Game We consder the followng determnstc economc lot szng settng wth multple retalers. A set of retalers : {1,..., n} faces determnstc demand for a sngle product over a fnte dscrete tme horzon. In partcular, each retaler needs to satsfy demand d t n perods t r,..., T ; we start n perod r nstead of perod 1 because we wll eventually vary the startng perod. In every perod t r,..., T, each retaler can order x unts at a total cost of c t x; the functon c t s concave and nondecreasng wth c t 0 0. Each retaler can also hold one unt of product n nventory n perod t r,..., T at a cost of h t. Fnally, each retaler has an ntal nventory of ŝ r 1. For notatonal convenence, we defne d [j,k] : k tj d t for, r j k T, ŝ r 1 : ŝ r 1. Here and throughout, we take a summaton n whch the ntal ndex s greater than the endng ndex to be vacuous and equal to zero; for example, d [j,j 1] 0. The economc lot szng problem for a subset of retalers seeks to satsfy the demands d r,..., dt wth ntal nventory ŝr 1 n a way that mnmzes the total orderng and nventory cost. The economc lot szng game s a transferable utlty cooperatve game, f r, ŝ r 1 n whch each retaler corresponds to a player, and the cost f r, ŝ r 1 to a subset of players s the optmal value of ts economc lot szng problem. Followng standard termnology n the cooperatve game theory lterature, we refer to a subset of players as a coalton, and the set of all players as the grand coalton. 2
3 It s well-known that any nstance of the economc lot szng problem can be transformed nto an equvalent nstance wth zero nventory by usng the ntal nventory to greedly satsfy demand n the begnnng tme perods Zabel It s also well-known that when the ntal nventory s zero, there exsts an optmal soluton that satsfes the zero-nventory property n whch orders only occur n perods when the nventory level s zero Wagner and Whtn 1958, Wagner Therefore, when ntal nventores are of the form ŝ r 1 d [r,τ 1] for for some τ r, we can model the economc lot szng problem for coalton wth the followng lnear program Chen and Zhang 2006: τ 1 f r, ŝ r 1 h t d [t1,τ 1] tr mn x s.t. T T [ jτ kj t jr kt c j d [j,k] k tj ] h t d [t1,k] x jk 1a T x jk 1 for t τ,..., T, 1b x jk 0 for j, k : τ j k T, 1c where x jk s a decson varable that ndcates an order n perod j to meet demands n perods j,..., k, for all j, k such that τ j k T. Although the decson varables are contnuous, ths nterpretaton s well-defned: Chen and Zhang 2006 showed that there always exsts an optmal soluton to 1 such that x jk {0, 1} for all j, k. The frst term of the objectve 1a s the cost of greedly usng the ntal nventory ŝ r 1 to satsfy demand n perods r,..., τ 1, and the coeffcent of x jk n the second term of the objectve s the cost of satsfyng demand n perods j,..., k wth an order n perod j. The constrants 1b ensure that demand s satsfed n each perod τ,..., T. We slghtly abuse notaton and use f r, ŝ r 1 to refer to the model 1 as well as ts optmal value. The dual of f r, ŝ r 1 s τ 1 f r, ŝ r 1 h t d [t1,τ 1] tr max α s.t. T d t α t tτ k tj d t α t c j d [j,k] k tj h t d [t1,k] for j, k : τ j k T. 2a 2b Gopaladeskan et al proposed a polynomal-tme combnatoral algorthm that solves f r, ŝ r 1 and ts dual actually, an affne transformaton of ts dual, and used the prmal and dual optmal solutons n conjuncton wth some structural propertes establshed by Chen and Zhang 2006 to compute an allocaton n the core of the economc lot szng game, f r, ŝ r 1. We summarze these results n the lemma below. Lemma 1 Chen and Zhang 2006, Gopaladeskan et al Let x and α respectvely be optmal solutons to f r, s r 1 and ts dual, obtaned usng the algorthm by Gopaladeskan et al Then, x and α satsfy the followng propertes: a α t h t1 α t1 for t τ,..., T. 3
4 b For all d d r,..., d T such that d t dt for t r,..., T, k tj d t α t c j d [j,k] k tj h t d [t1,k] for j, k : τ j k T. c There exst replenshment ntervals {a 1, b 1,..., a m, b m } such that a 1 τ, b m T, a l b l and b l 1 a l1 for all l 1,..., m, and x a l b l 1 for l 1,..., m, d t α [al,bl] t c al d 3 Dynamc Cost Allocaton b l h t d [t1,bl] for l 1,..., m. ow suppose that the tme horzon starts n perod r 1, wth each player havng zero ntal nventory, and the players agree to mplement an optmal soluton x to f 1, 0 wth correspondng dual optmal soluton α and replenshment ntervals {a 1, b 1,..., a m, b m } wth a 1 1 and b m T, obtaned by the algorthm of Gopaladeskan et al In prevous approaches to economc lot szng games Chen and Zhang 2006, Gopaladeskan et al. 2012, van den Heuvel et al. 2007, the authors assume the entre cost of the optmal soluton s allocated up front at the start of the horzon. Instead, we make the more realstc assumpton that orderng costs must be allocated n the same perod an order takes place, and holdng costs for each perod must be allocated n that same perod. Fnally, we assume the players agree that at any pont n tme, each player owns the nventory that was ordered to meet ts own demand; n other words, for each l 1,..., m, the nventory of player s ŝ a l 1 0, 3a ŝ r 1 d [r,b l] for r a l 1,..., b l. 3b Defnton 2 Kranch et al. 2005, Torello and Uhan In a dynamc allocaton χ χ t,t1,...,t, player s allocated a cost of χ t n perod t. The strong sequental core of the economc lot szng game, f r r1,...,t s the set of dynamc allocatons χ that satsfy the followng condtons for some optmal soluton x to f 1, 0: a Stage-wse effcency: For each l 1,..., m, χ a l [a c al d l,b l ] hal d [a l1,b l ], 4a χ t h t d [t1,b l] for t a l 1,..., b l. 4b In other words, the costs ncurred n each perod must be fully dstrbuted among the grand coalton. b Tme-consstent stablty: For each perod r a l,..., b l for some l 1,..., m, tr T χ t f r, ŝ r 1 for, 4c 4
5 where the ntal nventores ŝ r 1 are as defned n 3. In other words, at any pont n the tme horzon, the cost allocated to any coalton from that pont forward does not exceed ts cost f t abandons the grand coalton and contnues on ts own. We defne the dynamc allocaton ˆχ as follows: ˆχ a l d t ˆχ t h t d [t1,b l] t 1 ua l h al d [a l1,b l ] for t a l 1,..., b l for l 1,..., m. 5 Theorem 3. The dynamc allocaton ˆχ defned n 5 s n the strong sequental core of the economc lot szng game, f r r1,...,t. Proof. Frst, we show that ˆχ s stage-wse effcent. For any l 1,..., m, ˆχ a l [ bl d t d t d t d t d t c al d t [al,b l ] t 1 t 1 ua l ua l t 1 ] h al d [a l1,b l ] h al d [a l1,b l ] d t h al d [al1,bl] ua l ut1 h t d u h al d [a l1,b l ] h t d [t1,bl] h al d [a l1,b l ] hal d [a l1,b l ], where follows from Lemma 1c. In addton, we have for all l 1,..., m: ˆχ t h t d [t1,b l] h t d [t1,b l] for t a l 1,..., b l. ext, we show that ˆχ s tme-consstently stable. Fx r {a p 1,...,, a p1 }, for some p {0, 1,..., m} gnorng a 0 1,..., b 0 and a m1. For any, we have that tr T ˆχ t tr tr tr h t d [t1,bp] h t d [t1,bp] h t d [t1,bp] m lp1 m lp1 m lp1 bl d t bl d t bl d t t 1 ua l t 1 ua l d t ut1 h t d [t1,bl] h t d u h t d [t1,bl] ut1 h t d u 5
6 tr h t d [t1,bp] f r, ŝ r 1 m lp1 d t α t where holds because Lemma 1b mples that α s a dual feasble soluton for f r, ŝ r 1. Acknowledgements The authors work was partally supported by the U.S. atonal Scence Foundaton va grant CMMI eferences G.P. Cachon and S. etessne. Game theory n supply chan analyss. In Tutorals n Operatons esearch: Models, Methods, and Applcatons for Innovatve Decson Makng, pages IFOMS, X. Chen and J. Zhang. Dualty approaches to economc lot szng games. IOMS: Operatons Management Workng Paper OM , Stern School of Busness, ew York Unversty, M. Gopaladeskan,.A. Uhan, and J. Zou. A prmal-dual algorthm for computng a cost allocaton n the core of economc lot-szng games. Operatons esearch Letters, 406: , L. Kranch, A. Perea, and H. Peters. Core concepts for dynamc TU games. Internatonal Game Theory evew, 7:43 61, M. agarajan and G. Sošć. Game-theoretc analyss of cooperaton among supply chan agents: evew and extensons. European Journal of Operatonal esearch, 187: , Y. Pochet and L.A. Wolsey. Producton Plannng by Mxed Integer Programmng. Sprnger, A. Torello and.a. Uhan. Dynamc lnear producton games under uncertanty. optmzaton-onlne.org/db_html/2013/10/4064.html, W. van den Heuvel, P. Borm, and H. Hamers. Economc lot-szng games. European Journal of Operatonal esearch, 176: , H.M. Wagner. A postscrpt to Dynamc Problems n the Theory of the Frm. aval esearch Logstcs Quarterly, 7:7 12, H.M. Wagner and T.M. Whtn. Dynamc verson of the economc lot sze model. Management Scence, 51:89 96, E. Zabel. Some generalzatons of an nventory plannng horzon theorem. Management Scence, 10 3: , P.H. Zpkn. Foundatons of Inventory Management. McGraw-Hll,
10) Activity analysis
3C3 Mathematcal Methods for Economsts (6 cr) 1) Actvty analyss Abolfazl Keshvar Ph.D. Aalto Unversty School of Busness Sldes orgnally by: Tmo Kuosmanen Updated by: Abolfazl Keshvar 1 Outlne Hstorcal development
More informationPerfect Competition and the Nash Bargaining Solution
Perfect Competton and the Nash Barganng Soluton Renhard John Department of Economcs Unversty of Bonn Adenauerallee 24-42 53113 Bonn, Germany emal: rohn@un-bonn.de May 2005 Abstract For a lnear exchange
More informationAmiri 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 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 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 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 informationCost Allocation for Joint Replenishment Models
Cost Allocaton for Jont Replenshment Models Jawe Zhang Frst Verson: December 2006 Revsed: May 2007, July 2007 Abstract We consder the one-warehouse multple retaler nventory model wth a submodular jont
More informationCollege of Computer & Information Science Fall 2009 Northeastern University 20 October 2009
College of Computer & Informaton Scence Fall 2009 Northeastern Unversty 20 October 2009 CS7880: Algorthmc Power Tools Scrbe: Jan Wen and Laura Poplawsk Lecture Outlne: Prmal-dual schema Network Desgn:
More informationprinceton univ. F 17 cos 521: Advanced Algorithm Design Lecture 7: LP Duality Lecturer: Matt Weinberg
prnceton unv. F 17 cos 521: Advanced Algorthm Desgn Lecture 7: LP Dualty Lecturer: Matt Wenberg Scrbe: LP Dualty s an extremely useful tool for analyzng structural propertes of lnear programs. Whle there
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 informationVARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES
VARIATION OF CONSTANT SUM CONSTRAINT FOR INTEGER MODEL WITH NON UNIFORM VARIABLES BÂRZĂ, Slvu Faculty of Mathematcs-Informatcs Spru Haret Unversty barza_slvu@yahoo.com Abstract Ths paper wants to contnue
More informationAn Economic Lot-Sizing Problem with Perishable Inventory and Economies of Scale Costs: Approximation Solutions and Worst Case Analysis
An Economc Lot-Szng Problem wth Pershable Inventory and Economes of Scale Costs: Approxmaton Solutons and Worst Case Analyss Leon Yang Chu, 1 Vernon Nng Hsu, 2 Zuo-Jun Max Shen 3 1 Department of Industral
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 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 informationDynamic Linear Programming Games with Risk-Averse Players
Dynamc Lnear Programmng Games wth Rsk-Averse Players Alejandro Torello H. Mlton Stewart School of Industral and Systems Engneerng Georga Insttute of Technology Atlanta, Georga 30332 atorello at sye dot
More informationWinter 2008 CS567 Stochastic Linear/Integer Programming Guest Lecturer: Xu, Huan
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
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 informationGames of Threats. Elon Kohlberg Abraham Neyman. Working Paper
Games of Threats Elon Kohlberg Abraham Neyman Workng Paper 18-023 Games of Threats Elon Kohlberg Harvard Busness School Abraham Neyman The Hebrew Unversty of Jerusalem Workng Paper 18-023 Copyrght 2017
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 informationEFFECTS OF JOINT REPLENISHMENT POLICY ON COMPANY COST UNDER PERMISSIBLE DELAY IN PAYMENTS
Mathematcal and Computatonal Applcatons, Vol. 5, No., pp. 8-58,. Assocaton for Scentfc Research EFFECS OF JOIN REPLENISHMEN POLICY ON COMPANY COS UNDER PERMISSIBLE DELAY IN PAYMENS Yu-Chung sao, Mng-Yu
More informationHeuristic Algorithm for Finding Sensitivity Analysis in Interval Solid Transportation Problems
Internatonal Journal of Innovatve Research n Advanced Engneerng (IJIRAE) ISSN: 349-63 Volume Issue 6 (July 04) http://rae.com Heurstc Algorm for Fndng Senstvty Analyss n Interval Sold Transportaton Problems
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 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 informationMMA and GCMMA two methods for nonlinear optimization
MMA and GCMMA two methods for nonlnear optmzaton Krster Svanberg Optmzaton and Systems Theory, KTH, Stockholm, Sweden. krlle@math.kth.se Ths note descrbes the algorthms used n the author s 2007 mplementatons
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 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 informationAppendix for Causal Interaction in Factorial Experiments: Application to Conjoint Analysis
A Appendx for Causal Interacton n Factoral Experments: Applcaton to Conjont Analyss Mathematcal Appendx: Proofs of Theorems A. Lemmas Below, we descrbe all the lemmas, whch are used to prove the man theorems
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
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 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 informationCS : Algorithms and Uncertainty Lecture 17 Date: October 26, 2016
CS 29-128: Algorthms and Uncertanty Lecture 17 Date: October 26, 2016 Instructor: Nkhl Bansal Scrbe: Mchael Denns 1 Introducton In ths lecture we wll be lookng nto the secretary problem, and an nterestng
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 informationConvexity preserving interpolation by splines of arbitrary degree
Computer Scence Journal of Moldova, vol.18, no.1(52), 2010 Convexty preservng nterpolaton by splnes of arbtrary degree Igor Verlan Abstract In the present paper an algorthm of C 2 nterpolaton of dscrete
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 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 informationPerron Vectors of an Irreducible Nonnegative Interval Matrix
Perron Vectors of an Irreducble Nonnegatve Interval Matrx Jr Rohn August 4 2005 Abstract As s well known an rreducble nonnegatve matrx possesses a unquely determned Perron vector. As the man result of
More informationA Hybrid Variational Iteration Method for Blasius Equation
Avalable at http://pvamu.edu/aam Appl. Appl. Math. ISSN: 1932-9466 Vol. 10, Issue 1 (June 2015), pp. 223-229 Applcatons and Appled Mathematcs: An Internatonal Journal (AAM) A Hybrd Varatonal Iteraton Method
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 informationChapter 2 A Class of Robust Solution for Linear Bilevel Programming
Chapter 2 A Class of Robust Soluton for Lnear Blevel Programmng Bo Lu, Bo L and Yan L Abstract Under the way of the centralzed decson-makng, the lnear b-level programmng (BLP) whose coeffcents are supposed
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 NOTE ON A PERIODIC REVIEW INVENTORY MODEL WITH UNCERTAIN DEMAND IN A RANDOM ENVIRONMENT. Hirotaka Matsumoto and Yoshio Tabata
Scentae Mathematcae Japoncae Onlne, Vol. 9, (23), 419 429 419 A NOTE ON A PERIODIC REVIEW INVENTORY MODEL WITH UNCERTAIN DEMAND IN A RANDOM ENVIRONMENT Hrotaka Matsumoto and Yosho Tabata Receved September
More informationThe Order Relation and Trace Inequalities for. Hermitian Operators
Internatonal Mathematcal Forum, Vol 3, 08, no, 507-57 HIKARI Ltd, wwwm-hkarcom https://doorg/0988/mf088055 The Order Relaton and Trace Inequaltes for Hermtan Operators Y Huang School of Informaton Scence
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 informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
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 informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationA new Approach for Solving Linear Ordinary Differential Equations
, ISSN 974-57X (Onlne), ISSN 974-5718 (Prnt), Vol. ; Issue No. 1; Year 14, Copyrght 13-14 by CESER PUBLICATIONS A new Approach for Solvng Lnear Ordnary Dfferental Equatons Fawz Abdelwahd Department of
More informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationTHE SUMMATION NOTATION Ʃ
Sngle Subscrpt otaton THE SUMMATIO OTATIO Ʃ Most of the calculatons we perform n statstcs are repettve operatons on lsts of numbers. For example, we compute the sum of a set of numbers, or the sum of the
More informatione - c o m p a n i o n
OPERATIONS RESEARCH http://dxdoorg/0287/opre007ec e - c o m p a n o n ONLY AVAILABLE IN ELECTRONIC FORM 202 INFORMS Electronc Companon Generalzed Quantty Competton for Multple Products and Loss of Effcency
More informationU.C. Berkeley CS294: Spectral Methods and Expanders Handout 8 Luca Trevisan February 17, 2016
U.C. Berkeley CS94: Spectral Methods and Expanders Handout 8 Luca Trevsan February 7, 06 Lecture 8: Spectral Algorthms Wrap-up In whch we talk about even more generalzatons of Cheeger s nequaltes, and
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 informationEstimation: Part 2. Chapter GREG estimation
Chapter 9 Estmaton: Part 2 9. GREG estmaton In Chapter 8, we have seen that the regresson estmator s an effcent estmator when there s a lnear relatonshp between y and x. In ths chapter, we generalzed the
More informationA New Algorithm for Finding a Fuzzy Optimal. Solution for Fuzzy Transportation Problems
Appled Mathematcal Scences, Vol. 4, 200, no. 2, 79-90 A New Algorthm for Fndng a Fuzzy Optmal Soluton for Fuzzy Transportaton Problems P. Pandan and G. Nataraan Department of Mathematcs, School of Scence
More informationThe Expectation-Maximization Algorithm
The Expectaton-Maxmaton Algorthm Charles Elan elan@cs.ucsd.edu November 16, 2007 Ths chapter explans the EM algorthm at multple levels of generalty. Secton 1 gves the standard hgh-level verson of the algorthm.
More informationn α j x j = 0 j=1 has a nontrivial solution. Here A is the n k matrix whose jth column is the vector for all t j=0
MODULE 2 Topcs: Lnear ndependence, bass and dmenson We have seen that f n a set of vectors one vector s a lnear combnaton of the remanng vectors n the set then the span of the set s unchanged f that vector
More informationFinding Dense Subgraphs in G(n, 1/2)
Fndng Dense Subgraphs n Gn, 1/ Atsh Das Sarma 1, Amt Deshpande, and Rav Kannan 1 Georga Insttute of Technology,atsh@cc.gatech.edu Mcrosoft Research-Bangalore,amtdesh,annan@mcrosoft.com Abstract. Fndng
More informationValuated Binary Tree: A New Approach in Study of Integers
Internatonal Journal of Scentfc Innovatve Mathematcal Research (IJSIMR) Volume 4, Issue 3, March 6, PP 63-67 ISS 347-37X (Prnt) & ISS 347-34 (Onlne) wwwarcournalsorg Valuated Bnary Tree: A ew Approach
More informationWelfare Properties of General Equilibrium. What can be said about optimality properties of resource allocation implied by general equilibrium?
APPLIED WELFARE ECONOMICS AND POLICY ANALYSIS Welfare Propertes of General Equlbrum What can be sad about optmalty propertes of resource allocaton mpled by general equlbrum? Any crteron used to compare
More informationSuggested solutions for the exam in SF2863 Systems Engineering. June 12,
Suggested solutons for the exam n SF2863 Systems Engneerng. June 12, 2012 14.00 19.00 Examner: Per Enqvst, phone: 790 62 98 1. We can thnk of the farm as a Jackson network. The strawberry feld s modelled
More informationEquilibrium with Complete Markets. Instructor: Dmytro Hryshko
Equlbrum wth Complete Markets Instructor: Dmytro Hryshko 1 / 33 Readngs Ljungqvst and Sargent. Recursve Macroeconomc Theory. MIT Press. Chapter 8. 2 / 33 Equlbrum n pure exchange, nfnte horzon economes,
More informationChapter - 2. Distribution System Power Flow Analysis
Chapter - 2 Dstrbuton System Power Flow Analyss CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load
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 informationWeek 5: Neural Networks
Week 5: Neural Networks Instructor: Sergey Levne Neural Networks Summary In the prevous lecture, we saw how we can construct neural networks by extendng logstc regresson. Neural networks consst of multple
More informationNP-Completeness : Proofs
NP-Completeness : Proofs Proof Methods A method to show a decson problem Π NP-complete s as follows. (1) Show Π NP. (2) Choose an NP-complete problem Π. (3) Show Π Π. A method to show an optmzaton problem
More informationIntegrated approach in solving parallel machine scheduling and location (ScheLoc) problem
Internatonal Journal of Industral Engneerng Computatons 7 (2016) 573 584 Contents lsts avalable at GrowngScence Internatonal Journal of Industral Engneerng Computatons homepage: www.growngscence.com/ec
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 informationInfinitely Split Nash Equilibrium Problems in Repeated Games
Infntely Splt ash Equlbrum Problems n Repeated Games Jnlu L Department of Mathematcs Shawnee State Unversty Portsmouth, Oho 4566 USA Abstract In ths paper, we ntroduce the concept of nfntely splt ash equlbrum
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 informationInventory Model with Backorder Price Discount
Vol. No. 7-7 ead Tme and Orderng Cost Reductons are Interdependent n Inventory Model wth Bacorder Prce scount Yu-Jen n Receved: Mar. 7 Frst Revson: May. 7 7 ccepted: May. 7 bstract The stochastc nventory
More informationRandić Energy and Randić Estrada Index of a Graph
EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 5, No., 202, 88-96 ISSN 307-5543 www.ejpam.com SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA 29 JUNE -02JULY 20, ISTANBUL
More informationEconomics 8105 Macroeconomic Theory Recitation 1
Economcs 8105 Macroeconomc Theory Rectaton 1 Outlne: Conor Ryan September 6th, 2016 Adapted From Anh Thu (Monca) Tran Xuan s Notes Last Updated September 20th, 2016 Dynamc Economc Envronment Arrow-Debreu
More informationOn a direct solver for linear least squares problems
ISSN 2066-6594 Ann. Acad. Rom. Sc. Ser. Math. Appl. Vol. 8, No. 2/2016 On a drect solver for lnear least squares problems Constantn Popa Abstract The Null Space (NS) algorthm s a drect solver for lnear
More informationEconomics 2450A: Public Economics Section 10: Education Policies and Simpler Theory of Capital Taxation
Economcs 2450A: Publc Economcs Secton 10: Educaton Polces and Smpler Theory of Captal Taxaton Matteo Parads November 14, 2016 In ths secton we study educaton polces n a smplfed verson of framework analyzed
More informationUsing T.O.M to Estimate Parameter of distributions that have not Single Exponential Family
IOSR Journal of Mathematcs IOSR-JM) ISSN: 2278-5728. Volume 3, Issue 3 Sep-Oct. 202), PP 44-48 www.osrjournals.org Usng T.O.M to Estmate Parameter of dstrbutons that have not Sngle Exponental Famly Jubran
More informationAn efficient algorithm for multivariate Maclaurin Newton transformation
Annales UMCS Informatca AI VIII, 2 2008) 5 14 DOI: 10.2478/v10065-008-0020-6 An effcent algorthm for multvarate Maclaurn Newton transformaton Joanna Kapusta Insttute of Mathematcs and Computer Scence,
More informationImplicit Integration Henyey Method
Implct Integraton Henyey Method In realstc stellar evoluton codes nstead of a drect ntegraton usng for example the Runge-Kutta method one employs an teratve mplct technque. Ths s because the structure
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
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 informationCHAPTER-5 INFORMATION MEASURE OF FUZZY MATRIX AND FUZZY BINARY RELATION
CAPTER- INFORMATION MEASURE OF FUZZY MATRI AN FUZZY BINARY RELATION Introducton The basc concept of the fuzz matr theor s ver smple and can be appled to socal and natural stuatons A branch of fuzz matr
More informationApplication of B-Spline to Numerical Solution of a System of Singularly Perturbed Problems
Mathematca Aeterna, Vol. 1, 011, no. 06, 405 415 Applcaton of B-Splne to Numercal Soluton of a System of Sngularly Perturbed Problems Yogesh Gupta Department of Mathematcs Unted College of Engneerng &
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 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 informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
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 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 informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
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 informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationAdditional Codes using Finite Difference Method. 1 HJB Equation for Consumption-Saving Problem Without Uncertainty
Addtonal Codes usng Fnte Dfference Method Benamn Moll 1 HJB Equaton for Consumpton-Savng Problem Wthout Uncertanty Before consderng the case wth stochastc ncome n http://www.prnceton.edu/~moll/ HACTproect/HACT_Numercal_Appendx.pdf,
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 informationAn (almost) unbiased estimator for the S-Gini index
An (almost unbased estmator for the S-Gn ndex Thomas Demuynck February 25, 2009 Abstract Ths note provdes an unbased estmator for the absolute S-Gn and an almost unbased estmator for the relatve S-Gn for
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 informationLATERAL TRANSSHIPMENTS IN INVENTORY MODELS
LATERAL TRANSSHIPMENTS IN INVENTORY MODELS MIN CHEN 11.2008 BMI THESIS Lateral Transshpments n Inventory Models MIN CHEN 11.2008 II PREFACE The BMI Thess s the fnal report to acqure the Master's degree
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 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 CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens
THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of
More informationA Local Variational Problem of Second Order for a Class of Optimal Control Problems with Nonsmooth Objective Function
A Local Varatonal Problem of Second Order for a Class of Optmal Control Problems wth Nonsmooth Objectve Functon Alexander P. Afanasev Insttute for Informaton Transmsson Problems, Russan Academy of Scences,
More informationMixed-integer vertex covers on bipartite graphs
Mxed-nteger vertex covers on bpartte graphs Mchele Confort, Bert Gerards, Gacomo Zambell November, 2006 Abstract Let A be the edge-node ncdence matrx of a bpartte graph G = (U, V ; E), I be a subset the
More informationMulti-product budget-constrained acquistion and pricing with uncertain demand and supplier quantity discounts
Unversty of Wndsor Scholarshp at UWndsor Mechancal, Automotve & Materals Engneerng Publcatons Department of Mechancal, Automotve & Materals Engneerng 11-2010 Mult-product budget-constraned acquston and
More information