Multi-objective Programming Approach for. Fuzzy Linear Programming Problems
|
|
- Phoebe Hudson
- 6 years ago
- Views:
Transcription
1 Applied Mathematical Scieces Vl HIKARI Ltd Multi-bective Prgrammig Apprach fr Fuzzy Liear Prgrammig Prblems P. Padia Departmet f Mathematics Schl f Advaced Scieces VIT Uiversity Vellre-4 Idia padia6@rediffmail.cm Cpyright 03 P. Padia. This is a pe access article distributed uder the Creative Cmms Attributi Licese which permits urestricted use distributi ad reprducti i ay medium prvided the rigial wrk is prperly cited. Abstract A ew methd amely level-sum methd based the multi-bective liear prgrammig ad the simplex methd is prpsed fr cmputig a ptimal fuzzy sluti t a fuzzy liear prgrammig prblem iwhich fuzzy rakig fuctis are t used. The level-sum methd is illustrated by umerical examples. Keywrds: Fuzzy liear prgrammig prblem Optimal fuzzy sluti Multi-bective liear prgrammig Level-sum methd.. Itrducti Liear prgrammig (LP) is e f the mst applicable ptimizati techiques. It deals with the ptimizatis f a liear fucti while satisfyig a set f liear equality ad/r iequality cstraits r restrictis. I practice a LP mdel ivlves a lt f parameters whse values are assiged by experts / decisi makers. Hwever bth experts ad decisi makers d t precisely kw the value f thse parameters i mst f the cases. Therefre fuzzy liear prgrammig (FLP) prblem [9] was itrduced ad studied. I the literature a variety f algrithms fr slvig FLP have bee studied based fuzzy rakig fucti ad classical liear prgrammig. Taaka et al. [7] Zimmerma [0] Buckley ad Feurig [3] Thakre et al.[8] ad Zhag et al. [] slved FLP prblems usig multi-bective liear prgrammig (MOLP) techique. Padia[6] has prpsed a ew apprach amely sum f bectives (SO) methd fr fidig a prperly efficiet sluti t multi-bective prgrammig prblems.
2 8 P. Padia I this paper we prpse a ew methd amely level-sum methd fr fidig a ptimal fuzzy sluti t fully FLP prblems which is a crisp LP techique t usig the fuzzy rakig fucti. First we cstruct a crisp MOLP prblem frm the FLP prblem ad the we establish a relati betwee a ptimal fuzzy sluti t the fully FLP prblem ad a efficiet sluti t its related MOLP prblem. Based the relati we develp the prpsed methd. With the help f umerical examples the level-sum methd is illustrated. The advatages f the prpsed methd are that fuzzy rakig fuctis are t used the btaied results exactly satisfy all the cstraits ad the cmputati ca be made by LP slver because it is based ly crisp LP techique.. Prelimiaries We eed the fllwig defiitis f the basic arithmetic peratrs ad partial rderig relatis fuzzy triagular umbers based the fucti priciple which ca be fud i [4 90 ]. Defiiti. A fuzzy umber a is a triagular fuzzy umber deted by ( a where a a ad a3 are real umbers ad its member ship fucti μ ( x) is give belw: a ( x a) /( a a) fr a x a μ a ( x) = ( a3 x) /( a3 a) fr a x a3 0 therwise Let F(R) be the set f all real triagular fuzzy umbers. Defiiti. Let ( a ad ( b b ) be i F (R). The (i) ( a ( b b ) = ( a + b a + b a3 + ). (ii) ( a Θ ( b b ) = ( a a b a3 b ). (iii) k ( a = ( ka k k fr k 0. (iv) k ( a = ( ka 3 k ka ) fr k < 0. ( ab ab a3 ) a 0 (v) ( a ( b b ) = ( a ab a3 ) a < 0 a3 0 ( a ab a3b ) a3 < 0. Defiiti.3 Let A = ( a a a3 ) ad B = ( b b ) be i F (R) the (i) A B iff a i = b i i = 3 ; (ii) A p B iff ai b i i = 3 ad (iii) A f B iff ai b i i = 3.
3 Multi-bective prgrammig apprach 83 Based the tatis f Magasaria [5] we defie the fllwig partial rder relati f as fllws. Defiiti.4 Let A = ( a a a3 ) ad B = ( b b ) be i F (R) the A f B iff ai b i i = 3 ad a r > br fr sme r { 3}. Csider the fllwig multi-bective ptimizati prblem (MP) Miimize f ( x) = ( f( x) f ( x)... fk ( x)) subect t g ( x) 0 x X where f i : X R i =... k ad g : X R where g = ( g... g m ) are differetiable fuctis X a pe cvex subset f R. Nw P = { x X : g ( x) 0 =... m} is the set f all feasible slutis fr the prblem (MP). Defiiti.5: A feasible pit x is said t be efficiet [6] fr (MP) if there exists ther feasible pit x i P such that f i ( x) fi( x ) i =... k ad f x) < f ( x ) fr sme r { Kk}. r ( r m 3. Fully Fuzzy Liear Prgrammig Prblem Csider the fllwig fully FLP with m fuzzy iequality/equality cstraits ad fuzzy variables may be frmulated as fllws: (P) Maximize z c T x subect t A x { p f } b x f 0 where a i c x bi F( R ) fr all ad i m c T = ( c ) A = ( a i ) m x = ( x ) x ad b = ( bi ) mx. Let the parameters z a c x ad b be the triagular fuzzy umbers i ( z z z3) ( p q r ) ( x y t ) ( a i bi ci ) ad ( bi gi hi ) respectively. The the prblem (P) ca be writte as fllws: (P) Maximize ( z z z3) ( p q r ) ( x y t ) = subect t ( ai bi ci ) ( x y t ) { p f } ( bi gi hi ) fr all i =... m = ( x y t ) f 0 = m. i
4 84 P. Padia Nw usig the arithmetic peratis ad partial rderig relatis we write the give FLPP as a MOLP prblem which is give belw: (M) Maximize Maximize Maximize 3 subect t lwer value f = z = lwer value f ( p q r ) ( x y t )) = middle value f = z = ( p q r ) ( x y t )) z = upper value f ( p q r ) ( x y t )) = middle value f = upper value f = z z ; z3 z ; y x 0 = m. ( a i bi ci ) ( x y t )){ = } bi ( a i bi ci ) ( x y t )){ = } gi ( a i bi ci ) ( x y t )){ = } hi x = m ; y t = m fr all i =... m ; fr all i =... m ; fr all i =... m ; Remark 3.: I the case f a fully FLP prblem ivlvig trapezidal fuzzy umbers ad / r trapezidal fuzzy decisi variables we get a MOLP prblem havig fur bectives. We w prve the fllwig therem which establish a relati betwee a ptimal fuzzy sluti t a fully FLP prblem ad a efficiet sluti t its related MOLP prblems. Therem 3.: Let X = { x y t ; = m} be a efficiet sluti t the prblem (M). The X = { ( x y t ) ; = m} is a ptimal sluti t the prblem (P). Prf: Nw sice X = { x y t ; = m} is a efficiet sluti t the prblem (M) X = { ( x y t ) ; = m} is a feasible sluti t the prblem (P). Assume that X = { ( x y t ) ; = m} is t ptimal t the prblem (P). The there exists a feasible sluti X = { ( x y t ) ; = m} t the prblem (P) such that Z ( X ) f Z( X ) that is zi ( x y t) zi( x y t ) i = 3 ad
5 Multi-bective prgrammig apprach 85 z ( x y t) z ( x y t r > r ) fr sme r {3 } where x = { x ; = m} y = { y ; = m} t = { t ; = m} x = { x ; = m} y = { y ; = m} ad t = { t ; = m}. This meas that X = { x y t ; = m} is t a efficiet sluti t the prblem (M) which is a ctradicti. Hece the therem is prved. Nw we prpse a ew methd amely level-sum methd fr fidig a ptimal fuzzy sluti t a FLP prblem which is based MOLP ad the simplex methd. The prpsed methd prceeds as fllws: Step : Cstruct a crisp MOLP prblem frm the give FLP prblem. Step : Fid a efficiet sluti t the MOLP prblem btaied i the Step. usig the SO methd [6]. Step 3: The efficiet sluti btaied frm the Step. t the MOLP prblem yields a ptimal fuzzy sluti t the FLP prblem by the Therem 3... Remark 3.: The prpsed methd ca be exteded t fuzzy iteger LP prblems by addig the iteger restrictis ad replacig the simplex methd by a iteger LP techique. The prpsed methd is illustrated by the fllwig examples. Example 3.: Csider the fllwig fully FLP prblem: t Maximize z ( 3 ) x ( 3 4 ) x subect t ( 0 ) x ( 3 ) x ( 0 4 ); ( 3 ) x ( 0 ) x ( 8 ); x ad x f 0. Let x = ( x y t ) x = ( x y t ) ad z = ( z z z3 ). Nw usig the Step. the MOLP prblem related t the give fully FLP prblem is give belw: (M) Maximize z = t + x Maximize z = y + 3y Maximize z 3 = 3t + 4t subect t 0x + x = ; y + y = 0 ; t + 3t = 4 ; x + 0x = ; y + y = 8 ; 3t + t = ; z z ; z3 z ; y x ; y x ; t y ; t y ; x x 0. Nw by the Step. we csider the fllwig LP prblem (S) related t the abve MOLP prblem: (S) Maximize Z = x + y + 3y + t + 4t subect t
6 86 P. Padia 0x + x = ; y + y = 0 ; t + 3t = 4 ; x + 0x = ; y + y = 8 ; 3t + t = ; x + y + 3y + t 0 ; y 3y + 3t + 4t 0 ; y x 0 ; y x 0; t y 0; t y 0 ; x x 0 ad slve it by simplex methd. The ptimal sluti t the prblem (S) is x = ; x = ; y = ; y = 4 ; t = 3 ad t = 6 with Z = 50. Thus ( x = x = y = y = 4 t = 3 t = 6) is a efficiet sluti t the prblem (M). Nw by the Step 3. x ( 3 ) x ( 4 6 ) ad z (6 33 ) is a ptimal fuzzy sluti t the give fully FLP prblem. Remark 3.3 : Fr the fully FLP prblem ( the Example 4.) Amit Kumar et al. [] by the rakig methd btaied the same ptimal fuzzy sluti. Example 3.: Csider the fllwig FLP prblem: t Maximize z ( 704 5) x ( ) x subect t ( 3 4 ) x ( 6 7 ) x p ( 835 ); ( ) x ( 60 ) x p ( ); x ad x 0. Let z = ( z z z3 z4 ). Nw usig the Step. the MOLP prblem related t the give FLP prblem is give belw: (M) Maximize z = 7x + 0x Maximize z = 0x + 5x Maximize z 3 = 4x + 35x Maximize z 4 = 5x + 40x subect t x + x 8 ; 3x + 6x 3 ; 4x + 7x 5 ; 3x + x 3 4x + 6x 7 ; 6x + 0x 9 ; z z ; z3 z ; z4 z3 ; x x 0. Nw by the Step. we csider the fllwig LP prblem (S) related t the abve MOLP prblem: (S) Maximize Z = 56x + 0x subect t x + x 8 ; 3x + 6x 3 ; 4x + 7x 5 ; 3x + x 3 4x + 6x 7 ; 6x + 0x 9 ; 3x + 5x 0 ; 4x + 0x 0 ; 9x + 5x 0 ; x x 0 ad slve it by simplex methd. The ptimal sluti t the prblem (S) is x = 0 ad x = 0. 9 with Z = 08. Thus ( x = 0 x = 0.9) is a efficiet sluti t the prblem (M). Nw by the Step 3. x = 0 x = 0. 9 ad z ( ) is a ptimal fuzzy sluti t the give FLP prblem. Remark 3.4 : I Thakre et al. [8] by the weighted methd the same ptimal fuzzy sluti t the FLP prblem (the Example 3..) is btaied.
7 Multi-bective prgrammig apprach Cclusi I this paper we prpse the level-sum methd t fid a ptimal fuzzy sluti t a FLP prblem satisfyig all cstraits. The mai advatage f the prpsed methd is that the FLP prblems ca be slved by ay LP slver usig the level-sum methd sice it is based ly simplex methd. The level-sum methd ca serve maagers by prvidig a apprpriate best sluti t a variety f LP mdels havig fuzzy umbers ad variables i a simple ad effective maer. I ear future we exted the level-sum methd t fuzzy MOLP prblems. Refereces [] Amit Kumar Jagdeep Kaur ad Pushpider Sigh Fuzzy ptimal sluti f fully fuzzy liear prgrammig prblems with iequality cstraits Iteratial Jural f Mathematical ad Cmputer Scieces 6 (00) [] R. E. Bellma ad L. A. Zadeh Decisi makig i a fuzzy evirmet Maagemet Sciece 7(970) [3] J. Buckley ad T. Feurig Evlutiary algrithm sluti t fuzzy prblems: Fuzzy liear prgrammig Fuzzy Sets ad Systems 09(000) [4] Gerge J. Klir ad B Yua Fuzzy Sets ad Fuzzy lgic: Thery ad Applicatis Pretice-Hall 008. [5] O.L.Magasaria Nliear prgrammig McGraw Hill New Yrk 969. [6] P. Padia A simple apprach fr fidig a fair sluti t multibective prgrammig prblems Bulleti f Mathematical Scieces & Applicatis (0) [7] H. Taaka T Okuda ad K. Asai O fuzzy mathematical prgrammig Jural f Cyberetics ad Systems 3(973) [8] P. A. Thakre D. S. Shelar ad S. P. Thakre Slvig fuzzy liear prgrammig prblem as multi bective liear prgrammig prblem Jural f Egieerig ad Techlgy Research (009) [9] L. A. Zadeh Fuzzy sets Ifrmati ad Ctrl 8 (965) [0] H. J. Zimmerma Fuzzy prgrammig ad liear prgrammig with several bective fuctis Fuzzy Sets ad Systems (978 ) [] G. Zhag Y. H. Wu M. Remias ad J. Lu Frmulati f fuzzy liear prgrammig prblems as fur-bective cstraied ptimizati prblems Applied Mathematics ad Cmputati 39( 003) Received: Jauary 5 03
A New Method for Finding an Optimal Solution. of Fully Interval Integer Transportation Problems
Applied Matheatical Scieces, Vl. 4, 200,. 37, 89-830 A New Methd fr Fidig a Optial Sluti f Fully Iterval Iteger Trasprtati Prbles P. Padia ad G. Nataraja Departet f Matheatics, Schl f Advaced Scieces,
More informationFourier Method for Solving Transportation. Problems with Mixed Constraints
It. J. Ctemp. Math. Scieces, Vl. 5, 200,. 28, 385-395 Furier Methd fr Slvig Trasprtati Prblems with Mixed Cstraits P. Padia ad G. Nataraja Departmet f Mathematics, Schl f Advaced Scieces V I T Uiversity,
More informationM.Jayalakshmi and P. Pandian Department of Mathematics, School of Advanced Sciences, VIT University, Vellore-14, India.
M.Jayalakshmi, P. Padia / Iteratioal Joural of Egieerig Research ad Applicatios (IJERA) ISSN: 48-96 www.iera.com Vol., Issue 4, July-August 0, pp.47-54 A New Method for Fidig a Optimal Fuzzy Solutio For
More informationIJISET - International Journal of Innovative Science, Engineering & Technology, Vol. 2 Issue 12, December
IJISET - Iteratial Jural f Ivative Sciece, Egieerig & Techlgy, Vl Issue, December 5 wwwijisetcm ISSN 48 7968 Psirmal ad * Pararmal mpsiti Operatrs the Fc Space Abstract Dr N Sivamai Departmet f athematics,
More informationK [f(t)] 2 [ (st) /2 K A GENERALIZED MEIJER TRANSFORMATION. Ku(z) ()x) t -)-I e. K(z) r( + ) () (t 2 I) -1/2 e -zt dt, G. L. N. RAO L.
Iterat. J. Math. & Math. Scl. Vl. 8 N. 2 (1985) 359-365 359 A GENERALIZED MEIJER TRANSFORMATION G. L. N. RAO Departmet f Mathematics Jamshedpur C-perative Cllege f the Rachi Uiversity Jamshedpur, Idia
More informationIntermediate Division Solutions
Itermediate Divisi Slutis 1. Cmpute the largest 4-digit umber f the frm ABBA which is exactly divisible by 7. Sluti ABBA 1000A + 100B +10B+A 1001A + 110B 1001 is divisible by 7 (1001 7 143), s 1001A is
More informationCopyright 1978, by the author(s). All rights reserved.
Cpyright 1978, by the authr(s). All rights reserved. Permissi t make digital r hard cpies f all r part f this wrk fr persal r classrm use is grated withut fee prvided that cpies are t made r distributed
More informationFloating Point Method for Solving Transportation. Problems with Additional Constraints
Internatinal Mathematical Frum, Vl. 6, 20, n. 40, 983-992 Flating Pint Methd fr Slving Transprtatin Prblems with Additinal Cnstraints P. Pandian and D. Anuradha Department f Mathematics, Schl f Advanced
More informationSuper-efficiency Models, Part II
Super-efficiec Mdels, Part II Emilia Niskae The 4th f Nvember S steemiaalsi Ctets. Etesis t Variable Returs-t-Scale (0.4) S steemiaalsi Radial Super-efficiec Case Prblems with Radial Super-efficiec Case
More informationStudy of Energy Eigenvalues of Three Dimensional. Quantum Wires with Variable Cross Section
Adv. Studies Ther. Phys. Vl. 3 009. 5 3-0 Study f Eergy Eigevalues f Three Dimesial Quatum Wires with Variale Crss Secti M.. Sltai Erde Msa Departmet f physics Islamic Aad Uiversity Share-ey rach Ira alrevahidi@yah.cm
More informationUNIVERSITY OF TECHNOLOGY. Department of Mathematics PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP. Memorandum COSOR 76-10
EI~~HOVEN UNIVERSITY OF TECHNOLOGY Departmet f Mathematics PROBABILITY THEORY, STATISTICS AND OPERATIONS RESEARCH GROUP Memradum COSOR 76-10 O a class f embedded Markv prcesses ad recurrece by F.H. Sims
More informationMean residual life of coherent systems consisting of multiple types of dependent components
Mea residual life f cheret systems csistig f multiple types f depedet cmpets Serka Eryilmaz, Frak P.A. Cle y ad Tahai Cle-Maturi z February 20, 208 Abstract Mea residual life is a useful dyamic characteristic
More informationThe generation of successive approximation methods for Markov decision processes by using stopping times
The geerati f successive apprximati methds fr Markv decisi prcesses by usig stppig times Citati fr published versi (APA): va Nue, J. A. E. E., & Wessels, J. (1976). The geerati f successive apprximati
More informationScholars Journal of Physics, Mathematics and Statistics
Jaalakshmi M. Sch. J. Phs. Math. Stat., 015 Vol- Issue-A Mar-Ma pp-144-150 Scholars Joural of Phsics, Mathematics ad Statistics Sch. J. Phs. Math. Stat. 015 A:144-150 Scholars Academic ad Scietific Publishers
More informationActive redundancy allocation in systems. R. Romera; J. Valdés; R. Zequeira*
Wrkig Paper -6 (3) Statistics ad Ecmetrics Series March Departamet de Estadística y Ecmetría Uiversidad Carls III de Madrid Calle Madrid, 6 893 Getafe (Spai) Fax (34) 9 64-98-49 Active redudacy allcati
More informationare specified , are linearly independent Otherwise, they are linearly dependent, and one is expressed by a linear combination of the others
Chater 3. Higher Order Liear ODEs Kreyszig by YHLee;4; 3-3. Hmgeeus Liear ODEs The stadard frm f the th rder liear ODE ( ) ( ) = : hmgeeus if r( ) = y y y y r Hmgeeus Liear ODE: Suersiti Pricile, Geeral
More informationChapter 3.1: Polynomial Functions
Ntes 3.1: Ply Fucs Chapter 3.1: Plymial Fuctis I Algebra I ad Algebra II, yu ecutered sme very famus plymial fuctis. I this secti, yu will meet may ther members f the plymial family, what sets them apart
More informationGrade 3 Mathematics Course Syllabus Prince George s County Public Schools
Ctet Grade 3 Mathematics Curse Syllabus Price Gerge s Cuty Public Schls Prerequisites: Ne Curse Descripti: I Grade 3, istructial time shuld fcus fur critical areas: (1) develpig uderstadig f multiplicati
More informationPortfolio Performance Evaluation in a Modified Mean-Variance-Skewness Framework with Negative Data
Available lie at http://idea.srbiau.ac.ir It. J. Data Evelpmet Aalysis (ISSN 345-458X) Vl., N.3, Year 04 Article ID IJDEA-003,3 pages Research Article Iteratial Jural f Data Evelpmet Aalysis Sciece ad
More informationBIO752: Advanced Methods in Biostatistics, II TERM 2, 2010 T. A. Louis. BIO 752: MIDTERM EXAMINATION: ANSWERS 30 November 2010
BIO752: Advaced Methds i Bistatistics, II TERM 2, 2010 T. A. Luis BIO 752: MIDTERM EXAMINATION: ANSWERS 30 Nvember 2010 Questi #1 (15 pits): Let X ad Y be radm variables with a jit distributi ad assume
More informationUnifying the Derivations for. the Akaike and Corrected Akaike. Information Criteria. from Statistics & Probability Letters,
Uifyig the Derivatis fr the Akaike ad Crrected Akaike Ifrmati Criteria frm Statistics & Prbability Letters, Vlume 33, 1997, pages 201{208. by Jseph E. Cavaaugh Departmet f Statistics, Uiversity f Missuri,
More informationAn Algebraic Elimination Method for the Linear Complementarity Problem
Volume-3, Issue-5, October-2013 ISSN No: 2250-0758 Iteratioal Joural of Egieerig ad Maagemet Research Available at: wwwijemret Page Number: 51-55 A Algebraic Elimiatio Method for the Liear Complemetarity
More informationA Study on Estimation of Lifetime Distribution with Covariates Under Misspecification
Prceedigs f the Wrld Cgress Egieerig ad Cmputer Sciece 2015 Vl II, Octber 21-23, 2015, Sa Fracisc, USA A Study Estimati f Lifetime Distributi with Cvariates Uder Misspecificati Masahir Ykyama, Member,
More informationCh. 1 Introduction to Estimation 1/15
Ch. Itrducti t stimati /5 ample stimati Prblem: DSB R S f M f s f f f ; f, φ m tcsπf t + φ t f lectrics dds ise wt usually white BPF & mp t s t + w t st. lg. f & φ X udi mp cs π f + φ t Oscillatr w/ f
More informationENGI 4421 Central Limit Theorem Page Central Limit Theorem [Navidi, section 4.11; Devore sections ]
ENGI 441 Cetral Limit Therem Page 11-01 Cetral Limit Therem [Navidi, secti 4.11; Devre sectis 5.3-5.4] If X i is t rmally distributed, but E X i, V X i ad is large (apprximately 30 r mre), the, t a gd
More informationThe Molecular Diffusion of Heat and Mass from Two Spheres
Iteratial Jural f Mder Studies i Mechaical Egieerig (IJMSME) Vlume 4, Issue 1, 018, PP 4-8 ISSN 454-9711 (Olie) DOI: http://dx.di.rg/10.0431/454-9711.0401004 www.arcjurals.rg The Mlecular Diffusi f Heat
More informationThis article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and
This article appeared i a jural published by Elsevier The attached cpy is furished t the authr fr iteral -cmmercial research ad educati use, icludig fr istructi at the authrs istituti ad sharig with clleagues
More informationAxial Temperature Distribution in W-Tailored Optical Fibers
Axial Temperature Distributi i W-Tailred Optical ibers Mhamed I. Shehata (m.ismail34@yah.cm), Mustafa H. Aly(drmsaly@gmail.cm) OSA Member, ad M. B. Saleh (Basheer@aast.edu) Arab Academy fr Sciece, Techlgy
More informationQuantum Mechanics for Scientists and Engineers. David Miller
Quatum Mechaics fr Scietists ad Egieers David Miller Time-depedet perturbati thery Time-depedet perturbati thery Time-depedet perturbati basics Time-depedet perturbati thery Fr time-depedet prblems csider
More informationComparative analysis of bayesian control chart estimation and conventional multivariate control chart
America Jural f Theretical ad Applied Statistics 3; ( : 7- ublished lie Jauary, 3 (http://www.sciecepublishiggrup.cm//atas di:.648/.atas.3. Cmparative aalysis f bayesia ctrl chart estimati ad cvetial multivariate
More informationPartial-Sum Queries in OLAP Data Cubes Using Covering Codes
326 IEEE TRANSACTIONS ON COMPUTERS, VOL. 47, NO. 2, DECEMBER 998 Partial-Sum Queries i OLAP Data Cubes Usig Cverig Cdes Chig-Tie H, Member, IEEE, Jehshua Bruck, Seir Member, IEEE, ad Rakesh Agrawal, Seir
More informationAn S-type upper bound for the largest singular value of nonnegative rectangular tensors
Ope Mat. 06 4 95 933 Ope Matematics Ope Access Researc Article Jiaxig Za* ad Caili Sag A S-type upper bud r te largest sigular value egative rectagular tesrs DOI 0.55/mat-06-0085 Received August 3, 06
More informationClaude Elysée Lobry Université de Nice, Faculté des Sciences, parc Valrose, NICE, France.
CHAOS AND CELLULAR AUTOMATA Claude Elysée Lbry Uiversité de Nice, Faculté des Scieces, parc Valrse, 06000 NICE, Frace. Keywrds: Chas, bifurcati, cellularautmata, cmputersimulatis, dyamical system, ifectius
More informationRecovery of Third Order Tensors via Convex Optimization
Recvery f Third Order Tesrs via Cvex Optimizati Hlger Rauhut RWTH Aache Uiversity Lehrstuhl C für Mathematik (Aalysis) Ptdriesch 10 5056 Aache Germay Email: rauhut@mathcrwth-aachede Željka Stjaac RWTH
More information[1 & α(t & T 1. ' ρ 1
NAME 89.304 - IGNEOUS & METAMORPHIC PETROLOGY DENSITY & VISCOSITY OF MAGMAS I. Desity The desity (mass/vlume) f a magma is a imprtat parameter which plays a rle i a umber f aspects f magma behavir ad evluti.
More informationMarkov processes and the Kolmogorov equations
Chapter 6 Markv prcesses ad the Klmgrv equatis 6. Stchastic Differetial Equatis Csider the stchastic differetial equati: dx(t) =a(t X(t)) dt + (t X(t)) db(t): (SDE) Here a(t x) ad (t x) are give fuctis,
More informationMATH Midterm Examination Victor Matveev October 26, 2016
MATH 33- Midterm Examiati Victr Matveev Octber 6, 6. (5pts, mi) Suppse f(x) equals si x the iterval < x < (=), ad is a eve peridic extesi f this fucti t the rest f the real lie. Fid the csie series fr
More informationProbabilistic linguistic TODIM approach for multiple attribute decision-making
Graul. Cmput. (07) : 4 DOI 0.007/s4066-07-0047-4 ORIGINAL PAPER Prbabilistic liguistic TODIM apprach fr multiple attribute decisi-makig Peide Liu Xili Yu Received: 9 April 07 / Accepted: 5 July 07 / Published
More informationResult on the Convergence Behavior of Solutions of Certain System of Third-Order Nonlinear Differential Equations
Iteratial Jural f Mer Nliear Thery a Applicati, 6, 5, 8-58 Publishe Olie March 6 i SciRes http://wwwscirprg/jural/ijmta http://xirg/6/ijmta655 Result the Cvergece Behavir f Slutis f Certai System f Thir-Orer
More informationTHE MATRIX VERSION FOR THE MULTIVARIABLE HUMBERT POLYNOMIALS
Misklc Mathematical Ntes HU ISSN 1787-2405 Vl. 13 (2012), N. 2, pp. 197 208 THE MATRI VERSION FOR THE MULTIVARIABLE HUMBERT POLYNOMIALS RABİA AKTAŞ, BAYRAM ÇEKIM, AN RECEP ŞAHI Received 4 May, 2011 Abstract.
More informationD.S.G. POLLOCK: TOPICS IN TIME-SERIES ANALYSIS STATISTICAL FOURIER ANALYSIS
STATISTICAL FOURIER ANALYSIS The Furier Represetati f a Sequece Accrdig t the basic result f Furier aalysis, it is always pssible t apprximate a arbitrary aalytic fucti defied ver a fiite iterval f the
More informationRMO Sample Paper 1 Solutions :
RMO Sample Paper Slutis :. The umber f arragemets withut ay restricti = 9! 3!3!3! The umber f arragemets with ly e set f the csecutive 3 letters = The umber f arragemets with ly tw sets f the csecutive
More informationThe Choquet Integral with Respect to Fuzzy-Valued Set Functions
The Choquet Itegral with Respect to Fuzzy-Valued Set Fuctios Weiwei Zhag Abstract The Choquet itegral with respect to real-valued oadditive set fuctios, such as siged efficiecy measures, has bee used i
More informationx 2 x 3 x b 0, then a, b, c log x 1 log z log x log y 1 logb log a dy 4. dx As tangent is perpendicular to the x axis, slope
The agle betwee the tagets draw t the parabla y = frm the pit (-,) 5 9 6 Here give pit lies the directri, hece the agle betwee the tagets frm that pit right agle Ratig :EASY The umber f values f c such
More information, the random variable. and a sample size over the y-values 0:1:10.
Lecture 3 (4//9) 000 HW PROBLEM 3(5pts) The estimatr i (c) f PROBLEM, p 000, where { } ~ iid bimial(,, is 000 e f the mst ppular statistics It is the estimatr f the ppulati prprti I PROBLEM we used simulatis
More informationBounds for the Positive nth-root of Positive Integers
Pure Mathematical Scieces, Vol. 6, 07, o., 47-59 HIKARI Ltd, www.m-hikari.com https://doi.org/0.988/pms.07.7 Bouds for the Positive th-root of Positive Itegers Rachid Marsli Mathematics ad Statistics Departmet
More informationDistributed Trajectory Generation for Cooperative Multi-Arm Robots via Virtual Force Interactions
862 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS PART B: CYBERNETICS, VOL. 27, NO. 5, OCTOBER 1997 Distributed Trajectry Geerati fr Cperative Multi-Arm Rbts via Virtual Frce Iteractis Tshi Tsuji,
More informationENGI 4421 Central Limit Theorem Page Central Limit Theorem [Navidi, section 4.11; Devore sections ]
ENGI 441 Cetral Limit Therem Page 11-01 Cetral Limit Therem [Navidi, secti 4.11; Devre sectis 5.3-5.4] If X i is t rmally distributed, but E X i, V X i ad is large (apprximately 30 r mre), the, t a gd
More informationSolutions to Midterm II. of the following equation consistent with the boundary condition stated u. y u x y
Sltis t Midterm II Prblem : (pts) Fid the mst geeral slti ( f the fllwig eqati csistet with the bdary cditi stated y 3 y the lie y () Slti : Sice the system () is liear the slti is give as a sperpsiti
More informationResearch Article A New Second-Order Iteration Method for Solving Nonlinear Equations
Abstract ad Applied Aalysis Volume 2013, Article ID 487062, 4 pages http://dx.doi.org/10.1155/2013/487062 Research Article A New Secod-Order Iteratio Method for Solvig Noliear Equatios Shi Mi Kag, 1 Arif
More informationDirectional Duality Theory
Suther Illiis Uiversity Carbdale OpeSIUC Discussi Papers Departmet f Ecmics 2004 Directial Duality Thery Daiel Primt Suther Illiis Uiversity Carbdale Rlf Fare Oreg State Uiversity Fllw this ad additial
More informationAn epsilon-based measure of efficiency in DEA revisited -A third pole of technical efficiency-
GRIPS Plicy Ifrmati Ceter Discussi Paper : 09-2 A epsil-based measure f efficiecy i DEA revisited -A third ple f techical efficiecy- Karu Te Natial Graduate Istitute fr Plicy Studies 7-22- Rppgi, Miat-ku,
More informationIP Reference guide for integer programming formulations.
IP Referece guide for iteger programmig formulatios. by James B. Orli for 15.053 ad 15.058 This documet is iteded as a compact (or relatively compact) guide to the formulatio of iteger programs. For more
More informationROOT FINDING FOR NONLINEAR EQUATIONS
ROOT FINDING FOR NONLINEAR EQUATIONS Tele Gemecu Departmet matematics, Adama Sciece ad Teclgy Uiversity, Etipia Abstract Nliear equatis /systems appear i mst sciece ad egieerig mdels Fr eample, e slvig
More informationA New Method for Solving Integer Linear. Programming Problems with Fuzzy Variables
Appled Mathematcal Scences, Vl. 4, 00, n. 0, 997-004 A New Methd fr Slvng Integer Lnear Prgrammng Prblems wth Fuzzy Varables P. Pandan and M. Jayalakshm Department f Mathematcs, Schl f Advanced Scences,
More informationNew Results for the Fibonacci Sequence Using Binet s Formula
Iteratioal Mathematical Forum, Vol. 3, 208, o. 6, 29-266 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/imf.208.832 New Results for the Fiboacci Sequece Usig Biet s Formula Reza Farhadia Departmet
More informationFourier Series & Fourier Transforms
Experimet 1 Furier Series & Furier Trasfrms MATLAB Simulati Objectives Furier aalysis plays a imprtat rle i cmmuicati thery. The mai bjectives f this experimet are: 1) T gai a gd uderstadig ad practice
More information5.1 Two-Step Conditional Density Estimator
5.1 Tw-Step Cditial Desity Estimatr We ca write y = g(x) + e where g(x) is the cditial mea fucti ad e is the regressi errr. Let f e (e j x) be the cditial desity f e give X = x: The the cditial desity
More informationLecture 21: Signal Subspaces and Sparsity
ECE 830 Fall 00 Statistical Sigal Prcessig istructr: R. Nwak Lecture : Sigal Subspaces ad Sparsity Sigal Subspaces ad Sparsity Recall the classical liear sigal mdel: X = H + w, w N(0, where S = H, is a
More informationFrequency-Domain Study of Lock Range of Injection-Locked Non- Harmonic Oscillators
0 teratial Cferece mage Visi ad Cmputig CVC 0 PCST vl. 50 0 0 ACST Press Sigapre DO: 0.776/PCST.0.V50.6 Frequecy-Dmai Study f Lck Rage f jecti-lcked N- armic Oscillatrs Yushi Zhu ad Fei Yua Departmet f
More informationMatching a Distribution by Matching Quantiles Estimation
Jural f the America Statistical Assciati ISSN: 0162-1459 (Prit) 1537-274X (Olie) Jural hmepage: http://www.tadflie.cm/li/uasa20 Matchig a Distributi by Matchig Quatiles Estimati Niklas Sgurpuls, Qiwei
More informationA Hartree-Fock Calculation of the Water Molecule
Chemistry 460 Fall 2017 Dr. Jea M. Stadard Nvember 29, 2017 A Hartree-Fck Calculati f the Water Mlecule Itrducti A example Hartree-Fck calculati f the water mlecule will be preseted. I this case, the water
More informationCOMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q n } Sang Pyo Jun
Korea J. Math. 23 2015) No. 3 pp. 371 377 http://dx.doi.org/10.11568/kjm.2015.23.3.371 COMPLEX FACTORIZATIONS OF THE GENERALIZED FIBONACCI SEQUENCES {q } Sag Pyo Ju Abstract. I this ote we cosider a geeralized
More informationEuropean Journal of Operational Research
Eurpea Jural f Operatial Research 232 (2014) 4 4 Ctets lists available at ScieceDirect Eurpea Jural f Operatial Research jural hmepage www.elsevier.cm/lcate/ejr Discrete Optimizati A aalytical cmparis
More informationALE 26. Equilibria for Cell Reactions. What happens to the cell potential as the reaction proceeds over time?
Name Chem 163 Secti: Team Number: AL 26. quilibria fr Cell Reactis (Referece: 21.4 Silberberg 5 th editi) What happes t the ptetial as the reacti prceeds ver time? The Mdel: Basis fr the Nerst quati Previusly,
More informationDesign and Implementation of Cosine Transforms Employing a CORDIC Processor
C16 1 Desig ad Implemetati f Csie Trasfrms Emplyig a CORDIC Prcessr Sharaf El-Di El-Nahas, Ammar Mttie Al Hsaiy, Magdy M. Saeb Arab Academy fr Sciece ad Techlgy, Schl f Egieerig, Alexadria, EGYPT ABSTRACT
More informationChapter 1: Fundamentals
Chapter 1: Fudametals 1.1 Real Numbers Irratial umbers are real umbers that cat be expressed as ratis f itegers. That such umbers exist was a prfud embarrassmet t the Pythagrea brtherhd, ad they are said
More informationPreliminary Test Single Stage Shrinkage Estimator for the Scale Parameter of Gamma Distribution
America Jural f Mathematics ad Statistics, (3): 3-3 DOI:.593/j.ajms.3. Prelimiary Test Sigle Stage Shrikage Estimatr fr the Scale Parameter f Gamma Distributi Abbas Najim Salma,*, Aseel Hussei Ali, Mua
More informationNew Inequalities of Hermite-Hadamard-like Type for Functions whose Second Derivatives in Absolute Value are Convex
It. Joural of Math. Aalysis, Vol. 8, 1, o. 16, 777-791 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.1988/ijma.1.1 New Ieualities of Hermite-Hadamard-like Type for Fuctios whose Secod Derivatives i
More informationConvergence of Random SP Iterative Scheme
Applied Mathematical Scieces, Vol. 7, 2013, o. 46, 2283-2293 HIKARI Ltd, www.m-hikari.com Covergece of Radom SP Iterative Scheme 1 Reu Chugh, 2 Satish Narwal ad 3 Vivek Kumar 1,2,3 Departmet of Mathematics,
More informationSolutions. Definitions pertaining to solutions
Slutis Defiitis pertaiig t slutis Slute is the substace that is disslved. It is usually preset i the smaller amut. Slvet is the substace that des the disslvig. It is usually preset i the larger amut. Slubility
More informationThe Complexity of Translation Membership for Macro Tree Transducers
The Cmplexity f Traslati Membership fr Macr Tree Trasducers Kazuhir Iaba The Uiversity f Tky kiaba@is.s.u-tky.ac.jp Sebastia Maeth NICTA ad Uiversity f New Suth Wales sebastia.maeth@icta.cm.au ABSTRACT
More informationON FREE RING EXTENSIONS OF DEGREE N
I terat. J. Math. & Mah. Sci. Vl. 4 N. 4 (1981) 703-709 703 ON FREE RING EXTENSIONS OF DEGREE N GEORGE SZETO Mathematics Departmet Bradley Uiversity Peria, Illiis 61625 U.S.A. (Received Jue 25, 1980) ABSTRACT.
More informationWeakly Connected Closed Geodetic Numbers of Graphs
Iteratioal Joural of Mathematical Aalysis Vol 10, 016, o 6, 57-70 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/101988/ijma01651193 Weakly Coected Closed Geodetic Numbers of Graphs Rachel M Pataga 1, Imelda
More informationReview for cumulative test
Hrs Math 3 review prblems Jauary, 01 cumulative: Chapters 1- page 1 Review fr cumulative test O Mday, Jauary 7, Hrs Math 3 will have a curse-wide cumulative test cverig Chapters 1-. Yu ca expect the test
More informationROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND
Pacific-Asia Joural of Mathematics, Volume 5, No., Jauary-Jue 20 ROLL CUTTING PROBLEMS UNDER STOCHASTIC DEMAND SHAKEEL JAVAID, Z. H. BAKHSHI & M. M. KHALID ABSTRACT: I this paper, the roll cuttig problem
More informationDefinitions and Theorems. where x are the decision variables. c, b, and a are constant coefficients.
Defiitios ad Theorems Remember the scalar form of the liear programmig problem, Miimize, Subject to, f(x) = c i x i a 1i x i = b 1 a mi x i = b m x i 0 i = 1,2,, where x are the decisio variables. c, b,
More informationA Study on Fuzzy Complex Linear. Programming Problems
It. J. Cotemp. Math. Scieces, Vol. 7, 212, o. 19, 897-98 A Study o Fuzzy Complex Liear Programmig Problems Youess, E. A. (1) ad Mekawy, I. M. (2) (1) Departmet of Mathematics, Faculty of Sciece Tata Uiversity,
More informationComparison Study of Series Approximation. and Convergence between Chebyshev. and Legendre Series
Applied Mathematical Scieces, Vol. 7, 03, o. 6, 3-337 HIKARI Ltd, www.m-hikari.com http://d.doi.org/0.988/ams.03.3430 Compariso Study of Series Approimatio ad Covergece betwee Chebyshev ad Legedre Series
More informationGusztav Morvai. Hungarian Academy of Sciences Goldmann Gyorgy ter 3, April 22, 1998
A simple radmized algrithm fr csistet sequetial predicti f ergdic time series Laszl Gyr Departmet f Cmputer Sciece ad Ifrmati Thery Techical Uiversity f Budapest 5 Stczek u., Budapest, Hugary gyrfi@if.bme.hu
More informationA Simplified Nonlinear Generalized Maxwell Model for Predicting the Time Dependent Behavior of Viscoelastic Materials
Wrld Jural f Mechaics, 20,, 58-67 di:0.4236/wj.20.302 Published Olie Jue 20 (http://www.scirp.rg/jural/wj) A Siplified Nliear Geeralized Maxwell Mdel fr Predictig the Tie Depedet Behavir f Viscelastic
More informationFull algebra of generalized functions and non-standard asymptotic analysis
Full algebra f geeralized fuctis ad -stadard asympttic aalysis Tdr D. Tdrv Has Veraeve Abstract We cstruct a algebra f geeralized fuctis edwed with a caical embeddig f the space f Schwartz distributis.
More informationModified Decomposition Method by Adomian and. Rach for Solving Nonlinear Volterra Integro- Differential Equations
Noliear Aalysis ad Differetial Equatios, Vol. 5, 27, o. 4, 57-7 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ade.27.62 Modified Decompositio Method by Adomia ad Rach for Solvig Noliear Volterra Itegro-
More informationPipe Networks - Hardy Cross Method Page 1. Pipe Networks
Pie Netwrks - Hardy Crss etd Page Pie Netwrks Itrducti A ie etwrk is a itercected set f ies likig e r mre surces t e r mre demad (delivery) its, ad ca ivlve ay umber f ies i series, bracig ies, ad arallel
More informationAuthor. Introduction. Author. o Asmir Tobudic. ISE 599 Computational Modeling of Expressive Performance
ISE 599 Cmputatial Mdelig f Expressive Perfrmace Playig Mzart by Aalgy: Learig Multi-level Timig ad Dyamics Strategies by Gerhard Widmer ad Asmir Tbudic Preseted by Tsug-Ha (Rbert) Chiag April 5, 2006
More informationResearch Article Approximate Riesz Algebra-Valued Derivations
Abstract ad Applied Aalysis Volume 2012, Article ID 240258, 5 pages doi:10.1155/2012/240258 Research Article Approximate Riesz Algebra-Valued Derivatios Faruk Polat Departmet of Mathematics, Faculty of
More informationFunction representation of a noncommutative uniform algebra
Fucti represetati f a cmmutative uifrm algebra Krzysztf Jarsz Abstract. We cstruct a Gelfad type represetati f a real cmmutative Baach algebra A satisfyig f 2 = kfk 2, fr all f 2 A:. Itrducti A uifrm algebra
More informationOn Generalized Fibonacci Numbers
Applied Mathematical Scieces, Vol. 9, 215, o. 73, 3611-3622 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.5299 O Geeralized Fiboacci Numbers Jerico B. Bacai ad Julius Fergy T. Rabago Departmet
More informationOn Edge Regular Fuzzy Line Graphs
Iteratioal Joural of Computatioal ad Applied Mathematics ISSN 1819-4966 Volume 11, Number 2 (2016), pp 105-118 Research Idia Publicatios http://wwwripublicatiocom O Edge Regular Fuzz Lie Graphs K Radha
More informationDeclarative approach to cyclic steady state space refinement: periodic process scheduling
It J Adv Mauf Techl DOI 10.1007/s00170-013-4760-0 ORIGINAL ARTICLE Declarative apprach t cyclic steady state space refiemet: peridic prcess schedulig Grzegrz Bcewicz Zbigiew A. Baaszak Received: 16 April
More informationResearch Article Invariant Statistical Convergence of Sequences of Sets with respect to a Modulus Function
Hidawi Publishig Corporatio Abstract ad Applied Aalysis, Article ID 88020, 5 pages http://dx.doi.org/0.55/204/88020 Research Article Ivariat Statistical Covergece of Sequeces of Sets with respect to a
More informationAP Statistics Notes Unit Eight: Introduction to Inference
AP Statistics Ntes Uit Eight: Itrducti t Iferece Syllabus Objectives: 4.1 The studet will estimate ppulati parameters ad margis f errrs fr meas. 4.2 The studet will discuss the prperties f pit estimatrs,
More informationVariations on the theme of slacks-based measure of efficiency in DEA
GRIPS Plicy Ifrmati Ceter Dicui Paper : 8-4 Variati the theme f lac-baed meaure f efficiecy i DEA Karu Te Natial Graduate Ititute fr Plicy Studie 7-22- Rppgi, Miat-u, Ty 6-8677, Japa te@gripacp Abtract:
More informationComputational Intelligence and Application of Frame Theory in Communication Systems
America Jural Eieeri ad Applied Scieces Oriial Research Paper Cmputatial Itelliece ad Applicati Frame Thery i Cmmuicati Systems Rajupillai, K., S. Palaiammal ad 3 K. Bmmuraju Departmet Mathematics, Gvermet
More informationStable solutions for optimal reinsurance problems involving risk measures
Ivative Applicatis f O.R. Stable slutis fr ptimal reisurace prblems ivlvig risk measures Alejadr Balbás, Beatriz Balbás, Ati Heras Uiversity Carls III f Madrid, CL. Madrid, 6, 93 Getafe, Madrid, Spai Uiversity
More informationGraph Expansion and the Unique Games Conjecture
Graph xpasi ad the ique Games Cjecture rasad Raghavedra MSR New glad Cambridge, MA David Steurer ricet iversity ricet, NJ ABSTRACT The edge expasi f a subset f vertices S V i a graph G measures the fracti
More informationChristensen, Mads Græsbøll; Vera-Candeas, Pedro; Somasundaram, Samuel D.; Jakobsson, Andreas
Dwladed frm vb.aau.dk : April 12, 2019 Aalbrg Uiversitet Rbust Subspace-based Fudametal Frequecy Estimati Christese, Mads Græsbøll; Vera-Cadeas, Pedr; Smasudaram, Samuel D.; Jakbss, Adreas Published i:
More informationLinear Programming and the Simplex Method
Liear Programmig ad the Simplex ethod Abstract This article is a itroductio to Liear Programmig ad usig Simplex method for solvig LP problems i primal form. What is Liear Programmig? Liear Programmig is
More informationMULTIDIMENSIONAL EXTENICS THEORY
U.P.B. Sci. Bull., Series A, Vl. 75, Iss. 1, 213 ISSN 1223-727 MULTIDIMENSIONAL EXTENICS THEORY Ovidiu Ilie ŞANDRU 1, Luige VLǍDAREANU 2, Paul ŞCHIOPU 3, Victr VLǍDAREANU 4, Alexadra ŞANDRU 5 Î această
More informationStatistica Sinica 6(1996), SOME PROBLEMS ON THE ESTIMATION OF UNIMODAL DENSITIES Peter J. Bickel and Jianqing Fan University of California and U
Statistica Siica 6(996), 23-45 SOME PROBLEMS ON THE ESTIMATION OF UNIMODAL DENSITIES Peter J. Bickel ad Jiaqig Fa Uiversity f Califria ad Uiversity f Nrth Carlia Abstract: I this paper, we study, i sme
More information