Scheduling Jobs with a Common Due Date via Cooperative Game Theory
|
|
- Kory Sparks
- 6 years ago
- Views:
Transcription
1 Amerca Joural of Operato Reearch, 203, 3, Publhed Ole eptember 203 ( chedulg Job wth a Commo Due Date va Cooperatve Game Theory Irel Draga Uverty of Texa at Arlgto, Mathematc, Arlgto, UA Emal: draga@uta.edu Receved October 8, 202; reved December 30, 202; accepted Jauary 7, 203 Copyrght 203 Irel Draga. Th a ope acce artcle dtrbuted uder the Creatve Commo Attrbuto Lcee, whch permt uretrcted ue, dtrbuto, ad reproducto ay medum, provded the orgal work properly cted. ABTRACT Effcet value from Game Theory are ued, order to fd out a far allocato for a chedulg game aocated wth the problem of chedulg job wth a commo due date. A four pero game llutrate the bac dea ad the computatoal dffculte. Keyword: chedule; Effcet Value; Egaltara Value; Egaltara Noeparable Cotrbuto; hapley Value; Cot Excee; Lexcographc Orderg; Cot Leat quare Preucleolu. A chedulg Game ad mple oluto A mache may proce job, J, J 2,, J, wth the completo tme p, p2,, p, all potve umber. No two job ca be multaeouly doe, ad for all job there a commo due date d potve. Ay chedule a equece of job, ad o preempto allowed. The chedule determed by the umber C, N, the completo date of the job J. Ay devato from the due date wll be pealzed, ether a early or a late completo relatve to the due date. The total tme devato a chedule gve by C d. () N The uual chedulg problem to: fd out the chedule * for whch the total devato mmal. I [], J. J. Kaet olved the problem for the cae whe the um of completo tme maller tha, or equal to d, ad gave a algorthm for computg a chedule wth a mmal devato. Of coure, th algorthm may be ued to fd the total pealty for th chedule ad alo to fd the total pealty for ay mmal devato correpodg to ay ubet of job. Th make ee the cae whe the cot of the devato, early or late, are proportoal to ther ze. I the followg, we aume that the cot are equal to the pealte. A more geeral cae other tha Kaet ha bee olved by M. U. Ahmed ad P.. udararaghava [2]. I [3], N. G. Hall ad M. E. Poer coder mlar problem. The lterature coected to more geeral cae huge, ad the cocluo obtaed the preet paper ca be appled to mot other cae. For the preet dcuo, the mplet cae offered by Kaet algorthm, wth the above aumpto, good eough to ugget mlar approache all other cae, coecto wth a ew problem to be troduced below. Aumg that the grad coalto ha bee formed ad the total pealty for early ad late devato, w N, ha bee computed by ome algorthm, a ew problem : how much hould be a far dvdual pealty for each job? To awer the queto, we ow buld the followg cooperatve game wth traferable utlte: let N, 2,, be the et of player, the player be the cutomer orderg the job J, for each, 2,, Coder ay coalto of cutomer,, N,, ad otce that f, the the mmal chedule tart the correpodg job at d p, ad there wll be o devato from the due date. Therefore, f we deote the devato for coalto, by w, we have w 0. A algorthm for computg the mmal devato, for example Kaet algorthm, wll provde the total devato w 0, whe 2. I th way, we get a cooperatve TU game Nw,, whch we wat to dvde farly w N. To make the paper elf cotaed let u ketch Kaet algorthm whch wll be ued the example how be- Copyrght 203 cre.
2 440 I. DRAGAN low. Let be ay coalto ad deote by B, (before ), a equece of job whch were already elected, wth o-creag proceg tme ad the lat job edg at d ; alo, deote by A, (after ), aother equece of job whch were already elected, wth odecreag proceg tme ad tartg at d. Now, aume that we have B A, B A, Kaet algorthm cotug to buld the partto of a follow: f B A, elect the o-elected elemet of wth a maxmal proceg tme ad take t a the lat job B, (ow we have B A ). Further, f wth the ew B, we have B A, the chooe the o-elected job wth a maxmal proceg tme ad take t a the frt job A, (that tartg at d ). Repeat the procedure, electg alteratvely player B, the A, utl exhauted; the, the tme devato computed by formula () for the correpodg chedule ad the ame way for ay ubet of player. Example : Let J, J2, J3, J4 be a et of job to be proceed o a gle mache, wth the proceg tme p 2, p2 0, p3 8, p4 5, ad the due date d 39, uch that the Kaet codto how above hold. We ca compute for the et of player N, 2, 3, 4, ad t ubet, the total pealte. Kaet algorthm wll geerate the game w w 2 w3 w 4 0, w, 2 0, w,3 w 2,3 8, w w, 4w2, 4 w, 2, 38, w, 2, 45, w w 3, 4 5, w,3,4 2,3,4 3,, 2,3,4 28. Th correpod to the total devato of all coalto, ad our problem to: fd out how we hould dvde farly wn 28 amog the player? We tart by howg two mple oluto: the Egaltara allocato ad the Egaltara o-eparable cotrbuto allocato. Deotg the frt by x *, we get x* 7,7,7,7. Deotg the ecod by y*, whch gve geeral by formula y * w N w N wn wnw N j, N, jn (2) we get the margal cotrbuto M j wnwn j, j N, equal to 5, 5, 3, 0, o that the um make 53, ad from each margal cotrbuto we hould ubtract 25/4, to obta y*,,, Lookg at the charactertc value of our game, how example, we ee that the player ad 2 eem to be equal, whle player 3 ad 4 are weaker, hece the lat two hould be aked to pay maller dvdual pealte. The frt oluto doe ot eem to how th, whle the ecod eem more far, we hall ee a method below to compare the fare of the oluto. 2. Idvdual Pealte: et oluto, hapley Value To evaluate the fare of a poble oluto z, we may ue the exce fucto; however, here t eem more approprate to ue ome mlar fucto that we hall call the cot exce fucto. For ay coalto, N,, ad ay allocato z, the cot exce fucto, z z w. (3) Thee are the egatve of uual exce fucto, obvouly, we have 2 2 uch fucto, becaue for the grad coalto, for ay allocato, by defto we have N, z 0. I word, the cot exce the dfferece betwee what the coalto wll pay z to cotrbute a cloe a poble to the total pealty for herelf, whle t alo cotrbutg to the total pealty for N. The, what we wat to do to chooe the allocato z whch mmze all cot excee o the et of allocato. Note that the um of all cot excee a cotat, becaue for ay allocato z we have, z 2 wn w. N N Moreover, we ca defe the average cot exce w, z, 2 N whch by formula (4) doe ot deped o the allocato z Th mea that f the allocato of ome coalto creaed, the the allocato of at leat oe other coalto wll be decreaed. How the cot excee are ued to compare the fare of two allocato llutrated below. Example 2: Retur to example, ad wrte the cot excee for that game Nw,, ad ay allocato z z z 2, 3, z z z 8,, z z, 2, z z, 2 3, z z, 4, z z, 3 4, 2, 0, 3 (4) (5) Copyrght 203 cre.
3 I. DRAGAN 44 4, 4, z z z 5, 2,3, z 2, 4, z z z 5, 3, 4, z 2 4, 2, 4, z, 3, 4, z, 2, 3, z z z z2 z 8, z z 5, z z z 3, 3 4 2,3, 4, z z z z z z 8, 2 3 z z 5, Our problem to mmze all cot excee, whle we are o the et of allocato, that the effcecy codto hold. I other word, we wat to mmze the maxmal cot exce, ubject to effcecy codto, or to ue aother method to olve a mult objectve lear programmg problem. Let u try to evaluate the fare of the two allocato offered utl ow. For the Egaltara oluto, we ca compute the cot excee ad put them a vector of o creag excee, whch may be called the vector of uhappe, a the compoet are take the order of o creag uhappe x * 9,9,9,8,8,7,7,7,7,6,6,6,4,3. It follow that the mot uhappy coalto are the two pero coalto whch oe of the player player 4. For the Egaltara o eparable cotrbuto, we fd the vector of uhappe y * ,,,,,,,,,,,,,, ad 2. Moreover, we have alo larger tha y ad the mot uhappy coalto are x * *, that the mot uhappy coalto x * more uhappy tha the mot uhappy coalto y *. We ca ay that y *. better tha x *, or more far. Note that the ame thg could be ad f ome par of correpodg compoet the two uhappe vector are equal, but the frt oe whch dfferet maller y * tha y * L x*, x*. I th cae, we alo wrte where L mea the lexcographc order, ad read y * better tha x *. Utl ow, we have ee two mple oluto belogg to Game Theory, they are oe pot oluto becaue each oe provdg a uque oluto. Oe of the et oluto from Game Theory the CORE, whch for a cot game Nw, lke our the et CON, w (6) zr : N, z 0,, z 0, N,. Ay elemet of the CORE codered a a good allocato, becaue uch a allocato cover the total pealty for each coalto. Lookg at the two mple oluto of example, whch a ee example 2 have all excee o egatve, we ee that both are the CORE, but, of coure, there are other wth the ame property. Moreover, we ca alo ee that the um of excee equal 96, that the worth obtaed formula (4) for 4. The mot famou oe pot oluto, whch may alo be the CORE, the hapley Value, troduced [4], ad defed by a et of axom, decrbg ome bac properte requred for a far oluto. The hapley Value wa proved there to be gve by the formula N, w H :!! vv, N.! (7) Example 3: For the game codered example, we get H N, w 8,8,7,5. Computg the cot excee ad orderg them, we obta H 8,8,8,8,7,7,7,7,7,7,6,6,5,5. H y x We get * *, L L 2 Mmze f w, z, z w, N becaue the frt compoet of the uhappe vector are th order, hece the hapley Value better tha the Egaltara o eparable cotrbuto, whch better tha the Egaltara oluto. Note that th may ot be the cae for other game. Note alo that f the game large, the the hapley Value may ot be eay to compute. A algorthm baed upo the o called Average per capta formula, gve by the author [5], may be ued, a t wll be explaed the ext ecto ad the algorthm allowg eve a parallel computato of the hapley Value. mlar tuato may occur coecto wth the other value. 3. The Cot Leat quare Preucleolu I the followg, we may coder a a oluto the Leat quare Preucleolu of the game, troduced by L. Ruz, F. Valecao ad J. Zarzuelo [6]. Th mlar to the Preucleolu, troduced coecto wth the Nucleolu, due to D. chmedler [7], except that th wa defed by mea of the followg quadratc programmg problem ubject to (8) N, z 0. (9) Copyrght 203 cre.
4 442 I. DRAGAN By ug the Kuh-Tucker codto, (8), (9), t ha bee how that our problem ha a uque oluto, that the author called the Leat quare Preucleolu, amely w N LN, w aw a jw, N, jn (0) where 2 a w w, N,. () : A the cot excee are replacg the excee, we called th value the Cot Leat quare Preucleolu, but the expreo (0) ad () are the ame eve our cae. Example 4: Computg by (0) ad () th oluto for our game, we get L N, w,,, The vector of uhappe L * (ee foot-ote). Notce that the um of cot excee alo 96. Now, checkg the comparo of the ew oluto wth the other three oluto, we obta, LNw, H Nw y* x*, L L L that the hapley Value eem to be more far tha the other three value. Of coure, the chmedler Preucleolu may alo be computed; the computatoal method, due to A. Kopelowtz [8], alo how [9]. However, th clude a log computato for olvg a equece of lear optmzato problem. The Preucleolu would be the bet, by the defto of the value. Aother prcple may be ued to chooe the approprate allocato: for each allocato avalable, compute the dfferece betwee the cot for the mot uhappy coalto ad the happet coalto ad chooe the allocato that gve the mallet dfferece. uch a allocato would ot gve a hgh dfferece of cot betwee the happy ad the uhappy coalto. I our cae, we have dh 3, dl, dy* 5, d x * Notce that by the lat prcple the four value are ordered the ame way. 4. Cocluo The techology put together the preet paper apple to other chedulg problem whch the aocated chedulg game ca tll be geerated by ome algorthm. ome of the followg remark may help: ) The above dcuo wa llutrated by example, 2, 3, 4 relatve to a four pero game. If we have 5 job, uder the ame codto lke above, Kaet algorthm tll apple whe the Kaet aumpto hold. If the objectve dfferet, for example to mmze a weghted combato of devato relatve to a commo due date, the the algorthm by Hall ad Poer hould be ued to get the chedulg game. 2) A oo a the game avalable, the problem of dvdg farly the worth of the grad coalto the problem of choog a effcet value from Game Theory, for whch the computato could be doe. A all gleto have a zero worth, the Ceter of the mputato et [9] become the Egaltara value, whch geeral ot far. The Egaltara o-eparable cotrbuto may be a alteratve allocato. The hapley Value, whch ha a lot of properte derved from the axom, the mot preferred by almot all cett, but for more tha te player t become dffcult to compute. For uch large game t may be better to ue the formula gve by the author [5], called the Average per capta formula w w H N, w, N, (2) where w the average worth of coalto of ze, ad w the average worth of coalto of ze, that do ot cota player, wth w 0, N. It obvou that the tak ca be performed by team, ad each oe compute oe rato for oe. 3) I the computato of the Cot Nucleolu by Kopelowtz method [8,9] the paage from oe LP problem to the ext ot decrbed detal mot ource. The complemetary codto how whch cot excee hould rema cotat o all optmal oluto of the curret problem, ad hould be kept cotat the ext problem. Thee equato hould replace the correpodg equalte of the curret problem ad rema atfed utl we meet a problem whch ha a uque oluto. The oluto of the quadratc programmg problem eem eaer to compute. The geeralzed ucleolu may be ued a a oluto of ay Multcrtera Lear Programmg problem, a how by the author [0], workg wth a three pero game. The bac dea appeared a former paper of the author [], a well a the more recet paper by E. March ad J. A. Ovedo [2] L *,,,,,,,,,,,,, Copyrght 203 cre.
5 I. DRAGAN 443 REFERENCE [] J. J. Kaet, Mmzg the Average Devato of Job Completo Tme about a Commo Due Date, Naval Reearch Logtc Quarterly, Vol. 28, No. 4, 98, pp do:0.002/av [2] M. U. Ahmed ad P.. udararaghava, Mmzg the Weghted um of Late ad Early Completo Pealte a gle Mache, IEEE Traacto, Vol. 22, No. 3, 990, pp do:0.080/ [3] N. G. Hall ad M. E. Poer, Earle-Tarde chedulg Problem, I: Weghted Devato of the Completo Tme about a Commo Due Date, Operato Reearch, Vol. 39, No. 5, 99, pp [4] L.. hapley, A Value for -Pero Game, Aal of Mathematc, Vol. 28, 953, pp [5] I. Draga, A Average per Capta Formula for the hapley Value, Lberta Mathematca, Vol. 2, 992, pp [6] L. Ruz, F. Valecao ad J. Zarzuelo, The Leat quare Preucleolu ad the Leat quare Nucleolu, Two Value for TU Game Baed o the Exce Vector, Iteratoal Joural of Game Theory, Vol. 25, No., 996, pp [7] D. chmedler, The Nucleolu of a Charactertc Fucto Game, IAM Joural o Appled Mathematc, Vol. 7, No. 6, 967, pp do:0.37/0707 [8] A. Kopelowtz, Computato of the Kerel of mple Game ad the Nucleolu of -Pero Game, RM 3, Hebrew Uverty of Jerualem, Jerualem, 967. [9] G. Owe, Game Theory, 3rd Edto, Academc Pre, New York, 995. [0] I. Draga, A Game Theoretc Approach for olvg Multobjectve Lear Programmg Problem, Lberta Mathematca, Vol. 30, 200, pp [] I. Draga, A Game Theoretc Approach for olvg Multobjectve Lear Programmg Problem: A Applcato to a Traffc Problem, Quader de Grupp d Rcerca CNR, Pa, 98 [2] E. March ad J. A. Ovedo, Lexcographc Optmalty the Multple Objectve Lear Programmg: The Nucleolar oluto, Europea Joural of Operatoal Reearch, Vol. 57, No. 3, 992, pp do:0.06/ (92)90347-c Copyrght 203 cre.
CS473-Algorithms I. Lecture 12b. Dynamic Tables. CS 473 Lecture X 1
CS473-Algorthm I Lecture b Dyamc Table CS 473 Lecture X Why Dyamc Table? I ome applcato: We do't kow how may object wll be tored a table. We may allocate pace for a table But, later we may fd out that
More informationLinear Approximating to Integer Addition
Lear Approxmatg to Iteger Addto L A-Pg Bejg 00085, P.R. Cha apl000@a.com Abtract The teger addto ofte appled cpher a a cryptographc mea. I th paper we wll preet ome reult about the lear approxmatg for
More informationReaction Time VS. Drug Percentage Subject Amount of Drug Times % Reaction Time in Seconds 1 Mary John Carl Sara William 5 4
CHAPTER Smple Lear Regreo EXAMPLE A expermet volvg fve ubject coducted to determe the relatohp betwee the percetage of a certa drug the bloodtream ad the legth of tme t take the ubject to react to a tmulu.
More informationSolving Constrained Flow-Shop Scheduling. Problems with Three Machines
It J Cotemp Math Sceces, Vol 5, 2010, o 19, 921-929 Solvg Costraed Flow-Shop Schedulg Problems wth Three Maches P Pada ad P Rajedra Departmet of Mathematcs, School of Advaced Sceces, VIT Uversty, Vellore-632
More informationROOT-LOCUS ANALYSIS. Lecture 11: Root Locus Plot. Consider a general feedback control system with a variable gain K. Y ( s ) ( ) K
ROOT-LOCUS ANALYSIS Coder a geeral feedback cotrol yte wth a varable ga. R( Y( G( + H( Root-Locu a plot of the loc of the pole of the cloed-loop trafer fucto whe oe of the yte paraeter ( vared. Root locu
More informationSimple Linear Regression Analysis
LINEAR REGREION ANALYSIS MODULE II Lecture - 5 Smple Lear Regreo Aaly Dr Shalabh Departmet of Mathematc Stattc Ida Ittute of Techology Kapur Jot cofdece rego for A jot cofdece rego for ca alo be foud Such
More informationInternational Journal of Pure and Applied Sciences and Technology
It J Pure Appl Sc Techol, () (00), pp 79-86 Iteratoal Joural of Pure ad Appled Scece ad Techology ISSN 9-607 Avalable ole at wwwjopaaat Reearch Paper Some Stroger Chaotc Feature of the Geeralzed Shft Map
More informationThird handout: On the Gini Index
Thrd hadout: O the dex Corrado, a tala statstca, proposed (, 9, 96) to measure absolute equalt va the mea dfferece whch s defed as ( / ) where refers to the total umber of dvduals socet. Assume that. The
More informationCollapsing to Sample and Remainder Means. Ed Stanek. In order to collapse the expanded random variables to weighted sample and remainder
Collapg to Saple ad Reader Mea Ed Staek Collapg to Saple ad Reader Average order to collape the expaded rado varable to weghted aple ad reader average, we pre-ultpled by ( M C C ( ( M C ( M M M ( M M M,
More informationEuropean Journal of Mathematics and Computer Science Vol. 5 No. 2, 2018 ISSN
Europea Joural of Mathematc ad Computer Scece Vol. 5 o., 018 ISS 059-9951 APPLICATIO OF ASYMPTOTIC DISTRIBUTIO OF MA-HITEY STATISTIC TO DETERMIE THE DIFFERECE BETEE THE SYSTOLIC BLOOD PRESSURE OF ME AD
More informationEuropean Journal of Mathematics and Computer Science Vol. 5 No. 2, 2018 ISSN
Europea Joural of Mathematc ad Computer Scece Vol. 5 o., 018 ISS 059-9951 APPLICATIO OF ASYMPTOTIC DISTRIBUTIO OF MA-HITEY STATISTIC TO DETERMIE THE DIFFERECE BETEE THE SYSTOLIC BLOOD PRESSURE OF ME AD
More information1. a. Houston Chronicle, Des Moines Register, Chicago Tribune, Washington Post
Homework Soluto. Houto Chrocle, De Moe Regter, Chcago Trbue, Wahgto Pot b. Captal Oe, Campbell Soup, Merrll Lych, Pultzer c. Bll Japer, Kay Reke, Hele Ford, Davd Meedez d..78,.44, 3.5, 3.04 5. No, the
More information1. Linear second-order circuits
ear eco-orer crcut Sere R crcut Dyamc crcut cotag two capactor or two uctor or oe uctor a oe capactor are calle the eco orer crcut At frt we coer a pecal cla of the eco-orer crcut, amely a ere coecto of
More informationEconometric Methods. Review of Estimation
Ecoometrc Methods Revew of Estmato Estmatg the populato mea Radom samplg Pot ad terval estmators Lear estmators Ubased estmators Lear Ubased Estmators (LUEs) Effcecy (mmum varace) ad Best Lear Ubased Estmators
More informationTrignometric Inequations and Fuzzy Information Theory
Iteratoal Joural of Scetfc ad Iovatve Mathematcal Reearch (IJSIMR) Volume, Iue, Jauary - 0, PP 00-07 ISSN 7-07X (Prt) & ISSN 7- (Ole) www.arcjoural.org Trgometrc Iequato ad Fuzzy Iformato Theory P.K. Sharma,
More informationREVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION
REVIEW OF SIMPLE LINEAR REGRESSION SIMPLE LINEAR REGRESSION I lear regreo, we coder the frequecy dtrbuto of oe varable (Y) at each of everal level of a ecod varable (X). Y kow a the depedet varable. The
More information( ) Thermal noise ktb (T is absolute temperature in kelvin, B is bandwidth, k is Boltzamann s constant) Shot noise
OISE Thermal oe ktb (T abolute temperature kelv, B badwdth, k Boltzama cotat) 3 k.38 0 joule / kelv ( joule /ecod watt) ( ) v ( freq) 4kTB Thermal oe refer to the ketc eergy of a body of partcle a a reult
More informationOn the energy of complement of regular line graphs
MATCH Coucato Matheatcal ad Coputer Chetry MATCH Cou Math Coput Che 60 008) 47-434 ISSN 0340-653 O the eergy of copleet of regular le graph Fateeh Alaghpour a, Baha Ahad b a Uverty of Tehra, Tehra, Ira
More informationFeature Selection: Part 2. 1 Greedy Algorithms (continued from the last lecture)
CSE 546: Mache Learg Lecture 6 Feature Selecto: Part 2 Istructor: Sham Kakade Greedy Algorthms (cotued from the last lecture) There are varety of greedy algorthms ad umerous amg covetos for these algorthms.
More informationOrdinary Least Squares Regression. Simple Regression. Algebra and Assumptions.
Ordary Least Squares egresso. Smple egresso. Algebra ad Assumptos. I ths part of the course we are gog to study a techque for aalysg the lear relatoshp betwee two varables Y ad X. We have pars of observatos
More informationECONOMETRIC THEORY. MODULE VIII Lecture - 26 Heteroskedasticity
ECONOMETRIC THEORY MODULE VIII Lecture - 6 Heteroskedastcty Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur . Breusch Paga test Ths test ca be appled whe the replcated data
More informationOn a Truncated Erlang Queuing System. with Bulk Arrivals, Balking and Reneging
Appled Mathematcal Scece Vol. 3 9 o. 3 3-3 O a Trucated Erlag Queug Sytem wth Bul Arrval Balg ad Reegg M. S. El-aoumy ad M. M. Imal Departmet of Stattc Faculty Of ommerce Al- Azhar Uverty. Grl Brach Egypt
More informationSome distances and sequences in a weighted graph
IOSR Joural of Mathematc (IOSR-JM) e-issn: 78-578 p-issn: 19 765X PP 7-15 wwworjouralorg Some dtace ad equece a weghted graph Jll K Mathew 1, Sul Mathew Departmet of Mathematc Federal Ittute of Scece ad
More informationFunctions of Random Variables
Fuctos of Radom Varables Chapter Fve Fuctos of Radom Varables 5. Itroducto A geeral egeerg aalyss model s show Fg. 5.. The model output (respose) cotas the performaces of a system or product, such as weght,
More information{ }{ ( )} (, ) = ( ) ( ) ( ) Chapter 14 Exercises in Sampling Theory. Exercise 1 (Simple random sampling): Solution:
Chapter 4 Exercses Samplg Theory Exercse (Smple radom samplg: Let there be two correlated radom varables X ad A sample of sze s draw from a populato by smple radom samplg wthout replacemet The observed
More informationMulti Objective Fuzzy Inventory Model with. Demand Dependent Unit Cost and Lead Time. Constraints A Karush Kuhn Tucker Conditions.
It. Joural of Math. Aalyss, Vol. 8, 204, o. 4, 87-93 HIKARI Ltd, www.m-hkar.com http://dx.do.org/0.2988/jma.204.30252 Mult Objectve Fuzzy Ivetory Model wth Demad Depedet Ut Cost ad Lead Tme Costrats A
More informationMultiple Choice Test. Chapter Adequacy of Models for Regression
Multple Choce Test Chapter 06.0 Adequac of Models for Regresso. For a lear regresso model to be cosdered adequate, the percetage of scaled resduals that eed to be the rage [-,] s greater tha or equal to
More informationTheory study about quarter-wave-stack dielectric mirrors
Theor tud about quarter-wave-tack delectrc rror Stratfed edu tratted reflected reflected Stratfed edu tratted cdet cdet T T Frt, coder a wave roagato a tratfed edu. A we kow, a arbtrarl olared lae wave
More informationOn the Interval Zoro Symmetric Single Step. Procedure IZSS1-5D for the Simultaneous. Bounding of Real Polynomial Zeros
It. Joural of Math. Aalyss, Vol. 7, 2013, o. 59, 2947-2951 HIKARI Ltd, www.m-hkar.com http://dx.do.org/10.12988/ma.2013.310259 O the Iterval Zoro Symmetrc Sgle Step Procedure IZSS1-5D for the Smultaeous
More information10.2 Series. , we get. which is called an infinite series ( or just a series) and is denoted, for short, by the symbol. i i n
0. Sere I th ecto, we wll troduce ere tht wll be dcug for the ret of th chpter. Wht ere? If we dd ll term of equece, we get whch clled fte ere ( or jut ere) d deoted, for hort, by the ymbol or Doe t mke
More informationQR Factorization and Singular Value Decomposition COS 323
QR Factorzato ad Sgular Value Decomposto COS 33 Why Yet Aother Method? How do we solve least-squares wthout currg codto-squarg effect of ormal equatos (A T A A T b) whe A s sgular, fat, or otherwse poorly-specfed?
More informationLebesgue Measure of Generalized Cantor Set
Aals of Pure ad Appled Mathematcs Vol., No.,, -8 ISSN: -8X P), -888ole) Publshed o 8 May www.researchmathsc.org Aals of Lebesgue Measure of Geeralzed ator Set Md. Jahurul Islam ad Md. Shahdul Islam Departmet
More informationSimple Linear Regression
Statstcal Methods I (EST 75) Page 139 Smple Lear Regresso Smple regresso applcatos are used to ft a model descrbg a lear relatoshp betwee two varables. The aspects of least squares regresso ad correlato
More informationINEQUALITIES USING CONVEX COMBINATION CENTERS AND SET BARYCENTERS
Joural of Mathematcal Scece: Advace ad Alcato Volume 24, 23, Page 29-46 INEQUALITIES USING CONVEX COMBINATION CENTERS AND SET BARYCENTERS ZLATKO PAVIĆ Mechacal Egeerg Faculty Slavok Brod Uverty of Ojek
More informationEVALUATION OF PERFORMANCE MEASURES OF FMS Bottleneck Model. Part mix Mix of the various part or product styles produced by the system
Natoal Ittute of Techology Calcut Deartmet of Mechacal Egeerg EVALUATION OF PERFORMANCE MEASURES OF FMS Bottleeck Model Provde tartg etmate of FMS deg arameter uch a roducto rate ad umber of worktato Bottleeck
More informationSummary of the lecture in Biostatistics
Summary of the lecture Bostatstcs Probablty Desty Fucto For a cotuos radom varable, a probablty desty fucto s a fucto such that: 0 dx a b) b a dx A probablty desty fucto provdes a smple descrpto of the
More informationGeneralization of the Dissimilarity Measure of Fuzzy Sets
Iteratoal Mathematcal Forum 2 2007 o. 68 3395-3400 Geeralzato of the Dssmlarty Measure of Fuzzy Sets Faramarz Faghh Boformatcs Laboratory Naobotechology Research Ceter vesa Research Isttute CECR Tehra
More informationQuiz 1- Linear Regression Analysis (Based on Lectures 1-14)
Quz - Lear Regreo Aaly (Baed o Lecture -4). I the mple lear regreo model y = β + βx + ε, wth Tme: Hour Ε ε = Ε ε = ( ) 3, ( ), =,,...,, the ubaed drect leat quare etmator ˆβ ad ˆβ of β ad β repectvely,
More informationOn Modified Interval Symmetric Single-Step Procedure ISS2-5D for the Simultaneous Inclusion of Polynomial Zeros
It. Joural of Math. Aalyss, Vol. 7, 2013, o. 20, 983-988 HIKARI Ltd, www.m-hkar.com O Modfed Iterval Symmetrc Sgle-Step Procedure ISS2-5D for the Smultaeous Icluso of Polyomal Zeros 1 Nora Jamalud, 1 Masor
More informationInvestigating Cellular Automata
Researcher: Taylor Dupuy Advsor: Aaro Wootto Semester: Fall 4 Ivestgatg Cellular Automata A Overvew of Cellular Automata: Cellular Automata are smple computer programs that geerate rows of black ad whte
More information1 Onto functions and bijections Applications to Counting
1 Oto fuctos ad bectos Applcatos to Coutg Now we move o to a ew topc. Defto 1.1 (Surecto. A fucto f : A B s sad to be surectve or oto f for each b B there s some a A so that f(a B. What are examples of
More informationIRREDUCIBLE COVARIANT REPRESENTATIONS ASSOCIATED TO AN R-DISCRETE GROUPOID
UPB Sc Bull Sere A Vol 69 No 7 ISSN 3-77 IRREDUCIBLE COVARIANT REPRESENTATIONS ASSOCIATED TO AN R-DISCRETE GROUPOID Roxaa VIDICAN Ue perech covarate poztv defte ( T ) relatv la u grupod r-dcret G e poate
More informationA Result of Convergence about Weighted Sum for Exchangeable Random Variable Sequence in the Errors-in-Variables Model
AMSE JOURNALS-AMSE IIETA publcato-17-sere: Advace A; Vol. 54; N ; pp 3-33 Submtted Mar. 31, 17; Reved Ju. 11, 17, Accepted Ju. 18, 17 A Reult of Covergece about Weghted Sum for Exchageable Radom Varable
More informationLecture 3 Probability review (cont d)
STATS 00: Itroducto to Statstcal Iferece Autum 06 Lecture 3 Probablty revew (cot d) 3. Jot dstrbutos If radom varables X,..., X k are depedet, the ther dstrbuto may be specfed by specfyg the dvdual dstrbuto
More informationEstimation of Stress- Strength Reliability model using finite mixture of exponential distributions
Iteratoal Joural of Computatoal Egeerg Research Vol, 0 Issue, Estmato of Stress- Stregth Relablty model usg fte mxture of expoetal dstrbutos K.Sadhya, T.S.Umamaheswar Departmet of Mathematcs, Lal Bhadur
More informationENGI 3423 Simple Linear Regression Page 12-01
ENGI 343 mple Lear Regresso Page - mple Lear Regresso ometmes a expermet s set up where the expermeter has cotrol over the values of oe or more varables X ad measures the resultg values of aother varable
More informationResearch Article A New Iterative Method for Common Fixed Points of a Finite Family of Nonexpansive Mappings
Hdaw Publshg Corporato Iteratoal Joural of Mathematcs ad Mathematcal Sceces Volume 009, Artcle ID 391839, 9 pages do:10.1155/009/391839 Research Artcle A New Iteratve Method for Commo Fxed Pots of a Fte
More information12.2 Estimating Model parameters Assumptions: ox and y are related according to the simple linear regression model
1. Estmatg Model parameters Assumptos: ox ad y are related accordg to the smple lear regresso model (The lear regresso model s the model that says that x ad y are related a lear fasho, but the observed
More informationInternational Journal of Mathematical Archive-5(8), 2014, Available online through ISSN
Iteratoal Joural of Mathematcal Archve-5(8) 204 25-29 Avalable ole through www.jma.fo ISSN 2229 5046 COMMON FIXED POINT OF GENERALIZED CONTRACTION MAPPING IN FUZZY METRIC SPACES Hamd Mottagh Golsha* ad
More informationPTAS for Bin-Packing
CS 663: Patter Matchg Algorthms Scrbe: Che Jag /9/00. Itroducto PTAS for B-Packg The B-Packg problem s NP-hard. If we use approxmato algorthms, the B-Packg problem could be solved polyomal tme. For example,
More informationThe Mathematical Appendix
The Mathematcal Appedx Defto A: If ( Λ, Ω, where ( λ λ λ whch the probablty dstrbutos,,..., Defto A. uppose that ( Λ,,..., s a expermet type, the σ-algebra o λ λ λ are defed s deoted by ( (,,...,, σ Ω.
More informationA New Method for Decision Making Based on Soft Matrix Theory
Joural of Scetfc esearch & eports 3(5): 0-7, 04; rtcle o. JS.04.5.00 SCIENCEDOMIN teratoal www.scecedoma.org New Method for Decso Mag Based o Soft Matrx Theory Zhmg Zhag * College of Mathematcs ad Computer
More informationCOMPROMISE HYPERSPHERE FOR STOCHASTIC DOMINANCE MODEL
Sebasta Starz COMPROMISE HYPERSPHERE FOR STOCHASTIC DOMINANCE MODEL Abstract The am of the work s to preset a method of rakg a fte set of dscrete radom varables. The proposed method s based o two approaches:
More informationSupport vector machines II
CS 75 Mache Learg Lecture Support vector maches II Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Learl separable classes Learl separable classes: here s a hperplae that separates trag staces th o error
More informationKR20 & Coefficient Alpha Their equivalence for binary scored items
KR0 & Coeffcet Alpha Ther equvalece for bary cored tem Jue, 007 http://www.pbarrett.et/techpaper/r0.pdf f of 7 Iteral Cotecy Relablty for Dchotomou Item KR 0 & Alpha There apparet cofuo wth ome dvdual
More informationMEASURES OF DISPERSION
MEASURES OF DISPERSION Measure of Cetral Tedecy: Measures of Cetral Tedecy ad Dsperso ) Mathematcal Average: a) Arthmetc mea (A.M.) b) Geometrc mea (G.M.) c) Harmoc mea (H.M.) ) Averages of Posto: a) Meda
More informationLecture 7. Confidence Intervals and Hypothesis Tests in the Simple CLR Model
Lecture 7. Cofdece Itervals ad Hypothess Tests the Smple CLR Model I lecture 6 we troduced the Classcal Lear Regresso (CLR) model that s the radom expermet of whch the data Y,,, K, are the outcomes. The
More informationSupport vector machines
CS 75 Mache Learg Lecture Support vector maches Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 75 Mache Learg Outle Outle: Algorthms for lear decso boudary Support vector maches Mamum marg hyperplae.
More informationComparison of Dual to Ratio-Cum-Product Estimators of Population Mean
Research Joural of Mathematcal ad Statstcal Sceces ISS 30 6047 Vol. 1(), 5-1, ovember (013) Res. J. Mathematcal ad Statstcal Sc. Comparso of Dual to Rato-Cum-Product Estmators of Populato Mea Abstract
More informationCS 1675 Introduction to Machine Learning Lecture 12 Support vector machines
CS 675 Itroducto to Mache Learg Lecture Support vector maches Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square Mdterm eam October 9, 7 I-class eam Closed book Stud materal: Lecture otes Correspodg chapters
More informationA Remark on the Uniform Convergence of Some Sequences of Functions
Advaces Pure Mathematcs 05 5 57-533 Publshed Ole July 05 ScRes. http://www.scrp.org/joural/apm http://dx.do.org/0.436/apm.05.59048 A Remark o the Uform Covergece of Some Sequeces of Fuctos Guy Degla Isttut
More information8 The independence problem
Noparam Stat 46/55 Jame Kwo 8 The depedece problem 8.. Example (Tua qualty) ## Hollader & Wolfe (973), p. 87f. ## Aemet of tua qualty. We compare the Huter L meaure of ## lghte to the average of coumer
More informationCHAPTER 3 POSTERIOR DISTRIBUTIONS
CHAPTER 3 POSTERIOR DISTRIBUTIONS If scece caot measure the degree of probablt volved, so much the worse for scece. The practcal ma wll stck to hs apprecatve methods utl t does, or wll accept the results
More informationHamilton Cycles in Random Lifts of Graphs
Hamlto Cycle Radom Lft of Grap K. Burg P. Cebolu C. Cooper A.M. Freze Marc 1, 005 Abtract A -lft of a grap K, a grap wt vertex et V K [] ad for eac edge, EK tere a perfect matcg betwee {} [] ad {} [].
More informationPart 4b Asymptotic Results for MRR2 using PRESS. Recall that the PRESS statistic is a special type of cross validation procedure (see Allen (1971))
art 4b Asymptotc Results for MRR usg RESS Recall that the RESS statstc s a specal type of cross valdato procedure (see Alle (97)) partcular to the regresso problem ad volves fdg Y $,, the estmate at the
More informationCHAPTER 4 RADICAL EXPRESSIONS
6 CHAPTER RADICAL EXPRESSIONS. The th Root of a Real Number A real umber a s called the th root of a real umber b f Thus, for example: s a square root of sce. s also a square root of sce ( ). s a cube
More informationLecture 25 Highlights Phys 402
Lecture 5 Hhlht Phy 40 e are ow o to coder the tattcal mechac of quatum ytem. I partcular we hall tudy the macrocopc properte of a collecto of may (N ~ 0 detcal ad dtuhable Fermo ad Boo wth overlapp wavefucto.
More informationMean is only appropriate for interval or ratio scales, not ordinal or nominal.
Mea Same as ordary average Sum all the data values ad dvde by the sample sze. x = ( x + x +... + x Usg summato otato, we wrte ths as x = x = x = = ) x Mea s oly approprate for terval or rato scales, ot
More informationArithmetic Mean and Geometric Mean
Acta Mathematca Ntresa Vol, No, p 43 48 ISSN 453-6083 Arthmetc Mea ad Geometrc Mea Mare Varga a * Peter Mchalča b a Departmet of Mathematcs, Faculty of Natural Sceces, Costate the Phlosopher Uversty Ntra,
More informationRegression. Chapter 11 Part 4. More than you ever wanted to know about how to interpret the computer printout
Regreo Chapter Part 4 More tha you ever wated to kow about how to terpret the computer prtout February 7, 009 Let go back to the etrol/brthweght problem. We are ug the varable bwt00 for brthweght o brthweght
More informationProblem Set 3: Model Solutions
Ecoomc 73 Adaced Mcroecoomc Problem et 3: Model oluto. Coder a -bdder aucto wth aluato deedetly ad detcally dtrbuted accordg to F( ) o uort [,]. Let the hghet bdder ay the rce ( - k)b f + kb to the eller,
More informationCompound Means and Fast Computation of Radicals
ppled Mathematc 4 5 493-57 Publhed Ole September 4 ScRe http://wwwcrporg/oural/am http://dxdoorg/436/am4564 Compoud Mea ad Fat Computato of Radcal Ja Šute Departmet of Mathematc Faculty of Scece Uverty
More informationBasic Structures: Sets, Functions, Sequences, and Sums
ac Structure: Set, Fucto, Sequece, ad Sum CSC-9 Dcrete Structure Kotat uch - LSU Set et a uordered collecto o object Eglh alphabet vowel: V { a, e,, o, u} a V b V Odd potve teger le tha : elemet o et member
More informationKernel-based Methods and Support Vector Machines
Kerel-based Methods ad Support Vector Maches Larr Holder CptS 570 Mache Learg School of Electrcal Egeerg ad Computer Scece Washgto State Uverst Refereces Muller et al. A Itroducto to Kerel-Based Learg
More informationComparing Different Estimators of three Parameters for Transmuted Weibull Distribution
Global Joural of Pure ad Appled Mathematcs. ISSN 0973-768 Volume 3, Number 9 (207), pp. 55-528 Research Ida Publcatos http://www.rpublcato.com Comparg Dfferet Estmators of three Parameters for Trasmuted
More informationJohns Hopkins University Department of Biostatistics Math Review for Introductory Courses
Johs Hopks Uverst Departmet of Bostatstcs Math Revew for Itroductor Courses Ratoale Bostatstcs courses wll rel o some fudametal mathematcal relatoshps, fuctos ad otato. The purpose of ths Math Revew s
More informationr y Simple Linear Regression How To Study Relation Between Two Quantitative Variables? Scatter Plot Pearson s Sample Correlation Correlation
Maatee Klled Correlato & Regreo How To Study Relato Betwee Two Quattatve Varable? Smple Lear Regreo 6.11 A Smple Regreo Problem 1 I there relato betwee umber of power boat the area ad umber of maatee klled?
More informationNon-uniform Turán-type problems
Joural of Combatoral Theory, Seres A 111 2005 106 110 wwwelsevercomlocatecta No-uform Turá-type problems DhruvMubay 1, Y Zhao 2 Departmet of Mathematcs, Statstcs, ad Computer Scece, Uversty of Illos at
More informationEntropy ISSN by MDPI
Etropy 2003, 5, 233-238 Etropy ISSN 1099-4300 2003 by MDPI www.mdp.org/etropy O the Measure Etropy of Addtve Cellular Automata Hasa Aı Arts ad Sceces Faculty, Departmet of Mathematcs, Harra Uversty; 63100,
More informationResearch Article Some Strong Limit Theorems for Weighted Product Sums of ρ-mixing Sequences of Random Variables
Hdaw Publshg Corporato Joural of Iequaltes ad Applcatos Volume 2009, Artcle ID 174768, 10 pages do:10.1155/2009/174768 Research Artcle Some Strog Lmt Theorems for Weghted Product Sums of ρ-mxg Sequeces
More informationA note on testing the covariance matrix for large dimension
A ote o tetg the covarace matrx for large dmeo Melae Brke Ruhr-Uvertät Bochum Fakultät für Mathematk 44780 Bochum, Germay e-mal: melae.brke@ruhr-u-bochum.de Holger ette Ruhr-Uvertät Bochum Fakultät für
More informationMedian as a Weighted Arithmetic Mean of All Sample Observations
Meda as a Weghted Arthmetc Mea of All Sample Observatos SK Mshra Dept. of Ecoomcs NEHU, Shllog (Ida). Itroducto: Iumerably may textbooks Statstcs explctly meto that oe of the weakesses (or propertes) of
More informationMu Sequences/Series Solutions National Convention 2014
Mu Sequeces/Seres Solutos Natoal Coveto 04 C 6 E A 6C A 6 B B 7 A D 7 D C 7 A B 8 A B 8 A C 8 E 4 B 9 B 4 E 9 B 4 C 9 E C 0 A A 0 D B 0 C C Usg basc propertes of arthmetc sequeces, we fd a ad bm m We eed
More informationBinary classification: Support Vector Machines
CS 57 Itroducto to AI Lecture 6 Bar classfcato: Support Vector Maches Mlos Hauskrecht mlos@cs.ptt.edu 539 Seott Square CS 57 Itro to AI Supervsed learg Data: D { D, D,.., D} a set of eamples D, (,,,,,
More informationMA/CSSE 473 Day 27. Dynamic programming
MA/CSSE 473 Day 7 Dyamc Programmg Bomal Coeffcets Warshall's algorthm (Optmal BSTs) Studet questos? Dyamc programmg Used for problems wth recursve solutos ad overlappg subproblems Typcally, we save (memoze)
More informationIt is Advantageous to Make a Syllabus as Precise as Possible: Decision-Theoretic Analysis
Joural of Iovatve Techology ad Educato, Vol. 4, 2017, o. 1, 1-5 HIKARI Ltd, www.m-hkar.com https://do.org/10.12988/jte.2017.61146 It s Advatageous to Make a Syllabus as Precse as Possble: Decso-Theoretc
More informationReliability and Cost Analysis of a Series System Model Using Fuzzy Parametric Geometric Programming
P P P IJISET - Iteratoal Joural of Iovatve Scece, Egeerg & Techology, Vol. Iue 8, October 204. Relablty ad Cot Aaly of a Sere Syte Model Ug Fuzzy Paraetrc Geoetrc Prograg Medhat El-Dacee P 2 2 P, Fahee
More informationIntroduction to local (nonparametric) density estimation. methods
Itroducto to local (oparametrc) desty estmato methods A slecture by Yu Lu for ECE 66 Sprg 014 1. Itroducto Ths slecture troduces two local desty estmato methods whch are Parze desty estmato ad k-earest
More informationSimple Linear Regression. How To Study Relation Between Two Quantitative Variables? Scatter Plot. Pearson s Sample Correlation.
Correlato & Regreo How To Study Relato Betwee Two Quattatve Varable? Smple Lear Regreo 6. A Smple Regreo Problem I there relato betwee umber of power boat the area ad umber of maatee klled? Year NPB( )
More informationNP!= P. By Liu Ran. Table of Contents. The P versus NP problem is a major unsolved problem in computer
NP!= P By Lu Ra Table of Cotets. Itroduce 2. Prelmary theorem 3. Proof 4. Expla 5. Cocluso. Itroduce The P versus NP problem s a major usolved problem computer scece. Iformally, t asks whether a computer
More informationLINEAR REGRESSION ANALYSIS
LINEAR REGRESSION ANALYSIS MODULE V Lecture - Correctg Model Iadequaces Through Trasformato ad Weghtg Dr. Shalabh Departmet of Mathematcs ad Statstcs Ida Isttute of Techology Kapur Aalytcal methods for
More informationCan we take the Mysticism Out of the Pearson Coefficient of Linear Correlation?
Ca we tae the Mstcsm Out of the Pearso Coeffcet of Lear Correlato? Itroducto As the ttle of ths tutoral dcates, our purpose s to egeder a clear uderstadg of the Pearso coeffcet of lear correlato studets
More informationMultiple Regression. More than 2 variables! Grade on Final. Multiple Regression 11/21/2012. Exam 2 Grades. Exam 2 Re-grades
STAT 101 Dr. Kar Lock Morga 11/20/12 Exam 2 Grades Multple Regresso SECTIONS 9.2, 10.1, 10.2 Multple explaatory varables (10.1) Parttog varablty R 2, ANOVA (9.2) Codtos resdual plot (10.2) Trasformatos
More informationMATH 247/Winter Notes on the adjoint and on normal operators.
MATH 47/Wter 00 Notes o the adjot ad o ormal operators I these otes, V s a fte dmesoal er product space over, wth gve er * product uv, T, S, T, are lear operators o V U, W are subspaces of V Whe we say
More informationNP!= P. By Liu Ran. Table of Contents. The P vs. NP problem is a major unsolved problem in computer
NP!= P By Lu Ra Table of Cotets. Itroduce 2. Strategy 3. Prelmary theorem 4. Proof 5. Expla 6. Cocluso. Itroduce The P vs. NP problem s a major usolved problem computer scece. Iformally, t asks whether
More informationLecture Notes 2. The ability to manipulate matrices is critical in economics.
Lecture Notes. Revew of Matrces he ablt to mapulate matrces s crtcal ecoomcs.. Matr a rectagular arra of umbers, parameters, or varables placed rows ad colums. Matrces are assocated wth lear equatos. lemets
More informationJohns Hopkins University Department of Biostatistics Math Review for Introductory Courses
Johs Hopks Uverst Departmet of Bostatstcs Math Revew for Itroductor Courses Ratoale Bostatstcs courses wll rel o some fudametal mathematcal relatoshps, fuctos ad otato. The purpose of ths Math Revew s
More informationThis lecture and the next. Why Sorting? Sorting Algorithms so far. Why Sorting? (2) Selection Sort. Heap Sort. Heapsort
Ths lecture ad the ext Heapsort Heap data structure ad prorty queue ADT Qucksort a popular algorthm, very fast o average Why Sortg? Whe doubt, sort oe of the prcples of algorthm desg. Sortg used as a subroute
More informationDepartment of Agricultural Economics. PhD Qualifier Examination. August 2011
Departmet of Agrcultural Ecoomcs PhD Qualfer Examato August 0 Istructos: The exam cossts of sx questos You must aswer all questos If you eed a assumpto to complete a questo, state the assumpto clearly
More information0/1 INTEGER PROGRAMMING AND SEMIDEFINTE PROGRAMMING
CONVEX OPIMIZAION AND INERIOR POIN MEHODS FINAL PROJEC / INEGER PROGRAMMING AND SEMIDEFINE PROGRAMMING b Luca Buch ad Natala Vktorova CONENS:.Itroducto.Formulato.Applcato to Kapsack Problem 4.Cuttg Plaes
More information