A Mathematical Optimization Approach for Resource Allocation in Large Scale Data Centers

Transcription

1 A Mathmatica Optimization Appoach fo Rsouc Aocation in Lag Sca Data Cnts Cipiano Santos, Xiaoyun Zhu, Haan Cowd Intignt Entpis chnoogis Laboatoy HP Laboatois Pao Ato HPL (R.1) Dcmb 12 th, 2002 E-mai: {cipiano_santos, iaoyun_zhu, mathmatica pogamming, Intnt data cnt, souc aocation, scaabiity In this pap, w addss th souc aocation pobm (RAP) fo ag sca data cnts using mathmatica optimization tchniqus. Givn a physica topoogy of soucs in a ag data cnt, and an appication with ctain achitctu and quimnts, w want to dtmin which soucs in th physica topoogy shoud b assignd to th appication achitctu such that appication quimnts and bandwidth constaints in th ntwok a satisfid, whi communication day btwn assignd svs is minimizd. W hav dcomposd this comp combinatoia optimization pobm into a sis of tactab mathmatica optimization mod that can b asiy sovd using commciay avaiab mathmatica pogamming sovs. Piminay tsts of this appoach dmonstatd consistnty that optima o good soutions fo pobms fom sma to ag scas can b found in sconds, which povs its btt scaabiity compad to a Layd Patitioning and Puning (LPP) agoithm in pvious wok. In addition, this Mathmatica Pogamming appoach can b tndd to mo gna pobms using th sam sovs, that is, tnsions of th oigina pobm do not qui th dvopmnt of nw agoithms. Intna Accssion Dat Ony Appovd fo Etna Pubication Psntd at th Infoms Annua Mting, Novmb 2002, San Jos, CA Copyight Hwtt-Packad Company 2002

2 A Mathm aticaoptim ization Appoach fo R souc Aocation in Lag Sca Data Cnts Cipiano Santos Xiaoyun Zhu Haan Cowd Hwtt Packad Labs Pao Ato, CA {cipiano_santos, iaoyun_zhu, haan_cowd}@hp.com Abstact In this pap, w addss th souc aocation pobm (RAP) fo ag sca data cnts using mathmatica optimization tchniqus. Givn a physica topoogy of soucs in a ag data cnt, and an appication with ctain achitctu and quimnts, w want to dtmin which soucs in th physica topoogy shoud b assignd to th appication achitctu such that appication quimnts and bandwidth constaints in th ntwok a satisfid, whi communication day btwn assignd svs is minimizd. W hav dcomposd this comp combinatoia optimization pobm into a sis of tactab mathmatica optimization mod that can b asiy sovd using commciay avaiab mathmatica pogamming sovs. Piminay tsts of this appoach dmonstatd consistnty that optima o good soutions fo pobms fom sma to ag scas can b found in sconds, which povs its btt scaabiity compad to a Layd Patitioning and Puning (LPP) agoithm in pvious wok. In addition, this Mathmatica Pogamming 1 appoach can b tndd to mo gna pobms using th sam sovs, that is, tnsions of th oigina pobm do not qui th dvopmnt of nw agoithms. Kywods: Intnt data cnt, souc aocation, mathmatica pogamming, scaabiity 1 Intoduction Intnt Data Cnts (IDCs) aim to povid highy avaiab, scaab, fib, and scu infomation tchnoogy (I) infastuctu and svics to both Appications Svic Povids (ASPs) --- companis that off individua o businsss Intnt-basd accss to appications and atd svics --- and ntpiss with intna I quimnts. h cnt shakout of high-pofi ASP and IDC povids incuding sva spctacua bankuptcis has shiftd th focus and businss mod of th suviving companis fom gowth and makt sha at any cost to mo consvativ, od fashion objctivs such as pofitabiity managmnt and optimizing th ffctivnss and fficincy of isting soucs. IDC povids can b cassifid into two catgois. Som a co-ocation svic povids that as scu IDC spac and ntwok connctivity to copoations o thid paty ASPs. Oths a fd to as Managd Svic Povids (MSPs), who own th svs in th IDC and povid vau-addd svics incuding scuity, pfomanc monitoing, contnt distibution, and capacity panning. In addition, many IDCs a thmsvs ASPs who manag not ony th infastuctu but ao th appications. As w mov towads th wod of utiity computing, MSPs a of paticua intst to us bcaus thy off th potntia fo futh shaing th infastuctu among diffnt customs and appications, thus incasing th souc utiization in th data cnts. Rcnty, IDCs hav gown significanty in both siz and compity. Som IDCs today consist of tns of thousands of svs with high-spd ntwok connctions fo both int- and inta-idc communications. Managing both infastuctu and appications in such ag and comp nvionmnts aiss many changs sinc IDC opating costs can scaat vy quicky, damaging th pofitabiity of ths businsss. h ky to ffctiv managmnt of ths ag data comps is to continuay vauat and 1 In this pap, w us th tms mathmatica pogamming and mathmatica optimization indistincty

3 fin businss mod and pocsss, and to mpoy dcision suppot and automation too fo assssing pfomanc and making good opationa and statgic dcisions. h businss mod w want to addss in this pap is th shaing of soucs among appications in an IDC. A ky div fo this mod is th nd fo an IDC to manag a suit of comp appications with divs quimnts and dynamic chaactistics basd on svic-v agmnts. On on hand, som Wb appications pinc highy busty taffic whos wokoad intnsity vais damaticay duing diffnt tim of day, o day of wk. Ov-povisioning, a common pactic today, tuns out to b a highy infficint way fo capacity panning. If soucs a aocatd basd on maimum usag, th wi b piods whn th systm activity v is tmy ow and th soucs a vy pooy utiizd. In addition, spcia pomotions com and go with pak dmands much high than th gua usag, which maks ov-povisioning vn infasib to impmnt. On th oth hand, th may b faiy pdictab batch pocssing jobs that can utiiz soucs whn thy a not ndd fo oth appications. hfo, wh appicab, an appication shoud not b bound to a paticua pic o st of hadwa dvics, but ath th appication-souc assignmnt shoud b mad dynamicay by IDC managmnt pocsss. his is oftn fd to as capacity on dmand. [5] his businss mod is dsiab bcaus it aows IDC soucs to b vagd acoss many appications, with significant conomis of sca. But th downsid is a significant incas in th compity of IDC opations that quis nw v of sophistication in both wokoad dmand and souc capacity managmnt. In this pap, w addss th Rsouc Aocation Pobm (RAP) that is inhnt in any IDC that shas soucs among appications. In simp tms, th RAP is: Givn a physica ntwok topoogy of nods consisting of switchs and svs in an IDC, and appication quimnts achitctu, w nd to dtmin which svs in th physica topoogy shoud b assignd to th appication achitctu. Any such assignmnts much oby two us: (1) th appication quimnts and bandwidth constaints of th nods in th physica topoogy must b satisfid, and (2) th avag communication day btwn th assignd svs is minimizd. h RAP pobm was oiginay fomuatd by Zhu and Singha, and a ayd patitioning and puning (LPP) mthod was poposd to sov this pobm [7]. Whi thi mthod captus th ssntia mnts of th pobm, thi computationa soution mthod is not scaab to contmpoay-sizd IDC achitctu. h mathmatica optimization appoach in this pap is basd on th co idas of th LPP agoithm, but th pocdua appoach of LPP has bn tansatd into a dcaativ appoach, a squnc of mathmatica optimization mod that can b sovd fficinty by commciay avaiab sovs such as CPLEX 2 and MINOS 3. h RAP pobm is psnt ith whn appications a initiay dpoyd in an IDC, o whn changs in wokoad intnsity qui that soucs b addd to o movd fom an appication. h mathmatica pogamming mod psntd in this pap da with th initia aocation pocss in paticua. Howv, th mod a asiy tnsib to hand dynamic incmnta souc aocations to an isting appication. h IBM Ocano pojct ao focuss on th concpt of capacity on dmand and addsss th pobm of dynamic souc aocation. [1] Howv, th has not bn any pubishd wok on this subjct using anaytica tchniqus. In th mantim, th AO (ansactions, Anaysis, and Optimization) pojct of HP Labs is invstigating th issus in intgating wokoad chaactization and souc capacity panning using a combination of anaytica too and an pimnta tst bd. h wok psntd in this pap is ssntiay compmntay to th sach conductd in th AO pojct. Fo mo dtai about th AO pojct s Gag t a [3]. 2 CPLEX is an ILOG softwa poduct fo soving Lina and Mid Intg pogamming pobms. Fo mo infomation visit th foowing wbsit, 3 MINOS is softwa dvopd by Pofssos Wat Muay and Micha Saunds fom Stanfod Univsity, and Pofsso Phiip Gi fom UC San Digo fo soving Non-ina pogamming pobms. Fo mo infomation visit th foowing sit,

4 2 RAP Dfinition and an Eamp Fo a givn physica ntwok topoogy of an IDC, a givn appication with a tid achitctu, and souc quimnts fo th appication, th pobm is to aocat svs in th topoogy into th tid achitctu such that th appication souc quimnts a satisfid and th avag communication day btwn svs (ao known as atncy) is minimizd. h foowing amp iustats an instanc of RAP. 2.1 Ntwok opoogy An IDC ntwok topoogy is iustatd in Figu 1. h topoogy is chaactizd by: h ayout of th ntwok topoogy is a hiachica t. h oot of th t is a msh switch, an abstaction of a tighty connctd goup of switchs acting as a sing dvic. Bow th oot a two dg switchs, abd Edg 1 and Edg 2. Bow th dg switchs a fou ack switchs: Rack 1 though Rack 4. At th bottom of th t a 12 svs, Sv 1 though Sv 12. Each sv has 3 attibuts: numb of CPUs, CPU spd, and mmoy siz. Sv attibuts a summaizd in ab 1. Fo amp, Sv 1 has 8 CPUs of 400 MHz ach and 4 GB of andom accss mmoy. (Not that oth instancs of RAP may qui mo sv attibuts.) Each sv, ack switch, and dg switch has an incoming and outgoing bandwidth capacity, masud in Gbps. hs capacitis a summaizd in ab 2. Fo amp, both th incoming and outgoing bandwidth capacity of Sv 1 is 2.0 Gbps. Msh Switch Edg Switchs Edg 1 Edg 2 Rack Switchs Rack 1 Rack 2 R ack 3 Rack 4 Svs Sv1 Sv3 Sv4 Sv6 Sv7 Sv9 Sv10 Sv12 Sv2 Sv5 Sv8 Sv11 Figu 1. Eamp of ntwok topoogy Svs CPU Spd (MHz) Numb of CPUs RAM (GB) ab 1. Eamp of sv attibuts Edg S. Rack Switchs Svs Input (Gbps) Output (Gbps) ab 2. Eamp of sv and switch bandwidth - 3 -

5 2.2 Appication Achitctu h appication achitctu fo th amp is iustatd in Figu 2 and is chaactizd by: A th-ti achitctu consisting of Wb, appication and databas svs. W hav abd ths tis L1, L2 and L3. is L1 and L2 qui two svs ach. L3 quis on sv. h svs in ach ti hav minimum and maimum vaus fo th quid attibuts. hs quimnts a summaizd in ab 3. Fo amp, th minimum and maimum CPU spd fo any sv in ti L1 is 200MHz and 500MHz, spctivy. At th top of Figu 2 a notations that indicat th maimum amount of data taffic, masud in Gbps, going fom ach sv in a ti to ach sv in a conscutiv ti. In this amp, th maimum amount of taffic fom ach sv in ti L1 to ach sv in ti L2 is 0.2Gbps Gbps 0.15 G bps 0.2 Gbps 0.2 Gbps 0.2 G bps 0.2 Gbps Apps Wb DB L3:Databas Svs Apps L2:Appication Svs Wb L1:W b Svs Figu 2. Eamp of tid appication achitctu is L1 L2 L3 min ma min ma min ma CPU Spd INF CPU Numb INF RAM INF Not:INF = no bound spcifid ab 3. Eamp of appication quimnts on svs Ou pobm is to idntify which svs in th physica topoogy shoud b aocatd to ach ti in th appication such that atncy is minimizd whi bandwidth constaints and Min/Ma sv attibut quimnts a satisfid. 3 Mathmatica Fomuation of RAP h mathmatica fomuation is basd on th foowing assumptions. 1. h physica topoogy is a hiachica t. 4 4 W dfin a hiachica t as th t wh th st of nods can b patitiond into v. Lv zo is th oot of th t, v 1 is th st of chidn of v 0, v 2 is th union of th sts of chidn of ach nod in v 1, tc

6 2. h appication has a tid achitctu. 3. Svs at th sam ti hav th sam functionaity. Consqunty, thy hav th sam attibut quimnts. 4. h amount of taffic gnatd by diffnt svs in th sam ti is simia. And th amount of taffic coming into ach ti is vny distibutd among a th svs in th ti. 5. No taffic gos btwn svs in th sam ti. h asoning bhind ths assumptions was dscibd in [7]. Whn it is ncssay to consid appications with mo gna achitctu o taffic chaactistics, th mathmatica mod psntd bow can b asiy tndd to da with ths vaiations. h foowing notation is usd to dscib th mathmatica fomuation of RAP. Sts and thi indics L : St of tis o ays, wh L psnts th numb of tis. [Atnativ ind i] s S : St of svs. S psnts th numb of svs. [Atnativ ind j] a A : St of attibuts fo svs. A psnts th numb of attibuts. R : St of ack switchs. R psnts th numb of ack switchs. [Atnativ ind q] E : St of dg switchs. E psnts th numb of dg switchs. h ntwok topoogy of th IDC can b captud using th foowing sts. SR S : St of svs connctd to ack switch. SE S : St of svs connctd to dg switch. R R : St of ack switchs connctd to dg switch. h attibuts of svs in th physica topoogy a psntd by th mati V, wh ach mnt psnts th vau of th attibut a of sv s. h bandwidth capacity of svs, ack and dg switchs in th physica topoogy a psntd by th foowing st of paamts: BSI : h incoming bandwidth of sv s. s BSO : h outgoing bandwidth of sv s. s BRI : h incoming bandwidth of ack switch. BRO : h outgoing bandwidth of ack switch. BEI : h incoming bandwidth of dg switch. BEO : h outgoing bandwidth of dg switch. h appication achitctu quimnts a psntd by th foowing paamts. h numb of svs to b aocatd to ti is dfind by N. h maimum and minimum attibut quimnts a psntd by two matics VMAX and VMIN, wh ach mnt VMAX a and VMIN a psnt th maimum and minimum v of attibut a fo any sv in ti. h mati is dfind to chaactiz th taffic pattn of th appication, wh th mnt i psnts th maimum amount of taffic going fom ach sv in ti to ach sv in ti i. h numbs 01 and 10 psnt th Intnt taffic coming into and going out of ach sv in ti 1. Using ths taffic paamts, w can cacuat th tota amount of incoming and outgoing taffic at ach sv in diffnt tis, dnotd by I and O, spctivy. V as - 5 -

7 So fa w hav dfind a th input paamts to RAP. W now dfin th dcision vaiabs. In th optimization pobm w nd to dcid which sv in th physica topoogy shoud b assignd to which ti. h foowing mati of binay vaiabs psnts this. 1 = 0 sv s assignd to ti othwis h dvopmnt of th constaints and th dfinition of th objctiv function can b found in Appndi 1. In addition, intstd ads may consut th pap by Zhu and Singha [7]. In summay, th mathmatica optimization pobm fo RAP is th foowing. Subjct to: Ma VMIN R j SR a O I O I s S i + = N, L (1) V O I as 1, s S E j SE VMAX a (2) BSO, s S s BSI, s S s i j SR i j SR i j SE i j SE,, a A, s S (4) (5) BRO, R BRI, R BEO, E BEI, E { 0,1 },, i D, s, j M his fomuation is fd to as th oigina mathmatica optimization pobm, abd as P0. Sinc th objctiv function is nonina and th a nonina constaints, th optimization mod is a nonina pogamming pobm with binay vaiabs, which cannot b sovd fficinty by commciay avaiab mathmatica pogamming sovs. i (6) (7) (8) (9) (3) 4 Mid Intg Pogamming Fomuation --- h Fist Attmpt In this sction, th oigina fomuation of RAP is visd and an quivant Mid Intg Pogamming (MIP) fomuation is dvopd, which can b sovd dicty using commcia sovs. h intstd ad may consut th book wittn by Woy [6] fo a vy comphnsiv dsciption of th thoy and pactic bhind Mid Intg Pogamming mod. his fomuation is fd to as MIP0. h ky ida of th MIP0 fomuation is basd on th foowing obsvation. h poduct of binay vaiabs is ao binay. Consqunty, w dfin a nw vaiab fo ach poduct of binay vaiabs in th oigina mod as foows. 0 if = 0 o = 0 y = 1 if = = 1-6 -

8 h foowing constaints a imposd to nsu that th nw vaiab bhavs as th poduct of binay vaiabs. Fist, to guaant that y 0 whnv = 0 o = 0, w nd = y, y, i L, s, j S. (10) Scond, to nsu that y 1 whnv = 1 and = 1, w nd = + y 1. W now pain why th nw y vaiabs can b non-ngativ continuous vaiabs instad of binay vaiabs. his is cucia, bcaus a fomuation with a ag numb of binay vaiabs is tmy had to sov du to combinatoia posion, vn fo sma pobms. h main ida woks as foows. Obsv that th vaiabs y wi appa in th nw objctiv function by pacing in th objctiv function of th oigina fomuation. Du to th fact that w want to maimiz th objctiv function, and a th y vaiabs hav positiv cofficints, at optimaity th y vaiabs wi tnd to hit its upp bounds. hfo, th constaints y, y wi b binding, which mans at optimaity th y vaiab wi tnd to b = = y = 1. Consqunty, th y vaiabs do not nd to b binay. Bcaus th numb of y vaiabs is much ag than th vaiabs, th combinatoia posion is gaty ducd by making y vaiabs continuous. In addition, th ogica constaints + y 1 that woud othwis constitut a fai numb of quations can b movd. o futh duc th numb of y vaiabs, obsv that, = impis y = y. hs constaints a addd to th mod so that th CPLEX ppocssing ngin can automaticay mak ths substitutions, thus ducing th numb of vaiabs in th mod without incasing th numb of quations. o duc th numb of binay vaiabs in MIP0, a fasibiity mati F is dfind as foows. 1 if = 1satisfis (3), (4) and (5) of P0; F = 0 othwis. It is usd to p-scn th svs that a infasib. o mak su that th vaiab appas in th MIP0 fomuation if and ony if F = 1, an additiona constaint 0, F } is imposd. { In summay, th MIP0 fomuation pssd in tms of binay vaiabs and non-ngativ continuous vaiabs y is dfind as foows. Subjct to: Ma R j SE i y + E j SE i y O I O I s S = N, L (1) j SR j SR j SE j SE 1, s S (2) i i i i y y y y BRO, R BRI, R BEO, E BEI, E (6) (7) (8) (9) - 7 -

9 y, y,, i L, s, j S (10) { 0, F }, L, s S; y 0,, i L, s, j S h abov appoach was appid to th amp in Sction 2, fd to as M12. Givn that th siz of th soution spac gows ponntiay with th numb of binay vaiabs in MIP0, and th numb of binay vaiabs is a function of th numb of svs in th physica topoogy, w us th numb of svs as a masu of compity of th pobm. h M12 pobm has 12 svs. h numb of svs quid by th appication is N=(2,2,1). h MIP0 mod fo M12 has th foowing dimnsions. 351 quations. 110 non-ngativ continuous vaiabs y. 21 binay vaiabs. h mod can b sovd in 0.3 sconds with GAMS (MIP0) 5 /CPLEX7.0. h Layd Patitioning and Puning (LPP) agoithm by Zhu and Singha [7] took sconds. h optima objctiv function vau found by both appoachs is 3.3. h optima sv aocation found by th MIP0 appoach is shown in Figu 3. Msh Edg 1 Edg 2 R ack 3 Rack 4 Sv 7 Sv 9 Sv 10 Sv 12 Sv 8 Sv 11 Sv 7 Sv 10 Sv 9 Sv 12 Sv 11 L3 L2 L1 Figu 3. Optima sv aocation fo th M12 pobm 5 GAMS is a mathmatica optimization moding anguag that aows coding th MIP fomuation. GAMS ca CPLEX Banch and Bound (Cut) sov to find th optima soution. Fo mo dtai about GAMS visit th foowing wbsit,

10 Mo pimnts a dscibd in Appndi 2. h suts a summaizd in Figu 4. Basd on ths suts, w concud that th MIP0 pobm, whn sovd with standad MIP sovs, dos not sca popy. h ason is that th LP aation of th MIP0 fomuation is too ag fo th ag sca pobms, which dgads th soution tim of th Banch and Bound appoach usd by CPLEX. In addition, th LP aation povids vy poo bounds; which ao dtioats th soution tim of th Banch and Bound agoithm. Givn ths facts, tying to impov th pfomanc of CPLEX by finding th ight options to un th Banch and Bound agoithm is hopss. Finay, th LPP agoithm finds an optima soution fast than MIP0 fo sma to mdium sizd pobms. Athough it has succssfuy sovd som ag sca pobms, th scaabiity is not aways guaantd. hfo, w nd to attack th pobm fom a diffnt ang. Figu 4. Summay of pimnta suts with th MIP0 fomuation 5 A Suit of Mathmatica Optimization Fomuations In this sction, w dvop an appoach that compiss a sis of mathmatica optimization fomuations that sovs RAP huisticay and fficinty. h sis of mathmatica optimization mod a vaiations and aations of th oigina fomuation P0. h appoach has th stps. Each stp mpoys a mathmatica optimization fomuation that is asy to sov by commcia sovs. 1. Find a good initia appoimat soution. An MIP pobm is fomuatd that minimizs th numb of ack switchs and dg switchs invovd. Concptuay, this objctiv function is a suogat of th oigina objctiv function wh atncy is minimizd. In this fomuation, w do not hav visibiity of th spcific svs that a fasib in tms of th appication quimnts, but ony hav visibiity of th numb of fasib svs at ach ack. h soution gnatd by this MIP fomuation dos not consid ack and dg switch bandwidth constaints, which is why it is ony an appoimat soution. his MIP pobm can b sovd using CPLEX. 2. Givn th abov appoimat soution as an initia soution, w sov a aation of th oigina pobm by fomuating a nonina optimization pobm in tms of th numb of fasib svs at ach ack switch aocatd to th appication. his fomuation is quivant to th oigina mathmatica optimization fomuation P0. Howv, in this cas th fomuation is a aation bcaus a th dcision vaiabs can b continuous vaiabs instad of intg vaiabs. Sinc this nonina fomuation povids ony oca optima instad of goba ons w nd stp 1. his nonina optimization pobm can b sovd using MINOS. 3. Givn th oca optima soution fom stp 2, find a good soution to th oigina pobm. W fomuat anoth MIP that ssntiay ounds th oca optima soution and idntifis th act svs that satisfy th quimnts of th appication. CPLEX sovs this MIP pobm

11 W stat with th nonina optimization fomuation, stp 2 of ou appoach, sinc this fomuation is at th co of th soution appoach. hn, w dscib th MIP fomuation that compiss stp 3, which chooss th spcific svs. Finay, w dscib th MIP fomuation in stp 1, which dtmins good initia soutions fo th nonina optimization pobm. 5.1 QP (Stp 2) --- Nonina Optimization and Binay Raation Fo combinatoia optimization pobms with binay vaiabs, it is somtims advisab, if possib, to fomuat th pobm in tms of continuous vaiabs ov an intva. his bings convity to th fomuation and hps th continuous aation to b stong, which mans that th aation wi giv tight bounds. Fo this pupos, a quadatic pogamming appoimation of th oigina pobm is fomuatd, fd to as QP. h intstd ad might consut th book wittn by Gi, Muay, and Wight [4] fo a comphnsiv dsciption of th thoy and pactic of Non-ina Pogamming mod. A nw dcision vaiab is dfind as foows. : Numb of fasib svs connctd to ack switch that a aocatd to ti. [ 0, ] N. Fo a givn ack, = F. h vaiab appas in th QP fomuation if and ony if F 1, which mans th ack switch has a fasib sv fo ti. o simpify th notation, w dfin a nw st FSR SR S, which is th st of svs connctd to ack switch that a fasib fo ti. W now fomuat ach constaint in th oigina pobm P0 in tms of. h dtai of th fomuation of th QP pobm a paind in Appndi 3. h suting QP fomuation foows. Subjct to: Ma ZQP = i i R E R R + = N, L (1) Constaints (2). W assum a 3-ti achitctu; tnsions to oth numb of tis can b asiy considd. Fo a R, FSR1 FSR 2 FSR FSR1 FSR FSR1 FSR FSR 2 FSR 3 1 FSR1 0 2 FSR FSR 3 O I i i q R R i i i BRO, R BRI, R (6) (7) O BEO, E (8) R q R R i I BEI, E (9) R i iq iq 0, L, R 1 F i (0)

12 Epimnta suts with th QP mod a psntd in Appndi 4. W not that th QP mod is significanty sma than th LP aation of th MIP0. h QP mod psnts a pomising appoach sinc it can b sovd vy fast and finds th optima soution in tms of th vaiabs if a good initia soution is povidd. 5.2 MIP2 (Stp 3) --- Intignt Rounding In this sction, w fomuat a Mid Intg Pogamming pobm, MIP2, to intignty ound th oca optima soution gnatd by th QP mod. h MIP2 mod dfins th actua svs to aocat to th appication. h dcision vaiabs a th sam as thos in th oigina pobm P0. 1 sv s assignd to ti = 0 othwis h fist two constaints of th mod a simia to thos fo th P0 pobm. s S = N, L (1) 1, s S Fo ach ack switch and ti, aocat as many svs as commndd by th oca optima soution,, fom th QP mod. (2) if > 0 (3) As w hav paind, constaints (3), (4) and (5) of th oigina pobm P0 a captud by th fasibiity mati F. Accodingy, w impos anoth constaint 0, F } appas to nsu that th vaiab { in th fomuation if and ony if F = 1. Incoming and outgoing bandwidth capacity constaints a not considd bcaus ths constaints a satisfid by th soution of th QP mod. h MIP2 mod is just ounding th QP soution without modifying tota taffic going though ack switchs and dg switchs. h objctiv function is simpy to minimiz th numb of svs aocatd. Min ( 0) s S Obsv that th abov objctiv function is a constant, N, du to constaint (1). h ason why it is imposd is not that th ounding mod nds it. Instad, it is bcaus th commcia sov that w us fo mid intg pogamming quis that an objctiv function b spcifid. Du to th fact that a fasib soutions hav th sam objctiv function vau, th minimization dos not nfoc anything, which is what w want in this cas. In som oth cass, w might want to us th objctiv function to minimiz th tota cost of aocating svs to th appication. In summay, th MIP2 fomuation is as foows. Subjct to s S Min ( 0) s S = N, L (1) 1, s S (2) if > 0 (3), { 0 } F

13 his fomuation is ik a tanspotation mod with sid constaints. h CPLEX sov poits this stuctu nicy and sovs it tmy fast. 5.3 MIP0 (Stp 1) --- Initia Soution As w obsvd fom th pimnts of th QP mod, this mod nds a good stating soution to find th goba optima soution, o at ast a good oca optimum fo th oigina pobm P0. Rca that th initia soution fo th QP mod dos not nd to b fasib. Hnc, in this sction, w fomuat a Mid Intg Pogamming pobm, MIP1, to gnat a good initia soution fo th QP mod. h MIP1 fomuation is basd on th foowing intuitions. Fist, if th is a fasib sv assignmnt und a sing ack switch that satisfis constaints (1) to (5) of th oigina pobm P0, thn this soution is most iky fasib fo th ack and dg switchs bandwidth constaints (6) to (9). Scond, this fasib sv assignmnt is optima fo P0. his is asy to show. Fom appndi 1, th objctiv function of P0 is fomuatd as minimizing th wightd avag of th numb of hops btwn ach pai of svs, i.., ) R E Min z = 2 F + 4F + 6, R E M wh F, F and F a th tota amounts of inn taffic at a ack switchs, dg switchs and msh R E M switch, spctivy. In addition, F + F + F = N N, which is a constant. Lt it b E E M M dnotd by C. Hnc, zˆ = 2C + 2F + 4F. Sinc F 0 and F 0, w hav zˆ 2C. h optimum is achivd if and ony if F R = C and F E = F M = 0, which is acty th cas whn ony svs und on ack switch a chosn. Futhmo, vn if mo than on ack switchs a ndd, th E M E intntion wi b to minimiz F and F as much as possib. Obsv that F is ag whn mo M ack switchs a invovd, and F is ag whn mo dg switchs a invovd. hfo, th main ida in th MIP1 mod is to ty to aocat svs that a in th sam ack o that a in cos acks, wh two acks a considd to b cos if thy a connctd to th sam dg switch. It woud b dsiab fo ach ack switch to hav an appopiat sv mi so that an appication with typica sv quimnts can b hostd using a sing ack switch o a minimum numb of ack switchs. A atd tactica optimization pobm woud b to idntify such sv mi fo a st of appications. On th oth hand, data cnt opatos may pf th ida of having th sam kind of svs und ach ack switch fo asy maintnanc. Basd on th abov discussion, th objctiv function of th MIP1 fomuation is a suogat function of th objctiv function of P0. Roughy spaking, th objctiv of MIP1 is to minimiz th tota wightd usag of ack and dg switchs. Consqunty, th MIP1 pobm is fomuatd as a Faciity Location Optimization Pobm. W dfin th foowing ocation vaiabs: Fo ach E, u Fo ach R, v 1 = 0 1 = 0 F M i if dg switch is usd othwis if ack switch is usd othwis h wights fo ths ocation vaiabs a chosn so that th minimization of th objctiv function muats th diction of optimaity in th oigina pobm P0. In paticua, w dfin th wight fo ach switch usd to b th atncy masu (numb of hops) fo that switch, i.., CR = 2, and CE = 4. h main issu with th oigina fomuation P0 is that w hav a combinatoia optimization pobm with binay vaiabs, quadatic constaints, and a quadatic objctiv function. Having movd th noninaity fom th objctiv function in MIP1 aady, w woud futh mov th quadatic bandwidth constaints fo ack and dg switchs to inaiz th pobm. hfo, th MIP1 fomuation is an appoimation of th oigina pobm P0. It is not guaantd to gnat a fasib soution fo P0. i

14 Howv, this is accptab sinc th goa of th MIP1 mod is to gnat good initia soutions fo th QP mod, which picity consids th quadatic constaints movd in th MIP1 fomuation. As in th QP mod, w assum a 3-ti achitctu fo th appication. Etnsions to oth numb of tis a asy to impmnt. Simia to th QP fomuation, w dfin as th numb of fasib svs in ack switch aocatd to ti. h appas in th fomuation if and ony if ack switch has a fasib sv fo ti. h constaints of th MIP1 fomuation a as foows. Constaint 1) h tota numb of svs aocatd to ti is N. = N, L (1) R Constaint 2) Aocat at most on sv to a ti and nsu that no sv aocatd is doub countd FSR1 FSR 2 FSR FSR1 FSR FSR1 FSR FSR 2 FSR FSR1, 0 2 FSR 2, 0 3 FSR 3 Constaint 3) hs a ogica constaints ov th binay vaiabs u and v that nsu that ths vaiabs bhav as intndd. If w want to aocat svs fom ack switch to ti thn ack switch nds to b usd. hat is, 1. v = 1 if > 0 ; 2. v = 0 if = 0. hfo, w dfin th foowing constaint: N v, FSR (3.1) Wh th cofficint of th vaiab v is an upp bound of th vaiab. Not that condition 1 is satisfid by constaint (3.1), and condition 2 is satisfid by this constaint and bcaus w a minimizing th wightd summation of v vaiabs, at optimaity v = 0 if = 0. Now, if w want to us ack switch, w nd to us th dg switch connctd to this ack switch. hat is, 1. u = 1 if v = 1 ; 2. u = 0 if v = 0. Hnc, w dfin th foowing constaint: u v, E, R (3.2) his constaint nsus condition 1 is satisfid, and condition 2 is satisfid at optimaity. h objctiv function of th MIP1 fomuation is to minimiz th tota cost of using ack and dg switchs, and is dfind as foows Min CE u + CR v (0) E R In summay, th fomuation of th MIP1 mod is Min CE u Subjct to: E + R CR v (0)

15 = N, L (1) R FSR1 FSR 2 FSR FSR1 FSR FSR1 FSR FSR 2 FSR 3 1 FSR1, 0 2 FSR 2 N v u v u, v 0,, 0 3 FSR 3, FSR (3.1), E, R { 0,1 },, L R, E, R and (3.2) F his typ of Faciity Location fomuation has a stuctu that CPLEX can poit vy ffctivy and can b sovd tmy fast. 1 6 Epimnta Rsuts of th MP Appoach his sction summaizs th Mathmatica Pogamming (MP) appoach w hav dvopd to tack th souc aocation pobm P0 and shows som pimnta suts. h MP appoach invovs th stps. Each stp psnts a mathmatica optimization fomuation that is asy to sov using commcia sovs Gnat a good initia appoimat soution fo th QP mod using th MIP1 fomuation, which can b sovd fficinty by CPLEX Us th appoimat soution as an initia soution fo th QP pobm; us MINOS as th sov to find a oca optima soution of th QP pobm,. W pct that this oca optima soution woud b a goba optima soution o a good oca optima soution. 3. With th oca optima soution of th QP pobm, us th MIP2 fomuation to idntify th spcific svs to b aocatd to ach ti in th appication, pobm vy fast.. CPLEX sovs th MIP2 Both th MP appoach and th LPP agoithm w appid to a st of pobms of diffnt sizs to compa th pfomanc. h pobms w dividd into two sts, on with 500 svs and th oth with 1000 svs. Within ach st, a fw oth paamts w vaid to infunc th compity of th pobm, such as th numb of svs quid in ach ti of th appication, th numbs of dg and ack switchs, and th amount of incoming o outgoing bandwidth at th dg and ack switchs. Fom ach st, th pobms a chosn as psntativs and th suts a summaizd in ab 4. Du to th fact that th two agoithms w unning on diffnt kinds of machins, th compaison of th absout numbs fo computation tim is ss maningfu than th vaianc btwn diffnt pobms. h sam data is psntd gaphicay in Figu 5. It is asy to s that th soution tim of th LPP agoithm haviy dpnds on th paamts of th pobm, whi th pfomanc of th MP appoach is much mo stab, spciay within pobms of th sam sv numb. Ova th MP appoach is mo obust and scaab and is ab to find good soutions quicky whn th pobm is fasib and not too tight in tms of th ack and dg switch bandwidth constaints. Howv, whn th pobm is tight, th MP appoach is not vy iab, sinc it may dca th pobm infasib whi in fact th ist fasib soutions. On th oth hand, th LPP agoithm is in favo of tight pobms, bcaus its fficincy is on th abiity to quicky pun infasib nods in th sach t, which is asi whn th bandwidth

16 constaints a tight. Consqunty, a hybid MP and LPP appoach is dsiab, wh th LPP appoach is usd to dtmin infasibiity o idntify a good fasib soution whnv th MP appoach fai to find a fasib soution. his hybid mthod wi b dscibd in th nt sction. ab 4. Compaison btwn th LPP and MP appoachs Figu 5. Compaison of th LPP and MP appoachs

17 Anoth advantag of th MP appoach is that it can b tndd to hand mo gna pobms that captu th opationa issus and conomica factos inhnt in th managmnt of ag IDCs (.g., 50,000 svs). hs tnsions do not qui th invntion of nw agoithms. Instad, th nw pobms nd to b fomuatd in an intignt way such that thy can b fficinty sovd using isting commcia sovs. 7 A Hybid MP and LPP appoach In this sction, w dscib a hybid MP and LPP appoach to tack th RAP. Pas f to Figu 6 to foow th fow of vnts of th hybid appoach. 1. h appoach quis th foowing inputs a. Rsouc physica topoogy and capacitis. b. Appication achitctu and souc quimnts. 2. h Faciity Location suogat pobm, MIP1, is sovd to gnat a good initia soution. 3. Givn th initia soution gnatd by MIP1, a aation pobm QP is sovd. 4. h soution of th QP mod can ith b a oca optimum o infasib. a. If th soution of QP is infasib, thn th LPP agoithm is usd to dtmin if th oigina pobm P0 is fasib o not. If a fasib soution is found, go to 5; othwis, i. if th LPP agoithm dcas th oigina pobm P0 as infasib, thn sov a fasibiity mod, which is a vaiation of th aation pobm QP. h pupos of this fasibiity mod is to idntify which ack and/o dg switchs nd to incas th bandwidth and th amount of this incas. W f to this fasibiity mod as th FMod. h FMod adds atificia vaiabs to th incoming/outgoing bandwidth constaints to th QP mod and minimizs th sum of ths atificia vaiabs. If bandwidth is incasd as commndd by th FMod, thn go to 1, and sov th visd mod. b. If th soution of QP is a oca optimum, thn sov th intignt ounding pobm, MIP2. h soution of MIP2 is a fasib soution of th oigina pobm P0. 5. Impmnt th fasib soution found. 8 Futu Rsach h foowing tnsions to RAP and som atd topics wi b pod in ou futu sach. 1. h RAP studid in this pap mpoys a muti-ti achitctu fo appications. Athough th muti-ti configuation is faiy common in cunt Wb appications, th a appications that do not ncssaiy qui a tid stuctu. W a fomuating a mathmatica optimization mod fo gna distibutd appications that qui a st of communicating svs. 2. W hav takn th appoach of aocating soucs to on appication at a tim, which povids a suboptima soution to fficinty shaing soucs by mutip appications. It wi b intsting to po th fomuation of RAP whn mutip appications a considd simutanousy. 3. It is ao dsiab to consid gna ntwok topoogis that do not hav a hiachica t stuctu. 4. Capacity at IDCs can b pandd by scaing out adding mo svs of th sam typ, o scaing up upgading cunt svs by instaing atst softwa o adding hadwa componnts. W a woking on a mod that wi captu th cost tad-off btwn scaing out and scaing up. 5. W a invstigating a mthodoogy that may futh substantiat th notions of Intignt Povisioning and Capacity on Dmand which aows to intgat dmand panning wokoad managmnt at IDCs, with capacity panning optima souc aocation to satisfy dynamic appication quimnts. Fo mo dtai s Gag t a [3]. 6. W attmpt to us mathmatica pogamming appoachs fo mo systmatic tun on invstmnt (ROI) and tota cost of ownship (CO) anaysis fo IDCs. Fo mo dtai s Cowd t a [2]

18 1a. Rsouc Physica opoogy Rsouc Capacitis 1b. Appication Achitctu Rsouc Rquimnts 2. Sov MIP1 Faciity Location Pobm Initia Soution 3. Sov QP Raation of P0 4. Is QP soution Fasib? NO 4a. Sov using LPP agoithm YES YES Is LPP soution Fasib? NO 4b. Sov MIP2 Intignt Rounding 4a.i. Sov FMod Fasibiity Mod Rcommndations fo Rsouc Capacity Chang 5. Impmnt Fasib Soution Figu 6. Fow Diagam of th hybid MP & LPP appoach

19 Bibiogaphy 1. K. Appby, S. Fakhoui, L. Fong, G. Gozmidt, M. Kaanta, S. Kishnakuma, D.P. Paz, J. Pshing and B. Rockwg. Océano SLA basd managmnt of computing utiity, Pocdings of 2001 IEEE/IFIP Intnationa Symposium on Intgatd Ntwok Managmnt (IM 2001), Satt, May H. P Cowd, P. K. Gag, C. A Santos and X. Zhu. UDC Financia Panning (UFP). Pow point psntation, Dcmb 7, P. K Gag, Ming Hao, C. A Santos, H. K. ang, A. X. Zhang. Wb ansaction Anaysis and Optimization (AO), woking pap. 4. P. E. Gi, W. Muay and M. H. Wight. Pactica Optimization, Acadmic Pss, J. Roia, S. Singha and R. Fidich, Adaptiv Intnt Data Cnts, Pocdings of SSGRR 2000 Comput and Businss Confnc, L'Aquia, Itay, Juy-August, L. A. Woy. Intg Pogamming, Wiy, X. Zhu and S. Singha. "Optima Rsouc Assignmnt in Intnt Data Cnts," Pocdings of th Ninth Intnationa Symposium on Moding, Anaysis, and Simuation of Comput and communications Systms, pp , Cincinnati, Ohio, August

20 Appndi 1 Mathmatica Fomuation of P0 his appndi contains dtai of th dvopmnt of th constaints and th objctiv function of P0. h foowing constaints a posd on th dcision vaiabs. h tota numb of svs aocatd to ti is N : = N, L (1) s S Each sv in th physica topoogy can ony b assignd no mo than onc to a paticua ti: 1, s S h attibut vaus fo ach sv assignd satisfy th minimum and maimum quimnts: VMINa Vas VMAX a, a A, s S (3) Not that th summation is ith 0 o 1, bcaus of constaint (2) and th tms in th summation a binay vaiabs. hfo, constaint (3) taks th foowing foms fo any fasib soution: VMIN V VMAX o a a a. h outgoing and incoming bandwidth constaints fo a th inks that connct th svs to th ack switchs a: O BSO, s S (4) I s (2) BSI, s S Sinc th summation is ith 0 o 1, thn constaint (4) taks th foowing vaus fo any fasib soution: O BSOs o 0 0. h sam agumnt can b givn fo constaint (5). hfo constaints (4) and (5) foc th taffic at th sv assignd to a ti to satisfy th sv outgoing o incoming bandwidth quimnt fo that ti. h outgoing and incoming bandwidth constaints fo a th inks that connct th ack switchs to th dg switchs a: O BRO, R (6) I i j SR s i j SR Consid constaint (6). Fo ach ti and ack switch, dfin (5) BRI, R = (7) as th numb of svs und ack switch aocatd to ti. hn, th outgoing taffic at ack switch is th tota amount of O, ducd by th outgoing taffic gnatd by a th connctd svs und this switch, amount of inn taffic 6 at this switch, i j SR. At th sam tim, th outgoing taffic at ack switch shoud b ss than th ack switch s outgoing bandwidth. h sam asoning can b givn fo constaint (7). 6 Inn taffic at a paticua switch is dfind as th sum of taffic btwn any pai of svs connctd by this sam switch

21 h outgoing and incoming bandwidth constaints fo a th inks that connct th dg switchs to th msh switchs a: O BEO, E (8) I i j SE i j SE Consid constaint (8). Fo ach ti and dg switch, dfin BEI, E = (9) as th numb of svs und dg switch aocatd to ti. hn, th outgoing taffic at dg switch is th tota amount of O, ducd by outgoing taffic gnatd by a th connctd svs und this switch, amount of inn taffic at this switch, i j SE. At th sam tim, th outgoing taffic at dg switch shoud b ss than th dg switch s outgoing bandwidth. h sam asoning can b givn fo constaint (9). h objctiv of th optimization pobm is to minimiz th communication day btwn svs fo ach st of appication quimnts. Consid th numb of hops 7 ndd fo two svs to communicat. R Consid th cas whn th physica topoogy is a hiachica t. Dfin F as th tota taffic btwn R a th sv pais that communicat ony though a ack switch. hat is, F is th sum of th inn taffic fo ach ack switch, i. hfo, j SR R F =. R j SR Dfin NHR as th numb of hops ndd fo two svs und th sam ack switch to communicat. Not that NHR=2. E Dfin F as th tota taffic btwn a th sv pais that hav to communicat though an dg E switch. hat is, F is th sum of th inn taffic fo ach dg switch, i, R ducd by th taffic that ony gos though a ack switch, F. Consqunty, E R F = F. E j SE i i j SE Dfin NHE as th numb of hops ndd fo ach data packt btwn two svs to go though an dg switch. Obsv that NHE=4. M Dfin F as th tota taffic btwn a th sv pais that hav to communicat though a msh M switch. hat is, N N, ducd by th sum of F is th tota taffic btwn a th sv pais, th inn taffic at a th dg switchs. Consqunty, M F = N N. i i E j SE Dfin NHM as th numb of hops ndd fo ach data packt btwn two svs to go though a msh switch. Not that NHM=6. i i i 7 h numb of hops is dfind as th numb of acs in th path of minima ngth that conncts two svs in th gaph that psnts th physica topoogy

22 W now hav dfind a th tms that dtmin th objctiv function. hfo, if w want to minimiz th communication day btwn svs fo ach st of appication quimnts, th objctiv function of th optimization mod is to minimiz th avag numb of hops fo ach sv pai wightd by th cosponding amount of taffic btwn ths two svs. hat is, R E M Min NHR F + NHE F + NHM F. Substituting and cancing tms, w hav ) Min z = 6 N N 2 2 i i R j SR i E j SE h fist tm is a constant, so it can b ignod fom th optimization mod. hfo, instad of minimiz z ), w can maimiz z, wh Ma z = + R j SR i E j SE i i Appndi 2 Epimnta Rsuts of th MIP0 Mod In this Appndi, w psnt th pimnta suts of using th MIP0 fomuation. Fist a pobm cad M30 with 30 svs was tstd. h goba paamts fo this pobm a: Fiv attibuts p sv Si dg switchs wo ack switchs h tis h sv quimnts by th appication a N = (4,5,3). h MIP0 fomuation has th foowing dimnsions: o 1682 quations o 526 non-ngativ continuous vaiabs y o 50 binay vaiabs h mod can b sovd in 8.7 sconds with GAMS (MIP0) /CPLEX7.0 h LPP agoithm dvopd by Zhu and Singha taks 1.7 sconds. h optima soution vau found by both appoachs is hn a pobm cad M500 with 500 svs was tstd. h goba paamts fo this pobm a: Fiv attibuts p sv Fiv dg switchs Numb of ack switchs is 25 h tis h sv quimnts by th appication a N = (6, 8, 5). h LPP agoithm can sov this pobm optimay in sc. h optima objctiv function found is h svs in th soution a a in ack switch 18. h svs in i 1 a (s341, s342,, s346) i 2 a (s347,, s354) i 3 a (s355,,s359) h MIP0 pobm has about 810 binay vaiabs and about 57,800 continuous vaiabs y. h Lina Pogamming Raation ( 0 1) has th foowing poptis: 179,000 quations 58,610 vaiabs 560,000 non-zo cofficints his aation sovs using GAMS/CPLEX6.0 Pima Simp in about 17 min 35 sc. his aation sovs using GAMS/CPLEX6.0 Dua Simp in about 5 min 18 sc. his aation is too big, and sovs too sowy

23 h optima objctiv function vau of th aation is 264. h LP aation givs poo bounds (264/62.4 = 4.23). It sms th y vaiabs os thi maning as a poduct of 2 binay vaiabs, whn is no ong binay but continuous in th intva [0,1]. h MIP0 pobm itsf cannot b sovd by GAMS/CPLEX6.0 aft unning fo amost a wk. Appndi 3 Mathmatica Fomuation of QP his appndi povids th dtai of th dvopmnt of th QP mod. Consid constaint (1) fom P0, which says th tota numb of svs aocatd to ti is N. Sinc F = F = s S R R R, w hav = N, L (1) Consid constaint (2) fom P0, which nsus that ach sv in th physica topoogy can b assignd no mo than onc to a paticua ti. Fo this constaint w nd to b mo cafu. Fo amp, consid th fasibiity mati fo th amp M12. Rack Sv s1 s2 s3 s4 s5 s6 s7 s8 s9 s10 s11 s ti 1 F= ti ti 3 Consid th foowing constaints to guaant that at most on sv is aocatd to a ti. Fo ach ack switch th summation of th numb of th fasib svs aocatd to ach ti shoud b ss than th tota numb of svs fasib fo a th tis in this ack. hfo, + + FSR FSR FSR W ao impos a constaint on th numb of svs aocatd to ach ti, that is, 0 FSR. h abov constaints a not sufficint to avoid doub counting. Consid ack switch 3, and th soution. his soution satisfis th constaints 1, 2, and = 2 3 = 3 3 = { s7} { s7, s8} { s8, s9} = { s7} = { s7, s8} = { s8, s9} = 2 Howv, this soution is doub counting sv s7. If w now impos a constaint to nsu that w do not aocat mo svs than th tota numb of fasib svs at ti 1 and ti 2, i.., { s7} { s7, s8} = 2, thn th poposd soution is no ong fasib. Consqunty, to avoid doub counting w nd to hav a constaint of fasib svs at a ack switch fo ach possib ti combination, which mans w nd 2 L 1 constaints ach psnting a ti combination. Sinc a 3-tid achitctu is th most common, w dfin constaints (2) assuming a 3-ti achitctu. hfo, fo ach ack switch, FSR1 FSR 2 FSR FSR1 FSR FSR1 FSR FSR 2 FSR

24 0 1 FSR1 0 2 FSR FSR 3 Constaints (3), (4) and (5) fom P0 a omittd bcaus thy a impicity considd in th gnation of th vaiabs. h outgoing and incoming bandwidth constaints fo a th inks that connct th ack switchs to th dg switchs a psntd by constaints (6) and (7) fom P0. Consid th fist tm of constaint (6), Now consid th scond tm of constaint (6), = O F = O O. i = i F j SR, j SR F, = Constaints (7) can b tatd in a simia mann. hfo w hav constaints (6) and (7) dfind in tms of as foows. O I i i i i BRO, R BRI, R h outgoing and incoming bandwidth constaints fo a th inks that connct th dg switchs to th msh switchs a psntd by constaints (8) and (9) fom P0. Consid th fist tm of constaint (8), O = O F = R R Now consid th scond tm of constaint (8), = i i F j SE, R q R j SRq F (6) (7) O = i i,, q R Constaints (9) can b tatd in a simia fashion. Consqunty, w hav constaints (8) and (9) dfind in tms of as foows. O BEO, E (8) q R R R q R R i I BEI, E (9) R Simiay, th objctiv function fomuatd in tms of foows: Ma ZQP = + i iq iq i i R Eq R R i iq (0) i. iq Appndi 4 Epimnta Rsuts of th QP mod In this appndi w psnt th pimnta suts fo th QP mod. Fo th pobm M12 : h initia soution fo th QP mod is 0 = 0. h sv quimnts by th appication a N=(2,2,1). 32 quations 11 vaiabs

25 91 Non-zo cofficints 50 Non-ina cofficints h QP mod oca optima soution is o ZQP = o 1 1 = 2, 21 = 1, 2 2 = 1, 3 2 = 1. o Soution tim is 0.2 sc. In this cas, th QP mod using GAMS/MINOS5.5 found th goba optima soution in tms of vaiabs, using 0 = 0 as an initia soution. Fo th pobm M30 : h initia soution fo th QP mod is 0=0. h sv quimnts by th appication a N=(4, 5, 3). 43 quations 15 vaiabs 127 non-zo cofficints 68 non-ina cofficints h QP mod oca optima soution is o ZQP = 15. 5, NO a goba optima sinc th vau of th optima soution is o 1 1 = 3, 1 2 = 1, 2 2 = 3, 23 = 2, 33 = 3. o Soution tim is 0.2 sc. In this cas, th QP mod using GAMS/MINOS5.5 found a oca optima soution using 0 = 0 as initia soution. Anoth initia soution was tstd 8, wh is qua to its upp bound, that is iq = ( i, q). o h QP mod oca optima vau is ZQP = 17. 1, and taks 0.2 sc to find it. W obsvd that th QP mod is vy snsitiv to th initia soution povidd. Fo th pobm M500, which cannot b sovd using th MIP0 fomuation. h initia soution fo th QP mod is 0=0. h sv quimnts by th appication a N= (6,8,5). 149 quations 45 vaiabs 397 non-zo cofficints 202 non-ina cofficints Obsv that instantiation of th QP mod fo this pobm is significanty sma than th LP aation of th MIP0 fomuation. h QP mod oca optima vau is o ZQP = 50. 4, ocay infasib!!! o Soution tim is 2.6 sc. Anoth initia soution was tstd, wh is qua to its upp bound, that is = FSR. o h QP mod oca optima vau is ZQP = h optima vau found by th LPP agoithm is o Soution tim is 2.6 sc. If instad, 18 = N, th soution found by th LPP agoithm, was usd as an initia soution fo th QP mod, thn o h QP mod oca optima vau is ZQP = o Soution tim 2.6 sc. 8 Not that th initia soution fo th QP mod dos not nd to b fasib