Hidawi Publishig Corporaio Mahemaical Problems i Egieerig Volume 215, Aricle ID 783149, 21 pages hp://dx.doi.org/1.1155/215/783149 Research Aricle Dyamic Pricig of a Web Service i a Advace Sellig Evirome Ehram Safari, 1 Masoud Babakhai, 2 Seyed Jafar Sadadi, 1 Kamra Shahaaghi, 1 ad Khadieh Naboureh 1 1 Deparme of Idusrial Egieerig, Ira Uiversiy of Sciece ad Techology, Narmak, Tehra 1684613114, Ira 2 Kara Islmic Azad Uiversiy, Narmak, Tehra 1684613114, Ira Correspodece should be addressed o Ehram Safari; ehram.safari@gmail.com Received 5 April 214; Revised 8 Augus 214; Acceped 18 Augus 214 Academic Edior: Padia Vasa Copyrigh 215 Ehram Safari e al. This is a ope access aricle disribued uder he Creaive Commos Aribuio Licese, which permis uresriced use, disribuio, ad reproducio i ay medium, provided he origial work is properly cied. Cosider a web service wih differe qualiy of service levels where users may purchase heir required web service hrough a reservaio sysem. The service provider aduss prices of web service classes over a prespecified ime horizo o maage demad ad maximize profi. Users may cacel heir services as log as hey pay a pealy. Oe of he impora challeges for service providers is capaciy limiaio of he resources employed i offerig he web service. Thus, akig his impora proposiio io accou makes pricig sraegies cosidered by he provider has more credi. Aoher impora facor i deermiig pricig sraegies discussed i he prese paper is he marke ifluece which ca icrease or decrease he price ha he provider offers. This paper develops a coiuous ime opimal corol model for ideifyig pricig sraegies for he web service classes. We sudy he opimaliy codiio of he cosidered model based o maximum pricipal ad propose a algorihm o obai he opimal pricig policy. Moreover, we perform umerical aalyses o evaluae he effec of some parameers o corol ad sae variables ad obecive fucio. I addiio, we compare he proposed algorihm wih geeic algorihm (GA ad simulaed aealig (SA available i Malab. 1. Iroducio I rece years, web services have achieved populariy as a effecive ad efficie echology for developig ad icorporaig web applicaios. Web services are loosely coupled ad reusable sofware compoes ha semaically ecapsulae discree fucioaliies ad are disribued ad programmaically accessible over he sadard Iere proocols [1 6]. Ulike developed or licesed packaged applicaios, web services iclude specific busiess fucioaliies ha ca be reed over he ework (such as Iere o users by payig a fee [7]. For example, Google s AdWords web service helps users o creae compuer codes o coec direcly wih is AdWords plaform. Usig web services for sysem developme has several advaages. I addiio o he uiversal ieroperabiliy ad availabiliy of web services, users acquire lower coss ad more freque sofware updaes. Furhermore, a provider o oly provides he service implemeaios, bu also procures he echical suppors [8]. Despie abuda beefis, here are serious cocers abou he deermiig pricig sraegies for web services. I is esseial for a service provider o deploy a appropriae pricig mehod. A pricig mehod is employed as a effecive access corol ool by offerig adequae moivaios ocosumerssohaheyselecproperservicelevels.i oher words, as users decide o buy a web service, hey cosider is qualiy of service (QoS as well as is price. As a resul, he provider usually offers differe service classes of he web service o saisfy users wih differe requiremes. Each web service class has paricular QoS, which is defied as a group of service measures represeig he degree of user agreeme of he service [9]. Commo measures of QoS are respose ime, reliabiliy, ad availabiliy [1]. Wihou a suiable pricig sraegy for he web service, ay QoS based web service class is uusable; if we deermie o price for ay class, all users will choose some classes wih high prioriy QoS. I oher words, ideifyig a appropriae price for ay
2 Mahemaical Problems i Egieerig web service class should give users a ecourageme o lik he righ web service class. Limiaio of he resources employed, like sorage hroughpu, ework badwidh, CPU ime, is oe of he mos crucial challeges he service provider ecouers i offerig he web service o he users. Thus, he provider should maage his/her limied resources i a way o ge he mos desirabiliy ou of his/her accessible resources. The provider may employ echical sraegies o maage his/her limied resources. This ca lead o some problems like dissaisfacio of he users wih differe eeds, iabiliy of he providers i sadig by heir commimes due o iabiliy i maagig demad, ad a accreio of echical complexiies i offerig he web service. As a resul, we ca reduce he effecs of such problems by caegorizig he web service i classes wih differe QoS for users wih differe eeds ad sellig i before cosumpio, ad adopig a dyamic pricig mechaism o maage demad for web service classes. Oe of he appropriae pricig mehods for ideifyig pricig sraegy for he web services is dyamic pricig i whichheprovideradusspricesohecosumersbased o he value ha he cusomers cosider o he service. The dyamic pricig models have bee used i may idusries, such as reailig, maufacurig, airlies, ad e-busiess where he capaciy of he resources is fixed i shor-erm. Accessibiliy of he ecessary iformaio for demad, ease of chagig prices, ad availabiliy of required decisio ools for evaluaighedemaddaaie-commerceseigsmoivae us o employ he dyamic pricig as a efficie mehod o obai price of he web service [11 14]. Oe of he commo mehods i sellig services is advace sellig. This mosly happes because i service eviromes here is he risk ha i he mome of offerig service some of he service capaciy may o be sold. By applyig his mehod, he provider ca maage his/her service capaciy beer. Moreover, havig a advace sellig, he service provider ca icrease his/her profi gaied from cacellaio reveue. Such sellig mehod ca also be applied o he web service ad sigificaly help he provider i icreasig profi ad beer maagig resource capaciy of he web service. Oe of he mos impora facors affecig he olie pricig is marke ifluece which is usually deermied by usig feaures such as he umber of compeiors, cosumer ivolveme, ad populariy of he produc iem amog cosumers. Accordig o he sudy of he olie markes, some researchers have foud ha he icreasig i he umber of he compeiors reduces he price charged by a provider [15, 16]. I coras o he suggesio of reduced price differeiaio wih icreased compeiio, i a segmeed marke, he prices charged by a reailer ca icrease wih he umber of compeiors [17]. Furhermore, Vekaesa e al. hypohesized ha o average, prices charged by a olie service provider would firs icrease wih he umber of compeiors, ad beyod a specific hreshold, a icrease i he umber of compeiors would cause o a decrease i he prices charged [18]. Cusomer search behaviour relaes o he cosumer ivolveme. For services wih higher average price levels, cosumers may carry ou icreased search effors because of higher perceived risk levels. Cosequely, he providers ca se lower prices where he cusomer searchisesimaedobehigher.ieresigly,someempirical evidece idicaes he opposie o be rue [19]. Popular produc iems are hose well acceped ad bough by may of he cusomers. I olie markes, he cusomers commoly have well commuicaio ad exchage iformaio quickly via may elecroic chaels. Therefore, we expec ha price for popular producs should be lower ha ha of opopular producs [19].Based o wha was saed before,cosiderig he marke ifluece o deermiig pricig sraegies is highly impora ad failig o cosider i makes he specified prices ucrediable. I he prese paper he marke ifluece is cosidered as a parameer which affecs demad of he web service classes. I oher words, he effecs of he meioed feaures for he marke are cosidered as he marke ifluece parameerihedemadfucio. Opimal corol heory ca be applied as a effecive ool for sudyig he price behaviour of he web service over he ime horizo. A coiuous ime mehod has he advaage of offerig he exac soluio for he real-world applicaios. Whe a discree ime mehod is used by someoe, i is ecessaryoselecreasoableimesepsadspecifyhe prices oly a he give imes. However, he web service provider may eed eve more flexibiliy. To deal wih his problem,oecachoosesmallimeseps.ifheimesep is small ad he ime horizo is relaively large, he problem size i erms of he umber of variables ad he umber of cosrais will become exremely large. Thus, a cosiderable amou of effor is required o fid he opimal soluio. I his paper, we develop ad sudy a oliear coiuous ime opimal corol model for he web service pricig problem i which users buy heir required web service over a reservaio sysem ad uilize i i he fuure. We oe ha he users who reserve he web service have he righ o cacel heir orders before receivig hem. Sice differe users usually have various requiremes, providers ypically offer muliple web service classes wih differe qualiy of service (QoS levels. The mai par of he proposed model is ha required resources are shared amog all web service classes. Aoher impora facor i deermiig pricig sraegies which is discussed i he prese paper is he marke ifluece which ca icrease or decrease he price ha he provider offers. Furhermore, we suppose ha a each ime, he demad rae for a web service class is a geeral fucio of is price, marke ifluece, ad ime. The caceled orders a ime uiformly spread across he ierval [, ]. Cacellaio pealy cos for caceled orders depeds o ime ad he provider dyamically alers his/her proposed prices o maage he arrivig demad effecively. Our paper is associaed he service wih various classes which have paricular QoS. Sice early 199, QoS sudy has sared o examie ecoomic-based ework resource allocaio ools ha suppor usage-based pricig [2 22]. MacKie-Maso ad Varia argued ha cosiderig pricig approaches o reduce he cogesio cos has desirable side effecs [23]. Parameswara e al. cocluded ha pricig is a esseial aciviy i a fas daa commuicaio ework ad cosidered ha a appropriae pricig paer would be impora o moivae users o selec suiable QoS levels
Mahemaical Problems i Egieerig 3 [24]. Ibrahim e al. proposed a ovel QoS-based chargig ad resource allocaio framework for wireless eworks. The framework assigs resources o clies accordig o heir QoS requiremes. They also allowed he ework operaors o follow a wide variey of sraegies, icludig maximizig reveue ad usig aucio or uiliy-based pricig [25]. Gupa e al. developed a spo pricig model for iradomai expeced badwidh coracs wih loss-based QoS guaraees. They creaed a oliear pricig scheme for he cos recovery from earlier work ad exeded i o price risk. A uiliybased opio pricig mehod was developed o deal wih he uceraiies i deliverig loss guaraees [26]. Zhag e al. preseed a complee sudy for compeiive pricig of packe-swichig eworks wih a QoS assurace i erms of a expeced per-packe delay. They offered a srucure i which service providers preseig muliclass prioriy-based services compee o maximize heir profi while saisfyig he expeced delay assurace i each class. They firs deal wih he price compeiio wih fixed delay assurace ad he exeded i o he codiio where providers compee o price as well as QoS [27]. This paper also cosiders advace sellig i he web service pricig problem. Advace sellig ake places while he service provider allows users o buy a imes prior o he spo period [28]. Nagle ad Holde used advace sellig for machig demad ad supply [29]. Desirau ad Shuga viewed advace sellig as a ool for implemeig price discrimiaio. They cosidered he muliperiod pricig of a service i which he price is a fucio of forecased capaciy o be used i he fuure [3]. Shuga ad Xie showed ha advace sellig ca be more profiable ha spo sellig especially i he case where cosumers are ucerai abou heir fuure cosumpio saes [31]. Mesak e al. employed mehods of he opimal corol heory ad calculus of variaios o solve a problem i which a service provider was o maximize he prese value of reveue/profi over a plaed horizo i coiuous ime. They sudied he role of he hrea of compeiive ery ad discou rae i describig he opimal allocaio sraegy of capaciy over ime. Their paper also demosraed he supremacy of advaceselligicoiuousimeadsudiedheeffecof chages i model parameers o addiioal capaciy [32]. Jig combied ecoomic aspec of grid compuig wih resource reservaio. He proposed a mahemaical model which ca ideify amou of resource bookig. He also cosidered repuaio idex of service supplier ad ecoomic ieres i his model ad used umeric aalysis o show ha here exiss a opimal poi for he advace bookig of resource [33]. Lucas e al. provided a ew way of bookig resources i which cloud users specify he miimum ad maximum umber of virual resources required. They deal wih periods of peak load usig a uiversal framework which coais addiios ad/or modificaios of some compoes a various levels of he cloud archiecure [34]. I order o solve he proposed model, we use ideas from he corol heory ad oliear opimizaio. Opimal corolheoryhasbeedevelopedofidopimalapproaches o corol dyamic sysems [35 37]. Noliear opimal corol models are maily helpful for he dyamic pricig ad oher maageme sciece applicaios. For isace, we ca see hese models i he lieraure of producio plaig [38 4], dyamic pricig [41 44], fluid eworks [45], ad fiace [46]. Esmaeilsabzali ad Day cosidered a web service provider wih limied capaciy ad requesed for his/her service.theyofferedaoliealgorihmforselecigfrom arrived requess based o maximum willigess o pay off requesers. They also preseed ha boh olie ad offlie algorihms for he problem are NP-hard [47]. The opimal pricig ad locaio sraegy of a web service iermediary (WSIwassudiedbyTagadCheg.Theyofferedhe opimal soluio i a liear ciy model ad he exeded he resuls o he more geeral ui circle model. They showed ha he opimal sraegy ca be compued i erms of usig delay cos, iegraio cos, ad prices of he cosiue web services [48]. A effecive mehod for compuig price of webservicesbasedohequaliyofservice(qosaduser experiece was proposed by Wu. The aalysis experime showshahewebservicepricigmodelworksverywellad makes he service price more realisic ad accurae [49]. Pa e al. provided a dyamic pricig sraegy for a provider who offers a web service wih differe QoS levels. They assumed ha he provider usually dyamically chages his/her prices ad suggess several class services o saisfy eeds of differe cusomers. They aalyically solved he model ad obaied opimal soluio for he web service class prices ad capaciy. Furhermore,heirmodelhadocosraioheweb service capaciy ad hereby obaiig he opimal soluio for heir model was very sraighforward [5]. Zhag e al. addressed compeiio bewee wo providers who provide similar web service. Each provider should offer a service level (sadard or premium ad charge a price for he chose service level o mee he QoS guaraee. They firs sudied he case where he providers choose he service level ad price simulaeously, ad he exeded i o a sequeial-move siuaio [51]. I geeral, surveyig he above meioed research shows ha researchers have doe o sudy o he web service pricig problem wih cosidered assumpios. Geerally, he coribuios of his paper are he followig. (i We firs develop a coiuous ime model for he dyamic pricig of he web service ad he he model is discreized. (ii We eed cerai resources o offer he web service. Ihispaperiishypohesizedhaforeachresource here is a defiie capaciy which is shared amog he web service classes. (iii We cosider marke ifluece as a effecive facor i ideifyig pricig sraegies of he web service. (iv Demad fucio for each web service class is a geeral fucio of ime, is price, ad marke ifluece. Cacellaio rae ad raio of pealy are geeral fucios of ime.
4 Mahemaical Problems i Egieerig (v We ivesigae he srucure of he opimal pricig sraegies for he model based o he maximum pricipal. (vi We provide a heurisic algorihm o obai he opimal pricig sraegies wihi he ime horizo. (vii We perform umerical aalyses o evaluae he effec of some parameers o he corol ad sae variables ad obecive fucio. I addiio, we compare he proposed algorihm wih GA ad SA. The remaider of his paper is orgaized as follows. Secio 2 defies he assumpios ad oaios we will use hroughou he paper ad he we explai he cosidered model. I Secio 3, we provide some resuls o he srucure of he opimal soluio. I Secio 4, we provide a heurisic algorihmoobaiheopimalcorolvariables(prices. I Secio 5, we perform umerical aalyses o sudy he effec of he demad fucio, shared resource capaciy, ad cacellaio rae o he corol ad sae variables ad obecivefucio.furhermore,heproposedalgorihmis compared wih GA ad SA. 2. Model Formulaio 2.1. Model Noaios T: Advace sellig period. :Numberofwebserviceclasses. m: Number of required resources. c i :Uiwebservicecosofih class. Ca : Shared capaciy of h resource. e i (: Cacellaio rae of ih class a ime. Mf i (: Marke ifluece o demad fucio of ih class a ime. p i (: Ui web service sellig price of ih class a ime (corol variable. θ i (: Raio of pealy o sales price whe a user cacels his reservaio a ime of sellig period. SR i (: Sale reveue fucio of ih web service class a ime (sae variable. RL i (: Reservaio level of ih web service class a ime(sae variable. cap i : Needed capaciy of resource for ui web service class i. d i (p i, Mf i,: Demad fucio for web service class i a price p i ad ime. (i We deoe by p(, SR(, RL(, θ(, e(, c, Mf( he vecors wih respecive compoes p i (, SR i (, RL i (, θ i (, e i (, c i, Mf i (,,...,. (ii p (, SR (, RL ( deoe he opimal soluio. 2.2. Assumpios. This paper cosiders a fiie ime horizo model. Alhough we focus o he web service dyamic pricig, our resuls ca be applied o oher service maageme eviromes. The developed model icludes he subseque feaures. (i The developed model is ime-coiuous. (ii Users ge heir eeded web service hrough he bookig sysem ad uilize i i he fuure. (iii The provider offers he web service classes wih various QoS levels. (iv For each web service class, he demad fucio is depede o he ime ad is price ad he marke ifluece. (v Because he demad fucio is usually decreasig proporioal o price, he erm ( d i / (p i, Mf i, < exiss. Moreover, i is hypohesized ha as he price icreases, variaios i he demad decrease which meas he erm ( 2 d i / p 2 i (p i, Mf i, exiss. This happes due o he sesiiviy of he cusomers o he variaios of he price i he higher prices of offerig he web service. (vi I order o offer he web service, we eed m resources. The provider shares he capaciy of each resource amog he web service classes. (vii Users ca cacel heir order hrough he bookig period. (viii The caceled orders a ime are spread uiformly i he ierval [, ]. (ix Users who cacel heir orders are asked o charge pealy. (x Cacellaio rae ad raio of pealy are geeral fucios of ime. (xi The obecive is o maximize he profi. 2.3. Model Descripio. Sice users commoly buy he web service before cosumpio ime, his makes he real-world assumpio ha users may cacel heir orders before receivig hem. Users who cacel heir orders are asked o charge he pealy. For he web service class i, cacellaioreveue over he ime horizo [, T] is give by T θ i ( e i ( SR i ( d. (1 Siceweassumehecaceledordersforeachwebservice class a ime uiformly spread across he ierval [, ], we ca muliply θ i ( by SR i (e i ( o fid he pealy of he cusomers who cacel heir reservaio for he web service class i a ime. Iegraigθ i (e i (SR i ( over he ierval [, T] clearly provides cacellaio reveue for he web service class i. As he use of he web service becomes geeral, i ca be prediced ha he provider will o be able o offer service o all he users. The mai cause for his is limiaio of resources like ework badwidh ad CPU ime i offerig he web
Mahemaical Problems i Egieerig 5 service o he users. So i is hypohesized ha here is a eed for m resources o offer he web service ad he employed resources are differe i each of he web service classes. Moreover, i is hypohesized ha he provider akes io accou limied capaciy for each of he cosidered resources which he differe classes of he web service use commoly. The proposed model ca be wrie as follows: max s.. SR i (T + c i RL i (T, T θ i ( e i ( SR i ( d (2 SR i ( =d i (p i (, Mf i (,p i ( e i ( SR i ( [, T],,...,, (3 RL i ( =d i (p i (, Mf i (, e i ( RL i ( [, T],,...,, (4 cap i RL i (T Ca =1,...,m, (5 d i (p i (, Mf i (, [, T],,...,, (6 p i ( [, T],,...,, (7 SR i ( =, RL i ( =,...,. (8 I his corol model, he obecive fucio (2 isheoal reveue (composed of he sales reveue ad cacellaio reveue of he web service classes a ime Tmiusheoal cos (composed of he cos of he oal reserved web service for differe classes a ime T. The sae equaios (3ad(4 illusrae he chage of he sales reveue ad reservaio level a ime, respecively. Sales reveue level chage for he web service class i a ime is equal o he reveue ha is obaied from sellig he web service class i o d i (p i (, users a price p i ( mius he reveue ha he provider misses due o users cacellaio. Sice i is supposed ha he caceled orders a ime uiformly spread across he ierval [, ],we apply expressio SR i (e i ( o calculae he provider missed reveue. Reservaio level chage rae for he web service class i a ime is equal o orders for he web service class i a ime mius he reserved orders ha are caceled by he users a ime. Cosrai (5isdefiedomakesurehahe sum of sold resource for all he web service classes is equal o or less ha he shared capaciy of he resource amog all he web service classes. Cosrais (6ad(7are used o cofirm ha demad fucios ad prices are oegaive, respecively. Iiial value of he sae variables is deoed by (8. 2.4. Discreized Formulaio. The model ca be rewrie by discreizig he ime horizo. We pariio he ime ierval [, T] io N subiervals wih leghs τ 1,τ 2,...,τ N ad we eed o obai he decisio variables of he model i imes 1,..., N such ha 1 = τ 1 ad for l = 2,...,N,wehave l = l 1 +τ l.replacig l, d = τ l, dsr i ( l /d = (SR i ( l SR i ( l 1 /τ l,addrl i ( l /d = (RL i ( l RL i ( l 1 /τ l io (2 o(7 provides he discreized formulaio of he model as follows: max N SR i ( N + θ i ( l e i ( l SR i ( l τ l l=1 c i RL i ( N, s.. SR i ( l SR i ( l 1 =(d i (p i ( l,mf i ( l, l p i ( l e i ( l SR i ( l τ l l=1,...,n,,...,, RL i ( l RL i ( l 1 =(d i (p i ( l,mf i ( l, l e i ( l RL i ( l τ l l=1,...,n,,...,, (9 (1 (11 cap i RL i ( N Ca =1,...,m, (12 d i (p i ( l,mf i ( l, l l=1,...,n,,...,, (13 p i ( l l=1,...,n,,...,, (14 SR i ( =, RL i ( =,...,. (15 Equaio (9 idicaes he discreized obecive fucio of he model. The cosrais (1, (11, (12, (13, (14, ad (15 discreize he cosrais (3, (4, (5, (6, (7, ad (8, respecively. 3. Soluio Approach 3.1. Maximum Priciple for Opimal Corol Problems wih Mixed Iequaliy Cosrais. I his secio, he maximum priciple for opimal corol problems wih mixed iequaliy cosrais will be expressed. For furher facs see [35, 52]. Le us cosider he followig opimal corol problem wih mixed iequaliy cosrais: T max F (x (,u(, d + M (x (T, (16 x ( =f(x (,u(,, (17 g (x (,u(,, (18 a (x (T,T, (19
6 Mahemaical Problems i Egieerig where T is plaed ime horizo, x( E 1 is he vecor of sae raecories a ime, u( E m1 is he vecor of corol raecories a ime, hefuciosf from E 1 E m1 E io E, M from E 1 Eio E, f from E 1 E m1 Eio E 1,g from E 1 E m1 Eio E d,ada from E 1 E io E l are coiuously differeiable wih respec o all heir argume. I is supposed ha each eleme of he fucios g depeds explicily o he corol u ad sae x. More exacly, hesubsequefullrakcodiiomusbeheld: rak [ g, diag (g] = d. (2 u Codiio (2 deoes ha he gradies wih respec o u of all he acive cosrais g mus be liearly idepede. To express he maximum priciple, i is esseial o defie he Hamiloia fucio as follows: H (x, u, λ1, =F(x, u, +λ1f(x, u,, (21 where λ1 E 1, whose compoes are ermed adoi variables. We also defie he Lagragia fucio as L (x, u, λ1, μ1, = H (x, u, λ1, + μ1g (x, u,, (22 where μ1 E d is a row vecor, whose compoes are ermed Lagragia mulipliers. Theorem 1 (ecessary codiio. The ecessary codiios for u wih he correspodig sae raecory x o be a opimal soluio are ha here should exis coiuous ad piecewise coiuously differeiable fucios λ1, piecewise coiuous fucios μ1 such ha he followig codiios hold: x ( =f(x (,u (,. (23 Saisfyig λ1 = L x (x,u, λ1, μ1,. (24 Wih he rasversaliy codiios λ1 (T =M x (x (T,T+αa x (x (T,T, α, αa(x (T,T =, (25 where α E l is cosa vecor. A each [,T]ad for all u saisfyig g(x,u,, he Lagrage muliplier is such ha L =( H u u=u u +μ1 g u = (26 u=u ad he complemeary slackess codiios are μ1, μ1g (x,u, =. (27 Theorem 2 (sufficiecy. Le (x,u, λ1, μ1, α saisfy he ecessary codiios i Theorem 1. If H(x, u, λ1, is cocave i (x, u a each [,T], M i (16 is cocave i x, g i (18 is quasi-cocave i (x, u, a i (19 is quasi-cocave i x,he (x,u is opimal. 3.2. Opimaliy Codiio for he Proposed Model. I his secio, we firs prove ha he mixed iequaliy cosrai qualificaio holds. The, we creae he Hamiloia ad Lagragia fucios ad apply he maximum pricipal o prese some impora resuls. Lemma 3. Themixediequaliycosraiqualificaioholds. Proof. The cosrai qualificaio illusraes ha he gradies of he eire acive mixed iequaliy cosrais (6ad (7 wih respec o p( mus be liearly idepede. The mixed iequaliy cosrais for his problem are as follows: z = (p(, d(p,.basedo(2, i is eeded o show ha he rak M = ( z/ p(, diag(z is 2.TheMarixM R 2 R 3 ca be wrie as follows: 1 p 1 ( d d 1 p ( d 1 M= d p 1 (p 1 (, ( 1 d d. (28 ( d p d (p (, To demosrae he rak M=2,wecaseehefollwoig: sice here are 2 rows i M, herakisamos2. Ifhere is o bidig cosrai o he cosrai ses (6 ad(7, he las 2 colums are ozero ad liearly idepede; as a resul he rak of marix is 2. Foreachi such ha p i ( =, colum+iis he zero vecor, however, i ca be subsiued wih colum i o gai a se of liearly idepede colums. Similarly, for each i such ha d i (p i (, =,sice
Mahemaical Problems i Egieerig 7 d i (p i (, Mf i (, / >, i ca replace colum +2iwih colum i. I is ecessary o oe ha he cosrai ses (6 ad (7 cao be boud simulaeously. The maximum priciple defies he ecessary codiios for opimum corol variables. Usig (21, he Hamiloia fucio for his problem ca be illusraed as follows: H(SR (, RL (,p(,λ(,λ (, = + + θ i ( SR i ( e i ( λ i ( (d i (p i (, Mf i (,p i ( e i ( SR i ( λ i ( (d i (p i (, Mf i (, e i ( RL i (. (29 The adoi variables λ( = (λ 1,...,λ ad λ ( = (λ 1,...,λ dualize, respecively, he sae equaios (3ad(4a ime. Furhermore, usig (22, he Lagragia fucio for his problem ca be wrie as follows: L(SR (, RL (,p(,λ(,λ (,μ,η, =H(SR (, RL (,p(,λ(,λ (, + μ i ( d i (p i (, Mf i (,+ η i ( p i (. (3 The Lagrage mulipliers μ( = (μ 1,...,μ ad η( = (η 1,...,η relax, respecively, he cosrais (6 ad(7 a ime. However, sice lef side of cosrais (6 ad(7 for each web service class is oly depede o is price, marke ifluece ad ime, we ca remove hese cosrais from Lagragia fucio ad use hem o creae feasible soluio space. Therefore, Lagragia fucio ca be rewrie as L(SR (, RL (,p(,λ(,λ (,μ,η, =H(SR (, RL (,p(,λ(,λ (,. (31 Accordig o he maximal pricipal Theorem 1,aheopimal soluio (i The sae raecories saisfy SR i ( =d i (p i (, Mf i (,p i ( e i ( SR i ( (32 [, T],...,, RL i ( =d i (p i (, Mf i (, e i ( RL i ( (33 [, T],...,, cap i RL i (T Ca =1,...,m, (34 SR i ( =, RL i ( =,...,. (35 (ii The opimalcorolo [, T] is calculaed as follows: p ( = arg max p( D( L(SR (, RL (,p (,λ(,λ (,, (36 where D( is he se of admissible corols p( such ha: p i (,...,, (37 d i (p i (, Mf i (,,...,. (iii Usig (24, for every [,T],hecoiuous vecor of he adoi variable λ( fulfills he followig differeial equaio: λ i ( = L SRi (SR (, RL (,p (,λ(,λ (, = e i ( (θ i ( λ i (,...,. (38 Furhermore, he followig differeial equaio holds for he adoi vecor λ (: λ i ( = L RL i (SR (, RL (,p (,λ(,λ (,μ(, =e i ( λ i (,...,. (39 (iv Usig (25, The rasversaliy codiios ca be formed as follows: λ i (T = ( =1 SR (T c RL (T SR =1,...,, i (4 λ i (T = ( =1 SR (T c RL (T RL i α, + m =1 = (c i + α (Ca cap i RL i (T RL i m =1 α (Ca α cap i,,...,, cap i RL i (T =, =1,...,m. (41 Proposiio 4. For each web service class i a ime, he opimal raecory λ i ( is give by T 1 λ i ( = mi ( (m i (T + θ i (s e i (s m i (s ds, (42 where m i ( = e e i(d.
8 Mahemaical Problems i Egieerig Proof. Equaio (38 is a liear firs-order differeial equaio. Is sadard form is λ i ( e i (λ i ( = θ i (e i (, basedoheavailablesadardsoluioforhefirsorder differeial equaio i [53], we have λ i (T m i (T λ i ( m T i ( = θ i (s e i (s m i (s ds, (43 where m i ( = e ei(d,recallig(4givesλ i ( = (1/m i ( (m i (T + T θ i (se i (sm i (sds. Proposiio 5. For each web service class i a ime, he adoi variable λ i ( is give by λ i ( = (c i + m =1 α cap i e T e (sds i, (44 where α, α (Ca cap i RL i (T=,=1,...,m. Proof. From differeial equaio (39, we have λ i (/λ i ( = e i(, iegraig wih respec o gives l(λ i ( T = T e i (sds, recallig (41 gives λ i ( = (c i + m =1 α cap i e T e i. Proposiio 6. A each ime [,T],forobaiedλ (, λ( ad α = (α 1,...,α from ecessary codiio, he uique opimum corol soluio ca be give by p op i ( ={ p i ( p i ( P i ( P i ( p i ( >P i (, (45 where p i ( ad P i( ca be obaied from he equaios d i (p i (, Mf i (, + (p i ( + (c i + m =1 α cap i /(1 + T θ i (e i (e T e (d i d( d i (p i (, Mf i (, / = ad d i (P i (, Mf i (, =,respecively. Proof. I order o obai he opimal corol variables as fucio of λ (, λ( ad α, he opimizaio problem (36 is solved. I is also oiced ha he Lagragia fucio is separabl eacross he web service classes ad i p i (. To fid opimal soluio of problem, he parial derivaive of Lagragia fucio wih respec o corol variable p i ( ca be obaied as L (SR (, RL (,p(,λ(,λ (, =λ i ( (d i (p i (, Mf i (, + d i (p i (, Mf i (, p ( i +λ i ( d i (p i (, Mf i (,. (46 To achieve he opimal soluio, hese parial derivaives are se o zero as follows: λ i ( d i (p i (, Mf i (,+(λ i ( p i ( +λ i ( d i (p i (, Mf i (, =. (47 Furhermore, by subsiuig (42ad(44io(47, oe ca ge d i (p i (, Mf i (, +(p i ( + (1+ T (c i + m =1 α cap i d i (p i (, Mf i (, =. θ i ( e i ( e T e (d i d (48 To prove uiqueess of p i ( saisfyig (6, (7, ad (48. I is sufficie o show ha 2 L/ p 2 i <.I(48, Sice d i (p i (, Mf i(, ad d i (p i (, Mf i(, / <,we ca verify ha p i ( + (λ i (/λ i(,he, 2 L p 2 i = 2 H p 2 i =2λ i ( d i (p i (, Mf i (, ( +(λ i ( p i ( +λ i ( 2 d i (p i (, Mf i (, p 2 i <. (49 Le he maximum price be P i ( a ime, such ha d i (P i (, Mf i (, =.Ifp i lies wihi he ierval [, P i (], he i is he opimal corol. Oherwise, p i locaes i he boudary of he se of feasible corols, ha is, or P i (, depedig o which value correspods o he higher value of he opimizaio problem (36. Noe, ha by seig e i ( =, [, T], i = 1,...,,we obai he opimaliy codiio for he correspodig model which igores he cacellaio assumpio: d i (p WC i (, Mf i (,+(p WC i ( (c i + d i (p WC i (, Mf i (, =. m α cap i =1 (5 Referrig o he opimal price of he web service class ip WC i a ime, ad igorig resource capaciy cosrai (5, i ca be cocluded ha igorig cacellaio assumpio leads o overpricig.
Mahemaical Problems i Egieerig 9 Proposiio 7. If he capaciy cosrai (5 is removed from he model, he oe has p WC i ( >p i (. (51 Proof. Removig he resource capaciy cosrai (5, he value of expressio m =1 α cap i willbezero.thus,comparig (47ad(5,icabeobservedha Nex, by deoig f(p i ( =d i (p i (, Mf i (, +(p i ( c i θ i ( e i ( e T e (d i d d i (p i (, Mf i (, (54 d i (p i (, Mf i (, +(p i ( d i (p i (, Mf i (, c i θ i ( e i ( e T e (d i d =d i (p WC i (, Mf i (,+(p WC i ( c i d i (p WC i (, Mf i (, =, whileakigioaccouha d i (p i (, Mf i (, / <, d i (p WC i (, Mf i (, (52 ad recallig (49, we coclude ha f(p i (/ <.Thus, from codiios (52ad(53, we have f(p WC i ( < f(p i (, which wih respec o he las iequaliy eeds ha p WC i ( > p i (. Accordig o Proposiio 7, he opimal prices derived from he model igorig he order cacellaio assumpio is o opimal ad he proposed model is required o accou properly for he effecs of order cacellaio. Proposiio 8. For he model wihou resource capaciy cosrai (5, whe he ui cos of he web service class i icreases, he opimal price of he web service class i icreases. Proof. Suppose ha c1 i ad c2 i are wo cos values for he ui web service class i such ha c1 i < c2 i.toprovehis proposiio, oe ca show ha he correspodig prices p1 i ad p2 i have he followig relaioship: +(p WC i ( d i (p WC i (, Mf i (, =d i (p WC i (, Mf i (, c i θ i ( e i ( e T e (d i d p1 i <p2 i. (55 Removigheresourcecapaciycosrai(5adusig(48, we have d i (p1 i (, +(p WC i ( c i +c i d i (p WC i (, Mf i (, c i θ i ( e i ( e T e (d i d +(p1 i ( d i (p1 i (, Mf i (, =d i (p2 i (, c1 i θ i ( e i ( e T e (d i d (56 =(c i c i θ i ( e i ( e T e (d i d d i (p WC i (, Mf i (, <. (53 +(p2 i ( c2 i d i (p2 i (, Mf i (, =. θ i ( e i ( e T e (d i d
1 Mahemaical Problems i Egieerig By assumig ha d i (p i (, Mf i (, / <,icabesaed d i (p1 i (, Mf i (, Usig (48, for he prices p1 i ( ad p2 i ( we have d i (p1 i (, Mf1 i (, +(p1 i ( d i (p1 i (, Mf i (, =d i (p1 i (, Mf i (, c2 i θ i ( e i ( e T e (d i d +(p1 i ( d i (p1 i (, Mf1 i (, =, d i (p2 i (, Mf2 i (, c i θ i ( e i ( e T e (d i d (6 =( +(p1 i ( d i (p WC i (, Mf i (, c1 i c2 i c2 i +c1 i c1 i θ i ( e i ( e T e (d i d θ i ( e i ( e T e (d i d d i (p WC i (, Mf i (, >. Nex by defiig f(p i ( =d i (p i (, Mf i (, (57 +(p2 i ( d i (p2 i (, Mf2 i (, =. c i θ i ( e i ( e T e (d i d (61 By assumig ha d i (p i (, Mf i (, / <, d i (p i (, Mf i (, / Mf1 i > ad d i (p i (, Mf i (, / Mf i >, subsiuig p1 i ( wih p2 i ( io (61, we have d i (p1 i (, Mf2 i (, +(p1 i ( ( d i (p1 i (, Mf2 i (,. c i θ i ( e i ( e T e (d i d (62 +(p i ( c2 i θ i ( e i ( e T e (d i d d i (p i (, Mf i (, (58 recallig (49, agai, we ca derive ha f(p i (/ <. Therefore, from (56, (57, ad (6, we have f(p2 i ( < f(p1 i (, which wih respec o he las iequaliy eeds ha p2 i ( > p1 i (. Proposiio 9. For he model wih o resource capaciy cosrai (5, if d i (p i (, Mf i (, / Mf i > ad d i (p i (, Mf i (, / Mf i >, he he opimal price of he web service class i a ime icreases by icreasig he marke ifluece. Proof. Suppose ha Mf1 i ( ad Mf2 i ( are wo marke iflueces for he web service class i such ha Mf1 i ( < Mf2 i (. Now,wemusprovehepricesp1 i ( ad p2 i ( correspodig o Mf1 i ( ad Mf2 i ( have he followig relaioship: p1 i ( <p2 i (. (59 Therefore, d i (p1 i (, Mf2 i (, d i (p1 i (, Mf1 i (, >. +(p1 i ( ( d i (p1 i (, Mf2 i (, Nex by describig f(p i ( =d i (p i (, Mf2 i (, +(p i ( c i θ i ( e i ( e T e (d i d d i (p i (, Mf2 i (,. d i (p1 i (, Mf1 i (, c i θ i ( e i ( e T e (d i d (63 (64
Mahemaical Problems i Egieerig 11 Similar o (58, agai, we coclude ha f(p i (/ <. Usig (61 ad(63 ad(64 wehavef(p2 i ( < f(p1 i (, which wih respec o he las iequaliy eeds ha p2 i ( > p1 i (. Proposiio 1. For he model wih o resource capaciy cosrai (5, if d i (p1 i (, Mf i (, / Mf i < ad d i (p1 i (, Mf i (, / Mf i <, he he opimal price of he web service class i decreases by icreasig he marke ifluece. Proof. The proof of his Proposiio is similar o Proposiio 9. Proposiio 11. For each web service class i a ime, he opimal sales reveue SR i ( is give by SR 1 i ( = m i ( d i (p i (s, Mf i (,s p i (s m i (s ds, (65 where m i ( = e e i(d. Proof. The sae equaio (32 is he firs-order differeial equaio ad we have SR i ( = d i(p i (, Mf i(, p i ( e i (SR i (,hus, SR i (sm i(s = d i (p i (s, Mf i (,s p i (s m i (s ds, (66 where m i ( = e e i(d, herefore SR i ( m i ( SR i ( m i ( (67 = d i (p i (s, Mf i (,sp i (s m i (s ds, sice R i ( =,hus SR 1 i ( = m i ( d i (p i (s, Mf i (,s p i (s m i (s ds. (68 Proposiio 12. For each web service class i a ime, he opimal reservaio level RL i ( is give by RL 1 i ( = m i ( d i (p i (s, Mf i (,s m i (s ds, (69 where m i ( = e e i(d. Proof. The proof of his proposiio is similar o Proposiio 11. 4. Heurisic Algorihm I wha follows, we describe he heurisic algorihm which is uilized o derive he opimal soluio for he problem based o Evere s approach [54] for ideifyig he mulipliers α, = 1,...,m. I employs (45 o fid he soluio for he deermied mulipliers. For he specified mulipliers α, = 1,...,m, if cosrai (5 ad codiio (41 are saisfied, he he obaied soluio is opimal ad he algorihm will ed. Oherwise, i is ecessary o updae he value of mulipliers α,=1,...,mad repea he algorihm from he begiig. A kh ieraio of he proposed algorihm, he soluio p k (, [, T], = 1,...,, is firs calculaed by usig (45 for give mulipliers α k (s, = 1,...,m.Laero,ifollows ha σ k = Ca cap i RL i(t, = 1,...,m. Accordig o he calculaed σ k, he value of he mulipliers αk,=1,...,m may be deermied as follows: (i For all {1,...,m},ifσ k he α k+1 = max{ε,(1 ρ k αk },whereε has a posiive value ear zero. I esures ha α k+1 says oegaive. (ii For all {1,...,m},ifσ k <he αk+1 =(1+ρ k αk. I his mehod, if σ k <, he cosrai (5 iso saisfied, ad he value of mulipliers mus be icreased i order o pealize he addiioal violaio. I he same way, if σ k, he cosrai (5 isoviolaedadhevalueof he mulipliers mus be decreased. Furhermore, he parameers ρ k+1 are updaed as follows: if σ k σk 1 if σ k σk 1 if σ k σk 1 >he ρ k+1 =ε 1 ρ k, <he ρ k+1 =ε 2 ρ k, =he ρ k+1 =ρ k, where he parameers ε 1 ad ε 2 are cosa, ad also ε 1 >1 ad ε 2 < 1. If he capaciy cosrai (5 isviolaedahe ieraios of k ad k 1, he he value of ρ k+1 is gradually elarged accordig o he fac ha he value of he muliplier σ k may be raher disa from is opimal value. If cosrai (5 varies bewee ifeasibiliy ad feasibiliy, he he value of ρ k+1 isdecreasedomovehemulipliervalueσ k owards is opimal value. Accordig o Theorem 2, sice he Hessia marix of Hamiloia fucio (29 wihrespeco(p, SR, RL is a egaive defiie marix, we clearly coclude ha he Hamiloia fucio is cocave. Furhermore, I ca be evidely saed ha he expressio SR i(t c irl i (T i (2 is cocave, cosrais (6 ad(7 arequasi-cocave i (p, SR, RL, cosrai (5 isquasi-cocavei(sr, RL. Therefore he obaied soluio saisfied he codiios of Theorem 1 is opimal. I oher words, he opimal soluio of he primal ad dual problems is equal. The proposed heurisic algorihm has he followig seps: Sep 1: Se Ipu parameers; Sep 2: Se α =, = 1,...,m ad calculae λ i (, λ i (,,...,usig (42ad(44;
12 Mahemaical Problems i Egieerig Table 1: Seleced value for he ipu parameers of he web service classes. d i (p i, c i e i cap i1 Mf i Web service class 1 (2 + 5 1 2 2 p 1 ( 8.2.1 Web service class 2 2 (2 + 5 1 2 2 2p 2 ( 16.2.2 Sep 3: Calculae he price of he web service class i, (p i,,...,usig (45; Sep4:Compuehereservaiolevelofheweb service classes usig (69; Sep 5: If for all {1,...,m},hecosrai(5holds he go o 16; Sep 6: For all {1,...,m},Seα 1 =ε ad ρ 1 =ε ; Sep 7: Se k=1; Sep 8: Calculae λ i (,,...,usig (44; Sep9:Calculaehepriceofhewebserviceclasses usig (45; Sep 1: Compue he reservaio level of he web service classes usig (69; Sep 11: Se σ k = Ca cap i RL i(t,=1,...,m; Sep 12: For all {1,...,m},ifσ k <he se αk+1 = max{ε,(1 ρ k αk };elseseαk+1 =(1+ρ k αk ; Sep 13: For all {1,...,m},ifσ k σk 1 ρ k+1 = ε 1 ρ k ;elseifσk σk 1 else ρ k+1 =ρ k ; Sep 14: If σ k <θ;goo16;elsegoo15; Sep 15: Se k=k+1adgoo8; Sep 16: Ed. 5. Numerical Resul > he < he ρ k+1 = ε 2 ρ k ; We impleme he proposed algorihm for a ime horizo [, T]. We pla o explore he effec of ipu parameers, ha is, he demad fucio, shared resource capaciy, cacellaio rae, ad marke ifluece o he price, sales reveue, reservaio level, oal reveue, oal cacellaio reveue, ad profi. Cosidered examples iclude wo web service classes ad oe resource. Furhermore, he parameers of he algorihm are cosidered as ε =.1, ε =.1, ε 1 = 1.2, ε 2 =.8, adθ = 1 5.Theproposedheurisichasbee codedimaple15oapcwihaamddualcore(2.31ghz CPU ad 1 GB of RAM. I order o aalyze he effec of various parameers, we firsly creae Example 1 ad he use i o form oher examples. I oher words, ex examples are disic from Example 1 i oly oe parameer which we wa o ideify is impac. Assume he demad fucio of he web service class i (i = 1,...,2 be liear i erms of price wih a ime-depede maximal demad ad marke ifluece, d i (p i (, = α i ( β i p i ( + Mf i. The demad requireme d i (p i (, / = β i <ad 2 d i (/ p 2 i =are saisfied. The maximal demad α i (, (i = 1, 2 i a similar maer i he referece [42] isaquadraicfucioofheimead icreasig i he firs half of ime horizo ad decreasig i he secod half of ime horizo o sudy he impac of demad peak o he problem. Furhermore, we assume ha QoS of hewebserviceclass2isbeerhahewebserviceclass1. As a resul, he cos ad required resource of he ui web service class 2 ad he maximal demad of web service class 2aregreaerhahewebserviceclass1.Furhermore,we suppose ha he cacellaio raios θ i (, i = 1, 2 are cosa hrough ierval [, T 1] advariableoierval(t 1, T] which ca be give as θ i ( = {.3 [, T 1] { (T 1,T], { T i = 1, 2. (7 Oher seleced ipus for Example 1 are summarized i Table 1. To se he capaciy of he cosidered resource i Example 1, a firs we solve i wihou cosiderig capaciy cosrai. The, accordig o he prices obaied, he opimal fial reservaio level for he web service classes is calculaed ad based o ha, he maximum required oal capaciy of he cosidered resource CAP = cap 1 I 1 (T + cap 2 I 2 (T is aaied. Evidely, if he shared capaciy of he resource remais greaer ha or equal o CAP, he sraegies obaied are opimal. We uilize.75cap as he oal shared capaciy of he cosidered resource for Example 1. 5.1. Impac of he Demad Fucio. We cosider wo followig examples which are differe from Example 1 i he demad fucios. I Example 2, d 1 (p 1, ad d 2 (p 2, of Example 1 have bee doubled, while i Example 3, we riple he demad fucios d 1 (p 1,ad d 2 (p 2,of Example 1. Idicaedcurvesifiguresarelabelledasfollows:pli opimal pricig pah for he example l ad web service class i; dli opimal demad pah for he example l ad web service class i; Ili opimal reservaio level pah for he example l ad web service class i; Rli opimal sales reveue pah for he example l ad web service class i. As we ca see i Figure 1, he price of web service classes icreases as demad fucio doubles or rebles. Durig he imehorizo,hepricesfirsicreasehedecreasedueo aure of he demad fucios. Also, he price of he web serviceclass2isequalforexamples2ad3aheedof he ime horizo. The mai reaso for ha is lack of eough capaciy for sellig he web service class 2. I oher words, as
Mahemaical Problems i Egieerig 13 35 35 3 3 25 25 Price 2 15 Demad 2 15 1 1 5 5 1 2 4 5 6 7 8 9 1 1 2 4 5 6 7 8 9 1 p11 P12 p21 P22 p31 P32 d11 d12 d21 d22 d31 d32 (a (b 12 25 1 2 Reservaio level 8 6 4 Sales reveue level 15 1 2 5 1 2 4 5 6 7 8 9 1 1 2 4 5 6 7 8 9 1 I11 I12 I21 I22 I31 I32 R11 R12 R21 R22 R31 R32 (c (d Figure 1: The opimal price, demad, reservaio level, ad sales reveue pah for Examples 1 3 over he cosidered ime horizo. sellig he web service class 1 gais more profi o he provider of he service a he ed of he ime horizo, he seller of he service refrais from sellig he web service class 2. Aoher poi here is ha if he demad curve for oe web service moves i a posiive direcio due o differe reasos such as a icrease i usig he web service ad is populariy, heproviderofheserviceicreaseshepriceofheweb service classes o maage he demad, is cosa capaciy, ad icrease he profi. Moreover, i is illusraed i Figure 1 ha as he demad fucio moves i he posiive direcio, a firs he sales of he web service classes icreases, bu decreases a he ed of he ime horizo. For isace, i Examples 2 ad 3 he amou of sales ges equal o zero a he ed of he ime horizo. The mai reaso for such a collapse i sales is ha hewebserviceissoldoverhecapaciyahebegiigof he ime horizo ad profi is gaied for he provider hrough he pealies paid for he cusomers cacellaio of heir orders. Aoher coclusio derived from Figure 1 is ha as he demad fucio moves i he posiive direcio, he level of reveue gaied from he sales ad reservaio level ofhewebserviceicreaseforclass1whileforclass2 hey icrease a he begiig of he ime horizo, he decrease. Table 2 shows he opimal oal sales reveue, cacellaio reveue, profi, raio of he profi o he oal reveue, ad raio of he oal cacellaio reveue o he profi for Examples 1, 2, ad 3. Accordig o ha able, as demad fucio doubles or rebles, here is, respecively, a accreio equal o 23% ad 36% for he oal sale reveue, 42% ad 66% for he oal cacellaio reveue, 24% ad 38% for he oal
14 Mahemaical Problems i Egieerig Table 2: The opimal oal sales reveue, oal cacellaio reveue, oal reveue, profi, raio of he profi o he oal reveue, ad raio of he oal cacellaio reveue o he profi for Examples 1 o 3. Example 1 Example 2 Example 3 Sales reveue 1891 2331 2572 Cacellaio reveue 118 168 197 Toal reveue 29 2499 2768 Profi 632 112 1391 Profi/oal reveue 31% 45% 5% Cacellaio reveue/profi 19% 15% 14% Table 3: The opimal oal sales reveue, oal cacellaio reveue, oal reveue, profi, raio of he profi o he oal reveue, ad raio of he oal cacellaio reveue o he profi for Examples 1, 4, ad 5. Example 1 Example 4 Example 5 Sales reveue 1891 2219 257 Cacellaio reveue 118 6 Toal reveue 29 2278 257 Profi 632 9 1192 Profi/oal reveue 31% 39% 46% Cacellaio reveue/profi 19% 7% % reveue,ad77%ad12%forheprofigaied.besides, accordig o his able, he raio of he profi o he oal reveueforexamples1o3is,respecively,equalo31%,45%, ad 5% which idicaes ha accreio of he demad resuls i cosiderable accreio of he profi. Furhermore, he raio ofheoalcacellaioreveueoheprofiis,respecively, 19%, 15%, ad 14% for Examples 1, 2, ad 3. 5.2. Impac of he Cacellaio Rae. To aalyze he effec of flucuaios i cacellaio raes, Examples 3 ad 4 are cosidered which are differe from Example 1 i merely he cacellaio rae. The cacellaio raes of he web service classes for Examples 3 ad 4 are, respecively,.1 ad.. Thus, o aalyze he effec of flucuaios i he cacellaio rae,examples1,4,ad5arecompared. I ca be cocluded from Figure 2 ha as he cacellaio rae decreases, he service provider icreases he price o augme his/her ow profi. I oher words, as he cacellaio rae icreases, he provider decreases his/her price o le more people purchase web service classes. I fac, as some of hem cacel heir orders, he oal cacellaio reveue icreases whichiurraisesheprofiofheprovider. The reservaio level grows as he cacellaio rae icreases hrough ime. Moreover, whe he cacellaio rae is zero, he service provider refrais from sellig he web serviceclass2ahebegiigadedofhesellighorizo becauseigaisgreaprofiforheprovider. As he cacellaio rae icreases, he reservaio level icreases for he web service class 1, bu decreases for he web service class 2. The mai reaso for his is cosacy of he resource capaciy for Examples 1, 4, ad 5. I oher words, as he cacellaio rae icreases for he web service provider i is prediced ha hey sell more expesive web service class sice i gais more profi for hem. As a resul, due o he cosacy of capaciy, here is a declie i he reservaio level for he web service class 1 as he sales of he web service icreases for class 2. Table 3 deoes he oal sales reveue, cacellaio reveue, profi, raio of he profi o he oal reveue, ad raio of he oal cacellaio reveue o he profi for Examples 1, 4, ad 5. Based o Table 3, for Examples 4 ad 5, respecively, here is a growh equal o 17% ad 36% for he oal sales reveue,13%ad28%forheoalreveue,ad42%ad 89% for he profi obaied. Furhermore, i ca be observed ha he raio of he profi o he oal reveue for Examples 4 ad 5 is equal o 39% ad 46% which idicaes ha accreio of he cacellaio rae resuls i cosiderable accreio of he profi. The raio of he oal cacellaio reveue o he profi is, respecively, 6% ad % for Examples 4 ad 5. 5.3. Impac of he Resource Capaciy. To ivesigae he effec of flucuaios i resource capaciy, Examples 5 ad 6 are ake io accou which are differe from Example 1 oly i resource capaciy. Resource capaciies for Examples 6 ad 7are,respecively,akeCAPad.5CAP. Figure 3 shows quie well ha by he accreio of he levelofresourcecapaciyhepriceforhewebserviceclasses decreases. I oher words, as he capaciy icreases, he provider raise his/her profi by decreasig his/her price. For all he cosidered examples he prices icrease a firs, he decrease.themaireasoforhiscabefoudiheypeof he demad fucio. The amou of sales reveue decreases as he capaciy decreases. I Example 6, which has he leas capaciy, he provider avoids sellig he web service 2 a he ed of ime
Mahemaical Problems i Egieerig 15 35 25 3 25 2 Price 2 15 Demad 15 1 1 5 5 1 2 4 5 6 7 8 9 1 1 2 4 5 6 7 8 9 1 p11 P12 p41 P42 p51 P52 d11 d12 d41 d42 d51 d52 (a (b 9 18 8 16 7 14 Reservaio level 6 5 4 3 2 1 1 1 2 4 5 6 7 8 9 1 I11 I12 I41 I42 I51 I52 Sales reveue level 12 1 8 6 4 2 1 2 4 5 6 7 8 9 1 R11 R12 R41 R42 R51 R52 (c (d Figure 2: The opimal price, demad, reservaio level, ad sales reveue pah for Examples 1, 4, ad 5 over he cosidered ime horizo. horizo. This, i ur, leads o a icrease i he opimal cosumpio of he desired resources. Ashelevelofheresourcecapaciyicreases,hereservaio level ad sales reveue level icrease. However, i should be meioed ha his icrease i he reservaio level ad sales reveue level coiues uil here is o uused capaciy. I case here is exra capaciy, accreio i he capaciy will acually have o effec o he flucuaios of he level of reservaio ad reveue. Table 4 summarizes he oal sales reveue, oal cacellaio reveue, oal reveue, profi, raio of he profi o heoalreveue,adraioofheoalcacellaioreveue oheprofiforexamples1,6,ad7.ashecapaciyof cosidered resource is, respecively, dropped by 25% ad 5% forexamples1ad6,hereisa15%ad36%decreasei he sales reveue, 1% ad 25% i he cacellaio reveue, 15% ad 35% i he oal reveue, ad 5% ad 11% i he profi. 5.4. Impac of he Marke Ifluece. To evaluae marke ifluece o he pricig sraegies of he web service, Examples 8 o 11 are cosidered which are differe from Example 1 oly i demad fucios. I geeral, for each example, he demad fucio for web service classes 1 ad 2 is show by 3 p 1 ( + Mf 1, 3 p 2 ( + Mf 2,respecively.Demad fucios of Examples 8 o 11 are differe oly i he value of
16 Mahemaical Problems i Egieerig 3 25 25 2 2 15 Price 15 Demad 1 1 5 Reservaio level 9 8 7 6 5 4 3 2 1 5 1 2 4 5 6 7 8 9 1 p11 P12 p61 P62 p71 P72 (a 1 2 4 5 6 7 8 9 1 I11 I12 I61 I62 I71 I72 (c Sales reveue level 5 1 2 4 5 6 7 8 9 1 d11 d12 d61 d62 d71 d72 (b 18 16 14 12 1 8 6 4 2 1 2 4 5 6 7 8 9 1 R11 R12 R61 (d R62 R71 R72 Figure 3: The opimal price, demad, reservaio level, ad sales reveue pah for Examples 1, 6, ad 7 over he cosidered ime horizo. Table 4: The opimal oal sales reveue, oal cacellaio reveue, oal reveue, profi, raio of he profi o he oal reveue, ad raio of he oal cacellaio reveue o he profi for Examples 1, 6, ad 7. Example 1 Example 6 Example 7 Sales reveue 2234 1437 1891 Cacellaio reveue 131 98 118 Toal reveue 2365 1535 29 Profi 65 54 575 Profi/oal reveue 26% 35% 31% Cacellaio reveue/profi 22% 18% 19% he marke ifluece. Therefore, he marke ifluece value i he demad fucio of he web service classes 1 ad 2forExamples8o11cabemeioedasiExample8 Mf 1 = (2+5 (1/2 2,Mf 2 = (2+5 (1/2 2 ;iexample 9Mf 1 = (2 + 5 (1/2 2 /2, Mf 2 = (2 + 5 (1/2 2 /2; i Example 1 Mf 1 = (2 + 5 (1/2 2 /4, Mf 2 = (2 + 5 (1/2 2 /4; iexample8mf 1 = (2 + 5 (1/2 2 /8, Mf 2 = (2 + 5 (1/2 2 /8. Figure 4 depics ha as he marke ifluece icreases, he service provider icreases he price o augme his/her owprofi.ioherwords,ashemarkeiflueceaugmes, he demad icreases ad he provider icreases
Mahemaical Problems i Egieerig 17 43 25 38 2 Price 33 28 Demad 15 1 23 5 18 1 2 4 5 6 7 8 9 1 1 2 4 5 6 7 8 9 1 p81 P82 p91 P92 p11 P12 p111 P112 d81 d82 d91 d92 d11 d12 d111 d112 (a (b 9 Reservaio level 8 7 6 5 4 3 2 1 Sales reveue level 3 25 2 15 1 5 1 2 4 5 6 7 8 9 1 1 2 4 5 6 7 8 9 1 I81 I82 I91 I92 I11 I12 I111 I112 R81 R82 R91 R92 R11 R12 R111 R112 (c (d Figure 4: The opimal price, demad, reservaio level, ad sales reveue pah for Examples 8 o 11 over he cosidered ime horizo. Table 5: The opimal oal sales reveue, oal cacellaio reveue, oal reveue, profi, raio of he profi o he oal reveue, ad raio of he oal cacellaio reveue o he profi for Examples 8 o 11. Example 8 Example 9 Example 1 Example 11 Sales reveue 4331 358 252 2269 Cacellaio reveue 286 197 158 139 Toal reveue 4616 3256 2678 249 Profi 3238 188 13 131 Profi/oal reveue 7% 58% 49% 43% Cacellaio reveue/profi 9% 1% 12% 14%
18 Mahemaical Problems i Egieerig his/her price for maagig demad ad maximizig resource uilizaio. Ashemarkeiflueceicreases,hereservaiolevel icreases for he web service class 1 bu decreases for he webserviceclass2.themaireasoforhisiscosacyof resource capaciy for Examples 8 o 11. I oher words, as he marke ifluece icreases for he web service provider i is aicipaed ha he/she sells more web service class 1 sice i gais more profi. As a resul, due o he cosacy of capaciy, hereisaicreaseihereservaiolevelforwebservice class 1 as he sales of he web service declies for he web service class 2. Table 4 showshesalesreveue,cacellaioreveue, oal reveue, profi, raio of he profi o he oal reveue, ad raio of he oal cacellaio reveue o he profi for Examples 8 o 11. Accordig o Table 5,forExamples8,9,1, ad 11, as he marke ifluece decreases, he sales reveue, cacellaio reveue, ad oal reveue declie. Addiioally, i ca be deeced ha he raio of he profi o he oal reveue for Examples 8, 9, ad 1 ad 11 is equal o 7%, 58%, 49%,ad43%.Theraioofheoalcacellaioreveueo heprofiis,respecively,9%,1%,12%,ad14forexamples 8, 9, 1, ad 11. 5.5. Compariso he Proposed Algorihm wih GA ad SA. This secio is provided for compariso he resuls of he proposed algorihm wih hose of he exisig meaheurisics such as GA ad SA. Furher iformaio abou GA ad SA ca be foud i refereces [55 62]. Examples 1 o 11 ca be firs modelled by he discreized model described i Secio 2.3 ad he solved by he exisig solvers for GA ad SA i MATLAB. I his performed experime, we cosider he value of parameers for he discreized model as follows: N = 1, τ 1 =τ 2 = = τ N = 1/N ad l =q l 1/N, l = 1,...,N. Furhermore, Primary experimes deoe ha selecig he value of he parameers for GA ad SA based o Tables 6 ad 7, respecively, geeraes beer resuls i compariso wih oher values. Table 8 deoes ha he performace of he proposed is beer ha ha of GA ad SA. The mai reaso is ha he proposed model cosiders he eire pricig sraegies wihi he ime horizo. Corarily, he oher approaches offer opimal prices i oly 1 pois. Furhermore, Table 7 shows ha SA geeraes iferior resuls wih respec o oher algorihms. 6. Coclusio I his paper, we have sudied a web service-pricig problem wih limied resources cosrai over a defiie ime horizo. We have cosidered a provider who offers access o a web service wih differe QoS where users may purchase heir required web service hrough a reservaio sysem. The service provider aduss he prices of he web service classes over a prespecified ime horizo o maage demads ad o maximize profi. Users have he righ wih o obligaio o cacel heir services as log as hey pay a pealy. Oe of he impora challeges for service providers is capaciy Table 6: The value of GA parameers. Parameer Value Populaio size 2 Scalig fucio Rak Elie cou 5 Selecio fucio Roulee Crossover fracio.85 Muaio fracio.15 Crossover fucio Two poi crossover Muaio fucio Gaussia Soppig crieria 1 geeraios Table 7: The value of SA parameers. Parameer Value Aealig fucio Bolzma aealig Temperaure updae fucio Expoeial emperaure updae Soppig crieria 5 ieraios Iiial emperaure 1 Accepace crieria Simulaed aealig accepace limiaio of he resources employed i offerig he web service like ework badwidh ad CPU ime, which fids meaig whe populariy of usig web services icreases. Thus, akig his impora proposiio io accou makes pricig sraegies cosidered by he provider has more credi. This paper has developed a coiuous-ime opimal corol model for ideifyig pricig sraegy for he web service classes where he demad of he service class depeds o is price, marke ifluece, ad ime. We have sudied he opimaliy codiio of he cosidered model basedohemaximumpricipaladproposeaalgorihm o obai he opimal pricig policy. Moreover, we have coduced umerical aalyses o evaluae he effec of he demad fucio, shared resource capaciy, ad cacellaio rae o he price, reservaio level, sales reveue of he web service classes, ad obecive fucio. I addiio, we have compared he proposed algorihm wih geeic algorihm (GA ad simulaed aealig (SA available i MATLAB. This problem ca be geeralized o a codiio where merely wo web service providers offer a similar web service. Moreover, i ca be supposed ha he price offered by oe of he providers affecs he price offered by he oher. This problem ca also be examied whe demad fucio ad oher parameers relaed o ohers is sochasic. The, i ca be solved usig robus opimizaio mehods. Fially, his ime-coiuous model ca be adaped for dyamic pricig i differe fields like grid compuig, cloud compuig, ad oher services ad accepable resuls be aaied.
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