A New Bargaining Game Model for Measuring Performance of Two-Stage Network Structures

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1 Int. J. Rsach n Inustal Engnng, pp Volu, Nub, 0 Intnatonal Jounal of Rsach n Inustal Engnng ounal hopag: A Nw Baganng Ga Mol fo Masung Pfoanc of Two-tag Ntwok tuctus F. Hossnzah Lotf *, G.R. Jahanshahloo,. Hat,. Gvhch patnt of Mathatcs, cnc an Rsach Banch, Islac Aza Unvsty, Than, Ian A R T I C L E I N F O Atcl hstoy : Rcv: May 0, 0 Rvs: August 0, 0 Accpt: ptb 5, 0 Kywos : ata nvlopnt analyss (EA), Nash baganng ga ol, two-stag pocss, ntat asu A B T R A C T ata nvlopnt analyss (EA) anly utlzs nvlopnt tchnology to plac poucton functon n coconocs. Bas on ths placnt, EA s a wly us athatcal pogang appoach fo valuatng th latv ffcncy of cson akng unts (MUs) n oganzatons. Evaluatng th ffcncy of MUs that hav two-stag ntwok stuctus s so potant n anagnt an contol. Th sultng two stag EA ol not only povs an ovall ffcncy sco fo th nt pocss, but also yls an ffcncy sco fo ach of th nvual stags. In ths Pap w vlops Nash baganng ga ol to asu th pfoanc of MUs that hav a two-stag stuctu. Un Nash baganng thoy, th two stags a vw as plays. It s shown that whn only on ntat asu xsts btwn th two stags, ou nwly vlop Nash baganng ga appoach yls th sa sults as applyng th stana EA appoach to ach stag spaatly. Wth a nw bakown pont, th nw ol s obtan whch by povng xapl, th sults of ths ols a nvstgat. Aong ths sults can b pont to th changng ffcncy by changng th bakown pont.. Intoucton ata nvlopnt analyss (EA), ntouc by Chans t al. [], s an ffctv tool fo asung th latv ffcncy of p cson akng unts (MUs) that hav ultpl nputs an ultpl outputs []. Rsachs vlop two-stag Ntwok stuctus that th output of stag s th nput of stag. Th outputs fo stag a f to as ntat asus. Fo xapl, fo an Zhu [] us a two-stag pocss to asu th poftablty an aktablty of U cocal banks. Hwang xpss two stag pocsss an b plnt n th bankng nusty [3]. Chlngan an han [4] scb a two-stag pocss n asung physcan ca. Kao an Hwang off a nw tho of asung th ovall ffcncy of such a pocss [5]. Chn t al. [6] us a wght Atv ol to suaton th two stags an copos th ffcncy of th ovall pocss. Moov Lang t al. vlop a nub of EA ols that us th concpt of ga thoy [7]. pcfcally, Lang t al. [7] vlop a la follow ol boow fo th noton of tacklbg gas, an a cntalz o coopatv ga ol wh * Cosponng autho E-al ass: Faha@hossnzah.

2 8 F. Hossnzah Lotf t al. th cobn stag s of ntst. In nxt scton so plnay sults a pov. In scton 3 w scb popos ol an ts popts. Nucal xapls a psnt n scton 4. cton 5 gvs th concluson of ths pap.. Nash baganng ga ol Cons a two-stag pocss shown n Fgu, w suppos th a n MUs an ach MU ( =,,, n) has nputs to th fst stag that not by X = x,..., x ) an ( outputs fo ths stag not by Z = z,..., z ). Ths outputs thn bco th nputs ( to th scon stag, whch a f to as ntat asus. Th s outputs fo th scon stag a not by Y = y, y,..., y ). Th constant tuns to scal (CR) ( s ol (Chans t al. []), th (CR) ffcncy scos fo ach MU ( =,,..., n) n th fst an scon stags can b fn by of two stag pocss, wth usng th CR ffcncy, w can fn = = = =. = =. = = = v x v x an, spctvly, to gt th total pfoanc. =, snc: Th abov ovall ffcncy fnton nsus that fo,, an th = ovall pocss s ffcnt f an only f =. Th ffcncy-valuaton pobl can b appoach fo two ga thoy pspctvs. On s to vw th two-stag pocss as a non-coopatv ga ol, n whch on stag s assu to b a la an solv fo ts CR ffcncy fst, an th oth stag a follow, whos ffcncy s coput wthout changng th la s ffcncy sco. Th oth appoach s to ga th pocss as a cntalz ol, wh th ovall ffcncy gvn n () s axz, an a coposton of th ovall ffcncy s obtan by fnng a st of ultpls poucng th lagst fst (o scon) stag ffcncy sco whl antanng th ovall ffcncy sco. Not that n fact, th two stags can b ga as two plays n Nash baganng ga thoy. Thfo, w can appoach th ffcncy valuaton ssu of two-stag pocsss by usng Nash baganng ga thoy ctly. Cons th st of two nvuals patcpatng n th baganng by N = {, }, an a payoff vcto s an lnt of th payoff spac R, whch s th two-nsonal Euclan spac. A fasbl st s a subst of th payoff spac, an a bakown o status quo pont s an lnt of th payoff spac. A baganng pobl can thn b spcf as th tpl (N,, ) consstng of patcpatng nvuals, fasbl st, an bakown pont. Th soluton s a functon F that s assocat wth ach baganng pobl (N,, ), xpss as F(N,, ). In ths pap, Zhu t al. [8], onstat th Nash baganng ga [9] an povs that th s on unqu soluton fo t an th soluton s Nash soluton, whch satsfs th abov- ()

3 9 A Nw Baganng Ga Mol fo Masung Pfoanc of Two-tag nton fou popts, an can b obtan by solvng th followng axzaton pobl: Max ( u b ) u s, u b = () wh s th paynt vcto fo th nvuals, an vcto,, spctvly. Not that th bakown pont o status quo psnts possbl payoff pas obtan f on cs not to bagan wth th oth play. Cons: { } n ax z = ax z, z = n z, y = n y, y = ax y ax n ax n = thn ( X, Z )(,...,, =,... ) shows th last al MU n th fst phas that pouc th gatst aount of nput an th last aount of ntat asu. ax n = b, u { } { } { a th th coponnts of th laly ( z, y ) (,...,, =,..., ) shows th last as MU pouc n th scon stag, th axu aount of ntat asu an th lowst output. Th CR ffcncy fo th abov two last al MUs s th wost aong th xstng MUs. W not th (CR) ffcncy scos of th two last al MUs n th fst an scon stag as θ n an θ n, spctvly, an us θ n an θ n as ou bakown pont. EA ol wth nput-ont, an usng th foula of Nash baganngg ga pov n ol () can b xpss as a ol (3): } Fgu. Two-stag pocss

4 30 F. Hossnzah Lotf t al. Max s. t. w zo u yo = = θ n θ n v xo w zo = = ( ).( ) = = = = = = = = o o o o n n, =,..., n, =,..., n v, u, w > 0, =,...,, =,..., s, =,..., v x v x (3) Zhu t al. [8] tyng to co up on wth lna ol by changng of vaabl, thy ach to a paatc lna ol whch was quvalnt wth a non-lna ol. 3. Nw Mol wth ffnt Bakown Ponts Zhu t al. w as th thoy of Nash baganng ga, fo MU whch has two stag pocss [8]. Thy us of latv ffcncy of MU an bult a vtual MU, whch n vy stag has th axu obsv nput an th lowst obsv output. Thn ts CR ffcncy of vtual MU was calculat at ach stp. Th ffcncs obtan n both stags, consttut th bakown pont. In ths pap, w wll b ang a paat to th bakown pont an vw th sults of th Nash baganng ga ol wth ths nw bakown pont. Cons th (CR) ffcncy scos fo ach MU ( =,,, n) n th fst an scon stags: w z u y = = v x w z = = =, = (4)

5 3 A Nw Baganng Ga Mol fo Masung Pfoanc of Two-tag It s asonabl to st w qual to w, snc th valu assgn to th ntat asus shoul b th sa galss of whth thy a vw as outputs fo th fst stag o nputs to th scon stag. Thn th total ffcncy can b wttn as a pouct of snc:. = wh an a th nvual ffcncy scos of th two-stag pocss, = = = =. = =. = = = v x v x Th abov ovall ffcncy fnton nsus that fo,, an th = ovall pocss s ffcnt f an only f =. Now w wll b ang th paat to th bakown pont b ( θn, θn ) (5) = obtan by Zhu t al. [8]. If w look at th ts of th tansacton, a sll wants to sll hs poucts, h gav a scount, th bakown pont s th sa as scounts, now w want to know whth can b a to ths as scount an s f t s possbl o not. o w hav: +, + (6) n n Thn th EA ol wth th nput-ont fo a spcfc MU o, usng quatons () an (6) a xpss as follows: M ax s. t. w z o u y o = = θ n θ n v xo w z o = = ( ).( ) = = = = = = = = v x v x o o o o + n + n, =,..., n, =,..., n v, u, w > 0, =,...,, =,..., s, =,..., (7)

6 3 F. Hossnzah Lotf t al. All constants whch a fn n ol (7) a shown wth th st, s th st of fasbl soluton fo pobl. o th pobl s fn as a tpl ({,},,{ θ, θ }). La: Th fasbl st s copact an convx. Poof: nc th fasbl st s boun an clos n Euclan spac, thn s copact. Nxt w wll pov that s also convx. uppos ( v,..., v, u,... u s, w,..., w ) an. Fo any [ 0,] ( v,..., v, u,... u, w,..., w ) s λ w hav λv + ( λ) v > 0, =,...,, λu + ( λ) u > 0, =,..., s an λw + ( λ) w > 0, =,...,. nc = w z > 0 fo all =,,n, th constants n, quvalnt to v x =,..., n an = = all =,,n, an th constants to hav w zo ( θn + ) v xo = = = = an v x o = = v x = = n = n v x > 0 an u y an = = a =,..., n, spctvly, fo + n an = o o = u yo ( θn + ) w zo = = o + n a quvalnt spctvly. Thn w [ λw + ( λ) w ] z = λ w z + ( λ) w z = = = λ v x + ( λ) v x = = = = [ λv + ( λ) v ] x an [ λu + ( λ) u ] y = λ u y + ( λ) u y = = = λ w z + ( λ) w z = = = [ λw z + ( λ) w ] z.

7 33 A Nw Baganng Ga Mol fo Masung Pfoanc of Two-tag laly, w hav [ λw + ( λ) w ] zo ( θn + ) [ λv + ( λ) v ] x = = an λu + λ u y n + λw z + λ w z = = [ ( ) ] ( ) [ ( ) ]. Thfo w hav ( λv + ( λ) v, λu + ( λ) u, λw + ( λ) w ), wh =,,, =,,s, =,,, o quvalntly, λ( v,..., v, u,... u s, w,..., w ) + ( λ)( v,..., v, u,... u s, w,..., w ). By changng vaabls, ol (7) can b convt to nto th followng ol (8). s o n o o n = = = Max α. µ y θ ω z ω z + θ s. t. s s n yo + n + n n yo + = = θ µ θ θ θ µ = s = = = ω o n o n z µ y γ ω x o z o = = α + ω z γ x 0, =,..., n = = s + = = µ y ω z 0, =,..., n µ = αµ =,..., s α, v, u, ω, µ, µ > 0, =,...,, =,..., s, =,..., Now wth ga to th abov pobl w hav: n = o = θn θn α θ + α = ω z γ x = (9) Thn < + whch povs both upp an low bouns onα, an ncats that th optal valu of a psnts th fst-stag ffcncy sco fo ach MU. Thus α can b tat as a paat wthn θ, n (8). As a sult, ol (8) can b solv as a paatc lna poga va sachng ov th possbl α valus wthn θ, n. In coputaton, w st th ntal valu fo α as th upp boun on, an solv th cosponng lna poga. Thn w bgn to cas a by a vy sall postv nub ε (=0.000 fo xapl) fo ach stp t, naly, αt = ε t,t =,,... untl th low

8 boun 34 F. Hossnzah Lotf t al. θ n s ach, an solv ach lna poga of ol (8) cosponng to α t an not th cosponng optal obctv valu byω. Not that not all valus takn by a wthn θ, n la to fasbl solutons fo poga (8). Lt that * Ω = Ω an not th spcfc t t ax t * Ω s assocat wth sval ol (8). o ( ) α assocat wth * α valus. Thn s * * * * * o = α = ω o o = µ o = = z, y an t * * Ω asα. Not that t s lkly * Ω assocat wth. * * * o o o * α s ou soluton to = MU s baganng ffcncy scos fo th fst an scon stags an th ovall pocss, spctvly. 4. A al xapl fo bank Mllat In ths scton, w apply th nw Nash baganng ga on st of al ata fo bank Mllat. Th ata st s nclung 30 banchs of bank Mllat wth th fou ntat asus. Th nputs to th fst stag a nub of ploys an bnft paynts. Th ntat asus connctng th two stags a typs of posts (shot-t posts, loan posts, cunt posts an long-t posts). Th outputs fo th scon stag a Faclts, pofts cv, cossons an ans. Th CR ffcncy scos fo th last al MUs n th fst an scon stags a calculat as θ n= an θ n= spctvly. W nxt bgn wth th ntal valu α= n ol (8), thn cas α by a sall postv nub ε = 0.000fo ach stp t, naly, α = t, t =,,... untl th low boun θ n= s ach. In ths xapl, w hav, = 0.00,0.00. Bank ata st s as follows: ( ) ( ) t Tabl. ata st fo bank Mllat. MU Faclts Long-t posts Cunt posts Loan posts hot-t posts Bnft paynts Pofts cv Cossons ans Eploys

9 35 A Nw Baganng Ga Mol fo Masung Pfoanc of Two-tag Tabl. Contnu

10 36 F. Hossnzah Lotf t al. Tabl. Contnu

11 37 A Nw Baganng Ga Mol fo Masung Pfoanc of Two-tag Tabl. Contnu Tabl. Rsults of bank Mllat wth bakown pont { θ n, θn } + +. MU Effcncy of stag Effcncy of stag Ovall ffcncy. α

12 38 F. Hossnzah Lotf t al. Not : Th optal valu of paat α psnts th fst-stag baganng ffcncy sco fo th cosponng MU. Not : In ths tabl w show possbl xapls wth th sybol "---". o ang to th pvous bakown pont s possbl. Th ffcncy of unts wth th nw bakown pont wll b lss than o qual th ffcncy of unts wth th Pvous bakown pont. Not that th bakown pont cannot b chos abtaly, fo xapl, f th CR ffcncy of ach stag b us as a bakown pont, lkly ol (8) wll b possbl. Ths possblty ay b u that so unts a volat a nub of constants n th ol (8). Fnally, f th bakown pont s chosn sall, th pfoanc of two-stag syst wll ncas ung ngotatons. 5. Conclusons W conclu wth so xapls that, by ang to th pvous bakown pont, th ffcncy of unts wth th nw bakown pont wll b lss than o qual th ffcncy of unts wth th Pvous bakown pont. Th chosn bakown pont cannot b abtaly, fo xapl f w us th CR ffcncy fo ach bakown pont, lkly t bcos possbl fo ol (8) an t ay b possbl u to nub of ponts a volat so constants n ol (8). If w ncas th aount of bakown pont n Nash baganng ga, th aount of ffcncy wll b sall o qual to th ffcncy of th pvous bakown pont, so th bakown pont (0.0) wll hav axu pfoanc of two-stag syst n ths ol. Th goal s not th bst syst pfoanc ung th ngotaton but th goal s fnng th ost ffcncy ung th ngotaton. Rfncs [] Chans, A., Coop, W.W. an Rhos, E. (978), Masung th ffcncy of cson akng unts, Euopan Jounal of Opatonal Rsach, Vol. 3, pp [] fo, L.M. an Zhu, J. (999), Poftablty an aktablty of th top 55 U cocal banks, Managnt cnc, Vol. 45, No. 9, pp [3] Wang, C. H., Gopaol, R. an Zonts,. (977), Us of ata nvlopnt analyss n assssng nfoaton tchnology pact on f pfoanc, Annals of Opatons Rsach, Vol. 73, pp [4] Chlngan, J. an han, H.. (004), Halth ca applcatons: Fo hosptals to physcan, fo pouctv ffcncy to qualty fonts. In: Coop, W.W., fo, L.M. an Zhu, J. (Es.), Hanbook on ata Envlopnt Analyss. png, Boston. [5] Kao, C. an Hwang,.N. (008), Effcncy coposton n two-stag ata nvlopnt analyss: An applcaton to non-lf nsuanc copans n Tawan, Euopan Jounal of Opatonal Rsach, Vol. 85, No., pp [6] Chn, Y., Cook, W.., L, N. an Zhu, J. (009), Atv ffcncy coposton n two-tag EA, Euopan Jounal of Opatonal Rsach, Vol. 96, pp

13 39 A Nw Baganng Ga Mol fo Masung Pfoanc of Two-tag [7] Lang, L., Cook, W.. an Zhu, J. (008), EA ols fo two- stag pocsss: Ga Appoach an ffcncy coposton, Naval Rsach Logstcs, Vol. 55, pp [8] Juan,., Lang, L., Yao, C., Wa,., Cook, J. an Zhu, (0), A baganng ga ol fo asung pfoanc of two-stag ntwok stuctus, Euopan Jounal of Opatonal Rsach, Vol. 0, pp [9] Nash, J.F. (950), Th baganng pobl, Econotca, Vol. 8, No., pp