Boletim Técnico da Petrobras (Bulletin of Technology) - volume 60, p , 2017 ISSN: ISBN: Jin Wang 1,2, Bo Li 1,2, Le

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1 Jn Wang,2, o L,2, Le Wang,2,3,, ShanLn Yang,2 School of Management, Hefe Unversty of echnology, nhu, Hefe , Chna 2 Key Laboratory of Process Optmzaton and Intellgent Decson-Makng (Mnstry of Educaton), Hefe Unversty of echnology, nhu, Hefe 23009, Chna 3 School of Management Engneerng, nhu Polytechnc Unversty, nhu, Wuhu 24000, Chna wo-sded matchng model based on cumulatve prospect theory for decson makng of two-way referral system / Modelo de correspondênca de dos lados com base em eora cumulatva das perspectvas para a decsão fazendo um sstema de referênca de duas vas abstract KEYWORDS: Decson makng, wo-way referral, wo-sded matchng, Cumulatve prospect theory Wth the development of herarchcal dagnoss and treatment mode, the decson-makng problem n two-way referral system has attracted consderable attenton n recent years. Most research of two-way referral manly focuses on the qualtatve research on the operaton mechansm and the management mode, but there s no consderaton of psychologcal behavor factor of decson makers for decson-makng problems of two-sded matchng. o solve these problems, t s urgent to propose an mproved two-sded matchng method. In ths paper, we use a two-sded matchng model based on cumulatve prospect theory (CPSM) to carry out a numercal example for solvng these problems. * Correspondng author: Le Wang, Emal: leo-wong@ahpu.edu.cn. Present address: School of Management Engneerng, nhu Polytechnc Unversty, nhu, Wuhu 24000, Chna. 203

2 wo-sded matchng model based on cumulatve prospect theory for decson makng of two-way referral system - Jn Wang et al. resumo PLVRS-CHVE: omada de decsão, Referênca bdreconal, Combnação de dos lados, eora cumulatva das perspectvas Com o desenvolvmento do dagnóstco herárquco e do modo de tratamento, o problema de decsão no sstema de referênca de duas vas atrau atenção consderável nos últmos anos. maora das pesqusas de referênca bdreconal centra-se prncpalmente na pesqusa qualtatva sobre o mecansmo de operação e o modo de gerencamento, mas não há consderação de pscologa fator de comportamento dos tomadores de decsão para os problemas de tomada de decsão de combnação de dos lados. Para resolver esses problemas, é urgente propor uma método de correspondênca de dos lados melhorado. Neste artgo, usamos um dupla face modelo de correspondênca baseado na teora cumulatva de perspectva (CPSM) para transportar um exemplo numérco para resolver esses problemas. secton ntroducton In the 3th Fve-Year Plan, Healthy Chna s proposed as an mportant development strategy of Chna, and the development of the medcal reform s regarded as the key pont for ts mplementaton. In January 207, the State Councl released 3th Fve-Year Plan to deepen medcal and health system reform. Herarchcal dagnoss and treatment mode n the plan s an mportant way to mprove medcal resources allocaton, rase general level of people s health, cut down rsng medcal expense and reduce contradctons between doctors and patents, whch s always regarded as the most mportant part n medcal reform. he man features of herarchcal dagnoss and treatment nclude: basc ntal dagnoss, two-way referral and treatment dstncton of acute and chronc dseases. mong these features, the two-way referral system s the most mportant pont of that to gude the basc dagnoss and treatment to communty hosptals. hrough the lnkage of nternal resources of medcal allance, the herarchcal dagnoss and treatment mode characterzed as mnor dseases n the communty, serous dseases to the hosptal, rehabltaton back to the communty s formed[]. In two-way referral system, not only some patents wth acute and dffcult dseases transfer from communty hosptals to urban hosptals, but also some patents wth stable and chronc dseases transfer from urban hosptals to communty hosptals. From the meanng of two-way referral, we can see that the choce of medcal nsttutons n progress of two-way referral s a typcal 204

3 two-sded matchng decson-makng problem n the herarchcal dagnoss and treatment mode. It s of great theoretcal and practcal value for two-sded matchng decson-makng problem of two-referral system, whch s helpful to mprove the effcency of the herarchcal dagnoss and treatment mode and mprove the level of medcal servce. Durng the past years, many scholars and research nsttutes have studed on the prospect theory, two-sded matchng and two-referral system. Moffett et al. appled prospect theory to assessment of health state and conducted an emprcal experment ndcatng that ndvdual preferences were mportant for the evaluaton of health utltes[2]. Han et al. proposed a novel method ndcatng the relatonshp of medcal servces acquston and expectatons and valdated effectveness of the model by case study[3]. Phllps et al. studed the mpact of end-of-lfe treatment preferences on medcal decsons[4]. Yao et al. appled preference orderng and deferred algorthm to mprove twosded matchng model for two-way referral decson makng[5]. Gulla et al. ntervewed some prmary care physcans and made a qualtatve analyss dentfyng user requrements for a chronc kdney dsease clncal decson tool to promote tmely referral[6]. ccordng to the survey of mplementaton of two-way referral system and the socal securty system platform, Shang et al. made a comprehensve analyss of two-way referral to reduce the cost of that[7]. Gan et al. studed the mpact factors of fnancng to twoway referral based on the survey data and relevant lterature[8]. hrough the lterature revew, we can fnd that the research of two-way referral manly focuses on the qualtatve research on the operaton mechansm and the management mode. lthough the research of two-sded matchng nvolves a wde range of felds, the man research ponts are applcablty of the method and evaluaton results of decson makers, and those research often suggest that the decson maker s completely ratonal, whch s contrary to the real decson-makng scenaros. In the process of actual matchng problems of two-way referral, the decson maker s often bounded ratonal, and ts psychologcal preference s very mportant to the mpact of decson-makng. herefore, ths paper ntroduces the prospect theory to study the nfluence of decson maker s expected values on two-sded matchng decson-makng n two-way referral process. he remander of ths paper s organzed as follows: Secton 2 descrbes the two-sded matchng problem. Secton 3 ntroduces the procedures of two-sded matchng model based on cumulatve prospect theory (CPSM). Secton 4 presents a numercal example. Secton 5 makes some conclusons and dscussons regardng our possble future work. secton descrpton of decson makng problem o make a further analyss of two-sded matchng model based on cumulatve prospect theory (CPSM), we gve a comprehensve descrpton of ths decson makng problem for two-way referral. In ths two-sded matchng problem, let = {, 2,, m } and = {, 2,, n } be the defnte sets of communty hosptals and superor hosptals, where and are the th agent of sde and, respectvely ( =,2,,m; =,2,,n ). Let us suppose m n.let O = ( o,, o,2,, o n, ) and = (,, 2,,, m, ) P p p p be utlty vectors, where o and p denote the evaluaton result of agent gven by agent and agent gven by agent, respectvely ( o, p [ 0,0] ). Let E = ( e, e2,, e ) m and F = ( f, f2,, f ) n be vectors of expectaton value whch are obtaned based on the gven nformaton (e.g. they can be computed by the known data, determned by experence of some experts, and some other hybrd methods), 205

4 wo-sded matchng model based on cumulatve prospect theory for decson makng of two-way referral system - Jn Wang et al. where e and f denote expectaton values gven by agent and. Defnton. one to one mappng: ì : denotes a two-sded matchng, f and only f, : ) ì ( ) { }, ) ì ( ) { }, ) ì ( ) = f and only f ì ( ) =, where ì ( ) = denotes that and are matched, and ì ( ) = or ì ( ) = denotes that or s unmatched. In Defnton, f ì ( ) =, then (, ) s denoted as a matchng par. Furthermore, we can easly fnd that f (, ) s satsfed then (, ) s also satsfed. For convenence of analyss, f ì ( ) = or ì ( ) =, (, ) or (, ) s stll denoted as a matchng par. herefore, a two-sded matchng problem would be denoted as a set of n matchng par ì = {(, ),,( n, n)}. ased on the mentoned analyss, a two-sded matchng decson makng based on prospect theory s represented as Fg.. nd a weght of a drected lne between and denotes the evaluaton result between and ; an undrected lne between and denotes that and are matched. he purpose of a two-sded matchng decson makng s to maxmze the sum of all weghts wth a stable matched relatonshp between sde and. e e 2 e m m Fgure : he problem of two-sded matchng consderng agents expected value 2... n- n f f 2 f n- o, p, Matched par f n secton the proposed method s decson makng based on prospect theory s more consstent wth decson makers behavor, t has been wdely used n decson makng problems[9]. hs paper solves a decson makng problem of two-way referral usng CPSM, whch consders the agents expectaton values. herefore, ths secton focuses on the process of prospect matrx calculaton and constructon of decson makng model. 3. Prospect matrx calculaton Snce gan and loss relatve to the reference ponts are mportant for prospect matrx calculaton, and expected values as the reference ponts can nhert the varous propertes of prospect theory, ths paper regards the expected values gven by agents as the correspondng reference ponts[0]. Frst, gans and losses of each agent based on the expected values s computed as: ) For agent : If p, > e, then > 0, p, > e d = p, e= 0, p, = e < 0, p, < e d s gan of, () p relatve to the reference pont e ; f p, = e, t means that there s no gan or loss relatve to the reference pont; f p, < e, then d s loss of p, relatve to the reference pont e. ) For agent : If o, > f, then > 0, o > f d o f o f < 0, o, < f, =, = 0,, = d s gan of o, (2) relatve to the reference pont f ; f o, = f, t means that there s no gan or loss relatve to the reference pont; f o, < f, then d s loss of o, relatve to the reference pont f. 206

5 Second, consderng dfferent rsk atttude of agents, we can obtan prospect matrxes ( V and V ) usng cumulatve prospect theory based on the gans or losses as follows: ) For agent ) ( d ) α, d > 0 : v = 0, d = 0 ( β λ d ), < 0 d where v s a prospect value computed by agent satsfacton wth ; α and β both denote coeffcents of rsk preference ( 0 α, β ); λ s the parameter of rsk averson whch only exsts when d < 0 ( ) ; and the prospect matrx of sde s computed as: V = v. For agent m n ( d ) α, d > 0 : v = 0, d = 0 ( β λ d ), d < 0 where v s a prospect value computed by agent satsfacton wth ; α and β both denote coeffcents of rsk preference( 0 α, ); λ s the parameter of rsk averson whch only exsts when d < 0 ( λ > ) ; and the prospect matrx of sde s computed as: V = v. m n 3.2 Decson-makng model ccordng to prospect matrxes ( V, V ) and matchng relatonshp between agent and, we consder constructng a decson-makng model to obtan the matchng results. Suppose ( ) ( ) 0, µ that x s a bnary varable, and x =. o, µ = maxmze the sum of comprehensve prospect values based on prospect matrxes ( V, V ), the optmzaton model can be construct as followng: = m n = = (3) (4) Max Z v x (5a) = m n Max Z v x (5b) n = = = s.t. x =, =, 2,, m (5c) m = x, =, 2,, n (5d) { } x 0,, =, 2,, m; =, 2,, n (5e) Where Eq. (5a) and (5b) are obectve functons of decsonmakng model, Eq. (5a) and (5b) denotes maxmzng sum of prospect values of all matched agents and, respectvely; Eq. (5c) denotes that every agent can only be matched wth one agent ; Eq. (5d) denotes that every agent can be matched wth no more than one agent. o facltate the calculaton of mult-obectve optmzaton model, ths paper transforms mult-obectve optmzaton model nto a sngle obectve functon by a lnear weghtng method. Suppose ω and ω denote weghts of agent and, respectvely, then optmzaton model (5) can be transformed to the followng model: m n m n m n vx vx cx = = = = = = Max Z = ω + ω = (6a) n = s.t. x =, =, 2,, m (6b) m = x, =, 2,, n (6c) { } x 0,, =, 2,, m; =, 2,, n (6d) Where c = ωv + ω v, 0 < ω, ω <, ω + ω = ; ω and ω represent the mportance of a two-sded matchng problem whch can be obtaned by ether obectve weght methods or subectve weght methods. 3.3 Procedure of CPSM In total, the procedure of CPSM problem for two-referral decson-makng s gven as followng: Step. Use Eq. () and (2) to obtan gans and losses relatve to the reference ponts based on the utlty vectors ( O, P ) and expected value of every agent (, ); Step 2. Calculate prospect values by Eq. (3) and (4), then obtan the prospect matrxes ( V and ); Step 3. Construct mult-obectve optmzaton model based on the prospect matrxes; Step 4. ransform nto a sngle obectve functon by a lnear weghtng method; Step 5. Calculate the optmzaton model, and obtan the matched results. 207

6 wo-sded matchng model based on cumulatve prospect theory for decson makng of two-way referral system - Jn Wang et al. egn Calculate gans and losses relatve to the reference ponts Calculate prospect values and obtan prospect matrxes Construct optmzaton model based on the prospect matrxes ransform nto a sngle obectve model Obtan the matched results End Fgure 2: he procedure of CPSM secton v numercal example In ths secton, we gve a numercal example to llustrate a decson-makng problem of two-way referral system usng the two-sded matchng model n the background of herarchcal dagnoss and treatment. In fve communty hosptals of Hefe(, 2, 3, 4, 5), there are some patents wth acute and dffcult dseases whch need to be transferred to the seven urban hosptals(, 2, 3, 4, 5, 6, 7 ). t the same tme, seven urban hosptals need to transfer ther patents wth stable and chronc dseases to these fve communty hosptals. he person responsble for transfer n every communty hosptal makes a comprehensve evaluaton of seven urban hosptals based on medcal level, charge standard, patent satsfacton and other ndcators. Furthermore, they can gve the utlty vector O = o, o, o, o, o, o, o of urban hosptal set (,,2,3,4,5,6,7) and expected value vector of every communty hosptal E = ( e, e2, e3, e4, e5) as able shows. he person responsble for transfer n every urban hosptal makes a comprehensve evaluaton of fve communty hosptals based on medcal faclty, convenence of treatment and other ndcators. Furthermore, they can gve the utlty vector of communty hosptal set Communty hosptal Utlty vector Urban hosptal Expected vector E O O O O O able : Utlty vector O and expected value vector E 208

7 ( p,, p2,, p3,, p4,, p5, ) P = and expected value vector of every urban hosptal F = ( f, f2, f3, f4, f5, f6, f7) as able 2 shows. Urban hosptal Utlty vector P P 2 P 3 P 4 P 5 P 6 P 7 Communty hosptal Expected vector F able 2: Utlty vector P and expected value vector F In ths secton, we take a numercal example to obtan the matched results of two-referral decson makng by CPSM. Step. ased on the utlty vectors ( O, P ) and expected values of every agent (, ), the gans and losses relatve to the reference ponts (expected values) by Eq. () and (2); Step 2. ccordng to the gans and losses, the prospect matrxes ( V and V ) are calculated by Eq. (3) and (4) and the results are represented n able 3 and able 4. ased on many experments by versky, the most sutable values of parameters reflectng the real experment results: α = = = β α β = 0.88, λ = λ = 2.25 []. v able 3: Prospect matrx V v able 4: Prospect matrx V Step 3. ccordng to decson makng problem and prospect matrxes, a mult-obectve optmzaton model can be proposed as model (5); Step 4. For the convenence of calculaton, let ω = 0.5, ω = 0.5. herefore, the model (5) can be transformed nto a sngle obectve model (6) by a lnear weghtng method. nd the coeffcent matrx s as followng: Step 5. ased on the coeffcent matrx C and model (6), the matched results consderng expected values of 209

8 wo-sded matchng model based on cumulatve prospect theory for decson makng of two-way referral system - Jn Wang et al. C able 5: Coeffcent matrx C agents can be obtaned usng Lngo software as follows: ) matched pars:{(, 6 ), ( 2, 4 ), ( 3, 2 ), ( 4, 7 ), ( 5, 3 )}; 2) unmatched pars: {(, ), ( 5, 5 )}. If we doesn t consder expected values of agents, then coeffcent matrx s changed and the matched results s also changed as follows: ) matched pars:{(, 3 ), ( 2, ), ( 3, 5 ), ( 4, 2 ), ( 5, 4 )}; 2) unmatched pars: {( 6, 6 ), ( 7, 7 )}.y ths way, we can easly fnd that the expected values of agents have an mportant nfluence on a two-sded matchng decson-makng problem. secton v conclusons In the model of herarchcal dagnoss and treatment, the decson-makng problem of two-way referral s wdely concerned. On the one hand, prevous research has focused on the runnng mode of two-way referral, qualtatve analyss of decson-makers and seldom conducted some research about decson-makng method; on the other hand, the decson-makng problem of two-sded matchng s the most common method to solve ths problem. herefore, ths paper proposes a CPSM to solve the decson-makng problem of two-way referral system, ths method consders the reference ponts of dfferent agents n a perspectve of psychologcal behavor factors. Frst, we obtan the gans and losses relatve to the reference ponts representng rsk atttude by cumulatve prospect theory; and then wth the method of lnear weghtng, mult-obectve optmzaton model s transformed nto a sngle obectve optmzaton and the matchng results of twoway referral can be obtaned n the end. hs method has strong operablty and practcablty, and t can be used for reference n the research of two-way referral. However, ths paper does not take nto account evaluaton ndcators of agents for obtanng evaluaton results and more scentfc methods for calculaton of expected values. In future work, we are plannng to nvestgate a evaluaton ndcator system to assess the agents. secton v acknowledgement hs work was supported by Research on the Key echnologes of Communty Servce and Management n Replcable Intellgent Vllage Communty under Grant No. SK20604, the Natonal Natural Scence Foundaton of Chna (Nos , and ), references [] Lu X.Y. and Deng Q.Z., Mutual referral polcy for coordnatng dfferent-level hosptals n the medcal assocaton. Industral Engneerng and Management, 206(04):

9 [2] Moffett, M.L. and M.E. Suarez-lmazor, Prospect theory n the valuaton of health. Expert revew of pharmacoeconomcs & outcomes research, (4): p [3] Han, J., et al., Selectng medcal servce scheme: novel method based on prospect theory. Flomat, (5): p [4] Phllps, L.L., et al., gng Prsoners reatment Selecton: Does Prospect heory Enhance Understandng of End-of-Lfe Medcal Decsons? he Gerontologst, 20. 5(5): p [5] Yao L. L., Cheng W, Man X W, et al. pplcaton of wosded Matchng heory n Dual Referral[J]. Chnese General Practce, 206. [6] Gulla, J., et al., User Requrements for a Chronc Kdney Dsease Clncal Decson Support ool to Promote mely Referral. INERNIONL JOURNL OF MEDICL INFORMICS, : p [7] Shang, H.., et al., UML-ased wo-way Referral MIS Requrement nalyss. ppled Mechancs and Materals, : p [8] Gan, X.Q. and G. Kuo, Impact analyss of Chnese townshp health centers fnancng on two-way referral. Vol [9] Lu, P., et al., Research on the mult-attrbute decsonmakng under rsk wth nterval probablty based on prospect theory and the uncertan lngustc varables. Knowledge-ased Systems, (4): p [0] Heath, C., R.P. Larrck and G. Wu, Goals as Reference Ponts. Cogntve Psychology, (): p [] versky,. and D. Kahneman, dvances n prospect theory: Cumulatve representaton of uncertanty (4): p

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