Multi-objective optimization in service systems

Transcription

1 Mult-objectve optmzaton n servce systems Tad Gonsalves, Ke Yamagsh, Kyosh Itoh Department of Informaton and Communcaton Scences Faculty of Scence & Technology Sopha Unversty, 7-1 Kocho Chyoda-ku, Tokyo, Japan {t-gonsal, yamag-k, tohkyo}@sopha.ac.jp Journal of Dgtal Informaton Management Abstract: Mut-objectve optmzaton deals wth the smultaneous optmzaton of two or more conflctng objectve functons n real-lfe systems. Ths paper deals wth the mult-objectve optmzaton n servce systems. The goal of servce systems s to provde cost-effcent servce to customers, whle at the same tme, reducng the customer watng tme for servce. In general, a low cost n system operaton leads to longer watng tmes, whle a hgher cost n system operaton leads to shorter watng tmes. The two objectves servce cost (operatonal cost) and watng tme (customer satsfacton) are, therefore, conflctng n nature. We use the novel Mult-Objectve Partcle Swarm Optmzaton (MOPSO) algorthm to optmze the two conflctng objectve functons smultaneously. MOPSO s a farly recent swarm ntellgence meta-heurstc algorthm known for ts smplcty n programmng and ts rapd convergence. The mult-objectve optmzaton procedure s llustrated wth the example of a practcal servce system. MOPSO produces a famly of well-spread Pareto fronts for the two objectve functons n the practcal servce system. Categores and Subject Descrptors E. 5 [Fles]; Optmzaton; I.3.5 Computatonal Geometry and Object Modelng General Terms: Mult object optmzaton, Swarm optmzaton, Evolutonary algorthms Keywords: Servce systems, Servce cost, Smulaton optmzaton, Mult-objectve Partcle swarm optmzaton Receved: 19 October 2001; Revsed: 21 December 2009; Accepted: 30 December Introducton The goal of practcal servce systems such as banks, hosptals, restaurants, theme parks, reservaton counters, etc. s to provde cost-effcent servce to customers takng nto consderaton customer satsfacton and the cost of provdng servce. In general, a low cost n system operaton leads to a poor customer satsfacton, whle a hgh level of customer satsfacton demands hgher cost n system operaton. The two objectves are conflctng, makng t an deal stuaton for the applcaton of mult-objectve optmzaton technques. Customer satsfacton, of late, has become a major ssue n marketng research and a number of customer satsfacton measurement technques have been proposed (Hanan, Karp, 1991). Increasng efforts have been made to analyze the causes of customer dssatsfacton and suggest remedes (Taylor, 1994). In servce systems, nothng can be as detrmental to customer satsfacton as the experence of watng for servce. Watng s frustratng, demoralzng, agonzng, aggravatng, annoyng, tme consumng, and ncredbly expensve (Master, 1985). For customers, watng for servce s a negatve experence (Scotland, 1991), leadng to a poor evaluaton of the qualty of servce (Clemmer, Schneder, 1989; Davs, Heneke, 1998). The level of customer satsfacton can be rased by reducng the watng tme; however, ths leads to the ncrease n cost due to the hrng of greater number of personnel to provde servce. We propose an evolutonary mult-objectve optmzaton scenaro to fnd the optmal trade-off between the two objectve functons. Evolutonary algorthms are nspred by bologcal processes whch are at work n nature. The Genetc algorthm (GA) (Davs, 1991; Goldberg, 1989; Koza, 1992; Mtchell, 1996) modeled on the Darwnan evolutonary paradgm, s the oldest and the best known Evolutonary Algorthm. It mmcs the natural processes of selecton, cross-over and mutaton to search for optmal solutons n massve search spaces. Another very recent algorthm belongng to the class of bologcally nspred methods s the Partcle Swarm Optmzaton (PSO) (Kennedy, Eberhart, 1995; 2001). PSO mtates the socal behavor of nsects, brds or fsh swarmng together to hunt for food. PSO s a populatonbased approach that mantans a set of canddate solutons, called partcles, whch move wthn the search space. Durng the exploraton of the search space, each partcle mantans a memory of two peces of nformaton: the best soluton (pbest) that t has encountered so far and the best soluton (gbest) encountered by the swarm as a whole. Ths nformaton s used to drect the search. PSO has been found to be successful n a wde varety of optmzaton tasks (Englebrecht, 2005; Kennedy, Eberhart, 2001). Evolutonary Algorthms are prmarly desgned for the optmzaton of a sngle objectve. However, n recent years, ther applcaton to mult-objectve optmzaton has gven rse to the Evolutonary Mult-Objectve Optmzaton (EMO) research dscplne (Ztzler, 1999). A varety of EMO algorthms have been developed. NSGA-II (an extenson of GA) (Deb, et al., 2000) and MOPSO (an extenson of PSO) are the two wellknown algorthms used n EMO. Evolutonary algorthms offer a partcularly attractve approach to mult-crtera optmzaton because they are effectve n hgh-dmensonal search spaces (Parsopoulos, Vrahats, 2002). PSO seems partcularly sutable for mult-objectve optmzaton manly because of the hgh speed of convergence that the algorthm presents for sngle-objectve optmzaton (Coelho, et al., 2004, Mostaghm, Tech, 2003). MOPSO algorthms are extensons of the sngleoptmzaton PSO for solvng mult-objectve optmzaton problems (Alvarez-Bentez, et al., 2005; Coelho, Lechunga, 2002; Coelho, et al., 2004; Feldsend, Sngh, 2002, Hu, Eberhart, 254 Journal of Dgtal Informaton Management Volume 8 Number 4 August 2010

2 2002; Hu, et al., 2003; Ztzler, 1999). In ths study, we show how to apply MOPSO n smultaneously optmzng servce cost and customer satsfacton due to watng. The number of personnel to be assgned to each of the servce statons n the servce system and the servce tme per customer are the decson varables. Although there are bounds on the decson varables, the number of combnatons even for a modest servce system s prohbtvely large. The problem, therefore, s a typcal combnatoral optmzaton problem that cannot be solved by an exhaustve search. The fast convergng MOPSO s a sutable algorthm for ths problem. The mult-objectve optmzaton procedure s llustrated wth the example of a practcal servce system. MOPSO produces a well-spread Pareto front (the curve representng a trade-off relatonshp between the two objectve functons) for the two conflctng objectve functons n the practcal servce system. 2. Modellng of Servce Systems In ths secton, we defne the smulaton model of a real-world servce system, the two objectve functons and the decson varables. 2.1 Statc model We llustrate the smulaton optmzaton technque usng MOPSO, by means of a practcal servce system example shown n Fgure. 1. The small clnc servce system s made up of seven dfferent contexts represented by rectangles n the dagram. Recepton, Dagnoss 1, Dagnoss 2, Prescrpton, Medcal tests, Prescrpton and Accounts are the respectve contexts. These contexts are essentally the actvtes, or the servce statons at whch servce s provded to the customers. The servce-provdng personnel nclude doctors, nurses, medcal techncans, physotherapsts, pharmacsts, etc. These, together wth the rest of the resources are labeled above and below the contexts. The modellng detals of servce systems are found n (Hasegawa, et al., 2001). Dscrete event smulaton of the system provdes wth statstcs lke average watng tme whch are used to compute the objectve functons. 2.2 Dynamc model The operatonal vew of the collaboratve system s provded by the dscrete event smulaton (DES). Dscrete event smulaton (Banks, Carson II, 1984; Fshman, 1978) s event-drven, an event beng an occurrence that changes the state of the system. In a dscrete event system, the events take place n dscrete steps of tme. Smulaton proceeds as tme-keepng of the seres of events. EXTEND SIM s a well-known DES package extensvely used n academa and ndustry ( It conssts of several lbrares contanng smulaton blocks. The smulaton model s bult by mportng the blocks and connectng them together. EXTEND SIM s also provded wth anmaton whch helps to vsually verfy the operaton of the model under consderaton. The operaton of the system can be smulated for any desred perod of tme. At the end of the stpulated tme, the smulaton stops and the software package comples a report of the smulaton. The report gves the operatonal parameters of the system, such as average queue length, average watng tme n the queue, server utlzaton, etc. These parameters are used n computng the watng costs and constrants. 3. Mult-Objectve Optmzaton Most real-world optmzaton problems have multple objectves whch are often conflctng. The goal of multobjectve optmzaton (MOP) s to optmze the conflctng objectves smultaneously. In ths secton, we defne the general form of a MOP and Pareto domnance for dentfyng optmal solutons. Towards the end of the secton, we descrbe the use of Evolutonary Algorthms to solve MOP problems. Fgure 1. Workflow model of a small clnc servce system Journal of Dgtal Informaton Management Volume 8 Number 4 August

3 3.1 MOP Formulaton In general, a mult-objectve mnmzaton problem wth M decson varables and N objectves can be stated as: Mnmze f (x)=1,..., N (1) where x = (x 1,..., x m ) X subject to : g j (x) = 0 j = 1,..., M h k (x) 0 k = 1,..., k (2) Here, f s the th objectve functon, x s the decson vector that represents a soluton and X s the varable or parameter space. The functons g j and h k represent the equalty and the nequalty constrants, respectvely. The desred soluton s n the form of a trade-off or compromse among the parameters that would optmze the gven objectves. The optmal trade-off solutons among the objectves consttute the Pareto front. MOP deals wth generatng the Pareto front, whch s the set of non-domnated solutons for problems havng more than one objectve. A soluton s sad to be non-domnated f t s mpossble to mprove one component of the soluton wthout worsenng the value of at least one other component of the soluton. The goal of multobjectve optmzaton s fnd the true and well-dstrbuted Pareto front consstng of the non-domnated solutons. 3.2 Pareto Domnance Most mult-objectve optmzaton algorthms use the concept of domnaton. In these algorthms two solutons are compared on the bass of whether one soluton domnates the other or not. Assume that there are M objectve functons to be optmzed and the problem s one of mnmzaton. A soluton x (1) s sad to domnate the other solutons x (2), f condtons (1) and (2) are both true (Deb, 2001). 1. The soluton x (1) s no worse than x (2) n all objectves, or f j(x (1) ) f j(x (2) ) for all j = 1, 2, M. 2. The soluton x (1) s strctly better than x (2) n at least one objectve, or f j(x (1) ) < f j (x (2) ) for at least one j belongng to {1,2,..M}. lf ether of the above condtons s volated, the soluton x (1) does not domnate the soluton x (2). The non-domnated solutons gve rse to a Pareto front. The ponts on the front are used to select a partcular combnaton of functons that are n a trade-off balance. Fgure 2 llustrates the concept of Pareto domnance for the mnmzaton of two objectve functons. Soluton S s domnated by soluton P n the f 1 objectve, whle soluton T s domnated by soluton R n the f 2 objectve. The solutons P, Q, R are Pareto-optmal solutons snce none of them s domnated by any other solutons. 4. MOP n servce systems 4.1 Servce Costs The servers n the system are often called Perspectves. Servce systems consst of dfferent groups of Perspectves employed to provde servce. For nstance, the Perspectves n a clnc nclude groups of doctors, nurses, laboratory techncans, therapsts, pharmacsts, etc. If the servce cost per unt tme per personnel n a group assgned to a context s S C, then the total servce cost of the context s: Fgure 2. Pareto Domnance (mn-mn problem) Servce cost = m NPjSCj (3) j=1 where N Pj s the number of Perspectves n the j th group, and, m s the number of groups of Perspectves provdng servce at the gven context. Summng over the costs of all the contexts n the system, the total servce cost (objectve functon f 1 ) s: n m f = N S (4) 1 Pj Cj =1 j=1 where n s the total number of contexts n the system. 4.2 Watng tme The customers (patents) queue n front of the respectve contexts tll the server becomes avalable. If Q L s the average number of customers n queue for Q T perod of tme, then the weghted average of the watng tme (objectve functon f 2 ) s: n n f = Q Q / Q (5) 2 L T L =1 =1 The two functons are opposed to one another. An ncrease n one decreases the other and vce-versa. The MOPSO algorthm fnds a trade-off between the two. 4.3 Decson varables The number of professonals workng at each context and the average servce tme of the context are the decson varables drectly related to the cost functon. In practcal servce systems, both these varables have lower as well as upper bounds. The capacty N P of the server represents the number of personnel or the number of peces of equpment that are assgned to a gven context. 256 Journal of Dgtal Informaton Management Volume 8 Number 4 August 2010

4 N P< N P NP> (6) where N P < and N P > are the lower and the upper bounds, respectvely. Further, each context has an approprate servce tme that s usually drawn from an exponental dstrbuton. The servce tme lmts can be expressed as: t< t t> (7) where t< and t> are the lower and the upper bounds, respectvely. 5. Evolutonary Algorthms for Mult-objectve Optmzaton Evolutonary Algorthms (EA) seem to be especally suted to MOP problems, due to ther abltes to search smultaneously for multple Pareto optmal solutons and to perform better global searches of the search space (Mchell, 1996). Many evolutonary algorthms have been developed for solvng MOP. Examples are: GA (Holland, 1992), NSGA-II (Deb, et al., 2000), a varant of NSGA (Non-domnated Sortng Genetc Algorthm) (Srnvas, Deb, 1994); SPEA2 (Ztzler, et al., 2001) whch s an mproved verson of SPEA (Strength Pareto Evolutonary Algorthm); and PAES (Pareto Archved Evoluton Strategy). These EAs are populaton-based algorthms that possess an n-bult mechansm to explore the dfferent parts of the Pareto front smultaneously. The Partcle Swarm optmzaton (PSO), whch was orgnally desgned for solvng sngle objectve optmzaton problems, s also extended to solve mult-objectve optmzaton problems. Among those algorthms that extend PSO to solve mult-objectve optmzaton problems are Multobjectve Partcle Swarm Optmzaton (MOPSO) (Alvarez- Bentez, et al., 2005; Coelho, Lechunga, 2002; Coelho, et al., 2004; Feldsend, Sngh, 2002, Hu, Eberhart, 2002; Hu, et al., 2003; Ztzler, 1999), Non-domnated Sortng Partcle Swarm Optmzaton (NSPSO) (L, 2003) and the aggregatng functon for PSO (Knowles, Corne, 2000). 5.1 Partcle Swarm Optmzaton (PSO) The Partcle Swarm Optmzaton (PSO) algorthm conducts a search usng a populaton of ndvduals. The ndvdual n the populaton s called the partcle and the populaton s called the swarm. The performance of each partcle s measured accordng to a predefned ftness functon. Partcles are assumed to fly over the search space n order to fnd promsng regons of the landscape. In the mnmzaton case, such regons possess lower functonal values than other regons vsted prevously. Each partcle s treated as a pont n a d-dmensonal space whch adjusts ts own flyng accordng to ts flyng experence as well as the flyng experence of the other companon partcles. By makng adjustments to the flyng based on the local best (pbest) and the global best (gbest) found so far, the swarm as a whole converges to the optmum pont, or at least to a nearoptmal pont, n the search space. The notatons used n PSO are as follows: The th partcle of the swarm n teraton t s represented by the d-dmensonal vector, x (t) = (x 1, x 2,, x d ). Each partcle also has a poston change known as velocty, whch for the th partcle n teraton t s v (t) = (v 1, v 2,, v d ). The best prevous poston (the poston wth the best ftness value) of the th partcle s p (t-1) = (p 1, p 2,, p d ). The best partcle n the swarm,.e., the partcle wth the smallest functon value found n all the prevous teratons, s denoted by the ndex g. In a gven teraton t, the velocty and poston of each partcle s updated usng the followng equatons: and v ( t ) = wv ( t - 1) + c r ( p ( t - 1) - x ( t - 1)) c r ( p ( t- 1) - x ( t-1)) g (8) x ( t ) = x ( t - 1) + v ( t ) (9) where, =1, 2,,N P ; t = 1, 2,,T. N P s the sze of the swarm, and T s the teraton lmt; c 1 and c 2 are postve constants (called socal factors ), and r 1 and r 2 are random numbers between 0 and 1; w s the nerta weght that controls the mpact of the prevous hstory. 5.2 Mult-objectve Partcle Swarm Optmzaton (MOPSO) The Mult-Objectve Partcle Swarm Optmzaton (MOPSO) s an extenson of the sngle objectve Partcle Swarm Optmzaton (PSO). In PSO, gbest and pbest act as gudes for the swarm of partcles to contnue the search as the algorthm proceeds. The man dffculty n MOPSO s to fnd the best way of selectng the gudes for each partcle n the swarm. Ths s because there are no clear concepts of pbest and gbest that can be dentfed when dealng wth a set of multple objectve functons. Our algorthm s smlar to the ones descrbed n (Alvarez-Bentez, et al., 2005; Coelho, Lechunga, 2002; Coelho, et al., 2004). It mantans an external archve A, contanng the non-domnated solutons found by the algorthm so far. The algorthm begns wth the ntalzaton of an empty external archve, A (lne 1 n Fgure 3). The postons (x ) and veloctes (v ) of a swarm of N partcles are ntalzed randomly (lne 2). The number of Perspectves assgned to each context and the servce tme of the contexts are randomly generated wthn the gven bounds mposed by the management. The operaton of the servce system s smulated by means of the dscrete event smulator and the values of f1 (servce cost) and f2 (average watng tme) for each partcle are computed usng equatons 2 and 3, respectvely. The ntal poston of each partcle s consdered to be ts personal best (P = x ) as well as the global best (G = x ). At each teraton t the veloctes of the partcles are updated. The updated veloctes are forced nto the feasble bounds, f they have crossed the lower or the upper bounds. The partcle postons are then updated usng the updated veloctes (lnes 5-8). Ths s followed by the evaluaton of the objectve functons f1 and f2 for each of the partcles (lne 9). Any solutons whch are not weakly domnated by any member of the archve are added to A (lne 12) and any elements of A whch are not domnated by x are deleted from A. The crucal parts of the MOPSO algorthm are selectng the personal and the global gudes. If the current poston of x weakly domnates P or f x and P are mutually non-domnatng, then P s set to the current poston (lnes 15-17). Members of A are mutually non-domnatng and no member of the archve s domnated by any x. All the members of the archve are, therefore, canddates for the global gude. 6. Smulaton optmzaton results Intally, the number of Perspectves n each of the Perspectve groups, and the servce tme per context s randomly generated. These values are wthn the bounds set by the Journal of Dgtal Informaton Management Volume 8 Number 4 August

5 1: A := 2: { x, v, G, P } = 1,,N 3: for t := 1 : G 4: for := 1 : N 5: for k := 1 : K 6: v k := wv k + r 1 (P k x k ) + r 2 (G k x k ) 7: x k := x k + v k 8: end 9: y := f (x ) 10: f x u u A 11: A := {u A u x } 12: A := A U x 13: end 14: end 15: f x P 16: P := x 17: end 18: G := select Gude(x, A) 19: End Fgure 3. MOPSO Algorthm management (hard constrants). The operaton of the servce system s smulated by means of the dscrete event smulator. The smulaton output produces varous queueng statstcs. The average number of customers n queues s used as the weghts to compute the weghted average of the watng tme. Several senstvty analyss expermental runs reveal that c 1 = 1, c 2 = 1, w(ntal) = 0.9 and w(fnal) = 0.4 yeld hghest performance. We expermented wth a populaton sze of 100 partcles because t s a standard n mult-objectve evolutonary algorthms that use an external archve. Fgure 4 shows the Pareto front generated for the average nter-arrval tme of 2 mnutes. Snce the arrval s rapd, the weghted average of the watng tme for servce s relatvely hgh. A smlar Pareto front for average nter-arrval rate of 7 mnutes s shown n Fgure 5. The weghted average of the watng tme s relatvely smaller. Ths s because the arrval s slower. The Pareto fronts obtaned after 100 teratons are well-spread. The ponts on the front represent the optmal trade-off solutons. The search space of our applcaton problem can be estmated as follows. Assume, on an average, that there are 3 personnel n each of the 3 types of groups. Further, assume that the average servce tme of the contexts n the clnc system ranges from 10 to 30 mnutes. Snce there are 7 contexts n all, the search space = 37 x 37 x 37 x 207 = 1.34 x Even wth a modest number of contexts n the servce systems wth not so long servce tme ranges, the search space explodes n sze. 7. Conclusons and future work In ths paper, we ntroduced the Mult-Objectve Partcle Swarm Optmzaton (MOPSO) Evolutonary Algorthm to optmze two conflctng objectves- servce cost and watng tme n servce systems. The number of personnel assgned to each servce staton, ther cost and the servce tme per customer are the decson varables. Although there are bounds on the decson varables, the number of combnatons even for a modest servce system s prohbtvely large. The problem, therefore, s a typcal combnatoral optmzaton problem that cannot be solved by an exhaustve search. Evolutonary algorthms are well suted for MOP problems because of ther populaton based nature. Our expermental results wth a practcal servce system show a near-perfect Pareto-front that represents a trade-off between the servce cost and the watng tme whch s drectly related to customer satsfacton. As an extenson of ths study, we would lke to compare the performance of MOPSO to those of other EMO algorthms. 8. Acknowledgements Ths research has been supported by the Open Research Center Project funds from MEXT of the Japanese Government ( ). References Fgure 4. Pareto-front for servce cost & watng tme (t a = 2 mnutes) Fgure 5. Pareto-front for servce cost & watng tme (t a = 7 mnutes) [1] Alvarez-Bentez, J.E, Everson, R.M. Feldsend, J.E (2005). A MOPSO algorthm based exclusvely on Pareto domnance concepts. Lecture Notes n Computer Scence, Evolutonary Mult-Crteron Optmzaton, Sprnger, Berln / Hedelberg, 410: [2] Banks, J., Carson II, J.S (1984). Dscrete-Event System Smulaton. Prentce-Hall, New Jersey [3] Clemmer, E.C., Schneder, B (1989).Toward understandng and controllng customer dssatsfacton wth watng. Workng paper , Marketng Scence Insttute: Cambrdge, MA [4] Coelho, C., Lechunga, M (2002). MOPSO: A proposal 258 Journal of Dgtal Informaton Management Volume 8 Number 4 August 2010

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