Cross-Selling in a Call Center with a Heterogeneous Customer Population

Transcription

1 OPERATIONS RESEARCH Vol. 57, No. 2, March Aprl 2009, pp ssn X essn nforms do /opre INFORMS Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at Itay Gurvch Kellogg School of Management, Northwestern Unversty, Evanston, Illnos 60208, -gurvch@kellogg.northwestern.edu Mor Armony Stern School of Busness, New York Unversty, New York, New York 10012, marmony@stern.nyu.edu Constantnos Maglaras Columba Busness School, New York, New York 10027, c.maglaras@gsb.columba.edu Cross-sellng s becomng an ncreasngly prevalent practce n call centers, due, n part, to ts unque capablty to allow frms to dynamcally segment ther callers and customze ther product offerngs accordngly. Ths paper consders a call center wth cross-sellng capablty that serves a pool of customers that are dfferentated n terms of ther revenue potental and delay senstvty. It studes the operatonal decsons of staffng, call routng, and cross-sellng under varous forms of customer segmentaton. It derves near-optmal controls n each of the settngs analyzed, and characterzes the mpact of a more refned customer segmentaton on the structure of these polces and the center s proftablty. Subject classfcatons: call centers; cross-sellng; queueng systems; revenue management; prcng. Area of revew: Manufacturng, Servce, and Supply Chan Operatons. Hstory: Receved September 2006; revsons receved May 2007, September 2007, December 2007; accepted December Publshed onlne n Artcles n Advance January 5, Introducton Many organzatons consder ther call centers to be one of the most mportant channels of nteracton wth ther customers, actng both as a servce center and a pont of sales an opportunty for the frm to generate extra revenue by offerng new or exstng products to ther customers. The sgnfcant revenue potental of ths cross-sellng strategy s underscored by the nature of the nteracton that takes place n a call center and the wealth of nformaton that s avalable through state-of-the-art customer relatonshp management (CRM) systems. Together, they enable frms to segment ther customer pools effectvely and to talor ther product offerngs to each such segment to ncrease the lkelhood of purchase and the assocated expected revenue. A famlar and successful example of cross-sellng practce s n the fnancal servces ndustry, where customers who call for servce, such as for account balance nqures, are often offered new fnancal products. 1 Alongsde ts potental benefts, cross-sellng may substantally ncrease the total workload that needs to be handled by the call center s agents, 2 whch may degrade the system s qualty of servce and, n turn, have an adverse effect on the overall customer experence, as well as the effectveness of cross-sellng tself. It s mportant to carefully select whch cross-sellng opportuntes to pursue and when to do so, and to account for the mpact of these decsons n determnng the staffng level of the call center. Ths paper consders a call center wth cross-sellng capabltes that serves a heterogenous pool of customers, and studes the operatonal decsons of staffng, call routng, and cross-sellng under varous forms of customer segmentaton. It derves near-optmal controls n each of the settngs analyzed, and characterzes the mpact of more refned customer segmentaton on the structure of these polces and the center s proftablty. In more detal, we consder a call center wth a sngle pool of fully flexble agents that frst handle nbound call servce requests, and subsequently decde whether or not to attempt to cross-sell to some of these customers a certan product or servce whenever such an opportunty arses. Cross-sellng attempts are handled by the same agent that has served the customer s orgnal request, upon completon of that task. Each cross-sellng attempt s preceded by an nstantaneous step that captures the customer s decson of whether or not to agree to lsten to the cross-sellng offer. The processng tmes for the orgnal servce request and the cross-sellng phase are exponentally dstrbuted wth potentally dfferent parameters. Fnally, the heterogeneous pool of potental customers comprses a dscrete set of types or segments. (The terms type and segment are used n ths paper nterchangeably.) Types dffer n terms of ther delay senstvty and revenue potental. These are captured through the probablty that a customer wll agree to lsten to a cross-sellng offer as a functon of the watng tme that he encountered, and through a demand relaton that specfes the probablty that a customer decdes to buy 299

2 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton 300 Operatons Research 57(2), pp , 2009 INFORMS INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at the offered product as a functon of the quoted prce and the watng tme. The ablty to segment the caller populaton allows the call center to customze the product offered to each caller segment. In ths paper, we assume that the degree of segmentaton s exogenously specfed, for example, as the output of an upstream marketng analyss. Dependng on the applcaton settng at hand, product customzaton may nvolve chargng a dfferent prce to dfferent segments for the same product, or could nvolve changng the attrbutes, as well as the prce of the product offered to each segment. In both settngs, the goal s to better explot the preferences of each caller segment so as to ncrease the expected proftablty from cross-sellng. The output of ths prcng and/or product attrbute customzaton process s summarzed by the segment-specfc expected revenue per crosssellng attempt. As we show, the latter s crucal n decdng to whom to cross-sell and how to staff the call center. For purposes of the analyss n ths paper, we consder the smpler of the two settngs mentoned above, n whch the call center only customzes the prce of the product offered to each segment, keepng all other characterstcs of that product common across segments. We acknowledge ths fact by usng the term prce customzaton as opposed to product customzaton, agan keepng n mnd that the essental consequence of the customzaton capablty s that t leads to dfferent expected revenues per cross-sellng attempt for each segment. As an example of prce customzaton, one may consder the prcng of CD (certfcate of depost) products offered by banks to dfferent customers. It s natural to thnk of the prce of the CD as ts assocated nterest rate, although two mportant product attrbutes are the mnmum captal contrbuton and the length of the tme over whch the promsed nterest rate s guaranteed. An ncreasngly mportant applcaton of quanttatve prcng and revenue management tools n the fnancal servces ndustry s n decdng the terms, and more mportantly the nterest rate, of the CD product that s offered to exstng customers to entce them to roll ther exprng CD contrbuton from one product to another. Although prcng to ntally attract customers who may be shoppng around for such a product s qute compettve, the subsequent reprcng decsons tend to be less constraned and, ndeed, an area of ntense actvty n that ndustry. We study three varants of ths model wth an ncreasng avalablty of nformaton regardng customer segmentaton and, as a result, ncreasng flexblty n terms of the aforementoned operatonal and prcng decsons. The smplest model s one where customers are not segmented, or equvalently, where ther types are not observable. In ths case, the manager s lmted to makng the cross-sellng decsons based solely on the aggregate load n the system, and chargng all customers the same prce. The second model s one where types are observed sometme durng ther servce, and ths nformaton can therefore be used together wth the actual watng tme experenced by the customer n decdng whether to cross-sell to a customer, and f so, what prce to charge. The thrd model s one where customer types are observable upon arrval, n whch case the manager can also decde how to route customers of dfferent types to the avalable agents. For each of these models, the call center manager s problem s to select ts staffng, routng, cross-sellng, and prcng polces to maxmze the center s expected proft rate, gven by ts revenues mnus the staffng cost mnus a lnear watng tme cost that s experenced by all customers and s ncurred by the center. The controlled two-stage servce sequence of each customer and the dependence of the cross-sellng phase on dynamc watng tme nformaton makes an exact analyss of ths model cumbersome and dffcult, even f customers are treated as one segment. Our approach consders a determnstc relaxaton of ths problem, whch s solved n closed form. Its soluton suggests dfferent staffng and cross-sellng polces for each of the model varants lsted above. In each case, we show that our proposed polcy s asymptotcally optmal n systems wth ncreasng call volume, and as such s approprate for call centers wth hgh demand volumes. Our contrbuton s twofold: From a practcal vewpont we propose a concrete, smple, and provably nearoptmal soluton for the complex problem of cross-sellng n envronments wth multple customer classes. Our soluton wll allow frms to extract the revenue potental embedded n ther CRM systems through smart operatonal management of ther marketng nterface. From a manageral vewpont our tractable determnstc analyss and the asymptotc performance guarantees of the proposed polces lead to several nsghts. The frst one s that the marketng decsons of customer segmentaton and prce customzaton are effectvely decoupled from the operatonal decsons of staffng, routng, and cross-sellng. Specfcally, once the set of customer segments has been dentfed through an approprate marketng and statstcal analyss, and ther respectve characterstcs have been dentfed usng observed data, 3 the frm can precompute ts prce customzaton strategy ahead of tme, nstead of dynamcally choosng the prce charged to each customer. In partcular, the prces are statc and are dentcal across customers of the same type. These prces are then fed nto the operatonal control problem that nvolves staffng, routng, and cross-sellng decsons. The avalablty of nformaton on customer segmentaton has many mportant consequences, whch can also be easly seen from our determnstc relaxaton. To start wth, roughly speakng, the center wll only cross-sell to customers that generate an expected revenue that exceeds the capacty cost nvolved n pursung ths attempt; the expected revenue s equal to the quoted prce tmes the probablty that ths customer wll buy the offered product, provded that hs watng tme was zero. If the center can segment ts customers, then t wll only cross-sell

3 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton Operatons Research 57(2), pp , 2009 INFORMS 301 INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at to ts proftable types; f no segmentaton capablty s n place, then t wll ether cross-sell to all customers or to none, dependng agan on the expected proftablty of these cross-sellng attempts. In each case, the center wll staff so as to handle all regular servce requests plus the addtonal nomnal workload generated by ts expected cross-sellng actvtes. Because the cross-sellng s controllable, t can provde enough flexblty n the use of the center s capacty, whch elmnates the need to add safety staffng as s typcally done accordng to the square-root rule to stablze the system and guarantee moderate congeston. It s possble that even though t s proftable to cross-sell n a system that segments ts customers, ths s not the case wthout segmentaton. Our analyss outlnes such cases. Overall, customer segmentaton ncreases the center s proftablty n two ways: frst, through a more effcent use of capacty acheved by reducng the volume of cross-sellng attempts that are unlkely to be proftable, and second, by customzng the prce for each customer type so as to maxmze the resultng expected revenue. Fnally, we note that the effect of observng the customer type upon arrval, as opposed to after servce has commenced, s small. Ths s explaned by the fact that even when the system does not dfferentate between types n ts routng decsons and handles all external calls through a common frst-comefrst-served (FCFS) queue for all these types, the resultng watng tmes are small; these are moderated through the dynamc cross-sellng decsons of the call center and are renforced by the customers delay averseness. The remander of ths paper s organzed as follows. Ths secton concludes wth a bref lterature survey. Secton 2 descrbes the two models wth observable types, emphaszng mostly the model where customer type s revealed once hs servce starts. These two models are analyzed n 3. Secton 4 shows how the prcng problem can be treated separately from all other decsons, whch s then used n 5 to analyze a model wth no customer segmentaton. Secton 6 provdes results from our numercal experments. Secton 7 contans concludng remarks. The electronc companon contans all of our proofs and s avalable as part of the onlne verson that can be found at Lterature Revew. The lterature on the operatonal aspects of call centers s extensve and has grown rapdly over the past decade. A survey of ths lterature and a tutoral on the subject can be found n Gans et al. (2003). Of partcular relevance to our work s the lterature on staffng of call centers. The most commonly used staffng rule n the lterature s the so-called square-root safety staffng rule, accordng to whch the number of servers requred to handle an offered load of sze R s R + R for some constant. The square-root safety staffng rule dates back to Erlang n hs 1923 paper (that appeared n Erlang 1948). Ths rule was formalzed by Halfn and Whtt (1981), who showed that ths square-root safety staffng rule guarantees very short delays n an approprate asymptotc regme, and was shown to be nearly optmal for a pure servce center that handles a homogeneous customer populaton n Borst et al. (2004). Square-root safety staffng has been observed to be farly robust wth respect to changes n model assumptons to nclude features such as customer abandonment (Garnett et al. 2002, Mandelbaum and Zeltyn 2009), multple customer classes (Armony and Maglaras 2004a, b; Gurvch et al. 2008), multple server pools (Armony 2005), and nonstatonary arrval rates (Feldman et al. 2008). In contrast to the above set of papers, our work shows that the ssue of safety staffng s of lesser mportance n call centers wth sgnfcant crosssellng actvty because by adjustng the latter the manager can also control congeston. There s a small but growng porton of the recent lterature on call centers that n broad terms studes how to best manage the cross-sellng capablty of such systems. In more detal, the cross-sellng control problem,.e., the queston of when and to whom the center should try to cross-sell, has been studed by several authors, ncludng Akşn n a seres of papers wth Akşn and Harker (1999), Güneş and Akşn (2004), and Örmec and Akşn (2007), and by Byers and So n two papers (Byers and So 2007a, b). These papers consder varous aspects of the above dynamc control problem under three assumptons: (a) the staffng levels are exogenously fxed; (b) the products and prces offered to the varous customers are homogeneous even though the center may be able to segment ts customer pool accordng to ther preferences; and (c) a smplfed model of the servce system that treats customers that go through the cross-sellng phase as a separate class of servce requests wth longer servce tmes, as opposed to as a two-phase servce. Ths latter restrcton mples that cross-sellng decsons have to be made n the begnnng of the nteracton wth the customer, and t cannot use updated state nformaton that may be avalable at the completon of a customer s nomnal servce request. The servce faclty s ether modeled as a sngle-server queue, a multserver queue, or a multserver loss system (.e., customers that do not fnd an dle server upon ther arrval are lost). For the sngle-server model, Byers and So (2007a) showed that the optmal cross-sellng polcy s of a threshold type; the center cross-sells as long as the number of customers n the system s below a certan threshold. The optmalty of the threshold polcy n the multserver case has not been establshed. Despte the restrctve assumptons lsted above, these papers made sgnfcant contrbutons to the lterature by beng the frst to address the mportant motvatng questons mentoned earler, and by dervng nsghts that seem to be farly ntutve and, to some extent, robust. They also rased nterestng questons: Are these nsghts robust to more representatve models of the servce delvery process? What s ther mpact on staffng decsons? In what way would the staffng decson affect the structure of the cross-sellng polcy and the proftablty of cross-sellng? And, fnally, what s the mpact of customer segmentaton on all of the above?

4 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton 302 Operatons Research 57(2), pp , 2009 INFORMS INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at Recently, Armony and Gurvch (2006) proposed a more realstc stochastc model for the cross-sellng process, whereby the servce tme of each customer comprses two dstnct phases the frst captures the handlng of the customer s nomnal servce request, and the second, whch s optonal, captures the duraton of the cross-sellng attempt. The man analytcal contrbuton of Armony and Gurvch (2006) s to rgorously show that a threshold-type crosssellng polcy s asymptotcally optmal for ths more complex servce model as the nomnal demand and the sze of the call center grow large. Armony and Gurvch (2006) also conducted a prelmnary analyss of the jont staffng and cross-sellng control problem for the case where the entre pool of customers s ether homogeneous n terms of ts preferences, or s treated as such by the system; the latter would correspond to settngs where the customers are heterogeneous, but the system does not have segmentaton capablty. Our paper apples the servce model proposed n Armony and Gurvch (2006) to a settng wth heterogeneous and delay-senstve customers to address the jont prce customzaton, staffng, and cross-sellng control problem. Our economc model s more general than those used n earler papers, and the consderaton of customer delay senstvty s new. Our model allows for an nsghtful analyss of the jont prcng, staffng, and cross-sellng problems, whch emphaszes the trade-offs among customer segmentaton, prce customzaton, staffng costs, and the system proftablty. Our work renforces the nsghts derved n the varous papers lsted thus far. It also hghlghts that the ablty to segment the customer pool and customze the respectve prces leads to sgnfcantly dfferent staffng and crosssellng polcy recommendaton from those derved n the papers mentoned above. An mportant ngredent of our soluton methodology hnges on the use of a determnstc relaxaton for the orgnal jont prcng, staffng, and cross-sellng dynamc optmzaton problem, whch s motvated from the work of Maglaras and Zeev (2005). Fnally, the economc model that we adopt and the noton of prce dscrmnaton that underles our work are related to a vast lterature n economcs, marketng, and revenue management. We refer the reader to the book by Tallur and Van Ryzn (2004) for an ntroducton to these subjects. 2. Model Formulaton We consder a call center wth a sngle pool of N fully flexble agents that serves a heterogeneous customer populaton, comprsng K dstnct segments, or types, or classes. We study three model varants dependng on the extent to whch the customer types are observable by the system. These are graphcally depcted n Fgure 1. Model (a) assumes that types are unobservable, or that the call center does not segment ts customers. In model (b), the type of a customer s observed when s/he s beng served, and Fgure 1. Three cross-sellng models. (a) (b) (c) ths nformaton s subsequently used n the center s crosssellng decsons. Fnally, model (c) s one where the customer type s mmedately observed upon arrval, e.g., by requrng customers to enter an account number, and can therefore be used n routng as well as n cross-sellng decsons. We wll focus on model (b), and treat model (c) as an extenson and model (a) as a one-segment specal case of ths multsegment model. Basc Servce. Type- customers call the center accordng to a Posson process, A t t 0, wth rate. Let At = K =1 A t, and defne = K =1 to be the total arrval rate nto the system. All customers requre the same type of servce and the processng requrement s exponentally dstrbuted wth rate s, ndependent of the customer type. Under the assumpton that types are unobservable before servce begns (model (b)), all customers jon a sngle queue and get processed n an FCFS manner. Cross-Sellng. Once regular servce s completed, a customer ether leaves the system or enters a cross-sellng phase that s handled by the same agent. A cross-sellng attempt s preceded by an nstantaneous step n whch the customer s asked to lsten to the actual offer. The length of tme requred for the cross-sellng attempt may depend on the customer segment and s assumed to be exponentally dstrbuted wth rate cs for type- customers. All processng tmes (regular servce and cross-sellng) and nterarrval tmes are assumed to be ndependent. The probablty that a type- customer wll agree to lsten to the cross-sellng offer after experencng watng tme w s gven by an arbtrary nonncreasng contnuous functon q w + 0 1, wth lm w q w = 0. We set q = q 0 and note that t s possble to have q < 1. Ths allows us to model cases where some customers may always declne to lsten to the cross-sellng offer. If a customer of class- agrees to lsten to a crosssellng offer, he wll be offered the product at a certan prce that mght depend on both hs class and hs actual watng tme. Class- customers have..d. valuatons for

5 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton Operatons Research 57(2), pp , 2009 INFORMS 303 INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at ths product, denoted by v, drawn from a contnuous dstrbuton functon F. The perceved cost of the offered product may also depend on the watng tme s/he has experenced. Ths dependence may arse n some practcal settngs, such as when sgnng up for help desk servces where the watng acts as a proxy for the future qualty of servce. In other applcatons, the cost of the offered product should not depend on the watng tme, and ths s also allowed by our model. Specfcally, we assume that class- customers have a delay-senstvty constant c 0. Then, condtonal on agreeng to lsten to a cross-sellng offer, a class- customer who has wated for w tme unts before startng her servce wll buy the product wth probablty F p w = F p + c w = Pv >p +c w. Applcatons where the cost of the offered product s ndependent of the watng tme are captured by settng c = 0. The resultng condtonal expected revenue from a customer of class- who wated w tme unts s gven by r p w = p F p +c w. For smplcty of notaton, we let r p = r p 0 = lm w 0 r p w. We wll also assume that the functons r p are unmodal n the p s for each ; ths s satsfed by many commonly used demand functons (see Tallur and van Ryzn 2004). For the frst few sectons, we wll assume a fxed vector of prces p = p 1 p K. Hence, we wll use the smplfed notaton r w nstead of r p w and the notaton r for r 0. We wll return to the more general notaton n 4, n whch we consder the prcng problem. Control Decsons. The call center manager selects the number of agents N for the system and has dscreton wth respect to the cross-sellng and prcng decsons. We wll consder polces,, that decde whether to cross-sell to the jth type- customer and whch prce to charge hm as a functon of all the nformaton avalable up to the decson pont. In partcular, the cross-sellng and prcng decsons are dynamc and may depend on the customer s type, the watng tme encountered by ths customer pror to hs servce, whch we denote by wj, the number of customers n the queue, and the number of customers of each type- that are currently n servce, denoted by Q t and Z t, respectvely. We let Q t = K =1 Q t be the total queue length at tme t under. To guarantee the exstence of steady state or at least the exstence of long-run averages for varous quanttes of nterest, we wll restrct the set of admssble controls as follows. Defnton 1 (Admssble Controls). Gven a staffng level N, and parameters 1 K s cs 1 cs K, we say that s an admssble polcy f t s nonpreemptve, nonantcpatve, and lm t EQ t/t 0. We denote the famly of admssble polces by 1 K s cs 1 cs K N. Loosely speakng, 1 K s cs 1 cs K N s the set of stablzng polces under the gven parameters. Defnton 1 takes nto account the fact that the set of admssble polces depends on the parameters of the model through the stablty condtons of the system. To smplfy notaton, we wll omt the parameters 1 K s and cs, = 1K, whenever these are exogenously fxed, and wrte N or smply whenever the staffng level s clear from the context. Note that the above defnton mples that our system must be able to handle all of the nomnal demand, at least when no cross-sellng s exercsed; that s, the staffng choce must satsfy the constrant N >R= / s. Performance Crteron. We frst defne two system quanttes that wll play an mportant role n the call center s cost and revenue terms, respectvely. Observe that a steady state need not exst for any N. Wth that n mnd, for some N and = 1K, we defne [ A t EW = E lm sup t A t [ x = E lm nf t and r x = E A t j=1 x j A t [ A t lm nf t j=1 w j ] ] j=1 r w j x j A t where xj s an ndcator that s set to one whenever the jth class- customer goes through a cross-sellng phase, and xj equals zero otherwse. The performance measure r x should be nterpreted as the long-run average revenue per class- customer under the polcy. When a steady state exsts, EW and x concde wth the expected steady-state watng tme experenced by type- customers, and the steady-state fracton of class- customers that are asked and agree to lsten to a cross-sellng offer under, respectvely. r x wll then concde wth the steady-state revenue from class- customers. Because customers are processed FCFS, t must be that EW = EWk for all, k, whch wll also be denoted by EW. The call center ncurs lnear staffng and watng tme costs per unt tme, gven by c N and hew, respectvely. The latter assumes that the watng tme cost s type ndependent. The watng tme cost can be thought of as a penalty that the system ncurs n terms of lost goodwll from the customers. The type ndependence of the watng cost can be relaxed wth no effect on any of our results. Under an FCFS dscplne t seems reasonable, however, to assgn a common cost to all customers. The call center manager s optmzaton problem s the followng: sup N + N =1 ] r x cn hew (1) Note that although t s not guaranteed that there exsts a control that actually acheves the optmal proft rate,

6 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton 304 Operatons Research 57(2), pp , 2009 INFORMS INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at t s easy to establsh the exstence of an optmal N because N s dscrete, the proft rate s bounded above by r c R, and t decreases to as N grows large. An alternate formulaton to (1) would replace the watng tme cost by an upper-bound constrant on the expected watng tme, typcally n the order of 30 seconds, and consder the followng problem: { K } sup r x cn EW W (2) N + N =1 Indeed, one can vew (2) as a more natural startng pont, and (1) as a dualzed verson of the problem that s perhaps smpler to address. We wll refer to (1) and (2) as the watng cost and constraned formulatons, respectvely. We wll also make the followng assumpton: Assumpton 1. Types are labeled so that r 1 c/ cs 1 r K c/ cs K and r 1 c/ cs 1 > 0. The labelng assumpton s nnocuous. The condton r 1 c/ cs 1 > 0 means that t s proftable to cross-sell to at least type-1 customers. As wll be shown later, r 1 c/ cs 1 s roughly the expected revenue from cross-sellng to a class-1 customer mnus the margnal staffng costs assocated wth t. In the absence of ths assumpton, t makes sense not to nvest n extra capacty for cross-sellng and to only attempt to cross-sell to a neglgble fracton of the customers. 3. Observable Types: Analyss Based on a Determnstc Relaxaton A drect analyss of the problems formulated above s very dffcult due to ther multclass nature and the dependence of the cross-sellng success probablty on state-dependent nformaton. Our approach looks at relaxatons of the above problems, where n addton to the staffng and cross-sellng decsons, the manager can also select the watng tmes experenced by ts callers, whch n realty are random varables that depend on the system dynamcs. These relaxatons are tractable, determnstc optmzaton problems that have nsghtful solutons and gve rse to near-optmal heurstcs. Focusng on model (b) (cf. Fgure 1) frst, 3.1 studes the watng cost formulaton of (1). These results are extended to the constraned formulaton of (2) n 3.2, whereas 3.3 extends our work to model (c), where the customer types are observable upon arrval. All proofs are relegated to the onlne appendx The Watng Cost Formulaton Throughout ths secton, we focus on model (b) and the watng cost formulaton (1). Determnstc Relaxaton. Startng wth (1), we formulate the followng lnear program: maxmze r w x c R1 + z h w =1 s.t. x q w =1 x cs Rz z 0 x 0 w 0 =1 for all = 1K where x s nterpreted as the fracton of class- customers that are beng asked and agree to lsten to a cross-sellng offer; w s the fcttous watng tme experenced by class- customers n ths formulaton; and z s the excess (normalzed) staffng level beyond the nomnal requrement of the offered load R (=/ s ) as a fracton of R. The condton z 0 mples that the staffng level s suffcently large to handle all basc servce requests (.e., N R). The name determnstc relaxaton comes wth a slght abuse of termnology. As to whether or not ths s ndeed a relaxaton for (1) the answer to ths queston depends on the actual form of the functon q and, more specfcally, on ts concavty or lack thereof. It s a matter of a smple observaton, however, that any optmal soluton to (3) wll have w = 0 for all and, consequently, that an optmal soluton to (3) s necessarly an upper bound for any optmal solutons to (1) f such solutons exst. Hence, we choose to refer to (3) as the determnstc relaxaton. Recall the labelng conventon n Assumpton 1. Denotng the optmal soluton to the knapsack problem n (3) wth an overbar, we have the followng: set w = 0 for all = 1K, { q k k x = and z = q (4) 0 otherwse R cs where k = max r c/ cs 0q 0 >0. In fact, we wll assume throughout that r k c/ cs > 0, whch s equvalent to assumng that the determnstc relaxaton has a k unque soluton. In the presence of multple solutons to the determnstc relaxaton, our approach mght lead to multple asymptotcally optmal solutons. By Assumpton 1, z s guaranteed to be strctly postve. The resultng staffng =1 (3) level s R + k =1 q / cs. Note that the structure of the determnstc relaxaton s such that as long as / s known and s kept constant (whch we wll assume henceforth), the normalzed quanttes x, z do not change wth. Therefore, the relevant proft depends on the entre vector 1 K through ther sum only. Specfcally, the proft rate assocated wth soluton (4) s = cr + = cr + k =1 =1 q r c/ cs q r c/ cs 0 (5)

7 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton Operatons Research 57(2), pp , 2009 INFORMS 305 INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at whch s an upper bound for the optmal proft n (1). (Here and elsewhere x y = maxx y.) AStaffng and Cross-Sellng Proposal. The nested structure of (4) s ntutve: we cross-sell to all types for whch ther margnal revenue contrbuton, r q, exceeds the ncrease n staffng cost, c q / cs, resultng from the addtonal cross-sellng workload; ths reduces to the condton r c/ cs > 0. The soluton to the determnstc relaxaton suggests the followng par of polces for the orgnal stochastc system: (S) Staffng: Staff wth N = R1 + z. (C) Cross-sellng: Gven a sequence of thresholds k k 1 1 : cross-sell to a customer of type k that completes servce at tme t f and only f Qt <. The cross-sellng polcy (C) follows the soluton of the determnstc relaxaton when the queue length s modest, and then starts to reduce the amount of cross-sellng actvty as the system gets ncreasngly congested. The asymptotc performance analyss that wll follow does not use the precse values of the above thresholds, and n fact only makes use of the smallest threshold k. Consequently, one may prefer to use a smpler polcy that uses only ths smallest threshold k. Ths sngle-threshold polcy always cross-sells to classes 1 k 1 and stops cross-sellng to class k when the queue length exceeds the threshold. In our settng, n whch the arrvals rates,, are known and statonary, ths sngle-threshold polcy wll be asymptotcally equvalent to (C) n terms of the profts t generates. Stll, we choose to present the results for the more elaborate control (C). We motvate the use of multple thresholds n a nonstatonary envronment n 7. Asymptotc Optmalty of (S)-(C). Despte ts smple structure, (S)-(C) performs very well n the stochastc system under consderaton, and s, n fact, asymptotcally optmal n large-scale systems,.e., where s large. As a startng pont, we wll establsh that the system s always stable under (S)-(C) and that t admts a unque statonary dstrbuton. We do that by showng the stronger result that the system wll be stable under (C) as long as N>R,even f N<R1 + z. Proposton 1 (Stablty). Fx and assume that C s used for some set of thresholds k k 1 1. Then, N>Rs a suffcent condton for stablty. Moreover, for any N>R, the underlyng Markov process admts a unque statonary dstrbuton that s also ts lmtng dstrbuton. Ths proposton llustrates the self-stablzng nature of the cross-sellng system. Note that the use of thresholds s not necessary for ths result to hold. Indeed, they may all be set equal to ; the stablzng force stems from the delay senstvty of the customers. Intutvely, when the system s heavly loaded, the queue and the resultng watng tme wll grow large. In turn, fewer customers wll agree to lsten to cross-sellng offers, thus reducng the load. The remander of ths subsecton wll characterze the asymptotc performance of the orgnal stochastc call center system under (S)-(C) n settngs wth large call volumes, as measured by. One naturally expects that wth a threshold polcy, the best threshold values wll be a functon of the system sze and n partcular of, the overall arrval rate. Let k 1 be the threshold values correspondng to a system wth arrval rate. Then, we wll show n our subsequent results that, ndeed, there s a dependence of the threshold values on the system sze and, moreover, that asymptotcally optmal performance mples that these threshold values scale accordng to =ˆ for = 1 k (6) and approprate constants ˆ k ˆ 1. Let N, x, and denote the (unknown) optmal staffng level, realzed long-run average cross-sellng rates, and the correspondng proft rate for (1), respectvely, when the aggregate demand s. Also, let be the proft obtaned when usng (S)-(C) n the stochastc system. In the sequel, we wll make use of the followng notaton: for two postve sequences we say that x s oy f x /y 0as. Theorem 1 (Asymptotc Optmalty). Let grow large, keepng / constant for all. Then, wth thresholds satsfyng 6, S-C s asymptotcally optmal n the sense that = o (7) Alternatvely, one could wrte (7) n the form / 1as. The proof of the above result follows by showng the stronger result that approaches, whch tself s an upper bound for. Because s sandwched between and, t must also be close to. Ths leads to a partal characterzaton of the unknown optmal polcy n large-scale systems. Theorem 2 (Estmates ofthe Optmal Soluton). Let grow large, keepng / constant for all. Then, (a) = o, (b) N = R1 + z ± o, and (c) x = x + o1. Theorems 1 and 2 together demonstrate how the soluton of the determnstc relaxaton captures the frst-order behavor of the optmal polcy for (1), both n terms of ts staffng and cross-sellng decsons as well as ts resultng profts. A key component of the asymptotc optmalty proof s the next lemma that shows that f the thresholds are of order (as n (6)), then the steady-state watng tmes that characterze the system are of order 1/ and n partcular of order o1; ths s the nomnal tme t takes an order servers to clear a queue length of order. Thresholds of smaller magntudes would result n even smaller watng tmes.

8 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton 306 Operatons Research 57(2), pp , 2009 INFORMS INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at Lemma 1. Let grow large, keepng / constant for all. Denote by EW the steady-state expected watng tme under polcy S-C. Then, wth thresholds satsfyng 6, EW = O1/, or equvalently, lm sup EW <. In partcular, EW 0 as. The next lemma then shows that, actually, t would be always optmal to staff and cross-sell so that the watng tmes are very small. We denote by EW the expected steady-state watng tme under the optmal control N x. Lemma 2. Let grow large, keepng / constant for all. If an optmal polcy N x exsts for all large enough, then lm sup EW = 0. Remark 1 (Strengthenng the Noton ofasymptotc Optmalty). The man techncal problem n provng Theorems 1 and 2 les n the so-called lmt nterchange problem. Specfcally, although t mght be relatvely smple to get performance guarantees on fnte tme ntervals, t s much harder to characterze the asymptotc performance, as, of the system s steady state. The techncal arguments n that respect are qute complex, as the onlne appendx llustrates. The nterested reader s referred to part B of the onlne appendx for a further dscusson of the underlyng complextes. Consequently, refnng the performance bounds by showng, for example, an O devaton from optmalty, s complcated even n much smpler settngs than the system we consder especally when one wants to establsh convergence of moments. Remark 2 (Choosng the Threshold Values). For the cost formulaton, the values of the thresholds can be selected va smulaton. In most call centers, however, the constraned formulaton (consdered n the next secton) s more natural. Fortunately, for the constraned formulaton we have a very smple rule to determne the threshold value The Constraned Formulaton Lemmas 1 and 2 llustrate that the watng tmes experenced n an optmally controlled call center wll be of order o1. Wth that n mnd, a watng tme constrant of the form EW W wll become rrelevant as grows large because the actual watng tmes wll be much smaller than the desred target W. A more approprate formulaton that s meanngful as grows large replaces the upperbound constrant by a quantty that tself changes wth such as W = W/ for an approprate choce of W. 4 Ths would result n the followng problem: { K sup r x cn EW W } (8) N + N =1 where W = W/ for an approprate choce of W. Along the lnes of (3), the followng s a determnstc relaxaton of (8): maxmze s.t. r x c R1 + z =1 =1 w W x q w =1 x cs Rz x 0 w 0 for all = 1K The lnear program descrbed above has the same optmal soluton as (3), makng our soluton nsenstve to the precse artculaton of the effect of customer watng tmes. The resultng staffng and cross-sellng heurstcs are agan the ones descrbed by (S)-(C) n the prevous subsecton. In the case of the constraned formulaton, one can also get a crude estmate for the threshold k to be k = W, whch s consstent wth (6). Intutvely, f the queue length s mantaned below that threshold, then by a heurstc applcaton of Lttle s law, one would expect the watng tmes to be below W. The next theorem establshes ths result n an asymptotc sense as grows large. Wth a slght abuse of notaton, we use and to denote the proft rate for the constraned formulaton under (S)-(C) and the optmal polcy, respectvely. Theorem 3 (Asymptotc Optmalty). Let grow large, keepng / constant for all. Then, wth thresholds satsfyng 6, and such that k = W, (a) = + o and (b) EW W + o W. Theorem 3 shows that the watng tme constrant wll be volated only by a neglgble amount f one sets k = W. Of course, f one s nterested n strct satsfacton of the threshold, one may start wth the recommended threshold and fne-tune t n real tme wth small perturbatons around the recommended value The Value of Customer Type Identfcaton Upon Arrval We complete the analyss of the model wth observable types by comparng the model analyzed thus far (model (b) n Fgure 1) wth the one where the type of each customer s observed at the tme of hs arrval to the system (model (c)). The latter could be acheved by requrng callers to dentfy themselves through a PIN or an account number. Routng Capablty. Once the call center observes the type of each arrvng customer, t can mantan dfferent (vrtual) queues for customers of each type, and use that added flexblty n routng calls to avalable agents. Ths wll eventually trade off the delay senstvty and watng tme cost of each type aganst ts potental revenue (9)

9 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton Operatons Research 57(2), pp , 2009 INFORMS 307 INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at contrbuton. It s clear that ths added element of control can only mprove the call center s proftablty. The queston s by how much. The man result of ths secton shows that the performance dfference between FCFS routng (used when types are unobservable upon arrval) and any other routng polcy that makes use of the type nformaton, ncludng the optmal one, s small and asymptotcally neglgble. The crude asymptotc analyss of ths subsecton uses a sandwch argument, smlar to the one appled n Theorem 2, and does not need a detaled artculaton of the set of admssble routng polces. We refer the reader to Bassamboo et al. (2006) for one possble defnton of these controls. We henceforth drop the dstncton between the watng cost and constraned formulatons. The results n the remander of ths secton as well as those n 4 and 5 hold for both formulatons. Let be the optmal achevable proft for the system where customer types are observable upon ther arrval, and note that. The key to our analyss s that the determnstc relaxatons for models (b) and (c) are dentcal. The routng capablty of model (c) can only serve to mprove the vector of expected watng tmes EW. Because the relaxaton treats these as free optmzaton varables, denoted by w, and sets them equal to zero, ts soluton wll concde wth that of (3). It follows that. From Theorem 2 we have that = o, whch leads to the followng concluson: Proposton 2. Let grow large, keepng / constant for all. Then, = o. Therefore, although routng control capablty may mprove the qualty of servce enjoyed by some types and potentally smultaneously ncrease the revenue extracted from them, t wll not lead to a sgnfcant overall proft gan. Moreover, the asymptotcally optmal staffng and cross-sellng recommendatons that emerge from our analyss are nsenstve (up to frst order) to the use of ths nformaton. The queston that arses s whether segmentaton at the cross-sellng stage leads to sgnfcantly dfferent results n comparson to no segmentaton at all. To address ths queston, we frst study the ssue of type-dependent prce customzaton n 4, and then assess the value of customer segmentaton n The Prce Customzaton Problem Customer segmentaton n a call center settng allows frms to customze ther products to better match the characterstcs of each customer type and extract hgher revenues. In our model, the product offered to all customers s assumed to be the same, but the frm can customze the prce quoted to each customer type. In ths secton, we show that the optmal prces can be computed separately from the operatonal decsons of staffng and cross-sellng. Towards ths end, note that due to the dependence of the wllngness to pay on the watng tmes of customers, one expects the true optmal prcng mechansm to be a dynamc one that takes nto account these realzed watng tmes. Hence, the prcng mechansm should be regarded as a mappng from watng tmes to prces. Specfcally, we assume that prces may assume values n the space = 1 2 K, where for = 1K, s assumed to be a compact nterval n +. The prcng mechansm s then a functon p = p 1 p K + ; welet be the space of these functons. Accordngly, we expand the notaton used earler to let p and N p be the optmal proft rate and staffng level, respectvely, for (1) for a gven and prcng functon pw. We then redefne = sup p p to be the optmal achevable proft rate when the call center s allowed to optmze over ts prce functon over the set. Let p = p be the optmal prce functon, whch s assumed to exst, and N the correspondng staffng level. We also let p be the proft rate acheved n the determnstc relaxaton of (3) for a gven constant value of p, = max p p be the proft rate when optmzng over the prce, and let p denote the correspondng optmzer, whch wll most lkely be dfferent than the functon p. Whereas dentfyng p s hard, the determnstc prce vector p s easy to characterze by rewrtng the objectve functon as p = cr + =1 q r p c/ cs 0 (10) where r p = p F p 0; ths expresson reflects the fact that the center only cross-sells to and receves revenue from types for whch r p c/ cs, and that t staffs accordngly. It follows that the correspondng optmal prce n (10) s statc (watng tme ndependent) and satsfes p = arg max p F p 0 (11) p and = cr + K =1 q r p c/ cs 0 = p. The correspondng staffng level s R1 + z p, where k p z p = q and =1 R cs k p = max r p c/ cs (12) the above expressons assume w.l.o.g that types are relabelled so that r 1 p 1 r K p K. We also assume that r 1 p 1 >c/ cs 1 and that r k p p k p>c/ cs, whch guarantee, respectvely, that Assumpton 1 holds and that the k p soluton of the determnstc relaxaton gven p s unque. It s straghtforward to show that p, z p, and k p jontly characterze the optmal soluton of the determnstc relaxaton, and that ths soluton does not change f one were

10 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton 308 Operatons Research 57(2), pp , 2009 INFORMS INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at to scale large, whle keepng / constant (ths s the asymptotc setup adopted thus far). Note that although p may be dfferent than p, p s stll an upper bound for p. Usng ths observaton and applyng Theorem 2 (wth the fxed-prce vector p), we fnd the followng: Proposton 3. Defne p, z p through 11 and 12, respectvely. Let grow large, keepng / constant for all. Then: (a) p = p o, (b) N p = R1 + z p ± o, and (c) p 0 = p + o1. Consequently, we recommend addng the statc prce vector p to the staffng and cross-sellng rules proposed n 3. By Theorem 1 and Proposton 3 above, the resultng jont prcng, staffng, and cross-sellng soluton s asymptotcally optmal for the orgnal stochastc system. Decouplng of Prcng and Staffng. An mportant consequence of the above result s that the prcng decsons can be made ndependently of the operatonal ones of staffng and cross-sellng. Ths nsght s vald n the system where types are observed upon arrval (model (c)), as well as n settngs where products are customzed along other nonprce attrbutes that do not nvolve capacty and qualtyof-servce specfcatons. Ths decouplng trvally follows n settngs where the perceved cost of a product s ndependent of the watng tme encountered by the customer, but need not be true n the more general model consdered n our paper. Moreover, because the watng tme of the customer s known to the agent, the center may want to nvoke a dynamc prcng polcy to optmze the expected revenue per customer. The fact that a statc prcng polcy s shown to perform very close to optmal s an appealng characterstc of our soluton that allows the system manager to make the prcng and operatonal decson n a herarchcal sequence. 5. The Effect of Customer Segmentaton Ths secton compares the proftablty and behavor of the system studed n 3 and 4 aganst one that does not use a segmentaton mechansm and nstead treats ts entre customer pool as one segment. The latter s offered a common product,.e., at the same prce, and cross-sellng decsons are made wthout the customer type nformaton; ths s model (a) n Fgure 1. ASystem wth No Customer Segmentaton. The characterstcs of ths combned segment are a sngle delay senstvty functon q and a correspondng wllngnessto-pay dstrbuton F that are approprate mxtures of the correspondng quanttes for the varous types. The delay senstvty functon, qw, sgvenby qw = =1 q w The mean cross-sellng tme for the combned segment s estmated by 1 = K cs =1 q K j=1 jq j 1 cs Ths s a reasonably precse estmate assumng that the watng tmes are small. Moreover, the comparson result n Proposton 4 below holds when one uses a more precse estmate that takes nto account the watng tmes. The combned wllngness-to-pay dstrbuton F s computed ndrectly as follows. Let Fpw be equal to the probablty that the wllngness-to-pay of a customer that agreed to lsten to the cross-sellng offer after a watng tme of w tme unts s less than or equal to p. Then, qw and Fpw satsfy the followng ntutve relaton qw Fpw= =1 q w F p w from whch we can solve for Fpw. The determnstc relaxaton for the combned segment s now easy to solve by specalzng the results of 3 to a sngle segment wth characterstcs qw and Fpw. Specfcally, t s agan optmal to set w = 0, whch together wth (10) gves the followng objectve: a p = cr + qp Fp0 c/ cs 0 (13) where the superscrpt a s meant to assocate ths expresson to model (a), and q= q and Fp0= q K j=1 F p0 (14) jq j =1 As shown n 4, one can study ths determnstc formulaton by separately optmzng over the prce p, and then consderng the resultng staffng and cross-sellng problem at that prce. The prcng decson. The optmal prce that the call center should use n ths determnstc relaxaton s gven by the soluton to the followng problem: max p Fp0 (15) p whch we denote by p a, and let r a = p a F p a 0 and = 1 2 K. Note that despte our assumptons regardng the unmodalty of p F p 0, p Fp0 need not be unmodal tself. However, one can always fnd ts optmzer through a sngle-parameter search (assumng that the set s not empty). The staffng and cross-sellng decsons. Pluggng p a nto (13) and usng the results of 3, the soluton of the determnstc relaxaton can be dvded nto two cases: Case. If r a c/ cs : the call center cross-sells to all customers and staffs wth R max = R1 + z a servers, where z a = q/r cs. =1

11 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton Operatons Research 57(2), pp , 2009 INFORMS 309 INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at Case. If r a <c/ cs : the call center wll not cross-sell to any customer and staff wth R servers. Usng (13) and (14), the resultng proft rate n the determnstc relaxaton s gven by cr + q p a F p a 0 c/ cs a =1 = (16) f r a >c/ cs cr otherwse whch s agan an upper bound for the optmal proft rate of the stochastc call center system. As n 3, the natural mplementaton of the above polces n case would be to cross-sell as long as the queue s below an approprate threshold that serves to lmt excessve delays. In case, the system may stll elect to crosssell, but only f ether the queue s very small or there are a suffcent number of agents that are dle. Moreover, n that case the staffng level should be nflated to R + x R for an approprate constant x to provde moderate delays. The asymptotc analyss of 3 does apply to the snglesegment model when the soluton of the determnstc relaxaton falls nto case, but t does not cover case, where the system exercses neglgble cross-sellng. That case was studed n detal n Armony and Gurvch (2006) and wll not be further revewed here. The Effect of Customer Segmentaton. The key dfferences between the two systems, wth and wthout segmentaton, are best llustrated through ther respectve determnstc relaxatons, whch are smple and accurate, n the sense that they capture the structure of the underlyng optmal polces and ther resultng performance asymptotcally. 1. Cross-sellng all-or-none versus selected types. For both models, the call center wll do sgnfcant cross-sellng only f the expected revenue from dong so exceeds the capacty cost nvolved n that actvty. Wth no segmentaton capablty n place, the system wll ether choose to cross-sell to all of ts callers f r a c/ cs, or to none. In the frst case, ths may nvolve cross-sellng to customer segments to whch t s strctly unproftable to do so, whereas n the second case, t nvolves forgong proftable cross-sellng opportuntes that cannot be sngled out from the larger pool of callers (the latter follows from Assumpton 1). Usng customer segmentaton, the system wll only cross-sell to types = 1 k for whch r p c/ cs,.e., for whch cross-sellng s proftable. Fnally, we note that although Assumpton 1 guarantees that the call center wll always choose to cross-sell to some subset of the customer types, f these can be segmented out, t does not guarantee that t s proftable to do so n a system wth no segmentaton capablty. 2. Staffng. The model wth no segmentaton wll ether staff wth R max = R1 + z a or R + x R servers, dependng on whether t wll cross-sell or not. In contrast, the model wth segmentaton wll staff wth R1 + z servers; z<z a, unless t s proftable to cross-sell to all customer types. 3. Unform versus customzed prcng and proftablty. Most structural dfferences between the two systems orgnate from the prcng polces adopted by the call center n each case, and the correspondng expected revenue that they wll generate per customer that agrees to enter the cross-sellng phase. As explaned earler, the system that segments ts customers wll customze ts prces, p, for each type accordng to (11), whereas the system wth no segmentaton wll use a unform prce, p a, defned through (15). An mmedate consequence of the above s that r a = q K j=1 p a F p a 0 q K jq j j=1 p F p 0 jq j =1 Premultplyng by K j=1 jq j and subtractng out the correspondng capacty cost, we fnd that ( K j q j )r a c/ cs q r p c/ cs j=1 =1 =1 =1 q r p c/ cs + The rght-hand sde (RHS) of the above expresson s equal to the proft contrbuton due to cross-sellng n the system wth segmentaton, whch s clearly nonnegatve. Ths allows us to strengthen ths nequalty to the followng: ( K j q j )r a c/ cs + q r p c/ cs + (17) j=1 =1 where, n turn, the left-hand sde (LHS) of (17) s the proft contrbuton due to cross-sellng n the system wth no segmentaton. The above nequalty s strct as long as there exsts a type for whch p a F p a 0< p F p, whch by the defnton of p and the unmodalty of p F p 0, reduces to 1K for whch p p a (18) or equvalently, to j 1K such that p p j (19) Unless customer types have trval dfferences wth respect to ther wllngness to pay, condtons (18) or (19) are lkely to be satsfed, n whch case the ablty to segment the customer pool would lead to sgnfcant proft gans. For example, f the wllngness-to-pay dstrbutons for the varous types were exponental wth parameters b, then the above condtons would requre that at least two of these types had dfferent parameters b b j. If the dstrbutons were logstc wth scale parameters b (these are commonly used n the lterature n modellng dfferent customer segments), then agan (18) would requre that the parameters of at least two segments are dfferent. A smple extenson of our prevous results yelds the followng characterzaton of the potental value of customer segmentaton n the underlyng stochastc call center systems.

12 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton 310 Operatons Research 57(2), pp , 2009 INFORMS INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at Proposton 4. Under Assumpton 1, f18 or equvalently 19 holds, then for all, a =, where s the dfference of the RHS and LHS of 17 normalzed by. Moreover, f we let grow large, keepng / constant for all, then a = + o where a are the optmal expected proft rates for the underlyng stochastc systems wth and wthout segmentaton, respectvely. The above proposton together wth the results of Theorems 1 and 3 suggest that the staffng and cross-sellng polces proposed n ths paper would realze most of the proft dfferental that can be attrbuted to customer segmentaton. Operatonally, the latter also leads to more effcent capacty utlzaton because call centers that do not segment ther callers, but try to cross-sell to them, end up pursung too many customer prospects that are unlkely to lead to a sale. Our stylzed yet nsghtful analyss can be used to assess the magntude of ths potental beneft, whch s useful n decdng the value proposton of an nvestment n technology and agent tranng that would be needed to support a sophstcated customer segmentaton and crosssellng strategy. 6. Numercal Results Our results are organzed n three categores. The frst offers a representatve numercal llustraton of the accuracy of our asymptotc analyss. The second examnes the qualty of the proposed polces, and n partcular shows the senstvty of the system performance to changes n staffng and threshold levels that are used n the cross-sellng decsons. The last one gves some examples of the potental value of usng customer segmentaton n such a call center. For smplcty, we assume throughout ths secton that cs = cs for all. The Accuracy of Large-Scale Asymptotcs. We llustrate the accuracy of the proposed (S)-(C) heurstc by expermentng on a system wth four customer classes. The servce rates are s = 1 and cs = 2; one may regard all subsequent parameters as normalzed wth respect to s. The arrval rates are 1 = 2 = 1 and 3 3 = 4 = 1, 6 whereas the aggregate arrval rate,, wll be vared over a range of values n our experment. The product prces are exogenously gven and result n expected revenues per type- customer who goes through cross-sellng, gven by r 1 = 7, r 2 = 5, and r 3 = r 4 = 04, regardless of the realzed watng tme. 5 For smplcty, we assume that the customers wllngness-to-lsten functons are common across types and gven by the lnear functon q w = 1 01w +. The staffng cost s normalzed to c = 1, and for concreteness we consder the constraned formulaton wth an upper bound for the watng tme equal to 1/6; f the natural tme unts are mnutes, then ths upper bound s 10 seconds. Under ths choce of parameters, we have that z = 1 3 > 0 and k = 2,.e., the center wll cross-sell to types 1 and 2 only. These values of z and k and the above set of revenue and cost parameters gve = 267 as an upper bound on the system s proft rate. We have smulated the system behavor under three varants of the polcy (S)-(C) for rangng from 40 to 200. The frst varant s a drect translaton of the soluton of the determnstc relaxaton, wth a threshold 2 = 1 6 chosen accordng to the recommendaton n 3.2; recall that type 2 s the least proftable type to whch the system cross-sells. (For smplcty, we set 1 =,.e., the system would always cross-sell to type 1 customers.) The other two polcy varants had 2 and the staffng level N further optmzed va exhaustve smulaton,.e, by performng a search over all possble values of N. The smulaton code was wrtten n c++. Each sample path contaned 800,000 customer arrvals from whch we formed tme averages of the queue length and of the fracton of customers of each type that were cross-sold to. The length of each smulated path ensured that our estmates were close to the actual steady-state behavor. Frst, we note from Fgure 2(a) that the absolute devaton between the profts acheved through the three Fgure 2. Proft Scaled proft Performance of (S)-(C). (a) Realzed proft = Π(Λ) Upper bound +Threshold and staffng fne-tunng +Threshold fne-tunng (S)-(C) proft Arrval rate (b) Scaled proft = Π(Λ)Λ Arrval rate O( Λ)

13 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton Operatons Research 57(2), pp , 2009 INFORMS 311 INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at canddate polces as well as ther dfference aganst the determnstc upper bound ncreases wth the scale of the system, as measured by the aggregate call volume. However, Fgure 2(b) llustrates that f normalzed by, whch s the multplcatve factor by whch the above quanttes are growng, then the respectve dfference decays to zero. In fact, ths decay s of order 1/. The above fndngs are representatve of many examples that we tested. Second, we observe that as the sze of the system ncreases, most of the proft gans from fne-tunng the cross-sellng threshold parameter and staffng level can be attrbuted to the former. Ths s practcally appealng because t makes the model more robust to forecastng errors, because adjustments can be made onlne. The next set of results that we present studes ths ssue n more detal, and also revews the watng tme constrant qualfcaton. Performance Senstvty wth Respect to the Cross- Sellng Threshold and the Staffng Level. Fgure 3 offers a more detaled look at the effect of these two parameters to the center s proftablty and the steady-state expected watng tme experenced by ts callers for the system examned above for = 120. The parameters extracted from the determnstc relaxaton are z = 1/3 and k = 2, whch would translate to a nomnal staffng of N = 160 servers, and a nomnal threshold of 2 = W = 20;.e., the center would stop cross-sellng to type 2 customers when there are more than 20 customers n queue. Specfcally, Fgure 3(a) shows the dstance between the realzed proft and ts upper bound for varous values of 2 and N. Fgure 3(b) depcts the expected watng tme for each of these parameter combnatons; the respectve constrant requres that ths falls below It s worth notng that the center s proftablty s farly nsenstve to the staffng level around ts nomnal value of 160 servers because the effect of the latter can be compensated for by approprately adjustng the cross-sellng threshold. As expected, the watng tme s decreasng n the staffng level and ncreasng n the value of the crosssellng threshold;.e., more servers reduce the overall load, whereas hgher thresholds mply that the system s wllng to tolerate longer watng tmes. In fact, as expected from an nformal applcaton of Lttle s law, the expected watng tme ncreases almost lnearly as a functon of the threshold. The effect of the threshold on the proft s less sgnfcant, whch s consstent wth our asymptotc results that showed that (S)-(C) wth practcally any threshold level performs very close to the upper bound n large systems. Taken together, the above comments suggest that call centers of reasonably large sze can use the nomnal staffng level extracted through the determnstc analyss, and subsequently select the cross-sellng threshold to acheve constrant qualfcaton and mprove profts. The Value of Market Segmentaton. We conclude ths secton through a set of numercal experments that Fgure 3. Performance as functon of staffng and threshold levels Threshold Threshold 16 (a) Proft under (S)-(C) Staffng (b) Watng tme under (S)-(C) Staffng strve to llustrate the potental value of market segmentaton. The analyss here s crude n the sense that t s lmted to the determnstc relaxaton. The asymptotc performance guarantees and the numercal results presented above suggest that the proft gap between the respectve determnstc relaxatons wll persst n the stochastc systems as well. To facltate the presentaton of our results, we wll mostly focus on a two-type system, for whch s = 1, cs = 2, c = 1, = 100, and 1 = 2 = 05. The watng cost d or the watng tme upper bound W do not play any role Dstance from upper bound Watng tme

14 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton 312 Operatons Research 57(2), pp , 2009 INFORMS INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at n the determnstc analyss, and hence there s no need to specfy them. It remans to specfy the customer choce behavor. As n the prevous examples, we assume that the delay preferences of both types are the same wth q w = 1 01w +. Type- customers are assumed to have an exponentally dstrbuted wllngness to pay wth parameter b for whch F p = e b p, = 1 2. We assume that prces can obtan values on the bounded nterval 0 20 n each case. For the system that segments ts customers, the optmal prces are gven by p = 1/b 20 (where x y = mnx y), for whch r p = 1/b 20e 20b 1. Note that the optmal prce 1/b n the absence of the prce bound of $20 s equal to the average of the dstrbuton F, and that r p s lnear n 1/b as long as b 005. The soluton to the determnstc relaxaton wll cross-sell to type- provded that r p c/ cs, whch n ths model translates to b 074 =2/e, and that 1/b 136. The optmal prce for the system that cannot segment the two customer types does not admt a closed-form soluton, and s computed numercally usng (14) and (15). To test specfc numercal system nstances, we have generated 250 ndependent realzatons of the par (b 1 b 2 ) by drawng each of the b s ndependently from a unform dstrbuton on 0 2. For each realzaton of (b 1 b 2 ), we solved the determnstc relaxatons wth and wthout segmentaton. Ths nvolved computng the optmal prces, decdng to whch types to cross-sell, f any, calculatng the correspondng staffng level, and fnally the proft rate. Fgure 4 dsplays the relatve ncrease n profts, a / a, versus the maxmum of the average wllngness to pay among the two types, gven by max1/b 1 1/b 2. The average proft ncrease through segmentaton n ths two-class experment was around 24%. We have repeated ths experment several tmes, and n all of the experments the average proft ncrease was above 20%. Fgure 4 s rather ntutve. There wll be no proft gap between the two systems f b 1 = b 2 or f the b s are dfferent, but are such that no system decdes to Fgure 4. Relatve proft ncrease (%) Proft comparson of systems wth and wthout customer segmentaton. Value of segmentaton Max average wllngness to pay = max(1/b 1,1/b 2 ) cross-sell to any customer. In settngs where at least one type has a very large average wllngness to pay, both systems wll be very proftable n ther cross-sellng actvtes, and the relatve dfference n proft wll be small (the RHS of the fgure). In settngs where both parameters 1/b are small, then agan the proft dfferental wll be small because cross-sellng s barely compensatng for the cost of capacty. The dfference between the two systems s more pronounced when 1/b 1 and 1/b 2 are of moderate sze, n whch case the relatve added value from (a) prce customzaton and (b) selectve cross-sellng (.e., the capablty to cross-sell to only one of the two types) s sgnfcant. For example, 20% of the 250 nstances that we generated are such that the system wth segmentaton wll choose to only cross-sell to one type, whereas the system wth no segmentaton capablty wll not cross-sell at all. Fnally, as the number of customer types and the avalablty of nformaton on potental segmentaton ncreases, the overall proft contrbuton due to segmentaton becomes more substantal. In a set of experments that we ran wth four customer types, the average relatve proft ncrease was 40% (up from 24% for the system wth two types). Also, as the number of types was ncreased, we observed more nstances where the cross-sellng recommendatons of the two systems would dffer sgnfcantly. 7. Concludng Remarks To summarze, ths paper proposes a tractable determnstc relaxaton for studyng the varous control problems that arse n call center systems wth cross-sellng capablty, payng partcular attenton to the effect of customer segmentaton on the structure of the staffng, cross-sellng, and routng polces that the system may choose to adopt. The polces that are generated through ths analyss are smple to mplement, ntutve, and acheve near-optmal performance. Our analyss can be extended n several drectons to better model the operatonal complexty of modern call center systems, as well as that of customer behavor. In the former, ths may nclude systems that have multple pools of agents wth dfferent processng capabltes, as well as more complcated servce requrements, that may need a sequence of steps to be handled by the same or dfferent agents. Wth respect to the latter, one could allow the customer s decson of whether to lsten to the cross-sellng offer to nclude nformaton from the ntal phase of servce experenced by the customer, such as hs servce tme, whether hs ntal request was successfully resolved, etc. Another extenson would be to allow for customers to abandon the queue f ther watng tme s excessve. All of the above generalzatons ncrease the complexty of the underlyng system substantally, but can be addressed usng our approxmate analyss wth lttle addtonal effort. Fnally, an nterestng extenson would examne the staffng and control decsons n the face of nonstatonary arrval patterns or parameter estmaton and forecastng

15 Gurvch et al.: Cross-Sellng n a Call Center wth a Heterogeneous Customer Populaton Operatons Research 57(2), pp , 2009 INFORMS 313 INFORMS holds copyrght to ths artcle and dstrbuted ths copy as a courtesy to the author(s). Addtonal nformaton, ncludng rghts and permsson polces, s avalable at errors. Our asymptotc optmalty results n ths paper apply only to the statonary case wth known arrval rates. For ths settng, our asymptotc analyss and experence wth numercal examples show that only the smallest threshold, k, has an mportant effect on the system performance. Stll, the control (C) wth ts multple thresholds was desgned wth more general settngs n mnd. Indeed, t seems plausble that n settngs wth nonstatonarty and estmaton errors, these larger threshold wll play an mportant role by provdng the system wth a sgnfcant level of adaptablty and robustness. 8. Electronc Companon An electronc companon to ths paper s avalable as part of the onlne verson that can be found at nforms.org/. Endnotes 1. A recent study by McKnsey & Co. (Echfeld et al. 2006) suggests that bank call centers can generate revenues that are equvalent to 10% of the revenue generated through the retal branch channels. 2. In a recent study, a Purdue Unversty research group (Anton 2005) has estmated that call centers may attempt to cross-sell to as many as 60% of all ts callers. 3. The frst step nvolves the dentfcaton of approprate attrbutes along whch to segment the customer pool. The accuracy of the estmaton of the customer-type characterstcs wll be greatly mproved f the center can keep track of data on customers that refused to lsten to the cross-sellng offer, and on those that lstened but dd not buy. Fnally, there s a trade-off between the number of customer segments and the accuracy of ths estmaton procedure, whch may result n coarse segmentaton, as opposed to segmentng down to the level of each customer. 4. For example, f the problem of orgnal nterest has = 100 and W = 20 seconds, then W s selected so that W = W/, whch n ths case would gve W = 200 seconds. One should then study an asymptotc verson of (2) as grows large and W s scaled accordng to 200/ ; note that the orgnal formulaton s recovered for = Ths s equvalent to assumng that n ths case the wllngness to pay s ndependent of the realzed watng tme. References Akşn, O. Z., P. T. Harker To sell or not to sell: Determnng the trade-offs between servce and sales n retal bankng phone centers. J. Servce Res. 2(1) Anton, J Best practces n cross-sellng and up-sellng. benchmarkportal.com. Armony, M Dynamc routng n large-scale servce systems wth heterogeneous servers. Queueng Systems 51(3 4) Armony, M., I. Gurvch When promotons meet operatons: Crosssellng and ts effect on call-center performance. Workng paper, New York Unversty and Columba Unversty, New York. Armony, M., C. Maglaras. 2004a. 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