Promotion Optimization in Retail

Transcription

1 Promoon Opmzaon n Real Maxme C. Cohen NYU Sern School of Busness, New York, NY 10012, maxcohen@nyu.edu Georga Peraks MIT Sloan School of Managemen, Cambrdge, MA 02139, georgap@m.edu 1. Inroducon Ths chaper presens some recen developmens n real promoons. In many real sengs such as supermarkes, promoons are a key drver for boosng profs. Promoons are ofen used on a daly bass n mos real envronmens ncludng supermarkes, drugsores, fashon realers, elecroncs sores, onlne realers, convenence sores ec. For example, a ypcal supermarke sells several housands of producs, and needs o decde he prce promoons for all he producs a each me perod. These decsons are of prmary mporance, as usng he rgh promoons can sgnfcanly enhance he busness boom lne. In oday s economy, realers offer hundreds or even housands of promoons smulaneously. Promoons am o ncrease sales and raffc, enhance awareness when nroducng new ems, clear lefover nvenory, bolser cusomer loyaly, and mprove he realer compeveness. In addon, prce promoons are ofen used as a ool for prce dscrmnaon among he dfferen cusomers. Surprsngly, many realers sll employ a manual process based on nuon and pas experence n order o decde promoons. The unprecedened volume of daa ha s now avalable o realers presens an opporuny o develop suppor decson ools ha can help realers mprove promoon decsons. The promoon plannng process ypcally nvolves a large number of decson varables, and needs o ensure ha he relevan busness consrans (called promoon busness rules) are sasfed (more deals can be found n Secon 3.2). In hs chaper, we dscuss how analycs can help realers decde he promoons for mulple ems whle accounng for many mporan modelng aspecs observed n real daa. In parcular, we consder praccal models ha are movaed from a collaboraon beween academa and ndusry. Mos of he maeral dscussed n hs chaper s nspred by he resuls n Cohen e al. (2017) and n Cohen e al. (2018). For more deals on he specfcs of he algorhms, he proofs of he analycal resuls, and on he manageral nsghs, we refer he reader o he papers. 1

2 2 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes Several recen advances n operaons managemen and markeng have focused on developng new mehods o mprove he process of decdng real promoons. The ulmae goal s o ncrease he oal prof by promong he rgh ems a he rgh me perods usng he rgh prce pons. A a hgh level, real promoons can be caegorzed as follows: () manufacurer versus realer promoons, () markdowns versus emporary prce dscouns, () argeed versus mass campagns, and (v) prce reducons versus alernave promoon vehcles. We nex dscuss hese four caegorzaons. Manufacurer versus realer promoons: In real sengs, he brand manufacurer (e.g., Coca-Cola, Kellogg s) can drecly offer a prce dscoun eher o he realer or o he endconsumer. These ncenves are ofen called rade funds, vendor funds or manufacurer coupons. Ths ype of promoons usually come from long-erm negoaons beween he manufacurer and he realer, and nvolve several conracual erms. For example, a manufacurer can offer a rebae o he realer f he cumulave sales durng he quarer exceed a ceran arge level. In exchange, he realer wll place he manufacurer s producs n preferred locaons (e.g., end-cap-dsplays). A second example s a shared promoon conrac n whch he manufacurer subsdzes some poron of he prce dscoun offered o he consumers. A hrd example occurs when a manufacurer offers a coupon o he end-consumers who hen need o clam he dscoun (a he sore, on he Inerne or oward fuure purchases). Typcally, realers have o decde when o accep such vendor funds and under wha condons. In many suaons, manufacurers end o be aggressve on he conracual erms by mposng long-erm commmens, hgh volumes, and somemes exclusvy resrcons (e.g., no allowng he promoons of compeng brands). Markdowns versus emporary prce dscouns: Markdowns ypcally refer o he pracce of decreasng he prce of an em a he end of he sellng season. The regular prce s decreased n order o clear he remanng nvenory. Noe ha n such a case, he prce may be reduced several mes bu canno be ncreased back o he regular prce. Ths s common pracce n he fashon and oursm ndusres as well as n he busness of sellng ckes for meda evens (e.g., concers). For example, an apparel from he summer collecon may be dscouned oward he end of he season f he remanng nvenory s hgher han ancpaed. On he oher hand, emporary prce dscouns are used n dfferen conexs. A well-known such conex s of he Fas-Movng Consumer Goods (FMCG). Examples of FMCG nclude processed foods and sof drnks, as well as household producs (e.g., laundry deergen and oohpase). Noe ha hese producs are usually non-pershable, and have a long shelf lfe. Such purchases are recurrng, and realers do no need o clear he remanng nvenory. In order o ncrease he prof, became common for mos realers o use emporary prce reducons (e.g., 20% off he regular prce durng one week).

3 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes 3 Targeed versus mass campagns: Realers can eher decde o send promoons o a few argeed cusomers or o smply decrease he prce of a parcular produc for all he poenal buyers. Targeed markeng campagns can be mplemened va emal redeemable coupons or by usng advanced geo-localzaon echnques. Onlne realers ofen use argeed promoons by rackng poenal cusomers usng cookes, and by sendng promoonal offers o seleced ses of cusomers (e.g., acve members ha made a recen purchase). On he oher hand, mass promoons are prce dscouns ha apply o all cusomers. Brck-and-morar realers such as supermarkes manly employ mass promoon campagns. Prce reducons versus alernave promoon vehcles: Realers can use dfferen ways o promoe a produc. The mos sraghforward mehod s o use a prce dscoun, n whch he em s emporarly prced below s regular prce. Oher opons nclude buy-one-ge-one, nsore flyers, coupons, asng sands, placng producs a he end of an asle (end-cap-dsplay), sendng ou flyers, broadcasng TV commercals, rado adversemens, ec. (hese are ofen called promoon vehcles). Typcally, a realer can choose among 5-40 dfferen promoon vehcles a each pon n me. In hs chaper, we focus on he mass prcng promoon opmzaon problem faced by a realer who sells FMCG producs. Namely, we consder a realer (e.g., a supermarke) who needs o decde whch ems o promoe, a whch prce pons, and when o schedule he promoons of he dfferen ems. The problems of seng he rgh manufacurer ncenves, opmzng markdowns, desgnng argeed promoons, and opmzng promoon vehcles are also mporan real quesons, bu are beyond he scope of hs chaper. We wll brefly refer o some of he relevan leraure on hese problems n Secon 2. The amoun of money spen on promoons for FMCG producs can be sgnfcan - s esmaed ha FMCG manufacurers spend abou $1 rllon annually on promoons (Nelsen 2015). In addon, promoons play an mporan role n he FMCG ndusry as a large proporon of he sales s made durng promoons. For example, real daa ndcaes ha 12 25% of supermarke sales n fve European counres were made durng promoons (Gedenk e al. 2006). The marke research group IRI found ha more han half of all goods (54.6%) sold o UK shoppers n supermarkes and major realers were on promoon. 1 The promoon plannng process faced by a medum o large sze realer s challengng for several reasons. Frs, one needs o carefully accoun for he cross-em effecs n demand (cannbalzaon and complemenary). When promong a parcular em, he demand of some oher producs may also be affeced by he promoon. Consequenly, one needs o decde he promoons of all 1 hps://

4 4 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes he ems n he caegory whle accounng for hose effecs ha can be drecly esmaed from daa. Second, real promoons are ofen consraned by a se of busness rules specfed by he realer and/or he produc manufacurers. Example of busness rules nclude prces chosen from a se of dscree values, lmed number of promoons (boh per me perod and for each em), and cross-em busness rules ha resrc he relaonshp beween he prces of he dfferen ems (more deals are provded n Secon 3.2). Thrd, he demand usually exhbs a pos-promoondp effec. Ths effec s nduced by he promoon fague (.e., repeang he same promoon may have a low margnal mpac), and by he sockplng behavor of consumers. More precsely, for ceran caegores of (non-pershable) producs, cusomers end o sockple durng promoons by purchasng larger quanes for fuure consumpon. Ths ulmaely leads o a reduced demand followng he promoon perod. Fourh, he problem s dffcul due o s large scale. As we menoned, an average supermarke offers several housands of SKUs (Sock Keepng Uns), and he number of ems on promoon a any me can be very large. Consequenly, hs leads o a large number of decsons ha need o be made by he realer. Real promoons can have a sgnfcan mpac on boosng sales, and on nfluencng cusomers. For example, a sudy from he Inernaonal Councl of Shoppng Ceners shows ha 90% of adul consumers clam o be nfluenced by promoons n erms of he amoun hey spend, and he ems hey purchase. 2 Despe he complexy of he promoon plannng process, s sll o hs day performed manually n many supermarke chans. Ths movaes us o desgn and sudy promoon opmzaon models ha can make promoon plannng more effcen and auomaed. The goals of hs lne of research nclude he followng: Formulae he promoon opmzaon problem for mulple ems (labeled as Mul-POP). Ths formulaon s drecly movaed from pracce, holds for general demand models (esmaed from daa), and can ncorporae he relevan busness rules. Dscuss how he formulaon capures several mporan economc facors whch are presen n real envronmens. These facors nclude he pos-promoon dp effec (due o he sockplng behavor of consumers), he cross-em effecs, and he demand seasonaly. Develop an effcen approxmaon soluon approach o solve he problem. We propose a dscree lnearzaon mehod ha allows he realer o solve a large scale nsance of he problem whn seconds. We also convey ha our soluon approach provdes a paramerc wors-case bound on he qualy of he approxmaon relave o he opmal (nracable) soluon. 2 hps://realleader.com/brck-and-morar-makes-grade-back-school-shoppng

5 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes 5 Presen a begnnng-o-end applcaon of he enre process of opmzng real promoons. We dvde he process n fve seps ha he realer needs o follow; from collecng and aggregang he daa o compung he fuure promoon decsons. Dscuss he poenal mpac of usng daa analycs and opmzaon for real promoons. We convey ha n our esed examples (calbraed wh real daa), usng he promoons suggesed by our model can yeld a 2-9% prof mprovemen. Such an ncrease s sgnfcan, as real busnesses ypcally operae under small margns. Ths chaper s organzed as follows. In Secon 2, we revew some of he relaed leraure. In Secon 3, we repor he noaon, assumpons, and problem formulaon. In Secon 4, we presen a class of approxmaon mehods o effcenly solve he promoon opmzaon problem. In Secon 5, we use our model and soluon approach o draw praccal nsghs on promoon plannng, and presen a summary of how o apply our model o real-world real envronmens. Fnally, we repor our conclusons n Secon 6. As menoned before, more deals on he echncal resuls and on he nsghs can be found n Cohen e al. (2017) and n Cohen e al. (2018). 2. Leraure Revew The opc of real promoons has been an acve research area boh n academa and ndusry. In parcular, our problem s relaed o several sreams of leraure, ncludng dynamc prcng, promoons n markeng, and real operaons. Dynamc prcng: Dynamc prcng has been an exensve opc of research n he operaons managemen communy. Comprehensve revews can be found n he books and revew papers by Bran and Caldeney (2003), Elmaghraby and Kesknocak (2003), Tallur and Van Ryzn (2006), Özer and Phllps (2012), as well as he references heren. A large number of recen papers sudy he problem of dynamc prcng under varous conexs and modelng assumpons. Examples nclude Ahn e al. (2007), Su (2010) and Levn e al. (2010), jus o name a few. In Ahn e al. (2007), he auhors propose a demand model n whch a proporon of cusomers sraegcally wa k perods, and purchase he produc once he prce falls below her wllngness o pay. They hen formulae a mahemacal programmng model, and develop soluon echnques. In Su (2010), he auhor sudes a model wh mulple consumer ypes who may dffer n her holdng coss, consumpon raes, and fxed shoppng coss. The auhor solves he dynamc prcng model by compung he raonal expecaon equlbrum, and draws several manageral nsghs. In Levn e al. (2010), he auhors consder a dynamc prcng model for a monopols who sells a pershable produc o sraegc consumers. They model he problem as a sochasc dynamc game, and prove he exsence of a unque subgame-perfec equlbrum prcng polcy. A very promnen opc n he dynamc prcng leraure s o sudy he seng n whch consumers are sraegc (or forwardlookng) (see, e.g., Avv and Pazgal 2008, Cachon and Swnney 2009, Levna e al. 2009, Besbes and

6 6 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes Lobel 2015, Lu and Cooper 2015, Chen and Faras 2015). The problem consdered n hs chaper s n he same spr as he dynamc prcng problem. Neverheless, we focus on a seng where he demand model s esmaed from hsorcal daa, and he opmzaon formulaon ncludes he smulaneous promoon decsons of several nerconneced ems. In addon, we requre he dynamc prcng decsons o sasfy several busness rules. Promoons n markeng: Sales promoons are an mporan area of research n markeng (see Blaberg and Nesln (1990) and he references heren). However, he focus n he markeng communy s ypcally on modelng and esmang dynamc sales models (economerc or choce models) ha can be used o draw manageral nsghs (Cooper e al. 1999, Foekens e al. 1998). For example, Foekens e al. (1998) sudy economercs models based on scanner daa o examne he dynamc effecs of sales promoons. I s wdely recognzed n he markeng communy ha for ceran producs, promoons may have a panry-loadng or a pos-promoon dp effec,.e., consumers end o purchase larger quanes durng promoons for fuure consumpon (sockplng behavor). Ths effec leads o a decrease n sales n he shor erm. In order o capure he pospromoon dp effec, many of he dynamc sales models n he markeng leraure pos ha he demand s no only a funcon of he curren prce, bu also of he pas prces (see, e.g., Alawad e al. 2007, Macé and Nesln 2004). Fnally, noe ha several prescrpve works n he markeng communy sudy he mpac of real coupons n he conex of sales promoons (see, for example Helman e al. 2002). The demand models used n hs chaper also consder ha he demand depends explcly on he curren and pas prces as well as on he prces of oher ems. Real operaons: Several academc papers sudy he opc of real promoons from an emprcal descrpve pon of vew. Van Heerde e al. (2003) and Marínez-Ruz e al. (2006) use panel-daa o emprcally sudy how real promoons nduce consumers o swch brands. The recen work by Felgae and Fearne (2015) uses supermarke loyaly card daa from a sample of over 1.4 mllon UK households o analyze he effec of promoons on he sales of specfc producs across dfferen shopper segmens. Anoher lne of research dscusses feld expermens on prcng decsons mplemened a realers. A classcal successful example s he mplemenaon a he fashon real chan Zara (see Caro and Gallen 2012). In her work, he auhors repor he resuls of a conrolled feld expermen conduced n all Belgan and Irsh sores durng he 2008 fall-wner season. They assess ha he new process has ncreased clearance revenues by approxmaely 6%. An addonal recen work can be found n Ferrera e al. (2015) n whch he auhors collaboraed wh Rue La La, a flash sales fashon onlne realer. The auhors propose a non-paramerc predcon model o predc fuure demand of new producs, and develop an effcen soluon for he prce opmzaon problem. They esmae a revenue ncrease for he es group by approxmaely 9.7%. Pekgün e al. (2013) descrbe a collaboraon wh he Carlson Rezdor Hoel Group. In hs sudy, he auhors

7 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes 7 show ha demand forecasng and dynamc revenue opmzaon conssenly ncreased revenue by 2-4% n parcpang hoels relave o non-parcpang hoels. Oher ypes of promoons: As menoned before, real promoons can be dvded n several caegores. Whle he models presened n hs chaper focus on he mass prcng promoon opmzaon problem faced by a realer who sells FMCG producs, oher sudes have consdered he alernave promoon ypes. Several papers consder he problem of vendor funds n he conex of promoon plannng (see, e.g., Slva-Rsso e al. 1999, Njs e al. 2010, Yuan e al. 2013, Baardman e al. 2017b). As menoned before, an addonal relaed opc s he one of markdown prcng, or markdown opmzaon. In hs problem, he seller needs o decde when o decrease he prce of he em(s) n order o clear he remanng nvenory by he end of he season. There s a large number of academc papers ha propose dfferen models and mehods o solve he markdown prcng problem. Examples nclude Yn e al. (2009), Mersereau and Zhang (2012), Zhang and Cooper (2008), Vakhunsky e al. (2012), and Caro and Gallen (2012), jus o name a few. As we explaned before, he promoon opmzaon problem for FMCG producs dffers from he markdown opmzaon problem by he srucure of he prcng polcy and by he lack of nvenory expraon. The opc of desgnng argeed promoons has recenly araced a lo of aenon. Gven ha sendng promoons o exsng or new cusomers can be expensve and ofen resuls n low converson raes, several frms am o develop quanave mehods ha explo he large hsorcal daa ses n order o desgn argeed promoon campagns. For example, realers ofen need o decde whch ypes of cusomers o arge, and wha are he mos mporan feaures (e.g., geo-localzaon, demographcs, and pas behavor). Targeed markeng campagns (emal and moble offers) have been exensvely suded n he academc leraure (see, e.g., Arora e al. 2008, Fong e al. 2015, Andrews e al. 2015, Jagabahula e al. 2018). Fnally, n addon o prce promoons, realers ypcally need o decde how o assgn he dfferen vehcles (e.g., flyers and TV commercals). The recen work n Baardman e al. (2017a) addresses he problem of opmally schedulng promoon vehcles for a realer. Mehodology: From a mehodologcal perspecve, he ools used n hs chaper are relaed o he leraure on nonlnear and neger opmzaon. We formulae he promoon opmzaon problem as a nonlnear mxed neger program (NMIP). Due o he general classes of demand funcons we consder, he objecve funcon s ypcally non-concave, and such NMIPs are generally dffcul from a compuaonal complexy sandpon. Under ceran specal srucural condons (see, e.g., Hemmecke e al. (2010) and he references heren), here exs polynomal me algorhms for solvng NMIPs. However, many NMIPs do no sasfy hese condons and are solved usng echnques such as Branch and Bound, Ouer-Approxmaon, Generalzed Benders and Exended Cung Plane mehods (Grossmann 2002).

8 8 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes In he specal nsance of he Mul-POP wh lnear demand and connuous prces, one can formulae our problem as a Cardnaly-Consraned Quadrac Opmzaon (CCQO) problem. I has been shown n Bensock (1996) ha such a problem s NP-hard. Thus, alored heurscs have been developed n order o solve hs ype of problems (see, for example, Bensock 1996, Bersmas and Shoda 2009). The general nsance of our problem has dscree varables, and consders a general demand funcon. Noe ha our problem was also shown o be NP-hard (Cohen e al. 2016). Our soluon approach s based on approxmang he objecve funcon by explong he dscree naure of he problem. Gven ha we consder general demand funcons, s no possble o use lnearzaon approaches such as n Sheral and Adams (1998). Our man approxmaon mehod resuls n a formulaon whch s relaed o he feld of Quadrac Programmng. Such problems were exensvely suded n he leraure (see, e.g., Frank and Wolfe 1956, Balnsk 1970, Rhys 1970, Padberg 1989, Nocedal and Wrgh 2006). 3. Problem Formulaon In hs secon, we formulae he promoon opmzaon problem (labeled as Mul-POP). We frs nroduce he noaon and our assumpons. We hen dscuss he varous busness rules ha he realer needs o sasfy when decdng prce promoons. Fnally, we presen he resulng opmzaon formulaon. Consder a realer who sells several FMCG producs. Very ofen, realers decde he prce promoons of her producs for each caegory separaely. Consequenly, we focus our presenaon on a sngle caegory (e.g., ground coffee, sof drnks) composed of N ems (or SKUs). The goal of he caegory manager s o maxmze he oal prof over a sellng horzon composed of T perods (for example, one quarer of 13 weeks). We denoe by p he prce of em a me. 3 We also denoe by c he (exogenous) cos of a sngle un of em a me. In oher words, we assume ha he cos of each em a each me s known, and ha he realer needs o decde he prces of all N ems durng all T me perods. A summary of our noaon can be found a he end of hs secon Assumpons To gan racably, we mpose he followng assumpons. Assumpon The realer decdes all he prce promoons a he begnnng of he season. 2. The realer carres enough nvenory o mee he demand for each em n each me perod The demand s expressed as a deermnsc me-dependen nonlnear funcon of he prces. 3 Throughou hs chaper, he subscrp (resp. superscrp) ndex corresponds o he me (resp. em). 4 We herefore use he words demand and sales nerchangeably.

9 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes 9 4. The demand funcon depends explcly on self pas and curren prces, and on cross curren prces. We nex brefly dscuss he valdy of he above assumpons. Assumpon 1.1 apples o a seng where he realer needs o comm upfron for he enre sellng season. For example, such resrcons can emerge from vendor funds or can be mposed by sendng ou flyers hrough dfferen adversng channels. Noe ha Assumpon 1.2. does no apply o all producs and real sengs (e.g., very ofen n he fashon ndusry, lmed nvenory s produced o nduce scarcy). Unlke fashon ems whch may be seasonal, FMCG producs are ypcally avalable all year round. These producs have a long shelf lfe, and cusomers have been condoned o always fnd hese producs n sock a real sores. Snce FMCG producs are usually easy o sore and have a hgh degree of avalably, FMCG realers ypcally do no sock ou. In Cohen e al. (2017), he auhors analyze wo years of supermarke daa for FMCG producs, and convey ha () he demand forecas accuracy for hs ype of producs s ofen hgh (good ou-of-sample R 2 and MAPE), and () he nvenory s no ssue as very few sock-ous occurred over a wo-year perod. Ths can be jusfed by he fac ha supermarkes have a long experence wh nvenory decsons, and colleced large daa ses allowng hem o develop sophscaed forecasng demand ools o suppor orderng decsons (see, e.g., Cooper e al. 1999, Van Donselaar e al. 2006). Fnally, grocery realers are aware of he negave effecs of beng ou of sock for promoed producs (see, e.g., Corsen and Gruen 2004, Campo e al. 2000). However, for sengs where nvenory s lmed, one needs o consder a dfferen formulaon han he one presened n hs chaper. Assumpon 1.3 ranslaes o denong he demand of em a me by d (p), where p s a vecor of curren and pas prces (see more deals below). We assume ha he demand s a deermnsc funcon as we observed a hgh ou-of-sample predcon accuracy usng our daa. Exendng our model when he demand s a sochasc funcon s an neresng drecon for fuure research (e.g., by usng learnng algorhms). Assumpon 1.4 mples ha he demand does no explcly depend on cross pas prces. In oher words, he demand of em does no depend on he pas prces of he oher ems n he caegory. Ths assumpon was valdaed by runnng demand predcon models usng real daases (more deals can be found n Cohen e al. 2018). Consequenly, he demand of em a me can ( ) be any nonlnear and me dependen funcon of he form: d p, p 1,..., p, p M, where M represens he memory parameer of em (.e., he number of pas prces ha affec he curren demand), and p denoes he vecor of prces of all he ems excep a me. Noe ha n pracce M s esmaed from he hsorcal daa, and can be dfferen across ems. Noe ha he demand of em a me depends on several facors:

10 10 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes The self curren prce p Ths capures he prce sensvy of he consumers oward he em. The self pas prces ( p 1,..., p M ) Ths capures he pos promoon dp effec (nduced by he sockplng behavor of consumers). The cross curren prces p and complemenary). Ths capures he cross-em effecs on demand (subsuon Oher poenal feaures such as demand seasonaly (weekly, monhly or quarerly), rend facor, sore effec, holday booss, ec. Concree demand models such as he log-log demand funcon can be found n Cohen e al. (2017). In mos produc caegores, a promoon for a parcular em affecs s own sales, bu also he sales of oher ems n he caegory. As menoned, we capure hese cross-em effecs by assumng ha he demand of em depends on he prces of he oher ems (a he same me perod). The sandard example of subsuable ems are compeng brands such as Coke and Peps. In hs case, s clear ha promong a Coke produc poenally ncreases Coke s sales bu may also decrease Peps s sales. Mahemacally, one can assume ha f ems and j are subsues, hen d 0 p j and dj 0 for some. Two producs and j are complemens f he consumpon of nduces p cusomers o purchase em j (and vce versa), e.g., shampoo and condoner. Mahemacally, one can assume ha f ems and j are complemens, hen d 0 and dj p j 0 for some. p 3.2. Busness Rules In he real seng we consder, here are ypcally wo classes of busness rules: () busness rules on each em separaely (called self busness rules); and () busness rules ha mpose jon prcng consrans on several ems (called cross-em busness rules). The self busness rules are dencal o he ones presened n Cohen e al. (2017), whle he cross-em busness rules are smlar o Cohen e al. (2018). Self busness rules 1. Prces are chosen from a dscree prce ladder. For each produc, here s a fne se of permssble prces. In parcular, we consder ha each em = 1,..., N can ake several prces: he regular prce denoed by q 0, and K = Q 1 promoon prces denoed by q k. The oal number of prce pons for em s called he sze of he prce ladder (denoed by Q ). 5 Consequenly, he prce of em a me can be wren as p = K k=0 qk γ k, where he bnary decson varable γ k s equal o 1 f he prce of em a me s seleced o be q k, and 0 oherwse. 5 For smplcy, we assume ha he elemens of he prce ladder are me ndependen, bu our resuls sll hold when hs assumpon s relaxed. In addon, we assume whou loss of generaly ha he regular non-promoon prce q 0 = q 0 s he same across all ems = 1,..., n and all me perods (hs assumpon can be relaxed a he expense of a more cumbersome noaon).

11 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes Lmed number of promoons. The realer may wan o lm he promoons frequency for a produc n order o preserve he mage of her sore, and no ran cusomers o be deal-seekers. For example, may be requred o promoe em a mos L = 3 mes durng he quarer. Ths requremen for em s capured by he followng consran: T =1 K k=1 γk L. 3. Separang perods beween successve promoons (no-ouch consran). A common addonal requremen s o space ou wo successve promoons by a mnmal number of separang perods, denoed by S. Ths consran also helps realers preserve her sore mage and dscourage consumers o be deal-seekers. In addon, hs ype of requremen may be dcaed by he manufacurer ha somemes resrcs he frequency of promoons n order o preserve he brand mage. Such a requremen for em ranslaes o addng he followng consran: +S Cross-em busness rules τ= K k=1 γk τ Toal lmed number of promoons. The realer may wan o lm he oal number of promoons hroughou he sellng season. For example, a mos L T = 20 promoons may be allowed durng he season. Mahemacally, one can mpose he followng consran: Noe ha L T should sasfy L T < N T K γ k L T. (1) =1 =1 k=1 N L for hs consran o be relevan. =1 2. Iner-em ordnal consrans. Several prce relaons can be dcaed by busness rules. For example, smaller sze ems should have a lower prce relave o smlar larger-szed producs, and naonal brands mus be more expensve when compared o prvae labels. These consrans can be capured by lnear nequales among he prces (e.g., f em should be prced no hgher han em j, one can add he consran: p p j ). 3. Smulaneous promoons. Somemes, realers requre parcular ems o be promoed smulaneously as par of a manufacurer ncenve or a specal promoonal even. If ems and j should be promoed smulaneously, one can mpose: γ 0 = γ 0j, where γ 0 ha s equal o 1 f em (resp. em j) s no promoed a me. (resp. γ 0j ) s a bnary varable 4. Lmed number of promoons n each perod. One can mpose a lmaon on he number of promoons n each me perod. For example, a mos C = N 10 only a mos 10% of he ems. Mahemacally, we have: =1 k=1 promoons may be allowed.e., N K γ k C. (2) 5. Cross no-ouch consrans. An addonal requremen can be o space ou he promoons of a se of smlar ems by a mnmal number of separang perods, denoed by S c. As before, hs s

12 12 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes movaed by he wsh o preserve he sore mage and o mgae he ncenves for consumers o be deal-seekers. In hs case, we need o separae successve promoons for wo (or more) producs. Mahemacally, one can mpose: +S c K γ k τ 1, where he sum on can be over any gven τ= k=1 subse of ems n he caegory. Noe ha when S c = 0, hs corresponds o never promong he ems smulaneously n order o mpose an exclusve offer (very common n pracce) Problem Formulaon In wha follows, we presen he promoon opmzaon problem for mulple ems: max γ k N =1 =1 K s.. p = T ( ) (p c )d p, p 1,..., p M, p k=0 K T =1 k=1 +S K τ= k=1 K k=0 N q k γ k γ k L γ k τ 1, γ k = 1, T K =1 =1 k=1 K N =1 k=1 γ k γ k L T γ k C {0, 1},, k (Mul-POP) In hs problem, he objecve s o maxmze he oal prof from all he N ems durng he sellng season. Noe ha n he formulaon above, we nclude all he self busness rules, as well as he consrans on he oal lmed number of promoons from (1), and on he lmed number of promoons n each perod from (2). One can naurally nclude addonal cross-em busness rules no he formulaon, dependng on he requremens. I s worh menonng ha even n he absence of cross-em busness rules, he N ems are lnked hrough he cross-em effecs presen n he demand funcons. Summary of Noaon: T - Lengh of he sellng season. N - Number of dfferen ems n he caegory. c - Cos of em a me (assumed o be known). p - Prce of em a me (decson varable).

13 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes 13 p d - Vecor of prces of all ems bu a me. ( ) p, p 1,..., p, p M - Demand of em a me, whch s assumed o be a funcon of he self curren and pas prces as well as of he cross curren prces (esmaed from daa). M - Memory parameer of em,.e., he number of pas prces ha affec he curren demand (esmaed from daa). L - Lmaon of he number of promoons for em. S - No-ouch perod for em,.e., he mnmal number of me beween wo successve promoons. K - Number of promoon prces n he prce ladder of em. q 0 - Regular prce (assumed o be he same across he dfferen ems). Q = K Toal number of possble prces for em. q k - Prce pon k for em (k = 1,..., K ). γ k - Bnary decson varable o ndcae f he prce of em a me s equal o q k. M P OP - Objecve funcon of he (Mul-POP) problem,.e., he oal prof generaed by all ems a all mes. SP OP - Objecve funcon of he problem for a sngle em. 4. Soluon Approach Our goal s o solve he opmzaon problem (Mul-POP). Snce he problem s a nonlnear Ineger Program, solvng he formulaon effcenly s no sraghforward. Consequenly, we develop an approxmaon soluon approach. The requremens are wofold: () he soluon mehod needs o be effcen and o run fas, and () he approxmaon soluon needs o be near opmal. In real sengs, realers ypcally solve he (Mul-POP) problem for a large number of ems. In addon, realers ofen solve several nsances of he problem n order o es he robusness of he soluon before mplemenng. More precsely, hese roune ess are called wha-f scenaros. They conss of solvng perurbed versons of he nomnal opmzaon problem, where some of he demand parameers and some of he busness are rules are slghly modfed (more deals are dscussed n Secon 5.2). In wha follows, we descrbe he soluon approaches developed n Cohen e al. (2017) and n Cohen e al. (2018) Sngle Iem Seng We frs presen an effcen soluon approach o solve he sngle em problem. Whle he mos neresng and relevan case s he problem wh mulple ems, he sngle em seng s used as a sarng pon for he presenaon, and s neresng n s own rgh. In some real caegores, he dfferen ems can be ndependen,.e., he demand of each em depends solely on s prces, and no on he prces of he oher ems. In hs case, he (Mul-POP) problem decomposes n N

14 14 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes ndependen sngle em problems (assumng ha here are no cross-em busness rules), and one can solve each problem separaely. Even n he case of a sngle em, he problem s hard o solve (he problem s shown o be NP-hard n Cohen e al. 2016). We observe ha he consrans n he (Mul-POP) formulaon are lnear. However, he objecve funcon s nonlnear, and usually neher concave nor convex, as we do no wan o mpose resrcons on he form of he demand funcons. Ths movaes us o propose a way o approxmae he objecve funcon by usng a lnear approxmaon, and by explong he dscree naure of he problem. In parcular, we approxmae he objecve funcon by he sum of he margnal conrbuons of havng a sngle promoon a a me. For example, f he em s on promoon a mes 2, 3, and 7, we approxmae he objecve by he sum of he margnal devaons of havng a sngle promoon a me 2, a sngle promoon a me 3, and a sngle promoon a me 7. We nex presen hs approach, called App(1), n more deal. The App(1) approxmaon mehod gnores he second-order neracons beween promoons and capures only he drec effec of each promoon. Snce we consder he same se of consrans as n he orgnal problem, he soluon remans feasble. We nex nroduce some addonal noaon. We consder a parcular em, and hence we drop he superscrp n he remanng of hs subsecon. For a gven prce vecor p = (p 1,..., p T ), we defne he correspondng oal prof (of he em under consderaon) hroughou he season: Nex, we defne he prce vecor p k SP OP (p) = T (p c )d (p ). =1 such ha he promoon prce q k s used a me, and he regular prce q 0 (no promoon) s used a all he remanng perods. We denoe he regular prce vecor by p 0 = (q 0,..., q 0 ), for whch he regular prce s se a all he me perods. We defne he coeffcens b k as follows: b k = SP OP (p k ) SP OP (p 0 ). (3) The coeffcens n (3) represen he unlaeral devaons n he oal prof by usng a sngle promoon. One can compue hese T K coeffcens before sarng he opmzaon procedure so ha does no affec he complexy of he mehod. The approxmaed objecve funcon s hen gven by: SP OP (p 0 ) + max γ k T =1 k=1 K b k γ k, (4) whle he se of consrans s he same as n he orgnal problem. Consequenly, he approxmaon opmzaon problem s lnear, and can be solved usng a solver. As menoned before, wo

15 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes 15 mporan requremens for our soluon approach are () a low runnng me, and () a close o opmal soluon. We nex summarze he properes (boh heorecal and praccal) for he sngle em seng. Summary for he sngle em seng: We solve he promoon opmzaon problem for a sngle em by usng he App(1) approxmaon. Ths approxmaon lnearzes he objecve soluon by compung he sum of he margnal conrbuons of each promoon separaely. The followng properes hold: The formulaon s negral,.e., one can solve he problem by consderng he Lnear Programmng (LP) relaxaon. Under wo general demand models whch are dscussed below (mulplcave and addve), we derve a paramerc wors-case bound on he qualy of he approxmaon relave o he opmal prof. In many esed nsances (calbraed wh real daa), he approxmaon yelds a soluon whch s opmal or very close o opmal. We nex dscuss he mplcaons of he above summary. Snce one can ge a soluon by solvng an LP, he approach s effcen (we can solve large nsances n mllseconds). Consequenly, he realer can use hs approach n praccal sengs. The approach works for general demand funcon, and for any objecve funcon. If we furher mpose some srucure on he demand funcon, we can derve a paramerc bound on he qualy of he approxmaon. We do so by consderng wo general classes of demand funcons: 1. Mulplcave demand: ( ) d p, p 1,..., p M = f (p ) g 1 (p 1 ) g 2 (p 2 ) g M (p M ), (5) where he demand (of he em under consderaon) can be wren as he produc of M funcons ha each depends on a sngle prce. Noe ha snce we consder a sngle em seng, he demand does no depend on he prces of he oher ems. The class of demand funcons n (5) ncludes he log-log and he log-lnear funcons, whch are commonly used n real. 2. Addve demand: ( ) d p, p 1,..., p M = f (p ) + g 1 (p 1 ) + g 2 (p 2 ) g M (p M ), (6) where he demand (of he em under consderaon) can be wren as he sum of M funcons ha each depends on a sngle prce. Noe ha he class of demand funcons n (6) ncludes he lnear funcon as a specal case. For he wo classes of demand funcons presened above, one can derve bounds on he qualy of he App(1) approxmaon. These bounds explcly depend on he problem parameers, and depc a very hgh performance on all he nsances we esed based on real daa. More deals can be found n Cohen e al. (2017).

16 16 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes 4.2. Mulple Iems In hs secon, we consder he more general seng where he realer needs o decde he prces of N nerconneced ems by solvng he (Mul-POP) problem. Recall ha n hs case, a promoon n em may have an effec on he demand of em j. The cross-em effecs on demand can be drecly esmaed from daa. A poenal smple approach can be he followng: Solve he (Mul-POP) problem by applyng he App(1) soluon approach,.e., approxmae he objecve by he sum of he margnal conrbuons of each em a each me perod (as dscussed n Secon 4.1). We esed hs approach, and observed a poor performance (especally n cases where he cross-em effecs are sgnfcan). In parcular, fals o accuraely capure he cross-em effecs, and may fnd a promoon sraegy far from opmal. For example, may sugges o promoe wo ems smulaneously, whereas hs par of ems hghly cannbalze each oher. As a resul, one needs o develop an alernave soluon approach ha can capure he cross-em effecs, and a he same me remans effcen. We nroduce he followng sequence of mehods, App(κ), for any gven κ = 1, 2,..., N. App(1) s he approxmaon appled o (Mul-POP) n a smlar fashon as n he sngle em seng dscussed n Secon 4.1. In parcular, approxmaes he objecve funcon by he sum of he margnal conrbuons of a sngle promoon for each em and me perod separaely. As we prevously dscussed, n he case of mulple ems, wll generally provde a poor performance guaranee relave o he opmal soluon. App(2) s an alernave approxmaon appled o (Mul-POP) ha ncludes he margnal conrbuons (same as App(1)), as well as he parwse conrbuons (.e., havng wo ems promoed a he same me). App(2) s descrbed n full deals below. App(N) s an alernave approxmaon ha ncludes he margnal conrbuons, he parwse conrbuons, and so on, up o all he possble combnaons of havng he N ems promoed smulaneously. One can also naurally consder any nermedae mehod for 2 < κ < N. Noe ha here s a clear rade-off beween smplcy (as well as speed) and performance (n erms of accuracy of he approxmaon relave o he opmal soluon). On one exreme, App(1) s a smple approach ha only requres compung he margnal conrbuons of havng a sngle promoon a a me, bu can perform poorly as does no capure he cross-em effecs a all. On he oher exreme, App(N) s clearly more accurae, as successfully capures all he cross-em effecs. Bu hs benef comes a he expense of beng more complex, as one needs o compue he margnal conrbuon of each possble combnaon of ems ha could be promoed smulaneously. In parcular, requres us o compue an exponenal number of coeffcens, and o solve an Ineger Program (IP) ha grows exponenally wh he number of ems. Noe ha when T = 1 or M = 0, App(N) s exac

17 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes 17 as capures accuraely all he cross-em effecs. Neverheless, for a general dynamc problem wh T > 1 perods and non-zero memory parameers, App(N) s sll no an exac algorhm, as approxmaes he me effecs nduced by he pas prces. We nex descrbe App(2) n more deals as wll be used n he sequel. As we prevously menoned, App(2) approxmaes he objecve of (Mul-POP) by he sum of unlaeral devaons (.e., havng a sngle promoon a a me) and he parwse conrbuons (.e., havng wo ems promoed smulaneously). More precsely, he approxmaed objecve s: where he coeffcens b k { N MP OP (p 0 ) + max γ =1 and b klj T K b k γ k + =1 k=1 N T K K j,j:>j =1 k=1 l=1 b klj } γ klj, (7) are formally defned n equaons (8) and (9) respecvely. We denoe he regular prce vecor by p 0 = (q 0,..., q 0 ), whch means ha he regular prce s se for all ems a all mes. The frs erm, denoed by MP OP (p 0 ), represens he oal prof generaed by all he ems hroughou he sellng season, whou any promoon. The second erm capures all he margnal conrbuons of havng a sngle promoon,.e., for one em a one me perod. More precsely, we defne he prce vecor p kj { In oher words, he vecor p kj (p kj ) τ = as follows: q kj ; f τ = and = j, q 0 ; oherwse. has he promoon prce q kj for em j a me, and he regular prce q 0 (no promoon) s used a all he remanng perods for em j, and for all he oher ems a all mes. The coeffcen b kj s hen gven by: b kj = MP OP (p kj ) MP OP (p 0 ), (8) and represens he margnal conrbuon n he oal prof by havng a sngle promoon for em j a me, usng prce q kj. The hrd erm n equaon (7) represens all he parwse conrbuons of havng wo ems on promoon a he same me. More precsely, we defne he prce vecor p klju j > u as follows: In oher words, he vecor p klju q kj ; f τ = and = j, ) τ = q lu ; f τ = and = u, q 0 ; oherwse. (p klju for any par of ems uses he promoon prce q kj for em j a me, he promoon prce q lu for em u a me, and he regular prce q 0 for ems j and u n all he remanng perods, and for all he oher ems a all mes. The coeffcen b klju b klju s gven by: = MP OP (p klju ) MP OP (p kj ) MP OP (p lu ) + MP OP (p 0 ), (9)

18 18 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes and represens he margnal parwse conrbuon n he oal prof by havng wo smulaneous promoons. Fnally, n order o make he formulaon conssen, we should ensure ha when boh ems and j are on promoon, we coun he parwse conrbuon bu also boh unlaeral devaons,.e., for each par of ems and j <, γ k = γ lj = 1 f and only f γ klj = 1 for each and k, l. One can encode hs se of condons by ncorporang he followng consrans o he formulaon for each par of ems, j <, each, and each promoon prces q k and q lj : γ klj γ klj γ k, γ lj, γ klj 0, γ klj γ k + γ lj 1. When maxmzng he objecve of he approxmaed problem n equaon (7), he decsons are he bnary varables γ. In parcular, here s one such varable for each em/me/prce (.e., NT (K + 1), assumng for smplcy ha K = K ), and one such varable for any par of ems > j a each me/prce (.e., N(N 1) 2 T K 2 ). As we prevously menoned, for App(N), hs number grows exponenally wh N and K and hence, may no be praccal o go beyond App(3) or App(4). We nex summarze he man resuls for he mulple em seng. Summary for he mulple em seng: We solve he promoon opmzaon problem for mulple ems by usng he App(2) approxmaon. The followng properes hold: Assumng ha he cross-em effecs for each em are addvely separable,.e., ( ) ( ) d p, p 1,..., p M, p = h p, p 1,..., p M + H j (p j ), (10) hen App(2) = App(3) =... = App(N). If we furher assume ha he funcon h ( p, p 1,..., p M ) s addvely separable for each em,.e., ( ) h p, p 1,..., p M = f (p ) + g1(p 1) g M (p M ), (11) hen he App(2) soluon s opmal. Consder he class of demand models n (10) and K = 1 (.e., he regular prce and a sngle promoon prce). For subsuable ems, he App(2) formulaon can be solved effcenly n he absence of busness rules. Under wo general demand models (mulplcave and addve prce dependence), we derve a paramerc bound on he qualy of he approxmaon relave o he opmal prof. j

19 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes 19 In many esed nsances, he approxmaon yelds a soluon whch s opmal or very close o opmal. We nex dscuss he mplcaons of he above summary. Ineresngly, for demand funcons wh addvely separable cross-em effecs (n pracce, several demand models sasfy hs propery), s suffcen o consder App(2) as opposed o nclude hgher order erms. In he specal case where each em can ake wo prces, he App(2) approxmaon can be solved effcenly when all he ems are subsuable. Havng wo prces s common n pracce as he promoon prce s ofen negoaed upfron wh he manufacurer. In he more general case, where he realer can choose among several promoon prces, we observed compuaonally ha one can sll solve he IP whn low runmes for realsc sze nsances. I s worh menonng ha for mos caegores of supermarke ems, he producs whn a caegory are eher ndependen (.e., no cross-em effecs) or subsuable. In parcular, for caegores such as coffee, ea and chocolae, we could no fnd any complemenary effecs n he daa we analyzed. Noe also ha even f some of he producs are complemen, we observed by exensve esng ha solvng he relaxaon of he App(2) formulaon yelds an opmal neger soluon very ofen. More deals on such compuaonal ess are presened n Cohen e al. (2018). 5. Insghs and Praccal Impac In hs secon, we summarze he nsghs we have been able o draw by solvng he (Mul-POP) usng our soluon approach. We hen descrbe how o concreely apply our model o a real-world real seng Insghs We brefly dscuss several nsghs ha were drawn by usng our promoon opmzaon model. Very ofen, realers wan o nfer he mpac of promong he dfferen ems a he dfferen me perods. Our soluon approach can easly be used o es varous promoon sraeges n order o reach a beer undersandng on he mpac of real promoons. As we menoned before, several economc facors are presen n he conex of our problem: he cross-em effecs on demand, he pos-promoon dp effec, he seasonaly, and he presence of busness rules. I s defnely valuable for he realer o learn he radeoffs beween hese dfferen effecs, and o undersand how hey mpac he promoon decsons. Our model can help realers o deepen her knowledge on he followng opcs: Undersandng he srucure of he cross-em effecs: In a gven caegory of ems, he realer needs o decde he prce promoons by accounng for he cross-em effecs on demand. Usng our model, he realer can nfer he mpac of promong a specfc em on he demand of each em n he caegory. Ths can ulmaely allow he realer o carefully decde

20 20 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes whch se of ems should be promoed smulaneously, and whch should no. For example, when wo (or more) ems have srong subsuon effecs (.e., promong an em ncreases s own sales, bu also sgnfcanly decreases he sales of he oher ems), he realer should no promoe hose ems. More deals on such nsghs can be found n Cohen e al. (2018). Inferrng he srengh of he pos-promoon dp effec: I s well known ha promong a FMCG produc nduces a boos n s curren demand as well as a poenal decrease n s fuure demand, due o he sockplng behavor of consumers and he promoon fague effec. The srengh of he pos-promoon dp effec can vary sgnfcanly dependng on he caegory under consderaon. For example, n Cohen e al. (2017), he auhors found ha he number of pas prces ha affec he curren demand (whch s one possble way o measure he pos-promoon dp effec) hghly depends on he em and on he caegory. For example, he pos-promoon dp effec ends o be weaker for pershable producs and for luxury/expensve brands, as expeced. Idenfyng he presence of a loss leader effec: The loss leader s a common phenomenon n whch one em s prced below s cos n order o exrac sgnfcan profs on complemenary ems (see, e.g., Hess and Gersner 1987). I s repored n Cohen e al. (2018) ha he model consdered n hs chaper can denfy he presence of a loss-leader effec. Ths can be a very mporan nformaon for he realer ha can use one (or more) ems n order o profably rgger a loss-leader sraegy. Learnng he mpac of he busness rules: As dscussed n Secon 4, he realer can easly solve several nsances of he problem, wh and whou he presence of some of he busness rules. Consequenly, hs allows he realer o quanfy he prof mpac of relaxng some of he busness requremens. Ths can ulmaely help realers decde whch vendor funds o accep and under wha erms. In pracce, realers ofen solve he (Mul-POP) for large scale nsances ha nvolve a large number of dfferen facors. I does no seem possble for managers, as experenced as hey are, o undersand and ancpae he mpac of all he conflcng radeoffs. Usng an opmzaon ool calbraed wh acual daa can ake no accoun all he dfferen radeoffs, and compue a close o opmal soluon for he promoon plannng problem Praccal Impac We nex consder a concree applcaon of he (Mul-POP) opmzaon problem. We propose a generc process ha can be used by any realer who seeks o mprove s promoon plannng decsons. Ths process consss of he fve followng seps: 1. Daa collecon, cleanng, and aggregaon,

21 Cohen and Peraks: Opmzng Promoons for Mulple Iems n Supermarkes Sore and produc cluserng, 3. Demand esmaon, 4. Opmzaon and sensvy analyss, 5. Quanfyng he mpac. We nex descrbe each sep n more deals. Daa collecon, cleanng, and aggregaon: The frs sep s o collec and sore he relevan daa. In our conex, realers need o smply collec he daa from he pas ransacons. Each observaon ypcally ncludes: he sore, he dae/me, he ems purchased, he prces, he promoon vehcles ha were used, as well as varous feaures of he em (brand, sze, flavor, ec.). Afer gaherng a large enough daase, one needs o carefully clean he daa, and perform he approprae aggregaons. Varous echnques exs for cleanng and aggregang daa bu hs s beyond he scope of hs chaper (see, e.g., he book by Han e al. 2011). A a hgh level, one wans o deal wh he mssng daa, remove he oulers, and perform some basc sascal ess. Once he daa s cleaned, one needs o decde he level of aggregaon. Dependng on he conex, one can perform he analyss a he brand, em, or caegory level. Smlarly, one can aggregae he daa a he week, day, or hour level. Once he daa s cleaned and aggregaed a he rgh level, one can sar usng for esmaon and predcon purposes. For example, n Cohen e al. (2017), he auhors decded o aggregae he daa a he brand-week level. Sore and produc cluserng: In many real sengs, he avalable hsorcal daa can be sparse. As a resul, one needs o combne he daa from mulple sources n order o oban more relable forecass. Two common echnques wdely used n real conss of mergng several sores ogeher or cluserng smlar producs. The dea s o use he daa from several sores ha share smlar feaures (e.g., geographcal locaon, sze, managemen eam). Smlarly, ems from he same brand (e.g., dfferen szes or flavors) can ofen be clusered ogeher so as o mprove he predcon accuracy of he models. Demand esmaon: Ths sep s he acual frs sage of usng our promoon opmzaon model. As an npu o he opmzaon, one frs needs o esmae he demand models. The modeler has several degrees of freedom: choce of he demand funcon (e.g., log-log, log-lnear), selecon of he dependen varables, choce of he nsrumenal varables (f any), and choce of he esmaon procedure. In many applcaons, one can smply run a lnear regresson (e.g., ordnary or weghed leas squares, rdge regresson, lasso). The ypcal process also ncludes splng he daa for ou-ofsample esng. The demand esmaon sep s compleed once he predcon models yelds a hgh accuracy ou-of-sample. In pracce, one needs o es dfferen models and assumpons n order o reach a good and robus predcon model. In Cohen e al. (2017), he auhors presen a predcon model for wo coffee brands based on usng ordnary leas squares o predc a log-log model ha