Optimizing the Alloation of Marketing Promotional Contats Keith Hermiz, Ph.D., IBM Helene Miller, IBM Dhanesh Padmanabhan, Marketis Tehnologies Abstrat The typial marketing environment in the business-to-business (B-to-B) world is very omplex. Companies selling in that environment initiate promotional ontats aross different hannels and in support of various produt lines targeted at many marketable points-of-ontat. A fundamental question is Given a budget of marketing ontats, how an these be alloated aross the available prospet ompanies and pointsof-ontat suh that the revenue return is optimal? In this paper, we will use a ase study to demonstrate a multi-faeted approah that an be used to answer this question, drawing upon tehniques inluding ausal analysis and mathematial optimization. Introdution The typial marketing environment in the business-to-business world is very omplex. Companies selling in that environment may utilize several marketing hannels and may market through multiple brands. Here we think of marketing hannels as those with diret ontat with the prospet: diret mail, outbound telemarketing alls, and so on. Companies that are the target of these selling ativities may have many marketable points-of-ontat. A fundamental question is Given a budget of marketing ontats, how an these be alloated aross the available prospet ompanies and points-of-ontat suh that the revenue return is optimal? A brute-fore approah to answering this question would be to build ontat-revenue response models for eah target ompany, and then use some mathematial optimization tehnique to alloate the ontat budget so as to maximize revenue. In this paper we outline a multi-faeted approah that answers this question, under a number of neessary simplifying assumptions, in what we believe is a more robust manner. We begin by using a method borrowed from epidemiology, whih utilizes propensity sores to estimate the ausal impat of an intervention. 1 After a brief disussion on the theoretial underpinning of this approah, we review the harateristis of the data used in this projet and the preparatory transformations that were made. In applying the ausal framework, we hoose to use ordinal variables that measure number of marketing ontats and number of marketable points-of-ontat. This hoie required the onstrution of two ordinal logisti models. 2 We disuss the reasons behind this hoie, and show how the linear oeffiients from the ordinal logisti models are used to reate a two-way sublassifiation of the target ompanies. The pratial result of this approah is that we arrive at several groupings of ompanies that share similar propensities to be ontated, and similar propensities to have a number of marketable points-of-ontat. This approah of using sublassifiation on the propensity sore is known to redue the bias due to eah of the ovariates, 3 and we demonstrate this result. On this theoretially sound foundation, we then onstrut unique, two-stage wallet-share models for eah group whose independent variables inlude ompany firmographis, number of marketable points-of-ontat, and marketing touhes. 4 We are then positioned to apply standard optimization tehniques at this point, using as our objetive the maximization of inremental revenue due to marketing
ontats. However, the non-linear two-stage models and the number of reords made the naïve version of the problem intratable. In order to proeed, we made two simplifiations to the two-stage models. First, we made a quadrati approximation to the log and logit representations. Seond, we used separable programming to linearize the quadrati form. 5 The resulting formulation an be optimized quikly using a standard linear programming module, and we show our formulation in SAS OPTMODEL. The result is a reommendation for the number of marketing ontats that should be made, by ompany, in order to maximize revenue return. Propensity Sore Methodology Like most B-to-B database marketing analyti pratitioners, we would typially not onern ourselves with issues of ausality in our day-to-day work. However, we reognized that, in a omplex environment suh as ours, it is diffiult for the business to have a truly effetive ross-organizational, over-arhing proess for determining whih ompanies are targeted for promotional ontats as various business units bring forward individual tatis. Our hypothesis was that if we looked at the aggregate number of ontats made aross our target population, we would find two things: first, we expeted to find a high orrelation between ustomer revenue and number of ontats. Folklore and some evidene support the idea that the usual suspets get the most ontats in spite of the fat that those ontats may not be driving inremental revenue. Seond, we expeted to find very similar ompanies having vastly different ontat experienes. Our assessment was that building a single ontat-revenue model aross all possible target ompanies would not be satisfatory in this ase. We would need to use some approah to produe more homogeneous subpopulations prior to modeling. We had reently seen literature on the appliation of ausal analysis to marketing 1 that seemed to be a promising alternative. While arefully ontrolled, randomized studies remain the gold standard in assessing the ausal effet of interventions, the propensity sore method (PSM), applied to observational studies, is now widely aepted as a redible alternative in fields ranging from epidemiology to eonomis. The basi notion behind this approah is that one develops a model that estimates the propensity of intervention. Subjets, treated and untreated, are then grouped together based on this sore. A ausal inferene of the impat of the treatment an then be alulated. Although this paper does not address estimating the ausal impat of ontats there is value in understanding the PSM from that vantage point. One variation of the PSM uses the onept of subjet lones. Under this sheme, treated and untreated reords are mathed based on their propensity sore. The propensity sore serves to balane all the ovariates that were used in its onstrution so that in narrow sore ranges one will find that the ovariates have essentially equivalent distribution for both the treated and untreated populations. For a treated subjet, the answer to the ounterfatual ausal question What would have been the outome if they had been left untreated? an reasonably be estimated by it s untreated lone. An alternative to subjet mathing is to use the PSM to sublassify subjets based on their sore. 4 The theoretial foundation is the same as the mathing method, whih is that ovariates within eah sublass will show similar distributions. An estimate of overall ausal effet is then omputed with a weighted estimate of the effet derived from eah sublass. In our ase, the objetive was to group ompanies together in a way suh that the ovariate distributions would be similar aross the range of possible ontat levels, from no ontat whatsoever to rather high levels. By using the PSM, revenueontat response models in these groups would have some rigor with respet to the ausal effet of the ontats on revenue. In the next setion, it beomes lear that we needed to bin the ontat level variable due to its highly skewed distribution. The resulting variable represented ordinal levels of ontat treatment. The PSM sublassifiation required the use of a single, salar sore. The use of an ordinal ategorial variable for propensity to be ontated fored some modeling onsiderations, 2 and ultimately required that we use the linear oeffiient from an ordinal logisti model as the sublassifiation value. As part of this projet, we atually used a bivariate sublassifiation, using the propensity of a ompany to reeive a given number of marketing ontats and to have a given number of marketable points-of-ontats (POC.) The initial thinking was that the number of marketable POCs is not a treatment per se, but investments an be made to inrease the number of POCs if they were sub-optimal. We did not further develop the idea of maximizing levels of marketable POCs in this work, but took advantage of the additional balaning of ovariates that the seond sublassifation provided during our modeling exerise.
Bakground and Data The data used for this work onsisted of following information for over 275,000 ompanies onsidered as part of this study. For eah ompany: the number of ontats made and number of POCs ontated over the study timeframe ategorized by marketing hannel and produt/servie the number of marketable POCs for eah ompany firmographi information suh as number of employees, number of business loations, sales revenue, and industry historial revenue and modeled wallet size in dollars over the study timeframe ategorized by produt/servie This data was assembled from various soure tables ontaining ustomer relationship, transational and firmographi information. Over 80% of the ompanies in this data had not been ontated in the study timeframe and about 45% of the ompanies did not have a marketable POC. The data had a small perentage of missing information in number of marketable POCs, historial revenue, and wallet size. For ompanies that had missing marketable POCs but had a number of POCs ontated, the latter was used to impute the former. Otherwise, the marketable POCs were taken as zero. The missing revenues and wallet estimates were taken to be zero. Wallet share of a ompany was derived as the ratio of its revenue and wallet size. The historial revenue is aggregated from a transational database. In some ases, redits an exeed revenue in whih ase a small perentage of historial revenues are negative. Given that the wallet size is modeled, there are ases where historial revenue is greater than the wallet size. As a onsequene of these anomalies, the wallet shares were onstrained to be between 0% and 100% and lipped if neessary. The distributions of the number of ontats and number of POCs were highly skewed, and we hose to transform the ontinuous values into ordinal ategories. We ensured that eah ategory had at least 1,000 observations, but beyond that, used our subjetive judgement to reate the boundaries for the resulting eight ategories for eah measure. One ategory in eah ase aptures the ase of zero ontats or zero POCs respetively. We derived new variables by transforming the firmographi variables to obtain our ovariates for the analysis. The ontinuous variables, suh as number of employees, number of business sites, and sales were log-transformed with the objetive being to make the distributions of the transformed variables normal. Levels of the ategorial variables suh as industry were redefined suh that the infrequently ourring levels were ollapsed into other levels. Dummy variables were reated for these redefined ategorial variables for the ordinal logisti and wallet share models disussed later. Sublassifiation Using Propensity Sores The theory behind the use of propensity sores for ausal analysis suggests that sublassifying subjets (in our ase, ompanies) based on the propensity of intervention (in our ase, a marketing ontat or the establishment of marketable POC) redues the bias of the underlying data within the sublasses, sine interventions are seldom random. For a binary intervention (for example, the ase of a single marketing ontat or no ontat), the situation is straightforward. For intervention that an take on a range of ordinal values, as we have here, the situation is more omplex, and a salar propensity sore will serve as a balaning sore (i.e., one that redues bias) only under speial irumstanes. In order for a salar propensity sore p( x ) to serve as a balaning sore of the ovariates x it must apture the entire distribution of interventions given x. For sublassifiation of ompanies, we used two ordinal logisti models for obtaining the propensity of a ompany to be ontated at a partiular level and propensity of a ompany to have its number of POCs at a partiular level. The dependent variables were the ordinal ategories for the number of ontats and number of POCs, and the independent variables were the ovariates x. We used the proportional odds assumption for the ordinal logisti models. Mathematially, the ordinal logisti models for the level of number of ontats and level of number of POCs an be written as follows, t P ln i t 1 P i = δ t + λ t x,where i =1,..,n t 1 i P j ln = δ 1 P j + λ x,where j =1,..,n 1 j t P i and P j are the umulative probabilities for ontat ategory i and POC ategory j, n t and n are the number of levels in ontat and POC ategories respetively, δ t i and are the onstant terms orresponding to ontat ategory δ j
i and POC ategory j, and λ t and λ are the parameter estimate vetors for the ovariates for the ontat and POC ategories. Note that only the onstant terms in the logit formulation vary with the levels of the ontat and POC ategories. The resulting propensity distribution within levels is fully determined by the λx terms. This formulation therefore meets the theoretial requirements stated above if we use λ t x and λ x as the values over whih we sublassify. To obtain the parameter estimates we utilized SAS s PROC LOGISTIC. We note that the proportional odds assumption was not satisfied by our data for either ase above. Our end objetive is to sublassify the ompanies in a way that redues bias in the ovariates, something we an test separately after applying the model. The model is a means to an end, and therefore it is an aeptable trade-off for it to fail the proportional odds assumption, provided it redued ovariate bias through the resulting sublassifiation. We determined ontat quintiles and POC quintiles based on the two sores. This gave us quintile-wise groupings of ompanies, whih are shown in the five-by-five grid shown in Figure 1. orner of the grid exhibit lower propensity to be ontated and are less likely to have a large number of POCs. In order to determine if the sublassifiation was effetive in reduing the seletion bias, we devised a metri to apture bias redution aross all sublasses. We omputed the varianes in the means of ovariates aross the ontat ategories for the number of ontat sublasses (quintiles) and the overall ase with no sublassifiation. We then omputed an aggregate measure of bias redution for eah ovariate as 100 ( 1 B s / B o ) where B o is the overall variane and B s is the average of varianes in the sublasses. Table 1. shows these results. The analysis revealed that there was a redution of 80-92% in bias for these variables. Covariates Number of Number Employees of Sites Sales Quintile 1 0.5281 0.3847 0.8288 Quintile 2 0.0278 0.0033 0.3234 Quintile 3 0.0117 0.0010 0.4296 Quintile 4 0.0155 0.0008 0.4984 Quintile 5 0.0402 0.0002 0.1127 Overall 1.5712 0.7624 2.2219 Bias Redution (%) 92.07 89.77 80.26 Table 1: Seletion Bias Redution in Covariates Wallet Share Models Figure 1: Sublassifiation of Companies It an be seen that that there is a high onentration of ompanies in the diagonal of the grid. This is due to the underlying orrelation between the number of ontats and POCs. The ompanies in the upper left orner of the quintile grid typially onsisted of ompanies that have a high propensity to be ontated and high propensity to have a large number of POCs. Similarly, the ompanies in the lower right After sublassifiation, we onstruted wallet share models for eah of the thirteen populated sublasses out of the possible twenty-five. These wallet share models were essentially two stage models. The first stage model was a propensity-to-buy (PTB) model that predited the probability of a purhase event,p( buy ). The seond stage was an ordinary least squares (OLS) model that predited the expeted wallet share given a purhase event,e( wallet share buy ). The net expeted wallet share is then obtained as the produt of the PTB and OLS estimates. This an be mathematially represented as follows: E( wallet share ) = P( buy ) E( wallet share buy ) For the first stage models, we used all the data in eah sublass for training eah of the thirteen PTB models. The purhase event was defined to be any ase where for wallet share w, w >0. For the seond stage models, we used the data with w >0 in eah sublass for training eah of the OLS models.
Figure 2: Two-Stage Model Fators Figure 3: Two-Stage Model Fators 1.1 0.9 0.9 0.7 0.7 0.5 Fator Value 0.3 PTB OLS Comb Quad 0.5 0.3 PTB OLS Comb Quad 0.1 0.1-0.1-0.1 Contats Contats The ovariates x, the number of ontats t and the number of POCs were used as the independent variables for these models. After some experimentation, the log transformed version of wallet share, ln( w +1) was used as the dependent variable for the OLS model in order to remove the skew in data and ensure a better normal fit. We used linear terms in t,, and x for obtaining these models and, additionally, the quadrati terms, t 2 and 2 to apture nonlinear returns to inreasing number of ontats and POCs. An initial analysis of the underlying data used for the models revealed that these quadrati effets existed for many ases. Stepwise regression was used for seletion of the ovariates x, but the linear and quadrati terms in t and were fored into the models. Mathematially, the funtional forms for log-odds ratio for the PTB model and OLS model an be written as follows: PTB( t,,x ) ln 1 PTB( t,,x ) = a +a x +a 2 +bt +t 0 x ln ( OLS( t,,x ) +1) =l 0 +l x x +l +mt +nt 2 SAS s PROC LOGISTIC was used for estimatinga 0,a x,a, b, and in the PTB model. SAS s PROC REG was used for estimatingl 0,l x,l, m, and n. After these models were built, we had thirteen sets of OLS and PTB oeffiients that were assoiated with the thirteen sublasses of ompanies. The PTB models had signifiant parameter estimates for t and t 2 with lear maxima in t for all the sublasses, i.e. the PTB inreases as the number of ontats inrease until a point at whih it starts to derease. The maximum ontat point was a degree of magnitude higher for the sublass of ompanies in the upper left orner than the other sublasses in Figure 1. This shows that this group of ompanies needed many more ontats to obtain maximum PTB than the other sublasses. The OLS models on the other hand, revealed several insignifiant and small parameter estimates in t and t 2. For the sublass of ompanies in upper left orner, there was a lear maximum in the OLS model with respet to the number of ontats t, but for the rest of sublasses the OLS models exhibited minima in t. Despite the potentially ounterating effets of PTB and OLS models, the influene of number of ontats on OLS estimates was in general muh weaker than on the PTB estimates. Hene, we expeted the net expeted wallet share to be signifiantly influened by the PTB model and exhibit a maximum loser to the maxima of the PTB. We maximized the sum of net expeted wallet share for all the ompanies for the optimization problem that will be desribed later. For the optimization studies, the numbers of ontats t were the only optimization variables. Hene, for eah ompany, the onstant term and linear terms in x and in the PTB and OLS models were aggregated to net onstants, and the PTB and OLS estimates were rewritten as analyti expressions that were quadrati in number of ontats t. These an be written as, PTB( t ) ln 1 PTB( t ) = a +bt +t 2,where a = a 0 +a x x +a ln ( OLS( t ) +1) =l +mt +nt 2,where l =l 0 +l x x +l
Simplifiation of the Nonlinear Problem We were suessful in running the SAS OPTMODEL optimization proedure with small samples of the data using the nonlinear forms of the two stage models desribed above. However, as we moved to solve the omplete problem aross all ompanies, it beame apparent that the nonlinear form of the problem would be intratable. In examining the models, we found the situations in Figure 2 and Figure 3 to be typial. As ontats inreased from 0, the PTB would rise quikly, and would either sustain for range of ontats or ollapse. The OLS might have a shape reminisent of a quadrati, or perhaps tail upwards at a modest rate. We noted the apparent quadrati shape of the ombination of the two fators (Comb.) While one approah to simplifiation might have been to do a series expansion and look for a dominant quadrati form, we felt that our objetives ould be more simply met by fitting a quadrati that aptured the key points of the ombined model. The PTB and OLS models, depending on the number of ontats t have the form PTB(t) =( 1+e A(t) ) 1 where A(t) = a +bt +t 2 OLS( t ) =( e L ( t ) 1)where L( t ) =l +mt +nt 2 We wanted to approximate f ( t ) = PTB( t )* OLS( t ) by f(t) = α +βt + γt 2 suh that f( 0 ) = PTB( 0 )* OLS( 0 ), f(t max ) = PTB(t max )* OLS(t max )where PT B (t max ) = 0 and f ( t max ) =0. After some algebrai manipulation we get that: α =( e l 1) / ( 1+e -a ) β = 2 γt max γ =( α PTB(t max )* OLS(t max )) / t max 2. Using this quadrati estimate (Quad) appears to be very satisfatory, as seen in Figure 2 and Figure 3 It aptures the essene of the ombined nonlinear sores, espeially for the ontat range from 0 tot max. Given that optimization of ontats would prevent touhes exeeding t max for any ompany, auray of the estimate beyond this range is of no onern. There are pathologial ases where the quadrati is onave-up, so the resulting revenue would derease with any positive ontat. Sine f (t) = 2 γ we exluded from onsideration any ompany that had γ >0. The quadrati form might have been a suffiient approximation to allow us to omplete the optimization runs in aeptable times. However, given our needs, we deided to make one additional simplifiation to a linear form before proeeding. We felt that a linear approximation would be aurate enough, provided we ould apture an infletion in the revenue urve as the number of ontats inreased. To aomplish this, we used a tehnique known as separable programming 5 to build a piee-wise representation of the up-slope portion of the quadrati approximation. Separable programming substitutes a piee-wise linear approximation to a urve, parameterizing eah segment with a new variable that beomes another independent variable in the optimization. Our strategy was essentially to split the up-slope portion of the quadrati in half and use the end points f ( 0 ), f ( M ) and f ( t max ) as end points for two line segments, wheret M =t max / 2. These line segments would have slopes of s 1 = ( f ( 0 ) f ( t M ))/ t M and s 2 = ( f ( t max ) f ( t M ))/ t M and would be parameterized byt 1 and t 2 respetively. The simplified formulation for the optimization is now: Optimize f(t) = s 1 t 1 +s 2 t 2 Subjet to t 1 t M 0 t 1 +t 2 t max t 1,t 2 0 Optimization of Marketing Contats The simplifiations we desribed above led to a very straightforward linear optimization problem, as an be seen in Figure 4 (right). These linear problems ran in a matter of minutes with the full quarter-million reords. The optimization results passed heks for reasonableness. They generally moved ontats to ompanies with higher potential wallet shares. One interesting finding was that the optimization signifiantly redued the number of ompanies 8
that would be ontated, while inreasing the number of ontats per ompany. We have had anedotal onfirmation that this rebalaning of ontats from shallow-and-broad to narrow-and-deep has been onsidered, and thought to be potentially more effetive from a marketing perspetive. The modeling and optimization seems to bear this out. We repeated the entire proess desribed in this paper for several brand and marketing hannel ombinations. To get a sense for the projeted improvement in revenue assoiated with the optimization, we ompared historial versus modeled and optimized average revenue per ontat. Aross five of the six ases we optimized, we find signifiant improvement in the Figure 4: Optimization Sample Code
average revenue per ontat signifiant enough to suggest that the realloation sheme absolutely has merit. Conlusion We believe that in most ompanies, alloating marketing ontats happens organially and as a result of many individual tatial onsiderations, rather than being derived from a holisti, objetive driven proess. Our aspiration in doing this work, and in exhibiting it in this paper, was to demonstrate that analytis an be used in a logial and rigorous way to arrive at a ontat alloation sheme that has the potential to signifiantly improve total revenue return to the enterprise. We are enouraged by the interest and support we have reeived within our ompany, and hope to see ontrolled deployment of these results in the oming months. We hope that readers found our use of a ausal analysis framework and the separable programming tehnique novel, and will be motivated to apply these, either in the setting of ontat optimization or in other appliations, as appropriate. n REFERENCES 1 Rubin, D.B., Waterman, R.P., Estimating the Causal Effets of Marketing Interventions Using Propensity Sore Methodology, Statistial Siene, Vol.21, No.2, 206-22 (2006) 2 Joffe, M.M., Rosenbaum, P.R., Invited Commentary: Propensity Sores, Amerian Journal of Epidemiology, Vol.150, No.4, 327-333 (1999) 3 Rosenbaum, P.R. and Rubin, D.B., Reduing Bias in Observational Studies Using Sublassifiation on the Propensity Sore, Journal of the Amerian Statistial Assoiation, Vol.79, No.387 (1984) 4 Imai, K., Van Dyk, D.A., Causal Inferene With General Treatment Regimes: Generalizing the Propensity Sore 5 Hillier, F.S., Introdution to Mathematial Programming, 2nd ed., 635-641, MGraw-Hill (1995) 1 Rubin, D.B., Waterman, R.P., Estimating the Causal Effets of Marketing Interventions Using Propensity Sore Methodology, Statistial Siene, Vol.21, No.2, 206-22 (2006) 2 Joffe, M.M., Rosenbaum, P.R., Invited Commentary: Propensity Sores, Amerian Journal of Epidemiology, Vol.150, No.4, 327-333 (1999) 3 Rosenbaum, P.R. and Rubin, D.B., Reduing Bias in Observational Studies Using Sublassifiation on the Propensity Sore, Journal of the Amerian Statistial Assoiation, Vol.79, No.387 (1984) 4 Imai, K., Van Dyk, D.A., Causal Inferene With General Treatment Regimes: Generalizing the Propensity Sore 5 Hillier, F.S., Introdution to Mathematial Programming, 2nd ed., 635-641, MGraw-Hill (1995) 10