Optimal Customized Pricing in Competitive Settings

Transcription

1 Optmal Customzed Prcng n Compettve Settngs Vshal Agrawal Industral & Systems Engneerng, Georga Insttute of Technology, Atlanta, Georga vshalagrawal@gatech.edu Mark Ferguson College of Management, Georga Insttute of Technology, Atlanta, Georga Mark.Ferguson@mgt.gatech.edu In ths paper, we study prcng stuatons where a frm provdes a prce quote n the presence of uncertanty n the compettve landscape and the preferences of the buyer. We revew two possble customzed-prcng bdresponse models used n practce whch can be developed from the hstorcal nformaton avalable to the frm based on prevous bddng opportuntes. We show how these models may be used to explot the dfferences n the market segments to generate optmal prce quotes gven the characterstcs of the current bd opportunty. We also show how the models may be adjusted dependng on the amount of hstorcal bd nformaton avalable to the user. Fnally, we test the two methods on two ndustry datasets to compare ther performance and estmate the percent mprovement n expected profts that may be possble from ther use. Key words: bd-response functons, customzed prcng, prce optmzaton, bd-prcng. 1. Introducton Whle the majorty of the prevous lterature n the prce-optmzaton area focuses on the prcng of consumer goods or the optmal desgn of auctons, a large percentage of frms face prcng decsons n a busness-to-busness settng where a customer requests bds from a small set of competng frms and the frms vyng for the customer s busness respond wth a sngle prce quote for the product or servce. When the total dollar amount of the customer does not justfy a dedcated sales person on behalf of the frm respondng to the bd, many frms use bd-response models to provde customzed prcng recommendatons on what prce to offer for the busness beng bd upon. Customzed-prcng bd-response models (CPBRMs) provde a probablty of wnnng for every possble prce response, allowng a frm to balance a decreasng margn wth an ncreasng wn probablty needed n a prce optmzaton model. Examples of frms usng

2 CPBRMs nclude Unted Parcel Servce (UPS) when respondng to bds for from ther small to medum sze customers (Knple, 2006), banks replyng to nterest rate requests for personal and busness loans (Phllps, 2005a), and BlueLnx, the largest buldng products dstrbutor n the U.S, respondng to requests for products from constructon companes (Dudzak, 2006). The fnancal mpact from usng CPBRMs can be sgnfcant. UPS reported an ncrease n profts of over $100 mllon per year by optmzng ther prce offerngs usng CPBRMs (Boyd et al. 2005). In determnng the wnnng bd probablty, CPBRMs effectvely determne the prce segment the current bd falls n. Prce segments are defned as sets of transactons, classfed by customer, product, and transacton attrbutes, whch exhbt smlar relatve prcng level and prce senstvty. Customer attrbutes may nclude customer locaton, sze of the market the customer s n, type of busness the customer s n, the way the customer uses the product, customer purchase frequency, customer sze, and customer purchasng sophstcaton. Product attrbutes may nclude product type, lfecycle stage, and the degree of commodtzaton. Transacton attrbutes may nclude order sze, other products on the order, channel, specfc compettor, when the order s placed, and what the urgency s of the bdder. In addton, some models assume knowledge of the hstorcal and current bd-prce of competng frms partcpatng n the bd. A common characterstc of stuatons where frms employ CPBRMs s when the bdder wth the lowest prce does not always wn the bd. Thus, markets are characterzed by product dfferentaton where a gven frm may command a postve prce-premum over ts compettors; dependent upon the partcular customer offerng the bd. Even assumng a frm collects enough hstorcal data to perfectly derve ts prce premum for a gven customer, there may stll be some nherent amount of uncertanty n the bd wnnng probablty due to the bd-requestng frm randomly allocatng ts busness to dfferent compettors to ensure a compettve market for future bds. Therefore, a frm wll never be able to remove all uncertanty from the bd-prce response process and must work wth probablstc models.

3 Another common characterstc of stuatons where frms have used CPBRMs s when the sze of the bd opportuntes s not large enough to justfy a dedcated sales person for each bd opportunty. Thus, the most common alternatves to usng CPBRMs s ether to charge a fxed prce to all customers or to have a sales agent respond to each separate bd opportunty wth a customzed prce. Chargng a fxed prce leads to mssed opportuntes to prce dscrmnate between dfferent customer segments, a practce that has been well publczed for sgnfcantly ncreasng a frm s proft n many dfferent ndustres. The other alternatve, relyng on a sales agent to respond to multple bd opportuntes, s also problematc. Theoretcally, the sales agent should have knowledge of the market, based on a hstory of former bd-responses wth the customer requestng the bd, allowng the sales agent to respond wth a customzed prce that optmzes ths nherent trade-off between decreasng margns, due to lowerng the prce, and ncreasng probabltes of wnnng the bd. In realty, sales agents often do not make good tradeoffs n these stuatons, ether because of lack of hstorcal knowledge, the nablty to process ths hstorcal knowledge nto probablty dstrbutons, or ms-algned ncentves (Garrow et al., 2006). The judcous use of CPBRMs allows frms to capture hstorcal bd nformaton, process t, and present non-based prce recommendatons to bddng opportuntes. If there s addtonal nformaton avalable regardng the bddng opportunty that can not be captured n the CPBRMs, the CPBRM s recommended prce may serve as one of possbly many nputs to the person responsble for makng the bd-response decson. To summarze, CPBRMs apply to stuatons where a frm sellng a non-commodty product must respond to frequent request for small to medum szed bds from a number of dfferent customers where the bd-wnnng crtera s not always the lowest prce. To use a CPBRM, a frm must have access to ther hstorcal bd hstory that ncludes, as a mnmum, the prce the frm bd at each opportunty and the correspondng bd result (wn or loss). Other useful hstorcal nformaton used n developng CPBRMs s, for each hstorcal bd opportunty, the customer, the length of the relatonshp wth the customer, the sze of the order, delvery date

4 requrements, compettors bds, and any other pertnent nformaton useful for market segmentaton. When CPBRMs are used as an nput to a prce optmzaton model, there s also the mplct assumpton the actons of the compettors can be determned probablstcally and ndependently of the decson maker s acton. If all compettors have smlar analytc capabltes and jontly optmze aganst each other, compettve response modelng technques such as game theory must be used. In ths paper, we evaluate two CPBRMs, namely the Logt and Power functons, whch model the response of the buyer subject to the segmentaton crtera descrbed above. Based on numercal comparsons usng two ndustry datasets, we provde observatons on where each functon s preferred. We fnd the Logt functon s preferred when there s less hstorcal bd data avalable or lttle data about each bd s avalable for determnng customer segmentatons. When detaled nformaton s avalable about each former bddng opportunty such as the compettors prces and the sze of the order, the Power functon outperforms the Logt on our test datasets. We demonstrate how to modfy the functons to ncorporate varous degrees of segmentaton data avalable to the frm. We then test both functons usng the data avalable from the two ndustry datasets to analyze the relaton between the nature of nformaton avalable to the frm and the mprovements generated by our approach. As expected, we fnd the percent mprovement n expected profts ncreases n the amount of hstorcal bd data avalable and n the amount of nformaton recorded from each bd opportunty. Yet, even under the worst case condtons of lttle hstorcal data, we contnue to see sgnfcant mprovements n expected profts over the unoptmzed strateges the companes were followng before usng CPBRMs. The rest of the paper s organzed as follows. In 2 we revew the academc lterature and ndustry practces related to the modelng of bd-prce responses. In 3 we present two CPBRMs that are used n practce and show how they can be modfed to use under three dfferent levels of avalablty of hstorcal and compettve nformaton. In 4 we present the results from applyng the two CPBRMs to two ndustry datasets where we measure the percent mprovement n

5 expected profts under dfferent nformaton levels. In 5 we summarze our observatons from the numercal comparson and conclude wth some lmtatons and manageral mplcatons of usng CPBRM s. 2. Lterature Revew In ths secton, we dscuss the academc lterature on bd-prce response models and how CPBRMs are unque. We also dscuss the motvaton from ndustry practces related to such compettve prcng settngs. Several papers develop bd-prce response models where prce s the only attrbute of the model. Fredman (1956) and Gates (1967) both develop models whch use the hstorcal bd nformaton avalable. Morn and Clough (1969) buld on ther work by dentfyng key compettors and capturng temporal senstvty to changes n strategy by gvng recent data more mportance. However, these models consder prce as the sole crteron for wnnng a bd and only consder the objectve of maxmzng profts. Chapter 4 n Llen et al. (1992) provdes an overvew of competton orented prcng where the frm makes a trade-off between margn and probablty of wnnng the bd. Ths s the same trade-off the frm makes n our models, the dfference, however, occurs n the estmaton of the wnnng probabltes. In ther model, the lowest bd always wns so the probabltes are based on the number of compettors and each compettor s estmated bd-to-cost rato. Kng and Mercer (1991) dscuss estmaton methods for determnng the dstrbutons for these ratos. The models we revew are more general; they nclude non-prce factors such as order sze and contnue to hold when factors other than just prce are ncluded n the buyer s decson. In addton, prce optmzaton usng CPBRMs can accommodate several strategc frm objectves such as ncreasng or mantanng market share. Papaoannou and Cassagne (2000) provde a detaled revew of bd-prce response models and develop a ServPrce model whch, lke CPBRMs, accommodates several frm objectves and accounts for both prce and non-prce attrbutes. However, ther model reles only

6 on sales or prcng agents to make tradeoffs and analyze hstorcal nformaton. On the other hand, CPBRMs help the frm obtan a non-based nput to the bd-response decson by processng the relevant hstorcal nformaton statstcally. Lawrence (2003) develops a prescrptve model for quotes wth hgh potental revenues, whch predcts the outcome of a bd as a functon of ts attrbutes. Hs model requres more extensve bd hstory than a typcal CPBRM and uses a machne-learnng approach. It also only uses the outcomes of known transactons and doesn t explot any addtonal nformaton that s avalable to the frm. CPBRM s can explot much more extensve nformaton, f avalable, pertnent to a partcular market segment. We also study the dfference n mprovements when CPBRMs are used wth dfferent levels of nformaton knowledge. Ths paper s most closely related to the work presented n Chapter 11 of Phllps (2005b) and the U.S. patent of Boyd et al. (2005), who dscuss the use of CPBRMs n ndustry and develop models usng a Logt functon as a bd-response functon. These models capture the nherent preference uncertanty and non-prce factors whch play a crtcal role n wnnng a bd. The contrbutons of our work over the methods suggested by Phllps and Boyd et al. are as follows: We extend the Logt bd-response functon to nclude the compettor s prce whch helps to capture the compettve dynamcs. We also present another CPBRM, the Power functon (sometmes found n practce) whch ncludes a parameter of the rato of the bdder s prce to the expected bd prce of the bdder s compettor(s). We numercally test both models on two ndustry data sets representng best and worst case scenaros of where CPBRMs may be appled. Based on the performance of each model on the data-sets, we gve observatons on when each model may be preferred. To our knowledge, ths s the frst academc paper to present CPBRMs and test them on actual ndustry data.

7 3. Customzed-Prce Bd-Response Models In ths secton we descrbe what CPBRMs are, present two CPBRMs used n practce, and dscuss how they may be used n a prce optmzaton model. CPBRMs calculate the probablty of wnnng a bd opportunty for each possble prce response gven the market characterstcs and compettve dynamcs for a partcular customer segment. The parameter values for these models are statstcally estmated from hstorcal bd nformaton and nclude, at a mnmum, the bddng frm s hstorcal bds and the outcome from each bddng opportunty (wn or loss). Intutvely, f the prce quoted by a frm s very low compared to ts compettor s prce, the probablty of wnnng the bd should be close to 100%. If t s very hgh by comparson, the probablty of wnnng should be close to zero. Ths probablty of wnnng the bd should monotoncally decrease wth an ncrease n prce (or prce rato). Also, the slope of the response curve should be steeper for prces close to the compettor s prce as compared to prces far hgher or lower than the compettor s prce. Hence, the bd response curve s generally S-shaped n nature. In a sngle compettor settng wth no prce premum enjoyed by ether frm, prcng equal to the compettor s prce should result n a 50% chance of wnnng the bd opportunty. In practce, however, one of the frms usually enjoys some prce-premum over the other. Determnng what ths prce-premum s for each customer segment s one of the uses of CPBRMs.

8 Bd-Response Curve Probablty of Wnnng (1.00, 0.41) Prce Rato Fgure 1 Fgure 1 shows a CPBRM curve appled to one of our test case datasets. The prce rato on the x-axs s the rato of the frm s prce relatve to ts compettor s prce. For the partcular frm correspondng to ths bd-response curve, a prce equal to ts compettors prce (prce rato = 1) results n a probablty of wnnng the bd of 40.91% (a prce rato =.98 equates to a 50% wn probablty for ths frm). Thus, ths frm has a negatve prce-premum and must prce below ts compettor s prce for an equal opportunty of wnnng the busness of the frm offerng the bd. Before presentng the two CPBRM functons revewed n ths paper, we frst ntroduce some notaton. p Table 1: Notaton Unt prce quoted by the frm for bd opportunty (our decson varable) ρ ( p ) Bd-Response Functon,.e. probablty of wnnng bd opportunty gven a prce of p a, α j b, γ j Index for bd opportuntes, = 1, 2 j j Parameters related to non-prce factors for a segment j Parameters related to prce factors for a segment j j Index for segments, j = 1, 2 c c Parameter for the prce quote of the compettor p c, Unt prce quoted by the compettor(s) for bd opportunty

9 c q Q Parameter for the order sze Order sze for bd opportunty r( p ) Prce rato = p p, c x j Indcator varable for segment j (bnary) ε ( p ) Elastcty of the bd- response functon c p W Margnal costs for the frm Wn/Loss ndcator varable for a bd opportunty ( bnary) 3.1 Two Common CPBRMs In ths secton we descrbe the two CPBRMs (both commonly used n practce) compared n ths paper and dscuss how they can be adjusted to nclude segmentaton and compettve prcng nformaton. The ncluson of segmentaton and compettve prce parameters has been conjectured to sgnfcantly enhance the predctve power of a CPBRM. Bd-responses may dffer based on customers, channels, or product attrbutes such as warranty or payment terms. We capture these possble aspects n our models through a sngle countng varable j, where j = 1, 2, represents the number of dstnct, dscrete customer segments. Other factors such as the sze of the order or the compettve prce can often be modeled (dependng on the CPBRM) on a contnuous scale and may sometmes be treated separately. Logt Bd-Response Functon Phlps (2005b) & Boyd et al. (2005) both present the Logt functon as ther representaton of a CPBRM. As dscussed n Phlps (2005b, pg. 289), for a dataset wth j dstnct segments, the general form for the Logt functon s: 1 ρ( ) = j 1+ e p a + b p j j j.

10 One of the man advantages of the Logt functon s the ease of addng addtonal segmentaton factors such as the sze of the order, Q and the compettor s prce quote, p c,. If the segmentng varables can be used as contnuous varables, the model may be adjusted to nclude these segmentatons by addng the parameters c q to measure the effect of order quantty segmentaton and c c to measure the effect of the compettor s prce segmentaton. These parameters are multpled by Q and c, p respectvely: ρ( p Q, p ) = j 1+ e 1 c, a + b p + c Q + c p j j q c c, j. Note that a relatve prce rato may also be used n the Logt functon by replacng c c p c, wth c p n the equaton above. In our performance test, we found lttle dfference c( p /, ) c between these two representatons so we present the smpler form wth just the compettor s prce for the remander of ths paper. Usng the smplest form of the Logt functon: 1 ( p) =, the slope ρ ( p) and elastcty ε ( p) of the Logt functon s (Phlps a bp 1 + e ρ b pg. 284): ρ ( p) = bρ( p)(1 ρ( p)) and ε ( p) = bp(1 ρ( p)). Power Bd-Response Functon An alternate CPBRM sometmes used n practce s the Power functon, defned n ts general form as: ρ( p ) = α + r( p ) γ j j α j. The man advantage of the Power functon s that, compared to the Logt functon, compettve prce dynamcs are explctly captured. The man dsadvantage s that t s more cumbersome to adjust the model for non-prce, contnuous varable attrbutes. Segmentaton parameters can be added to the Power functon but only through a dscrete characterzaton. Thus, a varable such as order sze must be broken nto dscrete

11 ntervals and captured through the parameterγ, where the subscrpt j now represents the dscrete j ntervals of the order sze. Usng the smplest form of the power model wth no segmentaton, α ρ( p) =, the slope and elastcty of the Power functon s α + r( p) γ γ ρ ( p) = ρ( p)(1 ρ( p)) and ε ( p) = γ(1 ρ( p)). p For a CPBRM to be a strctly decreasng functon n p, the prce dependent parameters must be strctly greater than zero. More specfcally, for the Power functon: γ >0. The parameter γ s a measure of the prce senstvty of the buyer wth hgher values mplyng greater prce senstvty. The effect of the parameter γ on the probablty of wnnng s shown n Fgure 2. Prce Senstvty Probablty of Wnnng Gamma=5 Gamma =15 Gamma = Prce Rato Fgure 2 The parameter α s a measure of the prce premum the frm enjoys, wth a hgher value mplyng a hgher premum on the market. Thus, an ncreasng value ofα allows the frm to charge a hgher prce for the same probablty of wnnng. The effect of the parameter α on the probablty of wnnng s shown n Fgure 3.

12 Prce Premum Probablty of Wnnng Alpha=0.1 Alpha=1 Alpha= Prce Rato 3.2 Estmaton of Parameter Values Fgure 3 The parameter values of a CPBRM can be estmated statstcally by fttng a curve to the avalable bd-hstory data based on mnmzng the squared errors or usng maxmum-lkelhood estmates. We brefly descrbe each method below, usng the Power functon as the CPBRM of reference. (Phlps 2005b, pg. 285) descrbes how each method s appled to the Logt functon. Estmaton by Mnmzng the Squared Errors The frst method for estmatng parameter values s by mnmzng the squares of the error terms from the curve of the CPBRM to the actual wns and losses of each hstorcal bd opportunty. The procedure s as follows. Start by assgnng the ndcator varables bd was won and W =1 f the W = 0 f the bd was lost for each hstorcal bd opportunty. Remember that a CPBRM provdes a probablty of wnnng a bd for a gven prce. Thus, to determne the best estmates of the parameter values for a CPBRM, we want the bd-response to be as close to possble. Ths can be acheved by solvng the unconstraned optmzaton problem W as αγ, 2 [ ρ( α, γ) ]. Mnmze p W

13 Maxmum-Lkelhood Estmaton The second method for estmatng the parameter values s to choose parameter values that mmc, as close as possble, the same pattern of wns and losses as the actual outcomes. Assumng all bds are ndependent, the probablty of realzng the actual outcome for a partcular bd s L( α, γ) = ρ( p α, γ) W + [1 ρ( p α, γ)](1 W). Therefore, the parameters can be chosen n such a way that they maxmze the probablty of realzng the actual outcomes over all observatons. Ths can be acheved by solvng Maxmze [ ρ( p α, γ) W + [1 ρ( p α, γ)](1 W)]. αγ, A drawback of the method above, however, s when the amount of hstorcal bd data s small, the lkelhood of even a good predctor s also very small. Workng wth the product of very small numbers often creates computatonal problems. To avod ths problem, we nstead maxmze the natural logarthms of the lkelhood. Snce the logarthm s also an ncreasng functon of the probablty, the parameter values that maxmze the orgnal problem wll also maxmze the natural logarthm expressed by Maxmze ln[ ρ( p α, γ) W + [1 ρ( p α, γ)](1 W)]. αγ, The number of bd attrbutes (segments) that can be accurately estmated depends on the amount of hstorcal bd-nformaton avalable. If extensve nformaton s avalable, greater degrees of segmentaton can be acheved wthout compromsng the accuracy and robustness of the statstcal estmaton of the parameter values. After estmatng the parameter values of a CPBRM usng hstorcal bd data, the CPBRM can now be used to determne the optmal bdresponse prce for an upcomng bd opportunty. Ths process s descrbed n the next secton. 3.3 Use of a CPBRM n Prce Optmzaton We now look at how CPBRM curves can be used n prce optmzaton. For the followng dscusson, we use the objectve of maxmzng expected profts. However, other strategc or

14 operatonal objectves can be easly accommodated such as ncreasng market shares or ncludng constrants on capacty, nventory, prce or margn. The prce optmzaton problem for bd opportunty s Maxmze π( p ) = ρ( p ) ( p c ) Q. p p Note, the margn ( p c ) s strctly ncreasng n prce (Fgure 4) but the probablty of wnnng p the bd s strctly decreasng n prce (Fgure 5). Therefore, the expected proft s a unmodal functon as shown n Fgure 6. Margnal Deal Contrbuton Probablty of Wnnng Unt Prce Unt Prce Fgure 4 Fgure 5 Expected Proft Unt Prce Fgure 6

15 Determnng the optmal prce nvolves fndng a global maxma for the expected proft whch s unmodal n nature. The proft-maxmzng prce occurs where the elastcty of the expected proft functon s equal to the nverse of the margnal contrbuton rato, ε ( p) = * p * p c p. The dervaton s avalable from the authors by request. We have descrbed two CPBRMs and explaned how they can be used to fnd an optmal prce response for a specfc bd opportunty. In the next secton we demonstrate how to apply the CPBRMs to hstorcal bd data and test them on two ndustry datasets correspondng to two extremes of nformaton avalable to the user. 4. Numercal Comparsons of CPBRMs on Industry Data In ths secton, we compare the performance of the two CPBRMs descrbed n the prevous secton usng two bd-hstory ndustry datasets. The frst dataset contans a snglecompettor settng where extensve bd hstory s avalable ncludng the compettor s prce at each bd opportunty. Ths dataset represents a very favorable settng for applyng CPBRMs. The second dataset contans a mult-compettor settng where very lmted bd hstory s avalable, representng a more challengng envronment to mplement CPBRMs. Such dsparate datasets allow us to contrast the mprovements acheved from usng CPBRMs at the extremes of ther sutablty and compare the effectveness of the two competng models under dfferent envronments. 4.1 Test Case Scenaros We test the two CPBRMs under a wde set of scenaros pertanng to: 1) the amount of hstorcal bd data avalable, 2) the amount of knowledge of the compettors prce response to the current bd request, and 3) the amount of segmentaton ncluded n the models based on the sze of the order n each bd opportunty. We capture senstvty to the amount of hstorcal bd data

16 by testng the models on two ndustry datasets, one wth a large amount of hstorcal bd data (Dataset 1) and the other wth a small amount (Dataset 2). We descrbe the two datasets below. Bd Hstory Datasets Dataset 1: Sngle Compettor and Extensve Hstorcal Data The frm provdng our frst dataset manufactures and sells medcal testng equpment to laboratores at hosptals, clncs, and unverstes across North Amerca. One of ther popular products s a gas chromatograph refll cartrdge that has a lst prce of $ The margnal cost assocated wth each unt s $6.00. The refll cartrdges are ordered n batches rangng n sze from 100 to over Orders for fewer than 200 unts are handled through the company s webste or through resellers wth no assocated dscount from the lst prce. At the other extreme, the company receves about 100 orders per year for more than 1000 unts. These large deals are negotated by a natonal account manager, usually as part of a much larger sale. Orders for unts are handled by regonal sales staff that has consderable leeway wth regard to dscountng. We only look at ths mddle-sze segment to apply the CPBRMs. The requested sze of the order for each bd opportunty s also recorded, allowng us to test both segmented and unsegmented versons of the CPBRMs. Because of the specalzed nature of the product, the frm has only one sgnfcant compettor and they are able to capture ther compettor s prce after each bd opportunty. Ther bd hstory nformaton s exhaustve, wth approxmately 2400 records of prevous bd opportuntes. A snapshot of ths dataset s shown n Table 2. Table 2: Bd Hstory for a Medcal Devce Company Bd Number Wn Frm s Bd Compettor s Bd Order Sze 1 Y $8.44 $ N $11.88 $ N $11.29 $ Y $9.78 $ Y $9.28 $

17 For ths applcaton, the probablty of wnnng the bd at a prce equal to the compettor s prce (.e. prce rato s one) s 51% (Fgure 7). Ths percentage mples the frm doesn t enjoy any sgnfcant postve or negatve prce premum compared to ts compettor. Bd-Response Curve Probablty of Wnnng (1, 0.51) Prce Rato Fgure 7: Bd-response Curve for Dataset 1 Dataset 2: Several Compettors and Lmted Hstorcal Data Our second applcaton s hstorcal bd data from a federal agency (Foregn Food Assstance Program) acceptng bds from food mlls to provde bulk food commodty tems such as wheat, soybeans, and vegetable ol. Unlke the dataset from the medcal devce company, the amount of hstorcal bd data avalable to us was very lmted (approxmately 50 hstorcal bd opportuntes). For our test, we choose one of the food mlls that partcpated n the majorty of the bd opportuntes. Unlke n the frst dataset, n ths dataset there were multple bds from competng food mlls for each bd opportunty. Thus, we used an average of the compettors bds to represent the compettve bd parameter n our CPBRMs. We chose an average of the compettors bds rather than a mnmum or the wnnng bd because ths bddng stuaton s truly a case where the lowest bd does not always wn. Delverng food supples to foregn natons nvolves both purchasng the

18 food from the mlls and transportng the food to the naton n need. Whle the food mlls can control the prce they respond to the bd request, they have no control over the bds the transportaton companes also respond wth for delverng the food from the mll to the foregn country. Thus, there s an nherent degree of randomness n these bd opportuntes. We assumed a margnal per-unt cost (cost of provdng 1000 pounds) of $ for our test mll. A snapshot of the data (after substtutng n the average compettors bd) s provded n Table 3. Table 3: Bd Hstory for Bulk Food Provders Bd Number Wn Frm s Bd Average of Compettors Bds Order Sze (000) lbs 1 N $ $ N $ $ Y $ $ N $ $ Y $ $ In both datasets, the compettors prces were recorded for each bd opportunty. Thus, the data avalable s how the compettors prced on past bd opportuntes. The Power functon (and the Logt functon wth a compettors prce segmentaton parameter) requres an nput of the compettors prce for the current bd opportunty. In some cases, frms may have very lmted knowledge of how ther compettors wll prce n an upcomng bd opportunty whle, n other cases, frms may have substantal knowledge. We descrbe how we capture these dfferng knowledge levels below. Knowledge Level of Compettors Prcng We tested the two CPBRMs under three dfferent levels of knowledge a frm may posses regardng ts compettors prcng,.e. worst, medum, and best cases. Hstorcal compettve bdprce nformaton s often avalable n many B2B applcatons through ether formal or nformal channels, dependng on the relatonshp the bdder shares wth the buyer. UPS, for example,

19 obtans compettors bds n approxmately 40% of the parcel shppng bd opportuntes they partcpate n (Knple, 2006). In some busness-to-busness scenaros, a frm may even be provded wth the compettors bds and asked to respond wth a quote of ther own (note that for reasons explaned earler, provdng the lowest bd does not always guarantee a wn n these stuatons). In many busness-to-consumer markets such as loan and nsurance quotes, nformaton about the compettor s prce may be avalable from a smple web-page search. Worst Case: No Prce Informaton Case: In ths case, the frm has no hstorcal prce nformaton on ts compettors, nor does t have any nformaton about how ts compettors wll prce for the current bd opportunty. Ths scenaro s rare n practce but, for our analyss, serves as a lower bound on the knowledge of compettors prcng. Wth no compettor prce nformaton, the Logt functon s the only CPBRM avalable, as the Power functon requres an estmate of the compettor s prce n the current perod (va the prce rato). Medum Case: Naïve Prce Estmaton Case: In ths medum case of compettve prcng knowledge, the frm has no nformaton about how ts compettors wll prce n the current perod except for the prce hstory of ts compettors on past bddng opportuntes. Thus, the frm can estmate ts compettors prces for the current perod through some type of forecastng or regresson model. In our analyss we use a smple 10-perod movng average to predct the compettors prces n the current perod. Whle a movng average of the hstorcal prces s most lkely not the best forecastng method for ths applcaton, t was chosen because t represents a technque that s common n practce. We expermented wth movng averages of dfferent numbers of perods but found the 10-perod movng average resulted n the most accurate and least based estmates for the future compettor s bd. Now that we have an estmate for the compettor s prce, we can test both the Logt and Power functons as we now have an estmate for the prce rato requred to use the Power functon. Best Case: Perfect Compettve Prce Knowledge: In ths best case of compettve prcng knowledge, the frm knows exactly what ts compettors bds wll be n the current

20 perod. Ths can be consdered an upper bound on the frm s forecastng capabltes. It also apples to cases where the buyer provdes compettors bds before requestng a bd from the frm or n applcatons where a frm can check ts compettors prces (possbly va ther web pages) before respondng wth ts own prce quote. The chart below summarzes whch knowledge levels were tested for each CPBRM. Logt Power Worst Case Medum Case Best Case The next secton descrbes the procedure we used to test the two CPBRMs on the datasets and scenaros descrbed above. 4.2 Procedure for Testng CPBRMs The approach we used for both datasets s as follows: 1. We dvded the datasets nto two dstnct sets; the frst for estmaton of the model parameter values and the second for performance evaluaton (smlar to a holdout sample commonly used n forecast method evaluatons). We used the frst 90% of the hstorcal bd records as our estmaton data and the remanng 10% as our performance test data. Whle the choce of 10% for the performance test may seem arbtrary, t s a common choce for holdout samples n forecast methods evaluatons. Senstvty test wth dfferent percentages of the hstorcal data used for measurng performance were also performed. The changes n the parameter estmates and performance results on dataset 1 were nsgnfcant when tested over a range of 10% - 20% for the performance test dataset. The same was not true for the second dataset however, due to the small sample of hstorcal bds avalable. Thus, we decded to stay wth the 10% sample to provde the best opportunty for obtanng good parameter value estmates.

21 2. Usng the estmaton data, we calculated the parameter values for both the Logt and Power functons usng ordnary least squares and maxmum lkelhood estmators. We found lttle dfference n the ft of the models between the two estmaton methods so we present the values found usng maxmum lkelhood estmators for the remander of ths procedure. The parameter values from the un-segmented analyss of Dataset 1 are: Logt Model Power Model c α γ a B c Worst Case NA NA NA Medum Case Best Case and the parameter values from the un-segmented analyss of Dataset 2 are: Logt Model Power Model c α γ a B c Worst Case NA NA NA Medum Case Best Case Because the Power functon requres an estmate of the compettors prce, t can not be used under the Worst nformaton case. The reason the parameter values are the same for the Medum and Best nformaton cases s because past bd opportuntes are used for estmatng the parameter values when the compettors prce s known wth certanty (note the nformaton cases pertan to knowledge of the compettors prce n the current perod; the past prces are assumed to be known wth certanty). In the next secton, we descrbe how we also used the order sze as a segmentaton varable. 3. After estmatng the parameter values for each model, we used the CPBRMs to optmze the bd-prces for all the bds n the performance test data subset. It s mportant to test the models on a dfferent set of data than was used to buld the models; else the ft of the models wll be msleadngly hgh.

22 4. Fnally, we computed the percent mprovements over expected profts and over actual profts as explaned n secton 4.3. Ths provded us wth two metrcs of performance for each of our cases. The ncorporaton of compettors prces s only one possble nput to CPBRMs (although for the Power functon t s a requred nput). Another common nput s the sze of the order request. It s reasonable to assume the prce senstvty of customers only orderng a few unts wll dffer sgnfcantly from customers orderng large quanttes. Thus, we descrbe how we ncorporated dfferent levels of order sze segmentaton below. Segmentaton Based on Order Sze In Dataset 1, order quanttes range between 200 and For segmentaton based on the order sze usng the Logt functon, we used the order sze as a contnuous varable and estmated a statstcal constant dependent on the order sze Q as follows: 1 ρ ( p Q, p ) = e c a bp cqq ccpc 1 For dscrete segmentaton, a separate parameter must be estmated for each segment but for a contnuous varable, the estmaton of only one parameter s requred. Therefore, t s easer to use a contnuous varable f the model allows t. For segmentng based on the order sze usng the Power model however, we had to use a dscrete approach. We chose the eght segments below. Order Sze Between Segment To estmate the parameter value for each segment, we used a bnary ndcator varable x j, j = 2, 3,..9, whch was assgned a value of one for the order sze segment a specfc bd fell under. Ths classfcaton scheme s demonstrated n the table below: Bd Wn Frm's Compettor's Order Indcator Varables for Order Sze Segments Number Bd Bd Sze $8.44 $ $11.88 $

23 The bd response for the Power functon was calculated by estmatng a dfferent value of γ (our j estmates for α dd not change so we held t constant) for each order sze segment q as follows: ρ( p q) = α ( ) j x α + r p γ j. Based on our maxmum lkelhood fts, the estmated parameter values for the segmented analyss of Dataset 1 are: Knowledge of Comp Prce Logt functon a b c c c q Worst Case NA Medum Case Best Case Power functon α Medum Case Best Case The amount of hstorcal bd data avalable n Dataset 2 was nsuffcent to segment the data based on the order sze. Therefore, we dd not perform a segmented analyss on ths dataset. In summary, we compared the performance of the Logt and Power CPBRMs usng two ndustry datasets, three levels of knowledge of the compettors prces, two levels of segmentaton on the order sze, and usng two performance measures. Thus, we had a total of 24 scenaros to base our observatons. Table 4 summarzes the varous scenaros. Table 4: Summary of Test Scenaros Hstorcal Bd Data Avalable Knowledge of Compettors Prce Segmentaton on Order Sze Performance Measure Large (Dataset 1) Worst Case Segmented Actual Profts Small (Dataset 2) Medum Case Unsegmented Expected Profts Best Case In the next secton we descrbe how we measured the performance of the two CPBRMs. γ

24 4.3 Measures of Performance To test the mpact of usng CPBRMs on the ndustry datasets, we used two performance metrcs: percent mprovement n profts over un-optmzed actual profts and percent mprovement n profts over un-optmzed expected profts. To understand the dfference between the two performance metrcs, consder the followng numercal example from Dataset 1: Bd Wn Order Sze Orgnal Bd Optmal Bd Probablty of Wn Probablty of Wn at Orgnal Bd at Optmzed Bd 1 Y 353 $ 8.44 $ Applyng the unsegmented, worst nformaton case, Logt CPBRM wth the parameter values 1 obtaned through the procedure outlned n secton 4.2 we get: ρ( p) = p 1 + e Substtutng n the orgnal bd prce of $8.44, we calculate the probablty of wnnng for the unoptmzed bd = *$8.44 1/(1 e + ) + =0.79. Applyng the optmzaton procedure descrbed n secton 3.3, we calculate the optmal bd prce for ths bd opportunty should have been $9.35. Substtutng n ths prce results n a probablty of wnnng for the optmzed bd = *$9.35 1/(1 e + ) + = The actual proft from ths bd opportunty s = (Orgnal Unt Prce- Margnal Cost)* Order Sze* Wn/Loss Indcator Varable = $(8.44-6)*353*1 = $ If the orgnal bd had resulted n a loss, the actual proft would be zero. The orgnal bd expected proft = (Orgnal Unt Prce-Margnal Cost)* Order Sze * Probablty of Wn at the Orgnal Bd Prce = $(8.44-6)*353*0.79 = $ Note the expected proft s always smaller than the actual proft when the bd was won and s always larger when the bd was lost. The optmzed bd expected proft = (Optmzed Prce- Margnal Cost)*Order Sze* Probablty of Wn at the Optmzed Bd Prce = $ (9.35-6)*353*0.64= $ We now compare the percent mprovement of the latter case over the frst two:

25 Percent Improvement n Optmzed Bd Expected Profts over Un-Optmzed Bd Actual Profts = ($ $861.32)/$ = = % Percent Improvement n Optmzed Bd Expected Profts over Un-Optmzed Bd Expected Profts = ($ $680.44)/$ = = 11.20%. The calculatons above were performed for every bd opportunty n the performance test data subset and the average of each measure (over each bd opportunty n the performance test data subset) was used as the performance metrcs presented n the next secton. 4.4 Results from Numercal Comparsons Performance on Dataset 1 For Dataset 1, the un-optmzed bd actual and expected profts over the performance test subset of our dataset were exactly the same. Therefore, we only present one comparson of the percent mprovements n expected profts for each scenaro (segmented versus unsegmented, three levels of nformaton, and two CPBRMs). Fgure 8 presents the comparson of the models when the models were not adjusted for order sze segmentaton. For ths comparson, the Logt functon outperforms the Power functon n the absence of segmentaton. Performance on Dataset 1 ( Unsegmented) 40% 30% 20% Logt Pow er 10% 0% Worst Case Medum Case Best Case Fgure 8: Percent Improvement n Expected Profts (Unsegmented)

26 Fgure 9 presents the comparson wth an adjustment for order sze segmentaton. In the presence of order sze segmentaton, the Power functon outperforms the Logt functon. Performance on Dataset 1 (Segmented) 40.00% 30.00% 20.00% s Logt Pow er 10.00% 0.00% Worst Case Medum Case Best Case Fgure 9: Percent Improvement n Expected Profts (Segmented) In both the unsegmented and segmented comparsons, the ablty to perfectly forecast nformaton about the compettors prcng results n the largest mprovements. Havng access to hstorcal data on compettors bds can sometmes make a company worse off however, as evdenced by the worst nformaton case (no hstorcal compettors prce data) outperformng the medum nformaton case (10 perod movng average of compettors prces). A quck check of our 10-perod movng average forecast whch we used to provde the compettor s prce estmate showed the forecast was unbased but had a large varance n the forecast error. Thus, for ths dataset, the past hstorcal compettor s bds were poor ndcators of how the compettor wll bd n the future. The use of these poor estmates led to worse performance usng the CPBRMs than f no estmate of the compettor s prce was used at all.

27 Performance on Dataset 2 For ths applcaton, the probablty of wnnng the busness at a prce equal to the average compettors prce (.e. a prce rato of one) was 40.91% (Fgure 1 provdes the actual bd-response curve for the Power functon). Due to the small number of hstorcal bd records avalable to us n ths dataset, we performed our analyss for the un-segmented scenaro only. The percent mprovements from usng the optmzed prce from the CPBRMs on the performance test part of the datasets are shown n Fgures 10 and 11 compared aganst actual and expected profts respectvely. Smlar to our comparsons n the frst dataset, the case of perfect compettors prce nformaton results n the largest mprovements for both models and the Logt functon outperforms the Power functon n all nformaton cases. Unlke n the prevous dataset however, there were dfferences n actual and expected profts usng the un-optmzed prce and the medum nformaton case resulted n larger mprovements than the worst nformaton case. Performance on Dataset 2 (Actual Profts) 30.00% 20.00% 10.00% 0.00% % % % % Worst Case Medum Case Best Case Logt Power Fgure 10: Percent Improvement n Actual Profts (Unsegmented)

28 Performance on Dataset 2 (Expected Profts) % % 80.00% 60.00% 40.00% Logt Pow er 20.00% 0.00% Worst Case Medum Case Best Case Fgure 11: Percent Improvement n Expected Profts (Unsegmented) Fgure 10 shows the mprovements when compared to the actual profts are negatve for the two lower cases of nformaton on the compettors prce. We suspect ths s the result of nsuffcent data avalable for the performance test data as mprovements over expected profts were postve for all three nformaton cases as shown n Fgure Observatons from Numercal Comparsons and Conclusons In ths secton, we summarze our observatons based on our numercal comparsons and attempt to answer the queston: Gven a partcular set of condtons, whch CPBRM should a frm use to optmze prces? We then summarze our work and provde areas for future research. Our observatons come wth the followng caveats; they are based purely on the performance of the models on our two avalable datasets and may not be generalzable to applcatons dfferent than the ones tested. Thus, a frm should rgorously test the models usng ther own bd hstory data before drawng conclusons on the sutablty of a partcular model for ther specfc applcaton. Our prmary purpose s to descrbe a testng procedure for frms who wsh to do so. Based on the performance on our two ndustry datasets, we provde three man observatons:

29 Observaton 1. If enough hstorcal bd data s avalable to segment based on the order sze, the Power functon, adjusted for each dscrete customer segment, outperforms the Logt functon. Ths observaton s evdent n Fgure 9 whch compares the models developed wth a large amount of hstorcal bd data and segmented based on the order sze. If the frm has no knowledge about the compettors prces for the current bd opportunty however, the Power functon can not be used. Observaton 2. If suffcent hstorcal bd data s not avalable to a frm or the frm does not segment based on the order sze, the Logt functon outperforms the Power functon. Ths observaton s evdent n Fgures 8, 10, and 11. Fgure 8 compares the models developed wth a large amount of hstorcal bd data but not segmented based on the order sze. Fgures 10 and 11 compare the models developed usng the much smaller dataset where order sze segmentaton s not an opton. Observaton 3. Incorporatng hstorcal compettor prces nto a CPBRM does not ensure better performance. Ths observaton s evdent n Fgures 8 and 9 whch compare the models developed wth a large amount of hstorcal bd data. For both the unsegmented and segmented versons of the model, the worst case of compettor prcng nformaton (estmated compettors prces are not ncluded n the model) outperforms the medum case (compettors prces estmated through a 10 perod movng average). Thus, frms who have access to hstorcal compettors bd prces but are not profcent n usng ths data to forecast future prces may be better off leavng ths varable out of ther CPBRMs. A frm adoptng CPBRMs for prce optmzaton needs to be aware of ther lmtatons. CPBRMs assume the bd opportuntes are exogenous and are not affected by the bd responses suggested through the optmzaton model. In realty, a frm s prcng strategy may have a sgnfcant mpact on customer retenton, especally f the optmzaton model recommends

30 consstently prcng hgher than the competton for a partcular customer class. Also, CPBRMs, and ther correspondng optmzaton models, do not assume any response from the frm s compettors. Instead, they assume the actons of compettors can be determned probablstcally and ndependently of the decson maker s actons. In realty, compettors may react to a frm s new prcng strategy causng the hstorcal bd opportunty data to be unrepresentatve of future bd prce responses. To help detect these possbltes, mechansms should be put n place to montor and evaluate the performance of the CPBRMs over tme. If compettors change ther bd-prcng behavor due to the mplementaton of a CPBRM, more nvolved models usng concepts from game theory should be employed. In summary, we present two CPBRMs used n practce and explan how they may be used to calculate optmal bd-response prces and dscuss how they may be adjusted to accommodate segmentaton based on the dfferent levels of nformaton avalable. We present a numercal analyss on two ndustry datasets and, based on these results, offer a set of recommendatons about the type of CPBRM a frm should use dependng on the avalablty of past nformaton and the level of compettve knowledge avalable to a frm. Acknowledgments: The authors wsh to thank Loren Wllams for hs nsghtful comments, John Trestral of Nextwave for provdng the second ndustry data set, and Robert Phlps for provdng the frst ndustry data sets and for nformng us of the Power model. We also thnk the partcpants from presentaton gven at Unversty of Notre Dame and the Unversty of Maryland for ther helpful suggestons.

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