Recommender Systems in Food Retail: Modeling Repeat Purchase

Decisions on Transaction Data of a Stationary Food Retailer

Thomas Neifer

1,2

, Dennis Lawo

1,2

, Gunnar Stevens

1,2

, Alexander Boden

and Andreas Gadatsch

Verbraucherinformatik Research Group, University of Siegen, Siegen, Germany

Institut f

ur Verbraucherinformatik, University of Applied Sciences Bonn-Rhein-Sieg, Sankt Augustin, Germany

Keywords:

Recommender Systems, Food Retail, Repeat Purchase Recommendations, Bayesian Hierarchical Model.

Abstract:

In the course of growing online retailing, recommendation systems have become established that derive rec-

ommendations from customers’ purchase histories. Recommending suitable food products can represent a

lucrative added value for food retailers, but at the same time challenges them to make good predictions for

repeated food purchases. Repeat purchase recommendations have been little explored in the literature. These

predict when a product will be purchased again by a customer. This is especially important for food recommen-

dations, since it is not the frequency of the same item in the shopping basket that is relevant for determining

repeat purchase intervals, but rather their difference over time. In this paper, in addition to critically reﬂecting

classical recommendation systems on the underlying repeat purchase context, two models for online prod-

uct recommendations are derived from the literature, validated and discussed for the food context using real

transaction data of a German stationary food retailer.

1 INTRODUCTION

In times of digital transformation, enormous impor-

tance is attached to data. This manifests itself not least

in current trend topics in literature such as Big Data

and Data Science (Loebbecke and Picot, 2015). Food

retailers have also long since discovered such oppor-

tunities for themselves: A well-known example is the

insight into the connection between the purchase of

beer and diapers on the weekend. The information ob-

tained about consumer behavior has been used to op-

timize advertising and pricing mechanisms (Fu et al.,

2000). However, this requires a broad database (Chen

et al., 2012), which has led to asymmetric business

models such as ”Payback”, which collect user data

through discounts or loyalty programs and make it

available to cooperation partners in anonymized form

(Hofman-Kohlmeyer, 2016; Stevens et al., 2017).

But food retailing is not only experiencing change

from a digital perspective: consumers’ lifestyles are

also currently transforming strongly. Health, ecologi-

cal, ethical, social and culinary issues are gaining im-

portance. Nutrition and eating habits should no longer

merely satisfy hunger, but be an expression of the con-

sumer’s individual personality. A growing health con-

sciousness among consumers is also having an impact

on the demand for fresh and healthy foods. In the

course of this, the regionality and origin of products

in terms of quality, environmental awareness and eth-

ical aspects are coming into the focus of buyers in or-

der to eat healthy and climate-conscious (Hutapea and

Malanowski, 2019; Lawo et al., 2019; Stevens et al.,

2017). This leads to growing demands on food and

an increasing need for product variety in food retail

(Hutapea and Malanowski, 2019).

Especially in online retail, which is predestined

for the collection of user data (Jakobi et al., 2020),

transaction, behavioral, and rating data is used to en-

sure a personalized experience for customers by pro-

viding them with relevant content through recommen-

dation systems (Talasu et al., 2017). In online food

retailing, Amazon Prime Pantry, for example, also re-

lies on the use of recommendation systems to design

customer-centric marketing activities (Dokras, 2017).

However, recommendation systems are not yet

omnipresent in food and online retail. While click

pattern and user preference analyses are still relatively

easy to integrate (Poggi et al., 2013; Xu et al., 2011),

the more complex modeling of customers’ repeat pur-

chase behavior is particularly important in the food

sector. This is due to the fact that purchase deci-

sions there are often habitualized and therefore the

question is not what to buy but when to buy it (Kaas

and Dieterich, 1979; Ehrenberg, 2000). While estab-

Neifer, T., Lawo, D., Stevens, G., Boden, A. and Gadatsch, A.

Recommender Systems in Food Retail: Modeling Repeat Purchase Decisions on Transaction Data of a Stationary Food Retailer.

DOI: 10.5220/0010553600250036

In Proceedings of the 18th International Conference on e-Business (ICE-B 2021), pages 25-36

ISBN: 978-989-758-527-2

lished studies and models in the context of marketing

research mostly deal with the consideration of spe-

ciﬁc products or brands (Fader et al., 2005; Morri-

son and Schmittlein, 1988), there are, however, ﬁrst

approaches that discuss recommendation systems for

repeat purchase behavior using online tracked data

(Bhagat et al., 2018; Dey et al., 2016).

This paper critically derives the problems of exist-

ing recommender systems for food retail, and builds

on them to validate current models for repeat purchase

recommendations for this domain. However, in con-

trast to prior research, our work examines real-world

transaction data from stationary supermarket termi-

nals of a large German food retailer, as purchases

in food retail mainly take place ofﬂine (Pitts et al.,

2018). This represents a major difference to online

tracked data, which can usually be collected in a more

structured and traceable form (Jakobi et al., 2020).

The paper concludes with a critical discussion of the

results and identiﬁcation of possible improvements to

make the models more suitable for both online and

ofﬂine food retailing.

2 RECOMMENDER SYSTEMS

2.1 General

Recommender systems are designed to support users

in their (future) decisions based on their previous us-

age history and that of other users. In principle,

they can be differentiated into non-individual, col-

laborative, content-based, knowledge-based, demo-

graphic and hybrid ﬁlters (Aggarwal, 2016; Bobadilla

et al., 2013). While non-individual recommenda-

tions are the same for all users and thus lack per-

sonalization of the respective products (e.g., the most

clicked products) Bobadilla et al. (2013), collabora-

tive ﬁltering (CF) examines the preferences of dif-

ferent users based on their consumption and usage

patterns to identify similar individuals or items (e.g.,

movies on Netﬂix). Recommendations are made ei-

ther based on the similarity of two items in terms

of their user ratings (”item-based”) or on their rat-

ings of similar users (”user-based”) (Sarwar et al.,

2001; Zhang et al., 2017; Linden et al., 2003). In

contrast to collaborative approaches, content-based

ﬁlters generate their recommendations based on the

characteristics and content of the items already con-

sumed. For example, items similar to a user’s previ-

ous ones (e.g., articles about science and technology)

are recommended (Van Meteren and Van Someren,

2000; Bobadilla et al., 2013; Miranda et al., 1999).

Knowledge-based ﬁlters are based on satisfying cus-

tomer needs through speciﬁcally deﬁned product fea-

tures. Here, explicit rules are used to generate rec-

ommendations (e.g., speciﬁcations such as size, min.

and max. price, or zip code when buying a house)

(Aggarwal, 2016). Demographic ﬁltering makes rec-

ommendations based on a socio-demographic proﬁle

(e.g., age, gender, nationality) of a user (Thorat et al.,

2015). Hybrid approaches represent a combination of

different ﬁlters. These are often used to circumvent

problems of individual methods as well as to increase

the accuracy and efﬁciency of the ﬁltering (Aggarwal,

2016; Thorat et al., 2015).

In recent years and mainly driven by online re-

tailing, collaborative and content-based recommenda-

tion methods have become established (Breese et al.,

2013; Linden et al., 2003). However, these systems

have fundamental problems that are problematic for

recommending food products for repeated purchase.

2.2 Issues for Food Retail

2.2.1 Data Distribution

A fundamental problem of recommender systems is

the cold start problem, which leads to inaccurate pref-

erence capture for new users or products due to a

sparsely populated customer-product matrix (Thorat

et al., 2015). This is particularly problematic in food

retailing given that the ”long tail” of items (e.g., niche

items) has only sporadic ratings and therefore will

be difﬁcult to predict (Clement et al., 2019). This

is especially affecting the collaborative approaches,

as they are based on historical customer preferences.

Since user-based collaborative ﬁltering is built on

comparing item scores from different users, many

neighborhood scores are needed for a speciﬁc item

(Brusilovsky, 2007). Content-based methods are not

affected as much, since they determine similarities

based on item descriptions and thus recommend prod-

ucts for which there are no reviews yet (Thorat et al.,

2015). Due to dimension reduction, the matrix factor-

ization (MF) approach can lead to better results (Do

et al., 2010; kumar Bokde et al., 2015). In the con-

text of probabilistic model-based methods, such as the

Bayes classiﬁer, missing values are ignored in proba-

bility estimation (Isinkaye et al., 2015). However, in

content-based systems, the so-called portfolio prob-

lem occurs, which ensures that only items are rec-

ommended due to overspecialization, which are very

similar to already recommended products (Tintarev

and Masthoff, 2006).

For the transaction data in food retail, this means

that a recommendation system would need to have

many purchases of the same product in order to form a

ICE-B 2021 - 18th International Conference on e-Business

neighborhood of similar customers. This is especially

relevant regarding the cold start problem, so that new

customers can be quickly assigned to a suitable cus-

tomer group to be able to make appropriate recom-

mendations. However, when it comes to sparse pur-

chasing data with only a few comparable purchases,

this leads to problems in generating neighborhoods

and the model does not work accurately.

2.2.2 Scalability

Another problem arises from the resource intensity of

the algorithms for computing the optimal neighbor-

hood. The demand for time and memory increases

linearly with the number of users and scores (Zhang

et al., 2017; Brusilovsky, 2007). For a food retailer

with a broad product and customer base that wants

to make recommendations in a split second, such

an algorithm would create time and cost pressures.

This can be countered by means of subsampling and

model-based methods. With subsampling, only a sub-

set of users is selected at a time, which is intended

to relieve the storage capacity. However, the compu-

tation of the neighborhood remains ﬁxed (Lee et al.,

2012). In the context of model-based clustering meth-

ods, users are grouped into clusters based on com-

mon properties. The active user is now compared to a

group of users rather than individuals, so that the clos-

est neighborhood can be quickly identiﬁed. However,

problems with missing data also arise here when the

distance functions lead to non-intuitive and unstable

clusters (Lam and Riedl, 2004; Johnson, 1967; Lin-

den et al., 2003). Bayesian classiﬁcation also provides

an advantage here, as it is a probabilistic model that

represents the historical data and therefore can per-

form classiﬁcations without having to retrieve the en-

tire customer-product matrix (Aggarwal, 2016). Fur-

thermore, MF approaches also provide acceleration

of recommendations by reducing dimensionality (Sar-

war et al., 2002).

As many transactions are needed to calculate

neighborhoods, this leads to a high computational ef-

fort and accordingly raises resource issues. Memory-

based methods therefore seem rather unsuitable for

this type of recommendation, which is why prob-

abilistic (Bayesian) models in particular can offer

added value here.

2.2.3 Inherent Meaning

Collaborative ﬁltering is based on the assumption that

users have common preferences. The more homo-

geneous the preferences of different user groups, the

more functional the model will be. Furthermore, col-

laborative ﬁltering is particularly suitable for subjec-

tive characteristics (e.g., musical taste) that inﬂuence

a selection decision. If, on the other hand, there are

predominantly objective quality criteria (such as price

changes), which do not have to be weighed against

each other, other models should be used. Accord-

ingly, a homogeneity of the items is also desirable.

Thus, they are similar with regard to their objec-

tive criteria and differentiate only by subjective char-

acteristics (e.g., music albums usually have a simi-

lar price, a similar length and similar sale channels)

(Brusilovsky, 2007; Linden et al., 2003; Zhang et al.,

2017).

In the case of purchasing data, this homogeneity

is usually not given, since it is precisely the objective

characteristics that cause behavioral changes in cus-

tomers and are therefore often addressed by market-

ing (e.g., weekly offers). Due to the resulting trade-

off between subjective and objective criteria, collabo-

rative ﬁlters can be seen as problematic for the gener-

ation of recommendations in food retailing.

2.2.4 Data Persistence

The temporal validity and relevance of the data should

also be assessed. From the requirements of data dis-

tribution, the problem arises that items that are only

relevant for a short period of time (e.g. daily news) are

rather less suitable for collaborative ﬁltering, since in

principle few ratings can be expected. Historical rat-

ings are also less helpful when users’ tastes change

quickly (e.g., preferences for clothing items that have

since gone out of style) (Zhang et al., 2017; Linden

et al., 2003; Lee et al., 2012; Brusilovsky, 2007).

In the food sector, the temporal validity of the data

could play a subordinate role, since the tastes of the

customers are mostly habitualized or develop there

(Kaas and Dieterich, 1979).

2.2.5 Synonomy

Many recommender systems have problems when

similar and closely related products have different

names (e.g. clothes and dresses). Collaborative ﬁlters

are often not able to ﬁnd a match between such items

and therefore do not calculate their similarity. This

problem is solved, for example, by the automatic term

expansion (Liphoto et al., 2016; Alani et al., 2000), as

well as the singular value decomposition (SVD) in the

context of MF (esp. Latent Semantic Indexing) (Sar-

war et al., 2001).

Again, food transaction data cause a problem,

because there are often many very similar names

for an almost identical product. For example, dif-

ferent names for the same mineral water (”Still”,

”Sparkling”) could lead to model inaccuracies. Here,

Recommender Systems in Food Retail: Modeling Repeat Purchase Decisions on Transaction Data of a Stationary Food Retailer

extended methods of term expansion or machine

learning methods for word embeddings, i.e. the re-

construction of linguistic contexts based on words,

should be used (Gong, 2010; Lawo et al., 2020).

3 INTEGRATION OF THE TIME

FACTOR

Most recommendation systems are based on static

principles, as they only take into account information

about whether a user buys a product or not. How-

ever, the time factor is also already integrated in some

recommendation systems. In principle, two types of

time-based data are distinguished in the literature:

The product introduction time and the time of pur-

chase (implicit) or evaluation (explicit) (Park and Lee,

2006). Tang, Winoto, and Chan integrate temporal

characteristics of items to reduce relevant candidate

sets in the context of movie recommendations (pro-

duction year), improving the accuracy of recommen-

dations (Tang et al., 2003). Ding, Li, and Orlowska

consider user rating time to optimize an item-based

collaborative ﬁltering system. Here, weights are cal-

culated based on the rating times of different items

(Ding et al., 2006). Lee, Park, and Park developed

a time-based collaborative ﬁltering system using im-

plicit data (transactions). It is based on a time-based

pseudo-rating matrix that takes into account product

introduction time and purchase time under the as-

sumption that a user’s current preferences are dis-

proportionately inﬂuenced by more recent purchases

and that more recent items exhibit higher user interest

(Lee et al., 2008).

Lathia, Hailes, and Capra (2009) focus on the ef-

fect of weekly retraining within a CF algorithm as

a time-dependent predictive model. As an adaptive

temporal CF method, it adjusts the neighborhood size

of a k-Nearest Neighbor approach based on perfor-

mance measured up to the current time (Lathia et al.,

2009). Koren sees an inevitable need to incorporate

temporal changes into a recommender system. He de-

ﬁnes an MF model that analyzes temporal change be-

havior over the entire data history and validity (Koren,

2009). Cho, Cho, and Kim consider customer pur-

chase sequences over time to optimize recommenda-

tion quality. They use an extended customer-product

matrix, which shows the purchase times in addition

to the products. Furthermore, they cluster the trans-

actions into homogeneous subclusters using the self-

organizing map (SOM) technique. A change of the

cluster membership by each individual transaction of

a customer deﬁnes its purchase sequence. The time-

dependent change in a customer’s cluster membership

also enables prediction of a customer’s future pur-

chases. A recommendation is generated by the system

following the identiﬁcation of the most similar pur-

chase sequences compared to the current customer,

generating a set of products that the active customer

has is most likely to purchase based on the N most fre-

quently purchased products in the cluster (Cho et al.,

2005).

The ”Eigentaste” algorithm of Nathanson et al.

analyzes the time-related changes in customer prefer-

ences when selecting a product to recommend based

on the last evaluation (Nathanson et al., 2007). Chu

and Park describe a machine learning algorithm that

can improve the recommendations of new products

by continuously updating time-based features (e.g.,

popularity, freshness) in relevant content proﬁles per-

sonalized (Chu and Park, 2009). The time factor is

also being integrated into initial Deep Learning ap-

proaches for recommender systems. Most companies,

which do not have access to long usage histories, have

so far had to resort to item-to-item recommender sys-

tems. Hidasi et al. use recurrent neural Networks

(RNN) to address the problem of only short session-

based datasets in a memory-based approach (Hidasi

et al., 2015).

4 REPEAT PURCHASE

RECOMMENDATIONS

The aspect of repeat purchases addressed in this pa-

per is dealt with only sparsely in the literature on rec-

ommender systems. Repeat purchases describe any

situation in which a customer buys more than one

unit of a product (Bhagat et al., 2018). A key pub-

lication on the topic of repeat purchase decisions for

brands is Ehrenberg’s Repeat Buying Theory. Ehren-

berg describes that most aspects of brand buying be-

havior can be explained in terms of just two variables.

These are the market penetration and the average pur-

chase frequency of a product, whereby even these two

variables correlate with each other. Furthermore, a

product purchase decision essentially depends on the

timing of the purchase of a speciﬁc product class as

well as the brand choice. Accordingly, almost all in-

ﬂuences can be adequately explained if purchase fre-

quency processes are speciﬁed per brand (see Fig. 1)

(Ehrenberg, 2000; Silver, 1989).

Ehrenberg’s theory looks at the purchase histories

of individual users from consumer panels, i.e. the

chronological sequence of purchases in all of a cus-

tomer’s shopping baskets over a speciﬁc period of

time and a speciﬁc point of sale, to study the purchase

frequencies of a particular brand. It turned out that the

ICE-B 2021 - 18th International Conference on e-Business

Figure 1: Purchase Events as Independent Stochastic Pro-

cesses according to Ehrenberg (2000).

analysis should focus in particular on purchase occa-

sions, i.e. the frequency of purchase of one or more

items of a product at a speciﬁc time in a store, and not

on quantity or price. Repeat purchases can therefore

be described for a speciﬁc item by its market penetra-

tion and purchase frequency, where penetration rep-

resents the proportion of people who buy a particular

product in the ﬁrst place and purchase frequency rep-

resents the average number of these customers who

buy at least one product in the period under consid-

eration. The average purchase frequency here repre-

sents the basic measure of repeat purchases (Ehren-

berg, 2000).

Ehrenberg distinguishes between three types of re-

peat purchases: 1) A customer may buy an item in

more than one purchase in a given period. Different

customers can be characterized by the number of their

respective repeat purchases of the product. 2) A cus-

tomer can buy an item in more than one period. 3)

A customer can buy several units in the same pur-

chase. For the problem deﬁnition of this paper, the

ﬁrst point in particular comes into consideration. The

frequency distribution of (repeat) purchases and thus

the number of consumers who have made 0 or 1 or, in

the repeated case, 2, 3, 4, etc., purchases, can be de-

scribed by the Negative Binomial Distribution (NBD)

or the Logarithmic Series Distribution (LSD) (Bhagat

et al., 2018; Geyer-Schulz et al., 2001). Most products

in everyday life have a certain time interval between

the purchase of one product and the next (e.g. weekly

purchase of mineral water). However, certain branded

products in particular are rarely bought again, even

over a longer period of time. Many customers of a

brand buy a product only once, some twice, etc. Due

to a thus relatively small share of buyers of a branded

product in the population, the highest frequency is ob-

served among non-buyers (Ehrenberg, 2000). This re-

sults in a very skewed frequency distribution (Forbes

et al., 2011), which can theoretically be described by

a mathematical function. The underlying function to

ﬁt the above problem is the NBD, with the LSD being

a simplifying approximation to the NBD (Ehrenberg,

2000).

Bhagat et al. (2018) further discuss different ap-

proaches for repeat purchase decisions. In contrast to

the models above, they aim to generate individual rec-

ommendations for repeat purchases. In doing so, they

shed light on the temporal dependencies between re-

peat purchases of products, as these depend on when

a product was last purchased and how quickly the cus-

tomer runs out of it (Bhagat et al., 2018).

Therefore, they assume that a customer who has

bought a product frequently in the past will buy it

again (Repeat Customer Probability Model, RCP).

The recommendations are ranked in descending order

according to the number of repeat purchases. How-

ever, the problem arises that frequently purchased

items are not relevant for a certain recommendation

period, but are still considered due to their high num-

ber of repeat purchases. Even the integration of a

certain time decay, which models repeated purchases

based on a speciﬁc half-life, would be problematic

due to the assignment of the highest rank directly af-

ter a repeat purchase, as this would increase the rank

in the recommendation list. It can be assumed that

the attractiveness of a product for repeated purchase

immediately following the purchase of this product

is rather low. In order to include the temporal rele-

vance of products, it is also discussed that the pur-

chase of items represents a periodic phenomenon and

is therefore subject to a certain time interval (Ag-

gregate Time Distribution model, ATD). Other ap-

proaches based on this idea are the Poisson-Gamma

model (PG) and the Modiﬁed Poisson-Gamma model

(MPG), which perform better due to the personal-

ization of purchase rates by a Bayesian Hierarchi-

cal Model (Bhagat et al., 2018; Chu and Park, 2009;

De Oliveira, 2013; Gopalan et al., 2015).

5 METHODOLOGY

5.1 Data

The data set includes eleven features: CustomerId,

ShoppingCartId, MarketId, Date, ShoppingCart-

Value, ItemId, ItemName, ItemQuantity, ItemPrice,

CategoryId, and CategoryGroupLevel. Some prod-

ucts with negative item price and negative item quan-

tity are included. They were mainly marked as NaN

(Not a Number) and describe items such as deposits

and empties, which are refunded to the customer.

Negative item prices and quantities as well as NaNs

are already removed from the data set before data

preparation, as they would distort the data analysis

due to their negative values. In addition, other items

with the designation ”deposit” or ”delivery” existed,

Recommender Systems in Food Retail: Modeling Repeat Purchase Decisions on Transaction Data of a Stationary Food Retailer

which were also deleted.

The reduced and cleaned dataset includes

49,920,981 transaction records. The transactions

took place between December 28, 2018, and April

29, 2019. Overall, while the data set has a wide

range of analysis capabilities with approximately

50 million transactions and 200 thousand products,

these already reveal likely weaknesses with respect

to modeling repeat purchase decisions. For example,

the average number of purchases per customer across

all products is 5.1 with a standard deviation of 5.6.

This indicates a high number of customers who also

purchase fewer products in the complete period under

consideration. The average number of products in

a shopping cart also conﬁrms this impression with

only 1.27 items. The maximum number of items in a

shopping cart is only 22 for 200 thousand different

products. Accordingly, although there are many

individual purchases of speciﬁc products, there are

no frequent repeat purchases by a speciﬁc customer.

The number of purchases per customer is in the range

of one to two purchases in the considered period. The

number of one-time repeat buyers (loc) per product

is less than ﬁve in 50% of cases. The number of

repeat buyers (moc) is even less than three. These

key ﬁgures result in the Repeat Customer Probability

– i.e. the probability of a product being purchased by

a repeat buyer.

Since this is a time-dependent classiﬁcation prob-

lem, the data is divided on the basis of a reference

date. The test data thus represent (actual) future trans-

actions of the customers, which are to be predicted by

the model and used for validation within the frame-

work of a confusion matrix with their speciﬁc key ﬁg-

ures. Here, the due date was set at March 31, 2019,

and the last month of the data set thus serves as test

data. Accordingly, the training data set comprises

37,289,860 rows (74.7%), and the test data set in turn

12,631,121 rows (25.3%). In addition, it is assumed

that a customer’s purchase will only occur in the test

period if that customer purchased a product in both

training and test data (Bhagat et al., 2018).

Due to sparsely purchases at customer level (data

distribution), two models are to be evaluated for

the present data. These are the ATD model and

the MPG model. While the ATD model addresses

this weakness by considering the aggregated repeat

purchases per product, the MPG model analyzes

the personalized purchase rates of a product using

the Bayes approach as a probabilistic model-based

method.

5.2 Modeling

The problem of repeat purchase recommendations is

described as estimating the probability of a repeat pur-

chase as a function of time since his last purchase

of the item under consideration, given the customer’s

previous purchase history. Accordingly, the associ-

ated purchase probability density P

is to be esti-

mated for a future time interval t

k+1

assuming that a

customer C

has purchased an item A

k times in the

past with time intervals t

, ..., t

. Thus:

k+1

, ..., t

) (1)

It is supposed that the customer’s purchase times

for different products are independent (Bhagat et al.,

2018).

Furthermore, it is assumed that the above purchase

probability density is composed of two components.

represents the probability of a repeat purchase of

a customer who buys a product for (k + 1)

h times

with k previous purchases. R

deﬁnes the probability

distribution of t

k+1

, which depends on the repeated

purchase of the item by the customer (A

= 1).

k+1

, ..., t

) ≈ R

k+1

, ..., t

) ·Q

(2)

Moreover, the time distribution R

k+1

, ..., t

) is supposed to be asymptotic to

(t | t

, ..., t

) (Bhagat et al., 2018; De Oliveira,

2013; Trinh et al., 2014).

k+1

, ..., t

) ≈ R

(t |t

, ..., t

)

where :

∞

(t)dt = 1;

∞

(t)dt ≤ 1

(3)

The above mentioned RCP model serves as a ba-

sis for the following models in order to consider only

products that are suitable for repeat purchase. It is de-

ﬁned by analyzing aggregate repeat purchase behav-

ior as the ratio of the number of customers who have

purchased a product A

more than once (moc) to the

number of customers who have purchased a product

at least once (loc). The derived repeat customer

rate RCP

approximates Q(A

) and without consider-

ing the time intervals between purchases also P

ac-

cording to the following formula (Bhagat et al., 2018;

Fader and Hardie, 2009):

RCP

moc

loc

k+1

= t | t

, ..., t

) ≈ Q(A

) ≈ RCP

(4)

Only products with an RCP

> r

threshold

are taken

into account further.

ICE-B 2021 - 18th International Conference on e-Business

5.2.1 Aggregate Time Distribution Model

If there are only a few repeat purchases at the cus-

tomer level, but a large number of customers at the

product level who have bought the product repeat-

edly, a model is suitable which analyzes the aggre-

gated and time-based repeat purchase behavior across

all repeat purchasers of a product. This models the de-

termination of the probability distribution of the time

intervals (t) of repeat purchase of a speciﬁc product

across all repeat purchase customers. For this pur-

pose, Baghat et al. examined various distributions

in the context of determining mean time intervals for

each customer in a sample of repeat-purchased items,

and the log normal distribution achieved the best ﬁt

(Heyde, 1963; Bhagat et al., 2018).

(t) =

√

2πt

exp



−

(lnt − ¯µ

)



,t > 0 (5)

Accordingly, the ATD model estimates the pa-

rameters of the log-normal distribution for each suit-

able repeat purchase product by ﬁtting them to the

different repeat purchase time intervals t of all re-

peat customers. Here, Q(A

) represents a ﬁxed con-

stant q for all products A

of a given time t. Recom-

mendations are made based on the descending order

of probability density P

(t) at a given time t using

k+1

, ..., t

) ≈ R

k+1

, ..., t

) ·Q(A

) (Bha-

gat et al., 2018; Heyde, 1963).

5.2.2 Modiﬁed Poisson-Gamma Model

A Bayesian model is assumed whose evidence is Pois-

son distributed and the prior on λ is a gamma prior

(PG model). It is subject, on the one hand, to the

assumption that successive repeat purchases are un-

correlated and that repeat purchases follow a homo-

geneous Poisson process with repeat purchase rate λ.

On the other hand, λ across all customers follows

a gamma distribution of the form α with an inverse

scale parameter β. The parameters of the product-

speciﬁc gamma distributions are estimated by ﬁtting

them to the maximum likelihood estimators of the

purchase rates of repeat purchase customers. This is

followed by a Bayesian estimate of a customer’s re-

peat purchase rate based on the combination of the

prior distribution and the individual’s past purchase

history (Bhagat et al., 2018; Trinh et al., 2014):

k + α

t + β

,t > 0 (6)

In addition to the shape and scale parameters α

and β

of the gamma prior of product A

, k describes

the number of purchases of the speciﬁc product A

by customer C

. The elapsed time between the ini-

tial purchase of A

by C

and the current time, is ex-

pressed by t. Regarding recommendations, R

is as-

sumed to be Poisson distributed, with the inverse scale

parameter estimated using λ

and the likelihood

function estimated using the following equation R

(Gopalan et al., 2015; De Oliveira, 2013).

(t) =

∞

∑

m=1

exp(λ

)

,t > 0 (7)

Here m represents the number of expected future

purchases and Q

is considered as a ﬁxed constant

for all products A

. Recommendations are made by

classifying all items previously purchased repeatedly

by the customer based on their estimated probability

density P

in descending order at a given time t using

the equation 2 (Bhagat et al., 2018).

Unlike the classical Bayesian methods, the Bayes

estimation of the a priori distribution here is done em-

pirically using the underlying data, rather than deﬁn-

ing it ﬁxedly in advance without taking the data into

account. Therefore, this method is also titled Empir-

ical Bayesian Method. The PG model is thus a para-

metric Empirical Bayesian Model whose likelihood

and a priori distribution take simple parametric forms.

It approximates a Hierarchical Bayesian Model. The

Bayes theorem allows the individual purchase deci-

sions of a customer to be combined with the aggregate

purchase behavior of a product, thereby personalizing

this model. The PG model has already been used in

the past, but not yet in the context of stand-alone re-

peat purchase recommendations (Bhagat et al., 2018;

De Oliveira, 2013; Fader et al., 2005; Morrison and

Schmittlein, 1988; Sichel, 1982).

If λ is the purchase rate to be estimated at Y ∈

{0, 1, 2, ...}∈N

purchases occurred in period N, then

E(Y ) = Nλ describes the expected number of pur-

chases. Because λ follows a gamma distribution at

the product level, a maximum likelihood estimator

(MLE) of the gamma parameters Γ(α, β) for each

product of the respective repeat purchase customers

is established as

λ =

. Because Y represents the

number of purchases with the above expected value

E(Y ) = Nλ, the likelihood assumption is Y | λ ∼

Γ(α + Y, β + N). The Bayes estimator (a posteriori

mean) can be expressed as (Bhagat et al., 2018; Con-

sul and Jain, 1973; Sichel, 1982):

posterior

α +Y

β + N

(8)

Due to the estimation of the posterior distribution

of the purchase rate λ under the a priori assumption

Recommender Systems in Food Retail: Modeling Repeat Purchase Decisions on Transaction Data of a Stationary Food Retailer

of a gamma distribution, whose parameters are ap-

proximated via the MLE using the actual purchases

for each product, this is a parametric empirical Bayes

model, which is an approximation of a Bayesian Hier-

archical Model. The difference is that scale and shape

parameters (α, β) of the gamma prior are estimated

from the actual data rather than from additional hy-

perprior parameters (see Fig 2).

Figure 2: Empirical Bayesian Model.

However, the assumption of a homogeneous Pois-

son distribution is not valid for any type of product

since the purchase events can theoretically represent a

time-independent constant. Therefore, the PG model

needs to be modiﬁed (MPG model). This is due to

the fact that a Poisson process represents a limiting

case of the sequence of Bernoulli processes in the

boundary between a large sample and a small con-

stant probability and is memoryless. However, this

is not expected for purchase behavior for a number

of products, since a customer’s need to repurchase an

item after purchasing it is initially small but variable

as time progresses (Bhagat et al., 2018;

Ozekici and

Soyer, 2003; De Oliveira, 2013).

The MPG model assumes that a customer’s pur-

chases are correlated and repeat purchases follow a

modiﬁed Poisson process, which uses a single param-

eter λ as the repeat purchase rate. λ depends on the

last purchase of a product by the customer under con-

sideration. Thus, it differs by the homogeneous Pois-

son process in the PG model. Furthermore, a gamma

prior is assumed on λ. Thus, λ follows a gamma dis-

tribution of the form α across all customers with an

inverse scale parameter β (De Oliveira, 2013).

In analogy to the PG model, the estimates of the

parameters are made via a parametric empirical Bayes

model that ﬁts them to the MLEs of customers’ re-

peat purchase rates. The estimation is optimized per

customer by estimating the mean time interval for re-

peated purchases of a speciﬁc item based on the ﬁrst

and last purchases. The model assigns the observed

mean value to the associated highest repeat purchase

rate. This is achieved by making modiﬁcations to the

PG model: t

buy

denotes the elapsed time interval be-

tween the ﬁrst and last purchase of product A

by cus-

tomer C

, t represents the elapsed time interval be-

tween the last purchase of A

by C

and the current

time. t

mean

represents the estimated mean time inter-

val of repeat purchases of A

by C

. For t < 2 ·t

mean

the estimation of the repeat purchase rate of the MPG

model is done according to (Bhagat et al., 2018):

k + α

buy

+ 2 ·|t

mean

−t|+ β

(9)

and β

are shape and inverse scale parame-

ters of the gamma prior of A

. k represents the num-

ber of purchases of A

by C

. If t ≥ 2 ·t

mean

, λ

is determined via the separate formula of the MPG

model. This ensures that λ

increases from t = 0 to

t = t

mean

and then decreases until t = 2 ·t

mean

. At this

point, the MPG model is equivalent to the PG model.

A Poisson distribution is further assumed for R

whose likelihood function is estimated using R

(t)

of the PG model. Moreover, Q(A

) can be determined

via RCP

using equation 4. Recommendations are

generated by ranking all products in descending order

based on their estimated probability density P

(t) at

a given time t using equation 2 (Bhagat et al., 2018;

Consul and Jain, 1973; Gopalan et al., 2015).

6 RESULTS

The average RCP is 6.7% with a standard deviation of

12.7%. The median is 3.0%, which means that 50%

of the products have a higher and 50% a lower RCP.

However, products with a repeat customer probabil-

ity of 100% also occur and are excluded in the fol-

lowing. These are mostly products that have a very

low but equally high loc and moc frequency. Figure 3

provides information about the distribution of repeat

customer rates for all products. It can be seen that the

most frequent values are concentrated in the range of

a RCP < 0.2. Outliers are considered for all products

with a RCP > 0.13.

Figure 3: Distribution of Repeat Customer Probability.

Determining the threshold for products to be con-

sidered suitable for repeat purchase is therefore difﬁ-

cult. One study outlines that products are to be clas-

ICE-B 2021 - 18th International Conference on e-Business

siﬁed as suitable for repeat purchase if they have a

RCP > 0.27 (McEachern, 2021). However, different

thresholds are tested to show the impact of RCP ﬁlter-

ing, as this is expected to have a large impact on the

model results.

6.1 ATD Model

Since many customers in the data set make only a few

repeat purchases, the ATD model draws on the aggre-

gate and time-dependent repeat purchase behavior of

all repeat purchasers of a product. Figure 4 shows

that the aggregated mean repeat purchase periods of

e.g. mineral water are approximately log-normally

distributed. Therefore, R

is determined over the in-

terval of the log-normal distribution.

Figure 4: Distribution of time intervals (t) for repeated pur-

chases of mineral water over all customers.

Since the purchase rates at the individual level are

often unrealistic due to sparse data, looking at the ag-

gregated purchase rates of products across all repeat

purchasers ensures considerable results (see Table 1).

Increasing the RCP further improves these. It can be

seen that an RCP of 25% provides the best results, a

further increase would lead to a model degradation.

6.2 MPG Model

At the beginning, the individual purchase rates of the

products are determined for each customer. An exem-

plary distribution of the purchase rates of mineral wa-

ter can be seen in the following Fig 5a). It shows that

the normalized a posteriori distribution of λ achieves

a good ﬁt with respect to the data. The normalization

is stronger the closer the last purchase (t) is to the cur-

rent date (cf. Fig. 5b).

A conversion of the Bayesian estimators

posterior

to the actual observed purchase rates is still evident

(see Fig. 5 a, b). Even for long periods since the

last purchase, the purchase rates are well reproduced.

However, there is a general trend in that the closer

the ﬁrst purchase of a product is to the cutoff date,

the more the purchase rates diverge. Figure 5 b) re-

veals that the estimate is more in line with the true

rate when the ﬁrst purchase of a product was made

recently and a customer is simultaneously is a ﬁrst-

time repeat buyer. Figure 5 c) shows the distributions

of the probabilities P

(Y = 1) of all repeat buyers.

Table 1 shows that the MPG model outperforms

the ATD model. This is due to the fact that Bayesian

model-based methods can perform well with sparse

data (Isinkaye et al., 2015) and personalized pur-

chase rates are considered here. Also for the MPG

model, the best results are obtained with an RCP of

25%. Compared to the ATD model, the MPG model

leads on average to an improvement in precision@k

of 26.3%, recall@k of 21.1% and f1@k of 23.9%.

These are particularly driven by the increases under a

relatively small RCP of 7%.

7 DISCUSSION

7.1 Model Results

With regard to the problems of traditional recom-

mender systems, the model-based methods consid-

ered can reduce many weaknesses. They are suitable

for sparse data, which is achieved in the ATD model

by aggregations of the mean repeat purchase intervals

per product and complemented in the MPG model by

Bayesian personalization of purchase rates. The high

computational effort regarding memory-based meth-

ods is also improved here by using probabilistic mod-

els. The results show that the MPG model can outper-

form the ATD model. In particular, when a low RCP

is applied, the potential of the MPG model becomes

apparent (see Table 1).

Furthermore, in the food context, the model as-

sumptions show better suitability than similar models

discussed for repeat purchases in the marketing lit-

erature. These focus on repeat purchases of brands,

where the assumption is that an increasing time dif-

ference between the current date and last purchase re-

sults in a decreasing probability for a repeated pur-

chase (Miglautsch, 2000). Especially for food (e.g.

staple foods), the assumption is contradictory. Here,

customers are more likely to buy a product if this time

difference increases. This is, in particular, true as

brand loyalty is rather low, but products need to be

purchased frequently.

Nevertheless, MPG model errors often occured,

when a customer was rarely a repeat buyer and usu-

ally just before the set cutoff date, so a small number

of purchases were combined with a small time gap

from the ﬁrst purchase of the product. This is pre-

cisely the case if the customer does not appear again

Recommender Systems in Food Retail: Modeling Repeat Purchase Decisions on Transaction Data of a Stationary Food Retailer

Figure 5: MPG-Model – a) Likelihood and a posteriori distribution of mineral water, b) residuals between true purchase rate

and the Bayes estimator and c) Distribution of probabilities of all customers for a purchase of mineral water within 30 days.

Table 1: Model Evaluation.

0.07 0.20 0.25 0.30

model k p@k r@k f1@k p@k r@k f1@k p@k r@k f1@k p@k r@k f1@k

ATD

1 0.14 0.16 0.15 0.22 0.36 0.27 0.23 0.37 0.28 0.09 0.14 0.11

5 0.14 0.22 0.17 0.23 0.38 0.29 0.24 0.38 0.29 0.10 0.15 0.12

10 0.15 0.22 0.18 0.23 0.38 0.29 0.24 0.38 0.29 0.10 0.15 0.12

MPG

1 0.17 0.24 0.20 0.24 0.40 0.30 0.27 0.42 0.33 0.12 0.16 0.14

5 0.22 0.27 0.24 0.24 0.42 0.31 0.28 0.44 0.34 0.14 0.20 0.16

10 0.22 0.27 0.24 0.25 0.42 0.31 0.28 0.44 0.34 0.14 0.20 0.16

where: p@k: precision@k, r@k: recall@k, f1@k: f1score@k

in the test data set and not regularly as a repeat buyer

of the product. This can be explained by the shift-

ing behavior of consumers, who frequently change

the point of purchase (e.g. supermarket and discount

store) or adopt different consumption behaviors (e.g.

going vegan) (Lawo et al., 2019; Stevens et al., 2017).

Further research should therefore address the integra-

tion of changing preferences and shopping habits.

7.2 Inﬂuence of the RCP

Since there is no explicit information on RCP in food

retailing in the literature, different thresholds were

tested here. The results show that the RCP has a

strong inﬂuence on the model quality. With increas-

ing ﬁltering of the products regarding their probabil-

ity that a customer buys the product repeatedly, the

model quality increases signiﬁcantly. As a study sug-

gests (McEachern, 2021), in this case an RCP of 25%

provides the best results. However, this is related to

a trade-off between model goodness and the product

variety considered, which – especially with sparsely

populated data – also ﬁlters products that normally

follow a habitualized and regular purchase process.

If an RCP of 0.3 is reached, there is a decline of

the model, because now mostly only products are con-

sidered, which combine low loc and moc values, re-

sulting in a tendentially high RCP.

8 CONCLUSION

This paper has focused on validating a time-based

recommender system for repeat purchase decisions

in food retailing. For this purpose, an introduction

to recommender systems was given at the beginning

and classical problems for transaction data of food re-

tail were derived. The consideration of state-of-the-

art solutions for the integration of a temporal com-

ponent into recommendations created a transition to

current models in the marketing literature. Two spe-

ciﬁc models were identiﬁed that deal speciﬁcally with

repeat purchase decisions. These were applied to a

real data set of a stationary German food retailer and

discussed with respect to the previously derived prob-

lems of recommender systems. It is shown that the

Modiﬁed Poisson-Gamma model is well suited for the

sparse data situation at hand, but that model inaccura-

cies occur due to different consumption and shopping

habits. Therefore, future research should focus on the

integration of these consumption patterns.

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