MMRCDISCUSSION PAPER SERIES
No. 259

One-to-one demand forecasting model and its application for
inventory management
Sotaro Katsumata
Graduate School of Economics,the University of Tokyo
Masato Itohisa
Graduate School of Economics,the University of Tokyo
April 2009

東京大学ものづくり経営研究センター
Manufacturing Management Research Center (MMRC)
Discussion papers are in draft form distributed for purposes of comment and discussion. Contact the author for
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http://merc.e.u-tokyo.ac.jp/mmrc/dp/index.html

One-to-one demand forecasting model and its application for
inventory management

Sotaro KATSUMATA(Graduate School of Economics, the University of Tokyo)
Masato ITOHISA(Graduate School of Economics, the University of Tokyo)

Abstract:
In this study, we propose a demand forecasting model for inventory management and apply
it to low-involvement and repeat-purchase products, particularly to households’ demand
for alcoholic beverages. We constructed a model involving household heterogeneity, commonality and time-trend eﬀects. The results revealed that the proposed model maintained
a high forecasting performance for long periods. This model can estimate household-level
demand and potential tendencies of product consumption. We also discuss the wide range
of applications of the model.

Key word: Inventory Management, Marketing Demand Forecasting Model

1One-to-one demand forecasting model and its application for
inventory management
Sotaro KATSUMATA(Graduate School of Economics, the University of Tokyo)
Masato ITOHISA(Graduate School of Economics, the University of Tokyo)

1

Introduction

Demand forecasting is one of the most important issues in the ﬁeld of supply chain management (Willemain, Smart and Schwarz, 2004). Understanding the demand for ﬁnal
consumption is a fundamental aspect of production planning, including inventory management for each stage of the supply chain, which involves dealers but also OEM (original
equipment manufacturers), suppliers and material handlers. In addition, the proﬁt of eachcompany depends on the accuracy of demand forecasting. Although it is almost impossible
to predict demand with perfect accuracy, the longer the period, the more diﬃcult it is toforecast demand.
Two streams of research have attempted to solve the above problem. One stream ofresearch aims to develop as accurate a model as possible by utilizing detailed purchase data,for instance, POS (point of sales) data and data obtained through the use of advanced
mathematics (e.g. Brown, 1960; Berry and Linoﬀ, 1997; 2000). The other stream ofresearch is based on the BTO (build to order; Holweg and Pil, 2004) approach, which
involves initiating production after an order is taken from a customer, as a result of
which, there is no need to accurately predict uncertain demand. However, BTO cannot
be applied to all products. For example, BTO is a realistic solution for European luxurycar makers but for low-involvement products like food and beverages, if customers do
not ﬁnd the products on store shelves, they go to other shops. Therefore, increasing
the accuracy of demand forecasting for low-involvement products is regarded as the ﬁrst
solution. Regardless of the preciseness of a model, it remains a fact that predicting thedemand with accuracy is a diﬃcult task because the variance in the demand of the market
produces the inventory cost.
Our research will focus on low-involvement products and develop a forecasting modelon which the one-to-one marketing (Peppers and Rogers, 1993) method is applied toaccumulate individual demands. The models developed in the past view the market asa whole, whereas the new model directs attention to the commonality and heterogeneityof individuals and proposes two views: raising the accuracy of demand forecasting andeﬃciently promoting the sale of unsold goods to individuals. For instance, providing
discount oﬀers through direct mail to those who have latent purchasing power might bean eﬃcient method to sell unsold goods.
2In this paper, we begin by conducting a literature review of the demand forecastingmodel and of one-to-one marketing. We then proceed to construct the one-to-one demand
forecasting model and test it using validation data. After that, we discuss the limitations
and expansions of the model. Finally, we present the conclusion along with the directions
for future research.

2

Literature Review

2.1

Demand Forecasting

Since early times, companies have recognized the importance of demand forecasting. This
has led to the development of numerous forecasting methods. In the earliest years, ﬁrms
believed that demand volume could be predicted from past sales records. Na?ve forecasting, the moving average method and exponential average are commonly used primitive
forecasting methods (e.g. Brown, 1959). Na?ve forecasting uses the nearest sales as thepredicted value. The moving average and exponential average methods use the data ofan appropriate past period with some assigned weighting value. After that, many quantitative models were developed and proposed; some examples of these models are thetime-series analysis methods, of which the ARMA (autoregressive moving average) model
is well known, and the state-space model (e.g. Hamilton, 1994).
In addition to the above-mentioned methods, regression models that considered market
structure and customer characteristics were also developed. These methods are actively
researched in the ﬁeld of marketing. Eﬀorts to achieve predictive accuracy gave riseto the concept of market segmentation. Many segmentation methods were developed
to analyse the market segments, for example, mixture regression models (e.g. DeSarbo
and Cron, 1988; Kamakura and Russel, 1989; Wedel and Kamakura, 1998). Mixture
models divide the whole market into heterogeneous segments, based on which the market
aggregate demand can be derived from the sum of each heterogeneous segment, which
is a primitive method of one to one demand forecasting. In recent years, data mining
techniques like the neural network (Berry and Linoﬀ, 1997; 2000) and database marketing
methods (Blattberg, Kim and Neslin, 2008) propose a way to deal with vast amounts ofdata. In demand forecasting, the volume of data is increasing and the structure of theforecasting models is more complicated.

2.2

One-to-One Forecasting Model

Recently, these market segments have become more fractionizing and are being developed
into one-to-one marketing (Rogers and Peppers, 1993) and CRM (Customer RelationshipManagement; e.g. Reinartz et al., 2004), which assume customer heterogeneity and aim tocapture individual demand. The validity of these marketing activities is conﬁrmed on thebasis of two reasons. First, the progress of information systems enables us to accumulate
enormous quantities of customer information. Second, the introduction of the Bayesianinference allows us to estimate complex models (Rossi, Allenby and McCulloch, 2005).
The one-to-one forecasting model is often expressed by the hierarchical Bayes model (e.g.
Rossi, McCulloch and Allenby, 1996; Katsumata, 2008). The hierarchical structure isalso applied to other objects. Ansari, Essegaier and Kohli (2000) constructed a model3

that considers customer and product heterogeneities and applied it to movie rating data.
Ansari and Mela (2003) and Danaher, Mullarkry and Essegaier (2006) also used thismulti-heterogeneous model.
Although one-to-one models have been actively studied in the ﬁeld of marketing, few
researches have applied this model to long-term demand forecasting. In this paper, weconstruct a model that contains customer heterogeneity and commonality, and then apply
it to long-term forecasting. After the estimation, we discuss a wide range of applications
for inventory management.

3

Data Overview and Model Construction

Based on the discussion above, in this section, we show the construction and the applicationof the demand forecasting model. In particular, we forecast the usage of alcoholic beverages
by each household. At the same time, we use the ordinary method of forecasting for the
purpose of comparing the forecasting of demand by the two methods.

3.1

Data and Target Products

We use the food usage record of households obtained from the 2008 Data Analysis Competition hosted by the Joint Association Study Group of Management Science. These
records were collected over a period of one year.
In this research, the analysis and forecast focus on alcoholic beverages. We chose
this product category for the following three reasons. First, since the fermentation of
alcohol takes a long time, this category of products has PLT (production lead time).
Therefore, alcoholic beverages belong to a product category that needs high accuracy indemand forecasting because manufacturers often require a production plan considering
PLT. Second, this product category has some ﬂexibility regarding production volume.
Therefore, the price of alcohol is stable compared to the prices of vegetables and otherraw foods. Third, the products under this category have an expiration date. Even thoughsome alcoholic products such as wine and spirits do not have an expiration date, ordinary
brewed beverages have a short shelf life. Since these products with an expiration dateface the risk of being discarded, ﬁrms have to adjust the production and volume of stock.
These reasons show that there is a high requirement for demand forecasting methods inﬁrms producing and selling alcoholic beverages. In the following section, we construct amodel to forecast the usage of alcoholic beverages by each household.

3.2
3.2.1

Model Construction
Divide Multiple Decision-Making

In household h, if a product j is used on 20 out of 100 days, we can estimate the usage
probability of product j as 20/100 = 0.2. However, this observed probability contains thefollowing two factors:• Eating meals at home but not using product j• Not eating meals at home (eating out)4

Here, let us deﬁne yht and zhtj as follows:• zhtj = 1 if household h uses product j at time t and 0 otherwise.
• yht = 1 if household h makes a meal at time t and 0 otherwise.
Then, we can deﬁne the observed food usage probability as a joint probability of zhtj andyht
Observed product usage probability = P (zhtj = 1, yht = 1)
(1)
When we try to estimate P (yht = 1, zhtj = 1) directly, we need to consider two factors.
It is reasonable to estimate each factor. We decompose this probability into two parts.
In the above situation, if household h ate out 20 out of 100 days, the observed eat-in
probability P (yht = 1) = 80/100 = 0.8, and the conditional food usage probability is
P (zhtj = 1|yht = 1) = 20/80 = 0.25. These probabilities have the following relations:
P (zhtj = 1, yht = 1) = P (yht = 1)P (zhtj = 1|yht = 1)(2)This means that the observed product usage probability can be decomposed into themultiplication of eat-in probability and conditional food usage probability. However, wecannot observe P (zhtj = 1|yht = 0); therefore, in this paper, we assume zhtj and yht tobe independent. Then, we obtain that the observed usage probability of product j is thesimple multiplication of eat-in probability and unconditional usage probability of productj.
Next, to estimate the demand volume, we have to estimate the number of attendanceat the dinner table. We can interpret this variable as an outcome of the collective decisionmaking of the family. Here, we formulate the number of attendance at the dinner table.
Let Mh be the size of household h and λht be the number of attendance at time t inhousehold h. When we observe Mh attendance above, we treat it as Mh + 1. Then, we
can deﬁne the range of λht as follows:
λht ∈ {1, 2, · · · , Mh , Mh + 1}

(3)

We can estimate the number of attendance by using the discrete ordered choice modelwith Mh + 1 numbers. Let P (λht = m) be the probability of attendance of m individuals;
then, we can calculate the expected number of attendance ash∑1
λ̄ht =P (λht = m)
Mh + 1

M +1

(4)

m=1

Assuming that all decision-making is independent, the demand volume dhtj can be expressed by multiplying three factors: the expected number of attendance, eat-in probability and unconditional usage probability of product j.
dhtj = λ̄ht P (yht = 1)P (zhtj = 1)

(5)

Assuming that the number of households is H, we can express the whole market demand
as follows:
H
∑
Dtj =
dhtj
(6)
h=1

5

3.2.2Deﬁne Multiple Decision-Making

Based on the settings mentioned above, we can construct a model using a probit-typemodel. The number of attendance is estimated by using the ordered probit model, while
the eat-in probability and product usage probabilities are estimated by using the binomial
probit model. Let us bundle these observations as uht ,′ ′uht = (yht , λht , zht)(7)where, zht = (zht1 , · · · , zhtJ )′ . Let the number of products be J; then, the size of uht
becomes J + 2. Then, we set J ∗ = J + 2. In the probit model, Abler and Chib (1993)
introduced the latent variable applied by data augmentation (Tanner and Wong, 1987).
In this paper, we use latent variables in the same manner as that used in Albert and Chib
(1993).
We introduce latent variable for observed vector . When u∗ht = 1 and uht2 · · · uhtJ ∗ areobservable, the corresponding relation is as follows:u∗htj > 0 if j = 1
chm ≤ u∗htj < ch,m+1 , if uhj2 = m, j = 2{> 0 if uhtj = 1∗
uhtj, j = 3, · · · , J ∗≤ 0 if uhtj = 0(8)where cs are the threshold parameters, and ch1 = −∞、ch,Mh +1 = 0、ch,Mh +2 = ∞ is ﬁxed
(Koop, 2003). If uht1 = 0, since we cannot observe production usage and the number ofattendance, we complement the missing variables stochastically using data augmentation.
{ ∗uhtj ≤ 0, j = 1
if uht1 = 0(9)2 ), j = 2, 3, · · · , J ∗uhtj ∼ N (µhtj , σhtj
If eat-in probability, product usage probabilities and the number of attendance are independent, we can compute each probability using latent variable u∗ht and the probability
density function of normal distribution ϕ(·).
∫ u∗
ht1
Estimated eat-in probability : P (uht1 = 1) =ϕ(µ)dµ
(10)
−∞

∫ u∗

ht,j+2Estimated usage probability of j-th product : P (uht,j+2 = 1) =Estimated number of attendance : λ̄ht =
3.2.3

M +1

∫ ch,m+1

m=1

chm

h
∑
1
m
Mh + 1ϕ(µ)dµ(11)ϕ(µ)dµ(12)−∞Construction of Proposed ModelIn this section, we build the eﬀect factors into the model. At ﬁrst, we consider household
speciﬁc factors. Let xht be the K-dimension explanatory variables of household h =1, · · · , H at t = 1, · · · , T . We can obtain the following hierarchical model.
u∗ht = xht Bh + εht , εht ∼ NJ ∗ (0, IJ ∗ )βhk = whk ∆k + ζhk , ζhk ∼ NJ ∗ (0, Θk )6(13)

where Bh = (βh1 , · · · , βhK ). In this paper, we assume that each factor is independent.
Let the covariance matrix be the identity matrix IJ ∗ . Again, considering the time-trend
factor, we can estimate u∗ht as a state space model.
u∗ht = F γt + εt , εt ∼ NJ ∗ (0, IJ ∗ )γt = Gγt−1 + νt , νt ∼ NJ ∗ (0, Φ)(14)Combining the structures above, we construct a model considering household heterogeneities and time trend. In this paper, we assume the state space term to be a local-level
model, F = G = IJ ∗ .
u∗ht = xht Bh + γt + εht , εht ∼ NJ ∗ (0, IJ ∗ )βhk = whk ∆k + ζhk , ζhk ∼ NJ ∗ (0, Θk )γt = γt−1 + νt , νt ∼ NJ ∗ (0, Φ)(15)Since this model has a hierarchical structure, we estimate the parameters using the MCMC(Markov Chain Monte Carlo) method. The detailed procedures of estimation are describedin the Appendix.
3.2.4

Explanatory Variables

In this paper, we use the following explanatory variables: xht and whk ,Intercept
xht =  Wifeś Holiday (D)  ,Husbandś Holiday (D)InterceptIncome




Full
Time
Housewife
(D)




Wife has a collage education (D)
whk = 
 Husband has a collage education (D) log(Age of housewife) Age diﬀerence between husband and wife 
Number of elderly persons

(16)

(17)

where (D) denotes dummy variables and xht is a dummy variable vector. In household h,if time t is a holiday of the wife or husband, the corresponding element becomes 1.

3.3

Target Period

In this paper, although we have data for one year, to estimate the annual trend, we divide
the data into two parts and treat it as data for two years. To be more precise, there are 194
households in the data. We select 100 households to estimate parameter γ, and estimate
the residual 94 households using the estimated γ as the given parameter. In addition, in
these 94 households, we estimate the one-to-one parameter Bh using data for the ﬁrst halfof the year and, for forecasting, we use the data for the latter half of the year as validation
data. Therefore, we accomplish the following:
1. The full year estimation of 100 households
72. The estimation of the ﬁrst half of the year for 94 households using γ of the full year
estimation as the given parameter
3. The forecasting of the latter half of the year of 94 households
Figure 1 provides the details of these calculations.

First half of the year
1 January ~ 30 June
(182 days)

Latter half of the year
1 July ~ 31 December(184 days)
Time

Householdid 1,…,100

Calibration Set 1
Estimate {Bh}, {Θk}, ∆, Φ, γ
Substitute γ as
the givenparameter.

Householdid 101,…,194

Calibration Set 2
Estimate {Bh}, {Θk}, ∆, Φ

Validation Set

Figure 1: Calibration and Validation Period

3.4

Comparison Models

To compare the forecasting performance of the proposed model, we use the following twomodels.
Simple Trend Model
As mentioned above, there are many cases wherein time-series analysis methods areused as a demand forecasting model. Therefore, we use a simple time-trend model as thecomparison model. This model estimates the aggregate demand.
At time t, we will estimate the aggregated market demand Dt = (Dt1 , · · · , DtJ )′ :Dt = αt + εt , εt ∼ NJ (0, IJ )αt = αt−1 + νt , νt ∼ NJ (0, Ψ)

8

(18)

In this model, we use only Calibration Set 1 of Figure 1 for full year estimation. After
estimation, we forecast the demand of the validation set using trend parameter α.
94
αt
100

D̃t =

(19)

Naı̈ve Forecasting
As a primitive method, we use naı̈ve forecasting as our second comparison model. We
consider the aggregate demand of the target set at 30 June (one day prior to the calibration
period) as the future demand.

4

Result4.1

Trend and Regression Coeﬃcients

Figure 2 shows the estimated time trend. For beer and sake, we plot the observed
aggregate-demand movement on a thin line.

Number of attendanceEat-in
0.02

1

0

0.5
-0.02

1/1

2/1

3/1

4/1

5/1

6/1

7/1

8/1

9/1 10/1 11/1 12/1

-0.04

0
1/1

2/1

3/1

4/1

5/1

6/1

7/1

8/1

9/1 10/1 11/1 12/1

-0.06

-0.5
-0.08
-0.1

-1

-0.12

-1.5
-0.14
-0.16

-2

Beer

Sake

80

60

1.2

0.6

70

1
50

0.8

60
0.1
50

0.6

40

0.4

40

30

0.2

-0.4
0

30

20

20

-0.2

-0.9

-0.4

10

10

-0.6

0

-1.4
1/1

2/1 3/1

4/1

5/1

6/1

7/1

8/1

9/1 10/1 11/1 12/1

0

-0.8
1/1

2/1 3/1

4/1

5/1

6/1

7/1

Figure 2: Posterior Mean of Trend Parameter γ

9

8/1

9/1 10/1 11/1 12/1

The eat-in trend shows that there are some periods in which demand declines. The
ﬁrst such period is during the New Year holidays. We also observe a decline in demandin early May, which is when schools are closed and people go on trips. Another period
during which a decline in demand is observed is mid-August (Bon period), which is whenfamilies take days oﬀ and visit their hometowns. Compared to eat-in probabilities, there
is no obvious trend with regard to the number of attendance. With regard to the trends
for beer and sake, beer consumption rises in summer, while sake consumption falls in thesame season. Considering all these trends, we ﬁnd that this model divides the demand
factors of usage. For example, although the observed beer demand falls around 15 August,the estimated beer trend rises in this season. This decline in demand can be explained by
the eat-in trend.
Hereafter, we discuss the estimation result of commonality parameter ∆s. ∆s indicatesthe relation between eat-in probability, number of attendance and product usage probability, and demographic variables. Since we set xht = (intercept, wife’s holiday, husband’s
holiday), we can obtain ∆ for each variable. Table 1 shows the posterior mean of ∆s.
∆1 (Intercept)
Intercept
IncomeFull Time Housewife
Wife has a collage education
Husband has a collage education
Age of housewife
Age diﬀerence between husband and wife
Number of elderly persons∆2 (Wife’s Holiday)
Intercept
IncomeFull Time Housewife
Wife has a collage education
Husband has a collage education
Age of housewife
Age diﬀerence between husband and wife
Number of elderly persons∆3 (Husband’s Holiday)
Intercept
IncomeFull Time Housewife
Wife has a collage education
Husband has a collage education
Age of housewife
Age diﬀerence between husband and wife
Number of elderly persons

Eat-in
-1.02
0.09
-0.50
-0.31
0.13
0.60
0.00
-0.06
Eat-in
-0.39
-0.04
0.36
0.14
0.02
0.21
0.04
0.07
Eat-in
-2.60
-0.10
-0.37
0.16
-0.23
0.84 *
0.00
0.25# of attendance
-0.60
-0.02
0.09
0.01
0.00
0.11
0.00
0.00
# of attendance
-0.53
-0.01
0.02
-0.01
0.01
0.11
0.00
-0.01
# of attendance
-0.41
0.00
0.05
-0.01
0.02
0.08
0.00
-0.02

Beer
-2.28
0.23
1.36 *
-0.51
0.50
-0.67
0.04
0.39
Beer
-2.12
0.38 *-0.09-1.33 *
0.30
-0.12
-0.01
0.29
Beer
2.06
0.35 *
0.97 *-0.98
1.13 *
-1.41
0.05
0.15

Sake
-12.26 *-0.07
1.95 *
-0.15
0.06
2.04
0.05
0.08
Sake
-10.99 *
0.22
-0.16
-1.01
0.11
2.36
-0.01
-0.07
Sake
-10.35 *0.06
1.57 *
-0.55
0.76
1.79
0.06
-0.07

Table 1: Posterior Mean of Commonality Parameter ∆sNote that if a parameter is signiﬁcant at 95%, we add ’*’ to the right side of the ﬁgure.
It can be seen that households with full-time housewives tend to consume both beer andsake more than other households. Furthermore, this tendency becomes stronger when
their husbands take holidays. In households with older housewives, the eat-in probability
increases when husbands take holidays. We can infer the household tendency of productusage from the relation between holiday and demographic information. This result can beapplied to marketing activity.

10

4.2

Predictive Performance Comparison

As shown in Figure 1, we predict the product demand of 94 households for the latter halfof the year.
(n)
We obtain the demand volume directory from the MCMC samples; let dˆhtj be theestimated demand calculated from the n-th sample of N samples obtained by the following
procedures.
(n)First, from the following integration, we deﬁne eat-in probability P (uht1 = 1), usage(n)(n)
probability of the j-th product P (uh,j+2 = 1) and the mean number of attendance λht :(n)
P (uht = 1) =(n)
P (uht,j+2 = 1) =∫ (u∗ )(n)ht1ϕ(µ)dµ

(20)

−∞

∫ (u∗

(n)
ht,j+2 )ϕ(µ)dµ(21)∫ c(n)
M
h +1
∑
h,m+1
1
(n)
λht =
ϕ(µ)dµm(n)
Mh + 1
chm
m=1

(22)

−∞

From the values provided above, we obtain the estimated demand.
(n)
(n)
(n)
(n)
dˆhtj = λht P (uht1 = 1)P (uht,j+2 = 1)

(23)

The estimated demand of product j of household h at time t is derived from the mean of
the samples.
N
1 ∑ ˆ(n)ˆdhtj =
dhtj
(24)
N
n=1

By summing this individual demand for all H households, we can obtain the estimateddemand at time t.
H1 ∑ˆ
D̂htj =
dhtj
(25)
H
h=1

We also use comparison models (simple trend model and naı̈ve forecasting) for forecasting.
Figure 3 shows the cumulative volume of the prediction results of each model and therealized demand. It seems that the predictive performances of the proposed model andthe simple trend model are superior to that of na?ve forecasting. In the case of beerdemand, the proposed model ﬁts the realized demand for the next four months, after
which the performance of the simple trend model is better than that of the proposedmodel. In the case of sake demand, among the three models, the proposed model has
the highest performance throughout the period. Particularly for the next two months,
the proposed model has a fairly good forecasting performance. The predictive abilityof the proposed model is satisfactory, especially for recent periods. In practice, many
manufacturers and retailers routinely make updates on a short term basis through the useof the rolling horizon-type prediction adjustment. In these circumstances, the proposedmodel is more feasible.

11

Beer

Cumulative Volume7000
Observed Value

6000

Observed Value
5000

Proposed Model

Proposed Model

Naïve

5000

Sake

Cumulative Volume
6000

Naïve
4000

Time Trend

Time Trend

4000
3000
3000
2000
2000
1000

1000

0

0
7/1

8/1

9/1

10/1

11/1

12/1

7/1

8/1

9/1

10/1

11/1

12/1

Figure 3: Prediction Performance5

Discussion

5.1

Applicable Scope of the ModelIn this section, we discuss the applicable scope of the proposed model. As mentioned
above, the proposed model can be applied to low-involvement and repeat purchase goods.
Further, we assume that these products can be purchased through more than one channel.
Therefore, we can apply this model to almost all commodity goods.
Firms dealing in these products face severe competition in customer acquisition. They
try various expedients to acquire and enclose customers to the greatest extent possible.
These ﬁrms have to sell their products not only by forecasting passive demand but also
by actively selling their products. Although many researchers aim to improve forecasting
performance of demand forecasting methods, it is impossible for ﬁrms to eliminate unsold
goods that face uncertain demand. Even if the distribution, mean and variance of demand
are completely known to us, the dead stock volume would increase when we aim to raise
the safety inventory level to decrease opportunity loss. Previously, to solve this unsold
goods problem, ﬁrms would adjust the subsequent order quality. However, in the following
discussion, we will discuss how to actively adjust demand through sales promotion.

5.2

TargetingIf household-level demand volume could be predicted, it would help implement marketing activities such as sales promotion. Hereafter, we introduce a method of customer
discrimination that targets and that is applicable to validation data.
We use joint-usage probability q̂htj to rank customers. This is obtained by multiplying
the unconditional product-usage probability and eat-in probability. We exclude the number of attendance because it is reasonable to assume that when housewives are contacted,
they cannot adjust the number of attendance.
1 ∑
(n)
(n)
P (uht = 1)P (uht,j+2 = 1)NNq̂htj =

(26)

n=1

In this research, we set the threshold as 0.5. If qitj is greater than or equal to 0.5, we
12determine that household h will purchase the jth product. We use hit rate to measureperformance. Hit rate indicates the proportion of households that match prediction andobservation. This indicator ranges from 0 to 1. Since aggregate models such as the simple
trend model and naı̈ve forecasting cannot make predictions at the household level, we
provide only the result of the proposed model. We calculate household scores from 1 June
Sake

Beer
1

1.02
1

0.95
0.98
0.9

0.96
0.94

0.85
0.92
0.8

0.9
0.88

0.75

Beer

Sake

0.86

0.7

0.84
7/1

8/1

9/1

10/1

11/1

12/1

7/1

8/1

9/1

10/1

11/1

12/1

Figure 4: Hit Rateuntil 6 months after. Figure 4 shows the hit rate of each day. The proposed model can beapplied as a customer determination method as this ﬁgure shows that the proposed model
maintains a high performance throughout the period. This implies that a household thathas a high estimated usage probability actually uses the products. Based on this result,
we can implement sales promotion for customers who have high probability of using the
product but do not actually use the product. Additionally, the proposed model can beused to realize customer retention and improve customer value.

5.3

PED (Perfect Enclosed Demand)

As mentioned above, the proposed model estimates the market demand from the accumulated demand of all households. Although we determined that the proposed model maintains a high predictive performance even for demand forecasting at the market-aggregate
level, this model can also predict the demand at the household level. Furthermore, since
this model divides household eat-in probability and product usage probabilities, we canestimate the demand when a household eats dinner on all the days. We term this indicatoras PED (perfect enclosed demand). PED can be derived from estimate demand restricted(n)
as P (uht = 1) = 1; therefore,
1dˆPhtjED =
N

N
∑

(n)

(n)

λht P (uht,j+2 = 1)

(27)

n=1

Figure 5 depicts the estimated demand at the household level, observed demand and
PED. The upper two ﬁgures show the ﬁndings for beer, while the lower two ﬁgures show
the ﬁndings for sake. The estimated demand at the household level can predict observed
demand with high accuracy. PED increases dramatically in households with a low eat-in
probability, while it increases only marginally in households with a high eat-in probability.
13

Beer (Household ID = 102)

Cumulative Volume
200
180

Beer (Household ID = 150)

Cumulative Volume300

Observed Value

Observed Value
250

160

Proposed Model

140

Proposed Model

PED

PED

200

120
100

150

80
100

60
40

50

20
0

0
7/1

8/1

9/1

10/1

11/1

12/1

7/1

Sake (Household ID = 176)

Cumulative Volume
250

350

Proposed Model

9/1

10/1

11/1

12/1

Sake (Household ID = 182)

Cumulative Volume400

Observed Value
200

8/1

Observed Value
Proposed Model

300

PED

PED
250

150

200
100

150
100

50
50
0

0
7/1

8/1

9/1

10/1

11/1

7/1

12/1

8/1

9/1

10/1

11/1

Figure 5: Estimated Demand, Observed Demand, PED

14

12/1

This implies that we can improve demand until PED is attained. We can use PED as thebasic information for customer share (Verhoef, 2003) and customer LTV (lifetime value;
Blattberg, Getz and Thomas, 2001; Blattberg, Kim and Neslin, 2008; Abe, 2008).

6

Conclusion

This research proposes a model that is a hierarchical Bayes model containing one-to-one
prediction and a time-trend factor. This model estimates market demand from accumulated demand at the household level and maintains high predictive accuracy. We haveshown that this model has an adequate ability to be used as a demand forecasting model.
From this model, we can estimate household-level usage probability, which is also highly
accurate. Further, we can apply the model not only to passive inventory control but also
to basic information concerning sales promotion such as product recommendation. Furthermore, since decision-making is divided into various factors, PED and the degree of
enclosure for each household can be estimated. This information will serve as meaningful
information that can be used by ﬁrms implementing CRM programmes.
With regard to the application of the model, we raise the following two issues. The
ﬁrst issue is that the model can be expanded and we can propose more relaxed models.
In this research, we assume independence between the number of attendance and productusage probability, although it is possible to assume that these factors are correlated. While
it is necessary to consider the speciﬁc covariance structure to maintain the independenceof eat-in probability, we can estimate this matrix using the M-H algorithm proposed byManchanda, Ansari and Gupta (1999). In this research, we ﬁx the state-space term as
the simple trend model (local-level model). We can expand other forms of the state-spacemodel that can express the time-series behaviour more appropriately.
The second issue is the application of this model to other products. This model isapplicable to many ﬁrms and stores that record POS data. If the ﬁrms and stores haveadditional data such as visit records of other stores, they can estimate PED for every
customer of electronics retail stores and supermarkets. In the future, we need to validatea model that considers long-term economic conditions. In this paper, the proposed model
uses one-year records. However, it is better to use two-year records and test the predictive
accuracy of the model. Additionally, it is necessary to validate the robustness of the modelusing other products and data.
Acknowledgement
The authors would like to thank Yanyan Jiang and Mark Tan for advice and feedback.
Also, Joint Association Study Group of Management Science for providing data, andthe University of Tokyo Manufacturing Management Research Center (MMRC) for theirresearch support.

15

A

Appendix

A.1

Detailed Formulation

The proposed model can be expressed asu∗ht = xht Bh + γt + εht , εht ∼ NJ ∗ (0, IJ ∗ )βhk = whk ∆k + ζhk , ζhk ∼ NJ ∗ (0, Θk )γt = γt−1 + νt , νt ∼ NJ ∗ (0, Φ)(28)

In this paper, household demographic information whk can be updated within the periodt = 1, · · · , T . To consider the above condition, we redeﬁne the model. First, we explain
the case of household h. If household h updates its demographic information at time t∗ ,there are two demographic variables covering time 1, · · · , t∗ − 1 and time t∗ , · · · , T . Let
the demographic variables be wh1 k , wh2 k , k = 1, · · · , K and the range of these variables
be Th1 andTh2 . We can say that Th1 = {1, · · · , t∗ − 1}, Th2 = {t∗ , · · · , T }. Let a set ofhousehold information∪ variables collected from h be Ch = {h1 , h2 }. When we aggregate
these sets, we get i∈Ch Ti = {1, · · · , T }. Therefore, we can cover the entire period formutually exclusive and collectively exhaustive situations.
Hereafter, we will provide the formulation in general. Let any household information be
i = 1, · · · , L and the range of information be Ti . Suppose that the information is collectedfrom household h and let the set of household information of h be Ch ; then, i ∈ Ch .
When all informationcollected from h is aggregated, the range of time equals {1, · · · , T }.
∪Therefore, i∈Ch Ti = {1, 2, · · · , T }. Considering these discussions, the proposed model
can be redeﬁned asu∗it = Bi xit + γt + εit , εit ∼ NJ ∗ (0, IJ ∗ ), t ∈ Ti
βik = ∆k wik + ζik , ζik ∼ NJ ∗ (0, Θk )γt = γt−1 + νt , νt ∼ NJ ∗ (0, Φ)(29)

Let the set of household information that contains time t be It , where the element
count of It , n(It ) = H, ∀t ∈ {1, · · · , T }.

A.2

Prior Distributions

We set prior distributions as follows:
βik ∼ NJ ∗ (∆k wik , Θk ), k = 1, · · · , K(30)∆k ∼ NJ ∗ ×P (∆0 , Ψ0 ), k = 1, · · · , K(31)

Θk ∼ W(ν0 , S0 ), k = 1, · · · , K(32)

γ0 ∼ NJ ∗ (γ0 , Φ0 )(33)

γt ∼ NJ ∗ (γt−1 , Φ)(34)

Φ ∼ W(p0 , P0 )(35)

where, ∆0 = OJ ∗ ×P , Ψ0 = 1000IJ ∗ , ν0 = P, S0 = 1000IJ ∗ , γ0 = 0, Φ0 = 1000IJ ∗ , p0 =J ∗ , P0 = 1000IJ ∗ .

16

A.3

Posterior Distributions

In this section, we show posterior distributions. Detailed explanations of the MCMC
method can be found in Gamerman (1998) and Rossi, Allenby and McCulloch (2005).
Rowe (2002) discusses the generation of random variables from a matrix distribution.
Geweke (1991) shows an eﬃcient generation method of truncated normal random variables.
• Latent variable u
We draw samples for each element,
– j=1

{

u∗it1 ∼ T N (0,∞) (Bi1· xit + γt1 , 1), if uit1 = 1
u∗it1 ∼ T N (−∞,0] (Bi1· xit + γt1 , 1), if uit1 = 0

(36)

– j=2
{

u∗it2 ∼ T N (ci,m ,cim+1 ) (Bi2· xit + γt2 , 1), if uit2 = m ∧ uit1 = 1
u∗it2 ∼ N (Bi2· xit + γt2 , 1), if uit1 = 0– j = 3, · · · , J ∗ ∗ uitj ∼ T N (0,∞) (Bij· xit + γtj , 1), if uitj = 1 ∧ uit1 = 1u∗ ∼ T N (−∞,0] (Bij· xit + γtj , 1), if uitj = 0 ∧ uit1 = 1
 itj
u∗itj ∼ N (Bij· xit + γtj , 1), if uit1 = 0

(37)

(38)

• Threshold parameter c
For household information i, ci1j , ciMi j and ci,Mi +1,j of cimj , j = 1, · · · , J ∗ are ﬁxed
as ci1j = −∞, ciMi j = 0 and ci,Mi +1,j = ∞. Then, we draw samples for m =2, · · · , Mi − 1 . An estimate algorithm of the ordered probit model is found in Koop
(2003).
cim ∼ U (c̄i,m−1 , c̄i,m+1 )(39)• Parameter Bi = (βi1 , · · · , βiK )βik ∼ N (M, V )
where
V
M

(∑
)
−1 −1
′ x
=
x
+
Θ
itk
t∈T
itk
k
i
)
(∑
−1
′
∗
= V
t∈Ti xit (uit − Bi,−k xit,−k − γt ) + Θk ∆k wik

(40)

(41)

Bi,−k is a matrix that eliminates the k-th element from Bi . xit,−k is a vector thateliminates the k-th element.
• Parameter ∆
where
V
M

∆k ∼ N (M, V, Θk )(42)

= (Wk′ Wk + Θk0 )−1= V (Wk′ Bk + Θk0 ∆0 )

(43)

17

• Parameter ΘΘk ∼ IW(v1 , V1 )

(44)

where
v1
V1

= v0 + L
(∑
)−1
L
′
=
i=1 (βik − ∆k wik ) (βik − ∆k wik ) + S0

(45)

• Parameter γ
We reconstruct the model for γt :u∗it − Bi xit = γt + εit , εit ∼ NJ ∗ (0, IJ ∗ )(46)

We sum the above expression for household information It , which contains time t.
∑∑∑⇐⇒(u∗it − Bi xit ) = Hγt +εit ,εit ∼ NJ ∗ (0, H 2 IJ ∗ )
(47)
i∈It

i∈It

i∈I

Dividing both sides by H, we get
1 ∑ ∗
1 ∑
1 ∑
1
(uit − Bi xit ) = γt +εit ,εit ∼ NJ ∗ (0, 2 H 2 IJ ∗ )
H
H
H
H
i∈It

i∈It

(48)

i∈I

∑
∑
Let ũ∗it = H1 i∈It (u∗it − Bi xit ) , ε̃it = H1 i∈It εit ; then, we obtain the following
state space model:{ ∗ũt = γt + ε̃t , ε̃t ∼ NJ ∗ (0, IJ ∗ )(49)γt = γt−1 + νt , νt ∼ NJ ∗ (0, Φ)
From this model, we sequentially draw γt , using the Kalman ﬁlter and smoothing.
• Parameter ΦΦ ∼ IW(v1 , V1 )(50)

v1 = p0 + T
∑T
−1
′
V1 =
t=2 (γt − γt−1 ) (γt − γt−1 ) + P0

(51)

where

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