Retail Analytics (Anna-Lena Sachs)
Retail Analytics (Anna-Lena Sachs)
Retail Analytics (Anna-Lena Sachs)
Anna-LenaSachs
Retail
Analytics
680
Anna-Lena Sachs
Retail Analytics
Integrated Forecasting and Inventory
Management for Perishable Products
in Retailing
123
Anna-Lena Sachs
Department of Supply Chain Management & Management Science
University of Cologne
Cologne
Germany
Abstract
Based on real data from a large European retail chain, we analyze newsvendor
decisions for perishable products. We suggest a data-driven approach that integrates
forecasting and inventory optimization. The approach is distribution-free and takes
external factors such as price and weather into account. Using Linear Programming,
the model fits a linear inventory function to historical demand observations. It
accounts for unobservable lost sales and stockout-based substitution by considering
the timing of sales occurrences. We show that the model captures real-world retail
characteristics well, in particular, if the assumptions of existing standard models
are violated or if demand depends on external factors and is highly censored.
Additionally, we determine the optimal policy for an aggregated service level target
in a multi-product context. We investigate whether order quantities determined by
a real decision maker consider the elements of the optimal policy. We analyze
behavioral causes to explain suboptimal decision-making and find that behavioral
biases observed in laboratory experiments are also present in the real world.
Acknowledgements
First and foremost, I am very grateful to my supervisor Prof. Dr. Stefan Minner
for his invaluable support, advice, and encouragement throughout the course of my
Ph.D. studiesboth in Vienna and in Munich. Thanks to him, I had the chance to
pursue research projects in different fields and I enjoyed working on these topics
very much. I am also very thankful to Prof. Dr. Martin Grunow for giving me useful
feedback and many insightful comments on my research as well as for being part of
the examination committee. I would like to express my gratitude to Prof. Dr. Rainer
Kolisch who took over the position as chairman of the examination committee.
I want to thank all current and former members of the chair of Logistics and
Supply Chain Management at TU Mnchen and University of Vienna: Christian
Bohner, Maximilian Budde, Alexandra Ederer, Pirmin Fontaine, Evelyn Gemkow,
Miray Kzen, Dr. Nils Lhndorf, Katharina Mariel, Dr. Arkadi Seidscher, Dr.
Lena Silbermayr, Dr. Martin Stlein, Dariush Tavaghof Gigloo, and Michael
Weingrtner. Thank you all for the many fruitful discussions.
I would also like to thank Prof. Dr. Mirko Kremer, Dr. Christian Lang, and Prof.
Dr. Sandra Transchel for their guidance throughout the course of my Ph.D. studies.
Furthermore, I am very grateful for the fruitful cooperations during the past
years. I would like to thank Prof. Ulrich Thonemann, Ph.D., and Dr. Michael
Becker-Peth for our cooperation in several interesting projects in the field of
behavioral operations management. I would like to express my deepest gratitude
to Prof. Philip Kaminsky, Ph.D., for inviting me to work on our joint project on
data-driven inventory management at UC Berkeley. I thank Dr. Birgit Lhndorf
and Prof. Dr. Rudolf Vetschera for our joint work on probability effects in utility
elicitation. I gratefully acknowledge the cooperation and most valuable input from
Bernd Binder, Jens Daniel, and Madeleine Hirsch. Furthermore, I appreciate the
great support by Achim Frauenstein and Leschek Swiniarski without whom the data
collection would not have been possible. I also thank Walter Gossmann from the
Bakery Guild for the insights I gained on bakery business in Germany. Thanks to
Dr. Stefan Appel for providing technical support on the number crunching.
vii
viii
Acknowledgements
Finally, I would like to thank my parents for their critical comments and
proofreading, my brother Johannes for sharing his expertise in Econometrics, my
husband Kai for his patience and encouragement, as well as my daughter Elisa for
bringing so much joy to my life.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.1 Motivation .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.2 Problem Statement .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1.3 Outline .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
1
1
2
3
2 Literature Review .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.1 Unobservable Lost Sales . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.2 Assortment Planning .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
2.3 Assortment Planning with Stockout-Based Substitution .. . . . . . . . . . . . .
2.4 Stockout-Based Substitution in a Fixed Assortment . . . . . . . . . . . . . . . . . .
2.5 Joint Pricing and Inventory Planning with Substitution .. . . . . . . . . . . . . .
2.6 Behavioral Operations Management . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .
5
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Contents
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Contents
xi
List of Tables
24
25
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28
29
30
32
33
45
46
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48
73
Table 3.1
Table 3.2
Table 3.3
Table 3.4
Table 3.5
Table 3.6
Table 3.7
Table 3.8
Table 4.1
Table 4.2
Table 4.3
Table 4.4
Table 4.5
49
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xiii
xiv
List of Tables
83
91
93
95
98
99
List of Figures
22
23
31
38
61
66
Fig. 6.1
Fig. 6.2
Fig. 6.3
Fig. 6.4
xv
Acronyms
AIC
ARIMA
BIC
BLUE
CR
cv
EM
FIFO
Known
LIFO
LP
LR
MCS
MM
MNL
MSE
Non
OLS
OOS
Par
POS
RMSE
SI
SKU
SL
xvii
Chapter 1
Introduction
1.1 Motivation
Recent studies on food losses have caught attention to the large amounts of food
discarded in industrialized countries due to consumer behavior and lack of supply
chain coordination which leads to expired excess inventories (Gustavsson et al.
2011). A study by the United States Department of Agriculture found that 11.4 %
fresh fruit, 9.7 % fresh vegetables, and 4.5 % fresh meat and seafood are wasted
annually (Buzby et al. 2009). In Germany, 11 million tons of food are wasted per
year (Kranert et al. 2012). The large amount of discarded food not only represents
a waste of natural resources but causes also a monetary loss to consumers and
retailers.
Despite the large amounts of excess inventories, retailers experience frequent
out-of-stock (OOS) situations. According to Corsten and Gruen (2003) OOS rates
amount to 8.3 % worldwide considering non-perishable product categories. OOS
situations are estimated to occur even more frequently for perishable products due to
their short shelf-lives that make full product availability less desirable (ECR 2003).
Food waste and OOS situations are caused by mismatches between supply and
demand. Supply chain management is especially challenging for food products
due to demand uncertainty, short shelf lives, highly demanding customers, and
low profit margins (Akkerman et al. 2010; Hbner et al. 2013). In out-of-stock
situations, Corsten and Gruen (2003) estimate that about 50 % of the customers
leave the store without buying the product, while the other half opts for substitution.
Substitution leads to an increase in sales of other products. Empirical studies of
customer reactions to stockouts indicate that customers finding poor availability in
a store on a regular basis will not only incur short-term lost sales but will also decide
not to return to this store in the long-run (Anderson et al. 2006). If stockout situations
are ignored and ordering decisions are only made based on past sales data, a store
will most likely also be understocked in the future if there was poor availability in
1 Introduction
the past. Furthermore, sales of the substitute products will be overestimated thus
incorporating demand switching behavior even though it may be more beneficial to
stock more of the first choice product.
Another reason for suboptimal decision making eventually causing waste and
stockouts are the decision makers themselves. Human decision makers are subject
to behavioral biases. By anchoring on reference points such as mean demand or
chasing demand, the decision makers systematically deviate from the optimal order
quantities (Schweitzer and Cachon 2000; Bendoly 2006; Bolton et al. 2012).
Retail companies have access to large amounts of data collected by point-of-sale
(POS ) scanner systems, but lack the tools to process the data and improve their
current practices (Fisher and Raman 2010). Retail analytics aim to achieve a better
alignment of supply and demand by leveraging this data.
This work is motivated by challenges faced when solving an inventory optimization problem for perishable products at a large European retail chain. We have
collected POS data for fruit, vegetables, and bakery products of more than 60 stores
at the retail chain starting 2008. Based on the real data, we analyze behavioral causes
of suboptimal decision-making in practice and develop novel models to account for
the prevalent retail characteristics.
1.3 Outline
1.3 Outline
This research aims to use the data available to retailers in order to better balance
excess inventory and out-of-stock situations for perishable products. Using real
data containing daily and hourly sales from 2008 on, the inventory replenishment
decisions for fruit, vegetables, and bakery products of more than 60 stores were
analyzed.
Its contribution is to overcome limitations of existing research such as assumptions on the theoretical demand distribution not fitting the demand observations,
to improve order decisions by taking additional information into account and
to analyze the ordering behavior of a real decision maker from a behavioral
perspective.
We first review related work on unobservable lost sales estimation, assortment
planning and behavioral operations management in Chap. 2.
We then develop a data-driven model for single-period problems that integrates demand forecasting and inventory optimization in Chap. 3. The model is
distribution-free and takes external variables such as price and weather into account
that influence demand. A linear inventory function of the external variables is
fitted to historical demand observations to determine its coefficients using Linear
1 Introduction
Chapter 2
Literature Review
2 Literature Review
2 Literature Review
Mahajan and van Ryzin (2001b) extend the model of Smith and Agrawal (2000)
by introducing dynamic consumer substitution where the number of substitution
attempts is not restricted and substitution rates depend on the availability of
substitutes in a given assortment. They model demand and substitution as a general
choice process. The profit-maximization problem is solved with a stochastic sample
path gradient algorithm which is compared to heuristic policies. The setting with
dynamic substitution is compared to static substitution where demand is independent of the current on-hand inventory levels. An important finding of their analysis is
that the profit function is not quasi-concave in inventory levels. Furthermore, larger
amounts of popular items and fewer amounts of unpopular items should be stocked
in an inventory system with substitution compared to a traditional newsvendor.
Kk and Fisher (2007) develop a practice-motivated approach to determine
the optimal assortment from sales data. Given their focus on products with long
shelf life and high service level, the demand function is obtained from loglinear regression, ignoring unobservable lost sales. Parameters for assortment-based
substitution are estimated from stores with varying assortments calibrated on fullassortment stores. The approach is then extended to possible out-of-stock situations.
Stockout-based substitution rates are derived from individual store sales data using
the expectation-maximization (EM) algorithm. Input data required include time of
purchase, customer arrivals at different levels and number of product units sold.
Other factors influencing purchase behavior such as price, weather and promotional
activities are also incorporated. Finally, an iterative heuristic combined with a local
search algorithm is applied to solve the assortment optimization problem. They find
that stores should aim at higher inventory levels of goods with high demand variance
thus hedging against potential lost sales. The amount of inventory to be carried of
products with large case sizes depends on the available shelf space.
Hopp and Xu (2008) formulate an attraction model with a factor for each
product that depends on quality and price. Multiple substitution attempts are
modeled by a static approximation as a simplification of the dynamic substitution
approach of Mahajan and van Ryzin (2001b). Different settings of price, service and
assortment competition are studied. In a duopoly with price, service and assortment
competition, product variety diminishes compared to a monopoly in order to avoid
price competition whereas the total number of products and thus inventory level
increases.
Honhon et al. (2010) consider the assortment planning problem with stockoutbased substitution. Demand is classified into different customer types whereas each
type has a certain ranking of purchase preferences. Prices remain fixed in this model.
The optimal assortment is determined for a fixed proportion of each customer type
and a heuristic is provided for the more general case with random proportions. They
find that the optimal set of assortment possesses a certain structure in terms of
newsvendor fractiles and underage cost.
Ycel et al. (2009) combine assortment planning with the supplier selection
problem in the presence of quality issues and dynamic substitution behavior. For
each of these aspects, a cost function is included in the overall objective function
that is furthermore subject to shelf space constraints and constraints on the quantity
of each product that can be supplied. In their analysis, they show that ignoring one
of the factors substitution, supplier selection or shelf space limitations results in a
significant loss of profit.
10
2 Literature Review
11
bias also persists after many rounds and (almost) no learning takes place (Bolton
and Katok 2008; Ho et al. 2010). The pull-to-center bias is no matter of experience
since it is not only observed in experiments with students but also with managers
who already have experience with similar ordering tasks (Bolton et al. 2012).
Several potential causes for this pull-to-center bias have been identified: One
potential explanation by Schweitzer and Cachon (2000) is demand chasing where
the decision maker anchors on his order quantity and adjusts towards observed
demand.
A second potential explanation is ex-post inventory error minimization which
describes the minimization of anticipated regret from not matching demand
(Schweitzer and Cachon 2000). Kremer et al. (2014) find additional evidence
for decision makers preference to avoid ex-post inventory errors. If participants in
an experiment are offered additional information on demand which would help to
reduce the mismatch between demand and supply, i.e. the ex-post inventory error,
they are willing to pay a price that exceeds the benefit from eliminating risk based
on several risk utility functions.
A third potential explanation is anchoring on mean demand and insufficient
adjustment (Schweitzer and Cachon 2000; Schiffels et al. 2014). It describes the
tendency of human decision makers to facilitate the decision making process by
choosing mean demand as reference point that is then adjusted.
Another potential explanation was developed by Su (2008). Su (2008) assumes
that the newsvendor decision maker chooses the order quantity from a set of
alternative solutions where more attractive alternatives (i.e. order quantities yielding
higher profits) are chosen with higher probability. This decision process is modeled
as logit choice model where the decision makers make random errors.
Kremer et al. (2010) show that ex-post inventory minimization, anchoring and
adjustment, and demand chasing are valid in the presence of framing. They present
a newsvendor situation in two different ways: one group of participants is aware
of facing a standard newsvendor problem whereas the other group plays a lottery
experiment. The first group shows a significantly stronger tendency towards mean
anchoring and demand chasing than the second group. This result contrasts Su
(2008)s findings since the explanation of random errors should hold for both cases
and not depend on the frame.
Building on the ex-post inventory error minimization framework and prospect
theory (Kahneman and Tversky 1979), Ho et al. (2010) include reference-dependent
preferences in a multilocation newsvendor model. They model the referencedependent preferences as psychological costs which the decision makers associate
with leftovers and stockouts. Decision makers associate higher psychological costs
with leftover inventory than with stockouts.
Ren and Croson (2013) identify overconfidence as an explanation for the pull-tocenter bias. If the newsvendor is too confident in his estimates, he underestimates
the variance which results in an order quantity closer to mean demand than optimal.
The authors suggest corrective measures based on coordinating contracts.
Becker-Peth et al. (2013) show that knowledge about behavioral biases helps to
improve contract design. They find that order quantities under a buyback contract
12
2 Literature Review
do not always lie between mean demand and the optimal order quantity. Further,
they study how decisions under a buyback contract are affected by the contract
parameters finding that the chosen order quantities not only depend on the critical
ratio. The decision makers value revenues from sales differently than from returning
items to the supplier. Becker-Peth et al. (2013) use these findings and the data
collected in their experiments to fit parameters of adjusted buyback contracts. They
test these adjusted buyback contracts in another set of experiments and show that
they capture the decision making process better than without behavioral parameters.
Other studies investigate, for example, the effect of censoring (Feiler et al. 2013;
Rudi and Drake 2014) or forecasting (Kremer et al. 2011). For more general reviews
on behavioral operations not only in the newsvendor context refer to Katok (2011)
and Bendoly et al. (2010).
Chapter 3
3.1 Introduction
Forecasting demand is undoubtedly one of the main challenges in supply chain
management. Inaccuracy of forecasts leads to overstocks and respective markdowns
or shortages and unsatisfied customers. To secure supply chain performance against
Springer International Publishing Switzerland 2015
A.-L. Sachs, Retail Analytics, Lecture Notes in Economics and Mathematical
Systems 680, DOI 10.1007/978-3-319-13305-8_3
13
14
15
other forecasting methods were required. Econometrics provides a huge toolbox for
estimation and statistical analysis, especially with respect to forecast errors. These
methods allow decreasing safety stock by being able to explain a larger portion of
the demand variability.
The contribution of this work is to discuss and promote integrated causal
demand forecasting and inventory management in addition to the mainstream time
series based approach. Our inventory planning approach accounts for the causal
relationship between demand and external factors in order to explain larger portions
of demand variability. We extend the data-driven approach to causal forecasting
and compare it to existing methods such as regression analysis and the method
of moments (MM). The data-driven approach directly estimates optimal inventory
and safety stocks from historical demands and external factors whereas regression
analysis and the method of moments is divided into demand prediction and
inventory level determination. Demand variation is thereby separated into explained
variation (with no need for protection by safety stock) and remaining (unexplained)
variation which requires safety inventory.
In the following, Sect. 3.2 reviews the required basics from safety stock
planning and the ordinary least squares framework and Sect. 3.3 presents the datadriven Linear Programming approach for integrated demand estimation and safety
inventory planning. An illustrative example and a numerical comparison study
including real data show the advantages and disadvantages of the application of
the two proposed approaches in Sect. 3.4.
(3.1)
16
v
.v C h/
Z
1 1
P2 D 1
.x B/f .x/dx
B
F .B/ D P1 D
(3.2)
(3.3)
The target inventory level B thus consists of two components: mean demand
and safety inventory SI (3.1) and depends on the target P1 or P2 service level or,
equivalently, on penalty and holding costs. Equations (3.2) and (3.3) represent the
standard formulas to set target inventory levels under service level constraints (see,
e.g., Silver et al. 1998). For the widely used case of normally distributed demands
(forecast errors), letting f0;1 and F0;1 denote the respective standardized normal
density and distribution, the required safety factors under specified service level
constraints become
B D C k
1
1
.P1 / D F0;1
k1 D F0;1
k2 D G
1
.1 P2 /
v
vCh
(3.4a)
(3.4b)
(3.4c)
(3.4d)
(3.5)
17
Given the observations (Xi , Di ), the OLS estimators for 0 and 1 are (see, e.g.,
Gujarati and Porter 2009):
O1 D
Pn
i D1 .Xi X /.Di
Pn
2
i D1 .Xi X /
D/
(3.6)
O0 D D O1 X
(3.7)
where D denotes the average observed demand and X the average value observed
for the explanatory variable.
For the purpose of inventory management and safety stock planning, the
inventory for a given situation with known value of the explanatory variable X0
(e.g., the next days sales price) is required. For this purpose, the estimated standard
deviation of D.X0 / is of most interest.
s
i D1 .Di
O D
V ar .D.X0 // D O
Pn
O0 O1 Xi /2
n2
1
.X0 X /2
1 C C Pn
2
n
i D1 .Xi X/
(3.8)
!
(3.9)
Expression (3.9) consists of two parts, an estimator for the standard deviation of
the error term u, as commonly used in inventory models with theoretical (normal)
demand distribution, and the standard deviation of the estimator which accounts for
the additional risk of the sample. Obviously, for n ! 1 the latter term converges
to zero. Compared to using the method of moments for estimating the standard
deviation, the percentage reduction in safety stock by using regression can be stated
by R2 as the fraction of explained variation of total variation in observations.
In an idealized setting where the regression function perfectly explains demand
fluctuations, R2 would be equal to 1 and no safety stock would be required. Any
demand variation could be explained by changes in the exogenous variables that
are known in advance. In the opposite case where the portion of explained variation
approaches 0, results obtained by regression analysis are identical to those generated
by the method of moments.
According to the classical OLS assumptions, we assume that the error term ui
is normally distributed with mean zero and constant variance 2 (homoscedasticity
assumption). Furthermore, error terms of different observations are independent.
Given that these assumptions hold, the least squares estimator O is the best linear
unbiased estimator (BLUE) and linear in Di . Its expected value equals the true value
of , i.e. it is unbiased. Furthermore, the estimator for the coefficient has the
smallest variance compared to other linear unbiased estimators, i.e. it is efficient
or best. For a detailed overview on assumptions and methods, see Gujarati and
Porter (2009).
18
m
X
j Xji C ui
(3.10)
j D1
D D X C u
(3.11)
(3.12)
(3.13)
E.u0 u/
nm
(3.14)
(3.15)
O 2 D
19
(3.16)
with ri as the error in estimating the variance of the original error term u. Then,
using uO 2i as an approximation for the unknown i2 ,
ln uO 2i D ln 2 C ln Xi C ri
(3.17)
m
X
j Xj 0 :
(3.18)
j D1
The decision variables are the parameters j and indirectly the resulting inventory levels yi and satisfied demands si for each demand observation i . The problem
can be formulated as a newsvendor model with a cost minimization objective or
with a service level constraint.
20
n
X
.hyi C v.Di si //
(3.19)
i D1
s.t.
yi
m
X
j Xij Di
i D 1; : : : ; n
(3.20)
i D 1; : : : ; n
(3.21)
i D 1; : : : ; n
(3.22)
i D 1; : : : ; n
(3.23)
j D 1; : : : ; m
(3.24)
j D0
si Di
si
m
X
j Xij
j D0
si ; yi 0
j 2 <
The objective function (3.19) is the sum of holding costs and penalty costs over
all individual demand observation and their positioning above (inventory yi ) or
below (shortage Di si ) the linear target inventory level function. Equation (3.20)
together with the minimization objective determines excess inventory yi corresponding to each demand Di . Equations (3.21) and (3.22) enforce that the sales
quantity si under a given demand is equal to the minimum of demand Di and supply.
n
X
i D1
hyi
(3.25)
21
s.t.
yi
m
X
j Xij Di
i D 1; : : : ; n
(3.26)
i D 1; : : : ; n
(3.27)
j Xij
i D 1; : : : ; n
(3.28)
j Xji
i D 1; : : : ; n
(3.29)
j D0
si Di
si
m
X
j D0
Di i M
m
X
j D0
n
X
i n.1 P1 /
(3.30)
i D1
n
X
si P2
i D1
n
X
Di
(3.31)
i D1
si ; yi 0
i D 1; : : : ; n
(3.32)
i 2 f0; 1g
i D 1; : : : ; n
(3.33)
j 2 <
j D 1; : : : ; m
(3.34)
In (3.29) M is a large number (e.g., the maximum demand observation). If a nonstockout probability is required, then (3.29) ensures that the satisfaction indicator
i for demand Di becomes one if demand exceeds supply. Summing over all
observations n, (3.30) states that a maximum of .1 P1 /n observations are allowed
to result in a stockout. Note that because of integer n, certain values of the required
service level P1 will result in an overachievement of service. Under a fill-rate service
constraint expressed in (3.31), total sales have to exceed the fraction P2 of total
demand.
22
see Petruzzi and Dada (1999). In practice, the parameters a, b, and 2 are unknown
and need to be estimated. For each randomly generated problem instance, we
normalize the price range to p 2 0; 1 and draw the known parameters a, b and
as follows:
market size a U.1; 000; 2; 000/,
slope b U.500; 1; 000/,
demand volatility is generated such that the coefficient of variation (cv) at mean
price p equals 0.3 or 0.5, i.e. cv D = D 0:3 or 0.5, respectively.
For a given instance, we sample n 2 f50; 200g normally distributed demand
observations where the price is uniformly chosen from the interval 0; 1. Figure 3.1
shows an illustrative example with 200 demand observations drawn for an instance
with a D 1471:8; b D 702:72, and D 360:7.
From all n observations, the models are estimated and inventory levels are set
such that a desired service level (either non-stockout probability or fill rate) of either
90 % or 95 % is met. For the cost model, we use a unit penalty cost of v D 9 instead
of a non-stockout probability of 90 % and v D 19 for the 95 % case such that these
two parameter settings yield the same critical fractile. Further, h D 1.
The safety stocks SI (except for the LP method) are set according to (3.1)(3.4).
For known parameters, given the price, mean demand is determined and the required
safety stocks are obtained from (3.4). Ignoring demand dependency from prices and
estimating demand parameters from the sample only (method of moments), and
in (3.1)(3.4) are replaced by the respective estimators O and O . Using the OLS
method, O D O0 C O1 p and O is estimated by (3.9). For the data set shown in Fig. 3.1
and P1 D 90 % this yields the following results:
1500
1000
0
500
Demand
2000
2500
.2
.4
.6
Price
.8
23
2000
1600
1400
1000
1200
Order
1800
.2
.4
.6
.8
Price
Known
LP
OLS
MM
.p 0:5/2
1
C
200
16:45
24
SL (%)
P1 D 90
0.3
P1 D 95
0.5
P1 D 90
0.5
P1 D 95
0.3
P2 D 90
0.3
P2 D 95
0.5
P2 D 90
0.5
P2 D 95
n D 50
cv
0.3
SL (%)
P1 D 90
0.3
P1 D 95
0.5
P1 D 90
0.5
P1 D 95
0.3
P2 D 90
0.3
P2 D 95
0.5
P2 D 90
0.5
P2 D 95
OLS
0.8904
(0.0353)
0.9419
(0.0252)
0.8868
(0.036)
0.9386
(0.0261)
0.8969
(0.0188)
0.9469
(0.0141)
0.893
(0.0238)
0.9435
(0.0176)
LP Service
0.8441
(0.0481)
0.9129
(0.0375)
0.8465
(0.0479)
0.9155
(0.0375)
0.8932
(0.0204)
0.9433
(0.0155)
0.8895
(0.0259)
0.9397
(0.0204)
LP Cost
0.8817
(0.0458)
0.945
(0.0375)
0.8821
(0.0453)
0.9302
(0.037)
OLS
444.7
(132.4)
557
(164.5)
724.3
(217.9)
905.7
(270.2)
163.1
(49.1)
263.7
(79.7)
379.6
(114.6)
546.5
(165.3)
LP Service
392
(123.2)
507.8
(156.3)
649
(207.2)
844.1
(263.9)
157.3
(51.1)
255.3
(81.0)
373.2
(118.8)
537.4
(169.5)
LP Cost
443.1
(139)
548
(171.2)
729.8
(230.7)
905.3
(284.2)
25
SL (%)
P1 D 90
0.3
P1 D 95
0.5
P1 D 90
0.5
P1 D 95
0.3
P2 D 90
0.3
P2 D 95
0.5
P2 D 90
0.5
P2 D 95
n D 200
cv
0.3
SL (%)
P1 D 90
0.3
P1 D 95
0.5
P1 D 90
0.5
P1 D 95
0.3
P2 D 90
0.3
P2 D 95
0.5
P2 D 90
0.5
P2 D 95
OLS
0.8982
(0.0175)
0.9483
(0.0118)
0.895
(0.018)
0.9454
(0.0125)
0.8996
(0.0097)
0.9494
(0.007)
0.8967
(0.0121)
0.9469
(0.0087)
LP Service
0.8805
(0.0225)
0.9333
(0.0177)
0.881
(0.022)
0.9339
(0.0174)
0.899
(0.0103)
0.9487
(0.0077)
0.8982
(0.0128)
0.9478
(0.0095)
LP Cost
0.896
(0.0217)
0.9455
(0.0164)
0.8964
(0.0213)
0.9456
(0.0163)
OLS
453.3
(124.5)
567.5
(155.2)
737.4
(205.8)
922.1
(256.3)
165.2
(44.2)
272.1
(82.1)
384.4
(106)
554.4
(153.6)
LP Service
431.1
(119.2)
536
(149.0)
710.4
(198.2)
885.5
(249)
161.3
(45.8)
267.8
(83.6)
383.8
(110)
556.5
(158.5)
LP Cost
452.6
(125.6)
566.1
(158.8)
746.3
(209.2)
934.8
(264.8)
26
OLS
0.9255
(0.0501)
0.9386
(0.0261)
0.9437
(0.0185)
0.9454
(0.0125)
0.946
(0.0072)
LP Service
0.8541
(0.0766)
0.9155
(0.0375)
0.9231
(0.0265)
0.9339
(0.0174)
0.9419
(0.0097)
LP Cost
0.9107
(0.0639)
0.9302
(0.037)
0.9421
(0.0240)
0.9456
(0.0163)
0.9484
(0.0094)
OLS
900.8
(311.5)
905.7
(270.2)
907.2
(256.9)
922.1
(256.3)
902.6
(248.2)
LP Service
716
(282.6)
844.1
(263.9)
841.1
(236.9)
885.5
(249)
891.9
(245.8)
LP Cost
922.2
(387.3)
905.3
(284.2)
916.9
(263.5)
934.8
(264.8)
918.9
(253.3)
27
(3.35)
28
n D 200
SL (%)
P1 D 90
P1 D 95
P2 D 90
P2 D 95
LP Service
0.8801
(0.0229)
0.9323
(0.0181)
0.8984
(0.0095)
0.9484
(0.0071)
LP Cost
0.896
(0.0219)
0.945
(0.0166)
LP Service
431.4
(120.2)
535
(147.4)
137.0
(43.4)
236.7
(71.5)
LP Cost
453.8
(126.7)
565.8
(156.4)
(3.36)
(3.37)
29
P1 D 95
P2 D 90
P2 D 95
n D 200
SL (%)
P1 D 90
P1 D 95
P2 D 90
P2 D 95
LP Service
LP Cost
0.8857
(0.0231)
0.8951
(0.0226)
0.9383
(0.018)
0.9458
(0.0169)
0.8994
(0.0101)
0.9493
(0.0071)
LP Service
LP Cost
150.4
(44.2)
176.6
(52.3)
196.4
(58.8)
223.8
(65.5)
20
(6.8)
51
(16.2)
30
P1 D 95
P2 D 90
P2 D 95
n D 200
SL (%)
P1 D 90
P1 D 95
P2 D 90
P2 D 95
LP Service
LP Cost
0.8816
(0.0208)
0.8961
(0.0205)
0.9342
(0.0163)
0.9456
(0.0149)
0.8965
(0.0147)
0.9464
(0.0114)
LP Service
LP Cost
719.9
(218.2)
767.3
(231.5)
965.6
(292.1)
1038.4
(316.1)
415.5
(127.3)
654.8
(198)
31
4000
0
-4000
-2000
Error term
2000
.2
.4
.6
.8
Price
and thus and can be derived. For a coefficient of variation (cv) of 0.5, we
compute required safety stock for normally and gamma distributed errors with
safety factors based on Strijbosch and Moors (1999) (see Table 3.6). A comparison
of the resulting inventory and service level shows that OLS also works well with
gamma demand if the resulting inventory function is adjusted by a gamma safety
factor (see Table 3.6, marked with an asterisk). Otherwise, if no attention is paid to
the type of demand distribution, OLS underachieves the required service level. In
contrast, the LP model does not require any prior knowledge on the type of demand
distribution. Again, the LP Cost model proves to be superior to the service level
model for P1 concerning achievement of the target service level. For P2 , the LP
Service approach performs well and produces results close to the target service level
without accumulating large excess inventories.
In summary, the impact of violations of the standard OLS assumptions varies
not only with the type of assumption, but also with the kind of service level used.
While poor performance of OLS can be compensated by applying problem-specific
remedies, the LP approach still outperforms these countermeasures in several cases
in terms of inventory level.
32
Table 3.7 Example of the resulting order functions for one store
MM
OLS
LP Service
LP Cost
Order function Q
84.01
83:55 37:07p 0:25w
6:91d1 C 13:98d2 C 16:63
82:61 25:57p 0:15w
10:31d1 C 31:72d2
95:03 22:5p 19:25d1 C
13:25d2
P1 service level
0.87
0.89
Inventory level
34.42
28.75
0.83
21.69
0.87
28.46
future demand. There are no structural breaks, seasonality or other factors during
the time horizon considered. Therefore, it is reasonable to assume that customer
demand is alike during the course of the experiment. Consequently, the functions
established are then used to generate forecasts for the following days. For each
one of these approximately 220 days (depending on product availability), predicted
order quantities are compared to the observed demand realizations.
The explanatory variables are price, weather and weekdays. Prices recorded
range from 0.29 e to 1.49 e and daily temperatures (w) lie between 6 C and
32 C. Furthermore, due to variations in demand during the week, weekdays are also
included into the model. We segment weekdays into three categories: Tuesdays and
Wednesdays generally exhibit the lowest demand levels (denoted by d1 ), Fridays
and Saturdays when maximum demand is observed (denoted by d2 ) and Mondays
and Thursdays. d1 and d2 are binary variables. Note that Mondays & Thursdays do
not have an indicator variable to avoid collinearity.
Table 3.7 shows an example of a single store with the respective order functions
for MM, OLS, LP Service and LP Cost and P1 D 90 %. The last two columns
contain the P1 service level achieved and the resulting inventory levels.
Estimation of the multi-variable regression model yields goodness of fit measures
R2 between 0.25 and 0.74 with an average of 0.44. All estimated regression models
are statistically significant. Results of our analysis are shown in Table 3.8. We
observe that the method of moments best achieves the target P1 service level of
90 %, but incurs inventory levels more than 30 % above those attained with the other
approaches. The LP Cost approach exhibits the lowest variability compared to OLS
and LP Service. Its constant service level comes at the cost of a slightly higher
inventory level. In terms of underachievement, the LP Service model only achieves
a P1 10 % below target. This can be explained by sample size effects. Since the
sample contains 50 observations, the LP Service model can be expected to perform
worse than the other approaches due to the feedback mechanism. Using a larger
sample size would solve this problem, but add new issues such as seasonality and
trend.
The target fill rate is achieved equally well by all three approaches. The MM
approaches hedge against uncertainty by accumulating larger inventory levels since
it does not take any factors into account that allow to explain demand fluctuations.
3.5 Conclusions
Table 3.8 Results for real
data
33
n D 50
SL (%)
P1 D 90
P2 D 90
n D 50
SL (%)
P1 D 90
P2 D 90
LP Cost
0.8722
(0.064)
LP Cost
38.5
(14.7)
For example, instead of stocking less on the low-demand weekdays and thereby
reducing average inventory levels, the MM approach holds the same amount of
inventory as on Fridays & Saturdays when demand is high. The LP Service and
OLS approach take these effects into account which leads to overall lower inventory
levels.
3.5 Conclusions
This work presents an integrated framework for demand estimation and safety stock
planning in environments where demand depends on several external factors such
as price and weather where a time series model would thus be inadequate. Using
basic results from econometrics for causal demand forecasting and error estimation
as well as a Linear Programming, data-driven method, target inventory levels to
minimize cost or achieve desired service levels are calculated.
We illustrate and compare the proposed methods in a numerical experiment and
a retail case application. As expected, the method of moments can achieve required
service levels but at the cost of significantly higher inventory levels. The data-driven
approach provides a robust method for inventory level determination. However,
especially under non-stockout probability constraints, the data-driven approach
exhibits an underachievement of required service levels if the size of the available
data sample is too small. This problem can be avoided by applying the cost-service
equivalence and using a cost minimization data-driven approach instead. The OLSapproach shows the best performance as long as the assumptions are valid. In case
of misspecifications (heteroscedasticity, gamma distributed residuals), the robust
approach dominates. Misspecification, however, can be overcome and appropriately
addressed by adjusting the demand estimation or inventory level determination.
Chapter 4
Motivated by data from a large European retail chain, we tackle the newsvendor
problem with censored demand observations by a distribution-free approach based
on a data-driven approach. For this purpose, we extend the model introduced in
Chap. 3. To improve the forecast accuracy, we simultaneously estimate unobservable lost sales, determine the coefficients of the exogenous variables which drive
demand, and calculate the optimal order quantity. Since demand exceeding supply
cannot be recorded, we use the timing of (hourly) sales occurrences to establish
(daily) sales patterns. These sales patterns allow conclusions on the amount of unsatisfied demand and thus the true customer demand. To determine the coefficients of
the inventory function, we formulate a Linear Programming model that balances
inventory holding and penalty costs based on the censored demand observations. In
a numerical study with data generated from the normal and the negative binomial
distribution, we compare our model with other parametric and non-parametric
estimation approaches. We evaluate the performance in terms of inventory and
service level for (non-)price-dependent demands and different censoring levels.
We find that the data-driven newsvendor model copes especially well with highly
censored data and price-dependent demand. In most settings with price-dependent
demand, it achieves similar or higher service levels by holding lower inventories
than other benchmark approaches from the literature. Finally, we show that the nonparametric approaches are better than the parametric ones based on real data with
several exogenous variables where the true demand distribution is unknown.
4.1 Introduction
In a lost sales inventory system, demand exceeding supply is usually not
recorded. Studies have shown that OOS rates amount to 8.3 % of stock-keeping
units (SKUs) per category worldwide considering non-perishable products
Springer International Publishing Switzerland 2015
A.-L. Sachs, Retail Analytics, Lecture Notes in Economics and Mathematical
Systems 680, DOI 10.1007/978-3-319-13305-8_4
35
36
(Corsten and Gruen 2003). OOS situations are estimated to occur even more
frequently for perishable products, which is due to their short shelf-lives that make
full product availability less desirable (ECR 2003). This leaves the store manager
without sufficient knowledge on additional sales that could have been made had
the inventory level been higher. Ignoring excess demand and sticking to the same
order-up-to level results consecutively in demand misspecification and more lost
sales (Nahmias 1994).Empirical studies of customer reactions to stockouts indicate
that customers finding poor availability in a store on a regular basis will not only
experience short-term lost sales but will decide not to return to this store in the
long-run (Anderson et al. 2006). Retailers often apply rules of thumb to determine
the optimal inventory level, underestimating forecast errors and levels which may
affect the whole supply chain (Wagner 2002; Tiwari and Gavirneni 2007; Hosoda
and Disney 2009).
Consequently, unobservable lost sales estimation is a key factor in inventory planning when it comes to determining optimal order quantities. Existing
approaches can be categorized into parametric and non-parametric approaches.
Parametric approaches assume some kind of underlying demand distribution. It
is often questionable, whether this demand distribution is appropriate in practice.
It is thus useful to formulate a non-parametric estimation approach that relies on
data readily available to retailers and takes external factors with a strong impact
on demand into account. One such factor is price which is usually tracked and
linked to the sales quantity. Even though the inventory planning literature has
accounted for price-dependent demand in numerous settings (Petruzzi and Dada
1999; Khouja 2000), it has not yet been considered in lost sales estimation. If
we assume that demand is some function of the sales price, we can incorporate
additional information into our approach to better explain demand variations
(Fildes et al. 2008).
We suggest a novel approach based on data-driven optimization which overcomes the limitations of existing parametric and non-parametric approaches. There
are no prior distribution assumptions and it only requires POS scanner data as
typically available in retail stores. Observations including explanatory variables
obtained from store-level scanner data are directly incorporated in inventory
optimization.
37
10
0
15
38
10
11
12
13
14
15
16
17
18
19
Hourly intervals
Prices
0.35
0.65
0.99
placing an order. Excess demand is lost so that sales data contain incomplete demand
information. Any demand occurring after the product is out-of-stock remains
unobserved.
Assuming that demand per day follows a similar pattern across all observations,
we base our approach to estimate unobserved lost sales on Lau and Lau (1996).
We assume that the level of the demand before the stockout occurs also reflects the
influence of the external variables on demand. Figure 4.1 contains data on average
hourly sales quantities of lettuce at one retail store as an example for a product with
a short life-cycle, although this is not a newsvendor-product in the strong sense. The
level of demand increases with lower prices, but the overall sales pattern is similar
for low and high prices. Demand reaches its peak in the morning hours and around
noon, then decreases, and in the late afternoon increases again.
39
one coefficient 0 for X0i D 1 for the intercept. As a result, we obtain the target
inventory level Bi as a product sum of the external factors with their respective
coefficients
Bi D
m
X
j Xj i :
(4.1)
j D0
The coefficients of this linear inventory function are set such that a cost minimization objective or a target service level objective is achieved. The idea of fitting
the coefficients to past demand observations for cost objectives or service level
constraints (in-stock probability or fill-rate) is discussed in Chap. 3. The models
therein assume that the retailer observes complete demand. This assumption may be
violated if a retailer cannot track unobservable lost sales as is typically the case with
perishable products such as fruits and vegetables.
In the following, we assume that stockouts are possible and lost sales are unobserved. The retailer then only observes sales, not demand. The decision-makers
objective is to minimize the sum of the penalty cost for unmet demand and inventory
holding costs, but unmet demand is only known for days with complete demand
observations. The sample of sales observations is grouped into full (F D f1; : : : ; cg)
and censored (C D fc C 1; : : : ; N g) demand observations. Each observation in
i D 1; : : : ; N is either assigned to F or C , depending on whether a stockout occurs
on day i or not. If a product stocks out, then demand can only be observed until the
time of the last sale which is usually recorded by point-of-sale scanner systems. If
no stockout occurs, then demand can be observed for the entire day.
For all observations in set C (censored demand observations), the product is sold
out before the store closes. Cumulative sales Ht i and sales ht i at times t (usually
per hour) are recorded by the POS scanner system until the product stocks out. The
time point of the last sale, that is when the product stocks out, is recorded as t D ki .
This is an approximation because part of the demand in ki is already lost.
Based on the complete demand observations in set F , sales patterns are defined
according to the following procedure. Each day i is divided into t D 1; : : : ; T
discrete time intervals, e.g., hours. For each time interval, sales ht i are recorded.
Cumulative demand Ht i at the end of the day matches Di when t D T , i.e., HT i
Di . Therefore, we know that the ratio of the mean of cumulative sales HN T to mean
demand DN of all demand observations is:
HN T
D1
DN
(4.2)
40
The ratios of the preceding time intervals t can then be obtained recursively as
HN t
HN t C1
HN t C1 HN t C1 hN t C1
D
D
DN
DN
DN HN t C1
DN
hNt C1
1
HN t C1
1
Kt
(4.4)
with KT D 1:
(4.5)
It follows that
Kt C1
Kt D
hN
1 HNt C1
t C1
The cumulative demand for all days with censored observations can then be
estimated by interpolating the corresponding ratios before and after the stockout
occurred in ki . Since cumulative demand is multiplied with a term greater than one,
it inflates the sales of the respective day.
Dik D Hki i
.Kki C Kki 1 /
2
(4.6)
N
X
hyi C
i D1
c
X
v.Di si /
i D1
N
X
(4.7)
s.t.
si Di ; 8i 2 F
Hki i .Kki C Kki 1 /
2
m
X
si
j Xj i
si
(4.8)
8i 2 C
(4.9)
8i 2 F [ C
(4.10)
8i 2 F
(4.11)
j D0
yi
m
X
j D0
j Xj i Di
yi
m
X
j D0
j Xj i
41
8i 2 C
(4.12)
si ; yi 0
(4.13)
j 2 <
(4.14)
The objective function (4.7) consists of inventory holding costs for all observations i D 1; : : : ; N and penalty costs for unmet demand. The penalty costs for unmet
demand is split up into complete (i D 1; : : : ; c) and censored (i D c C 1; : : : ; N )
demand observations. In the case of censored demand observations, an estimator
based on (4.6) accounts for the unobservable lost sales.
For days in set F (full demand observations), we introduce a constraint (4.8)
that the newsvendor can only meet incoming demand, i.e., sales si may not exceed
demand Di . Constraint (4.9) ensures that sales may not exceed the lost sales
estimate. The newsvendor cannot sell more than the order quantity (4.10).
The leftover inventory yi , which is discarded at the end of the day, is the
difference between the order quantity and the incoming demand (4.11). As part
of the objective function, the retailer has to consider potential penalty costs for
the estimated unobservable lost sales in set C . The lost sales estimate reduces the
leftover inventory as in (4.12).
p
O D 0:5.S D c /.z C z2 C V 2 / with
D c D .1=c/
c
X
Di
(4.15)
(4.16)
(4.17)
i D1
Therefore, the authors derive an estimator for V and implicitly for z according to
the following equations:
Pr
V2 D 4 1C
2
i D1 .Di D c /
r.S D c /2
!
(4.18)
42
p
p
2
2c z C z2 C V 2
c
.1= 2
/e 0:5z
zC
p
D
R1
N c
N c
V2
.1= 2
/ z e 0:5D 2 dD
(4.19)
For a more detailed description and proofs on how to derive these parameters,
see the maximum likelihood estimators for normal demand in Nahmias (1994) and
Halperin (1952).
Agrawal and Smith (1996) investigate empirical retail data and claim that the
negative binomial distribution provides a better fit. They argue that the negative
binomial distribution is appropriate for retail data that shows high variability due to
external influences such as weather or promotions. The observed frequency values
for each sales quantity smaller than the order up-to-level are unbiased estimators of
the true frequency values. The observed frequency of demand j is denoted fjobs .
The censored sample mean xN s is calculated according to:
xN s D
S 1
X
.j S /fjobs C S:
(4.20)
j D0
The aim is then to match the censored sample mean to the mean computed from
the negative binomial distribution with unknown rO and q:
O
.Or ; q/
O D
S 1
X
.j S /fj .Or ; q/
O C S:
(4.21)
j D0
L
X
fj .Or ; q/
O
(4.22)
j D0
assigns a unique success probability q.r/ to r for any value L < S . According to the
procedure by Agrawal and Smith (1996), L should be chosen such that FLobs 0:1.
In a non-parametric approach, Lau and Lau (1996) first determine the fractiles of
the left-hand tail of the demand distribution according to Kaplan-Meiers Product
Limit method. Based on complete demand observations, daily-sales patterns with
Rt D Rt C1
hN t C1 HN t C1
HN t C1 DN t C1
(4.23)
2S
.Rki C Rki 1 /
(4.24)
43
which allow to calculate the fractiles of the right-hand tail of the distribution.
Finally, a Tocher-curve with parameters .a; b; c; d; e/ is fitted to the fractiles q using
regression analysis. Fractiles are selected according to subjective elicitation sets.
The number of fractiles depends on the portion of the demand distribution included
in the data. Given the inverse cumulative demand distribution function with:
FT1 .q/ D a C bq C cq 2 C d.1 q/2 ln.q/ C eq 2 ln.1 q/;
(4.25)
optimal order quantities can be determined (Lau and Lau 1997). Additionally,
we correct for external factors by adding a term for the coefficients gj of the
independent variables when fitting the Tocher curve:
FT1 .q/ D a C bq C cq 2 C d.1 q/2 ln.q/ C eq 2 ln.1 q/ C
m
X
gj Xj
(4.26)
j D1
We then calculate the optimal inventory level from the Tocher-curve. This
adaptation allows for a fair comparison with lower inventories in a setting where
demand dependency on external factors can be observed.
(4.27)
We compare two settings concerning price pi . In the first setting, price is constant
at pi D 0:5 and in the second one, price is uniformly distributed on the interval
0I 1.
For each type of distribution, we choose a common parametric estimation
approach (Par) from the literature and the nonparametric approach (Non) according
to Lau and Lau (1996) as a comparison. For the normal distribution, we follow
the approach of Nahmias (1994), and for the negative binomial Agrawal and Smith
44
(1996). The column Known contains the results for a decision maker who has full
information on the true parameters of the distribution. The results of the Linear
Programming model if demand was fully observable are contained in LP. Since we
artificially censor the demands, we know the true level of demand and use these
values in the model named LP, because the model in Chap. 3 works only with
full demand observations and does not account for any censoring. The model with
censoring based on (4.7)(4.14) is named LPc in the following.
To measure the performance, we draw a sample of N observations to estimate
the parameters of the respective model and then calculate the resulting inventory and
service level for an out-of-sample size of 100,000 observations. The following two
sections deal with samples of N D 200 observations, whereas Sect. 4.4.3 compares
the results for varying sample sizes. The experiment is repeated for 500 randomly
generated instances.
45
cv
S (%)
0.3
50
0.3
75
0.3
90
0.3
95
cv
S (%)
0.3
50
0.3
75
0.3
90
0.3
95
LPc
271.26
(71.19)
432.76
(113.46)
601.18
(157.69)
710.87
(186.82)
Par
271.05
(71.15)
431.17
(113.20)
595.82
(156.65)
702.05
(184.68)
Non
269.54
(70.72)
429.82
(112.70)
596.81
(156.29)
704.17
(184.24)
LPc
274.58
(69.88)
437.91
(110.98)
607.32
(153.95)
718.83
(182.38)
Par
322.85
(57.85)
512.88
(92.81)
704.36
(130.71)
824.83
(155.61)
Non
324.85
(58.37)
517.89
(94.38)
711.04
(133.21)
831.56
(158.51)
The fewer demands are censored (and the larger the tail of the distribution that
can be observed), the more does the parametric approach outperform the other
models in terms of inventory, since estimators of mean and standard deviation can be
calculated from a greater number of observations. However, with increasing demand
variability (cv D 0:5), the parametric approach has more difficulty in achieving the
target service level.
The non-parametric approach is closest to the target service level for all censoring
levels, which is compensated by higher overall inventory levels (up to 3.2 % higher
than the Known for S3 ). Comparing coefficients of variation, the average inventory
levels increase due to the additional amount of safety stock required to hedge against
increasing demand uncertainty.
For samples with uniformly distributed price on the interval 0I 1 (see Table 4.3),
demand variability can be partly explained by price changes. The parametric
approach does not take the price information into account and adjusts for the
additional variability by holding higher inventory. The adapted non-parametric
approach still incurs higher inventory levels than the LPc model, but at the same
time overachieves the required service levels of 75 %, 90 %, and 95 %. The results
show that the LPc model copes very well with all censoring levels and adjusts the
optimal order quantity according to the selling price.
46
cv
S (%)
0.3
50
0.3
75
0.3
90
0.3
95
0.5
50
0.5
75
0.5
90
0.5
95
cv
S (%)
0.3
50
0.3
75
0.3
90
0.3
95
0.5
50
0.5
75
0.5
90
0.5
95
LPc
0.4508
(0.0146)
0.7398
(0.0430)
0.9026
(0.0297)
0.9513
(0.0203)
0.4508
(0.0146)
0.7354
(0.0355)
0.8990
(0.0254)
0.9492
(0.0181)
Par
0.4521
(0.0100)
0.7186
(0.0176)
0.8841
(0.0148)
0.9400
(0.0119)
0.4493
(0.0096)
0.7136
(0.0172)
0.8797
(0.0147)
0.9367
(0.0122)
Non
0.4801
(0.0146)
0.7540
(0.0238)
0.9061
(0.0211)
0.9522
(0.0157)
0.4744
(0.0141)
0.7460
(0.0228)
0.8985
(0.0212)
0.9481
(0.0159)
LPc
115.00
(30.66)
272.19
(79.48)
459.34
(130.94)
577.92
(165.50)
186.97
(49.89)
441.85
(124.61)
746.24
(207.02)
940.94
(262.87)
Par
115.47
(30.48)
254.20
(67.60)
423.11
(112.52)
535.05
(142.39)
185.93
(49.10)
413.02
(109.87)
689.52
(183.34)
872.67
(232.31)
Non
126.54
(33.72)
281.59
(75.85)
462.05
(125.63)
574.92
(156.36)
202.36
(53.89)
453.75
(121.89)
742.43
(201.60)
931.06
(252.82)
47
cv
S (%)
0.3
50
0.3
75
0.3
90
0.3
95
0.5
50
0.5
75
0.5
90
0.5
95
cv
S (%)
0.3
50
0.3
75
0.3
90
0.3
95
0.5
50
0.5
75
0.5
90
0.5
95
LPc
0.5174
(0.0168)
0.7672
(0.0202)
0.9059
(0.0161)
0.9509
(0.0132)
0.5162
(0.0151)
0.7660
(0.0197)
0.9053
(0.0162)
0.9509
(0.0134)
Par
0.5086
(0.0282)
0.7567
(0.0226)
0.9043
(0.0147)
0.9531
(0.0097)
0.5046
(0.0222)
0.7551
(0.0177)
0.9011
(0.0123)
0.9497
(0.0087)
Non
0.5433
(0.0303)
0.7912
(0.0240)
0.9197
(0.0169)
0.9595
(0.0115)
0.5403
(0.0226)
0.7884
(0.0198)
0.9141
(0.0160)
0.9560
(0.0117)
LPc
144.89
(37.91)
297.50
(77.80)
469.65
(120.71)
583.95
(158.30)
233.97
(62.49)
487.55
(129.12)
775.04
(202.00)
966.73
(265.45)
Par
165.89
(33.80)
341.60
(65.75)
546.82
(104.61)
682.49
(130.36)
239.77
(56.19)
503.42
(117.18)
807.72
(185.57)
1008.56
(232.23)
Non
184.63
(38.26)
377.56
(74.35)
582.95
(117.01)
710.96
(143.88)
268.44
(63.28)
555.13
(128.99)
853.13
(200.75)
1049.51
(248.19)
48
(4.28)
2
:
2
(4.29)
rO denotes the rth success and qO the probability of success on a single trial. We
then draw the demands and perform the censoring. A cost comparison for negative
binomial data with cv D 1 in Table 4.4 shows that total costs for highly censored
data are always lower (0.32.2 %) with the LP and LPc model, even for constant
prices.
cv
S (%)
50
75
90
95
cv
S (%)
50
75
90
95
LPc
4.2588
(1.0266)
8.3712
(2.0512)
13.7643
(3.4294)
17.8829
(4.4978)
Par
4.2756
(1.0338)
8.3898
(2.0639)
13.7761
(3.4466)
17.8404
(4.4961)
Non
4.2643
(1.0244)
8.3791
(2.0528)
13.7719
(3.4288)
17.8857
(4.4958)
LPc
4.2597
(1.0445)
8.4195
(2.0736)
13.9170
(3.4477)
18.1398
(4.5147)
Par
4.3574
(1.0077)
8.5376
(2.0271)
13.9621
(3.3868)
18.0505
(4.4137)
Non
4.3528
(1.0045)
8.5174
(2.0276)
13.9002
(3.4014)
17.9740
(4.4486)
49
Tables 4.5 and 4.6 contain data at the less aggregate level, displaying average
service and inventory levels for constant and varying prices, respectively. We
investigate whether increasing the level of dispersion has an effect on the accuracy
of the estimates by using coefficients of variation of 1, 1.5, and 2.
For constant prices, the non-parametric approach incurs the lowest average
inventory levels together with a general service underachievement. The other
models tend to overestimate demand for high censoring. Other than for normally
distributed demand, increasing the coefficient of variation greater than one yields
lower inventory levels for highly censored data. Ridder et al. (1998) prove that larger
variances may also result in lower costs in the newsvendor model. In our setting, the
effect only occurs for low and is reversed for high service levels. The data reveals
that the number of days with no demand increases strongly with higher coefficient
of variations. Consequently, low service levels can easily be achieved with smaller
order quantities which results in lower average inventory levels. Compared to the
Table 4.5 Numerical results for the negative binomial distribution (p D 0:5)
cv
S (%)
50
75
90
95
1.5
50
1.5
75
1.5
90
1.5
95
50
75
90
95
LPc
0.5365
(0.0409)
0.7686
(0.0323)
0.9069
(0.0207)
0.9535
(0.0146)
0.5354
(0.0369)
0.7615
(0.0314)
0.9031
(0.0212)
0.9516
(0.0150)
0.5480
(0.0396)
0.7596
(0.0296)
0.9018
(0.0212)
0.9509
(0.0154)
Par
0.5457
(0.0527)
0.7710
(0.0358)
0.9077
(0.0220)
0.9526
(0.0144)
0.5408
(0.0411)
0.7644
(0.0333)
0.9020
(0.0221)
0.9470
(0.0158)
0.5471
(0.0394)
0.7615
(0.0327)
0.8977
(0.0228)
0.9401
(0.0176)
Non
0.4848
(0.0429)
0.7393
(0.0347)
0.8912
(0.0227)
0.9423
(0.0165)
0.4852
(0.0416)
0.7385
(0.0333)
0.8905
(0.0228)
0.9421
(0.0163)
0.4768
(0.0441)
0.7425
(0.0330)
0.8913
(0.0225)
0.9423
(0.0162)
(continued)
50
cv
S (%)
50
75
90
95
1.5
50
1.5
75
1.5
90
1.5
95
50
75
90
95
LPc
1.27
(0.40)
4.01
(1.14)
8.56
(2.40)
12.53
(3.62)
0.96
(0.35)
4.56
(1.41)
12.03
(3.56)
19.28
(5.81)
0.48
(0.25)
4.10
(1.42)
14.24
(4.52)
25.08
(8.07)
Par
1.33
(0.49)
4.02
(1.24)
8.56
(2.50)
12.21
(3.49)
1.00
(0.40)
4.57
(1.49)
11.82
(3.53)
17.92
(5.12)
0.47
(0.24)
4.12
(1.59)
13.48
(4.29)
21.24
(6.12)
Non
1.19
(0.36)
3.85
(1.10)
8.15
(2.25)
11.63
(3.22)
0.88
(0.31)
4.30
(1.34)
11.27
(3.31)
17.52
(5.13)
0.38
(0.21)
3.88
(1.36)
13.15
(4.15)
22.49
(6.97)
other approaches, the LPc model best matches the target service level for data with
high variability.
The results for data with price variability in Table 4.6 show that the nonparametric approach matches the target service levels well. This comes at the
cost of high inventories. The LPc model also suffers from this drawback for high
variability that cannot be explained by price .cv D 2; S4 /. For lower coefficients of
variation, the LPc model achieves service levels similar to the non-parametric model
while building up fewer inventories. Compared to the dataset with constant prices,
inventory levels of the parametric and non-parametric model are higher. In the case
of the parametric model this is due to the unexplained demand variability resulting
from varying prices. In contrast, the LP and LPc model capture the price effects and
thus incur less inventory than with constant price for cv D 1 and cv D 1:5.
51
cv
S (%)
50
75
90
95
1.5
50
1.5
75
1.5
90
1.5
95
50
75
90
95
LPc
0.4984
(0.0340)
0.7477
(0.0311)
0.8987
(0.0216)
0.9487
(0.0152)
0.5028
(0.0331)
0.7479
(0.0308)
0.8980
(0.0212)
0.9480
(0.0151)
0.5132
(0.0336)
0.7481
(0.0305)
0.8961
(0.0223)
0.9437
(0.0238)
Par
0.5505
(0.0456)
0.7803
(0.0361)
0.9157
(0.0209)
0.9586
(0.0129)
0.5381
(0.0436)
0.7643
(0.0327)
0.9011
(0.0212)
0.9448
(0.0147)
0.5454
(0.0368)
0.7633
(0.0327)
0.8959
(0.0244)
0.9357
(0.0244)
Non
0.5018
(0.0410)
0.7535
(0.0321)
0.9041
(0.0179)
0.9515
(0.0121)
0.4936
(0.0374)
0.7469
(0.0322)
0.9044
(0.0172)
0.951
(0.0124)
0.4893
(0.0464)
0.7502
(0.0308)
0.9027
(0.0198)
0.9467
(0.0232)
(continued)
52
cv
S (%)
50
75
90
95
1.5
50
1.5
75
1.5
90
1.5
95
50
75
90
95
LPc
1.24
(0.37)
3.92
(1.09)
8.50
(2.50)
12.34
(3.63)
0.95
(0.31)
4.43
(1.35)
11.96
(3.76)
19.22
(8.01)
0.51
(0.25)
4.28
(1.52)
17.72
(13.96)
34.18
(28.03)
Par
1.40
(0.43)
4.31
(1.18)
9.13
(2.36)
13.02
(3.33)
1.01
(0.42)
4.71
(1.42)
12.20
(3.43)
18.42
(4.89)
0.50
(0.27)
4.51
(1.72)
14.70
(4.54)
23.31
(7.02)
Non
1.33
(0.38)
4.17
(1.10)
8.92
(2.38)
12.66
(3.47)
0.92
(0.32)
4.56
(1.37)
12.60
(3.53)
19.56
(5.56)
0.44
(0.24)
4.46
(1.73)
17.16
(7.93)
29.64
(13.71)
approach. Service deviates from the target values by up to 21 % for small sample
sizes, but results are reasonably accurate for 500 observations (Table 4.7).
53
Prices lie between 0.09 e and 1.79 e for two sorts of lettuce and 0.85 e and
1.79 e for mushrooms. Maximum temperatures range between 11 and 38 C.
We only include days with full availability in order to have information on the
true demand values. Availability strongly varies with product and store. We then
artificially censor demand at order-up-to levels S and pretend not to know hourly
sales as recorded by the POS scanner. We then divide all observations into a sample
Table 4.7 Sample size effects for the negative binomial distribution (p 0I 1, cv D 1:5)
S (%)
50
75
20
90
95
50
75
50
90
95
50
75
100
90
95
50
75
500
90
95
LPc
0.5168
(0.0987)
0.7358
(0.0901)
0.8775
(0.0688)
0.9239
(0.0593)
0.5110
(0.0674)
0.7408
(0.0630)
0.8908
(0.0446)
0.9365
(0.0341)
0.5055
(0.0505)
0.7453
(0.0447)
0.8940
(0.0308)
0.9447
(0.0219)
0.5033
(0.0230)
0.7498
(0.0199)
0.9005
(0.0134)
0.9503
(0.0096)
Par
0.5330
(0.1023)
0.6892
(0.0951)
0.7410
(0.1036)
0.7503
(0.1092)
0.5465
(0.0741)
0.7551
(0.0654)
0.8600
(0.0577)
0.8919
(0.0594)
0.5386
(0.0531)
0.7630
(0.0479)
0.8911
(0.0332)
0.9309
(0.0276)
0.5406
(0.0311)
0.7651
(0.0225)
0.9046
(0.0135)
0.9500
(0.0087)
Non
0.4743
(0.1064)
0.7189
(0.0959)
0.8643
(0.0683)
0.9049
(0.0629)
0.4867
(0.0679)
0.7400
(0.0635)
0.8902
(0.0386)
0.9346
(0.0314)
0.4888
(0.0532)
0.7440
(0.0459)
0.8991
(0.0256))
0.9456
(0.0194)
0.4943
(0.0310)
0.7487
(0.0216)
0.9082
(0.0109)
0.9559
(0.0077)
(continued)
54
S (%)
50
75
20
90
95
50
75
50
90
95
50
75
100
90
95
50
75
500
90
95
LPc
1.32
(1.06)
5.11
(3.21)
13.57
(8.51)
21.83
(14.96)
1.09
(0.63)
4.57
(2.06)
12.53
(6.08)
18.95
(9.68)
0.99
(0.44)
4.48
(1.63)
12.01
(4.90)
19.20
(7.82)
0.94
(0.27)
4.43
(1.22)
11.94
(3.33)
19.02
(6.01)
Par
1.20
(1.34)
3.24
(1.88)
4.64
(2.74)
5.15
(3.57)
1.20
(1.01)
4.72
(2.25)
9.44
(3.70)
12.29
(4.79)
1.04
(0.51)
4.80
(1.87)
11.47
(3.68)
16.29
(4.86)
1.00
(0.32)
4.68
(1.22)
12.40
(3.14)
19.32
(4.88)
Non
0.97
(0.83)
4.42
(2.62)
10.70
(5.62)
15.19
(8.52)
0.95
(0.53)
4.59
(1.99)
11.79
(4.36)
17.37
(6.48)
0.92
(0.41)
4.55
(1.63)
12.26
(3.83)
18.68
(5.99)
0.92
(0.28)
4.55
(1.21)
12.87
(3.31)
20.27
(5.22)
of size 250 for parameter estimation and out-of-sample observations. The out-ofsample size depends on the individual availabilities of each product at a store.
Table 4.8 shows a summary of the results: the non-parametric approach captures
low target service levels very well and has the lowest inventory of all. However,
it accumulates more inventory to achieve high service levels. Both parametric
approaches underachieve target service levels, especially for high censoring. The
parametric approaches have a major advantage in the examples with simulated
4.5 Conclusions
55
S (%)
50
75
90
95
S (%)
50
75
90
95
LP
0.4326
(0.1571)
0.6976
(0.1589)
0.8729
(0.1021)
0.9354
(0.0631)
Par
Neg.Bin.
0.3972
(0.1388)
0.6681
(0.1277)
0.8444
(0.0787)
0.9064
(0.0539)
Non
0.4559
(0.1449)
0.6841
(0.1494)
0.9216
(0.0447)
0.9760
(0.0198)
LP
4.04
(2.90)
10.27
(6.35)
19.89
(10.38)
27.92
(13.31)
Par
Neg.Bin.
3.19
(2.28)
9.52
(5.74)
18.93
(9.58)
25.46
(11.74)
Non
4.40
(3.27)
11.42
(8.10)
28.71
(14.53)
43.46
(18.42)
data from a theoretical demand distribution as they are targeted at this specific
type of distribution. According to the Kolmogorov-Smirnov test, the assumption
that demand follows the normal distribution can be rejected for all datasets. The
assumption of the negative binomial distribution can be rejected for 65 of the 192
datasets at the 0.1 level. Since the real data does not always follow a specific
theoretical distribution, the parametric approaches make assumptions on the form
of the distribution that might not be true. The LPc model performs well for all levels
of censoring and service. The inventory level is only slightly higher than the results
of the non-parametric approach for low service levels, but achieves high service
levels of 90 % (95 %) with 23 % (30 %) less inventories.
4.5 Conclusions
In retailing, out-of-stock situations are a common phenomenon. By taking this
effectas well as additional information that explains parts of the demand
variabilityinto account, we extended the data-driven newsvendor model to
56
Chapter 5
We extend the data-driven inventory model with censored demand to the twoproduct case with stockout-based substitution. If one product stocks out, a fraction
of the demand that cannot be satisfied is shifted to the substitute. As a result, sales
of the substitute are inflated by the additional demand. The amount of substituted
demand as well as unobservable lost sales are estimated based on the timing of
stockout events. Similarly to the model in Chap. 4, we establish sales patterns based
on the hourly sales observations before a stockout occurs. Our numerical study and
data from a large European retail chain shows that the data-driven model achieves
higher average profits than an existing approach from the literature. Investigating the
trade-off between learning about substitution behavior from highly censored data
versus learning about demand from little censored data, we find that more learning
about substitution yields slightly better results in terms of profits.
5.1 Motivation
Sales and operations planning in retail can considerably benefit from the availability
of big data provided by point-of-sale demand information. Nevertheless, the solution
of the trade-off between ordering too many and too few is still a challenge
for operations practice, in particular when managing perishable products. This
challenge results in a considerable amount of product waste on the one hand
(Gustavsson et al. 2011) and stockouts on the other hand (Aastrup and Kotzab
2010). The practical retail decision problem is further complicated by the fact that
part of the required data for decision making is not readily available from the large
amount of sales data, that is lost sales and customer substitution behavior is typically
unobserved when products are out-of-stock. On average, 8.3 % of all products are
typically found to be out-of-stock in retailing (Corsten and Gruen 2003). This
number is usually even higher for perishable products where leftover inventories
Springer International Publishing Switzerland 2015
A.-L. Sachs, Retail Analytics, Lecture Notes in Economics and Mathematical
Systems 680, DOI 10.1007/978-3-319-13305-8_5
57
58
can only be carried for a short period of time (ECR 2003). If customers do not find
their first choice available, they may be willing to compensate this lack by buying a
substitute instead or otherwise do not make or postpone the purchase.
In the existing retail operations literature, these particular challenges are
addressed, but usually distributional and prior assumptions on demands and
substitution rates are made. In our research, we follow a more recent approach of
data-driven optimization (Bertsimas and Thiele 2006; Huh et al. 2011; Beutel and
Minner 2012) to provide an integrated forecasting and inventory level optimization
framework. The general idea of this approach is to use the historical data (typically
multivariate data including sales prices, weather and other demand influencing
environmental variables) directly to optimize a (linear) target inventory level as
a function of the respective environmental variables and then use this functional
relationship to provide ordering decisions. The problem to optimize the inventory
level function in a multi-product assortment is modeled and solved as a MixedInteger Linear Program and therefore can rely on the large-scale solution capabilities
of standard solvers which have successfully been applied for many other operations
problems in practice. Following this approach does not require the analysis and
parameterization of distributional assumptions which usually will also vary between
products and stores and therefore does not provide a robust framework for proposing
inventory level decisions.
We collected 4 years of hourly sales data including prices of perishable products
of 66 stores of a large European retail chain. This data set is used to illustrate
the functionality of our data-driven framework and to quantify the benefits and
robustness compared to existing approaches from the literature.
From a customers point of view, willingness to substitute depends on the
products characteristics and availability. From a retail companys point of view,
holding large amounts of stock is costly and a store manager often prefers not to
satisfy all the demand but rather shift some demand to other products. In order
to control demand, a store manager needs a clear understanding of the factors
influencing demand as well as unobservable lost sales and substitution behavior.
59
unobservable lost sales, such as the normal (Nahmias 1994) or the negative binomial
distribution (Agrawal and Smith 1996). Substitution probabilities can be derived
based on similar assumptions such as the negative binomial in Smith and Agrawal
(2000) or the Poisson process in Anupindi et al. (1998).
Another important aspect is to include substitution into inventory optimization.
One stream of research focuses on the decision which products to offer in an
assortment (van Ryzin and Mahajan 1999; Mahajan and van Ryzin 2001a; Gaur
and Honhon 2006; Caro and Gallien 2007; Hopp and Xu 2008; Ulu et al. 2012;
Saur and Zeevi 2013) while another stream of research including this work
studies inventory optimization decisions given a fixed assortment (Nagarajan and
Rajagopalan 2008), or a combination of both (Topaloglu 2013).
The literature on inventory optimization with consumer-driven demand substitution goes back to the works by McGillivray and Silver (1978) and Parlar and
Goyal (1984). McGillivray and Silver (1978) solve the two-product problem with
substitution by heuristic policies. Parlar and Goyal (1984) show that the profit
function for two products with substitution is concave. Netessine and Rudi (2003)
extend their work by analyzing the n-product case in a competitive and a noncompetitive environment.
While Anupindi et al. (1998) focus on estimating substitution probabilities
without inventory optimization, Smith and Agrawal (2000) also study a newsvendor
setting with substitution. As opposed to our approach, demand substitution and
inventory optimization are calculated sequentially. Chen and Plambeck (2008)
suggest an inventory management approach with learning about the distribution
function. Information about the demand distribution and substitution probabilities
is continuously updated using Bayesian learning. Vulcano et al. (2012) estimate
primary demand and lost sales from sales data by determining preference weights
and customer arrival rates. Their approach requires, however, information on a
retailers aggregate market share.
Several approaches working with the multinomial logit choice model to analyze
demand require data on customer arrivals (Kk and Fisher 2007; Karabati et al.
2009) which is not always available. A number of other approaches with focus
on inventory optimization assume that substitution probabilities are known (Caro
and Gallien 2007). Netessine and Rudi (2003) study the multiple product inventory
control problem with stockout-based substitution for centralized and decentralized
decision-making. The focus of their work lies on the inventory optimization problem. Demand substitution occurs with deterministic rates. Our approach integrates
inventory optimization and estimation of demand including unobservable lost sales
and substitution behavior.
Gilland and Heese (2013) analyze the importance of the sequence of customer
arrivals. In a retail setting with two mutual substitutes and shortage penalty costs,
the profit gained depends on whether a product satisfies a customers first choice
(no shortage penalty) or whether it serves as a substitute (shortage penalty for not
satisfying the customers first choice incurs).
Aydin and Porteus (2008) consider the joint inventory and pricing problem for
multiple products with price-based, but no stockout-based substitution. Karakul and
60
Chan (2008) study the joint inventory and pricing problem for multiple products
with stockout-based substitution. Even though they account for price-dependent
demand behavior, they do not consider that substitution itself may be pricedependent as well. We study an inventory problem for multiple products with
price-based and stockout-based substitution, but without price optimization.
For a comprehensive review on perishable inventory control with substitution
and related aspects refer to Kk et al. (2008), Pentico (2008), and Karaesmen et al.
(2011).
There is still a large gap between models existing in the literature and decisionmaking in practice which is mostly based on subjective judgements (Tiwari and
Gavirneni 2007; Fisher 2009). This is partly due to the fact that store-level POS
scanner data does not contain all information required to apply these models or that
distributional assumptions cannot be generalized to all products and retail environments in practice. Additionally, demand and substitution estimation are often separated from inventory optimization which is essentially one task for a store manager.
As opposed to the above literature, we extend the inventory optimization problem
by introducing stockout-based substitution rates in a data-driven multi-product
newsvendor setting. Our model does not require any data on customer-specific
shopping behavior such as individual departure times. Our methodology is based on
the data-driven newsvendor model (Bertsimas and Thiele 2006; Beutel and Minner
2012; Sachs and Minner 2014) where optimal order quantities are calculated directly
from the data and which is distribution-free.
5.3 Model
A retailer offers two products within a category in a fixed assortment. Depending on
the personal preferences, a customer may choose to purchase a substitute product
if the first choice product is out-of-stock. We assume that a fixed proportion
(rate) of unsatisfied demand is shifted to the substitute product which is common
in the literature, e.g., Parlar and Goyal (1984), Netessine and Rudi (2003), and
Nagarajan and Rajagopalan (2008). We consider only single-attempt substitution.
If the substitute product is also out-of-stock, the sale is lost. If a stockout situation
occurs and the customer is not willing to purchase another product instead, lost
sales occur which are not recorded. We assume no shortage penalty costs so that
the sequence of customer arrivals does not affect the model (see Gilland and Heese
2013 for an analysis).
5.3.1 Data
A retailer has collected POS scanner data for two newsvendor-type products (i D
1; 2) over several periods t D 1; : : : ; T . The respective substitute of product i is
denoted as j with j D 1; 2. The store is open Z hours per day (z D 1; : : : ; Z).
5.3 Model
61
80
70
60
50
40
30
20
10
0
1
Demand product i
10 11 12 13 14 15 16 17 18 19 20
Day
Sales product i
Demand product j
Sales product j
There are two possible settings concerning the retailers demand information: In
c
one setting, the retailer observes only sales. He observes daily sales Di;t
and hourly
i
sales hz;t . If a product is out-of-stock, demand is unknown and sales consist of
demand censored at order-up-to level Si . In the other setting, the retailer additionally
has full demand information Di;t (in hindsight). If a product is out-of-stock, he
knows the amount of lost sales. Figure 5.1 shows an example of the data for 20 days.
Furthermore, the retailer is aware of a causal relationship between price and
demand and has collected data on the selling price pi;t as external variable.
5.3.2 Decisions
The retailers objective is to maximize the profit of a given assortment by determining the optimal order quantities of two products that may be mutual substitutes.
To determine the optimal order quantity, he takes the selling prices pi;t and unit
purchasing costs ci;t into account. The selling price also explains a part of the
demand variability. Therefore, he determines the optimal order quantity as a linear
inventory function Bi;t that depends on the price and reflects the price-demand
relationship. An individual linear inventory function is determined per product i
and substitute j . A linear relationship is postulated in order to enable solving the
problem with Linear Programming. We obtain the following inventory function for
product i :
Bi;t D ai C bi i pi;t C bj i pj;t
8t 2 T I i D 1; 2I j D 1; 2I i j:
(5.1)
62
The first part of the function determines the order quantity to satisfy primary
demand and therefore depends on the selling price of the first-choice product. The
second part of the function determines the order quantity to satisfy demand that
is shifted from the out-of-stock product to its substitute and therefore depends on
the selling price of the out-of-stock product. The coefficient ai represents the yintercept of the function. The influence of the external variables on the optimal
inventory level is determined by the coefficients bii and bij which are set individually
for each product based on the past sales observations. Setting the coefficients from
past sales data requires that causal relationships from the past also hold for the
future. bii is the coefficient that reflects the price-dependency of the demand and is
negative for most products (exceptions due to snob effects are possible and would
not affect the model). bji (and bij , respectively) is a coefficient for the substituted
demand which is shifted from product j to i (and i to j , respectively).
We do not consider reference-price effects. The coefficients of the inventory
function are decision variables. Further decision variables are the resulting sales
from primary demand (si;t ) and from stockout-based substitution (sji;t ). These are
determined according to the constraints that will be outlined in the following
sections.
T
X
pi;t si;t C sj i;t ci;t Bi;t C pj;t sj;t C sij;t cj;t Bj;t
(5.2)
t D1
5.3 Model
63
8t 2 T
(5.3)
Since there was no substitution behavior from product i to j in the past, the
retailer cannot learn about the customers potential willingness to substitute (sij;t D
0). Consequently, primary sales of product j are limited by the inventory function
without accounting for potential substitution according to:
sj;t Bj;t
8t 2 T
(5.4)
Further, the retailer can only sell as many units as the customer demands.
Customer demand Dj;t for product j is either satisfied from the first-choice
product resulting in sales of product j or from the secondary choice sji;t due to
substitution (5.5). For product i only primary demand occurs (5.6).
sj;t C sji;t Dj;t
8t 2 T
(5.5)
si;t Di;t
8t 2 T
(5.6)
64
c
If substitution occurs, sales observations (Di;t
) increase by the amount of demand
shifted from j to i . Therefore, the difference between sales observations and
primary demand for i reflects the willingness of customers to substitute and is an
upper limit (5.7).
c
sji;t Di;t
Di;t
8t 2 T
(5.7)
Note that it is not possible to learn about substitution from historical observations
if no substitution occurred. If the order quantities set by the LP model are lower than
the historical inventory level with full availability and the product stocks out so that
substitution could occur, the potential substitution effects are still not considered
by the model due to the lack of information on the customers willingness to
substitute.
The following additional constraints for all decision variables apply:
si;t ; sj;t ; sij;t ; sji;t ; Bi;t ; Bj;t 0
ai ; bii ; bji ; aj ; bjj ; bij 2 <:
(5.8)
(5.9)
5.3 Model
65
and the sum of all demand occurring during the day Di;t is one. This holds for the
mean over all observations T , denoted by HN iZ and DN i :
i
D
HZ;t
Z
X
hiz;t
(5.10)
zD1
HN Zi
D 1:
DN i
(5.11)
Our aim is to obtain a multiplier Kzi that can be used to inflate unobservable lost
sales and deflate sales with substitution. Recursively, we can calculate the ratios of
mean cumulative sales and demand for all days with full availability based on mean
hourly sales hNiz by
i
HN Z1
hN iZ
D
1
DN i
DN i
i
i
i
hN izC1
HN zC1
HN zC1
HN zC1
HN zi
D
D
i
DN i
DN i
DN i HN zC1
DN i
(5.12)
1
hN izC1
HN i
zC1
!
D
1
Kzi
(5.13)
so that
Kzi D
i
KzC1
1
hNizC1
HN i
(5.14)
zC1
In order to estimate demand, we replace the hourly time intervals z with the
stockout time ki;t . If product i stocks out at time ki;t on day t, we have to correct
for unobservable lost sales which is accomplished by
Fkii ;t D Hkii;t
(5.15)
Replacing ki;t by kj;t in (5.15) allows to calculate the deflated demand of the
substitute i if product j is out-of-stock.
Figure 5.2 illustrates demand estimation for the two-product case where product
j is sold out. Demand for product j is filled until the order-up-to level (horizontal
line) is reached. The time of the stockout is recorded as kj;t . Any demand occurring
after kj;t cannot be satisfied from product j s inventory and is unobserved. The
shaded area of product j is an estimate for the unobserved demand and depends
j
on kj;t . The total demand for product j is estimated as Fkj ;t . Cumulative sales
of product i before j stocks out corresponds to primary demand for i . After kj;t ,
demand for i is inflated by substitution (shaded area). A fraction of the unsatisfied
demand for product j is shifted to product i . Fkij ;t is an estimate for the primary
demand of product i deducting substitution.
66
Demand
Product i
Product j
Stockout Observations of One Product Constraints (5.3) and (5.4) are not
affected by censoring and remain unchanged. Constraints (5.5) and (5.6) have to
be adapted since demand cannot be observed. They are therefore replaced by an
estimate of demand [see (5.15)] in constraints (5.16) and (5.17).
si;t Fkij ;t
j
8t 2 T
(5.16)
8t 2 T
(5.17)
8t 2 Fi \ Cj
(5.18)
8t 2 Fi \ Fj
(5.19)
(5.20)
(5.21)
5.3 Model
67
8t 2 T
(5.22)
If both products stock out, the sequence of stockouts is important. Only sales
observations before the first stockout occurs reflect primary demand. Therefore, the
time of the first stockout kt is obtained as the minimum of the stockout time of
product i and j :
kt D min.ki;t I kj;t /:
(5.23)
In addition to constraints (5.3) and (5.16) to (5.19), the cases when product i or
product i and j stock out have to be covered. If product i stocks out and j is fully
available, constraints (5.24) ensures that substitution does not exceed its estimate.
In this case, no substitution from j to i can be observed (5.25).
j
c
Fki ;t
sij;t Dj;t
8t 2 Ci \ Fj
(5.24)
sji;t D 0
8t 2 Ci \ Fj
(5.25)
8t 2 Ci \ Cj
(5.26)
j
Fkt
8t 2 Ci \ Cj
(5.27)
c
sij;t Dj;t
(5.28)
t D1
s.t.
One product is out-of-stock:
c
Fkij ;t
sji;t Di;t
8t 2 Fi \ Cj I i D 1; 2I j D 1; 2I i j
(5.29)
sij;t D 0
8t 2 Fi \ Cj I i D 1; 2I j D 1; 2I i j
(5.30)
68
No stockouts:
sij;t D 0
8t 2 Fi \ Fj I i D 1; 2I j D 1; 2I i j
(5.31)
8t 2 Ci \ Cj I i D 1; 2I j D 1; 2I i j
(5.32)
All observations:
si;t C sji;t Bi;t
8t 2 T I i D 1; 2I j D 1; 2I i j
(5.33)
8t 2 T I i D 1; 2I j D 1; 2I i j
(5.34)
(5.35)
(5.36)
69
(5.37)
uj uij ui C uj
(5.38)
Customers arriving at each product can be calculated from the overall customer
arrival rate as follows:
i D u i
8i D 1; 2
(5.39)
ji D uji
8i D 1; 2I j D 1; 2I i j
(5.40)
As long as both products are available, demand for products i and j follows
independent Poisson processes (P oi s). In the presence of stockouts, the purchase
process for the substitute is independent from pre-stockout purchases. Hence, interarrival times satisfy the i.i.d. (independent and identically distributed) criterium.
Therefore, the likelihood function can be described as a product of the purchase
processes for the five cases and transformed into a log-likelihood function:
LD
Pois1 .Si ; Sj /
t 2Fi \Fj
t 2Ci \Cj
Pois2 .Si ; Sj /
t 2Ci \Fj
Pois4 .Si ; Sj /
Pois3 .Si ; Sj /
t 2Cj \Fi
Pois5 .Si ; Sj /
(5.41)
t 2Cj \Ci
(5.42)
70
the presence of stockouts are denoted as Hi .j /, which are the sales of product i if
product j is out-of-stock. By maximizing logL subject to the constraint (5.44), we
obtain the following maximization problem:
max logL
(5.43)
s.t.
i ji i C j
8i D 1; 2I j D 1; 2I i j
(5.44)
8i D 1; 2
(5.45)
8i D 1; 2I j D 1; 2I i j
(5.46)
8i D 1; 2I j D 1; 2I i j:
(5.47)
ji i
j
t Dpi;t
xD0
1
X
xDqi;t
qi;t 1
C pi;t
Bt
X X
xD0 yDqj;t
C pi;t
1
X
71
xD0 yDBt C1
qj;t 1
C pj;t
At
X X
yD0 xDqi;t
qj;t 1
C pj;t
1
X
yD0 xDAt C1
qj;t 1
C pj;t
1
X
(5.48)
yDqj;t
yD0
with At and Bt :
k
y
C qi;t
a
k
jq x
i;t
C qj;t :
Bt D
b
At D
jq
j;t
(5.49)
(5.50)
Parlar and Goyal (1984) show that the function is concave if the following
relationship between selling prices and substitution rates holds:
bpi;t pj;t pi;t =a:
We assume that customers arrive according to a Poisson process with arrival rates
i and j :
f .x/ D
xi i
e
x
(5.51)
g.y/ D
j
e j
(5.52)
(5.53)
(5.54)
(5.55)
(5.56)
72
8i D 1; 2
(5.57)
For each instance, we randomly draw the parameters from the uniform distribution according to:
a1 7I 12; b1 2I 5; a2 5I 10 and b2 1I 3:
The daily arrival rate can be obtained from the sum of the hourly arrival rates
since the sum of independent Poisson random variables is also Poisson distributed
(Ross 2010). We assume that demand follows a homogeneous Poisson process.
We artificially censor demand by assuming different order-up-to levels. In the first
setting, we assume that the order-up-to level of product 1 is always high enough to
cover demand, i.e. no stockouts. We set the order-up-to level of product 1 so that
it can fill all the demand for product 1 and substituted demand from product 2 in
hindsight. In the second setting, we set order-up-to levels at the 75th percentile of
demand for product 1. The order-up-to levels of product 2 are at the median in both
settings.
Any demand exceeding the order-up-to level is shifted to the substitute product
with a certain rate. The substitution rate if product 1 is out-of-stock is a D 0:8 and
if product 2 is out-of-stock b D 0:4.
We fit the parameters of the different models to a sample of 100 observations.
Given the parameters, we compare the resulting profits of an out-of-sample size
of 100 observations. The experiment was repeated for 100 randomly generated
instances.
5.5 Results
5.5.1 Known Demand with Stockout Observations of One
Product
We first analyze the simple setting from Sect. 5.3.4 where demand is known and
product 1 is always available. We compare the results for the data-driven approach
with the optimal solution (Parlar and Goyal 1984) given the customer arrival rates of
the Poisson process and substitution rates for both products. We denote the optimal
solution as parametric approach. The price difference between both products is
fixed such that p2;t D gp1;t with g D 1:4.
5.5 Results
73
In-stock probability
product 1
In-stock probability
product 2
Total leftover
inventory
Total profit
Mean
(Standard deviation)
Data-driven approach
0.9141
(0.0441)
0.6247
(0.0687)
18.4
(2.8)
59.62
(11.4)
Known values
0.8834
(0.0342)
0.5476
(0.0501)
16.0
(2.7)
59.65
(11.5)
Table 5.1 shows the mean results in terms of the total profit, leftover inventory
and in-stock probabilities of both products for the out-of-sample observations.
Standard deviations are in parentheses. The mean total profits of the data-driven
approach are slightly lower than those of the parametric approach. The parametric
approach leads to slightly more demand substitution with 3.5 % of the total demand
of product 2 being satisfied by product 1 compared to 3.3 % for the data-driven
approach. According to t-tests, the differences between profits are not significant
for 82 % of all instances (p > 0:1). The parametric approach makes use of
the distributional assumptions and takes the customer arrival rates of each day as
given. Therefore, this approach serves as a benchmark for the optimal solution
in the following. The standard deviation is slightly lower and the mean in-stock
probabilities of both products are higher for the data-driven approach. The higher
in-stock probabilities result in higher revenues which are compensated by the costs
of larger leftover inventories.
74
In-stock probability
product 1
In-stock probability
product 2
Total leftover
inventory
Total profit
Mean
(Standard deviation)
Data-driven
Parametric approach with
approach
Known values
Estimated values
0.8064
0.8834
0.771
(0.0690)
(0.0342)
(0.0736)
0.4819
0.5476
0.3618
(0.0696)
(0.0501)
(0.0675)
12.2
16.0
18.6
(2.4)
(2.7)
(3.5)
59.59
59.65
58.7
(11.4)
(11.5)
(11.7)
From Table 5.2, we observe that the data-driven approach achieves slightly
higher profits than the parametric one with estimated values, but achieves lower
mean total profits and in-stock probabilities than with the known demand values.
The slightly lower profits of the parametric approach with estimated values are
due to lower in-stock probabilities resulting in fewer revenues and at the same time
higher leftover inventories. Furthermore, more substitution from product 2 to 1 takes
place with the parametric one with estimated values (8.3 % of the total demand for
product 2) than with the data-driven approach (4.3 %). Conducting t-tests on the
instances shows that the differences between the profits of the data-driven approach
and the parametric approach with estimated values are significant (p < 0:1) for
93 % of the instances.
5.5 Results
75
In-stock probability
product 1
In-stock probability
product 2
Total leftover
inventory
Total profit
Mean
(Standard deviation)
Data-driven
Parametric approach with
approach
Known values
Estimated values
0.6952
0.8834
0.6801
(0.0679)
(0.0342)
(0.0797)
0.4741
0.5476
0.5411
(0.0663)
(0.0501)
(0.0829)
9.3
16.0
15.9
(1.5)
(2.7)
(3.5)
59.5
59.6
58.5
(11.4)
(11.5)
(11.7)
sample size effects. The results are shown in Table 5.4. The difference between the
profits from the parametric approach with known values and from the data-driven
approach increases and is significant (p < 0:1) for all instances. The parametric
approach with estimated values achieves already similar results with a small sample
of only 20 observations compared to the one with 100 observations, but slightly
lower profits than the data-driven approach. However, from a customers point of
view, the parametric approach with estimated values should be preferred since the
in-stock probabilities of the two products are higher. The difference increases with
larger sample sizes where the data-driven approach improves profits. But overall,
the increase of profits is small. The number of instances for which the difference is
significant at the 10 % level increases as well from 58 to 78 % with larger sample
sizes.
In the previous analyses we assumed that the in-sample observations of product
2 are highly censored. There is a trade-off associated with censoring: If demand is
highly censored, less can be learnt about the true demand values, but at the same
time, more can be learnt about the substitution behavior. We now vary the censoring
level by adding percentiles from 75 to 95 %.
For both approaches, we observe that the effect of learning about substitution if
censoring is high, results in slightly higher profits than if there is only few censoring
and more can be learnt about demand. The effect is due to the fact that we are able to
estimate unobservable lost sales if demand is censored, but we are not able to learn
about substitution behavior if only few substitution occurs (Table 5.5).
76
Sample
size
20
In-stock probability
product 1
In-stock probability
product 2
Total leftover
inventory
Total profit
50
In-stock probability
product 1
In-stock probability
product 2
Total leftover
inventory
Total profit
100
In-stock probability
product 1
In-stock probability
product 2
Total leftover
inventory
Total profit
Mean
(Standard deviation)
Data-driven
Parametric approach with
approach
Known values
Estimated values
0.6167
0.8964
0.7161
(0.1380)
(0.0284)
(0.0916)
0.4958
0.2917
0.5357
(0.0613)
(0.0468)
(0.1338)
10.4
15.6
16.9
(3.1)
(2.6)
(4.0)
54.5
55.9
54.4
(9.9)
(10.0)
(10.0)
0.6706
0.7051
(0.0947)
(0.0890)
0.5477
0.5363
(0.0554)
See
(0.1205)
11.2
above
16.4
(2.2)
(4.0)
54.7
54.4
(9.9)
(10.0)
0.681
0.6996
(0.0764)
(0.0761)
0.4745
0.541
(0.0641)
See
(0.0887)
10.1
above
16.2
(1.9)
(3.5)
54.9
54.4
(9.9)
(10.0)
sample of 250 days per store on which we fit the parameters of the models. We then
compare the performance on the remaining observations as an out-of-sample. Since
the retailer only observes sales if a product is out-of-stock, we include only days
with no stockouts in the out-of-sample so that we have full demand information.
The size of the out-of-sample therefore varies with the frequency of stockouts and
contains between 404 and 610 days (on average 522 days).
In addition to daily and hourly sales, we collect information on prices and
daily temperatures. The price of product 1 (2) lies between 0.29 e (0.09 e ) and
1.79 e (1.49 e ). Since we have no information on costs, we assume that c1;t D
0:1p1;t and c2;t D 0:25p2;t . The substitution rates are unknown. We assume that
a D 0:1 and b D 0:7 according to expert opinions from the managers of the
retail chain. We add information on weekdays. d1;t is a binary variable if it is a
5.5 Results
77
75
In-stock probability
product 1
In-stock probability
product 2
Total leftover
inventory
Total profit
90
In-stock probability
product 1
In-stock probability
product 2
Total leftover
inventory
Total profit
95
In-stock probability
product 1
In-stock probability
product 2
Total leftover
inventory
Total profit
Data-driven
approach
0.6865
(0.0835)
0.5343
(0.0632)
10.9
(2.2)
54.8
(9.9)
0.6861
(0.0810)
0.5398
(0.0629)
11.0
(2.1)
54.8
(9.9)
0.6872
(0.0779)
0.5406
(0.0589)
11.0
(2.0)
54.8
(9.9)
Tuesday or Wednesday when demand is typically lowest and d2;t for Fridays and
Saturdays when peak demand occurs. Mondays and Thursdays are not included
to avoid collinearity. The temperatures (wt ) during the time horizon considered
range between 11:1 C and 36:4 C. We include price, weekday and temperature
as external factors in the inventory functions (5.58 and 5.59) to determine the
optimal order quantities according to the data-driven approach. The coefficient for
Tuesdays/Wednesday is denoted bid1 , for Fridays/Saturdays bid2 and temperature biw .
B1;t D a1 C b11 p1;t C b21 p2;t C b1d1 d1;t C b1d2 d2;t C b1w wt
8t 2 T
(5.58)
B2;t D a2 C b22 p2;t C b12 p1;t C b2d1 d1;t C b2d2 d2;t C b2w wt
8t 2 T:
(5.59)
78
In-stock probability
product 1
In-stock probability
product 2
Total leftover
inventory
Total profit
Mean
(Standard deviation)
Data-driven approach
0.8241
(0.0620)
0.7004
(0.1042)
45.9
(14.7)
51.0
(11.4)
Estimated values
0.7065
(0.0761)
0.6158
(0.1192)
34.3
(9.3)
48.5
(10.5)
The results in Table 5.6 show that the data-driven approach also outperforms the
parametric approach with estimated values for real data. According to t-tests, the
differences between profits are significant (p < 0:1) for the majority of stores
(91 %). One reason for the lower profit of the parametric approach with estimated
values is that the approach assumes that the data is Poisson distributed. But the
Chi-Square Goodness of Fit and the Kolmogorov-Smirnov Test reject that the data
is Poisson distributed (p D 0:0000). In contrast, the data-driven approach is
distribution-free. Another reason is that the parametric approach with estimated
values does not take the external variables which are considered by the data-driven
approach.
5.6 Conclusions
We suggest a novel approach based on data-driven optimization which integrates
demand forecasting, substitution and inventory optimization. There are no prior
distribution assumptions and it only requires POS scanner data as it is readily
available in most stores. Observations obtained from store-level scanner data are
directly incorporated in inventory optimization, thus taking substitution effects and
unobservable lost sales into account. Additionally, we account for other variables
affecting demand such as price, weekdays or weather. These factors also influence
the willingness of customers to substitute which is reflected in our model. The model
fits a linear function of external variables to the available data. We build our model
based on data for perishable products from a large European retail chain.
The model is limited in that it is only able to account for substitution behavior that
was observed in the past. It does not reflect additional substitution behavior if inventory levels set by the LP approach are lower than the historical ones. In addition,
shortage penalty costs are not considered. Another limitation of the LP approach
is the assumption that the relationship between the inventory level and the external
variables is linear. In both cases, the model would otherwise become non-linear.
Chapter 6
We analyze the order decisions of a manufacturer who supplies several retail stores
with bakery products in a newsvendor context. According to a service level contract
between the manufacturer and the retailer, the manufacturer has to achieve an
in-stock service level measured over multiple products. We derive the optimal
inventory policy for an aggregate service level contract and compare the optimal
solution to the decisions made by the manufacturer. Further, we address the question
whether the results of experimental studies from the laboratory can be transferred to
real-world decisions. Our findings indicate that real decision makers show similar
decision biases as students in laboratory environments.
6.1 Introduction
In recent studies, many researchers show that decision making in newsvendor
settings does not follow profit-maximizing predictions if human decision makers are
involved: In various lab experiments, students and managers face the newsvendor
problem, but their behavior consistently differs from normative profit-maximizing
textbook solutions. But how do decision makers facing newsvendor problems
behave in real-world situations? Are they also biased in their decisions? Do they
show similar deviations for expected profit-maximizing behavior? In this chapter,
we will address this issue. We analyze empirical data from newsvendor-like
decisions in order to compare the observed deviations in the lab with real-world
decisions.
To analyze real-world newsvendor-type decisions, we investigate data from a
company selling bakery items at several supermarkets. Bakery items are typical
newsvendor products, since all leftovers are discarded at the end of the day. In
contrast to most of the recent behavioral newsvendor studies, our decision maker is
not facing a single-product cost-minimization problem, but sells various products.
Springer International Publishing Switzerland 2015
A.-L. Sachs, Retail Analytics, Lecture Notes in Economics and Mathematical
Systems 680, DOI 10.1007/978-3-319-13305-8_6
79
80
81
observed in the laboratory, such as the pull-to-center effect and demand chasing,
can also be observed in practice. This implies that the concepts that have been
developed by behavioral operations researchers using laboratory experiments, are
likely to generalize to practice.
For practitioners, the results of our research provide the means to optimally
manage inventory under aggregated service level constraints. Our results indicate
that substantial profit improvements can be achieved by differentiating service levels
by unit revenue, unit cost, and demand uncertainty. The manufacturer who motivated
our research, uses elements of the optimal inventory policy and differentiates the
service levels between make and buy products. This policy allows to achieve a profit
that is 0.9 % higher than the profit he would achieve under a policy that does not
differentiate service levels. However, if the full optimal policy were implemented,
i.e., if service levels were differentiated based on the unit cost and unit revenue of
individual products and based on demand uncertainty by product and location, then
the manufacturer could improve profits by an additional 7.1 %.
The remainder of this chapter is structured as follows. In Sect. 6.2, we describe
the setting we analyze. In Sect. 6.4, we present our model and derive the optimal
solution. The optimal solution determines the service levels (strategic solution) and
daily order quantities (operational solution) by product and location. In Sect. 6.5, we
compare the strategic solution of the model with the solution the manufacturer has
implemented. We show that the manufacturer differentiates service levels between
make and buy products, but ignores unit revenue and unit cost differences within
each of these segments. He also differentiates service levels based on demand
uncertainties. We build a model based on this observation and show that the model
captures actual behavior well. In Sect. 6.6, we compare the operational solution of
the model with the solution the manufacturer has implemented. We show that the
manufacturer over-reacts to changes in the demand, an effect that has been observed
in many laboratory experiments on newsvendor-type decision making. The literature
on behavioral operations management offers various explanations for this behavior.
We analyze the most popular explanations and show that a demand-chasing model
describes actual operational ordering better than the other models we analyze. In
Sect. 6.8, we summarize our results and discuss the implications for theory and
practice.
82
The sequence of events is as follows: Late in the morning, the decision maker
observes sales of the previous day and analyzes historical sales data to forecast
demand for each product in each store for the next day. The decision maker then
determines the production quantity for each product. During the day, items are
produced and production is completed early in the morning on the next day. The
finished items are then delivered to each store before the stores open. Upon delivery,
the decision maker collects leftover items from the previous day and discards them.
Given that the decision maker knows the production quantities of the previous day,
he can draw conclusions on how many items were sold and which products stocked
out from the amount of leftover inventory. The decision maker then returns to his
production facility to determine the forecast and the production quantities for the
next upcoming day.
The decision maker has a service level contract with the retailer for the bakery
products and has to achieve a service level of 70 % (in-stock probability) across
all products in each store. Since the decision maker has to achieve the service level
target at each store, the stores are treated individually in the subsequent calculations.
83
External products
Internal products
CR
0.49
0.71
0.31
0.33
0.70
0.52
0.41
0.96
0.58
0.26
0.79
0.67
0.30
1.00
0.70
0.55
1.18
0.53
0.55
1.19
0.54
0.31
0.68
0.55
0.64
0.72
0.11
10
0.37
0.42
0.12
11
0.60
0.68
0.11
12
0.59
0.75
0.21
13
0.59
0.81
0.27
14
0.26
0.63
0.58
15
0.45
0.51
0.12
16
1.00
1.35
0.26
17
1.00
1.52
0.34
18
0.69
1.16
0.40
19
0.60
1.04
0.42
20
0.75
1.19
0.37
21
0.73
0.82
0.11
22
0.90
1.09
0.18
23
0.89
1.08
0.18
Avg.
21.4
(10.5)
10.1
(5.3)
10.8
(4.8)
8.5
(4.5)
5.7
(3.5)
5.5
(3.1)
12.7
(6.6)
5.4
(3.7)
11.1
(5.5)
17.0
(8.1)
6.7
(4.0)
4.8
(3.1)
7.9
(5.1)
9.7
(4.8)
11.0
(5.1)
6.9
(3.9)
Mo
22.1
(10.2)
12.6
(6.7)
11.8
(5.1)
10.2
(5.3)
6.8
(4.2)
6.3
(3.5)
15.0
(7.2)
4.6
(2.8)
11.3
(5.5)
19.6
(8.9)
7.9
(4.5)
3.6
(2.0)
5.3
(2.9)
9.5
(4.3)
9.7
(4.2)
5.4
(2.7)
Mean sales
(Standard deviaon)
Tu
We
Th
18.2
18.5
22.3
(8.9)
(9.3) (10.6)
8.9
8.6
9.7
(4.9)
(4.4)
(4.6)
9.0
9.2
10.9
(3.8)
(4.2)
(4.5)
7.5
7.3
8.1
(3.9)
(4.0)
(3.7)
4.9
4.8
5.7
(3.1)
(3.0)
(3.3)
4.8
4.8
5.7
(2.7)
(2.7)
(3.1)
10.8
10.9
12.4
(5.8)
(6.0)
(6.1)
4.1
4.6
5.1
(2.5)
(3.9)
(3.1)
9.4
9.7
12.2
(4.7)
(5.0)
(5.6)
15.4
15.2
18.9
(7.5)
(7.5)
(8.3)
5.8
5.8
7.4
(3.5)
(3.6)
(4.1)
3.5
3.8
5.2
(2.0)
(2.6)
(3.2)
5.0
5.8
8.2
(2.6)
(4.0)
(4.4)
8.0
8.2
10.2
(3.7)
(3.9)
(4.6)
8.6
9.4
12.5
(3.8)
(4.4)
(5.2)
4.8
5.2
7.0
(2.4)
(3.3)
(3.5)
9.5
(7.8)
17.6
(11.0)
15.0
(7.8)
38.7
(20.4)
23.2
(10.7)
11.0
(9.3)
9.1
(6.7)
6.4
(4.5)
19.8
(13.4)
17.3
(8.6)
38.2
(19.9)
22.3
(9.7)
10.4
(10.4)
8.9
(7.2)
6.1
(4.0)
14.8
(11.1)
12.7
(6.7)
32.6
(16.7)
19.1
(9.0)
8.6
(7.5)
7.4
(5.5)
7.0
(5.7)
15.0
(8.7)
12.7
(6.7)
34.3
(18.4)
19.8
(9.6)
8.3
(5.7)
7.2
(4.4)
8.6
(6.9)
16.5
(8.2)
15.0
(6.8)
39.6
(19.0)
25.8
(10.8)
11.6
(9.4)
9.9
(7.2)
Fr
21.8
(10.0)
9.3
(4.4)
10.8
(4.3)
7.8
(3.9)
5.7
(3.2)
5.5
(3.0)
12.4
(6.2)
5.8
(3.5)
11.6
(5.2)
16.5
(7.6)
6.5
(3.6)
6.2
(3.4)
10.5
(4.9)
9.9
(4.6)
12.5
(5.4)
8.9
(4.1)
Sa
25.6
(12.0)
11.5
(5.6)
13.1
(5.5)
10.1
(5.0)
6.3
(3.8)
6.2
(3.4)
14.6
(6.8)
7.9
(4.7)
12.4
(6.0)
16.6
(7.7)
6.8
(4.0)
6.2
(3.3)
12.5
(5.6)
12.5
(5.8)
13.2
(5.5)
9.8
(4.2)
Avg.
22.1
(11.4)
10.5
(5.7)
11.2
(5.3)
8.9
(4.8)
6.0
(3.8)
5.8
(3.4)
13.1
(6.9)
5.7
(4.1)
11.5
(5.9)
17.6
(8.7)
7.0
(4.3)
5.0
(3.3)
8.3
(5.5)
10.2
(5.2)
11.5
(5.7)
7.2
(4.3)
12.0
(7.9)
17.0
(9.9)
14.8
(7.0)
38.4
(18.2)
26.0
(10.9)
12.5
(9.5)
10.3
(7.3)
16.8
(9.7)
22.8
(11.5)
17.9
(9.1)
49.3
(25.0)
26.7
(11.6)
14.7
(10.8)
10.9
(7.2)
9.9
(8.3)
18.4
(12.1)
15.6
(8.3)
40.1
(21.7)
24.1
(11.6)
11.4
(9.7)
9.4
(6.9)
Mo
22.8
(10.9)
13.0
(7.0)
12.3
(5.7)
10.7
(5.6)
7.1
(4.5)
6.6
(3.8)
15.5
(7.5)
4.9
(3.1)
11.7
(5.9)
20.3
(9.6)
8.2
(4.8)
3.8
(2.2)
5.6
(3.3)
10.0
(4.8)
10.2
(4.6)
5.6
(2.9)
Mean demand
(Standard deviaon)
Tu
We
Th
18.7
19.1
23.0
(9.4) (10.0) (11.3)
9.2
8.8
10.1
(5.1)
(4.6)
(4.9)
9.3
9.5
11.3
(4.1)
(4.5)
(4.9)
7.7
7.6
8.5
(4.1)
(4.2)
(4.0)
5.1
5.0
6.0
(3.2)
(3.1)
(3.5)
4.9
5.0
6.0
(2.9)
(2.9)
(3.3)
11.1
11.3
12.8
(6.0)
(6.4)
(6.4)
4.3
4.9
5.4
(2.6)
(4.2)
(3.3)
9.7
10.0
12.7
(5.0)
(5.3)
(6.0)
15.8
15.7
19.6
(7.9)
(7.9)
(8.9)
6.0
6.2
7.7
(3.7)
(4.0)
(4.6)
3.6
4.0
5.4
(2.1)
(2.7)
(3.4)
5.2
6.1
8.5
(2.8)
(4.2)
(4.7)
8.3
8.5
10.6
(3.9)
(4.2)
(5.0)
9.0
9.8
13.1
(4.1)
(4.7)
(5.8)
5.0
5.4
7.3
(2.6)
(3.4)
(3.7)
Fr
22.5
(10.6)
9.7
(4.7)
11.2
(4.8)
8.2
(4.2)
6.0
(3.5)
5.7
(3.2)
12.7
(6.4)
6.1
(3.7)
12.0
(5.7)
17.1
(8.3)
6.8
(3.9)
6.4
(3.8)
10.9
(5.4)
10.4
(5.1)
13.0
(5.8)
9.2
(4.5)
Sa
26.9
(13.5)
12.1
(6.2)
13.9
(6.3)
10.7
(5.6)
6.7
(4.3)
6.6
(3.8)
15.1
(7.2)
8.5
(5.6)
13.0
(6.8)
17.5
(8.7)
7.1
(4.3)
6.5
(3.7)
13.2
(6.5)
13.2
(6.5)
14.0
(6.5)
10.3
(4.8)
6.7
(4.8)
20.6
(16.1)
17.9
(9.1)
39.5
(21.0)
23.1
(10.5)
10.8
(10.6)
9.1
(7.3)
6.3
(4.1)
15.4
(11.4)
13.1
(7.0)
33.6
(17.6)
19.7
(9.6)
8.8
(7.6)
7.6
(5.6)
12.4
(8.2)
17.7
(10.4)
15.4
(7.5)
39.5
(19.2)
26.8
(11.7)
12.9
(9.9)
10.6
(7.5)
17.7
(10.9)
24.1
(12.9)
18.8
(10.0)
51.6
(27.0)
27.9
(12.7)
15.2
(11.3)
11.3
(7.5)
7.3
(5.8)
15.5
(9.4)
13.1
(7.0)
35.3
(19.2)
20.5
(10.2)
8.6
(6.0)
7.4
(4.5)
9.0
(7.0)
17.2
(8.7)
15.6
(7.2)
41.0
(19.9)
26.8
(11.7)
12.1
(10.1)
10.2
(7.5)
of information, i.e. daily versus hourly sales data. For our forecast and inventory
optimization we use the approach by Bell (1981) and Wecker (1978) which makes
distributional assumptions, but works with daily sales observations which the
decision maker is able to observe. For the analysis of the performance of the
different models and the decision maker, we use the approach by Lau and Lau
(1996). This approach takes hourly sales into account which improves the forecast
accuracy, but the decision maker is not able to observe. Furthermore, it does not
require any assumptions on the demand distribution, thus avoiding any potential
misspecification issues. Table 6.1 contains the demand estimated according to Lau
and Lau (1996).
Figure 6.1 shows the achieved service level for all 23 products per store. Each
boxplot spans over 64 stores per product.
84
Service Level
.8
.6
.4
.2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
85
0.6
Autocorrelaon
0.5
0.4
0.3
0.2
0.1
0
1
7 8
Lag
10 11 12 13 14
For Si;t D qi;t , we can only observe censored demand, but conclude that
yi;t qi;t . Furthermore, assumptions about the distribution of complete demand
observations also hold for censored demands, but we have to estimate the truncated
part of the distribution function. Therefore, we can estimate the expected demand
conditional on the out-of-stock situation E.Yi;t jYi;t qi;t / based on O i;t and O i;t as
R1
(6.1)
(6.2)
where i;t is normally distributed white noise, with E.i;t / D 0 and Var.i;t / D i2 .
Single exponential smoothing is the mean squared error (MSE) minimizing
forecast method for ARIMA(0,1,1) (Chatfield 2001).Using the single exponential
smoothing forecast with a smoothing factor
, we optimally update our mean
demand forecast O with each new demand observation:
O i;t D
i Yi;t 1 C .1
i /O i;t 1
(6.3)
86
same weekday in the previous week in our dataset (lagD 6). We choose productweekday specific
in our analysis since they have the best out-of-sample predictive
power. We use the first 10 weeks to initialize our model. Each week, we update the
smoothing parameter.
We calculate the standard deviation of the forecast as root mean square error
(RMSE):
v
u
u
O i D t
T
X
1
.Yi;t O i;t /2
Ti df i t D1
(6.4)
(6.5)
where qi is the stocking quantity of product i . Therefore, the expected non-demandweighted service level for a store N with N products is:
N D
N
1 X
i
N i D1
(6.6)
87
N
X
qi
with
Z
Ei .qi / D
s:t:N O
(6.7)
(6.8)
EYi i .qi /
i D1
being the expected profit of the stocking quantity qi for the demand yi .
A simple approach for achieving the service level constraint (based on an item
approach) is to stock at least as many units of each product that Pr.ISi D 1/ ,
O so
Fi .qi / D i O for all products. It can be shown that this solution is not generally
maximizing expected profits. A differentiation of service levels between products is
favorable, e.g., it is beneficial to achieve a higher service level for cheap products,
and lower the service level of expensive products, keeping the aggregated service
level constant (Thonemann et al. 2002).
Solving the model in Eq. (6.7) we can derive the following Proposition concerning the optimal ordering decision:
Theorem 6.1 (Optimal Order Quantities Under System Approach) The optimal product-specific service level targets i fulfill the following first order condition:
Zj
Zi
D
N i
N j
cjo Fj .qj / cju .1 Fj .qj //
cio Fi .qi / ciu .1 Fi .qi //
D
fi .qi /
fj .qj /
8i; j
(6.9)
8i; j
(6.10)
qi
0
N
X
Fi .x/dx/ C
Fi .qi /
N i D1
(6.11)
i D 1; : : : ; N
(6.12)
88
0 D .
N
X
Fi .qi //
(6.13)
i D1
If the service level constraint is not binding, i.e., D 0, then all products achieve
i
their profit optimal level, i.e. Fi .qi / D ri c
ri . Otherwise,
ri ci ri Fi .qi /
D
N
fi .qi /
!
N
X
r
c
N
i
i
D N
PN
r
i
i D1 fi .qi /
i D1
8i D 1; : : : ; N
(6.14)
(6.15)
t
u
From Theorem 6.1 we can derive the following corollary concerning the optimal
service level targets:
Corollary 6.1 (Service Level Differentiation) Keeping all other factors constant,
(a)
(b)
(c)
(d)
We note that this corollary is a ceteris paribus analysis just changing one
factor of one product. Changing attributes of multiple products, the relative change
determines the change of SL per attribute of a product. Changing multiple attributes
of one product, the final direction of the change is not clear, as the attributes interfere
with each other.
We have analyzed the optimal service level determination for given demand
distributions. Using this model, we can model the expected profit-maximizing
behavior of a decision maker.
Finding the normative predictions of the setting, we will analyze the empirical
data next.
89
Internal Products
1
0.9
0.9
0.8
0.8
Empirical Service Level
External Products
1
0.7
0.6
0.5
0.4
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0
0
0.2
0.4
0.6
0.8
Opmal Service Level
0.2
0.4
0.6
0.8
Opmal Service Level
90
Looking at the achieved service levels in more detail, we observe that the service
levels differ between internal and external products (Fig. 6.3). But the differentiation
within the groups seems to be too small. A regression analysis reveals that the
decision maker does not systematically differentiate within product categories: For
internal products, the differentiation is significant, but very small (SLOpt D 0:052,
p D 0:005). For external products, there is also a significant differentiation (p D
0:000), but in the opposite direction (SLOpt D 0:263).
We take this as a first hint for a simpler optimization model the decision maker
might be using. The question remains why actual service levels do not vary enough
compared to the normative benchmark. According to Theorem 6.1, the service level
should decrease in demand variability, and decrease in overage cost, but increase in
underage costs. Therefore, we test whether the achieved empirical service levels are
in line with this Theorem.
We use a random-intercept model (Eq. 6.16) to analyze the effect of different
factors on empirical service levels. We model the products and stores as random
effects:
i;s D0 C 1 Internali C 2 i;s C 3 i;s C 4 cio C 5 ciu C i C i;s
(6.16)
91
Internal
OPT
0.876
(0.033)
0:020
(0.001)
0.000
(0.000)
0:101
(0.011)
0:081
(0.01)
EMP
0.743
(0.054)
0:016
(0.003)
0.000
(0.000)
0:024
(0.018)
0:037
(0.020)
# obs
# groups
Log-likelihood
AIC
BIC
1472
23
3450
6886
6849
1472
23
2377
4740
4703
Factor
Constant
FC
FC
cio
ciu
ADJ
EMP
0.648
(0.021)
0:016
(0.003)
0.000
(0.000)
0:128
(0.024)
1472
23
2386
4761
4729
ADJ2
EMP
0.699
(0.038)
0:016
(0.003)
0.000
(0.000)
0:014
(0.013)
0:039
(0.021)
0:170
(0.03)
1472
23
2382
4749
4707
(LR-test: p < 0:01) and also the smallest AIC/BIC for the empirical models
(controlling for the additional parameter). In the last two columns we separated our
analyses for internal and external products. There is a differentiation within external
products, but partly in the wrong direction, i.e., underage costs have a significant
negative impact.
Summarizing the results in this section, we find empirical evidence for suboptimal decision making. The empirical service levels differ between internal and
external products, but not between actual cost differences within the groups, and
too little for different demand variabilities. Using internal and external products
for differentiation is actually a proxy for the correct cost-differentiation. But the
cost differences within categories and the impact of demand variability are not
considered adequately.
We use these observations to derive alternative (simpler) decision models in the
next section.
92
Group
.Ri
Group
min.qi ; yi / Ci
(6.17)
Group
where Ri
is the average revenue for internal product, for products 1; : : : ; 16 and
the average revenue for external products for products 16; : : : ; 23. The same holds
Group
for production costs concerning Ci
.
Ignoring demand variability differences the first order condition for the optimal
target service level is:
o
u
u
C o Ext CExt
Int CInt
.1 Int /
.1 Ext /
CInt
D Ext
;
fN.0;1/ .z.Int //
fN.0;1/ .z.Ext //
(6.18)
where z.Int / is the value of the inverse cumulative distribution function of the
standard normal distribution, and f . / is the corresponding probability density
function.
Using the second constraint
O
16
X
pD1
Int C
23
X
Ext
(6.19)
pD17
we can derive the optimal service level for internal and external products.
Note: The target service levels are similar for all internal (external) products, as
we use identical cost parameters within the groups and standard deviations even
between groups (mean demand is irrelevant for optimal service level targets).
Using detailed, i.e. product-specific, cost information for all products results in
Model 3. The different models are summarized in Table 6.3.
93
Demand variability
Individual
Cost differentiation
None
Group level
Model 1 Model 2
9.42
9.25
Individual
Model 3
9.94
42
41
40
39
38
37
36
35
34
0.66
0.68
Model 0
0.7
Empirical
0.72
Model 1
Model 2
0.74
Model 3
94
meaning that he is better than the naive newsvendor, but not as good as our best
fitting alternative decision models suggest. The question is why this is the case.
Recent behavioral operations literature found additional aspects of decision making,
such as demand chasing, mean chasing, and others. We will test these factors in the
next section to analyze whether these aspects can be found in our empirical data
and might be an explanation for the suboptimal performance of the decision maker
compared to our simplified Model 2.
(6.20)
(6.21)
Parameters
70
Log-likelihood
BIC
(1)
0.582
1,000,799
2,001,662
95
(2)
0:532
1,003,703
2,007,471
(3)
0:405
0:268
1,000,275
2,000,665
results in a worse fit than anchoring on the overall SL target. In general, both
types of anchoring are possible, as service level targets can be below mean demand
in extreme cases. Including both anchoring parameters into the regression shows
similar results as the separate analyses.
o
u
.CExt
C o / Ext .CExt
C u / .1 Ext /
:
7=23fN.0;1/.z.Ext //
(6.22)
96
(6.23)
where
i;t D i;t i;t 1 ; and i;t D i;t i;t 1 :
In Theorem 6.2, z D Fi1 .i / denotes the z-value of the normal distribution for
the optimal product-specific target service level from Eq. (6.10). As a result, we see
that the decision maker might react to demand realizations based on the forecast
updating, as we do not have independent demand over time.
As a first analysis on reactions to demand realizations Fig. 6.5 shows the changes
in order quantities between weeks. Adjustment towards demand realizations might
be a good strategy according to forecast updating.
If the decision maker is subject to demand chasing, he over-adapts his order
quantities. Therefore, we have to take optimal order adjustments into account. To
test the demand chasing effect for our data we conduct a regression for Eq. (6.24).
Formally, we estimate
qt;p;s D a0 qt;p;s
C
dt ;p;s qt ;p;s C t;p;s C usp
(6.24)
97
Repeat choice
By Products
.8
.6
.4
.2
0
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17* 18* 19* 20* 21* 22* 23*
.5
.4
.3
.2
.1
.25
.2
.15
.1
.05
98
Parameter
qt;p
.d q/t6
Ind. stockoutt6
Log-likelihood
Opt M2
0.189
(0.001)
0.265
(0.007)
506,803
Baker
0
0.695
(0.002)
0.406
(0.013)
727,842
Baker M2
1
0.480
(0.002)
0.274
(0.016)
732,136
Note that dt ;p qt ;p 0, as we have censored demand data. Therefore,
we additionally test the effect of a stockout indicator which is 1 for a stockout and
zero otherwise. Our model generally also enables a normative increasing of order
quantities, e.g., in cases of stockouts, qt;p
is most likely greater than zero, as we
have an average SL 70 % > 50 % and the updated forecast tends to increase in
stockout situations and so most likely qt;p
> 0.
We analyzed different models to investigate the effect of demand realizations.
As a baseline we regress the optimal order adaptations using the stockouts and the
leftovers of the previous week. As we use these factors to calculate the optimal
order quantities, this yields a simplified model. We see that the decision maker
should adopt his order quantities downwards in case of left-over inventory (0:189)
and upwards in cases of stockouts (C0:265). Model Opt M2 shows the overall
effect of inventory and stockouts on order quantities. Using a0 D 0 in Model
Baker, we see that the decision maker decreases the order quantity of a product
by 0:695 units for each item left over in the previous week and increases his order
quantity if a product is out-of-stock in the previous week (C0:406). Comparing this
to our optimal baseline model we see that the decision maker adapts stronger than
expected in both cases. Model Baker M2 analyzed this in detail and tests whether
these adaptations are rational, setting a0 D 1. We also tested the other lags (15)
which partly have a significant impact, but the size of these effects is negligible.
We see that 6 is significantly different from 0 for all models and we conclude
that the decision maker is actually chasing demand with respect to the same weekday
of the previous week. This chasing is not based on rational changes (Model Baker
M2) and we conclude that the decision maker is chasing demand.
In a recent study Lau and Bearden (2013) analyze which method to use in
order to test demand chasing. They show that several methods are prone to false
detection of demand chasing, showing demand chasing where there is actually no
chasing. They show that a simple correlation analysis is not prone to this failure.
Therefore, we also conduct a correlation analysis between last period demands and
recent order quantities. In order to account for the autocorrelated demand series
we do not compare the correlation coefficients against 0, but against the demand
correlation. Table 6.6 shows the results of this analysis. We find that the correlation
between last period demand and recent orders are significantly higher than the
99
Product
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
.dt ; dt6 /
0.55
0.51
0.48
0.41
0.32
0.32
0.50
0.58
0.61
0.55
0.62
0.47
0.56
0.29
0.66
0.30
0.63
0.53
0.53
0.45
0.73
0.53
0.47
.qt ; dt6 /
0.75
0.71
0.67
0.66
0.54
0.56
0.74
0.76
0.77
0.71
0.73
0.68
0.74
0.69
0.75
0.52
0.75
0.70
0.72
0.61
0.83
0.81
0.70
demand correlation (paired t-test, p < 0:0001). This also shows the demand chasing
behavior of our empirical decision maker.
As a third test of demand chasing we analyze the correlation of last periods
inventory and the change of order quantities between periods (Rudi and Drake
2014). Figure 6.6 shows the changes of empirical and optimal orders and sales
depending on inventories of the last week. We see a strong positive correlation
between inventories of the last week and the changes between weeks . D 0:63/.
Opposed to that optimal changes have a significant lower correlation . D 0:32/
and even slightly negative for sales . D 0:19/.
Our results show that demand chasing is not only a laboratory artifact, but also
occurs in the real world. But the chasing is asymmetric, as the decision maker reacts
more strongly on inventory than on stockouts.
100
Delta sales
50
-50
-50
L6(d-q)
-50
L6(d-q)
-50
L6(d-q)
6.8 Conclusions
Fig. 6.7 Profits of different
decision models (interpolated
to achieve empirical SL of
decision maker)
101
1.00
0.95
0.90
0.85
Model 0
Empirical
Model 1
Model 2
Model 3
6.8 Conclusions
In this chapter, we study the situation of a newsvendor-like decision maker facing
an aggregated service level contract. For this setting, we analyze the relevant cost
drivers and derive the profit-maximizing order policy integrating product-specific
costs and demand variability. We show that expected profits can be increased
significantly if these factors are incorporated compared to a simple item approach.
Additionally, we analyze the decision-making process in a real setting. While most
of the recent behavioral operations studies focus on the cost-minimizing single
product newsvendor in laboratory settings, we extend the research in two directions.
Firstly, we analyze field data from a real decision maker. Secondly, the decision
maker focuses on an aggregated service level contract.
Our findings show that the decision maker strategically differentiates service
levels between internal and external products, although not perfectly maximizing
his expected profit. While this differentiation is based on differences in profitability
between products, the decision maker ignores additional aspects such as demand
variability, and product-specific cost differences within these categories.
On the operational level, we find some additional explanations for the suboptimal
performance: The decision maker is chasing demand. Adapting the order quantities
to demand realizations is correct, because we do not necessarily have stationary
demand. But our analyses show that the decision maker is over-reacting on demand
realizations compared to normative changes predicted by the newsvendor model.
A limitation of our study is that our analyses use a simple exponential smoothing
forecast ignoring additional external factors and substitution. This could be an area
for possible extensions for future research.
Chapter 7
Conclusions
7.1 Summary
Point-of-sale scanner systems collect large amounts of data that can be used to make
better informed decisions. The present work shows how available information such
as selling prices and the timing of sales occurrences can be leveraged to better align
supply and demand with retail analytics. For this purpose, we have collected data
from a large European retail chain. We develop several models to improve stocking
decisions and to analyze decision-making in the real world.
We suggest a novel approach that works directly with the data by analyzing
causal relationships between demand and external variables. This data-driven
approach integrates forecasting and inventory optimization. It is distribution-free
and can also be applied if the underlying assumptions for other methods such as OLS
regression analysis are violated. The problems are solved with Linear Programming.
As a result, the decision maker obtains optimal order quantities by fitting a linear
inventory function to historical demand observations and external variables affecting
demand such as price, weather and weekdays.
The first model determines order decisions where the retail manager has full
demand observations. We formulate the model for a cost-minimization objective
and service level targets. Comparing the results of different approaches shows that
the approaches taking causal relationships into account outperform the time-series
forecasting method. The regression analysis yields better estimates for small sample
sizes and in-stock target service levels. This holds only if the data meets the OLS
assumptions, otherwise the LP approach achieves more robust inventory levels than
regression analysis.
In the next model, the retailer is not able to observe demand, only sales. If a
stockout occurs, demand is censored at the order-up-to level and the lost sales are
unobserved. We establish sales patterns from days with full demand observations
to estimate the unobservable lost sales. We integrate this aspect into our datadriven model. Based on demand data that follow the normal and the negative
Springer International Publishing Switzerland 2015
A.-L. Sachs, Retail Analytics, Lecture Notes in Economics and Mathematical
Systems 680, DOI 10.1007/978-3-319-13305-8_7
103
104
7 Conclusions
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