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Fight Inventory Shrinkage: Simultaneous Learning of Inventory Level and Shrinkage Rate

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In 2020, inventory shrinkage eroded $61.7 billion profit in the U.S. retail industry. Unfortunately, fighting inventory shrinkage to protect retailers' already slim profits is challenging due to unknown shrinkage rates and invisible inventory levels. While the latter has been studied in the literature, the former has not. To deal with this challenge, we introduce two new features to the Bayesian inventory models: (1) interleaving customer and theft arrival processes that contribute to actual sales and shrinkages, respectively, and (2) learning of both inventory level and shrinkage rate. We first derive the learning formulae using the triple-censored sales data (invisible lost sales, shrinkages and ``lost shrinkages") and then use them to construct a POMDP (Partially Observable Markov Decision Process) model for making inventory and loss prevention decisions. For different level of information deficiency, we analyze the model property and design heuristic order policies to capture the benefit of learning. Through a numerical study, we show that our estimated shrinkage rate converges quickly and monotonically to the actual value. For products with high shrinkage rates (5%-12%), our heuristic policy can help seize 82%-94% of the ideal profit retailers could earn under full information. We note that feature (1) of our model is crucial. It not only reflects the actual arrival order but also allows us to learn the unknown shrinkage rate, which, in turn, can prevent serious under-ordering and vicious inventory cycles and can increase the profit by 108% in some cases. Our approach thus enables both effective inventory management and early identification of ineffective loss prevention strategies, reducing shrinkage and increasing sales and profit.
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Forthcoming in Production and Operations Management
Fight Inventory Shrinkage: Simultaneous Learning of
Inventory Level and Shrinkage Rate
Rong Li
The Whitman School of Management, Syracuse University, 721 University Ave, Syracuse, NY 13244;
Corresponding author; rli138@syr.edu
Jing-Sheng Jeannette Song
The Fuqua School of Business, Duke University, 100 Fuqua Drive, Durham, NC 27708; jingsheng.song@duke.edu
Shuxiao Sun
Newhuadu Business School, Minjiang University, No.200 Xiyuangong Road, Fuzhou, Fujian, China 350108;
sunshuxiao@nbs.edu.cn
Xiaona Zheng
Guanghua School of Management, Peking University, No.5 Yiheyuan Road Haidian District, Beijing, China 100871
xzheng@gsm.pku.edu.cn
In 2020, inventory shrinkage eroded $61.7 billion profit in the U.S. retail industry. Unfortunately, fighting
inventory shrinkage to protect retailers’ already slim profits is challenging due to unknown shrinkage rates
and invisible inventory levels. While the latter has been studied in the literature, the former has not. To
deal with this challenge, we introduce two new features to the Bayesian inventory models: (1) interleaving
customer and theft arrival processes that contribute to actual sales and shrinkages, respectively, and (2)
learning of both inventory level and shrinkage rate. We first derive the learning formulae using the triple-
censored sales data (invisible lost sales, shrinkages and “lost shrinkages”) and then use them to construct a
POMDP (Partially Observable Markov Decision Process) model for making inventory and loss prevention
decisions. For different level of information deficiency, we analyze the model property and design heuristic
order policies to capture the benefit of learning. Through a numerical study, we show that our estimated
shrinkage rate converges quickly and monotonically to the actual value. For products with high shrinkage
rates (5% 12%), our heuristic policy can help seize 82% 94% of the ideal profit retailers could earn under
full information. We note that feature (1) of our model is crucial. It not only reflects the actual arrival
order but also allows us to learn the unknown shrinkage rate, which, in turn, can prevent serious under-
ordering and vicious inventory cycles and can increase the profit by 108% in some cases. Our approach thus
enables both effective inventory management and early identification of ineffective loss prevention strategies,
reducing shrinkage and increasing sales and profit.
Key words : inventory shrinkage; Bayesian learning; interleaving arrival processes; data-driven heuristic
History : Received: June 2021; accepted: January 2022 by Albert Ha after one revision.
1
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
2Forthcoming in Production and Operations Management
1. Introduction
Inventory shrinkage is the loss of inventory in the selling season. The National Retail Federation
(NRF) survey shows that the inventory shrinkage in the US retail industry has been growing every
year; it was worth $61.7 billion in 2020, while the net profit was only less than $109 billion1(NRF
Survey 2020). To survive, retailers actively examine the causes of shrinkage and invest heavily in
loss prevention. It is shown that 74.7% of the shrinkage is caused by theft and 18.8% driven by
administrative error; the implemented loss prevention strategies include security tags, warehouse
entry/exit security, and employee training (NRF Survey 2018).
Inventory shrinkage not only erodes away retailers’ slim profit, but also renders their actual
inventory level invisible, complicating inventory management. Although inventory counts can help
straighten inventory records, they are generally performed only once a year, due to high cost,
disruption to operations and counting errors (Cafferty 2012). Moreover, in reality inventory shrink-
age is unobservable by nature and interwoven with sales, rendering its distribution unknown. As
a result, retailers often have to operate facing severe information deficiency—unknown shrinkage
distribution and invisible inventory level—within a very long counting cycle2. Therefore, it is vital
to find out how to model shrinkages as interleaving with sales, how to estimate or learn the missing
information from heavily censored sales data, and, more importantly, how to use the estimate to
improve decision-making on inventory and loss prevention investment.
While the existing literature has studied invisible inventory level, unknown shrinkage distribution
has not been addressed. To deal with this challenge, we introduce two new features to the Bayesian
inventory models: (1) interleaving customer and theft arrival processes (which contribute to actual
1This is calculated by the revenue times the net profit margin for the retail industry in 2020, where the revenue was
less than $3.9 trillion (Thomas 2020) and the net profit margin was about 2.79% (Damodaran 2021).
2A counting cycle is the time period between two consecutive physical inventory counts.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 3
sales and shrinkages, respectively) and (2) learning of both shrinkage rate3and inventory level. Note
that feature (1) allows us to not only reflect the actual arrival order, but also learn the unknown
shrinkage rate. The learning of both shrinkage rate and inventory is based on triple-censored sales
data (invisible lost sales, thefts and “lost thefts”). We first derive the learning formulae and then
use them to construct a POMDP (Partially Observable Markov Decision Process) model for making
inventory and loss prevention investment decisions.
Different from the MDP (Markov4Decision Process) with on-hand inventory levels as the states
and order quantities as the actions or decisions, when the on-hand inventory level (state) is invisible
or partially observable, the decision-making problem must be formulated as a POMDP, which is
an MDP with the states (inventory levels) replaced by the belief states (probability distribution of
the states); see more discussions of POMDP in Mersereau (2013).
Due to the well-known analytical difficulty of POMDP, for different level of information defi-
ciency, we analyze the model property and design easy-to-implement heuristic order policies to
capture the benefit of learning. We prove that our new feature (1), the interleaving arrival pro-
cesses, can help avoid under- or over-stocking5. Through a numerical study, we demonstrate the
effectiveness of our model in terms of learning. Our main numerical findings are as follows.
1. Even with less information than the existing learning models, our heuristic policy for the case
with unknown shrinkage rate and invisible inventory is surprisingly effective: It achieves 82%94%
of the ideal profit (achieved with full information) for retail products with high shrinkage rates
(5 12%); and it is more effective when the shrinkage rate is lower.
2. Our estimated shrinkage rate converges monotonically to 95% of the actual value in only
ten periods. Thus, our model can help quickly identify an ineffective loss prevention strategy and
3As defined in Section 3.2, the shrinkage rate is the probability at which inventory shrinkage occurs.
4The Markov property is for the state transitions and it is that given the on-hand inventory level (state) and order
quantity (action) of the current period, the next period’s on-hand inventory level (state) is independent of all previous
periods’ states and actions.
5Under- (over-) stocking means the use of a base-stock level lower (higher) than the optimal.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
4Forthcoming in Production and Operations Management
terminate it, improving the effectiveness of loss prevention and profit. Setting up the value-range
for the estimated shrinkage rate is critical, and it is indeed better to set this range wide enough,
which may seem counter-intuitive.
3. It is crucial to model the interleaving arrival order of customers and thefts (feature (1) of our
model). For high-profit products with high shrinkage rates (that retailers typically focus on), an
alternative model that assumes customers arrive before thefts can result in under-ordering and a
vicious cycle. This is because it leads to not only under-stocking even under full information but
also over-estimating the on-hand inventory under censored sales data, which only exacerbates the
problem. We find that the inclusion of feature (1) can result in 108% profit increase in some cases.
The rest of the paper is organized as follows. In Section 2, we review the relevant literature
and position our paper. In Section 3, we introduce the dynamics of our Bayesian inventory model.
We then analyze the model with a known shrinkage rate in Section 4and the model with an
unknown shrinkage rate in Section 5. Section 6evaluates these models numerically. Finally, Section
7concludes the paper.
2. Relationship with the Existing Literature
Canonical inventory models ignore inventory shrinkage and assume that customer-demand distri-
bution is known and inventory level is visible, i.e., assume full visibility, and they use random
variables to represent uncertain demands. To focus on the costly and complex inventory shrink-
age in the retail industry, our model relaxes these assumptions. Specifically, we consider unknown
shrinkage distribution and invisible inventory level and apply Bayesian learning on them using
triple-censored sales data, and we use interleaving counting processes to depict stochastic customer
and theft arrivals (rather than modeling the aggregated arrival if each type as a random vari-
able). While similar arrival processes (Poisson processes, a special type of counting processes) are
adopted by Atali et al. (2009), ok and Shang (2014) and Bassamboo et al. (2020), these works do
not consider learning; and the works that consider learning do not use stochastic processes. Our
work is closely related to two streams of Bayesian inventory research: learning of unknown demand
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 5
distribution (UD) and learning of invisible inventory level (II), where demand here can come from
customers and other sources like thefts. Note that these two streams of work have no overlap; that
is, unlike ours, the existing Bayesian inventory works either consider UD or II, but not both. The
general insights obtained from these works are: lack of visibility is costly and Bayesian learning
can substitute visibility and hence improve inventory management.
For the UD stream, some works develop learning using uncensored data from sales (e.g., Scarf
1959,Scarf 1960,Azoury 1985,Miller 1986 and many others reviewed by Bensoussan and Guo 2015)
or other sources (e.g., stockout times used by Bensoussan and Guo 2015 and machine degradation
signals used by Li and Ryan 2011). Other models, like ours, use censored (invisible lost sales or/and
shrinkage) sales data; see Chen and Mersereau (2015) for a comprehensive review and Table 1for
some representative works.
Table 1 The position of our paper in Bayesian inventory models using censored data
Not modeling the arrival process of demand*
Modeling the arrival
process of demand*
Learning unknown demand*
distribution (UD)
Lariviere and Porteus (1999), Ding et al. (2002),
Lu et al. (2008), Chen and Plambeck (2008), Chen (2010),
Bisi et al. (2011), Bensoussan and Guo (2015),
Jain et al. (2015), Mersereau (2015), Luo et al. (2021)This paper
Learning invisible
inventory level (II)
DeHoratius et al. (2008), Bensoussan et al. (2008),
Bensoussan et al. (2010), Huh et al. (2010),
Mersereau (2013), Bensoussan et al. (2014),
Bensoussan et al. (2016), Chen (2021)
*Demand here is general, coming from not only customers but also other sources like thefts.
With censored data, it is difficult to obtain structural results, unless for special demand distri-
butions like Weibull (Lariviere and Porteus 1999 and Bisi et al. 2011). To capture the impact of
censored data on inventory decisions, some works focus on perishable goods, for which the analysis
is easier as tracking inventory is not necessary; and they identify the effect of stock-more (than the
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
6Forthcoming in Production and Operations Management
case with uncensored data) for improved demand learning (Ding et al. 2002,Lu et al. 2008 and Jain
et al. 2015). For nonperishable goods, such stock-more effect also holds (Chen and Plambeck 2008
and Bensoussan and Guo 2015). But, due to the difficulty to solve, bounds and heuristic policies
are usually proposed (Chen 2010). While most works consider censored data (invisible lost sales),
some works consider less censored data, e.g., visible stockout times by Bensoussan and Guo (2015)
and visible sales times by Jain et al. (2015). Some works consider multiple products and also learn
stockout substitution probabilities (Chen and Plambeck 2008 and Luo et al. 2021). Some works
also consider invisible inventory levels, but do not apply Bayesian learning for them (Mersereau
2015). In contrast, our work learns both unknown shrinkage distribution and invisible inventory
using more (triple) censored data.
For the II stream, the invisible demand is considered and often modeled separately from the
customer demand as either one-sided (non-negative causing inventory shrinkage) or two-sided (non-
negative or negative causing inventory shrinkage or increase, respectively); see Chen and Mersereau
(2015) for a comprehensive review. Some works do not model the invisible demand separately
when studying the backlogged demand case (Bensoussan et al. 2010). Most works model the invis-
ible demand separately as independent of the customer demand and consider the case with lost
sales. Typically, the customer and invisible demands are assumed to occur in a specific order to
avoid the complexity of different accounting of lost sales and lost invisible demand. The most
common assumption is that the customer demand occurs first (DeHoratius et al. 2008,Huh et al.
2010,Bensoussan et al. 2014,Mersereau 2015); instead of assuming the order, Bensoussan et al.
(2016) adopt a random demand ratio to represent the real sales. It is acknowledged by DeHoratius
et al. (2008) that “In reality, we might expect that visible and invisible demand arrivals would be
interwoven throughout a day. Such a situation, however, is difficult to model under the lost sales
assumption.” Besides heuristic inventory policies, some works also design heuristic audit policies
and demonstrate their effectiveness numerically (DeHoratius et al. 2008,Huh et al. 2010,Chen
2021). In contrast, our work models the customer and (one-sided) invisible/theft demand arrivals
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 7
as interwoven and also learns the unknown shrinkage distribution; see Table 1for the position of
our work.
We note that there are many non-Bayesian inventory models that also study the impact of
censored sales data with no learning (e.g., Iglehart and Morey 1972,Lee and ¨
Ozer 2007,ok and
Shang 2007) or with non-parametric learning (e.g., Huh and Rusmevichientong 2009,Shi et al.
2016,Chen and Chao 2020). There are also some non-Bayesian inventory models that involve
learning on random capacity (e.g., Chen et al. 2020 and Song et al. 2020) and some capacity models
that involve Bayesian learning on demand (e.g., Qi et al. 2017).
3. Model Dynamics
We consider a retailer who orders inventory of a nonperishable product periodically. For ease of
exposition, we focus on the case where the retailer never performs inventory counts due to high cost
and operations disruption. We model it as a stationary infinite-horizon problem with unobservable
inventory. (We note that the case with a finite counting cycle can be similarly modeled as a finite-
horizon problem.)
Let [0,) denote the planning horizon and assume the length of each period is 1. At the start of
each period n,n1, the retailer needs to order at unit cost cbefore demand occurs, and the order
arrives instantaneously. Then two types of random demand arrive within the period: the customer
demand Dnand the theft demand Vn. The retailer “fills” both types of demand with available
inventory, but earns unit revenue ronly for the customer demand filled. The theft demand filled
is the inventory shrinkage, which is invisible itself and also makes the inventory level invisible.
Any unfilled demand is lost, but the retailer incurs a unit penalty cost ponly for the unfilled
customer demand. Any excess inventory is carried over to the next period at a unit holding cost h.
Across periods, the customer demands Dn,n1, are independent and identically distributed (iid)
with a cumulative distribution function (cdf) FD(·); the theft demands Vn,n1, are also iid with
cdf FV(·), where Dand Vare iid to Dnand Vn,n1, respectively. The retailer’s objective is to
choose the order quantity Qn0, based on the invisible inventory levels, to maximize his expected
profit-to-go.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
8Forthcoming in Production and Operations Management
3.1. Benchmark 1: The Canonical Inventory Model
For the decision-making scenario mentioned above, canonical inventory models assume that (A1)
thefts are negligible (i.e., assume Vn0 for all n) and hence the inventory level is visible and
(A2) the customer demand distribution is known (i.e., assume FD(·) is known). Under these two
assumptions, the literature has shown that an order-up-to policy is optimal.
At the start of period n, facing his on-hand inventory level in, the retailer decides his order
quantity Qn0 and uses his total inventory zn:= in+Qnto fill the customer demand Dn. This
way, the retailer generates sales znDnat the cost of ordering cQn, penalty p(Dnzn)+and
holding h(znDn)+, where ab= min{a, b}for any a, b lR. A similar decision is made in every
future period. That is, we have the following Dynamic Programming (DP) formulation for the
canonical (c) inventory model which is infinite-horizon and stationary. The retailer chooses the
order quantity Qn0 to maximize his expected profit-to-go uc(in), i.e.,
uc(in) = max
Qn0cin+πc(in+Qn) + αEuc((in+QnDn)+), n 1,(1)
where α(0,1) is the discount factor and
πc(zn) = Er(znDn)cznp(Dnzn)+h(znDn)+(2)
is the retailer’s current-period profit.
It is well known that we can rewrite the DP given zero on-hand inventory as uc(0) =
maxz0{πc(z) + αcE[(zD)+]+αuc(0)}, where recall that Dis iid to Dn. This implies that a
myopic policy is optimal, and the optimal order-up-to level is zc:= F1
D(r+pc
r+p+hαc ) and the optimal
expected profit-to-go given zero on-hand inventory is
uc(0) = 1
1α(rc)zcpE[(Dzc)+](r+hαc)E[(zcD)+].(3)
Thus, the optimal order quantity for period nis
Qc
n= min Q0 : FD(in+Q)r+pc
r+p+hαc=F1
Dr+pc
r+p+hαcin+
,(4)
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 9
where F1
Dis the inverse function of FD, which also works for discrete distributions with F1
D(y) :=
min{x0 : FD(x)y}for y[0,1]. (For other nonnegative random variables, the inverse func-
tions of their distributions can be similarly defined.)
In practice, however, the assumptions (A1) and (A2) used in these canonical inventory models do
not hold and hence the DP should be modified accordingly. For example, when (A1) does not hold,
the actual inventory levels inin (1) become invisible and hence (1) needs to be redefined with some
states that are visible. When the assumption (A2) does not hold, the distribution of Dnbecomes
unknown and hence the expectation taken over Dnin (1) needs to be redefined. In this paper, we
build a new Bayesian inventory model free of both assumptions. To facilitate understanding, we
will introduce our model gradually and start with how we take thefts into consideration (Vn6≡ 0
for all n) in the next subsection.
3.2. Theft Demand: Interleaving with Customer Demand
We assume each customer buys one and only one unit of the product. Similarly, each thief steals one
and only one unit of the product. Different from the customer-first (CF) model in the literature,
which assumes customers always arrive before thefts, we model interleaving (IL) customer and
theft arrivals to reflect the reality. Please see Figure 1for an illustration and comparison of these
models in a special case with Poisson arrival processes (a special type of counting processes).
Let N(t) denote the total number of customers and thieves arrived in time interval [0, t) and the
aggregated arrival process {N(t); t0}be a counting process6with stationary and independent
increments7. We then have ˜
Dn:= N(n)N(n1) = Dn+Vnrepresent the store traffic to the
product or the aggregated demand of customer and theft in period n; and ˜
D1,˜
D2,...,are iid random
variables with a known8cdf F˜
D(·), where ˜
Dis iid to all ˜
Dn,n1.
6A counting process is a stochastic process {N(t); t0}with values that are non-negative integers (N(t)Z+) and
non-decreasing (N(s)N(t) for any st).
7Examples of such counting processes are renewal processes and Poisson processes. Note that the stationary and
independent increments are only required for analyzing the infinite-horizon model, not for the finite-horizon model.
8This is reasonable as retailers can rely on sensors like Aurora to accurately forecast the store traffic to each product.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
10 Forthcoming in Production and Operations Management
n-1 ntime
Dn~ Poisson(ld) Vn~ Poisson(lv) CF Model
(ldand lv are known)
time
Dn+ Vn= 𝐷
"#$~ Poisson(l)
Dn~ Poisson((1-r)l)
Vn~ Poisson(rl)
customer thief
r1-r
Our IL Mod el
(lis known, but ris unkno wn)
n-1 n
Figure 1 The illustration of the CF and our IL models in the special case with Poisson arrivals
As shown in Figure 1, each arrival of the aggregate demand ˜
Dnis either a customer, with
probability (1 ρ), or a thief, with probability ρ. Let Ynk indicate the type of the kth demand,
i.e., Ynk = 1 or 0 if the kth arrival is a customer or a thief, respectively, k1. Then we can write
the customer demand and theft demand as Dn=P˜
Dn
k=1 Ynk and Vn=P˜
Dn
k=1(1 Ynk ), respectively.
In the special case where the aggregated arrival {N(t); t0}is a Poisson process with arrival rate
λ, it can be shown that Dnand Vnare independent and follow Poisson distribution with mean
(1 ρ)λand ρλ, respectively. Since the probability ρis the mean rate at which inventory shrinkage
occurs, we refer to it as the shrinkage rate.
3.3. Benchmark 2: The Full-Visibility Model with Interleaving Demand
Using the above IL (interleaving) demand model, we now incorporate the theft-consideration in the
canonical inventory model defined by (1)-(2). Assuming the retailer observes the on-hand inventory
level inand knows the shrinkage rate ρ, we have a full-visibility model. At the start of period n,
facing his on-hand inventory in, the retailer decides his order quantity Qn0 and uses his total
inventory zn=in+Qnto fill the aggregated demand ˜
Dn=Dn+Vn. This way, the retailer generates
the sales Sn:= Pzn˜
Dn
k=1 Ynk (from filling the customer demand Dn) at costs, including the ordering
cost cQn, the penalty cost p(DnSn) and the holding cost h(in+Qn˜
Dn)+. Note that since
SnDn, the penalty cost p(DnSn)+=p(DnSn). A similar decision is made at the start of
every future period.
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Forthcoming in Production and Operations Management 11
We have the following DP for this full-visibility (f ) model. The retailer decides the order quantity
Qn0 to maximize his expected profit-to-go uf(in), i.e.,
uf(in) = max
Qn0ncin+πf(in+Qn) + αEhuf((in+Qn˜
Dn)+)io, n 1,(5)
where where α(0,1) is the discount factor and
πf(zn) = EhrSncznp(DnSn)h(zn˜
Dn)+i(6)
is the retailer’s current-period profit. Here, the sales Sn,n1, are iid and so are the customer
demands Dn,n1, and the aggregated demands ˜
Dn,n1.
It is important to note the following differences between the above full-visibility model and the
canonical model defined by (1)-(2). First, the retailer’s inventory is now used to fill the aggre-
gated demand ˜
Dn, rather than the customer-demand Dn. Second, due to the interleaving demand
consideration, the sales quantity Sn=Pzn˜
Dn
k=1 Ynk now is the number of customers arrived before
inventory runs out; that is, Snhas a different expression from most inventory models, i.e., Sn6=
znDn. Third, due to the theft-demand consideration, the penalty cost now applies only to the
unmet customer-demand and the holding cost now applies only to the actual inventory held.
Solving the above DP (5)-(6), we obtain the following results.
Proposition 1. For the full-visibility model, a myopic base-stock policy is optimal; the optimal
order-up-to level is zf:= F1
˜
D(r+p)(1ρ)c
(r+p)(1ρ)+hαc , where recall that ˜
Dis iid to ˜
Dn, and the optimal
expected profit-to-go given zero on-hand inventory is
uf(0) = 1
1α(r(1 ρ)c)zfp(1 ρ)E˜
Dzf+
(r(1 ρ)+(hαc)Ezf˜
D+.(7)
Different from the canonical model with zc=F1
D(r+pc
r+p+hαc ) and the expected profit-to-go shown
in (3), we note that when inventory shrinkage is considered, the order-up-to level zfis based on
the aggregate demand ˜
Dand the shrinkage rate ρappears in the critical fractile. Similar changes
are also observed in the expected profit-to-go. Using this result, we next derive the optimal order
quantity of each period and analyze how it is affected by the shrinkage rate.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
12 Forthcoming in Production and Operations Management
Proposition 2. The optimal order quantity of period nis Qf
n:= (zfin)+,n1. As the
shrinkage rate ρdecreases, zfand hence Qf
nincreases; when ρdrops to 0,zfand Qf
nconverge to
zcand Qc
n, respectively, i.e., the full-visibility model converges to the canonical model.
As ρdecreases, more store visitors are customers, so the customer-demand increases. Conse-
quently, the retailer should raise the order-up-to level and hence the order quantity. When the
shrinkage rate drops to 0, all the store visitors are customers ( ˜
DnDn), and hence the full-visibility
model reduces to the canonical model.
3.4. Benchmark 3: The Full-Visibility Model under the CF Assumption
Rather than modeling the actual interleaving demand as above, most relevant literature simpli-
fies the analysis under the CF (customer-first) assumption. That is, the customer demand Dnis
assumed to occur before the theft demand Vnin every period, as shown in Figure 1. (Note that
we can similarly analyze the case with the theft-first assumption, which however may be unlikely
to happen in practice given many loss prevention programs in place.) Under the CF assumption,
we rewrite the sales quantity as Sn=znDn, and the DP for the full-visibility model given in the
previous subsection becomes:
uf|CF (in) = max
Qn0ncin+πf|CF (in+Qn) + αEhuf|C F ((in+Qn˜
Dn)+)io, n 1,(8)
where
πf|CF (zn) = Ehr(znDn)cznp(Dnzn)+h(zn˜
Dn)+i(9)
is the retailer’s current-period profit.
We solve the above DP (8)-(9) and obtain the following results.
Proposition 3. For the full-visibility model under the CF assumption, a myopic base-stock
policy is optimal; the optimal order-up-to level is zf|CF := ((r+p)FD+ (hαc)F˜
D)1(r+pc)
and the optimal expected profit-to-go given zero on-hand inventory is
uf|CF (0) = 1
1αnrEzf|CF Dczf|CF pE(Dzf|C F )+(hαc)E(zf|CF ˜
D)+o.(10)
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Forthcoming in Production and Operations Management 13
Comparing to the results of the same model without the CF assumption (shown in Proposition
1), we note that this assumption makes both zf|CF and uf|CF (0) not only depend on ˜
D, but also D.
Despite this difference, we find that the CF assumption does affect the convergence of the model
to the canonical one as shown below.
Proposition 4. The optimal order quantity of period nis Qf|CF
n:= zf|CF in+,n1. As the
shrinkage rate ρdecreases, zf|C F and hence Qf|CF
nincreases; when ρdrops to 0,zf|CF and Qf|CF
n
converge to zcand Qc
n, respectively, i.e., the full-visibility model with the CF assumption converges
to the canonical model.
We next compare zf|C F with zfin detail to further examine the impact of the assumption.
Proposition 5. If the critical fractile is large enough such that zf|CF x0(holds for most retail
products), the CF-assumption leads to under-stocking, i.e., zf|C F zf, where x0:= ( ¯
FD(1
ρ)¯
F˜
D)1(0). Otherwise, it leads to over-stocking, i.e., zf|CF > zf.
This result shows that the CF-assumption has an interesting asymmetric impact on the inventory
decision. That is, it will make the retailer under-stock for products with high critical fractile (e.g.,
products with a relatively low shrinkage rate), but over-stock for products with low critical fractile
(e.g., products with a relatively high shrinkage rate). The product segmentation is shown in Figure
2below. According to Damodaran (2021) and NRF Survey (2018), the gross margin in the retail
industry is about 25% (general) and 45% (online) and the shrinkage rate is generally below 15%.
This means that almost all retail products fall into the “under-stocking” region. Also, as the
penalty cost pincreases, e.g., from p= 0.5(rc) to p= 2(rc) as shown in Figure 2(a) and 2(b),
respectively, the critical fractile increases and hence the retailer is more likely to under-stock than
optimal for more valuable products.
Intuitively, assuming the customer demand arrives first implies that all the inventory is used
to fill the customer demand, i.e., the inventory is “more effective” in filling customer demand.
This would naturally lead to under-estimating the lost sales and therefore the under-stocking
tendency (zf|CF < zf). It is, however, often overlooked that the more-effective inventory would
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
14 Forthcoming in Production and Operations Management
ρ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
(rc)/r
5%
10%
20%
30%
40%
50%
60%
70%
80%
Under-stocking(zf|C F < zf)
Over-stocking(zf|CF > z f)
ρ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
(rc)/r
5%
10%
20%
30%
40%
50%
60%
70%
80%
Under-stocking(zf|C F < zf)
Over-stocking(zf|CF > z f)
(a) p= 0.5(rc) (b) p= 2(rc)
Figure 2 The CF-assumption’s impact on stocking (λ= 15,c= 30,h= 1)
also result in over-estimating the sales and hence the over-stocking tendency (zf|C F > zf). For
most retail products (which have a high critical fractile), lost sales are more important such that
the corresponding under-stocking tendency outweighs the over-stocking tendency. As a result, the
retailer will stock less than optimal. The opposite, over-stocking, holds true for the products with
a low critical fractile for which lost sales are less important.
In practice, as most retailers focus on high-valuable products for loss prevention, we know that
they would stock less than optimal if they were to follow the solutions derived under the CF
assumption. The cumulative negative effect of such under-stocking over time can be significant.
It can be even more pronounced in practice as retailers often do not have full visibility (i.e., do
not know inventory level and shrinkage rate). Thus, our model and results based on the actual
interleaving arrivals are essential for retailers to improve their inventory management.
3.5. Learning inand ρfrom Triple-Censored Sales Data
In reality, retailers do not have full visibility over the inventory information. First, inventory shrink-
age is invisible by nature, and so is the actual inventory level in, as shown in Figure 3(a). Second,
the shrinkage rate ρis often unknown. To face reality, we estimate inand ρusing Bayesian learning
from the observed demand data, which are triple-censored as illustrated in Figure 3(b). And these
estimates will help re-define the state variable inand the expectation over Dnin the full-visibility
model, given in (5)-(6).
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 15
In-1+Qn-1: Starting inventory
sn-1 kn-1
Sales
Shrinkages
In=In-1+Qn-1sn-1kn -1
Ending
inventory
Invisible InvisibleVisible
(a)
Visible
Invisible
𝑑
"#: 𝐷
%#s realization
dnvn
sn(dn– sn)+
Invisible
Visible
Sales
Lost sales
Customers
Thieves
kn(vn– kn)+
Invisible
Invisible
Shrinkages
Lost thefts
(b)
Figure 3 The illustration of visible and invisible information in our IL model
To learn the shrinkage rate ρ, we treat it as a random variable with any prior distribution.
Since ρaffects the customer-demand and hence the sales, we apply Bayesian learning to update its
distribution periodically using the observed sales data. That is, we re-define the expectation over
Dnin (6) as the expectation over ˜
Dnand ρ.
To learn the actual on-hand inventory in, we treat it as a random variable, In. As shown in
Figure 3(a), Inis also the ending inventory of the previous period, period (n1). Without loss
of generality, we assume the planning horizon starts with no inventory (I10). Leveraging the
relationship between Inand the observed sales of the previous period, sn1, we can update the
distribution of Inusing the sales data and replace the state variable inin (5) by the distribution
of In(see the belief state in Sections 4.2 and 5.2).
As explained above, both ρand inneed to be learned using the observed sales data. Unlike in
the existing literature, the sales/demand data in our problem is triple-censored by invisible lost
sales, inventory shrinkages and lost thefts, as shown in Figure 3(b). This may reduce the efficiency
of learning and complicate the analysis. Our problem is how to use the observed sales data to best
learn Inand ρ, which, in turn, improves the order decision in the current period.
4. Invisible Inventory and Known Shrinkage Rate
To ease exposition and facilitate understanding, we start by introducing our IL inventory model
with invisible inventory but known shrinkage rate (ρ). This model is useful in its own right for
products with a stable shrinkage rate or a stable shrinkage rate estimation.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
16 Forthcoming in Production and Operations Management
We study how much to order based on the learning of on-hand inventory in(the state variable).
Since this state is invisible and estimated by In, we replace it by the distribution of In,bn(·) =
P{In=·|~on}, learned from the observed past sales data ~on= (s1,...,sn1). This distribution bn(·)
represents the retailer’s belief of the state Inand hence is referred to as the belief state. This makes
our decision-making process a POMDP (Partially Observable Markov Decision Process). We first
derive the expression for the belief state bnand then introduce the details of the POMDP.
4.1. Expression of bn
At the start of period n, we derive bn(·) = P{In=·|~on}using the on-hand inventory In’s dependence
on the previous period’s on-hand inventory In1and sales quantity Sn1. As shown in Figure 3(a),
Inis also the ending inventory of the previous period after filling the demand from both customers
and thefts, i.e.,
In= (In1+Qn1)(Sn1+Kn1).(11)
We first identify the range for Inand In1. Since I10, we know that Inranges from 0 to
Mn:= Pn1
j=1 QjPn1
j=1 sj, where the upper limit Mnis reached when no shrinkage occurs in the
past periods and hence all the stock (Pn1
j=1 Qj) is only consumed by customers (Pn1
j=1 sj). Similarly,
the range of In1is [0, Mn1] which can be refined using the additional information: the observed
sales quantity sn1. Since the shrinkage quantity Kn1(= In1(In+sn1Qn1)) is always
non-negative, we know that In1[(In+sn1Qn1)+, Mn1].
After identifying the range for Inand In1, we can follow the Bayes’ rule and probability theory
to obtain the following expression for bn: for i[0, Mn],
bn(i|ρ) = PMn1
j=(i+sn1Qn1)+j+Qn1i
sn1(1 ρ)sn1ρj+Qn1isn1F˜
Dn1(j, i)bn1(j|ρ)
PMn
k=0 PMn1
j=(k+sn1Qn1)+j+Qn1k
sn1(1 ρ)sn1ρj+Qn1ksn1F˜
Dn1(j, k)bn1(j|ρ),(12)
where F˜
Dn1(j, k) is defined in the proof of Proposition 6. Note that we use bn(·|ρ) to emphasize
the fact that the distribution of Independs on the known value of the shrinkage rate ρ.
Although the expression of bn(i|ρ) looks quite complex, it is easy to compute numerically. To
understand its expression, we note that the numerator in (12) represents the probability that the
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 17
retailer sells sn1units of inventory in period (n1) and ends the period with exactly iunits.
To calculate this probability, we need to condition on the retailer’s on-hand inventory of period
(n1), j. For example, for each given j, we know that in period (n1), the retailer starts with
(j+Qn1) units and ends with iunits. This means it costs the retailer (j+Qn1i) units of
inventory to generate sn1units of sales. Therefore, the probability translates to the likelihood
of seeing sn1customers among the first (j+Qn1i) store visitors. The denominator in (12)
represents a bigger probability that the retailer sells sn1units in period (n1) and ends of the
period with any units of inventory.
4.2. The POMDP Model
Using the above expression of the belief state bn(·) = P{In=·|~on}, learned from the past sales
observations ~on= (s1,...,sn1), the retailer can determine the order quantity Qn0 by solving
the following DP on the expected profit-to-go, i.e.,
u(bn) = max
Qn0nπn(bn, Qn) + αX
bn+1Bn+1
u(bn+1)P(bn+1 |bn, Qn)o, n 1,(13)
where
πn(bn, Qn) = EhrSncQnp(DnSn)h(In+Qn˜
Dn)+i(14)
is the retailer’s current-period profit and bn+1 is the next period’s belief state of In+1 = (In+Qn
˜
Dn)+defined in space Bn+1. Note that πn(bn, Qn) differs from πf
n(in+Qn), defined by (6) for the
full-visibility model, in that the on-hand inventory inis replaced by its estimate, In, which, in turn,
affects the random sales quantity Snand the order decision Qn.
Note that to expand the expression for the DP (13)-(14), we need an expression for both the
current belief state bn, given by (12), and the transition probability P(bn+1 |bn, Qn), derived below.
Proposition 6. When the shrinkage rate ρis known, the belief state bn+1 can be derived itera-
tively using (12) and b1(0|ρ) = 1, and the transition probability is uniquely determined by the sales
quantity Sn, i.e., P(bn+1|bn, Qn) = P(Sn|bn, Qn).
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
18 Forthcoming in Production and Operations Management
Using this result, we are able to have the following complete expression for the DP:
u(bn) = max
Qn0πn(bn, Qn) + αESn|bn,Qn[u(bn+1)],(15)
where πn(bn, Qn) and bnsatisfy (14) and (12), respectively. Note that bndepends on Qn1, the
order decision of the previous period, in a complex way. This makes the above DP analytically
intractable.
We next explain how to implement our model in practice. The retailer starts the first period with
no on-hand inventory and solves the DP (15) to decide the order quantity Q1. During this period,
he sells s1(Q1) units to the customers and loses K1units to the thefts. At the start of period 2,
since the shrinkage quantity K1is invisible, the retailer does not know his exact on-hand inventory
i2. He therefore estimates it by the belief state b2=P{I2|~o2=s1}using (12) with observed s1units
of sales and his estimate of i1(b1(0|ρ) = 1). Based on the belief state b2, the retailer then calculates
his expected profit-to-go and solves the DP (15) to decide the order quantity Q2. The retailer
repeats this in every future period.
4.3. Heuristic Order Policies and Comparison
It is well recognized in the literature that POMDPs are challenging to solve (Russell and Norvig
2010). For our POMDP model, the intractability arises from that the decision Qnaffects the
distribution of both Snand In+1 in a complex way. We therefore propose a heuristic order policy
that captures the learning benefits. We numerically compare it against the case with full information
in Section 6.
4.3.1. Heuristic Order Policies: We start by proposing the following heuristic policy that
captures the learning benefit of our IL (interleaving) model.
IL Heuristic Policy with Known ρ:ˆ
Qn(ρ) := zfˆ
in(~on)+
, where recall that zf=
F1
˜
D(r+p)(1ρ)c
(r+p)(1ρ)+hαc is the optimal order-up-to level under full-visibility and is shown decreasing
in ρ, and ˆ
in(~on) = E[In|~on] is the Bayesian learning-based estimator of the on-hand inventory level.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 19
Note that this heuristic policy has the following traits. First, rather than using the entire dis-
tribution bn=P{In|~on}, we use its mean E[In|~on], the minimum mean square error (MMSE) esti-
mator, to estimate the on-hand inventory level in. This estimator not only is easy-to-calculate and
implement, but also will converge to the actual inventory level as the shrinkage rate diminishes.
Lemma 1. As the shrinkage rate ρdecreases to 0, the estimator E[In|~on]converges to in.
Second, the order quantity ˆ
Qn(ρ) is designed based on the true optimal policy of an “equivalent”
full-visibility problem, which shares our model parameters, but has our estimate of in(E[In|~on]) as
the observed on-hand inventory level. (Please see the proof that ˆ
Qn(ρ) is optimal for this equivalent
problem and a similar result holds for ˆ
QCF
n(ρ) defined below at the end of the online supplement.)
To quantify our contribution of modeling the interleaving demand, we compare the performance
between our IL heuristic policy and the following comparable heuristic policy for the CF (customer-
first) model. Please find the details of In’s distribution under the CF assumption in the proof of
Proposition 7.
CF Heuristic Policy with Known ρ:ˆ
QCF
n(ρ) := zf|CF ˆ
iCF
n(~on)+
, where recall that
zf|CF =((r+p)FD+ (hαc)F˜
D)1(r+pc) is the optimal order-up-to level under full-visibility
and ˆ
iCF
n(~on) = EC F [In|~on] is the Bayesian learning-based estimator of the on-hand inventory level
under the CF-assumption.
4.3.2. Comparison and Impact of the CF Assumption: We first compare the two heuris-
tics introduced above and note the following two differences between ˆ
Qn(ρ) and ˆ
QCF
n(ρ). First, the
order-up-to levels, zfand zf|CF , are different as shown in Proposition 5. That is, for most retail
products, the CF-assumption leads to under-stocking, i.e., zf|CF < zf. Second, the estimators of
the on-hand inventory level, E[In|~on] and EC F [In|~on], are also different as shown below.
Proposition 7. Given that bC F
n1=bn1and the aggregated arrival {N(t); t0}is a Poisson
process, when sn1< Qn1, the retailer will over-estimate its on-hand inventory under the CF
assumption, i.e., ECF [In|~on]>E[In|~on], iff the shrinkage rate ρis not too large.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
20 Forthcoming in Production and Operations Management
To understand the above results, let us consider the first inventory estimation which occurs at
the start of period 2. The retailer estimates its current on-hand inventory (I2) with or without the
CF-assumption, facing the same amount of realized sales s1. Since the sales is certainly less than
the order quantity (s1< Q1), our results show that the CF-assumption will lead to over-estimation
(ECF [In|~on]>E[In|~on]) for products with a relatively low shrinkage rate. As illustrated in Figure 4,
such over-estimation always occurs for products with shrinkage rates up to 12.5% (which holds for
most retail products). For products with higher shrinkage rates, over-estimation occurs only when
the observed sales are low enough, where the sales level threshold decreases in the shrinkage rate.
ρ
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8
s1
0
2
4
6
8
10
12
14
16
18
20
ECF [I2|~o2]> E[I2|~o2]
ECF [I2|~o2]< E[I2|~o2]
Figure 4 The CF-assumption’s impact on the on-hand inventory estimation (λ= 15,c= 30,h= 1,I1= 0,Q1= 20)
As mentioned above, the CF-assumption results in under-stocking (zf|CF < zf). Such over-
estimation of the on-hand inventory will only exacerbate the problem. That is, the retailer will
not only target a lower stocking level, but also over-estimate its current on-hand inventory. As
a result, the retailer will significantly under-order (ˆ
QCF
n(ρ)<ˆ
Qn(ρ)). Such a compound-deviation
effect accumulates over time and quickly depletes the retailer’s actual inventory and drives the
sales down to 0. Our results offer an alternative explanation for the declining sales observed in
practice. That is, it may be caused by the inventory decisions based on simplifying assumptions
such as the CF-assumption.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 21
5. Invisible Inventory and Unknown Shrinkage Rate
This section extends the above POMDP model to consider the general case where the shrinkage
rate is unknown. Since ρis unknown, the key change is that the belief state should expand to include
the estimate (or the belief) of ρ. This expanded belief state estimates both the on-hand inventory
inand the shrinkage rate ρ. This state is referred to as the joint belief state βn(i, ρ) = bn(i|ρ)fn(ρ),
which is the joint pdf of Inand ρ. Note that in this joint pdf, bn(·|ρ) is the probability mass function
(pmf) of Inif the unknown shrinkage rate takes value ρ, given by (12), and fn(·) is the posterior
pdf of ρfor ρ[0, a] and f1(·) is the prior of ρ. (Note that later in Section 6.4 we will discuss the
selection and impact of the support of ρ, [0, a].)
5.1. Expressions of fn(ρ)and βn
To estimate the shrinkage rate ρ, the retailer starts with any prior distribution f1with support
[0, a], a(0,1). Once the sales in period (n1) are observed, the estimate will be Bayesian-updated
by the posterior distribution fnusing sn1and fn1,n2. That is,
fn(ρ) = f(ρ|~on) = P{Sn1=sn1|ρ, ~on1}fn1(ρ)
Ra
0P{Sn1=sn1|ρ,~on1}fn1(ρ)
=PMn1
j=(sn1Qn1)+P{Sn1=sn1|In1=j, ρ}bn1(j|ρ)fn1(ρ)
PMn1
j=(sn1Qn1)+Ra
0P{Sn1=sn1|In1=j, ρ}bn1(j|x)fn1(x)
| {z }
βn1(j,x)
dx,(16)
where bn1(j|ρ) = P{In1=j|ρ,~on1}is the pmf of In1given by (12) and
P{Sn1=sn1|In1=j, ρ}=
j+Qn1
X
k=sn1k
sn1(1 ρ)sn1ρksn1P{˜
Dn1=k}
+j+Qn1
sn1(1 ρ)sn1ρj+Qn1sn1P{˜
Dn1> j +Qn1},(17)
which follows from the fact that Sn1=sn1|In1=jis equivalent to that the aggregate demand
˜
Dn1is at least sn1and out of ˜
Dn1there are exactly sn1customers before inventory runs out.
As shown above, we can derive fniteratively using formulae (16)-(17), the prior f1and the sales
quantity sn1. Note that these formulae of fnalso apply to the case with visible inventory level
(In1in1), where P{Sn1=sn1|In1=i, ρ}in (16) becomes P{Sn1=sn1|In=i, ρ}= 1 for
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
22 Forthcoming in Production and Operations Management
i=in1and = 0 for all i6=in1. These formulae for fn, together with (12) for bn, can then be used
to express the joint belief state βn=bnfn. Note that similar to bn,fnalso depends on the decision
Qn1in a complex way, making the DP (20) intractable. Moreover, fnand hence βn=bnfnis also
uniquely determined by the observed sale quantity sn1as stated in Proposition 8.
5.2. The POMDP Model
Given the expression we have derived for the joint belief state βn, we now determine the order
quantity Qn0 by solving the following DP on the expected profit-to-go. Note that we can also
add decisions such as whether to terminate the currently implemented loss prevention strategies if
the estimated shrinkage rate exceeds a preset level, and all the results still hold.
u(βn) = max
Qn0nπn(βn, Qn) + αX
βn+1∈Bn+1
u(βn+1)P(βn+1 |βn, Qn)o, n 1,(18)
where
πn(βn, Qn) = EhrSncQnp(DnSn)h(In+Qn˜
Dn)+i(19)
is the current-period profit and βn+1 =bn+1fn+1 is the joint belief state for the next period defined
in space Bn+1. Note that πn(βn, Qn) defined above is an expectation taken over ρ, which differs
from πn(bn, Qn), defined by (14) for the case with a known shrinkage rate.
To derive a complete expression for the above DP, we apply the posterior distribution of ρ,fn(ρ),
the belief state βn=bnfnand the transition probability P(βn+1|βn, Qn) as shown below.
Proposition 8. When the shrinkage rate is unknown, the belief state βn+1 can be derived iter-
atively using (12) and (16)-(17), and the transition probability is uniquely determined by the sales
quantity Sn, i.e., P(βn+1|βn, Qn) = P(Sn|βn, Qn).
Using this result, we obtain the following complete expression for the DP:
u(βn) = max
Qn0πn(βn, Qn) + αESn|βn,Qn[u(βn+1)],(20)
where πn(βn=bnfn, Qn), bnand fnsatisfy (19), (12) and (16)-(17), respectively.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 23
5.3. Heuristic Order Policies
Compared to the known shrinkage rate case, this general POMDP model is even more intractable
because not only bn, but also fndepends on Qnin a complex way. We thus propose a similar
Bayesian heuristic policy that captures the benefit of learning both inand ρ. We do so by estimating
both the critical fractile and the actual on-hand inventory inby the best Bayesian learning-based
estimators.
IL Heuristic Policy: ˆ
ˆ
Qn:= F1
˜
DEρ|~onh(r+p)(1ρ)c
(r+p)(1ρ)+hαc iˆ
ˆ
in(~on)+
, where
Eρ|~onh(r+p)(1ρ)c
(r+p)(1ρ)+hαc iand ˆ
ˆ
in(~on) = E[In|~on] are the Bayesian learning-based estimators of the
critical fractile and the on-hand inventory level, respectively.
6. Performance Evaluation
In this section, we perform a comprehensive numerical study to evaluate the performance of our IL
model, which includes the shrinkage rate estimate and the heuristic order policy for different cases.
We can thus assess how well our model performs in terms of using Bayesian learning to substitute
information deficiency (the value of learning). Specifically, we evaluate the value of learning the
shrinkage rate only, the inventory level only, and both the shrinkage rate and inventory level.
We consider a 3.5-day reorder period, representing semiweekly replenishment, and set the dis-
count factor α= 0.98. In addition, we use the uniform distribution U[0, a] as the prior for the
unknown shrinkage rate ρ. We simulate each instance 2,000 10,000 times and use the parameters
appropriate for the retail industry. Since the recent gross and net profit margin of the retail indus-
try is about 24.79% and 2.44%, respectively (Damodaran 2021), we set r= 40, c = 30 (such that
the gross margin rc
r24.79%), h= 1, and p= 15,20,...,40. We consider ρ= 0.01,0.02,...,0.3
(NRF Survey 2020).
6.1. Value of Learning Shrinkage Rate ρ
We first evaluate our IL model’s value of learning the shrinkage rate ρalone. We do so by applying
the model with unknown ρin a special case with visible inventory levels. For this case, we can
obtain the distribution of ρusing (16)-(17) in Section 5with a modified belief state bn(in) = 1
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
24 Forthcoming in Production and Operations Management
n
2 4 6 8 10 12 14 16 18 20
Estimated Shrinkage Rate
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
ρ=0.30
ρ=0.25
ρ=0.20
ρ=0.15
ρ=0.10
Figure 5 The estimated shrinkage rates change over time (a= 0.4,λ= 15,r= 40,c= 30,h= 1,p= 25)
and bn(i) = 0 for i6=in. Specifically, we examine the convergence performance of our estimated
shrinkage rates (E[ρ|~on]).
As illustrated in Figure 5, we first note that our estimated shrinkage rate does converge to the
actual value, and it reaches 95% of the actual value in only ten periods. Second, although the
estimates of different shrinkage rates (ρ= 0.1,...,0.3) all start at the same initial estimate (the
prior mean a
2= 0.2), they quickly separate (increase or decrease) to reach their respective actual
values with bigger and bigger differences as time goes by. Such a monotone convergence property
can help quickly identify an effective loss prevention strategy. Third, the convergence is faster if the
initial estimate (a
2) is closer to the actual value. Thus, it is important for the retailers to analyze
their historical sales and shrinkage data to provide a better initial estimate.
6.2. Value of Learning Inventory Level inand Comparison to the CF Model
We next evaluate our IL model’s value of learning the on-hand inventory level alone. To do so, we
apply the model with known ρstudied in Section 4. Specifically, we compare our IL model (using
ˆ
Qn) and the CF model (using ˆ
QCF
n) on how close they bring the average inventory level and the
profit to the ideal ones (using ˆ
Qf
n) in the full-visibility (Full) model studied in Section 3.3.
As shown in Figure 6, our IL model is more effective than the CF model: the average order
quantity and inventory level after replenishment in the IL model are much closer to the ideal ones
in the Full model. This is because the CF (customer first) assumption exaggerates how much the
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 25
observed sales represent the customer-demand. For example, low sales may be mis-interpreted as
low customer-demand. This leads to customer-demand under-estimation and hence under-stocking,
which, in turn, will reduce the future sales, forming a vicious cycle.
n
5 10 15 20 25 30
Average Order Quantity
13
14
15
16
17
18
19
20
21
Full
IL
CF
n
5 10 15 20 25 30
Average Inventory Level After Replenishment
17
18
19
20
21
22 Full
IL
CF
(a) Average order quantity over time (b) Average inventory after replenishment over time
Figure 6 Comparison between the Full, IL and CF models (λ= 15,ρ= 0.1,r= 40,c= 30,h= 1,p= 25)
We define the value of inventory-learning in terms of efficiency as VI L
I:= P rof it under IL model
P rof it under F ull model
for our IL model and VCF
I:= P rof it under CF model
P rof it under F ull model for the CF model, where the profit is for the first
Nperiods. As illustrated in Figure 7(a), both VIL
Iand VCF
Idecrease in ρ(shrinkage rate) and N
(duration of learning). Intuitively, as ρincreases, inventory shrinkages increase and inventory level
becomes less visible, hence inventory-learning becomes more difficult and its value decreases. As
Nincreases, the retailer suffers from a longer period of information loss. Since the negative impact
of information loss accumulates over time, the value of inventory-learning decreases.
Figure 7(a) also shows that VIL
I90% for ρ0.15 and VIL
I80% for 0.15 < ρ 0.2. This means
that our model can help seize 80 98% of the ideal profit. The comparison between the IL and
CF models is better shown in Figure 7(b). Note that our model always outperforms the CF model
with more than 108% additional profit in some cases (e.g., the case with ρ= 0.20 and N= 30).
The benefit of switching from the CF to our IL model increases significantly in the duration of
information loss (which is the same as the duration of learning) and the shrinkage rate.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
26 Forthcoming in Production and Operations Management
N
10 15 20 25 30
30%
40%
50%
60%
70%
80%
90%
100%
ρ=0.10, CF/Full
ρ=0.15, CF/Full
ρ=0.20, CF/Full
ρ=0.10, IL/Full
ρ=0.15, IL/Full
ρ=0.20, IL/Full
N
10 15 20 25 30
0%
20%
40%
60%
80%
100%
120%
ρ=0.10
ρ=0.15
ρ=0.20
(a) VIL
I(IL/Full) and VCF
I(CF/Full) (b) (VIL
IVCF
I)/V CF
I
Figure 7 Comparison of VIL
Iand VCF
I(λ= 15,r= 40,c= 30,p= 25,h= 1)
6.3. Value of Learning Both ρand in
We next quantify our IL model’s value of learning both the shrinkage rate (ρ) and on-hand inven-
tory level (in) in two aspects: (i) the overall efficiency Vρ,I := P rof it under our model
P rof it under F ull model and (ii) the
convergence of our estimator E[ρ|~on] to the actual shrinkage rate, n2.
N
1 5 10 15 20
50%
55%
60%
65%
70%
75%
80%
85%
90%
95%
100%
ρ=0.05
ρ=0.08
ρ=0.10
ρ=0.12
n
2 4 6 8 10 12 14 16 18 20
Estimated Shrinkage Rate
0.05
0.1
0.15
0.2
Actual Shrinkage Rate
Estimated shrinkage rate with known inventory level
Estimated shrinkage rate with unknown inventory level
(a) Vρ,I changes over time (b) The estimated ρchanges over time
Figure 8 Our model’s value in learning ρand in(λ= 15,a= 0.40,r= 40,c= 30,p= 25,h= 1)
For (i), as shown in Figure 8(a), the efficiency Vρ,I is generally quite high: Vρ,I [82%,94%] for
N= 20 learning periods and a typical shrinkage rate ρ[0.05,0.12] (for the retail industry). This
means that facing severe information loss, our IL model can still effectively substitute information
visibility to help order the right quantity, improving sales and profit. Note that for any shrinkage
rate, the efficiency always starts at 100% as inventory levels are visible at that time (period 1) and
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 27
hence no learning is needed. We also note that Vρ,I decreases in the shrinkage rate ρand learning
period N. This means that learning is less effective for retailer products with a higher shrinkage
rate or/and a longer period of invisible demand (or period of learning).
For (ii), as shown in Figure 8(b), our estimator of ρconverges quickly: it monotonically reaches
95% of the actual value in only ten periods. Surprisingly, the convergence is as fast as if the inventory
level were visible. Facing severe information loss, our model can still help retailers quickly learn
the actual shrinkage rate to identify an effective loss prevention strategy early, reduce shrinkages
and loss prevention costs, and improve sales and profit.
As demonstrated in our numerical study, our IL model has an excellent performance in substi-
tuting the information loss with Bayesian learning. Apparently, the performance is better when
negative impact of information loss is less, for example, when the duration of information loss is
shorter or the shrinkage rate is smaller. Comparing to the existing models, our IL model outper-
forms the CF model, and the performance improvement significantly increases as the impact of
information loss goes up. This is because the IL model can avoid the vicious cycle triggered by
the under-stocking in the CF model. Therefore, it is vital to model the arrival process and model
the arrival order correctly (as IL). We have also observed that the initial estimated range of the
shrinkage rate (ρ) affects the learning of ρdirectly (e.g., the convergence speed) and the learning
of inventory level indirectly. It is important to set up a good range.
6.4. Importance and Guidance of Initial Estimated Range of ρ
In this subsection, we provide some guidance on how to set the estimated range of the shrinkage
rate (ρ). Recall that in our IL model, when ρis unknown, we estimate its range as ρ[0, a], which
is used (as the support) for not only the prior f1(·), but also all the posteriors fn(·). Intuitively,
it is better to set this range narrower and closer to the actual shrinkage rate, which is however
impractical as we do not know the actual value. On the contrary, we find that it is indeed better to
set this range wide enough. Since 0 is lower bound for any shrinkage rate, we performed numerical
study on the upper bound a. The convergence of the estimated shrinkage rate (to the actual rate
ρ= 0.15) under different values of ais illustrated in Figure 9.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
28 Forthcoming in Production and Operations Management
n
2 4 6 8 10 12 14 16 18 20
Estimated Shrinkage Rate
0.05
0.1
0.15
0.2
0.25
0.3
0.35 a=0.70
a=0.50
a=0.40
a=0.30
a=0.25
a=0.20
a=0.15
a=0.13
Figure 9 The estimated shrinkage rates change over time with different a(ρ= 0.15,λ= 15,r= 40,c= 30,h= 1,
p= 25)
Recall that our initial estimator of ρis a
2, where all the curves start in the above figure. We
analyze the convergence behavior for three ranges of a. First, when ais large (a0.3, i.e., a2ρ),
i.e., the initial estimator a
2ρ), we find that our estimated shrinkage rate quickly converges down
to the actual rate (ρ), and it reaches about 87% of the actual rate within only five periods even if a
is “way off” (e.g., a= 0.7>4.5ρ). This is because although the initial estimator (a
2) deviates largely
from the actual rate, our learning and heuristic order policy can quickly correct the estimation;
and the more the deviation, the faster the correction. Second, when ais small (a(0.15,0.3), i.e.,
a(ρ, 2ρ)), the estimator converges up to the actual rate slowly compared to the case of large a; it
can reach only about 80% of the actual after twenty periods. Third, when ais too small (a0.15,
i.e., aρ), the range [0, a] barely covers the actual ρand hence the estimator never even converges
up to the actual rate.
In summary, it is not only safe, but also better for the retailers to set the range [0, a] wide
enough or alarge enough. In practice, we recommend the retailers to first estimate the shrinkage
rate using the inventory counting data (estimator = inventory shrinkage/ (inventory shrinkage +
sales) during a counting cycle) and then set the parameter agreater than 2-3 times the maximum
estimator across different counting cycles in the past.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 29
7. Conclusions
With today’s advanced information technology, although more and richer sales data have become
available to retailers, lost sales and inventory shrinkages in the physical stores are still prevalent
and invisible, creating challenges in inventory management and loss prevention investment. To
help retailers overcome these challenges, in this paper, we developed a new modeling approach
for learning the missing information from sales data. The previous relevant literature assume a
known shrinkage rate and do not model the demand arrival processes. In contrast, we considered
an unknown shrinkage rate and modeled the customer and theft demand arrivals as a general
interleaving (IL) counting process (e.g., Poisson, Renewal, etc.), which captures the reality more
accurately.
Even though our IL model is more complex, we were able to derive Bayesian formulas to learn
the shrinkage rate and on-hand inventory simultaneously. We also developed a POMDP model
to incorporate such learning, supporting the real-time decision-making on inventory and loss pre-
vention. Surprisingly, the easy-to-calculate heuristic order policies we proposed were numerically
shown quite effective and can be easily incorporated into automated order systems in practice.
We also proved that the CF (customer-first) assumption can result in under-stocking (even with
full information) and under-estimating of on-hand inventory for high-profit products with high
shrinkage rates. As both consequences lead to under-ordering, inventory would deplete quickly and
then sales would drop quickly, sending a false signal of low or no demand to the retailer. Therefore,
using the IL model is critical to inventory management.
Our numerical results showed that our learning model has a promising potential to practice. The
estimated shrinkage rate converged monotonically to 95% of the actual value in only ten periods.
This means that our model can help quickly identify an effective loss prevention strategy, reduce
shrinkages, and improve profit. In addition, our approach helped seize 82% 94% of the ideal profit
for retail products with high shrinkage rates (5% 12%), demonstrating the effectiveness of our
heuristic policies in fighting inventory shrinkage and dealing with severe information efficiency. It
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
30 Forthcoming in Production and Operations Management
also outperforms the CF model with more than 108% additional profit in some cases; the perfor-
mance improvement increases significantly in the duration of information loss (which is the same
as the duration of learning) and the shrinkage rate. We also demonstrated the importance of the
initial estimated range of the unknown shrinkage rate and suggested using a wider range rather
than a narrower one. This seems a bit counter intuitive from learning’s perspective, but a wider
range in our model may lead to a faster learning. In sum, our study contributes to both theory
and practice.
Acknowledgments
The authors thank the Department Editor, the Senior Editor, and the two reviewers for their extremely
helpful comments and suggestions that have significantly improved the study.
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Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 1
Online Supplement: Proofs and Technical Details for “Fight Inventory
Shrinkage: Simultaneous Learning of Inventory Level and Shrinkage
Rate”
Proof of Propositions 1and 2:For the full-visibility model, we start by deriving the following
Bellman equation from the DP.
uf(0) = max
z0
rE
z˜
Dn
X
k=1
Ynk
cz pE
˜
Dnz˜
Dn
X
k=1
Ynk
(hαc)Ez˜
Dn+
+αuf(0)
uf(0) = 1
1αmax
z0r(1 ρ)Ehz˜
Dnicz p(1 ρ)E˜
Dnz+
(hαc)Ez˜
Dn+
uf(0) = 1
1αmax
z0(r(1 ρ)c)zp(1 ρ)E˜
Dz+
(r(1 ρ)+(hαc)Ez˜
D+
| {z }
H(z)
,
where recall that ˜
Dis iid to ˜
D1,˜
D2, . . .. We know that the optimal order-up-to level zfthat
maximizes the above expected profit-to-go is the solution to H0(z) = 0, which is equivalent to (r+
p)(1ρ)c= (r+p)(1ρ)F˜
D(z)+(hαc)F˜
D(z). This implies that that zf=F1
˜
D(r+p)(1ρ)c
(r+p)(1ρ)+hαc
and the optimal order quantity is Qf
n=F1
˜
D(r+p)(1ρ)c
(r+p)(1ρ)+hαc in+
, for all n1. Note that zf=
F1
˜
D(r+p)(1ρ)c
(r+p)(1ρ)+hαc =F1
˜
D1(1α)c+h
(r+p)(1ρ)+hαc ; since (1α)c+h
(r+p)(1ρ)+hαc increases in ρ, we know
that zfdecreases in ρ.
It is not difficult to verify that as the shrinkage rate ρdecreases to 0, zfconverges to zc=
F1
Dr+pc
r+p+hαc . (Note that when ρ= 0, we have D˜
D.) That is, the full-visibility model converges
to the canonical model.
Proof of Propositions 3and 4:Under the CF (customer-first) assumption, the sales quantity
becomes Sn=zDn, where Dn=P˜
Dn
k=1 Ynk. We first derive the following Bellman equation from
the DP.
uf|CF (0) = 1
1αmax
z0rE[zDn]cz pE(Dnz)+(hαc)Ez˜
Dn+
=1
1αmax
z0(rc)zrE(zD)+pE(Dz)+(hαc)Ez˜
D+
| {z }
H(z)
,
where recall that D(˜
D) is iid to D1, D2, . . . (˜
D1,˜
D2, . . .). We know that the optimal order-up-to level
zf|CF is the solution to H0(z) = 0, which is equivalent to r+pc= (r+p)FD(z)+(hαc)F˜
D(z).
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
2Forthcoming in Production and Operations Management
Solving it, we obtain that zf|CF =((r+p)FD+ (hαc)F˜
D)1(r+pc) and the optimal order
quantity is Qf|CF
n= (zf|CF in)+, for all n1. Note that F˜
D(z) is independent of ρ, while FD(z) =
P{P˜
Dn
k=1 Ynk z}is dependent of ρ, where Ynk = 1 and 0 with probability (1 ρ) and ρ, respectively.
When ρincreases, Ynk is more likely to be 0 and hence P˜
Dn
k=1 Ynk zis more likely to happen.
This means that FD(z) increases in ρand therefore zf|CF decreases in ρaccording to its equation
r+pc= (r+p)FD(z)+(hαc)F˜
D(z).
It is not difficult to verify that as the shrinkage rate ρdecreases to 0, FDincreases to F˜
Dand thus
zf|CF converges to zc. (Note that when ρ= 0, we have D˜
D.) That is, under the CF assumption,
the full-visibility model also converges to the canonical model.
Proof of Proposition 5:To compare zf|CF and zf, we rewrite them in a similar formate as
zf|CF =(r+p)¯
FD(hαc)F˜
D1(c) and zf=(r+p)(1 ρ)¯
F˜
D(hαc)F˜
D1(c). This means
that it suffices to compare ¯
FDand (1 ρ)¯
F˜
Dand we analyze their difference ∆(x) := ¯
FD(x)
(1 ρ)¯
F˜
D(x), x0. We can show that as xincreases from 0 to , ∆(x) starts from ρ, drops
below 0, and then goes back up and converges to 0. This is because ∆(0) = ρ, ∆0(x) = fD(x) +
(1 ρ)f˜
D(x) = λx
x!eλ[(1 ρ)(1 ρ)xeλρ]and ∆0(0) <0 and ∆0(x)>0 for all x > 1λρ
ln(1ρ), and
limx→∞ ∆(x) = 0. The fact that ∆(x) will drop below 0 is because
X
x0
∆(x) = X
x0
¯
FD(x)(1 ρ)X
x0
¯
F˜
D(x) = E[D](1 ρ)E[˜
D] = 0.
Note that x0= ∆1(0), i.e., ¯
FD(x0)(1 ρ)¯
F˜
D(x0) = 0. If the critical fractile is small enough
such that zf|CF < x0, since ∆(x)>0 for x < x0, we have zf|CF > zfby simply comparing their
expressions given above. Note that (r+p)(1 ρ)¯
F˜
D(hαc)F˜
D, which determines zf, is a
decreasing function. Otherwise (if zf|CF x0), since ∆(x)0 for xx0, we have zf|CF zf.
Proof of Proposition 6:We start by showing how to derive the expression of bn(i|ρ) as shown
in (12). Following the Bayes’ rule and probability theory, for i[0, Mn], we have
bn(i|ρ) = P{In=i|~on}=
X
j=0
P{In=i, In1=j|~on}
=P
j=0 P{In=i, In1=j, Sn1=sn1|~on1}
P{Sn1=sn1|~on1}
=PMn1
j=(i+sn1Qn1)+P{In=i, Sn1=sn1|In1=j,~on1}bn1(j|ρ)
PMn
k=0 PMn1
j=(k+sn1Qn1)+P{In=k, Sn1=sn1|In1=j, ~on1}bn1(j|ρ),(21)
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 3
where
P{In=i, Sn1=sn1|In1=j,~on1}
=P{Sn1=sn1|In=i, In1=j,~on1}P{In=i|In1=j, ~on1}.(22)
To obtain an expression for bn, we need to write out the two probability terms in (22). For the
first term, the given condition implies that the consumed inventory in the previous period is
(j+Qn1i) units, out of which sn1units are sold to customers and the rest are stolen. This
means the first (j+Qn1i) aggregate arrivals in the previous period consist of sn1customers
and (j+Qn1isn1) thieves, i.e.,
P{Sn1=sn1|In=i, In1=j,~on1}=j+Qn1i
sn1(1 ρ)sn1ρj+Qn1isn1.(23)
For the second term in (22), In=i|In1=jis equivalent to that there is enough aggregate demand
(˜
Dn1) such that inventory drops from (j+Qn1) units to iunits in the previous period. That is,
P{In=i|In1=j,~on1}
| {z }
denoted by F˜
Dn1(j,i)
=P{˜
Dn1j+Qn1}1{i=0}+P{˜
Dn1=j+Qn1i}1{i>0}.(24)
Using (22)-(24), we obtain the expression for bnas shown in (12). Note that for i > 0, P{In=
i, Sn1=sn1|In1=j,~on1}=P{Dn1=sn1}P{Vn1=j+Qn1isn1}, where Vn1=˜
Dn1
Dn1is the theft demand.
As (12) shows, bn(i) is derived for every single realization of Sn1,Sn1=sn1, i.e., bn(i|ρ) is
uniquely defined for every single realization of Sn1. This implies that the transition probablity
P(bn+1|bn, Qn) = P(Sn|bn, Qn).
Proof of Proposition 7:Under the CF-assumption, the sales quantity Snis expressed as Sn=
(In+Qn)Dninstead of Sn=PIn+Qn
k=1 Znk. Using (21), we obtain the distribution of Inas:
bCF
n(i|ρ) = PMn1
j=(i+sn1Qn1)+PCF {In=i, Sn1=sn1|In1=j, ~on1}bCF
n1(j|ρ)
PMn
k=0 PMn1
j=(k+sn1Qn1)+PCF {In=k , Sn1=sn1|In1=j,~on1}bC F
n1(j|ρ),(25)
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
4Forthcoming in Production and Operations Management
where
PCF {In=i, Sn1=sn1|In1=j, ~on1}
=
P{Dn1=sn1}P{Vn1j+Qn1sn1}i= 0, j > sn1Qn1, j 0,
P{Dn1j+Qn1}i= 0, j =sn1Qn10,
P{Dn1=sn1}P{Vn1=j+Qn1sn1i}i > 0.
(26)
Comparing the above equation to (22)-(24), we find that the CF-assumption only affects the first
two cases above. The first case happens if the starting inventory (j+Qn1) is more than enough to
cover the customer-demand (Dn1) and hence the sales equals the customer-demand; the second
case happens otherwise and hence the sales equals the starting inventory level.
For the first case (i= 0, j > sn1Qn1, j 0), we rewrite the two probabilities as:
P{In= 0, Sn1=sn1|In1=j,~on1}=(1 ρ)sn1ρj+Qn1sn1eλ
sn1!
X
k=j+Qn1
λk
k!
(j+Qn1) !
(j+Qn1sn1) !
PCF {In= 0, Sn1=sn1|In1=j, ~on1}=(1 ρ)sn1ρj+Qn1sn1eλ
sn1!
X
k=j+Qn1sn1
λk+sn1
k!ρk(j+Qn1sn1).
Comparing their expressions above in the summation term, we find that iff ρis small enough, we
will have PCF {In= 0, Sn1=sn1|In1=j,~on1}<P{In= 0, Sn1=sn1|In1=j, ~on1}.
Similarly, for the second case (i= 0, j =sn1Qn10), we rewrite the two probabilities as:
P{In= 0, Sn1=sn1|In1=j,~on1}=
X
k=sn1
λkeλ(1 ρ)sn1
k!
PCF {In= 0, Sn1=sn1|In1=j, ~on1}=
X
k=sn1
λkeλ(1 ρ)sn1
k!eλρ(1 ρ)ksn1.
Comparing their expressions above, we find that for any ρ, the difference term (eλρ (1 ρ)ksn1)
decreases from eλρ to 0 as kgoes from sn1to infinity. A smaller ρresults in the difference
term starting at a lower value (eλρ), but decreasing to 0 slower. Therefore, in general the relative
magnitude of these two probabilities cannot be determined by ρand it is possible to have PC F {In=
0, Sn1=sn1|In1=j,~on1}>P{In= 0, Sn1=sn1|In1=j, ~on1}for very small ρ.
Li, Song, Sun, and Zheng: Fight Inventory Shrinkage
Forthcoming in Production and Operations Management 5
Suppose bn1=bCF
n1. When sn1< Qn1, the second case never happens. The above results imply
that that iff ρis small enough, we will have bC F
n(0|ρ)< bn(0|ρ), bCF
n(i|ρ)> bn(i|ρ) for i > 0, and
finally ECF [In|~on]>E[In|~on]. When sn1Qn1, the second case will happen and it is possible to
have ECF [In|~on]<E[In|~on] for very small ρ.
Proof of Proposition 8:As shown by their expressions, (12) and (16)-(17), the belief states
bn(i|ρ) and fn(ρ) are both uniquely derived for every single realization of Sn,Sn=sn. Thus, the
joint belief state βn(i, ρ) = bn(i|ρ)fn(ρ) is uniquely defined for every single realization of Sn. This
implies that P(βn+1|βn, Qn) = P(Sn|βn, Qn), where βn(i, ρ) = bn(i|ρ)fn(ρ) for all iand ρ.
Proof for the IL and CF Heuristic Policies with Known ρ:We first prove that ˆ
Qndefined for
the IL heuristic policy is the optimal order quantity of the equivalent problem (EP) defined in
Section 4.3. Recall that the EP assumes that the visible inventory level in=E[In|~on]. Applying the
infinite-horizon results in Zipkin (2000), we know that the unique optimal order quantity of the
EP, Q
n, satisfies (r+p)(1 ρ)c(1 α) = ((r+p)(1 ρ) + h)F˜
D(E[In|~on] + Q
n). That is, Q
n=
ˆ
Qn(ρ) = F1
˜
D(r+p)(1ρ)c(1α)
(r+p)(1ρ)+hE[In|~on]+
. Similarly, for the CF heuristic policy, ˆ
QCF
n(ρ) is the
optimal order quantity of the EP similarly defined under the CF assumption and assumes the visible
inventory level in=ECF [In|~on]. Following the above analysis, we know that the unique optimal
order quantity of this EP, Q∗∗
n, satisfies r+pc=(r+p)FD+ (hαc)F˜
D(ECF [In|~on] + Q∗∗
n);
that is, Q∗∗
n=ˆ
QCF
n(ρ).
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