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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), K¨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,K¨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,n≥1, 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,n≥1, are independent and identically distributed (iid)

with a cumulative distribution function (cdf) FD(·); the theft demands Vn,n≥1, are also iid with

cdf FV(·), where Dand Vare iid to Dnand Vn,n≥1, respectively. The retailer’s objective is to

choose the order quantity Qn≥0, 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 Vn≡0 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 Qn≥0 and uses his total inventory zn:= in+Qnto ﬁll the customer demand Dn. This

way, the retailer generates sales zn∧Dnat the cost of ordering cQn, penalty p(Dn−zn)+and

holding h(zn−Dn)+, where a∧b= 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 Qn≥0 to maximize his expected proﬁt-to-go uc(in), i.e.,

uc(in) = max

Qn≥0cin+πc(in+Qn) + αEuc((in+Qn−Dn)+), n ≥1,(1)

where α∈(0,1) is the discount factor and

πc(zn) = Er(zn∧Dn)−czn−p(Dn−zn)+−h(zn−Dn)+(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) =

maxz≥0{πc(z) + αcE[(z−D)+]+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:= F−1

D(r+p−c

r+p+h−αc ) and the optimal

expected proﬁt-to-go given zero on-hand inventory is

uc(0) = 1

1−α(r−c)zc−pE[(D−zc)+]−(r+h−αc)E[(zc−D)+].(3)

Thus, the optimal order quantity for period nis

Qc

n= min Q≥0 : FD(in+Q)≥r+p−c

r+p+h−αc=F−1

Dr+p−c

r+p+h−αc−in+

,(4)

Li, Song, Sun, and Zheng: Fight Inventory Shrinkage

Forthcoming in Production and Operations Management 9

where F−1

Dis the inverse function of FD, which also works for discrete distributions with F−1

D(y) :=

min{x≥0 : 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); t≥0}be a counting process6with stationary and independent

increments7. We then have ˜

Dn:= N(n)−N(n−1) = 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,n≥1.

6A counting process is a stochastic process {N(t); t≥0}with values that are non-negative integers (N(t)∈Z+) and

non-decreasing (N(s)≤N(t) for any s≤t).

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, k≥1. 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); t≥0}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 Qn≥0 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(Dn−Sn) and the holding cost h(in+Qn−˜

Dn)+. Note that since

Sn≤Dn, the penalty cost p(Dn−Sn)+=p(Dn−Sn). A similar decision is made at the start of

every future period.

Li, Song, Sun, and Zheng: Fight Inventory Shrinkage

Forthcoming in Production and Operations Management 11

We have the following DP for this full-visibility (f ) model. The retailer decides the order quantity

Qn≥0 to maximize his expected proﬁt-to-go uf(in), i.e.,

uf(in) = max

Qn≥0ncin+πf(in+Qn) + αEhuf((in+Qn−˜

Dn)+)io, n ≥1,(5)

where where α∈(0,1) is the discount factor and

πf(zn) = EhrSn−czn−p(Dn−Sn)−h(zn−˜

Dn)+i(6)

is the retailer’s current-period proﬁt. Here, the sales Sn,n≥1, are iid and so are the customer

demands Dn,n≥1, and the aggregated demands ˜

Dn,n≥1.

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=

zn∧Dn. 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:= F−1

˜

D(r+p)(1−ρ)−c

(r+p)(1−ρ)+h−αc , where recall that ˜

Dis iid to ˜

Dn, and the optimal

expected proﬁt-to-go given zero on-hand inventory is

uf(0) = 1

1−α(r(1 −ρ)−c)zf−p(1 −ρ)E˜

D−zf+

−(r(1 −ρ)+(h−αc)Ezf−˜

D+.(7)

Diﬀerent from the canonical model with zc=F−1

D(r+p−c

r+p+h−αc ) 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:= (zf−in)+,n≥1. 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 ( ˜

Dn≡Dn), 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=zn∧Dn, and the DP for the full-visibility model given in the

previous subsection becomes:

uf|CF (in) = max

Qn≥0ncin+πf|CF (in+Qn) + αEhuf|C F ((in+Qn−˜

Dn)+)io, n ≥1,(8)

where

πf|CF (zn) = Ehr(zn∧Dn)−czn−p(Dn−zn)+−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+ (h−αc)F˜

D)−1(r+p−c)

and the optimal expected proﬁt-to-go given zero on-hand inventory is

uf|CF (0) = 1

1−αnrEzf|CF ∧D−czf|CF −pE(D−zf|C F )+−(h−αc)E(zf|CF −˜

D)+o.(10)

Li, Song, Sun, and Zheng: Fight Inventory Shrinkage

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+,n≥1. 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(r−c) to p= 2(r−c) 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

(r−c)/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

(r−c)/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(r−c) (b) p= 2(r−c)

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-1–sn-1–kn -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 (n−1). Without loss

of generality, we assume the planning horizon starts with no inventory (I1≡0). Leveraging the

relationship between Inand the observed sales of the previous period, sn−1, 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,...,sn−1). 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 In−1and sales quantity Sn−1. 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= (In−1+Qn−1)−(Sn−1+Kn−1).(11)

We ﬁrst identify the range for Inand In−1. Since I1≡0, we know that Inranges from 0 to

Mn:= Pn−1

j=1 Qj−Pn−1

j=1 sj, where the upper limit Mnis reached when no shrinkage occurs in the

past periods and hence all the stock (Pn−1

j=1 Qj) is only consumed by customers (Pn−1

j=1 sj). Similarly,

the range of In−1is [0, Mn−1] which can be reﬁned using the additional information: the observed

sales quantity sn−1. Since the shrinkage quantity Kn−1(= In−1−(In+sn−1−Qn−1)) is always

non-negative, we know that In−1∈[(In+sn−1−Qn−1)+, Mn−1].

After identifying the range for Inand In−1, we can follow the Bayes’ rule and probability theory

to obtain the following expression for bn: for ∀i∈[0, Mn],

bn(i|ρ) = PMn−1

j=(i+sn−1−Qn−1)+j+Qn−1−i

sn−1(1 −ρ)sn−1ρj+Qn−1−i−sn−1F˜

Dn−1(j, i)bn−1(j|ρ)

PMn

k=0 PMn−1

j=(k+sn−1−Qn−1)+j+Qn−1−k

sn−1(1 −ρ)sn−1ρj+Qn−1−k−sn−1F˜

Dn−1(j, k)bn−1(j|ρ),(12)

where F˜

Dn−1(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 sn−1units of inventory in period (n−1) and ends the period with exactly iunits.

To calculate this probability, we need to condition on the retailer’s on-hand inventory of period

(n−1), j. For example, for each given j, we know that in period (n−1), the retailer starts with

(j+Qn−1) units and ends with iunits. This means it costs the retailer (j+Qn−1−i) units of

inventory to generate sn−1units of sales. Therefore, the probability translates to the likelihood

of seeing sn−1customers among the ﬁrst (j+Qn−1−i) store visitors. The denominator in (12)

represents a bigger probability that the retailer sells sn−1units in period (n−1) 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,...,sn−1), the retailer can determine the order quantity Qn≥0 by solving

the following DP on the expected proﬁt-to-go, i.e.,

u(bn) = max

Qn≥0nπn(bn, Qn) + αX

bn+1∈Bn+1

u(bn+1)P(bn+1 |bn, Qn)o, n ≥1,(13)

where

πn(bn, Qn) = EhrSn−cQn−p(Dn−Sn)−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

Qn≥0πn(bn, Qn) + αESn|bn,Qn[u(bn+1)],(15)

where πn(bn, Qn) and bnsatisfy (14) and (12), respectively. Note that bndepends on Qn−1, 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) := zf−ˆ

in(~on)+

, where recall that zf=

F−1

˜

D(r+p)(1−ρ)−c

(r+p)(1−ρ)+h−αc 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+ (h−αc)F˜

D)−1(r+p−c) 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

n−1=bn−1and the aggregated arrival {N(t); t≥0}is a Poisson

process, when sn−1< Qn−1, 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 (n−1) are observed, the estimate will be Bayesian-updated

by the posterior distribution fnusing sn−1and fn−1,n≥2. That is,

fn(ρ) = f(ρ|~on) = P{Sn−1=sn−1|ρ, ~on−1}fn−1(ρ)

Ra

0P{Sn−1=sn−1|ρ,~on−1}fn−1(ρ)dρ

=PMn−1

j=(sn−1−Qn−1)+P{Sn−1=sn−1|In−1=j, ρ}bn−1(j|ρ)fn−1(ρ)

PMn−1

j=(sn−1−Qn−1)+Ra

0P{Sn−1=sn−1|In−1=j, ρ}bn−1(j|x)fn−1(x)

| {z }

βn−1(j,x)

dx,(16)

where bn−1(j|ρ) = P{In−1=j|ρ,~on−1}is the pmf of In−1given by (12) and

P{Sn−1=sn−1|In−1=j, ρ}=

j+Qn−1

X

k=sn−1k

sn−1(1 −ρ)sn−1ρk−sn−1P{˜

Dn−1=k}

+j+Qn−1

sn−1(1 −ρ)sn−1ρj+Qn−1−sn−1P{˜

Dn−1> j +Qn−1},(17)

which follows from the fact that Sn−1=sn−1|In−1=jis equivalent to that the aggregate demand

˜

Dn−1is at least sn−1and out of ˜

Dn−1there are exactly sn−1customers before inventory runs out.

As shown above, we can derive fniteratively using formulae (16)-(17), the prior f1and the sales

quantity sn−1. Note that these formulae of fnalso apply to the case with visible inventory level

(In−1≡in−1), where P{Sn−1=sn−1|In−1=i, ρ}in (16) becomes P{Sn−1=sn−1|In=i, ρ}= 1 for

Li, Song, Sun, and Zheng: Fight Inventory Shrinkage

22 Forthcoming in Production and Operations Management

i=in−1and = 0 for all i6=in−1. 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

Qn−1in a complex way, making the DP (20) intractable. Moreover, fnand hence βn=bnfnis also

uniquely determined by the observed sale quantity sn−1as 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 Qn≥0 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

Qn≥0nπn(βn, Qn) + αX

βn+1∈Bn+1

u(βn+1)P(βn+1 |βn, Qn)o, n ≥1,(18)

where

πn(βn, Qn) = EhrSn−cQn−p(Dn−Sn)−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

Qn≥0πn(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:= F−1

˜

DEρ|~onh(r+p)(1−ρ)−c

(r+p)(1−ρ)+h−αc i−ˆ

ˆ

in(~on)+

, where

Eρ|~onh(r+p)(1−ρ)−c

(r+p)(1−ρ)+h−αc 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 r−c

r≈24.79%), 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

I≥90% for ρ≤0.15 and VIL

I≥80% 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

I−VCF

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, n≥2.

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 (a≥0.3, i.e., a≥2ρ),

i.e., the initial estimator a

2≥ρ), 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 (a≤0.15,

i.e., a≤ρ), 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

z≥0

rE

z∧˜

Dn

X

k=1

Ynk

−cz −pE

˜

Dn−z∧˜

Dn

X

k=1

Ynk

−(h−αc)Ez−˜

Dn+

+αuf(0)

⇔uf(0) = 1

1−αmax

z≥0r(1 −ρ)Ehz∧˜

Dni−cz −p(1 −ρ)E˜

Dn−z+

−(h−αc)Ez−˜

Dn+

⇔uf(0) = 1

1−αmax

z≥0(r(1 −ρ)−c)z−p(1 −ρ)E˜

D−z+

−(r(1 −ρ)+(h−αc)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)(1−ρ)−c= (r+p)(1−ρ)F˜

D(z)+(h−αc)F˜

D(z). This implies that that zf=F−1

˜

D(r+p)(1−ρ)−c

(r+p)(1−ρ)+h−αc

and the optimal order quantity is Qf

n=F−1

˜

D(r+p)(1−ρ)−c

(r+p)(1−ρ)+h−αc −in+

, for all n≥1. Note that zf=

F−1

˜

D(r+p)(1−ρ)−c

(r+p)(1−ρ)+h−αc =F−1

˜

D1−(1−α)c+h

(r+p)(1−ρ)+h−αc ; since (1−α)c+h

(r+p)(1−ρ)+h−αc 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=

F−1

Dr+p−c

r+p+h−αc . (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=z∧Dn, where Dn=P˜

Dn

k=1 Ynk. We ﬁrst derive the following Bellman equation from

the DP.

uf|CF (0) = 1

1−αmax

z≥0rE[z∧Dn]−cz −pE(Dn−z)+−(h−αc)Ez−˜

Dn+

=1

1−αmax

z≥0(r−c)z−rE(z−D)+−pE(D−z)+−(h−αc)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+p−c= (r+p)FD(z)+(h−αc)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+ (h−αc)F˜

D)−1(r+p−c) and the optimal order

quantity is Qf|CF

n= (zf|CF −in)+, for all n≥1. 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+p−c= (r+p)FD(z)+(h−αc)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−(h−αc)F˜

D−1(c) and zf=(r+p)(1 −ρ)¯

F˜

D−(h−αc)F˜

D−1(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), x≥0. 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!e−λ[(1 −ρ)−(1 −ρ)xeλρ]and ∆0(0) <0 and ∆0(x)>0 for all x > 1−λρ

ln(1−ρ), and

limx→∞ ∆(x) = 0. The fact that ∆(x) will drop below 0 is because

X

x≥0

∆(x) = X

x≥0

¯

FD(x)−(1 −ρ)X

x≥0

¯

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−(h−αc)F˜

D, which determines zf, is a

decreasing function. Otherwise (if zf|CF ≥x0), since ∆(x)≤0 for ∀x≥x0, 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, In−1=j|~on}

=P∞

j=0 P{In=i, In−1=j, Sn−1=sn−1|~on−1}

P{Sn−1=sn−1|~on−1}

=PMn−1

j=(i+sn−1−Qn−1)+P{In=i, Sn−1=sn−1|In−1=j,~on−1}bn−1(j|ρ)

PMn

k=0 PMn−1

j=(k+sn−1−Qn−1)+P{In=k, Sn−1=sn−1|In−1=j, ~on−1}bn−1(j|ρ),(21)

Li, Song, Sun, and Zheng: Fight Inventory Shrinkage

Forthcoming in Production and Operations Management 3

where

P{In=i, Sn−1=sn−1|In−1=j,~on−1}

=P{Sn−1=sn−1|In=i, In−1=j,~on−1}P{In=i|In−1=j, ~on−1}.(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+Qn−1−i) units, out of which sn−1units are sold to customers and the rest are stolen. This

means the ﬁrst (j+Qn−1−i) aggregate arrivals in the previous period consist of sn−1customers

and (j+Qn−1−i−sn−1) thieves, i.e.,

P{Sn−1=sn−1|In=i, In−1=j,~on−1}=j+Qn−1−i

sn−1(1 −ρ)sn−1ρj+Qn−1−i−sn−1.(23)

For the second term in (22), In=i|In−1=jis equivalent to that there is enough aggregate demand

(˜

Dn−1) such that inventory drops from (j+Qn−1) units to iunits in the previous period. That is,

P{In=i|In−1=j,~on−1}

| {z }

denoted by F˜

Dn−1(j,i)

=P{˜

Dn−1≥j+Qn−1}1{i=0}+P{˜

Dn−1=j+Qn−1−i}1{i>0}.(24)

Using (22)-(24), we obtain the expression for bnas shown in (12). Note that for i > 0, P{In=

i, Sn−1=sn−1|In−1=j,~on−1}=P{Dn−1=sn−1}P{Vn−1=j+Qn−1−i−sn−1}, where Vn−1=˜

Dn−1−

Dn−1is the theft demand.

As (12) shows, bn(i) is derived for every single realization of Sn−1,Sn−1=sn−1, i.e., bn(i|ρ) is

uniquely deﬁned for every single realization of Sn−1. 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) = PMn−1

j=(i+sn−1−Qn−1)+PCF {In=i, Sn−1=sn−1|In−1=j, ~on−1}bCF

n−1(j|ρ)

PMn

k=0 PMn−1

j=(k+sn−1−Qn−1)+PCF {In=k , Sn−1=sn−1|In−1=j,~on−1}bC F

n−1(j|ρ),(25)

Li, Song, Sun, and Zheng: Fight Inventory Shrinkage

4Forthcoming in Production and Operations Management

where

PCF {In=i, Sn−1=sn−1|In−1=j, ~on−1}

=

P{Dn−1=sn−1}P{Vn−1≥j+Qn−1−sn−1}i= 0, j > sn−1−Qn−1, j ≥0,

P{Dn−1≥j+Qn−1}i= 0, j =sn−1−Qn−1≥0,

P{Dn−1=sn−1}P{Vn−1=j+Qn−1−sn−1−i}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+Qn−1) is more than enough to

cover the customer-demand (Dn−1) 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 > sn−1−Qn−1, j ≥0), we rewrite the two probabilities as:

P{In= 0, Sn−1=sn−1|In−1=j,~on−1}=(1 −ρ)sn−1ρj+Qn−1−sn−1e−λ

sn−1!

∞

X

k=j+Qn−1

λk

k!

(j+Qn−1) !

(j+Qn−1−sn−1) !

PCF {In= 0, Sn−1=sn−1|In−1=j, ~on−1}=(1 −ρ)sn−1ρj+Qn−1−sn−1e−λ

sn−1!

∞

X

k=j+Qn−1−sn−1

λk+sn−1

k!ρk−(j+Qn−1−sn−1).

Comparing their expressions above in the summation term, we ﬁnd that iﬀ ρis small enough, we

will have PCF {In= 0, Sn−1=sn−1|In−1=j,~on−1}<P{In= 0, Sn−1=sn−1|In−1=j, ~on−1}.

Similarly, for the second case (i= 0, j =sn−1−Qn−1≥0), we rewrite the two probabilities as:

P{In= 0, Sn−1=sn−1|In−1=j,~on−1}=

∞

X

k=sn−1

λke−λ(1 −ρ)sn−1

k!

PCF {In= 0, Sn−1=sn−1|In−1=j, ~on−1}=

∞

X

k=sn−1

λke−λ(1 −ρ)sn−1

k!eλρ(1 −ρ)k−sn−1.

Comparing their expressions above, we ﬁnd that for any ρ, the diﬀerence term (eλρ (1 −ρ)k−sn−1)

decreases from eλρ to 0 as kgoes from sn−1to 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, Sn−1=sn−1|In−1=j,~on−1}>P{In= 0, Sn−1=sn−1|In−1=j, ~on−1}for very small ρ.

Li, Song, Sun, and Zheng: Fight Inventory Shrinkage

Forthcoming in Production and Operations Management 5

Suppose bn−1=bCF

n−1. When sn−1< Qn−1, 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 sn−1≥Qn−1, 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) = F−1

˜

D(r+p)(1−ρ)−c(1−α)

(r+p)(1−ρ)+h−E[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+p−c=(r+p)FD+ (h−αc)F˜

D(ECF [In|~on] + Q∗∗

n);

that is, Q∗∗

n=ˆ

QCF

n(ρ).