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Risk and Decision Analysis 5 (2014) 189–210 DOI 10.3233/RDA-150110 IOS Press

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Inventory management with overlapping shrinkages and demands Alain Bensoussan a,b , Metin Çakanyıldırım a , Meng Li c,∗ and Suresh P. Sethi a a School

of Management, University of Texas at Dallas, Richardson, TX, USA of Engineering, The City University of Hong Kong, Hong Kong c College of Business, University of Illinois at Urbana-Champaign, Champaign, IL, USA b School

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Abstract. This article analyzes a discrete time lost sales inventory system with partially observed demand and unobserved shrinkages which happen both before and after the demand realization. When the demand exceeds the remaining inventory, the unmet demand is lost and unobserved. This problem in general has a nonlinear state evolution, and we use unmoralized probability to linearize the system. Despite this, the problem still has a complex-dimensional state space, and we therefore focus on two special cases where the shrinkage either happens before or after the demand realization. With dynamic programming and unnormalized probability, we formulate the Bellman equation. We obtain a lower bound on the cost analytically via the formulation of a fictitious inventory problem. We also develop an iterative algorithm, and compare its solution to a myopic solution as well as the lower bound. This comparison reveals that the solution obtained using the iterative algorithm performs significantly better than the myopic solution in many cases and, moreover, the achieved cost is close to the lower bound, thereby highlighting the value for our algorithm.

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Keywords: Inventory inaccuracy, shrinkage

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1. Introduction

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In inventory management, a point of sale (or a similar device) can always provide the recorded inventory level. When this level differs from the actual inventory level that is physically in the store, the inventory record is inaccurate. Reasons for inventory inaccuracy are spoilage, damage, pilferage or process failure/error, which are commonly referred to as shrinkage [1,2]. For example, perishable products deteriorate over time and become unfit for sale. Products can also be damaged by customer handling. Another example is pilferage, which can be committed by both customers (external) and employees (internal) by stealing small amounts over an extended period of time. At different retailer chain stores reported by Kang and Gershwin [20] and DeHoratius and Raman [12], 51% and 65% of StockKeeping Units (SKUs) have inaccuracy. The chain in the latter paper has 37 stores and uses advanced technologies but has some items whose inventory level inaccuracy is 50% or more of the average on-hand quan* Address for correspondence: Meng Li, College of Business, University of Illinois at Urbana-Champaign, Champaign, IL 61820, USA. E-mail: [email protected].

tity. Beck et al. [2] find that the shrinkage rates for fast moving consumer goods is 1.75% of sales (e258 million a week) for European retailers; these rates increased from 2.32% of sales in 2002 to 2.77% in 2005 according to Miller and Allen [24]. Hollinger [17] reports that the US retail losses from inventory shrinkage is 1.52% of sales ($702 million a week). Inventory shrinkage is an important factor [14,15,20] and hence deserves a rigorous treatment. This paper studies the inventory control problem on a single item with inventory inaccuracy. In a typical sales transaction for a specific item, a customer comes to a store and buys the quantity equal to his demand if the inventory is sufficient. The store clerk scans (bar codes of) purchased items, which enables the IM to record the transaction. If the inventory is insufficient, the customer may buy the entire inventory, and leave the store with a portion of his demand unmet. The IM updates the inventory record by deducting the quantity sold from the inventory level. The same (or a similar) system also processes order receipts from suppliers by increasing the inventory level. As in the current literature [5,6,10,11,16,20,23], we adopt a discretetime framework in which the inventory levels can be changed at the instants of discrete time in our model.

1569-7371/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved

A. Bensoussan et al. / Inventory management with overlapping shrinkages and demands

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performance of the myopic policy and the iterative algorithm are compared with each other and the lower bound provided by the fictitious inventory model under various parameter settings. These comparisons reveal that the iterative algorithm always beats the myopic policy, especially when the discount factor or the penalty cost are large. When the magnitude of shrinkage is small with respect to the demand, the performance of the iterative algorithm is not only better than the myopic policy, but also close to the lower bound. Our paper does not assume the independence of the inventory inaccuracy build up over periods and the inventory levels in periods, and maintains a distribution of the inventory level in a Bayesian fashion. Our paper contributes to the literature in a number of ways. We study multi-period (infinite horizon) models instead of a single period model and simplify them by introducing unnormalized probability. Shrinkage can happen both before and after demand realization in a period in our general model, whose special cases are later presented as demand-first and demand-last models. We present a fictitious inventory system which provides an easy-to-compute lower bound and a novel iterative approach to numerically obtain an upper bound for the model. The remainder of this paper is organized as follows. We introduce the our model in Section 2. We obtain the conditional probability of inventory in Section 3. In Section 4, we present the Bellman equation. In Section 5, we study the demand-first and demandlast models. We further develop a fictitious model to obtain a lower bound for the optimal cost of both models. In Section 6, we approximate the optimal cost by an iterative algorithm. Section 7 concludes this paper. All proofs are relegated to the Appendix.

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Note that, in reality, multiple demands and shrinkages could happen during a period. Thus, in contrast to the current literature assuming that the inventory shrinkage happens only once in each period, inventory shrinkage can occur both before and after the lump demand in our model. In our lost sales model, sales rather than demands are observed [8,13,18], and we show that the observation process in our model leads to a particular mix of discrete and continuous distributions that represents the conditional probability law of the inventory. This mixture is not trivial and is completely novel, and the analysis of the system involving such a mixture is more complicated. Since our model has a complex inventory evolution, the Bellman equation is difficult to solve. In particular, it has a nonlinear system evolution and an infinitedimensional state space and is hence intractable. We use the unnormalized probability to linearize the system evolution. However, the system’s state space is still infinite and there is no sufficient way to reduce the problem to finite-dimensional problem as in the related inventory literature [25]. Thus, we study the special cases by noting that companies can often determine the times of receipts from a supplier and shipments to customers, and can close the time gap between them. This significantly reduces the portion of the shrinkage before demand and leads to the demand-first model, where entire spoilage of a period takes place after the demand of that period. For completeness, we also study the demand-last model, which has the same inventory dynamics as the demand-first model. Despite this, the evolution of probability law of inventory is interestingly more complicated in the demand-last model compared to the demand-first model. We design a fictitious inventory model that provides a theoretical lower bound for the optimal cost of the demand-first model. The demand in this fictitious model equals to the sum of the demand and the shrinkage in the demand-first model. The shrinkage is zero in the fictitious model. The fictitious system records the sales in the same way as in the demand-first model, but the sales have different values. Given the sales in the fictitious model, we cannot recover the sales in the demand-first model or vice versa. So, neither model has more information than the other. The state of the dynamic program is a probability distribution and the analysis must be carried out in the infinite-dimensional space of probability distributions. To overcome the curse of dimensionality, we develop an implementable iterative algorithm, and characterize some features of the optimal feedback policy. The

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2. The model Our model is a single-item, lost-sales, periodicreview inventory model in which sales are recorded, but the unmet demands and shrinkages are not observed. Such a situation occurs commonly in retail stores where customers finding an empty shelf (a stockout) simply leave without revealing their demands. If there is inventory, an unobserved shrinkage can occur and cause the actual inventory to be different from the recorded inventory. Although IM does not observe the exact actual inventory, he can infer it from the orders, sales and the fact that the shrinkage cannot exceed actual inventory.


Inventory carried over Lost sales penalized



Construct πt

Construct πt+1

Order qt−1

Order qt

Order qt+1

Zt •

Zt+1 •

Dt−1 materializes z observed

f t St−1 materializes

but unobserved but unobserved Period t − 1

Zt+2 •

Dt materializes observed z f t+1 St materializes

l St−1 materializes

Unobserved actual inventory It−1 Observed recorded inventory Rt−1

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but unobserved -

Stl materializes

but unobserved

Period t

Unobserved actual inventory It Observed recorded inventory Rt

-

Dt+1 materializes observed z

f t+2 St+1 materializes

- but unobserved

Unobserved actual inventory It+1 Observed recorded inventory Rt+1

Fig. 1. The sequence of events.

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(1) Period t starts with actual and recorded inventories It and Rt , and the IM places the order qt that arrives instantaneously. f (2) First-shrinkage St reduces the actual inventory by min{It + qt , Stf }. (3) Demand Dt occurs and leads to sales zt+1 = min{Dt , It +qt −min{It +qt −Stf }}. The sales are recorded by the system, and the recorded inventory is reduced by the amount of the sales. (4) Last-shrinkage Stl reduces the actual inventory f by min{Stl , It + qt − min{It + qt − St } − zt+1 }. f

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We assume Dt (resp. St and Stl ), t = 1, 2, . . . , are independent and identically distributed (i.i.d.) and the sequences {Dt }, {Stf } and {Stl } are independent. F , Gf and Gl denote the cumulative distributions of the demand, the first-shrinkage, and the last-shrinkage, ¯ f (x) = and F¯ (x) = 1 − F (x) = P(Dt x), G f ¯ l (x) = 1 − Gl (x) = 1 − Gf (x) = P(S x) and G t

P(Stl x) denote the complementary cumulative distributions, f , gf and gl denote the corresponding density functions. Note that while inventory shrinkage generally results in the actual inventory being less than the recorded inventory, the companies can also experience a negative shrinkage. For example, process failure which includes inventory misplacement and transaction error may lead to a higher actual inventory than the recorded inventory. We therefore allow for negative shrinkage.

The actual inventory level in period t is It + qt immediately after the order is delivered; It + qt − min{Stf , It + qt } immediately after the first-shrinkage; It +qt −min{Stf , It +qt }−zt+1 after the demand; and It + qt − min{Stf , It + qt } − zt+1 − min{Stl , It + qt − min{Stf , It +qt }−zt+1 } after the last-shrinkage. Combining these, we obtain It+1 = It + qt − min{Stf , It + qt }−zt+1 −min{Stl , It +qt −min{Stf , It +qt }−zt+1 }. This equation can be rewritten as follows using the notation x+ = max{0, x}:

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We use the following notations for period t: It is the beginning actual inventory, Rt is the beginning recorded inventory, qt is the amount ordered, Dt is the demand, and the shrinkage includes the first-shrinkage Stf and the last-shrinkage Stl . Shown in Fig. 1, the sequence of events in period t is as follows:

It+1 =

+ f + It + qt − St − zt+1 − Stl .

(1)

The recorded inventory evolves as Rt+1 = Rt + qt − zt+1 and differs from the actual inventory because of the shrinkage. In this paper, both orders and costs are based on the actual inventory (or its distribution to be precise) rather than the recorded inventory. So only the actual inventory evolution (1) is considered and inventory without a qualifier refers to the actual inventory. Based on prior observations, the IM considers the conditional distribution of the (actual) inventory. He observes only the sales zt+1 = min{(It + qt − f f St )+ , Dt } for t 1. If Dt < (It + qt − St )+ , then zt+1 is the demand; otherwise, it is the inventory level after ordering and the first-shrinkage. The information available at the beginning of period t 2 is denoted by the σ-algebra Zt := σ(z2 , z3 , . . . , zt ) generated by the sales. We set Z1 to be the minimum σ-algebra {∅, Ω}. The order quantity qt is adapted to Zt . When such order quantities make up a policy q˜ = {qt , t 1}, the policy is said to be adapted to the filtration Zt , or simply admissible. When the demand is met entirely, an inventory holding cost is incurred on the remaining inventory. Other-


Theorem 1. With πt and the corresponding At and Bt (·) that satisfy (3), let

wise, there are lost sales resulting in a shortage cost. Inventory holding and shortage costs are captured by the cost function c(It , qt ) that depends on the inventory level It and the order size qt . Without the exact knowledge of the inventory, the IM evaluates this cost as an expected value based on the inventory distribution. That is, the expected cost in period t 1 is αt−1 E[c(It , qt )|Zt ], where α ∈ (0, 1) is the discount factor. Given an admissible policy q˜, an initial probability law π(·) on the initial inventory level, the total expected discounted cost can be written as J(π, q˜) := t−1 c(I , q ) = E ∞ αt−1 E[c(I , q )|Z ]. α E ∞ t t t t t t=1 t=1 The optimal cost function V (π) is given by V (π) = inf J(π, q˜),

At+1 ¯ l (0) = 1zt+1 =0 1At G¯ f (qt )+ G¯ f (λ+qt )Bt (λ) dλ=0 G + (1zt+1 >0 + 1zt+1 =0 1At G¯ f (q)+ G¯ f (λ+q)Bt (λ) dλ=0 ) ×

Bt+1 (η) = 1zt+1 =0 1At G¯ f (qt )+ G¯ f (λ+qt )Bt (λ) dλ=0 gl (−η)

and we need to show that there exists a policy q˜∗ such that J(π, q˜∗ ) = V (π), and we would like to obtain it, at least numerically. It is worth pointing out that the state π in the optimal cost function is the probability law.

+ (1zt+1 >0

+ 1zt+1 =0 1At G¯ f (q)+ G¯ f (λ+q)Bt (λ) dλ=0 )

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Starting the first period with a prior probability law π, the IM can derive the inventory probability for use in his decision process in period t 2. Let πt (·) be the conditional probability law of It given Zt , which can be expressed as πt (Γ) = E 1Γ (It )|Zt = At 1Γ (0) + Bt (η)1Γ (η) dη (3)

where At is a probability mass at zero in the distribution of It , Bt (·) is a probability measure for positive It values, and 1Γ is the indicator function of a subset Γ of [0, ∞), which is defined as 1Γ (x) = 1, if x ∈ Γ; otherwise, 1Γ (x) = 0. It suffices to consider only 0 At 1 and Bt (η) = 0 for η 0. As a convention, any integral without the limit is understood to be from 0 to ∞. Since πt is a probability law, we must have At + Bt (η) dη = 1. Using the Dirac measure δ0 (·), which takes the value of zero everywhere except zero and δ0 (η) dη = 1, we can formally write πt (dη) = At δ0 (η) + Bt (η) dη. As we can see below, πt+1 can be written also in the same form by using At+1 and Bt+1 . This validates (3) and points out that πt is sufficient for inferring It as it captures all of the sales observations in the previous periods. Then, πt can be viewed as the system state.

(5)

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3. Bayesian updating of the inventory level

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γ(η; qt , zt+1 ; At , Bt (·)) β(qt , zt+1 ; At , Bt (·))

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for Γ ⊆ [0, ∞),

(4)

and

(2)

q˜

α(qt , zt+1 ; At , Bt (·)) β(qt , zt+1 ; At , Bt (·))

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for η > 0, where α qt , zt+1 ; At , Bt (·) ¯ l (0)F¯ (zt+1 ) At gf (qt − zt+1 ) =G

gf (λ + qt − zt+1 )Bt (λ) dλ

+

qt −zt+1 gf (v) + f (zt+1 ) At −∞

¯ l (qt − zt+1 − v) dv ×G λ+qt −zt+1 + gf (v) −∞

¯ l (λ + qt − zt+1 − v) dvBt (λ) dλ , ×G

(6)

β qt , zt+1 ; At , Bt (·) = F¯ (zt+1 ) At gf (qt − zt+1 )


+

+ f (zt+1 ) At Gf (qt − zt+1 )

+

Gf (λ + qt − zt+1 )Bt (λ) dλ

(7)


vides a linear recursion of the probability law. The Zakai equation uses unnormalized probability, which is obtained from the likelihood functions and the initial prior probability law (without performing the normalization step). Analogous to (3), the unnormalized probability ρt (·) can be expressed as ρt (Γ) = φt 1Γ (0) + ψt (η)1Γ (η) dη by using the number φt and function ψt that are obtained recursively as follows:

and γ η; qt , zt+1 ; At , Bt (·) ¯ = F (zt+1 )gl (−η) At gf (qt − zt+1 )


qt −zt+1 + f (zt+1 ) At gf (v)

¯ l (0) φt+1 = 1zt+1 =0 1φt G¯ f (q)+ G¯ f (λ+q)ψt (λ) dλ=0 G

−∞

+ (1zt+1 >0

× gl (qt − zt+1 − v − η) dv λ+qt −zt+1 gf (v) +

+ 1zt+1 =0 1φt G¯ f (q)+ G¯ f (λ+q)ψt (λ) dλ=0 ) (9) × α qt , zt+1 ; φt , ψt (·)

−∞

× gl (λ + qt − zt+1 − v − η) dvBt (λ) dλ .

and

(8)

ψt+1 (η) = 1zt+1 =0 1φt G¯ f (q)+ G¯ f (λ+q)ψt (dλ)=0 gl (−η)

Then, πt+1 (·) can be written as in (3) with At+1 and Bt+1 (·) in (4)–(8).

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In the first terms on the right-hand sides of (4) and (5), we have the event [zt+1 = 0] that hapf pens when [St It + qt ] or [Dt = 0]. The event f ¯ [S t It + qt ] happens with probability At Gf (qt ) + ¯ f (λ + qt )Bt (λ) dλ. Because the demand has a denG sity, the event [Dt = 0] happens with zero probabilf ity. Therefore, when [zt+1 = 0], the event [St ¯ f (λ+ ¯ f (qt )+ G It +qt ] happens almost surely as At G

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qt )Bt (λ) dλ = 0. Then (It + qt − St )+ − zt+1 = 0, which implies It+1 = (−Stl )+ . That is, the event ¯ l (0). When [It+1 = 0] happens with probability G l It+1 > 0, we have It+1 = −St , which in turn implies Bt+1 (η) = gl (−η). The other terms associated with ¯ f (λ + ¯ f (q) + G events [zt+1 > 0] and [zt+1 = At G q)Bt (λ) dλ = 0] on the right-hand sides of (4) and (5) f correspond to the event [St < It + qt ]. Remark 1. The shrinkage in each period could depend on the sales and inventory of that period, e.g., when it is mostly due to damaged items and pilferage by the customers. This is because a high sales is a good indicator of high store traffic that causes more shrinkage. When the shrinkages depend on the sales z, then F , Gf and Gl in Theorem 1 should be replaced with Fz , Gf ,z and Gl,z , respectively. From Theorem 1, πt evolves according to a highly nonlinear equation akin to the Kushner equation [21] in our inventory context. The Zakai equation [29] pro-

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+ 1zt+1 =0 1φt G¯ f (q)+ G¯ f (λ+q)Bt (λ) dλ=0 ) (10) × γ η; qt , zt+1 ; φt , ψt (·) ,

starting initially with φ1 = A1 and ψ1 (·) = B1 (·). More importantly, as justified in the Appendix, πt (x) =

ρ (x) t . φt + ψt (x) dx

(11)

Since πt integrates to 1 while ρt does not, ρt is called the unnormalized probability. Equations (9) and (10) are linear in (φt , ψt (·)), unlike the nonlinear Eqs (4) and (5). Moreover, there is a one-to-one correspondence between the unnormalized probability ρt = (φt , ψt (·)) and the probability πt = (At , Bt (·)); see (11). We mainly use the unnormalized probability in the remainder.

4. The Bellman equation We now proceed to derive the dynamic programming equation for the problem with the initial probability law π1 (·) = π(·) and the associated (A, B(·)) satisfying (3). Our derivation is for the infinite horizon problem so the time index t does not encumber the notation. In the current period, the inventory level is characterized by (A, B(·)), and hence the cost incurred in the current period is Ac(0, q) + B(η)c(η, q) dη.

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When the sales is zero, which happens with probabil¯ f (x + q)B(x) dx, the next period’s ¯ f (q) + G ity AG ¯ l (0) and has a deninventory is zero with probability G sity gl (·) for positive inventory. Inthis case, the cost-to¯ f (q)+ G ¯ f (x+q)B(x) dx]. ¯ l (0), gl (·)))[AG go is V ((G When the sales is positive, the inventory probability law needs to be evolved using the operators α/β and γ/β for zero inventory and positive inventory as in (4) and (5), respectively. As formalized in the Appendix, these lead to the Bellman equation V

+α

W

¯ f (x + q)ψ(x) dx G

α q, z; φ, ψ(·) ,

γ ·; q, z; φ, ψ(·) dz .

(14)

¯ l (0), gl (·) + αV G

¯ f (x + q)B(x) dx ¯ f (q) + G × AG

A quick glance reveals that (14) is shorter than (12). More importantly, the update of ρ in (14) is linear, while the update of π in (12) is nonlinear. That is, (14) does not have a denominator that involves the control variable q. The linearity of (14) greatly facilitates our analysis. To calculate the optimal policy for our system, we only need to find W . From (13), W exactly matches V when theargument ρ = (φ, ψ) is a probability law, i.e., φ + ψ(x) dx = 1. Furthermore, the next theorem validates the unnormalized probability approach.

α(q, z; A, B(·)) γ(·; q, z; A, B(·)) , +α V β(q, z; A, B(·)) β(q, z; A, B(·)) × β q, z; A, B(·) dz . (12)

Theorem 2. If V and W and the optimal orders exist to satisfy (12) and (14), the orders are the same when A = φ/[φ + ψ(x) dx] and B(·) = ψ(·)/[φ + ψ(x) dx].

A, B(·) = inf Ac(0, q) + B(η)c(η, q) dη

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Although there are three events in the inventory evolution equations (4) and (5), the event with [zt+1 = ¯ f (λ + q)Bt (λ) dλ = 0] happens with ¯ f (qt ) + G At G probability zero and does not play a role in the Bellman equation. We therefore can ignore this event in the remainder. The presence of nonlinear operators α/β and γ/β makes a direct study of (12) difficult. If we use the unnormalized probability ρ1 = ρ and the associated (φ, ψ(·)) as the system state instead of π, then the analysis becomes easier, because ρ evolves linearly. To make the ideas concrete, we define

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¯ f (q) + × φG

ρ, ψ(·)

:= φ + ψ(x) dx V

ρ . φ + ψ(x) dx

(13)

Using this definition, the Bellman equation (12), and some algebra, we can show that W satisfies the Bellman equation (see the proof of Theorem 2 in the Appendix)

W (ρ) = inf φc(0, q) + q

+ αW

ψ(η)c(η, q) dη

¯ l (0), gl (·) G

5. Demand-first and demand-last models Companies can sometimes determine the selling time of their products to customers. For example, a food (vegetables, fruit or bakery) wholesaler can sell (ship) food to grocery stores at the starting time of a period. If this transaction happens immediately after receiving food orders from suppliers, there cannot be much spoilage S f between the order receipt and the demand fulfillment in the same day (period), i.e., S f = 0. When demand fulfillment happens after the order receipt in a time period, then S l = 0, since there is virtually no time left in a period after the demand fulfillment. The case of S f = 0 and S l = 0, respectively, lead to the demand-first and the demand-last models presented below. Compared to the demand-last model, receipt of supplier shipments is better synchronized with shipments to retailers in the demand-first model. Companies favoring just-in-time philosophy and its implementation with cross-docking are more likely to adopt the demand-first model. This explains why the demand-first model is more common and dominates the current inventory literature [11,19]. In addition to being of interest on their own, both models can be used to approximate the general model, whose Bellman equation (14) is complicated.


Retailers typically observe that the actual inventory level is below the recorded inventory level over a certain period of time as discussed in Section 1. That is, the positive shrinkage generally dominates the negative shrinkage in the industry, and the inventory shrinkage in a period thus can be assumed to be nonnegative. Because there is only one positive shrinkage in the demand-first model or the demand-last model, we ¯ to denote the shrinkage’s can use g(·), G(·) and G(·) distribution in the remainder. 5.1. The demand-first model f

Bt (λ) dλ

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zt+1 −qt

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and Bt+1 (η)

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∞ + f (zt+1 )

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zt+1 −qt

(15)

= 1zt+1 qt F¯ (zt+1 )ψt (zt+1 − qt ) + f (zt+1 )

∞ ¯ ψt (λ)G(λ + qt − zt+1 ) dλ ×

∞

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+ f (zt+1 )

∞ ¯ × Bt (λ)G(λ + qt − zt+1 ) dλ F¯ (zt+1 )Bt (zt+1 − qt )

In the first term on the right-hand side of (15), we have the event [zt+1 = qt ] that happens when [It +qt = qt ]. The event [zt+1 = qt ] happens when [It + qt = qt ] or [Dt = qt ]. Because the demand has a density, the event [Dt = qt ] happens with zero probability. Therefore, when [zt+1 = qt ], the event [It = 0] must happen with a positive probability in the prior distribution, i.e., At = 0. Starting with It = 0 and using zt+1 = qt , we obtain It+1 = 0. That is, when [It = 0, zt+1 = qt ], the event [It+1 = 0] happens almost surely, and this in turn implies At+1 = 1. The other terms (indicated by 1zt+1 qt ) on the right-hand side of (15) and (16), respectively, correspond to the events [Dt < qt ] and [Dt > qt ]. Since At = 1 − Bt (x) dx, we can concentrate on the Bt evolution in (16). The evolution of Bt+1 from Bt is depicted in Fig. 2. For an illustrative example, we set the inventory distribution πt equal to a censored normal distribution obtained from a normal distribution with mean 3 √ and standard deviation 2. Then At = 1/2 − erf(3/(2 2))/2 0.07 and √ √ 2 Bt (x) = e−(x−3) /8 /{ 2π[1 + erf(3/(2 2))]}, where √ 2 erf(x) = (2/ π) 0x e−u du. To sketch the distributions in Fig. 2, we assume that qt = 2. The figure on the top-right panel ∞ is Bt+1 (η) = 1η q]; see (19). When z < q, the current demand must equal z, and we obtain the cost-to-go as 0q f (z)V (Tq−z π) dz. When z > q, there are two relevant events ∞ whose total probability is ˜ z−q,z F¯ (z)B(z − q) + f (z) z−q B(η) dη, and then Θ applies, and V (π) = V A, B(·) = inf Ac(0, q) + B(η)c(η, q) dη

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+ 1zt+1 =0 1At G¯ (q)+ G¯ (λ+qt )Bt (λ) dλ=0 )

×

γ(η; qt , zt+1 ; At , Bt (·)) , β(qt , zt+1 ; At , Bt (·))

(25)

where α, β and γ are operators in (6)–(8) specialized for the demand-last model. The inventory evolution in the demand-last model is much more complicated than in the demand-first model. The system evolution (15) of the demand-first model does not include the terms related to the event [zt+1 = 0]. Moreover, At+1 and Bt+1 are linear in At and Bt in the demand-first model for the event [zt+1 qt ] as seen from (15) and (16), while both terms are highly nonlinear in the demandlast model. This is due to the demand-last model’s observation zt+1 = min{(It + qt − St )+ , Dt }, which includes the shrinkage, while the observation in the demand-first model does not. The demand-first model is therefore simpler than the demand-last model although these models seem to be symmetric. The complexity of the inventory evolution in the demand-last model leads to more complicated Bellman equations than in the demand-first model. But they can be obtained by following an approach similar to that of the demand-first model. In the rest of this paper, we focus on the more common and simpler demand-first model. The analysis of the demand-last model is similar.

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π2 = (A2 , B2 (·)) as follows:

Remark 2. Both the demand-first and demand-last models have the same inventory evolution equation. From (1), the demand-first model has It+1 = (It +qt − zt+1 − St )+ , where zt+1 = min{It + qt , Dt }. Therefore, It+1 = (It + qt − Dt − St )+ if Dt It + qt ; otherwise, It+1 = (It + qt − It − qt − St )+ = 0, which imply It+1 = (It + qt − Dt − St )+ . Similarly, for the demand-last model, (1) implies that It+1 = (It + qt − St )+ − zt+1 = ((It + qt − St )+ − Dt )+ , where zt+1 = min{(It + qt − St )+ , Dt }. Therefore, if St < It + qt , It+1 = (It + qt − St − Dt )+ ; otherwise, It+1 = (It + qt − It − qt − Dt )+ = (It + qt − St − Dt )+ = 0. Then, both models have the same inventory evolution It+1 = (It + qt − zt+1 − St )+ . However, they have different inventory distribution evolutions in (15)–(16) and (24)– (25) due to different sales observations. This difference underlines the importance of the observation process for systems that have the same inventory dynamics.

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5.3. Lower bound

For obtaining a lower bound of the optimal cost function, we shall assume that the initial inventory I1 is known. But as we move in time, the inventory level at any time t > 1 will not be known with certainty because of the unobservability of the demand and the shrinkage. Therefore, we are dealing with all the complexity of our inventory system. We start with π1 = δx , which is represented by A1 = 1 and B1 (·) = 0 if I1 = x = 0 and by A1 = 0 and B1 = δx if I1 = x > 0. Following the analysis in Section 5.1, we have the inventory probability law

(26)

B2 (η) = 1z2 0

+ 1zt+1 =0 1φt G¯ f (q)+ G¯ f (λ+q)ψt (λ) dλ=0 ) × α qt , zt+1 ; φt , ψt (·) +

γ η; qt , zt+1 ; φt , ψt (·) dη

+ 1zt+1 =0 1φt G¯ f (q)+ G¯ f (λ+q)ψt (λ) dλ=0 ) × F¯ (zt+1 ) At gf (qt − zt+1 )


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E φ(z2 )|I1 = E 1S f I +q φ(0) 1 1

f + 1S f