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Int. J. Production Economics 129 (2011) 86–101
Contents lists available at ScienceDirect
Int. J. Production Economics
0925-52
doi:10.1
n Corr
E-m
journal homepage: www.elsevier.com/locate/ijpe
A Markov model of liquidity effects in reverse logistics processes: The effectsof random volume and passage
William Wilcox a, Philip A. Horvath b, Stanley E. Griffis c,n, Chad W. Autry d
a Department of Accounting, Monfort College of Business, University of Northern Colorado, 2095-C Kepner, Greeley, CO 80639, USAb National City Bank/Robert T. Stevenson, Jr. Professor of Finance and Chair Finance and Quantitative Methods Department, Foster College of Business, Bradley University, 1501 West
Bradley Ave., Peoria, IL 61625, USAc Department of Supply Chain Management, Broad College of Business, Michigan State University, East Lansing, MI 48824, USAd Department of Marketing and Logistics, College of Business Administration, The University of Tennessee, Knoxville, TN 37996, USA
a r t i c l e i n f o
Article history:
Received 29 January 2010
Accepted 6 September 2010Available online 16 September 2010
Keywords:
Liquidity management
Financial management
Reverse logistics
Product returns
Markov modeling
73/$ - see front matter & 2010 Elsevier B.V. A
016/j.ijpe.2010.09.005
esponding author. Tel.: +1 517 432 4320.
ail address: [email protected] (S.E. Griffis).
a b s t r a c t
Firms at various levels of the supply chain are implementing reverse logistics systems to maximize the
value captured from products flowing backwards from customers to suppliers. However, due to the
sporadic and unpredictable cash outflows associated with returns, firms must take care to avoid
liquidity problems. Previous work addressing reverse logistics liquidity issues has considered long-term
expectations, uncertainty, and shock potential inherent in the retail reverse logistics process, but the
impact of the expected returns volumes and random return quantities within fixed-scale systems has
yet to be explored. The current paper addresses these concerns.
& 2010 Elsevier B.V. All rights reserved.
1. Introduction
The challenge of dealing with products moving backwardthrough the supply chain is significant for modern firms, andincreases in both difficulty and importance each year. The annualcosts of dealing with the nearly $100 billion of returned productsin the US market has been estimated as over $35 billion (Feuling,2009; CSCMP, 2009), with consumer returns representing slightlymore than half of the total (Angrick, 2009). Similarly, the value ofproducts remanufactured into saleable form is estimated toexceed $50 billion annually in the US market (Guide and VanWassenhove, 2003). In light of these actualities, and given therelevance of reverse logistics activities to both the firm’s financialposition and customer relations (e.g., Daugherty et al., 2005), theneed for further research addressing reverse logistics implicationsfor the firm has never been more vital.
Serrato et al. (2007) observe that an abundance of empiricalwork has already addressed reverse logistics topics from anoperational standpoint, but few analytical models have beenoffered that adequately represent the current state of reverselogistics practice. The few exceptions have limited their focus tothe implications of reverse logistics on production (Fleischmannet al., 2001; Nakashima et al., 2004) and on inventory policy (i.e.,Dobos, 2003; Minner, 2001). There is a relative lack of analytic
ll rights reserved.
research addressing how best to manage reverse logistics productflows, especially from a financial standpoint; additional work isrequired that models reverse logistics from a financial perspec-tive. The limited extant research aims to help managers betterunderstand how to best achieve cost reductions and profitmaximizations from reverse logistics activities (i.e., Kannanet al., 2009; Guide et al., 2006; Mukhopudhyay and Setoputro,2004). However, though these models address reverse logisticsoutcomes from an eventual profit-and-loss perspective, theygenerally fail to account for a more pressing concern of thereverse logistics financial process: reversed cash flows paid out inremuneration for product returns, which are problematic to thefirm due to their impact on firm liquidity in the short- to medium-term.
While the impact of liquidity as a constraint is well understoodwith regard to outbound inventory policy (Kashyap et al., 1994;Hendel, 1996; Carpenter et al., 1998), the relationship betweenreverse product flows and liquidity is less understood. WhileHorvath et al. (2005) examined uncertainty, shock, and long-termimpacts of potential illiquidity in retail reverse logistics system,no research has yet assessed random return volumes at differentsupply chain echelons, nor operational and financial systemdesign constraints. This omission in the literature is problematic.The current paper addresses these gaps in the financially orientedreverse logistics stream with a model designed to assist firms inaccounting for the unpredictable quantity of returns and proces-sing cash costs and inflows at each stage of the reverse logisticsprocess. The model helps managers synchronize the activities of
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–101 87
operations with finance to more proactively and accurately planfor short- and long-term liquidity needs, thereby better facilitat-ing the integration of the firm’s operations and finance functions.
2. Literature review: reverse logistics and firm liquidity
Reverse logistics has been termed 1 of the 5 key components ofreverse supply chains (Guide and Van Wassenhove, 2003). For thepurposes of this research, reverse logistics is considered to be ‘‘theprocess of moving a product from its point of consumption topoint of origin to recapture value or for proper disposal’’ (Rogersand Tibben-Lembke, 1999). While this definition emphasizes theoperational elements of reverse logistics for all firms, it isnoteworthy that customers and suppliers engage in the processfor different reasons. Customers benefit from a well-managedreverse logistics process through greater product availability andperceptions of heightened customer service, especially as theprocesses become more formalized (Autry, 2005; Carter andEllram, 1998). From the supplier perspective, a successful reverselogistics program offers the firm the ability to build a closerrelationship with the customer by providing additional opportu-nities to serve and satisfy them (Barsky and Ellinger, 2001). Anadditional supplier benefit is that worn out or obsolete productscan be remanufactured and resold, thus allowing for theachievement of additional margins (Heese et al., 2005, Stocket al., 2002).
The reverse logistics process was initially described by Rogersand Tibben-Lembke (1999) and is illustrated in Fig. 1. As can beseen from the figure, products pass through a number of statesupon their return, including a gatekeeping function, various formsof processing and/or sorting and processing, and an eventualdisposition state that is either internal or external to the firm.
However, the aforementioned benefits come with associatedchallenges, specifically the expenses associated with managingreverse logistics process complexities (Skinner et al., 2008, Aminiet al., 2005). Dowlatshahi (2005) identifies five strategic factorsessential for designing and implementing successful reverselogistics systems, one of which is financial costliness. Addressingshort-term costs with liquid assets is a primary concern in thecurrent research, given the importance of cash in facilitatingthe firm’s ability to shoulder short-term reverse logistics costs.The current article adopts a liquidity management perspective inexamining reverse logistics operations.
Liquidity management is defined as the ability of the firm tomanage its short-term resources and obligations, for the purposesof tracking the amount and timing of cash inflows and outflows(Gentry and De La Garza, 1990), and provides ‘‘early warning ofproblems ahead y to prevent illiquidity while creating corporatevalue’’ (Badell et al., 2005). Firm underinvestment in inventory
CollectionProduct ReturnsGatekeeping
Regular InventoInternal Disposi
Firm
Fig. 1. The firm-level reve
due to liquidity pressures can lead to stock-outs, therebynegatively affecting performance (Pirtilla and Virolainen, 1992).Calomiris et al. (1995) found that firms with limited access tofinancial instruments place greater pressure upon their inventoryto account for liquidity shocks, and Tribo (2007) shows thatdiffering ownership structures’ affect upon liquidity also impactsinventory levels. Sodhi and Tang (2009) incorporate liquidity insupply chain planning in the face of demand uncertainty but donot address the impact to liquidity of the reverse flows. Given thatreturns-based cash outflows can tie up liquidity just as other cashflows can, the impact of the reverse process upon liquidity meritsfurther examination.
Horvath et al. (2005) applied Markov chain analysis to assessthe liquidity impact of product returns on the retail reverselogistics process. However, they failed to address the randomnature of product returns, limited returns to levels matched tocapacity, and failed to consider cost variation due to returnsprocess utilization. This article addresses these shortcomings,leading to increased ability to plan for liquidity shocks that thereverse flows of product can create. In summary, although limitedwork has analyzed liquidity within the reverse logistics area, thecurrent model helps by assisting firms in making provisions forliquidity needs, by including provisions for uncertainties andpotentials for shocks not only in the reverse logistics process butalso for variations in number of returns, and cash outflows at eachstage of the process. This model allows firms to better plan forlonger-term liquidity needs, especially given the dynamic natureof the returns process and the varying nature of its associatedcosts.
3. Liquidity effects of reverse logistics model
3.1. Process and capacity
Our modeling of liquidity assessment includes an analysis ofthe number of returns, the process that these returns followthrough the reverse logistics system until they are finallyremoved, and the cash flows associated with these dynamics.We first model the reverse logistics process, then incorporate thenumber of returns, and finally include cash dynamics.
The process impacts of reverse logistics on firm liquidity,considered independent of the number of returns at any giventime, begin with a periodic classification of a returned unit asbeing in state i, si: i¼(1, 2,y,T), such that T is the number oftransient, or processing, states depending upon the unit’s status.Similarly, a unit may be classified in an absorbing state, sk: k¼(I,II,y,K), where K is the number of absorbing states or ways theunit can leave the process. Associated with these states aretransition probabilities: ri,k: i¼(1, 2,y,T), k¼(I, II,y K) are the
ExternalDispositionChannels
Sortation Disposition
ry/tion
rse logistics process.
Absorbing States I, II, IIITransient Returns States 1, 2, 3
- 0 +
-
0
+
-
0
+
0.009 0.018 0.018
0.140.02
0.06
0.320.02
0.41 0.41
Returns State 3 (e.g. Sortation)
Returns State 2 (e.g. Collection)
0.005
0.1
0.1
0.045
0.9
Returns State 1 (e.g. Gatekeeping)
0.9
0.72
0.135
0.07
0.07
0.0035
0.0105
0.0560.015
0.08
(ex: External -Disposal)
(ex: Internal -Remanufacture)
(ex: Internal - Restock
for Resale)
0.07
0.07
0.07
0.340.34
0.34
0.14
0.140.14
0.37
0.37
0.37
0.42
0.42
0.42
Fig. 2. Transition State Diagram.
1 The transition matrix is readily generalizable to any number of size-states
but is limited here to three for ease of exposition.
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–10188
probabilities of a unit moving from transient state i to absorbingstate k, and qi,j: i,j¼(1, 2,y,T) are the probabilities of a unitmoving from a transient state to the same or another transientstate. The probability of a unit moving from an absorbing state toa transient state, pk,i, is zero. And the probability of a unit movingfrom an absorbing state to the same absorbing state, pk,k, is 100%.Therefore, the Markov chains require that 0rqi,j, ri,kr1.0; pk,k¼1and pk,i¼0. These probabilities may be organized into transitionmatrix P, which in canonical form provides sub-matrices Q
containing all pi,j, R consisting of pi,k, I an identity matrix, and 0a null matrix.
P¼I 0
R Q
" #: R¼ frikgT�K ;Q ¼ fqijgT�T
In order to fully identify the impact of returns volume inconjunction with planned capacity, and liquidity through costs,we redefine our transition matrix. Via this model, each firm has aplanned scale, i.e., its design capacity. That is, the returns processis designed to efficiently process a given number of units. Shouldthe return volume materially exceed returns processing capacity,units could no longer be effectively or efficiently processed. As aresult, units may stay in a given state longer than expected due tothe system’s inability to reprocess them. In this case, transitioningunits may be misrouted to an absorbing state (for example,disposed of as waste when they should have been restocked).Conversely, if the number of returns is lower than the plannedcapacity, both the processing dynamics and the associated cashbehaviors may similarly differ. Our reverse logistics liquiditymodel reflects these dynamics.
Cyert and Thompson (1968), modeling accounts receivablerisks, establish that differing transition probabilities and riskcharacteristics could impact receivable accounts. To address thispossibility, they create a different transition matrix for eachdifferent risk class. Horvath et al. (2005) extend Cyert andThompson (1968) to the reverse logistics context but only within
the context of a simplistic returns management process. Thecurrent model accommodates the effects of randomly occurringdifferences in numbers of units returned at each stage. We expandon the previous research by including an augmented transitionmatrix. Each transient state is defined as follows. If the quantity ofreturned units entering a transient state is below a lowerthreshold, indicating a value that is less than capacity, it isdenoted as state S�i . If the quantity of returned units entering atransient state is above an upper threshold, indicating that thereis insufficient capacity, it will be denoted as state Sþi . Similarly, ifthe quantity of units is at capacity, it will be denoted as state S0
i .With each state: S�i , S0
i , Sþi there is an associated transitionprobability p�ij , p0
ij, pþij , respectively, portraying the probabilities ofmoving from one state to another. The number of transient statemay now be T0 ¼3T,1 and the number of augmented absorbingstates may be K0 ¼3K or K depending on whether the state/sizewould affect the absorbing state. A Markov state diagram of theexpanded model identifying the transient and absorbing states,with probabilities, appears in Fig. 2.
The creation of independent transition matrices as per Cyertand Thompson (1968) does not allow a straight-forward model ofchanges as flows proceed from one state to another. No previousmodels allow a one-period old account in a high risk class tobecome classified as a member of a different risk class as atwo-period old account. Our expansion of the transition matrixallows the model to accommodate the dynamics of transitionfrom one state to another as well as process transitions as theunits move from one period to the next. For instance, employeesdoing the sorting may misallocate product returns to absorbedstates, e.g., dispose of excess units as waste when they might beamenable to restocking. This mistake would reduce the number ofitems at the next processing state to be within capacity.Alternatively, the number of returned items may be well below
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–101 89
capacity within a period, and employees may for a variety ofreasons not process these. The effect would be similar—the nexttime period would require initial processing of newly returnedunits plus those held from the previous period, resulting in a statethat may be at or above capacity. In either case, the transitionmatrix we propose accommodates transitions from each proces-sing state/size such as s�i , s0
i , sþi (hereafter noted s�=0=þi for
convenience) to any other processing state/size such as s�j , s0j , or
sþj (hereafter noted s�=0=þj for convenience) in the subsequent
time period.The expanded transition matrix P with sub-matrices: Q, R, I,
and 0 are understood to include our state characteristics:
P¼I 0
R Q
" #: R¼ frikgT u�K u;Q ¼ fqijgT u�T u
Following Horvath et al. (2005) the appropriate characteristics of
the reverse logistics system can be determined as ðI�Q Þ�1¼NT u�T u
with NT u�T uxT u�1 ¼ tT u�1. NT u�T urepresents the average number ofperiods the unit will require for transition from si to sj. Each element,I, of tT0 �1 represents the number of periods in a unit in each transientstate, si, will, on average, be in the system before being absorbed. We
may find BT u�K u ¼ ðI�Q Þ�1T u�T uRT u�K u ¼ NT u�T uRT u�K u. BT u�K uis a matrix of
probabilities of a unit being absorbed in absorbed state k for unitscurrently in transient state i, bik. The variance of NT u�T u is2
Var{N}¼N(2Ndg� I)�Nsq; Ndg is a matrix containing values of N onthe diagonal and zeroes elsewhere; Nsq is a matrix with each elementdetermined as Var{nij}. The matrix of standard deviations of N,Stdev{N} is given by determining the square root of each element of
Var{N}: StdevðNÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVarfNg
pand Stdevfnijg ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVarfnijg
p. The variance
of t is: Var{t}¼{2N� I}t�tsq. The standard deviation of t is
Stdevftg ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiVarftg
pwith Stdevftijg ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiVarftijg
p. The skew of N,
Skew{N}, is given by: Skew{N}¼N[I�{3Ndg� I}2Ndg]�3NVar{N}�Ncu;with the skew of t, Skew{t}¼3{N� I}[{2N� I}t+t]+t�3tVar{t}�tcu.
3.2. Number of returns
Let tildes denote random variables and ~U1 denote the numberof units sold in all preceding periods returned reaching thegatekeeper (i.e. in state 1), s1, in any period, such thatPð ~U1Þ�probability that the actual number of returns, in s1, duringthat period will be U1. The distribution is characterized as aprobability mass function due to an assumed discrete nature ofthe reverse logistic process, i—given that units are measureddiscretely. ~U
þ
1 is the number of returned units, in s1, during theperiod in excess of normal design capacity ( ~U
þ
1 A ~U1); ~U0
1is thenumber of returned units, in s1, in the period that are withinnormal design capacity ( ~U
0
1A ~U1); ~U�
1 is the number of returnedunits, in s1, in the period that are materially less than normaldesign capacity ( ~U
�
1 A ~U1). These will be noted below as ~U�=0=þ
1 tomaintain some parsimony in subsequent equations. ~U
�=0=þ
1 isintended to be read as the random number of returns below, at, orabove planned capacity, respectively. These have associatedconditional probabilities as follows: Pð ~U
�=0=þ
1 ¼U�=0=þ1 Þ is the
conditional probability of U�1 , U01 , or Uþ1 units being returned,
reaching s1, in the period at materially less than (�), within (0), orin excess of (+) design capacity. These probabilities are alsocharacterized as probability mass functions due to an assumeddiscrete nature of the reverse logistic processes. With theexception of a de-novo firm on day one of operations there will
2 Subscripts of matrices indicating dimension are hereafter eliminated as
understood from the preceding unless otherwise noted in order to clarify notation.
be estimable numbers of units that have been returned and arestill in the reverse logistics system in different states and have notyet exited. Our observation of firms of all sizes supports thecontention that, like the potential returns entering the system,the numbers of units in any other state are not known withcertitude. These are random variables, albeit potentially with lessuncertainty.
Given that ~U�=0=þ
i and Pð ~U�=0=þ
i ¼U�=0=þi Þ are available for
all transient states, s�=0=þi , one can construct vectors: EfUgT u�1 ¼
fEð ~U�=0=þ
i ÞgT u�1 the vector of returns expected in si; VarfUgT u�1 ¼
fVarð ~U�=0=þ
i ÞgT u�1 the vector of variances, and SkewfUgT u�1 ¼ fSkew
ð ~U�=0=þ
i ÞgT u�1 the vector of skews (expected value of the cubed
deviations from the expected value) for each s�=0=þi : iAT u.
3.3. State and capacity contingent costs
When cash flows are incorporated into our assessment ofliquidity impact, these cash flows are applied as being knownwith certitude, though the cash flows used within our currentexample (see the Appendix) and in real-world scenarios may beestimates based upon historical accounting allocation schemes. Infact, we suggest that the firm does not know with certitude theactual cash flows of processing their reverse logistics units for anyof the d periods in the future. Thus, we treat cash flows as randomvariables and are noted with tildes as above.
Associated with each transient or absorbed state is a relatedcash flow. These cash flows result from movement from any statei to any other state j. In addition, the cash flows will differ forin-process units based on � , 0, or + capacity in both thebeginning and ending states. For example, the cash effects arecharacterized as: ~c�ij , ~c0
ij, ~c þij , for units moving from eachintermediate state in the process to another intermediate state(i.e., transition) and ~c�ik, ~c0
ik, ~c þik for movement from eachintermediate state to states in which the item(s) leave the system(i.e., absorption). These are denoted ~c�=0=þ
ij and ~c�=0=þik , respec-
tively. Each state/capacity combination per unit cost has statis-tics: EfCT ugT u�T u ¼ fEð~c
�=0=þij Þg
T u�T u, VarfCT ugT u�T u ¼ fVarð~c�=0=þ
ij ÞgT u�T u
,which represent the variance, as well as a skew measure givenby SkewfCT ugT u�T u ¼ fSkewð~c�=0=þ
ij ÞgT u�T u
. This information providesexpectations, uncertainties, and propensities for per unit costs ineach transient state/capacity.
Managers interested in the liquidity aspects of reverse logisticsalso value the ability to assess the same characteristics of cashflows from each transitional state/capacity combination to thosethat represent exit from the process. In a manner similar to thatfor the cash characteristics of transitions through the reverselogistics process, the expected value of each state/capacitycombination i to each absorbing state/capacity combination k is
given by EfCK ugT u�K u ¼ fEð~c�=0=þik Þg
T u�K u, the variance: VarfCK ugT u�K u ¼
fVarð~c�=0=þik Þg
T u�K u, and skew: SkewfCK ugT u�K u ¼ fSkewð~c�=0=þ
ik ÞgT u�K u
.
Thus, the total period cash flow effect is a random variable thatrepresents a joint distribution of those developed thus far.
4. Periodic cash flows under relative certainty
Thus far, we have developed measures of expectation, risk, andcatastrophic occurrences (skew) with respect to the number ofreturns and the per unit cash flows as each unit moves from eachstate combination to each other state combination, includingthose returned units exiting the reverse logistics system, as wellas the probabilities of movement of units from each intermediatestate to each other state including absorption. The primary
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–10190
interest here is the periodic (daily) cash flow effects from the jointoccurrences of these unknown, but estimable, elements.
The expected cash flow effects of returned items as theyproceed from transient state to absorbing state next period (day)can be found using the Einstein/Scalar/dot-product notation(Z ¼ X3Y , where each zij¼xijyij): EfCashK ugT u�K u ¼ EfUgT u�13
EfCK ugT u�K u3RT u�K u, with total cash generated by being absorbed inthe next period:
EfCashK ugT u�1 ¼ EfCashK ugT u�K uxK u�1 or:
EfCashK ug1�K u ¼ x1�T uEfCashK ugT u�K u
Therefore:
EðCashK uÞ ¼ x1�T uEfCashK ugT u�1 ¼ EfCashK ugxT u�1
The number of units in the reverse logistics system andmoving to another transient state in the subsequent periodprovide an expected cash of: EfCashT ugT u�T u ¼ EfUgT u�13EfCT ugT u�T u3
QT u�T u, with total cash for transitioning returns:
EfCashT ugT u�1 ¼ EfCashT ugT u�1xT u�1 Or:
EfCashT ug1�T u ¼ x1�T uEfCashT ugT u�1
Therefore:
EðCashT uÞ ¼ EfCashT ug1�T uxT u�1 ¼ xT u�1CfCashT ug1�T u
Finally, we can see that the total cash generated by units inthe RL system today over the next period is: EðCashRLÞ ¼
EðCashK uÞþEðCashT uÞ.
4.1. Uncertainty in periodic cash flows
Continuing with our treatment of uncertainty as variance, wecan also provide measures of the uncertainty surroundingthe cash flows associated with units in the reverse logisticsprocess and being absorbed as (per Goodman, 1960, 1962;Blumenfeld, 2001)
VarfCashK ugT u�K u ¼ ½VarfUg3VarfCK ugþVarfUg3EfCK ugsqþVarfCK ug3EfUgsq�Rsq
Resultantly, for state independence across all K, the variancecan be defined as VarfCashK ug1�K u ¼ VarfCashK ugT u�K uxK u�1. And, forstate/capacity combinations, the variance of cash being absorbedfrom each T0 state is VarfCashK ugT u�1 ¼ x1�T uVarfCashK ugT u�K u, withthe variance of cash being absorbed next period given as:VarðCashK uÞ ¼ VarfCashK ug1�K uxK u�1 ¼ x1�T uVarfCashK ugT u�1For thoseunits remaining in the system, variance is defined as
VarfCashT ugT u�T u ¼ ½VarfUg3VarfCT ugþVarfUg3EfCT ugsqþVarfCT ug3EfUgsq�Qsq
such that the variance of cash effects in the next period for eachdestination state is given by VarfCashT ug1�T u ¼ VarfCashT ugT u�T uxT u�1,and for departing state we model uncertainty surroundingthe cash flows as VarfCashT ugT u�1 ¼ x1�T uVarfCashT ugT u�T u, with thevariance of cash in transition to the next period beingVarðCashT uÞ ¼ VarfCashT ug1�T uxT ux1 ¼ x1�T uVarfCashT ugT u�1, where Var
(CashT0) is the uncertainty in the cash flows from a randomnumber of units entering the reverse logistics process above, at, orbelow design capacity. For uncertainty in cash effects in the nextperiod we therefore have
VarðCashÞ ¼ VarðCashT uÞþVarðCashK uÞ:
4.2. Skew in periodic cash flows
Finally, it is useful for us to model skew in the returnsmanagement process. For random numbers of units and cash
effects per unit entering the process and being absorbed, we haveskew defined as
SkewfCashK ugT u�K u ¼ ½SkewfUg3SkewfCK ugþSkewfUg3EfCK ugcu
þSkewfCK ug3EfUgcu�Rcu
Given this definition, and assuming state independence acrossall K, the skew of these cash effects can be defined asSkewfCashK ug1�K u ¼ SkewfCashK ugT u�K uxK u�1. Further, for each state/capacity combination the skew of cash being absorbed from eachT0 state is given as SkewfCashK ugT u�1 ¼ x1�T uSkewfCashK ugT u�K u, withthe skew of cash being absorbed next period therefore beingSkewðCashK uÞ ¼ SkewfCashK ug1�K uxK u�1 ¼ x1�T uSkewfCashK ugT u�1.
For those units remaining in the system, skew is defineddifferently, as
SkewfCashT ugT uxT u ¼ ½SkewfUg3SkewfCT ugþSkewfUg3EfCT ugcu
þSkewfCT ug3EfUgcu�Qcu
so that the skew of cash effects in the next period for eachdestination state is:SkewfCashT ug1�T u ¼ SkewfCashT ugT u�T uxT u�1, andfor departing state we model skew surrounding the cash flows asSkewfCashT ugT u�1 ¼ x1�T uSkewfCashT ugT u�T u, with the skew of cashin transition to the next period being SkewðCashT uÞ ¼ Skew
fCashT ug1�T uxT u�1 ¼ x1�T uSkewfCashT ugT u�1.In sum, the skew of cash flow for the next period is
SkewðCashÞ ¼ SkewðCashK uÞþSkewðCashT uÞ, where Skew(Cash) canbe considered the asymmetry in cash flows from a randomnumber of units entering the reverse logistics process above, at, orbelow design capacity resulting from random cash effects ofabsorption and of transition for one period.
5. Systematic cash flows
While it is certainly informative to have information concern-ing the periodic (next stage) cash characteristics as developedabove, questions concerning the amount of cash required/generated by the reverse logistics process in the long run are ofequivalent importance. It would be helpful to the cash manager toassign dollar values in the environment of random numbers ofunits in each state and random cash effects of transition toabsorption and/or transition. Fundamental questions partiallyaddressed by the past research are expanded to:
I.
What are the expectations, uncertainty, and skew of cashinvestment required to support random units with randomcash effects, remaining in the system a random number ofperiods prior to being absorbed; andII.
What are expectations, uncertainties, and skew in casheffects as random units with random cash effects areabsorbed after being in the system?Here we further extend the literature by applying their resultsrepeated above to our random cash developments: EfCashNgT u�T u ¼
EfUgT u�13EfCT ugT u�T u3N. Here, EfCashNgT u�T u provides the expectedcash effect of random numbers of returned units in eachbeginning state i, ~Ui, at random cash costs/returns ~cijof processingor storage in state i and moving to state j for an average number oftimes, ~nij, prior to being absorbed. For the long run expectedvalue, variance and skew of RL units in transition, prior to beingabsorbed, we find EfCashNg1�T u ¼ x1�T uEfCashNgT u�T u provides thecash effects on average for units transitioned to state j. Likewise,EfCashNgT u�1 ¼ EfCashNgT u�T uxT u�1 provides the expected long runcash effects of returned units in each beginning state. Finally,expected long run effects of random numbers of returns atrandom cash effects for random numbers of times are given byEðCashNÞ ¼ x1�T uEfCashNgT u�1 ¼ EfCashNg1�T uxT u�1.
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–101 91
The uncertainty or volatility is measured by the variance:
VarfCashNgT u�T u ¼ VarfNg3VarfUg3VarfCT ugþVarfNg3VarfCT ug3EfUgsq
þVarfNg3VarfUg3EfCT ugsqþVarfNg3EfUgsq3EfCT ugsq
þVarfUg3VarfCT ug3EfNgsqþVarfCT ug3EfUgsq3EfNgsq
þVarfUg3EfCT ugsq3EfNgsq
The uncertainty can be obtained from VarfCashNg1�T u ¼
x1�T uVarfCashNgT u�T u, which provides the variance of cash effectsfor units transitioned to state j. Likewise VarfCashNgT u�1 ¼
VarfCashNgT u�T uxT u�1 provides the variance of long run cash effectsof returned units in each beginning state. Variance of long runeffects of random numbers of returns at random cash effects forrandom numbers of times is given by VarðCashNÞ ¼ x1�T uVar
fCashNgT u�1 ¼ VarfCashNg1�T uxT u�1. And skew as
SkewfCashNgT u�T u ¼ SkewfNg3SkewfUg3SkewfCT ug
þSkewfNg3SkewfCT ug3EfUgcu
þSkewfNg3SkewfUg3EfCT ugcu
þSkewfNg3EfUgcu3EfCT ugcu
þSkewfUg3SkewfCT ug3EfNgcu
þSkewfCT ug3EfUgcu3EfNgcu
þSkewfUg3EfCT ugcu3EfNgcu
Similarly, SkewfCashNg1�T u ¼ x1�T uSkewfCashNgT u�T u provides theskew of cash effects for units transitioned to state j. Likewise,Skew CashN
� �T u�1¼ Skew CashN
� �T u�T u
xT u�1 provides the skew oflong run cash effects of returned units in each beginning state.Skew of long run effects of random numbers of returns at randomcash effects for random numbers of times is given bySkewðCashNÞ ¼ x1�T uSkewfCashNgT u�1 ¼ SkewfCashNg1�T uxT u�1.
The second question can be answered using a matrix ofprobabilities being in each absorbed state, provided it began eachstage in a transition state, B. When EfCashAgT u�K u ¼ EfUgT u�13
EfCK ugT u�K u3B is performed we provide the expected number ofdollars for each absorbing state by each transitory state. Subse-quently EfCashAg1�K u ¼ x1�T uEfCashAgT u�K u provides the cash effectson average for units absorbed in each absorbing state k. LikewiseEfCashAgT u�1 ¼ EfCashAgT u�K uxK u�1 provides the expected long runcash effects of returned units in each beginning state. Then, asabove we have:EðCashAÞ ¼ x1�K uEðCashAÞT u�1 ¼ EfCashAg1�K uxT u�1,with expected long run total cash of E(CashRL)¼E(CashN)+E(CashA).
Variance and skew are relatively straight-forward applicationof the processes for Var(CashN) and Skew{CashN}: VarfCashAgT u�K u
¼ ½VarfUg3VarfCK ugþ VarfUg3EfCK ugsqþ VarfCK ug3EfUgsq�Bsq; Var
fCashAg1�K u ¼ x1�T uVarfCashAgT u�K u; VarfCashAgT u�1 ¼ VarfCashAgT u�K u
xT u�1; and VarðCashAÞ ¼ x1�K uVarfCashAgT u�1 ¼ VarfCashAg1�K uxT u�1.And finally we find variance of cash for the reverse logisticsprocess as Var(CashRL)¼Var(CashN)+Var(CashA). Following the aboveprocess for skew we have SkewfCashAgT u�K u ¼ ½SkewfUg3 Skew
fCK ugþSkewfUg3EfCK ugcuþSkewfCK ug3EfUgcu�Bcu; SkewfCashAg1�K u ¼
x1�T uSkewfCashAgT u�K u; SkewfCashAgT u�1 ¼ SkewfCashAgT u�K uxT u�1; andSkewðCashAÞ ¼ x1�K uSkewfCashAgT u�1 ¼ SkewfCashAg1�K uxT u�1. Follow-ing the above we find Skew(CashRL)¼Skew(CashN)+Skew(CashA) forindependent cash dynamics between transition and absorbingstates, which is a property of Markov chains.
6. Managerial implications
Returns are known to have an impact upon firm liquiditythrough the cash outlays required to manage a returns process.While planned capacity is not alterable over the subsequentperiod, the firm can draw a number of conclusions based on themoments of the random elements described above. In general
these conclusions are helpful in at least two ways. Firstly,knowledge of numbers of units of future saleable inventory(either directly back to stock shelves or through remanufacturing)benefits a company in planning their purchases. Firm ignoringsaleable returns inventory run the risk of purchasing replacementstock to be positioned for sale to the customer when saleableproduct can be expected to be made available by the returnsprocess within the same period. The impact to firm liquiditybrought on by purchasing unnecessary inventory is an expensethat cannot be afforded. Secondly, firms with no warning ofupcoming cash outlays to customers returning product can becaught short of cash reserves, necessitating short-term loans atless than favorable rates. Awareness of these impending liquidityuncertainties under differing returns processing design capacitiesallows the firm to plan for these events ahead of time, therebyallowing them to acquire the best terms, reducing overallexpenses.
A number of specific returns processing issues are strategicallyimportant for operations and financial management, and areworthy of mention here. First, it can be anticipated that the cashflow will be, on average, of the value and sign given byEð ~C Þ, andthe firm can plan accordingly. Similarly, the firm may use Eð�Þ :
� ¼ ~U0, ~U I , ~Ui,. . . for any of the random return or cost elementsdescribed above to anticipate the levels of returns, cash flows, etc.,for each of these items, given that the scale of the returns processis known.
Using the variance measures that were previously determined,the firm can make further provisions for total cash flows that maybe needed for the next period’s activities. Since it is unlikely thatthe distribution of these cash flows is normal or approximately so,and without fitting the distribution to any particular form, werecommend the conservative Chebyshev estimation that at least(1�(1/k2))% of the outcomes occur within k standard deviationsof the mean. In this situation:
Pr Eð ~C Þ�kSð ~C ÞrCrEð ~C ÞþkSð ~C Þh i
Z 1�1
k2
� �
This implies that the probability that C, the cash flow nextperiod, will be between the expected cash flow minus k times thestandard deviation of cash flows, given by Eð ~C Þ�kSð ~C Þ, and theexpected cash flow plus k times the standard deviation of cashflows, Eð ~C ÞþkSð ~C Þ. This estimation is given by (1�(1/k2)). Thistechnique provides a confidence interval that allows more refinedplanning for liquidity needs over the next period. The expectationthe stochastic elements comprising ~C renders may also becharacterized by Pr½Eð�Þ�kSð�Þr�rEð�ÞþkSð�Þ�Z 1�ð1=k2Þ
� �,
where � is as noted above. Again, this provides a confidenceinterval that may be used by management for better assessmentof its liquidity needs, as well as returns process management, overthe subsequent period.
The information from the confidence interval for ~C and otherstate/capacity combination random variables provides onlylimited additional information. The usual assumption is that theoutcomes of ~C are symmetrically arrayed around Eð ~C Þ ormore generally E(�). This would be the case if the skew of ~C(or � more generally) were zero, i.e., Skew0ð
~C Þ ¼ 0 (orSkew0(�)¼0). However, the skew measure may be positive ornegative when its value is not zero, and these non-zero valueshave differing implications. If Skew0ð
~C Þo0, then values of ~C
(or � more generally) greater than the mean are less likely tooccur than if the distribution was symmetric, and values of ~C
(or �) less than Eð ~C Þ are more likely to occur than if symmetryprevailed. Conversely if Skew0ð
~C Þ40 then values of ~C (or �) lessthan the mean are less likely to occur than if the distribution wassymmetric, and values of ~C (or �) greater than Eð ~C Þ are more likely
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–10192
to occur than if ~C (or �) was symmetrical. While this addssomething to our ability to plan for actual outcomes of ~C (or �), itonly marginally improves on the Eð ~C Þ, kSð ~C Þ confidence intervalsabove.
Based on the above, we suggest operations and financialmanagers that estimates of the probabilities of the occurrence of~C (or �) over some interval {a,b}would be more valuable. Onemeans of generating this information would be to orderthe outcomes of ~C (or �) over the interval a to b and sum theproducts of the probabilities of joint occurrences of the compo-nents of ~C (or �). An alternative, perhaps more elegant, approachwould be to apply Cornish–Fisher Asymptotic Expansion (Hill andDavis, 1968) to the approximation of the cumulative distributionfunction, cdf, of ~C (or �). Cornish–Fisher Asymptotic Expansion iswidely used to provide asymptotic cdf probabilities for thedetermination of Value at Risk (see Holton (2003), for asummary). This technique asymptotically approximates quantilesof random variables. Its primary strength is that the approxima-tions may be made on the basis of only the first severalcumulants, which are similar to moments that we have developedabove and beyond. Using the three moments we have developedwithin the current analysis, we find
F�1x� ðpÞ �F�1
z ðpÞþ16 ðF
�1z ðpÞ
2�1Þk3þ
136ð2F
�1z ðpÞ
3�5F�1
z ðpÞÞk23
þ 1324ð12F�1
z ðpÞ4�53F�1
z ðpÞ2þ17Þk3
3:
For this equation, F�1z ðpÞ is, for p¼100, the percentile of a
zN(0,1), k3 ¼ Skewð ~C Þ
ffiffiffiffiffiffiffiffiffiffiffiffiS2
0ð~C Þ
q� �3
and x� ¼ ðC�Eð ~C Þ=S0ð~C ÞÞ for
the case of the total cash flow for the next period, given theestablished capacity. The k3,x * are transformations of cumulantsso that the requirement that the random variable for which
F�1x ðpÞ is obtained has mean zero and standard deviation of one is
satisfied.Similarly, by substituting � for ~C or C in k3,x * in F�1
z ðpÞ
(such that � ¼ ~U0, ~Ui,. . .), one then has the probabilities that ~C(or �) will be less than or equal to some C (or �) of interest,or that the probabilities that ~C (or �) will be greater than or
equal to some C (or �). Furthermore, the probabilities that ~C(or �) will lie in some interval a, b: b4a, can be found byF�1
x�bðpÞ�F�1
x�aðpÞ.
Taken together these findings provide management with morepowerful planning tools than the simple moments of the reverselogistics process. In this sense, firms can now better plan for thefinancial impacts of its reverse logistics activities on a short-termand intermediate-term basis.
7. Conclusions
This study contributes to the reverse logistics operationsliterature by providing additional analysis related to the financialmanagement of the reverse logistics/returns processes. Adoptinga liquidity-based perspective on reverse logistics operationsmanagement, the current paper better incorporates the con-tingencies associated with the operation of a reverse logisticssystem. This paper builds on past research while providing a morethorough evaluation of the short-term impacts and a morecomplete model of the dynamic reverse logistics process. Thebenefits of this model include the introduction of randomnumbers of units entering the system, as well as random numbersof units present within the various states of the system. Secondlywe consider the system states operating above, at, or belowdesign capacity, and the variable effects this will have on thereverse logistics system. Finally we consider the costs associatedwith each state behaving as a random variable, and the impacts itwill have on returns and liquidity management.
Ultimately the paper contributes to the management planningprocess by facilitating the development of confidence intervalsassociated with the distributional characteristics identified withthe Markov chain analysis. The different confidence intervalsoutlined in the managerial implications section allow managers toestimate the likelihood that next period’s results will occur withina particular range. This allows firms to better understand short-term needs and plan for longer-term liquidity needs, given thedynamic nature of the returns process and the varying nature ofthe associated costs.
Appendix A. Illustrative scenario
In order to illustrate the value of the proposed model, we operationalize its parameters within the context of the retail sporting goodsindustry. Making some reasonable extrapolations from the Horvath et al. (2005) actual data we illustrate the model as follows.
First, we construct transition matrices Q and R. As noted, Q describes transition probabilities for units moving from one transitive stateto another transitive states under three different capacity versus design constraints (with – indicating below design capacity, 0 indicatingat regular design capacity, and + indicating above design capacity). In the current case, transient state 1 refers to the returns‘‘gatekeeping’’ function (e.g., Rogers and Tibben-Lembke, 1999), while transient state 2 refers to item collection, and transient state 3refers to item sortation. For example, in the example of Q matrix, the transition probability for this retailer to move an item from thegatekeeping state at underutilized capacity to the collection state at regular design capacity is 0.328.
1� 10 1þ 2� 20 2þ 3� 30 3þ
Q ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
0:009 0:144 0:027 0:0205 0:328 0:0615 0 0 0
0 0:18 0 0 0:41 0 0 0 0
0 0 0:18 0 0 0:41 0 0 0
0 0 0 0:005 0:08 0:015 0:045 0:72 0:135
0 0 0 0 :1 0 0 0:9 0
0 0 0 0 0 :1 0 0 0:9
0 0 0 0 0 0 0:0035 0:056 0:0105
0 0 0 0 0 0 0 0:07 0
0 0 0 0 0 0 0 0 0:07
266666666666666664
377777777777777775
:
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–101 93
Similarly, the R matrix describes transition probabilities for units moving from transitive to absorbed states under the samethree design constraints. In the current case, the absorbed states under consideration are: (I) disposed externally as waste; (II)disposed externally for remanufacturing (outsourced); and (III) disposed internally as items to be restocked for sale. Thus, in the illustrativeR matrix, the transition probability for an item to be transitioned from the ‘‘gatekeeping’’ state to the ‘‘disposed externally as waste’’ stateis 0.07:
R¼
I II III
1�
10
1þ
2�
20
2þ
3�
30
3þ
0:07 0 0:34
0:07 0 0:34
0:07 0 0:34
0 0 0
0 0 0
0 0 0
0:14 0:37 0:42
0:14 0:37 0:42
0:14 0:37 0:42
266666666666666664
377777777777777775
In real-world scenarios, values for the Q and R matrices would necessarily come from internal research related to item trackingthrough the firm’s reverse logistics system over several periods.
Based on the values for Q and R derived from company data, it is possible to predict the expected mean number of returned units, thevariance of returned units, the standard deviation of returned units, and skew of returned units, under each design capacity, for eachtransient state. For example, at a given moment, one would expect there to be 5000 units in the gatekeeping function under regulardesign capacity, with a variance of roughly 4.6 M units, standard deviation of 2887 units, and skewness of �2225276759.
EfUgT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
1000
5000
10,000
21
3228
5640
10
3141
5462
266666666666666664
377777777777777775
, VarfUgT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
93,400
4,585,586
8,332,132
46
526,108
2,344,540
38
1,519,354
2,194,579
266666666666666664
377777777777777775
,
StdevfUgT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
306
2141
2887
7
725
1531
6
1233
1481
266666666666666664
377777777777777775
SkewfUgT ux1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
�4,940,000
�2,225,276,579
3,572,050,731
�50
�275,532,474
�4,229,547,600
�8
�581,841,476
�3,183,837,162
266666666666666664
377777777777777775
Once statistics describing the returned units under different design capacities can be identified, it is possible to begin con-ducting unitary cash flow analyses associated with transient state–transient state movements. For the current retailer, the negativecash flow associated with an item beginning a period in the collection function at regular design capacity, and remaining in thatstate, is �$0.48 (note: for the sake of comparative clarity, the average unit cost of a unit within this system was valued by the retailer at$15.00).
1� 10 1þ 2� 20 2þ 3� 30 3þ
EfCT ugT u�T u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$0 �$0:48 �$0:48 �$0:48 �$0:48 �$0:48 $0 $0 $0
$0 �$0:52 $0 $0 �$0:24 $0 $0 $0 $0
$0 $0 �$0:52 $0 $0 �$0:52 $0 $0 $0
$0 $0 $0 �$0:48 �$0:52 �$0:52 �$0:48 �$0:48 �$0:48
$0 $0 $0 $0 �$0:48 $0 $0 �$0:24 $0
$0 $0 $0 $0 $0 �$0:52 $0 $0 �$0:52
$0 $0 $0 $0 $0 $0 �$0:48 �$0:48 �$0:52
$0 $0 $0 $0 �$0:24 $0 $0 �$0:52 $0
$0 $0 $0 $0 $0 $0 $0 $0 �$0:52
266666666666666664
377777777777777775
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–10194
In cases of uncertainty, the variance, standard deviation, and skew associated with the same scenario are $0.0013, $0.0359, and 0.0001,respectively:
1� 10 1þ 2� 20 2þ 3� 30 3þ
VarfCT ugT u�T u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
0 0:0133 0:0133 0:0133 0:0133 0:0133 0 0 0
0 0:0013 0 0 0:0013 0 0 0 0
0 0 0:0164 0 0 0:0164 0 0 0
0 0 0 0:0133 0:0133 0:0133 0:0133 0:0133 0
0 0 0 0 0:0013 0 0 0:0013 0
0 0 0 0 0 0:0164 0 0 0:0164
0 0 0 0 0 0 0:0133 0:0133 0:0133
0 0 0 0 0:0013 0 0 0:0013 0
0 0 0 0 0 0 0 0 0:0164
266666666666666664
377777777777777775
1� 10 1þ 2� 20 2þ 3� 30 3þ
StdevfCT ugT u�T u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$0 $0:1155 $0:1155 $0:1155 $0:1155 $0:1155 $0 $0 $0
$0 $0359 $0 $0 $0:0359 $0 $0 $0 $0
$0 $0 $0:1281 $0 $0 $0:1281 $0 $0 $0
$0 $0 $0 $0:1155 $0:1155 $0:1155 $0:1155 $0:1155 $0
$0 $0 $0 $0 $0:0359 $0 $0 $0:0359 $0
$0 $0 $0 $0 $0 $0:1281 $0 $0 $0:1281
$0 $0 $0 $0 $0 $0 $0:1155 $0:1155 $0:1155
$0 $0 $0 $0 $:0359 $0 $0 $0:0359 $0
$0 $0 $0 $0 $0 $0 $0 $0 $0:1281
266666666666666664
377777777777777775
1� 10 1þ 2� 20 2þ 3� 30 3þ
SkewfCT ugT u�T u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
0 �0:003 �0:003 �0:003 �0:003 �0:003 0 0 0
0 �0:0001 0 0 �0:0001 0 0 0 0
0 0 �0:0044 0 0 �0:0044 0 0 0
0 0 0 �0:003 �0:003 �0:003 �0:003 �0:003 �:003
0 0 0 0 0:0001 0 0 �0:0001 0
0 0 0 0 0 �0:0044 0 0 �:0044
0 0 0 0 0 0 �0:003 �0:003 �:003
0 0 0 0 �0:0001 0 0 �0:0001 0
0 0 0 0 0 0 0 0 �:0044
266666666666666664
377777777777777775
Similar analyses can be undertaken that describe the cash flow effects of units moving from transient to absorbed states. For example,in a scenario where this retailer holds a unit in the gatekeeping function under regular design capacity, and is forced to dispose of thisitem as refuse, we would expect negative cash flow of �$15.24, with a variance of $0.0013, standard deviation of $0.0359, and skew of -0.0001, as shown:
I II III I II III
EfCK ugT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
�$15:48 $0 $8:52
�$15:24 $0 $8:76
�$15:52 $0 $8:48
$0 $0 $0
$0 $0 $0
$0 $0 %0
�$15:48 $0 $8:52
�$15:24 $0 $8:76
�$15:52 $0 $8:48
266666666666666664
377777777777777775
; VarfCK ugT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
0:0133 0 0:0133
0:0013 0 0:0013
0:0164 0 0:0164
0 0 0
0 0 0
0 0 0
0:0133 0 0:0133
0:0013 0 0:0013
0:0164 0 0:0164
266666666666666664
377777777777777775
;
I II III I II III
StdevfCK ugT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$0:1155 $0 $0:1155
$0:0359 $0 $0:0359
$0:1281 $0 $0:1281
$0 $0 $0
$0 $0 $0
$0 $0 $0
$0:1155 $0 $0:1155
$0:0359 $0 $0:0359
$0:1281 $0 $0:1281
266666666666666664
377777777777777775
; SkewfCK ugT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
�0:003 0 �0:003
�0:0001 0 �0:0001
�0:0044 0 �0:0044
0 0 0
0 0 0
0 0 0
�0:003 0 �0:003
�0:0001 0 �0:0001
�0:0044 0 �0:0044
266666666666666664
377777777777777775
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–101 95
At the aggregate level, for the total number of returns expected within the current period, we can see that the cash flow ramificationsare impactful. Using the same example as above, we would expect regular design/gatekeeping returns to cost the retailer �$5334 in cashflow this period. Interestingly, however, these costs are more than offset by the cash flows gained through restocking directly from thesame state, which gains the firm $14,892 for a total absorption of positive $9558 through regular state gatekeeping:
I II III
EfCashK ugT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
�$1084 $0 $2897
�$5334 $0 $14,892
�$10,864 $0 $28,832
$0 $0 $0
$0 $0 $0
$0 $0 $0
�$21 $0 $35
�$6702 $0 $11,556
�$11,868 $0 $19,453
266666666666666664
377777777777777775
; EfCashK ugT ux1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$1813
$9558
$17,968
$0
$0
$0
$14
$4855
$7586
266666666666666664
377777777777777775
Thus, the total periodic impact on cash flow from managing the entire reverse logistics process for this retailer under all designconditions is nearly $42,000:
I II III
EfCashK ug1�K u ¼ �$35,872 $0 $77,665 �
;
EðCashK uÞ ¼ $41,793
The total dollar variance associated with the system is $191,897,174:
I II III
VarfCashK ugT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
109,741 0 785,448
5,218,869 0 40,682,528
9,842,829 0 69,469,079
$0 0 0
$0 0 0
$0 0 0
178:1 0 485:7
6,916,764 0 20,569,339
10,371,034 0 027,930,878
266666666666666664
377777777777777775
; VarfCashK ugT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
895,189
45,901,397
79,311,908
0
0
0
663:7
27,486,104
38,301,912
266666666666666664
377777777777777775
;
I II III
VarfCashK ug1�K u ¼ 32,459,415 0 159,437,759 �
; VarðCashK uÞ ¼ 19,189,7174
The periodic standard deviation of the systemic cash flows is $12,397 and skewness is 1.236�1011, which serves as further indicationof this retailer’s risks associated with not managing the reverse logistics system:
I II III
StdevfCashK ugT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$331:27 0 $886:26
$2284:48 0 $6378:29
$3137:33 0 $8334:81
$0 0 0
$0 0 0
$0 0 0
$13:34 0 $22:04
$2629:97 0 $4535:34
$101:47 0 $304:39
266666666666666664
377777777777777775
; StDevfCashK ugT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$946:14
$6775:06
$8905:72
$0
$0
$0
$25:76
$5242:72
$320:8596
266666666666666664
377777777777777775
I II III
StdevfCashK ug1�K u ¼ $4700:92 $0 $11,471:68 �
StdevðCashK uÞ ¼ $12,397:51
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–10196
I II III
SkewfCashK ugT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
6,284,386 0 �1:2� 108
2:7� 109 0 �5:88� 1010
�4:58� 109 0 8:54� 1010
0 0 0
0 0 0
0 0 0
83:563 0 �378:41
5:85� 109 0 �2:9� 1010
3:09� 1010 0 �1:59� 1011
266666666666666664
377777777777777775
; SkewfCashK ugT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
�113,916,771
�56,092,670,707
80,859,667,906
0
0
0
�292:84
�23,326,666,369
�1:28� 1011
266666666666666664
377777777777777775
I II III
SkewðCashK uÞ1�K u ¼ 34,669,136,434 0 �1:811� 1011h i
; SkewðCashK uÞ ¼�1:2637� 1011
Next, we turn our attention to the total cash implications of the retailer allowing units to remain ‘‘stuck’’ within the system fromperiod to period, due to transient units received within the period not being processed to the point of absorption within the same period.For example, assuming that the number of units to be processed in the current period is regular, the retailer allowing goods to languishwithin the collection phase without sortation costs the retailer $852 in cash flow. Similarly, where the number of returns significantlyexceed design capacity, the systemic malfunction would result in an expected �$2933 cash flow deficit. The total systemic cash flowwould be �$8416 for units that lingered in transient states under all design capacities:
1� 10 1þ 2� 20 2þ 3� 30 3þ
EfCashT ugT u�T u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$0 �$69:12 �$12:96 �$9:84 �$157 �$29:52 $0 $0 $0
$0 �$468 $0 $0 �$492 $0 $0 $0 $0
$0 $0 �$936 $0 $0 �$2132 $0 $0 $0
$0 $0 $0 �$05 �$:88 �$:16 �$:46 �$7:29 �$1:37
$0 $0 $0 $0 �$154:96 $0 $0 �$697:32 $0
$0 $0 $0 $0 $0 �$293:28 $0 $0 �$2639:52
$0 $0 $0 $0 $0 $0 �$:02 �$:26 �$:05
$0 $0 $0 $0 $0 $0 $0 �$114:33 $0
$0 $0 $0 $0 $0 $0 $0 $0 �$198:92
266666666666666664
377777777777777775
EfCashT ugT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
�$279
�$960
�$3068
�$10
�$852
�$2933
$0
�$114
�$199
266666666666666664
377777777777777775
;
1� 10 1þ 2� 20 2þ 3� 30 3þ
EfCashT ug1�T u ¼ $0 �$537:12 �$948:96 �$9:89 �$805:28 �$2424:96 �$0:47 �$819:20 �$2839:76 �
EðCashT uÞ ¼�$8416
The systemic variance in cash flow for units remaining in transient states from period to period is $7.92�1011:
1� 10 1þ 2� 20 2þ 3� 30 3þ
Var CashT u
� �T u�T u¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
0 748:77 26:324 15:175 3884:83 136:58 0 0 0
0 41,408:78 0 0 50,805:87 0 0 0 0
0 0 130,560:8 0 0 677,385:1 0 0 0
0 0 0 :0004 :1217 :0043 0:035 8:91 :1938
0 0 0 0 1353:17 0 0 35,968:3 0
0 0 0 0 0 11,940:9 0 0 967,214:1
0 0 0 0 0 0 0:0001 :033 0:0013
0 0 0 0 0 0 0 2084:94 0
0 0 0 0 0 0 0 0 5481:5
266666666666666664
377777777777777775
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–101 97
VarfCashT ugT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
4811:67
92,214:65
807,945:9
9:26
37,321:46
979,155
0:034
$2084:94
5481:5
266666666666666664
377777777777777775
1� 10 1þ 2� 20 2þ 3� 30 3þ
VarfCashT ug1�T u ¼ 0 42,157:55 130,587 15:18 56,043:99 689,462:6 :035 38,062:17 972,695:6 �
VarðCashT uÞ ¼ 7:92� 1011
This results in a standard deviation of nearly $900,000 for the entire system, across all transient states and design conditions:
1� 10 1þ 2� 20 2þ 3� 30 3þ
StdevfCashT ugT u�T u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$0 $27:36 $5:13 $3:90 $62:33 $11:69 $0 $0 $0
$0 $203:4915 $0 $0 $225:4 $0 $0 $0 $0
$0 $0 $361:33 $0 $0 $823:03 $0 $0 $0
$0 $0 $0 $:02 $:33 $:07 $:2 $2:98 $:44
$0 $0 $0 $0 $36:79 $0 $0 $189:65 $0
$0 $0 $0 $0 $0 $109:28 $0 $0 $983:47
$0 $0 $0 $0 $0 $0 $:01 $:18 $:04
$0 $0 $0 $0 $0 $0 $0 $45:66 $0
$0 $0 $0 $0 $0 $0 $0 $0 $74:04
266666666666666664
377777777777777775
, StdevfCashT ugT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$68:38
$303:67
$898:86
$3:05
$193:19
$989:52
$:19
$45:66
$74:04
266666666666666664
377777777777777775
1� 10 1þ 2� 20 2þ 3� 30 3þ
StdevfCashT ug1�T u ¼ $0 $205:32 $361:37 $3:90 $236:74 $830:34 $:19 $195:10 986:25 �
StdevðCashT uÞ ¼ $889,937:40
And the system-wide skewness for the system is a robust �484,096,430:
1� 10 1þ 2� 20 2þ 3� 30 3þ
SkewfCashT ugT u�T u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
0 �7325:2 �48:29 �21:14 �86,566:9 �570:631 0 0 0
0 1,775,178 0 0 1,533,933 0 0 0 0
0 0 �28,456,887 0 0 �3:36� 108 0 0 0
0 0 0 �2:82� 10�6�:012 �7:1� 105
�0:0021 �8:42 �0:06
0 0 0 0 28,159:74 0 0 1,091,320 0
0 0 0 0 0 �169,342 0 0 �1:23� 108
0 0 0 0 0 0 �7:87� 10�8�0:0003 �1:8� 106
0 0 0 0 0 0 0 27,338:84 0
0 0 0 0 0 0 0 0 �85,463:4
266666666666666664
377777777777777775
SkewfCashT ugT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
�94,532:10
3,309,111
�34,752,686
�8:49
1,119,479:93
�123,619,670
�0:0003
27,338:84
�85,463:4
266666666666666664
377777777777777775
1� 10 1þ 2� 20 2þ 3� 30 3þ
SkewfCashT ug1�T u ¼ $0 1,767,853:15 �28,456,935 �21:14 1,475,526 �336,465,712 �0:0021 1,118,651 �123,535,792 �
SkewðCashT uÞ ¼ �484,096,430
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–10198
Finally, we concern ourselves with situations where the expected number of units returned is random. In the random returnssituation, transient–transient movements (or lack thereof) are seemingly more costly. For the focal retailer, the expected cash flow for asingle period under random conditions is �$27,240, which is significantly greater than the expectation for the more predictable situation(�$8416) shown previously in the more predictable scenario.
1� 10 1þ 2� 20 2þ 3� 30 3þ
EfCashNgT u�T u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$0 �$85:06 �$15:95 �$9:98 �$216:16 �$40:53 $0 $0 $0
$0 $3170:73 $0 $0 �$666:67 $0 $0 �$645:16 $0
$0 $0 �$6341:46 $0 $0 �$2888:89 $0 $0 $0
$0 $0 $0 �$10:17 �$:98 $:18 �$0:46 �$8:78 �$1:65
$0 $0 $0 $0 �$1721:79 $0 $0 �$833:12 $0
$0 $0 $0 $0 $0 �$3258:67 $0 $0 �$3153:55
$0 $0 $0 $0 $0 $0 �$4:68 �$0:28 $:06
$0 $0 $0 $0 $0 $0 $0 �$1756:23 $0
$0 $0 $0 $0 $0 $0 $0 $0 �$3054:02
266666666666666664
377777777777777775
1� 10 1þ 2� 20 2þ 3� 30 3þ
EfCashNg1�T u ¼ $0 �$3255:79 �$6357:41 �$20:15 �$2605:59 �$6188:27 �$5:14 �$2598:41 �$6209:27 �
EfCashNgT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
�$367:67
�$3837:40
�$9230:35
�$22:21
�$2554:91
�$6412:22
�$5:02
�$1756:23
�$3054:02
266666666666666664
377777777777777775
; EðCashNÞ ¼ �$27,240:03
The variance and standard deviation of the retailer’s system are similarly problematic, at $40,276,704 and $6346, respectively:
1� 10 1þ 2� 20 2þ 3� 30 3þ
VarfCashNgT u�T u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
0 60,725:54 12,488:23 5496:87 99,963:22 25,861:24 0 0 0
0 4,052,494:82 0 0 738,555:31 0 0 0 0
0 0 1,430,222:22 0 0 12,571,000:6 0 0 0
0 0 0 17:96 14:24 2:82 5:25 42:3 18:44
0 0 0 0 480,218:44 0 0 103,522 0
0 0 0 0 0 2,683,496:6 0 0 2,173,395:8
0 0 0 0 0 0 10:68 2:16 0:4864
0 0 0 0 0 0 0 742,302:35 0
0 0 0 0 0 0 0 0 2,036,846:26
266666666666666664
377777777777777775
1� 10 1þ 2� 20 2þ 3� 30 3þ
Stdev CashN
� �T u�T u¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$0 $246:43 $111:75 $74:14 �$316:17 $160:81 $0 $0 $0
$0 $2013:08 $0 $0 $859:39 $0 $0 $0 $0
$0 $0 $3782:89 $0 $0 $3570:85 $0 $0 $0
$0 $0 $0 $4:24 $3:77 $1:68 $2:29 $6:503 $4:29
$0 $0 $0 $0 $692:98 $0 $0 $321:75 $0
$0 $0 $0 $0 $0 $1638:14 $0 $0 $1474:24
$0 $0 $0 $0 $0 $0 $3:27 $1:47 $:70
$0 $0 $0 $0 $0 $0 $0 $861:57 $0
$0 $0 $0 $0 $0 $0 $0 $0 $1427:18
266666666666666664
377777777777777775
1� 10 1þ 2� 20 2þ 3� 30 3þ
VarfCashNg1�T u ¼ $0 4,113,220:4 14,322,710:5 5514:83 1,318,751:2 15,460,361:3 15:94 845,868:8 4,210,261 �
1� 10 1þ 2� 20 2þ 3� 30 3þ
StdevfCashNg1�T u ¼ $0 $2028:11 $3784:54 $74:26 $1148:37 $3931:97 $3:99 $919:71 $2051:89 �
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–101 99
VarfCashNgT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
909:30
2872:47
7353:74
22:78
1014:73
3112:38
5:44
861:57
1427:18
266666666666666664
377777777777777775
StdevfCashNgT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$30:16
$53:60
$85:75
$4:77
$31:85
$55:79
$2:33
$29:35
$37:78
266666666666666664
377777777777777775
VarðCashNÞ ¼ 40,276,704; StdevðCashNÞ ¼ 6346:39
The distribution continues to be highly skewed:
1� 10 1þ 2� 20 2þ 3� 30 3þ
SkewfCashNgT u�T u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
0 �36,670,494:7 �5,349,949:9 �2,444,457:8 �93,082,787:8 �8,698,940:5 0 0 0
0 �87,784,503,323 0 0 �1,955,086,667 0 0 0 0
0 0 �7:53� 1011 0 0 �1:7� 1011 0 0 0
0 0 0 �3241:18 �166:32 �26:45 �53:96 �2679:07 �258:8
0 0 0 0 �14,651,883,559 0 0 �1:56� 109 0
0 0 0 0 0 �1:01� 1011 0 0 �85865778086
0 0 0 0 0 0 �314:96 �7:86 �1:65
0 0 0 0 0 0 0 �1:55� 1010 0
0 0 0 0 0 0 0 0 �84678448820
266666666666666664
377777777777777775
1� 10 1þ 2� 20 2þ 3� 30 3þ
SkewfCashNg1�Tu ¼ $0 �87,821,173,818 �7:5� 1011�2,447,699 �16,700,053,180 �2:7� 1011
�368:91 �1:7� 1010�1:71� 1011
h i
SkewfCashNgT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
�146,246,631
�89,739,589,990
�9:2� 1011
�6425:77
�16,213,805,455
�1:87� 1011
�324:47
�15,481,846,852
�84,678,448,820
266666666666666664
377777777777777775
SkewðCashNÞ ¼�1:314� 1012
For the transient–absorbed case with multiple design capacities and random numbers of returns, the cash impactsare also significantly greater. Expected systemic cash flow from reverse logistics management is a positive $61551, primarily due tothe large inflows that can be recuperated from proper restocking decisions, including their tradeoff versus other absorptionoptions:
I II III
EfCashAgT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
�$2486:62 $0 $5456:55
�$12,240:36 $0 $28,051:30
�$24,930:50 $0 $54,309:36
$0 $0 $0
$0 $0 $0
$0 $0 $0
�$22:65 $0 $37:40
�$7205:95 $0 $12,426:00
�$12,761:11 $0 $20,917:70
266666666666666664
377777777777777775
; EfCashAgT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$2969:93
$15,810:94
$29,378:86
$0
$0
$0
$14:75
$5220:06
$8156:59
266666666666666664
377777777777777775
I II III
EfCashAg1�K u ¼ �$59,647:2 $0 $121,198:32 �
;
E CashK uð Þ ¼ $61,551:12
The cash flow variance yielded by these decisions is $549,577,582, with a standard deviation of $23,443 and skewnessof �1.98�1010. Thus, when the retailer is faced with a returns marketplace where the numbers of returns are less predictable,careful reverse logistics management has an even greater relative impact than in the more predictable scenario. Carefully managedreturns forecasting processes can aid in making key transient and absorption decisions, and result in better use of the firm’s
P.A. Horvath et al. / Int. J. Production Economics 129 (2011) 86–101100
cash resources:
I II III
VarfCashAgT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
577,897 0 2,786,874
27,482,669 0 144,347,016
51,832,533 0 246,485,522
$0 0 0
$0 0 0
$0 0 0
206 0 562
7,997,184 0 23,782,332
11,991,021 0 32,293,766
266666666666666664
377777777777777775
; VarfCashK ugT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
3,364,771
171,829,685
298,318,055
0
0
0
767:39
31,779,516
44,284,787
266666666666666664
377777777777777775
I II III
VarfCashAg1�K u ¼ 99,881,510 0 449,699,072 �
; VarðCashAÞ ¼ 549,577,582
I II III
StdevfCashAgT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$760 0 $1669
$5242 0 $12,014
$7199 0 $15,700
$0 0 0
$0 0 0
$0 0 0
$14 0 $24
$2828 0 $4877
$3463 0 $5683
266666666666666664
377777777777777775
; StDevfCashK ugT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
$1834
$13,108
$17,272
$0
$0
$0
$28
$5637
$6655
266666666666666664
377777777777777775
I II III
StdevfCashK ug1�K u ¼ $9994 $0 $21,206 �
StdevðCashK uÞ ¼ $23,443
I II III
SkewfCashAgT u�K u ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
75,942,797 0 �1:3� 107
3:3� 1010 0 �6:2� 109
�5:5� 1010 0 9:01� 109
0 0 0
0 0 0
0 0 0
103:89 0 �17:33
7:03� 109 0 �1:3� 109
4:06� 1010 0 �6:6� 109
266666666666666664
377777777777777775
; SkewfCashK ugT u�1 ¼
1�
10
1þ
2�
20
2þ
3�
30
3þ
63,266,577
2:64� 1010
�4:6� 1010
0
0
0
86:56
5:69� 109
�3:4� 1010
266666666666666664
377777777777777775
I II III
SkewfCashK ug1�K u ¼ 2:5� 1010 0 �5:2� 109h i
; SkewðCashK uÞ ¼�1:98� 1010
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