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This paper examines the financial impact of technological decisions firms face when introducing a new product which, as firms increasingly assign responsibility for the recovery of their used products, must be expanded to include their influence on available options on how to deal with used products. The integrated choice between a low (disposal) or high (remanufacturing) level of product recovery and, in the second case, at which time to start remanufacturing activities and whether the firm would dispose of initial product returns or store them until start time result in three generic options: (a) design for single use, (b) design for reuse, and (c) design for reuse with stock-keeping. After introducing a basic dynamic framework consisting of a product life cycle for the demand and a similar development for an external return stream, the Net Present Values of the relevant payments connected with each option are given and optimal policies for options (b) and (c) are derived. An extensive numerical study is used to examine the potential benefits from using the inventory and the applicability of simple heuristic rules for stock-keeping and for determining when to start remanufacturing [PUBLICATION ABSTRACT]
OR Spectrum 28:2151 (2006)DOI 10.1007/s00291-005-0011-6REGULAR ARTICLEPublished online: 22 November 2005
Springer-Verlag 2005Abstract This paper examines the financial impact of technological decisions
firms face when introducing a new product which, as firms increasingly assign
responsibility for the recovery of their used products, must be expanded to include
their influence on available options on how to deal with used products. The
integrated choice between a low (disposal) or high (remanufacturing) level of
product recovery and, in the second case, at which time to start remanufacturing
activities and whether the firm would dispose of initial product returns or store them
until start time result in three generic options: (a) design for single use, (b) design for
reuse, and (c) design for reuse with stock-keeping. After introducing a basic
dynamic framework consisting of a product life cycle for the demand and a similar
development for an external return stream, the Net Present Values of the relevant
payments connected with each option are given and optimal policies for options (b)
and (c) are derived. An extensive numerical study is used to examine the potential
benefits from using the inventory and the applicability of simple heuristic rules for
stock-keeping and for determining when to start remanufacturing.Keywords Reverse logistics . Technology choice . Product life cycle . Inventory
management1 IntroductionWhen developing new products and setting up production facilities, firms often
have the choice between different technologies to manufacture the product. BesidesThe author wishes to thank two anonymous referees for their suggestions that considerably
helped to improve the paper.R. Kleber
Faculty of Economics and Management, Otto-von-Guericke-University Magdeburg,
P.O. Box 4120, 39016 Magdeburg, GermanyE-mail: [email protected] KleberThe integral decision
on production/remanufacturing technology
and investment time in product recovery22 R. Kleberquality and service aspects, this decision has a major impact on direct (variable)
production costs and necessary capital expenditures in building and maintaining
new facilities or modifying existing ones. In the context of reverse logistics, an
additional issue has to be considered, since growing recycling and reuse legislation
and environmental awareness of customers forces firms to take back products from
its customers after use. At this point, the selected production technology also affects
the possible options on how to deal with returned used products, and it has to be
extended by the question of whether to produce a product for a single life time only,
or in such a way that facilitates its reuse after some recovery process (e.g. rework,
upgrading or remanufacturing, see de Brito and Dekker (2004) for an overview on
available options). This can yield an additional profit, since some of the added value
will not be lost under certain recovery options, as it would be the case with materialrecycling or disposing of the returned item. On the other side, there can be higher
expenses for setting up production facilities, as well as higher direct production unit
costs, that are caused by the necessity to add properties to the product in order to
make it recoverable, which is illustrated by the following real-life example.Case: CopyMagic (see Thierry et al. 1995; Thierry 1997). As a multinational copier
manufacturing company, CopyMagic sells its products in all segments of the copier market,
mainly by using leasing contracts. This creates a continuous flow of returning used products
made of off-lease copiers. Depending on the returned product one or more of the following
product recovery options is used: repair, cannibalization, remanufacturing and recycling. The
last two options particularly require a special product design which is different from a
classical single use product which can only be disposed of after use. Design for recycling
requires a reduction in the number of used materials, a replacement of non-recyclable with
recyclable materials. Design requirements for remanufacturing go even further. Here it must
be ensured that a product or its components are (in principle) capable to be used for more than
one life time of the product for it to be sold on a as-good-as-new basis. This results in higher
production expenses and together with outlays for product recovery the question has to be
answered which recovery option is preferable.Although there exists an enormous amount of literature on operative issues in
reverse logistics (for literature surveys see Dekker et al. 2004), aspects of financial
justification, being highly influential to investment decisions, have widely been
neglected so far. Models usually deal with stationary situations and focus on average
cost/profit. Debo et al. (2005) for instance consider the problem of technology
selection in connection with market segmentation. Thereby, they assume a situation
where remanufactured products are valued less than newly produced items, but both
compete on a market where customers have a heterogeneous willingness-to-pay. The
chosen technology is characterized by a level of remanufacturability influencing
variable production costs, but investment expenditures are not considered. The
objective is to select a remanufacturability level which maximizes average profit.A large stream of research on strategic issues in reverse logistics focuses on the
design of re-distribution networks in a single-period (see, e.g. Barros et al. 1998;
Louwers et al. 1999) and less commonly a multi-period (Realff et al. 2004) setting
with given product characteristics. Krikke et al. (2003) additionally include the
product design issue together with ecological aspects, yielding a very complex butThe integral decision on production/remanufacturing technology 23still static Mixed Integer Programming model. In all these approaches the decision
at which time to set up the respective facilities has already been made. Our more
aggregate approach focuses rather on the timing of investment decisions which
becomes important in a dynamic environment. By neglecting facility location and
detailed capacity acquisition, for instance expenses for setting up facilities are set in
such a way that a sufficient capacity is available, general insights can be obtained
using an analytical approach.Durable products tend to remain with the customer for a considerable amount of
time compared to the time horizon where they are sold. Demand may be subject to
dynamic processes like the product life cycle and thus, a static (equilibrium) analysis based on average costs is often not appropriate. A dynamic DCF framework is
required which also takes into account the time value of money and, especially, of
investments. In recent contributions to dynamic product recovery (see Kiesmller
et al. 2004, for an overview), it has been assumed that both, production and remanufacturing, facilities already existed. Since introducing a technology for remanufacturing may require additional investments, it remains to be seen if these
investments will pay off or not. Further, when considering more than a single
technology with different investments and variable unit costs, the optimal product
recovery technology has to be determined.Regarding the remanufacturing activities, it is seldom preferable to start it
immediately when production starts, as this often requires a considerable capital
commitment while not yet having a large number of returns available. For instance
in the case of engine remanufacturing in car industry (see, e.g., Seitz and Peattie
2004) the usual life time of an engine must be considered while specialized
equipment is required that sometimes must meet even higher standards than that
used for producing new engines. New production technologies like the use of
aluminum instead of steel for engine components necessitate high investment in the
remanufacturing process as well. Therefore, a decision has to be made when to
introduce the process. Another question is whether to hold returned items in a
strategic inventory for a later use or just to dispose of excess returns. Practitioners
often apply simplistic rules, like keeping all returns and disposing of none (see
Kiesmller et al. 2004). Both issues are related to the problem of timing a capacity
expansion known from production/inventory theory (see, e.g., Slack and Lewis
2002), where a (serviceables) inventory is used to postpone capacity expansion
compared to a strategy without stockkeeping where capacity expansion must lead
demand in order to avoid backorders. It is questionable whether a recoverables
inventory would influence the remanufacturing investment time in the same way.This paper intents to investigate the effects of integrating technology selection
with investment time decisions. Within this paper it is assumed that recovered
products or their main components can be regarded as a perfect substitute to newly
produced items, and thus be sold to the same market. Even though a simplistic
situation is analyzed by assuming deterministic demand and return dynamics which
are not to be influenced by the decision maker, i.e. used products automatically
return and cannot be rejected and all demand must be satisfied immediately, insights
into the technology selection process can be gained by minimizing the discounted
cash outflows caused by appropriate investment and return inventory building24 R. Kleberdecisions. An extension which also considers the influence of a stochastic environment can be achieved by performing a scenario analysis.The paper is organized as follows. In Section 2, we propose a dynamic environment for strategic decision making in the context of reverse logistics. Three
investment projects representing different environmental policies are introduced,
and for each one of them optimal policy parameters are determined. Main results of
a numerical study, in which effects of a strategic inventory were examined and a
comparison of simple heuristic rules with optimal strategies took place, are
presented in Section 3. The last section provides conclusions and further research
possibilities.2 A dynamic model for strategic decision makingIn this section we propose a generic dynamic environment for investment decisions of
product recovery based on simple assumptions regarding the product life cycle and
the availability cycle for returns. Afterwards, three basic investment projects, each of
which representing a different environmental policy, are introduced and the respective
optimal dynamic policies, that describe how production, remanufacturing, and
disposal decisions evolve over time, are characterized and individual policy parameters are derived. Finally, rules on how to determine the optimal policy parameters
are deduced from minimizing the Net Present Value (NPV) of the respective cash flow
series within the planning horizon which for analytical convenience is set to infinity.2.1 A generic dynamic environmentIn the following, we consider a demand/return scenario as illustrated in Fig. 1
which complies with the following assumptions:Fig. 1 Dynamic environment with unimodal demand and return functions satisfying required
assumptionsThe integral decision on production/remanufacturing technology 25A.1. Demand d(t) is assumed to be a deterministic continuously differentiable
function of time showing the typically unimodal shape of a product life
cycle with its maximum located at tdmax>0. At the end demand must vanish,i.e. limt!1 dt
0:A.2. Returns u(t) are not available prior to a time point >0 (u(t)=0t<) and
otherwise given by an unimodal function of time with u(t)>0t
representing the availability cycle of returns. Further, the return function
has a maximum at tumax and is continuously differentiable for t>.As
demand, returns finally vanish, i.e. limt!1 ut 0.A.3. There exists at most a single intersection point of demand and return
functions tImax{tdmax, } for which it holds u(t)<d(t)if t<tI and u(t)>d(t)if
t>tI. We thus presume that returns do not exceed demand during the growth
phase of the product life cycle. If demand always exceeds the return rate there
is no intersection, and tI is set to infinity.It is intuitively clear that at tI the product life cycle has already entered its
decline phase, i.e. demand is decreasing and intersects the return rate from above.
Using and tI, one can distinguish between three different regions. In Region I,
which ends at , there are practically no returns accessible. Demand has to be filled
completely by producing new items. Region II shows less returns than demand
(excess demand) and Region III is characterized by returns exceeding demand
(excess returns) and decreasing demand. If the remanufacturing option is available,
product returns can be used in both regions (II and III) to satisfy part or all of the
demand. Moreover, in the last region it makes no sense to keep more than currently
needed returns, because returns remain larger than demand until the end of the
products life cycle. Therefore, excess returns are to be disposed of.Since we do not assume an explicit relation between demand and return
functions, the return function might show a jump at , i.e. u() is much bigger
than 0. Such a situation is explicitly regarded during our analysis.2.2 Three investment projectsSeveral types of cash outflows are considered: investment expenditures for
production and remanufacturing processes, constant production, remanufacturing
and disposal per unit payments as well as out-of-pocket inventory holding costs.
The discount rate is denoted by . Revenues, as well as payments connected with
the take back of used products are not taken into account, since both demand as well
as returns are assumed to be given and thus not subject of our considerations.
Investment expenses include all discounted outlays for acquiring, maintaining,
extending, and (possibly) salvage revenues for selling facilities with sufficient
capacity. Therefore, restrictions on operative processes are supposed to never become binding.26 R. KleberRegarding the environmental policy of the firm and the existence of a strategic
inventory that keeps returns for a later use, the following capital investment
projects are considered:(a) Design for single use. Products are designed in such a way that they cannot be
remanufactured. Thus, all returns have to be disposed of at costs cw, which can
be positive, if actual payments are necessary to get rid of the used product, or
negative, if it has a positive salvage value. The corresponding investment into
the production process at t=0 is Kps>0. Direct unit production costs are given
by cps>0.(b) Design for reuse. Products are designed such that all returned products can
serve as perfect substitutes to newly produced items after remanufacturing. This
also implies that products might be remanufactured several times, i.e. we
assume unlimited (or at least sufficient) durability. See Geyer and Van
Wassenhove (2005) for a discussion on the impacts of limited remanufacturability. Investment expenditures for setting up production at t=0 amount to
Kpr>Kps, producing each item would cost cpr>cps. Both values exceed the above
expenses for single use design because of the additional requirements which are
needed for a later remanufacturing. Returning used products can be remanufactured at unit costs cr>0 after introducing a remanufacturing process at
time tr. This leads to a cash outlay of Kr>0. In order to assure that this
investment project constitutes a viable option a benefit must be realized if,
instead of simultaneous production of a new item and disposal of the returned
one, the latter is merely remanufactured, i.e. there exists a positive direct
recovery cost advantage cpr+cwcr>0. Otherwise, no remanufacturing would
take place. A possibility to store returns is not considered; all returns arriving
before tr must be disposed of.(c) Design for reuse with strategic inventory. In addition to (b), it is now possible to
keep returns for later use in a recoverables inventory, e.g. in order to put items into
the inventory at a time where remanufacturing is not yet possible. The inventory
level at t is denoted by yu(t). Out-of-pocket holding costs are assumed to be
proportional to the time and quantity of used products on stock. The respective
holding cost parameter is denoted by hu>0. A meaningful solution is assured if it
is not advantageous to hold unneeded returned products as opposed to disposing
of them, i.e. hu>cw. If this were not the case, disposal would not take place,
because delaying the disposal of an item saves interests on the required expenses
which would be higher than out-of-pocket costs incurred by holding the item.2.3 Valuation of investment project (a)design for single useLet p(t), r(t), and w(t) denote production, remanufacturing and disposal rates
at time t, respectively. When assuming a single use product, the optimal dynamic policy is obviously to dispose of all returns immediately upon receipt,The integral decision on production/remanufacturing technology 27i.e. w*(t)=u(t). Product requirements are satisfied by producing new items
(p*(t)=d(t)). This leads to the following expression for the Net Present ValueNPVa Kt cpsdt cwutdt: (1)NPVa can be considered as a benchmark against which the financial benefits of
the other investment projects have to be compared.2.4 Investment project (b)design for reuseDynamic policy The remanufacturing option is available after a capital expenditure
of Kr at time tr, subdividing the planning horizon into two parts. Before tr, only
production can be used to satisfy demand. Since returned items cannot be stored,
the same policy applies as for investment project (a)p* t
s
p Z 10edt
; r* t
0; w* t
ut
8t < tr:The optimal policy for ttr is to remanufacture as many units as possible, to produce
(if necessary) for excess demand, and to dispose of remaining returns, yieldingp* t
max; 0
fgdt
ut
; r* t
min; ut
fgdt
;w* t
max; 0
fg8t tr:The dynamic policy is shown in Fig. 2.Optimization of policy parameters The Net Present Value of investment project (b)
NPVb depends on the investment time tr. When choosing it, a trade-off has to beFig. 2 Optimal decisions in investment project (b)ut
dt
28 R. Kleberstruck between the lower discounted value of investment expenses Kr if the
introduction of the remanufacturing process is postponed and a larger realized
recovery cost advantage if it is placed earlier. The following non-linear optimization problem must be solved in order to determine trAlthough the properties of this function depend to a certain extent on the
underlying demand and return functions, it can be shown (for a derivation see the
Appendix) that based on assumptions A.1A.3 from Section 2.1, NPVb is strictly
decreasing and convex for tr<, and there exists a time point t after which it finally
becomes a strictly decreasing and convex function. Depending on the parameters,
either there exists a local minimum which is followed by a local maximum or the
function is decreasing during the whole planning period [0, ). The first derivative
of NPVb is a continuous function except for tr=, given a jump discontinuity of the
return function u(t)at . By exploiting the first and second order optimality
conditions local properties regarding the optimal investment time tr* are derived as
expressed by Proposition 1.Proposition 1 If it exists, a finite investment time tr* must be located within the
half open interval [, min{tumax, tI}), and one of the following situations must
apply(1) utr*
crp cw cr
Kr and dut
dtttr*> 0 if tr* > ;(2) utr*
crp cw cr
Kr if tr* :Proof For all proofs see the Appendix.Proposition 1 states that at tr*, the current cost advantage of remanufacturing
u(tr*)(cpr+cwcr) must at least earn interests on the investment, i.e. Kr. If (1) equality
holds, the remanufacturing rate must increase at tr* to start paying back the investment expenses. This is not possible at or later than either the time where returns reach
their maximum tumax or the intersection time of demand and return rate tI, because in
both cases the remanufacturing rate no longer increases. Thus, investment time tr*
must lie in an interval given by [, min{tumax, tI}). As a special case (2), at time
there may be a jump in the return rate (and consequently in the potential remanufacturing rate) from u()=0 up to a point u(+)>0 where a higher remanufacturing
cost advantage may be realized than interests on investment expenses require.minNPVb Krp Z
tret crpdt
cwut
hidt etr Krtr(2) Z10etcrp max dt
ut
; 0
fg cr min dt
; ut
fgtr
cw max ut
; 0
fg
dtdt
The integral decision on production/remanufacturing technology 29Resorting the first order condition in Proposition 1 leads to a critical value for the
return rate ucritucrit Kr
crp cw cr(3)which can be interpreted as a dynamic break even point. If the investment into a
remanufacturing facility takes place at all, then it is placed at a point in time where
an increasing return rate surpasses ucrit. Since the remanufacturing rate is limited to
the available returns, a finite investment time tr* does not exist if the return rate
never exceeds the critical return rate ucrit within Region II. In such a case, the initial
interest rate on the investment exceeds the maximum possible current cost advantage of remanufacturing (to be earned at min{tumax, tI}).So far, only local conditions ensuring the possibility to start paying back the
investment expenditures required for the remanufacturing facility have been considered. Of course, these investments have to be paid off completely. If the expenses are rather high, even in the optimal case, the cumulative cost advantage may
not be sufficient for amortization. In this case, the optimal investment time is
infinity. By comparing the values of the objective of the finite candidate satisfying
Proposition 1 with its limit as time approaches infinity, the following sufficient
(global) condition for optimality of a finite investment time tr* results.Proposition 2 For the optimal investment time tr* it must hold that the total
realized advantage of remanufacturing discounted to tr* at least equals the expenses needed for setting up the remanufacturing facilityKr Z 1
tr*e ttr*
crp cw cr
hi min dt
; ut
fgdt: (4)Comparison with investment project (a) There exists no simple rule for determining the best of the two investment projects, but by comparing (optimal) Net
Present Values it can be stated that investment project (b) is preferable to (a) if the
total discounted net advantage of remanufacturing Arb0Abr Z 1
tr*tr*Kr (5)exceeds the increase of the total discounted expenditures for the production process
Dpb>0Dbp Kr
pet crp cw cr min dt
; ut
fgdt eKsp
crp csptdt
dt: (6)It is easy to see that if there exists no finite optimal investment time for
investment project (b), then Arb=0 holds and investment project (a) should be
chosen. If Arb is positive, then preferability of the design for reuse depends on the
increase of initial investments in the production process compared with a single use
production as well as on the increase of direct production costs.Z 1e030 R. Kleber2.5 Investment project (c)design for reuse with strategic inventoryDynamic policy As an extension to investment project (b), returns can be stored in a
recoverables inventory. As such, the problem becomes truly dynamic, because the
decision to store a returned item influences future possibilities of remanufacturing. In
analogy to investment project (b) and given a value for tr and for the systems state at
this time yu(tr), the planning horizon can be subdivided into two parts. Prior to tr,the
question arises when to start keeping returns in order to achieve the desired stock.
Obviously, it is not useful to dispose of returns during the stock-keeping period,
because otherwise one could have started gathering later and thus, saved holding
costs. Therefore, to each value yu(tr) a corresponding time point tetr can be given
where disposal stops and all returns are put to stock, being defined byR tr
te usds yutr . Since stock-keeping can start earliest at,i.e. te, the maximum possible
quantity on stock at tr is given byyu tr
Z trus
ds: (7)Thus, optimal decisions in the first part are given byp* t
dt
; r* t
0; w* t
ut
8t < te;p* t dt; r* t
0; w* t
0 8te t < tr:The optimal solution of the second part is derived by using the optimal control
framework presented by Minner and Kleber (2001) and is given as follows. First, the
recoverables inventory is depleted by filling excess demand d(t)u(t) from
remanufacturing stored returns. This is completed at a time point txtr, characterized
byR tx
tr ds us
ds yutr: Completion time tx must not be larger than tI
because afterwards returns always exceed the demand rate and having an inventory
is no longer necessary. Beside Eq. (7), this gives another condition for yu(tr)yu tr
Z tI
trus
ds: (8)ds
After tx, the same policy is used as in investment project (b) because it is not
useful to build up stock again, yielding the following optimal decisionsp* t
0; r* t
dt
; w* t
0 8 tr t < tx;p* t max; 0
fgdtut; r* t
min; ut
fgdt
;w* t maxutdt; 0
fg 8t tx:The dynamic policy in investment project (c) is depicted in Fig. 3. Of course,
this policy requires sufficient capacity and a high flexibility both in the production
as well as in remanufacturing process. This is for instance the case if workers that
normally are employed to produce new items, can easily be transferred to remanufacture used products. In the presence of capacity constraints, a more complex model would be necessary for which only numerical results can be obtained.The integral decision on production/remanufacturing technology 31Fig. 3 Optimal decisions in investment project (c)Optimization of policy parameters In contrast to investment project (b), the
optimal solution to policy class (c) not only consists of investment time tr*, but also
the corresponding recoverables stock yu*(tr*) has to be determined. Equivalently,
and probably even more interesting than the actual stock value, starting time of
keeping returns te* may be decided upon. In the following, we restrict ourselves to
find the best finite solution candidate. Since the option of investing never, i.e.
te=tr=, still belongs to the set of possible solution candidates, it has to be
considered in order to find the global optimum.Restricting to finite solution candidates, we get the following optimization
problem Eqs. (9)(14).minte;tr NPVc Krp R te0 et crpdt
cwut
hidt R trte et crpdt
huyu t
hidt etr Kr R tx
tr crdt
huyu t
dt(9)1 Ret crp max dt
; 0
fgtx; 0
fg
hidt(9)ut
cr min dt
; ut
fgcw max ut
dt
withyu t; te; tr; tx
R t
te us
ds for t 2 te; tr8
<
:(10)R trte usds R ttr dss for t 2 tr; tx
0 otherwiseusdand tx being implicitly defined by a function f (te, tr, tx)tx : fte; tr; tx
Z tr
teus
ds Z tx
trus
ds 0 (11)ds
32 R. Klebersubject to the restrictionsThe objective function (9) incorporates all payments caused by applying
optimal policies in each of above distinguished parts. Function (10) is used to
determine the inventory level and Eq. (11) gives an implicit definition of the time
point tx where the inventory is depleted. Constraint (12) ensures continuity of the
objective by limiting the admissible set and Eq. (13) is needed in order to assure a
meaningful solution. Restriction (14) is equivalent to txtI but technically it is
easier to handle. This inequality represents remaining excess demand between tx
and tI, which must be non-negative.Due to the quite general assumptions on demand and return functions, objective(9) is neither a (quasi-)convex function in each nor in both decision parameters.
Moreover, even the admissible region is not convex because of Eq. (14). Hence,
conditions derived below by using standard methods of nonlinear programming are
only necessary for optimality. As a consequence, a solution candidate can represent
a local minimum, maximum, or a saddle point. Further, as there might exist several
solution candidates for a single problem instance, in order to find the optimal
solution the respective objective values need to be compared.In the following, four different types of solution candidates are distinguished.
For ease of representation, a candidate is given by a triplet (te, tr, tx), bearing in
mind that tx is a function of the other two time points.Proposition 3 (solution candidates) If (te*,tr*,tx*) is an optimal solution to
problem (9)(14), te*<tr*(<tx*) must hold, and one of the following four cases applies(1) <te*, tx*<tI (interior solution)(2) =te*, tx*=tI (complete use of interval [, tI])(3) =te*, tx*<tI (availability of returns is binding restriction)(4) <te*, tx*=tI (availability of excess demand is binding restriction)Now, by exploiting first order necessary conditions, properties of these cases are
discussed. Firstly, a general condition regarding the optimal holding time tx*te*is
given. te; (12)te tr; (13)Z tI
txus
ds 0: (14)ds
The integral decision on production/remanufacturing technology 33Proposition 4 (maximal holding time) If (te*, tr*,tx*) is an optimal solution to problem (9)(14), it must hold that tx*te* does not exceed a maximal holding time , i.e.t
xt
e1ln0@ 1
A : : crp cr hu(15)cw huMaximal holding time as defined in Proposition 4 comprises the same marginal
criterion as known from Kiesmller et al. (2004) which balances the cost advantage
of storing an otherwise disposed of returned item between te* and tx* in order to
replace production by remanufacturing at tx* and the required holding costs.The following Propositions 58 present results for each of the cases.Proposition 5 (Case (1)interior solution) Atriplet (te, tr, tx) with <te<tr<tx<tI
is a solution candidate to problem (9)(14) of Case (1), if it satisfies the following
equationstx te ; (16)
etr crp cr
hidtr
huZ tx
tretdt d tr
etx crp cr
hidtr
etr Kr:(17)Equations (16) and (17) follow from setting the first derivatives of the objective(9) to zero and can be interpreted as follows. Since txte equals the maximal
holding time , the decision maker must be indifferent between (1) disposing of a
(marginal) return unit arriving at te, and producing a new one to meet demand at tx
or (2) holding this item until tx when it is remanufactured to serve demand. Next, at
tr one needs to be indifferent between starting the remanufacturing process and
thereby realizing the direct cost advantage of remanufacturing immediately or to
postpone it which saves interests on the investment expenses. Then, a (marginal)
demand d(tr) is served from producing new items and the thus saved (marginal)
return is kept until tx which results in holding costs and lowers the discounted value
of the direct remanufacturing cost advantage.Using Eq. (16) together with the definition of the case requires tI>, which has
to be assured first in order to find a Case (1) solution candidate. Then, simultaneously
solving Eqs. (11), (16)and (17)for (te, tr, tx) yields the candidate, given it exists.Proposition 6 (Case (2)complete use of interval [, tI]) A triplet (te, tr, tx) with
=te<tr<tx=tI is a solution candidate to problem (9)(14) of Case (2), if the
following conditions are satisfiedecwdtr
huZ tretdt d tr
etr crp cr
hidtr
etr Kr; (18)
etr crp cr
hidtr
huZ tI
tretdt d tr
etI crp cr
hidtr
etr Kr: (19)34 R. KleberInequality (18) implies that it would be preferable to put additional returns in stock
at te for use at tr by simultaneously lowering te and tr, even at the cost of an earlier
investment. But this is not possible because =te. Likewise, using Eq. (19), the value
of the objective could be lowered by postponing investment time tr. Thisisalso
forbidden because we would need to increase tx=tI which again is not possible.A Case (2) candidate only may exist, if tI is satisfied. Thus, it is not possible
to have a planning situation where we could obtain solution candidates in both Case(1) and Case (2) simultaneously. If the Case (2) pre-requirement is fulfilled, from Eq.(11) one gets a value for tr which is verified if Eqs. (18) and (19)hold.Proposition 7 (Case (3)availability of returns is binding restriction) A triplet
(te, tr, tx) with =te<tr<tx<tI is a solution candidate to problem (9)(14) of Case(3), if the following conditions are satisfiedtx (20)
etr crp cr
hidtr
huZ tx
tretdt d tr
etx crp cr
hidtr
etr Kr:(21)From Eq. (20) we know that maximal holding time is not yet reached. But, in
contrast to Case (2), from Eq. (21) we are indifferent regarding the postponement of
tr. Placing it earlier is also not possible, because te is fixed to . A Case (3)
candidate is to be found by simultaneously solving Eqs. (11) and (21) for tr and tx
by assuming te=. The result is a solution candidate if inequality (20) is satisfied.Proposition 8 (Case (4)availability of excess demand is binding restriction) A
triplet (te, tr, tx) with <te<tr<tx=tI is a solution candidate to problem (9)(14) of
Case (4), if the following conditions are satisfiedtI te (22)ete cwdtr
huZ tr
teetdt d tr
etr crp cr
hidtr
etr Kr: (23)As before, but with tx fixed to tI, Eq. (22) implies to decrease te which is not
possible without changing tr. Choosing tr requires indifference between disposing a
(marginal) returned item at te or using it to lower tr which in turn causes an increase
in associated holding and interest expenses due to sooner investment but it also
replaces production by remanufacturing at tr. The determination of a Case (4)
candidate requires to simultaneously solve Eqs. (11) and (23) for te and tr assuming
tx=tI. The result is a solution candidate if inequality (22) is satisfied.Putting together the results of the parameter optimization, a solution algorithm
can be derived which is given in Fig. 4.The integral decision on production/remanufacturing technology 35Comparison with investment project (b) Since investment project (c) is a
generalization of (b), it leads to a reduced Net Present Value. Another interesting
question is how the possibility to hold returns for later use affects investment time tr.
Unfortunately, there is no unequivocal answer. An aspect that allows for postponing
the investment time is that it no longer has a direct effect on the remanufacturability of
returns since these also can be put to stock and remanufactured later. Other aspects
make it possible to start remanufacturing earlier, e.g. a higher direct cost advantage of
remanufacturing can be realized at tr because demand is sourced completely from
remanufacturing returns. In order to gain more insight into this question we performed
a numerical investigation which is presented in the next section.3 Numerical investigationThe purpose of this study was threefold. Firstly, a pre-test should show that all
types of (finite) solution candidates of investment project (c) as presented in
Proposition 3 are relevant for determining the optimal solution. Further, an
assessment of the potential benefit derived from permitting stock-keeping had to be
performed and finally, the influence of a strategic recoverables inventory on the
investment time tr was assessed. A second test was used to clarify, under which
conditions simple heuristic rules being relevant for practical application perform
sufficiently well. Since the effects of changes in the interacting parameters are
manifold, we decided to perform the study based on a large number of randomly
generated examples.In this pre-test study we used a demand function according to the continuous
version of the well-known Bass model2 : (24)According to Mahajan et al. (1993), this model and its revised forms have been
proven to have good predictive validity and have been successfully applied in retail
service, industrial technology, pharmaceutical and consumer durable markets.
M=100,000 is total market demand and the other two parameters were set toFig. 4 Solution algorithmdt
MP P Q
ePQ
t2P QePQ
t
36 R. KleberP=0.01, Q=0.3, i.e. values which lie in typical ranges for consumer durable
products (Sultan et al. 1990). Demand shows a maximal rate of 8,008 units per time
unit located at tdmax=11 The return function u(t) is given by
where >0 denotes the average duration of use and F(0, 1] stands for the fraction
of previous demands which becomes available for remanufacturing. Since we did
not have real-live data, the parameters for each one of 30,000 instances were
generated from uniform probability distributions over each of the following
ranges, partly including extreme values: Range(F)=[0; 1], Range()=[0; 12],
Range()=[0.05; 0.15], Range(cpr)=[4; 5], Range(cr)=[1; 3], Range(cw)=[1; 1],
Range(Kr)=[20,000; 100,000] and Range(hu)=[0; 1]. Since the cash outlays for
setting up the production facility were not relevant for our comparison, we set Kpr
equal to zero.In total, 10,658 examples showed finite optimal solutions for investment
project (c) according to Cases (1)(4). Of these, 4,685 (44.0%) belonged to Case(1), 90 (0.8%) to Case (2), 5,872 (55.1%) to Case (3), and only 11 (0.1%) examples
were Case (4) solutions. Although all cases are relevant for determining the optimal
solution these numbers show that in more than half of all considered instances it
was optimal to immediately start storing returns. Slightly less instances showed an
interior solution.Next, by comparing the optimal objective values for investment projects (b)
and (c), we found that the benefit from keeping stock averaged to about 2% but
the maximum difference was more than 11%, being found in a scenario with the
following parameters: F=0.63, =0.60, =0.14, cpr=4.23, cr=1.07, cw=0.93,
Kr=91,628, hu=0.16. Thus, taking into account that aside of operational expenses
also investment expenditures have been considered, savings amount to a remarkable
sum of money.Regarding the investment time tr, the results indicate that it is usually (i.e. in81.7% of all considered examples) postponed due to the strategic inventory, except
for cases where the optimal investment time is infinite in investment project (b) but
finite in (c) because of the additional benefit from storing returns. This happened
for 15.7% of all examples. But 280 instances (2.6%) including all Cases (1)(4)
exhibited the opposite behavior. Particularly noteworthy, all examples where the
availability of excess demand was a binding restriction (Case (4)), exhibited an
earlier investment time when allowing for stockkeeping. This result was confirmed
by another 2,000 instances, which were generated in order to increase the number
of Case (4) solutions, where we changed the ranges for the following parameters:
Range(F)=[0.8; 1], Range()=[0; 2], Range()=[0.1; 0.15], Range(Kr)=[60,000;
100,000], Range(hu)=[0.1; 0.3]. Thereby, 161 (11.5%) out of 1,396 finite optimal
solutions were of Case (4).In a second test we compared the performance of four simple heuristic rules,
which are described in the following. While the first two neglect the possibility of
keeping stock and just try to select an appropriate investment time, H3 and H4 use
more or less sophisticated methods to control storing of returns for later use by
following the most common investment project (c) cases identified before. Since
the change of the optimal investment time when allowing for stock-keepingut
:
(25)0 for t < F dt
otherwiseThe integral decision on production/remanufacturing technology 37averaged to just about 1.1 in the pre-test, all heuristics except of the first use as
investment time tr the value which is optimal for investment project (b). The
heuristics are now explained in more detail.H1: The remanufacturing facility is set up as soon as the first returns arrive, i.e. at
time tr=. Thus, there is no need for stock-keeping. This heuristic neglects the
decreasing time value of investment expenses due to discounting and is expected to
perform well if investment expenditures for the remanufacturing facility are low or
discount rate is small.H2: The optimal solution of investment project (b) is used as a second heuristic. It
should lead to good results in environments where the maximal holding time is
relatively small, e.g. where out-of-pocket holding costs are large.H3: The third heuristic combines H2 with a simple rule with respect to stock
keeping. Returns are kept starting at time and used up after tr. If there are any
items left on stock at tI, these are disposed of. This heuristic approximately
corresponds to a Case (3) solution candidate and might perform well if out-ofpocket holding costs are low because it disregards a possible limitation of stockkeeping in time.H4: In contrast to H3, this last heuristic only keeps stock that can be used up
before tI and stored no longer than the maximal holding time. Thus, building up the
anticipation inventory may start later than leading to a Case (1) candidate like
solution. Since this heuristic considers both, holding time and the time value of
investment expenses, its performance is expected to be superior to that of the other
presented heuristics.The heuristic rules are tested for a collection of classes of randomly generated
instances. The variety is based on a single demand and a number of return
scenarios, as well as on different levels of key parameters, namely direct recovery
cost advantage, discount rate, out-of-pocket holding cost rate, and investment
expenditures for setting up the remanufacturing facility. For each key parameter
two ranges representing comparably high and low values were defined. More
precisely, we used the following experimental design: Demand function is fixed as in the pre-test. Four return scenarios are used with a low/high return fraction F and small/large
duration of use as depicted in Table 1. Since after tI not all returns can be used,
the last column in Table 1 shows the maximal usable number of returns which
better expresses the potential benefit from remanufacturing.Table 1 Four considered return scenariosScenario F tI Total returns Usable returnsI 0.4 3 28.4 40,000 39,982
II 0.4 6 17.7 40,000 33,059
III 0.7 3 15.1 70,000 62,562
IV 0.7 6 15.3 70,000 46,06338 R. Kleber As in the pre-test, initial investment expenditures into production facilities are
set to zero (Kpr=0). The difference cprcr is normalized to 1. Objective values are
calculated by using cpr=1 and cr=0. Since cpr and cr are fixed, the recovery cost advantage only depends on the
disposal cost rate cw. Two intervals are considered, one with relative low
disposal costs, i.e. cw(1, 0), and another one with a comparably high cost rate
cw(0, 1). In the first case, the direct recovery cost advantage ranges between 0
and 1, and in the second it is located in an interval between 1 and 2. In order to find a possible influence of discounting, is assumed to belong to
one out of two intervals, being either low ((0.05, 0.1)) or high ((0.1, 0.15)).
Holding cost rate hu was assumed to be taken either out of an interval with a
relative low level, i.e. hu(0, 0.25), or from another with comparably high level
hu(0.25, 0.5). Thereby it has been ensured that only values are used which
satisfy our assumption regarding hu. For the investment expenditures Kr we chose the following two ranges: Kr(0,40,000) and Kr(40,000, 80,000). The upper border represents about the
maximal investment expenditures that can be earned from remanufacturing in
Scenario III where most usable returns are present.In total, there have been 64 combinations of scenario and parameter intervals
which correspond to a certain setting (42222 factorial design). Since we fixed
some of the parameters, we could suffice with only 200 examples for each setting
(12,800 in total) yielding enough material for statistical tests. For each example the
relative errors of the heuristic (H1H4) objective values were calculated. In order
to do a fair comparison, only those experiments were considered, under which
remanufacturing actually would be useful, i.e. where a finite investment project (c)
solution is optimal. With other words, it is assumed that the decision maker is able
to decide whether remanufacturing actually makes sense or not, and he is only
interested in ascertaining at which time to invest and whether and when to start
keeping returns. By appropriately grouping the examples, a sensitivity analysis of
the average performance of the heuristics with respect to return scenarios as well as
for the examined key parameters was performed.This analysis was complemented by statistical tests which ensured the
comparison of average performance of the heuristics one against each other,
where the results originated from the same experiments (matched pairs), but also
the change of the heuristics performance due to different settings (independent
group means), two different types of tests had to be performed. In the first case, a
paired t-test was carried out which, because of the large sample sizes, was
approximated by a Normal z Test. For comparing independent group means w.r.t.
the same heuristic in different settings, a single-sided version of the approximative
two groups Normal z Test was performed. Because of the large sample size, the
significance was tested on a 99% level. For a detailed treatment of the tests
performed see Kleber (2004). All comparisons of average errors are statistically
significant except otherwise stated. In spite of this procedure, since it is not
possible to generate a general setting which integrates all possible demand/return
situations and cash flow parameter combinations, all following statements should
rather be seen to express tendencies, which should be verified before applying to an
actual situation.The integral decision on production/remanufacturing technology 39The main results of the study are presented in Tables 2, 3, 4, 5, 6 showing
average and maximal relative errors of the heuristic solutions with respect to the
objective. Here also the fraction of finite optimal investment project (c) solutions in
the respective subset of all experiments can be found, being represented by the sign
#. This number expresses the relationship between the setting and the average
profitability of remanufacturing.Overall results and scenario comparison Considering all examples (see last row in
Table 2), H4 performs best in terms of the average error as was previously
expected. It also shows the smallest maximal deviation from the optimal solution.Table 2 Maximal and average NPV deviations of heuristics from optimal solution (in percent)
within the considered return scenariosScenario # H1 H2 H3 H4Avg Max Avg Max Avg Max Avg MaxI 27.8 8.9 49.2 3.1 13.6 1.5 13.7 1.2 13.6
II 25.1 5.3 27.7 2.1 8.1 1.0 6.5 0.8 6.5
III 44.4 11.7 59.0 4.2 15.9 2.1 33.3 1.4 14.5
IV 35.5 6.2 33.4 2.8 10.2 1.3 11.9 1.2 8.0
Overall 33.2 8.4 59.0 3.2 15.9 1.6 33.3 1.2 14.540 R. KleberTable 5 Maximal and average NPV deviations of heuristics from optimal solution (in percent)
with relative high and low out-of-pocket holding cost ratehu # H1 H2H3H4Avg Max Avg Max Avg Max Avg MaxLow 34.9 9.9 59.0 4.0 15.9 1.9 20.3 1.8 14.5
High 31.6 6.8 44.2 2.3 9.0 1.3 33.3 0.6 7.7Table 6 Maximal and average NPV deviations of heuristics from optimal solution (in percent)
with relative high and low investment expendituresKr #H1 H2 H3 H4Avg Max Avg Max Avg Max Avg MaxLow 59.0 6.9 46.8 2.9 15.9 1.5 33.3 1.1 13.6
High 7.4 20.7 59.0 5.4 15.4 2.6 14.5 2.4 14.5Also not unexpected, H1 performs worse than all other heuristics in both criteria.
Especially when comparing its performance with H2, a quite large benefit can be
obtained only by postponing the investment time. Introducing a simple rule for
stock-keeping (H3 instead of H2) yields in average an additional benefit, but under
circumstances described below it can also lead to a substantial performance loss.By comparing the results in the different scenarios and reconsidering the
corresponding number of usable returns, it can be seen that the higher this number
the higher also the potential benefit from remanufacturing. In such cases,
specifically the number of instances where remanufacturing takes place increases
(e.g. in Scenario III it is higher than in Scenario II). The performance of the
heuristic approaches decreases if either the return fraction F increases or returns
arrive (relatively) early. Since both cases allow for higher investment expenses, an
erroneous determination of investment time has a higher impact on the
performance.Low versus high recovery cost advantage (disposal cost rate) Having high
disposal costs or a high recovery cost advantage noticeably increases the number of
instances where remanufacturing makes sense as shown in Table 3. Performance of
the considered heuristics decreases except for H3 where the difference lacks
significance, because of its generally large variability of relative deviations from
the optimal solution. Another reason why this heuristic does not perform much
worse is the positive effect of the recovery cost advantage on the maximal holding
time . Thus, the profitability of using a recoverables inventory increases, but also
the possible error when neglecting decreases. This reasoning also explains why
H3 performs poorly compared with H2 if cw is low.The integral decision on production/remanufacturing technology 41Low versus high discount rate Although the effect can hardly be termed large (see
Table 4), a higher discount rate leads to a decreasing profitability of
remanufacturing, but it also lowers the precision of H1H3. This especially
holds for H1, which does not take any time value considerations into account. The
potential benefit of keeping stock increases (H2 performs worse), because it would
allow for a further postponement of the investment time which has a stronger effect
than would be with a lower discount rate. In contrast to H3 which performs worse
when increasing the discount rate, there is probably (although it lacks significance)
an improvement for H4, which is due to the fact that H4 reacts on a modification in
discounting both by changing the investment time and by correctly adapting for the
modified maximal holding time.Low versus high out-of-pocket holding cost rate Similarly to a high discount rate, a
high out-of-pocket holding cost rate decreases the profitability of remanufacturing,
especially of items that have been kept in stock for later use (see Table 5). As
intuition suggests, it improves the performance of those heuristics which do not
keep stock, but also H3 and H4 perform better. For H3 this appears at first glance
counter-intuitive, but it becomes reasonable because less stock-keeping also
yielded a smaller deviation of the approximated investment time from optimum
and an improvement of the investment time estimation overcompensated higher
holding costs.Low versus high investment expenditures for remanufacturing facility High
investment expenditures clearly lead to a strong decrease of the number of
instances where remanufacturing makes sense (see Table 6). All heuristics perform
worse, as they do not correctly reflect the potential of changing the investment time
due to stock-keeping, which has a larger effect if investment expenditures increase.
This can especially be seen from the average error H4 exhibited.Summary of results Summarizing the results it can be seen that dumb investment
time rule H1 should not be used, because there exists a considerable amount of
savings to be realized by applying H2. It is also clear that the use of an anticipation
stock not only becomes reasonable because of the additional remanufacturing but it
also can be used to change (mostly to postpone, as seen in the pre-test) the
investment time tr. This yields the biggest effect when having low out-of-pocket
holding costs or a high recovery cost advantage. The question on whether to apply
one of the heuristics (H2H4) depends (a) on the situation under consideration and(b) on the error which the decision maker is willing to accept. Especially, the
knowledge of data and computational requirements which H4 necessitates are
comparable to those needed for finding the optimal solution.42 R. Kleber4 ConclusionsIn this paper we used properties of a dynamic situation consisting of a product life
cycle and a returns availability cycle in order to find optimal dynamic policies for
three investment projects that differ with respect to the environmental policy. For
investment projects that incorporate remanufacturing, the time of the remanufacturing investment proved to be a crucial decision variable because it influences
both the time value of the expenses accompanied with it, but also the advantage
that can be obtained by replacing production of new products by remanufacturing
of returns. We have shown that improvements exist if returns can be kept in a
strategic inventory. From our experiments it looks that this option should not be
implemented as a rule of thumb e.g. by generally keeping all returns. That being
said, further research using real life data is required to quantify the performance
losses of heuristic approaches.A number of possibilities exist for further research. It would be interesting to
see how robust our results behave when assuming imperfect knowledge on future
demand and return developments. Capacity aspects have not been explicitly
considered. The dynamic policy especially in the investment project with remanufacturing and strategic inventory, however, shows a large variability in both
production and remanufacturing rates. As an extension, for instance the effects of a
limited remanufacturing rate could be analyzed, which would potentially lower the
benefit from stock-keeping. Furthermore, the effects of remanufacturing on the
required production capacity have been neglected. Consequently, in a generalized
model optimal production/remanufacturing capacity expansion paths could be
determined. Two types of trade-off have to be considered. As in pure production
models, building up capacity can be avoided by using a strategic serviceables
inventory. Since there are two options to fill the demand, the choice of capacity in
the remanufacturing shop influences the required production capacity and vice
versa. But in contrast to the simplified situation introduced above, in this setting
only numerical results can be derived.Remanufacturing of used products is usually connected with disassembly
processes which are known to be labor intensive. Thus, learning aspects may
further complicate the issue, as we may initially remanufacture with no (or even a
negative) direct recovery cost advantage, this may be compensated by lower direct
remanufacturing costs later on.We assumed that demand development does not depend on the chosen
technology. Marketing aspects like consumer awareness towards environmental
conscious products are neglected. Further, competition both on demand as well on
the return side are not considered. For instance, the easier products are recoverable,
the higher the possibility that other firms will want to participate and to carry out
remanufacturing in competition against the OEM. This can be seen for example in
the case of refilling toner cartridges for laser printers utilized by Majumder and
Groenevelt (2001) to motivate a two-period model which aims to explain how the
level of remanufacturability of a product influences competition.The integral decision on production/remanufacturing technology 43Appendix 1 ProofsProperties of NPVbWhen inserting u(t)=0 for t< and replacing the max/min operators in Eq. (2) the
objective is (omitting time indices) given by8
>
>
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>
>
: (26)Krp R0etcrpd dt etr Kr R
tIt crp d u
cru
hidt R1tItr < eet crd cw u d
dtKrp R0etcrpd dt R
tret crpd cwu
hidt etr KrNPVb R
tIt crp d u
cru
hidt R1tIeet crd cw u d
dt Krp Rtr0tcrpd dt R
trtr < tItI treet crpd cwu
hidt etr Kr1 Ret crd cw u d
dttrThe first derivative of Eq. (26) differs for each of the three regions as defined in
Section 2.1 and is given by@NPVb@tr
8
>
>
>
<
>
>
>
:etr Kr for tr < Region 1
undefined for tr etr crp cw crutr
Kr
hi for < tr < tI Region 2
e(27)tr crp cw crdtr
Kr
hi for tItr Region 3
:(27)Note that except for tr=, Eq. (27) is continuous. It has negative sign in Region
1 and from assumption A.1 there exists a time t>tI in Region 3 for which it holds
8<and thus the objective finally must decrease. A similarargumentation holds for u(tr)if tI=. Let _
x denote the first derivative of x with
respect to time. The second derivative of Eq. (26) is given bytr > t : dtrKr
cpcwcr@2NPVb@t2
r8
>
>
>
>
>
<
>
>
>
>
>
:2etr Kr for tr <
undefined for tr
etr crp cw crutr
2Kr
hi for < tr < tI
undefined for tr tIetr crp cw cr u tr
(28)dtr
2Kr
hi for tI < tr dtr
(28)44 R. KleberIt shows a positive sign in Region 1 and as time reaches infinity where d and _
d
approach zero. Hence if there exists a local minimum, it must be followed by a
local maximum.Proof of Proposition 1 Candidates for tr* are given by time points where Eq. (27)
changes its sign from negative to positive. This is not possible in Region 1, where
Eq. (27) is negative. Time point tr,1= is a candidate for tr*if limtr!0@NPVb
@tr0 .This yieldscrp cw cr
Kr: (29)
Further candidates, tr,2 and tr,3, are given by setting the first derivative in the two
remaining regions to zero, which gives the following conditionsutr;2
crp cw cr
Kr for <tr;2 < tI or (30)dtr;3
crp cw cr
Kr for tI tr;3: (31)
Inserting Eqs. (30) and (31) for Kr in the respective part of Eq. (28), conditions
for a local minimum can be derived. tr,2 is a local minimum if@2NPVb@t2
rjtrtr;2 u
e
tr crp cw cr_
utr;2
> 0 , _
utr;2
> 0 (32)and thus, tr,2<tumax. Analogously, tr,3 is a local minimum if _
d(tr,3)>0. Since demand
must decrease for any time point tr,3tI (by definition of tI), candidate tr,3 fails the
second order necessary conditions. From our assumptions about the return function
(unimodal) it follows that if inequality (29) holds, i.e. tr,1= is a candidate for an
optimal solution, there will be no candidate tr,2 and vice versa. Hence, there exists at
most a single finite solution, being located in a half open interval [, min{tumax, tI}).Proof of Proposition 2 Since NPVb decreases for sufficient high tr solution
candidate t1r(invest never) has to be considered. In order to find the best
alternative, the Net Present Value of the payment stream arising by assuming a
relevant finite candidate ~tr 2 ; min t1maxu ; tI
, i.e. NPVb ~tr, has to be comparedwith NPVb t1rt crpdt
cwut
hidt: (33)This gives an expression of the total discounted advantage of remanufacturing Arb, which is given byNPVb t1
r Kr
p Z 10eAbr NPVb t1
r NPVb etre~tr Kr R 1~
tr et crp cw cr min dt
; ut
fg
hidt (34)The integral decision on production/remanufacturing technology 45Therefore, t*r~
tr if Ab
r > 0 ,i.e.Kr Z 1~
tre t~
tr
crp cw cr
hi min dt
; ut
fg
dt: (35)Otherwise t*r1:Proof of Propositions 3 to 8 Propositions 3 to 8 are results of the following
optimization approach. Since both the objective function NPVc and constraint (14)
are in general not convex, in accordance with Sydster and Hammond (1995), p
608 the following solution method is used:(1) Determination of the partial derivatives of the objective function.(2) Identification of possible solution candidates (Steps 1 and 2 in Sydster and
Hammond 1995) using standard methods of Nonlinear Programming. This proofs
Proposition 3. Exploring a joint property of all valid cases proofs Proposition 4
while individual properties confirm results stated in Propositions 58.(3) Comparison of values of NPVc at candidate points against each other (Step 3)
and with the Net Present Value of investing never NPVb t1r as given in (33).Smallest value is the (global) minimal value of NPVc (Step 5). As this requires
actual data, we will omit this part.(1) Partial derivatives of the objective functionSubsequently, the partials of tx with respect to te and tr will be needed. This is
applied by using implicit differentiation rules leading to@tx@te @f@teute
; (36)@fdtx utx
@tx@f@tr @tx@tr@fdtr
: (37)dtx utx
@txThe first partial derivative of NPVc(te, tr) with respect to te is (after collecting
terms and inserting yu(te)=yu(tx)=0) given by@NPVc@te ete cwute
Z tx
teethu@yu t
@tedt
edt
@txtx crp cr
hi dtx
utx
Z tx
teethu@yu t
@tx:@te46 R. KleberSince the partial of yu(t) with respect to te equals@yu t
@te 8
>
>
<
>
>
:undefined for t te
ute fort 2 te; tx
undefined for t tx0 otherwise; (38)and by additionally replacing@yu t
@tx 0 and@tx@teby Eq. (36) it follows@NPVc@te tdt etx crp cr
hi
ute
: (39)Equation (39) can be interpreted as follows. If te increases, a marginal return u(te)
arriving at this time is no longer stored but disposed of, leading to additional unit
costs of ete cwute
ete cw huZ tx
tee: Since this (marginal) return is not available for later
remanufacturing, tx decreases and thus, costs for producing a (marginal) newproduct at tx are caused, given by etx crp cr: On the other hand, storingless returns reduces inventory holding costs per unit by huR tx
te etdt u te
ute
.Evaluating the integral in Eq. (39) finally yields@NPVc@te
ute
ete cw
etx crp cr huhu: (40)The first partial derivative of NPVc(te, tr) with respect to tr is (after collecting
terms and inserting yu(tx)=0) given by@NPVc@tr etr crp cr
hidtr
etr Kr Z tx
teethu@yu t
@trdtdt
@txetx crp cr
hi dtx
utx
Z tx
teethu@yu t
@tx:@trReplacing@yu t
@tr 8
>
>
<
>
>
:t 2 tr; tx
undefined for t tx0 otherwiseundefined for t tr
dtr for(41)and@tx@trby (37) yields@tr @NPVctdt
dtr
etretxcrp cr
hi huZ tx
trtr Kr: (42)A later investment time tr decreases the Net Present Value of the investment
expenses by etr Kr . A (marginal) demand d(tr) is no longer satisfied by
remanufacturing returns at tr, which instead are stored for a later use at tx. Therefore,eeThe integral decision on production/remanufacturing technology 47a cost reduction at tx by remanufacturing instead of producing faces an increase in
costs at tr. Additional holding costs are caused by storing the (marginal) return.Continuing in the same manner as above gives@NPVc@tr huetretx
dtr
crp cr etr Kr: (43)(2) Identification of solution candidatesBy introducing Lagrange multipliers i, i=1, 2, 3 which are associated with
constraints (12)(13) the Lagrangian Lte; tr;1;2;3is defined asLte; tr;1;2;3
NPVcte; tr
1 te
2 tr te
(44)3ds
:Z tI
txus
ds
The partials of Lte; tr;1;2;3
have to equal zero@te @L@NPVc@te 1 2 3ute
0; (45)@tr @L@tr @NPVc2 3dtr
0: (46)The complementary slackness conditions are given by@1 te 0;1 0;1 te
0; (47)@L@L@2 te tr 0;2 0;2 te tr
0; (48)@3 Z tI
tx@Lus
ds 0;3 0;3 Z tI
txds
us
ds
0:ds
(49)Having three constraints, each either being active or inactive, in total eight cases
have to be distinguished. It can be shown that all Cases where stockkeeping does
not occur (i.e. where te=tr) can be excluded (see Kleber 2004, for the
straightforward proof). Thus, for the optimal solution (te*, tr*) it holds te*<tr*. The
remaining cases (1)(4) constitute the different types of solution candidates as
stated in Proposition 3. This completes the proof to Proposition 3.48 R. KleberFor all remaining cases (1)(4) it holds te*<tr* which requires from Eq. (48) 2=0.
Inserting this value into Eq. (45) (reconsidering non-negativity of 1, 3, and u(te*)
necessitates@NPVc
@te0, yielding@NPVc@te
ut*
eet*e
et*xcw hucrp cr 0 (50)huBecause of u(t)>0 t this is equivalent toet*e
et*xcw crp cr hu
0huand solving for tx*te* finally yieldst*xcw hu
: : (51)This completes the proof to Proposition 4.Case (1) te < tr; < te;R tI
tx ds us
ds > 0 , < te < tx < tI : None of
the conditions is active. Thus, 1=0, 2=0 as well as 3=0. Inserted into Eq. (45),
this yields@NPVc@te t*e1ln cp cr hutdt etx crp cr
hi
ute
ete cw huZ tx
tee0:Proceeding as above leads tote : (52)Further, inserting Lagrange Multipliers into Eq. (46) requirestx
etr crp cr
hidtr
huZ tx
tretdt d tr
etx crp cr
hidtr
etr Kr:(53)This completes the proof to Proposition 5.Case (2) te < tr; te; R tItx dsus
ds 0 , te < tr < tx tI : Insert-ing 2=0 into (46) yields3 @NPVc@tr
1txtI:dtr
1etretIcrp cr
hi hu R tI
tr etdt etr Kr:dtr
30 requirestIetr crp cr
hidtr
huZetdt d tr
etI crp cr
hidtr
etr Kr: (54)trThe integral decision on production/remanufacturing technology 49Further, inserting 3 into Eq. (45) leads to1 @NPVc@tete; txtI @NPVc@trtxtI u
dtr
:
(55)ecw huR tr etdt etr crp cr
hiu
etr Kru
dtr
Since 10, Eq. (55) impliesecwdtr
huZ tretdt d tr
etr crp cr
hidtr
etr Kr: (56)ds > 0 , te < tr < tx < tI : Both
second and third conditions are inactive. Then, 2=0 and 3=0 from Eqs. (48) andThis completes the proof to Proposition 6.Case (3)te < tr; te; R tItx dsus
(49), respectively. Inserting both values into Eq. (46) again yields Eq. (53). This
completes the proof to Proposition 7.Case (4) te < tr; < te;R tI
tx ds
us
ds 0 , < te < tx tI : Both,
first and second conditions are inactive. Then, 1=0 and 2=0 from Eqs. (47)
and (48), respectively. Both values inserted into Eq. (45) yields the value for 313 @NPVc@tetxtI ete cw huZ tI
teetdt etI crp cr
hiute
Inserting 3 into Eq. (46) gives@NPVc@trtxtI dtr
ute
@NPVc@tetxtI 0(57),ete cwdtr
huR tr
te etdtd tr
etr crp cr
hidtr
etr Kr:This completes the proof to Proposition 8.Appendix 2 List of symbolsGeneric dynamic environmentt time index(.)(t) time dependence.* optimal values
d(t) demand rate at time t
tdmax time point of maximum demand rate
u(t) return rate at time t time delay of returnstumax time point of maximum return ratetI intersection point of demand and return functionsProcesses (states and variables)yu(t) physical stock recoverables
p(t) production rater(t) remanufacturing ratew(t) disposal rate50 R. KleberCash flow parameters discount rate or continuous interest rate
cps variable unit production cost at single use production
cpr variable unit production cost at reuse productioncr variable unit remanufacturing costcw variable unit disposal cost or negative salvage revenue
hu holding cost rate recoverablesKps initial investment expenditures for single use production
Kpr initial investment expenditures for reuse productionKr investment expenditures for remanufacturing facilityPolicy parameters and optimization oriented notationtr time of remanufacturing investment
te start time of storing collected returns
tx time where all stored returns are used up
ucrit critical return rate in investment project (b)
Arb total discounted net advantage of remanufacturing in investment project (b)
Dpb increase of total discounted expenditures for production in investment project (b)
maximal holding time of returns in investment project (c)Parameters used in numerical investigationM number of potential adopters (parameter in Bass model)
P coefficient of innovation (parameter in Bass model)Q coefficient of immitation (parameter in Bass model)
F fraction of demanded products being available for remanufacturing
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