Product recovery in stochastic remanufacturing systems with multiple reuse options

European Journal of Operational Research 133 (2001) 130±152

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Theory and Methodology

Product recovery in stochastic remanufacturing systems with multiple reuse options q K. Inderfurth a

a,*

, A.G. de Kok b, S.D.P. Flapper

b

Faculty of Economics and Management, Otto-von-Guericke University Magdeburg, P.O. Box 4120, D-39016 Magdeburg, Germany b Eindhoven University of Technology, School of Technology Management, P.O. Box 513, 5600 MB Eindhoven, Netherlands Received 18 June 1998; accepted 27 June 2000

Abstract In many product recovery situations, returned products can be reused in multiple ways. Under these circumstances, the problem arises in which quantities reusable items should be allocated to the di€erent remanufacturing options, especially in case of insucient stock of returns. For this problem a periodic review model is formulated which also includes a disposal option and incorporates uncertainties in returns and demands for the di€erent serviceable options. The structure of the optimal policy is analyzed, and it is shown that under speci®c allocation rules a near-optimal policy with a simple structure exists. An ecient computational procedure for determination of the optimal policy parameters is presented. This procedure is applied in a numerical investigation that gives interesting managerial insights into important product recovery issues. Ó 2001 Elsevier Science B.V. All rights reserved. Keywords: Product recovery management; Remanufacturing; Multiple reuse options; Optimal policies

1. Introduction Recently, more and more manufacturers reuse old products and incorporate product recovery activities in their regular production environment. The reason for this is two-fold. Firstly, reuse of old products, in the form of components, subassemblies or even completely after cleaning and upgrading, can be highly pro®table. This is especially the case with complex, high-tech products with a reasonably long product life cycle, like copiers and medical equipment. Secondly, legislation aimed at environment-benign production forces manufacturers to take back their products from end-users after they discard them.

q *

The editorial review process for this article was handled by Professor W.C. Benton. Corresponding author. Tel.: +49-391-6718798; fax: +49-391-6711168. E-mail address: [email protected] (K. Inderfurth).

0377-2217/01/$ - see front matter Ó 2001 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 7 - 2 2 1 7 ( 0 0 ) 0 0 1 8 8 - 0

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131

Reuse of old products raises a lot of Operations Management problems for manufacturers, starting with collection and ending with sale of products with used components. In this paper we focus on one of these new Operations Management problems. In many situations an old product can be used in di€erent ways, each yielding di€erent costs and pro®ts. Often besides di€erent ways of upgrading, downgrading alternatives or pure restoring are possible options when used products are remanufactured. Additional recovery options like disassembly and reuse of major modules may be reasonable ways to exploit most of the value a used product still contains. For example, a PC might be reused as such, by replacing its hard disk, its processor, by adding some extra work space memory or a modem for internet, but also via its screen, keyboard and its memory that is sold to a toy manufacturer. These options are mutually exclusive and di€er with respect to the costs incurred during the remanufacturing process and the revenues generated by each option. The selection of these options is one of the main Operations Management problems associated with remanufacturing. One main question in this context is how to allocate a possibly limited amount of reusable products to the di€erent remanufacturing options, taking into consideration all relevant cost and demand aspects. It has to be decided which stock level for the di€erent serviceable options should in general be achieved and how the desired level of reusables stock should be ®xed. The latter inventory cannot only be controlled by remanufacturing decisions, but also by determining to which extent used products should be disposed of in case of excessive returns. The complexity of this Operations Management problem is caused by the fact that the disposal, remanufacturing and allocation decisions have to be made simultaneously and that from both the demand and the returns side major uncertainties can come into the problem. In this paper we consider a special class of product recovery problems with multiple remanufacturing options. It is our objective to determine the structure of the optimal periodic review policy for both discounted costs over ®nite and in®nite horizons, and long-term average costs. It is shown that the optimal policy is of a fairly simple structure: Returns are discarded if the echelon stock of returns exceeds a control limit. The echelon stock of returns equals the sum of the stock of returns and the inventory positions of the serviceable options. The stocks of serviceable options are replenished according to order-up-to-policies. In case insucient returns are available, an allocation policy determines how much is remanufactured of each option. In order to compute near-optimal policies, we apply the so-called linear allocation rules introduced in Diks et al. (1996). Note that we assume that serviceable options cannot be procured from outside suppliers. In Clegg et al. (1995) and Uzsoy and Venkatachalam (1994) deterministic planning models considering more than one option for product recovery are presented. To our knowledge the present paper is the ®rst to address the stochastic remanufacturing problem with multiple reuse options. Several models exist in literature that describe the single option situation for a stochastic environment. In Inderfurth (1997) the optimal policy is determined for a periodic review model with remanufacturing and procurement. It turns out that in case procurement lead times and remanufacturing lead times are identical, the structure of the policy is simple. In case the lead times di€er, the structure of the optimal policy is not obvious. Van der Laan (1997) considers a number of continuous review single option models which include lotsizing and leadtime di€erences. Assuming di€erent variants of simply structured policies, Van der Laan derives expressions for costs and service levels, from which cost-optimal policies within the class of policies considered are derived. For a general overview of Operations Management problems associated with product recovery we refer to Thierry et al. (1995). A review of quantitative models in the ®eld of remanufacturing is given in Fleischmann et al. (1997). The paper is organized as follows. In Section 2 we de®ne the model in detail and derive the optimization problem to be solved. In Section 3 we derive the structure of the optimal policy for discounted and average cost criteria, both for ®nite and in®nite horizon. In Section 4 we concentrate on the in®nite horizon, average cost case, for which we derive near-optimal policies. An approximation algorithm is presented that computes these near-optimal policies. This algorithm is applied in Section 5 to various situations in order to gain an insight into di€erent aspects of interest for managerial decisions. Conclusions and topics for further research are presented in Section 6.

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2. The multiple remanufacturing options problem 2.1. Problem description and notation The speci®c problem addressed in this paper can be described as follows. In a periodic review system we face stochastic returns Rt of used products in each time period t. These products can either be disposed of or remanufactured using n alternative modes from a set N of di€erent options. A remanufacturing option i takes a processing leadtime of ki periods and has an outcome which is needed for satisfying stochastic demands Dit . Additional sources of procuring serviceable items of option i do not exist. Situations in practice for which this assumption holds include: situations where items were purchased, but the supplier got bankrupt or stopped, for some reason, producing these items, and situations where a company used to produce items itself, but stopped their production, e.g., because the company expected to be able to ful®l demand by items that it had to take back, or because the production facility had to be disposed of, because space had to be made available for other activities. The latter situation notably occurs at the end of the business economic life cycle of an item. For many second-hand markets, like for cars and refrigerators, the above holds. As still more producers get involved in these markets, the practical importance of the topic dealt with in this paper will further increase. Demands are assumed to be independent of time and across options. Returned products …RP † as well as serviceable items of each option i …SPi † can be stocked so that the structure of the multiple remanufacturing options systems can be depicted as in Fig. 1. Stocks on hand are charged with a per unit cost hR for returned products and hSi for serviceable items of option i, respectively. Demand for an item i which is not satis®ed immediately is backordered at a cost of vi . Di€erent amounts of the backorder costs also re¯ect di€erences in the selling prices of the respective serviceable options. Disposal and remanufacturing costs are proportional per unit, described by cost parameters cB and cRi , respectively. For sake of simplicity all model parameters are assumed to be time independent. Only for the steady-state analysis in the sequel of this paper, this assumption is absolutely necessary. The objective is to choose the quantities rit of items to be remanufactured for option i and the amount bt of units for disposal in each period t of a planning horizon T in such a way that the expectation of the total discounted costs of remanufacturing, disposal, stock holding and backordering is minimized. We also consider the case where the horizon T tends to in®nity.

Fig. 1. System with multiple remanufacturing options.

K. Inderfurth et al. / European Journal of Operational Research 133 (2001) 130±152

133

Notation N n ˆ jN j T rit bt xRt xNSit xSit xEt cRi cB hR hSi vi a ki Di R P SˆR i Di uDi …
† uR …
† uS …
†

set of remanufacturing options number of remanufacturing options number of planning periods remanufacturing quantity w.r.t. option i (i 2 N ) in period t disposal quantity in period t on-hand inventory of returned products in period t net inventory of serviceable option i in period t inventory position of serviceable option i in period t echelon inventory position in period t remanufacturing cost per unit w.r.t. option i disposal cost per unit inventory holding cost of returned products per unit and period inventory holding cost of serviceables option i per unit and period shortage cost of backordered serviceable option i per unit and period discount factor (0 6 a 6 1) remanufacturing leadtime w.r.t. option i stochastic demand per period w.r.t. option i (i.i.d.) stochastic returns per period (i.i.d.) stochastic net supply of remanufacturable products continuous density function of demand Di continuous density function of returns R continuous density function of net supply S

The sequence of state observations and decisions in each period is the following one. At the beginning of a period t (before arrival of returns and previous remanufacturing orders) we observe both the inventory on hand of returned products …xRt † and the net inventories (on hand minus backorders) of serviceables with option i…xNSit †. Based on these stock levels and on the information about in-process remanufacturing orders we decide on the number of returned items which are disposed of and remanufactured in period t. After these decisions are made, the stochastic returns and demands during the period are taken into account. Holding and backordering costs are charged to inventories at the end of the period. As aggregated stock information for decision making, we additionally de®ne the inventory position xSit for each serviceable option i (net Pinventory plus in-process remanufacturing orders) and the systems echelon inventory position xEt ˆ xRt ‡ i2N xSit . 2.2. The optimization model Following the above notation we can formulate the problem as a stochastic dynamic optimization model. The objective function is given by ( " ! T X X X Min E at 1 c B bt ‡ cRi rit ‡ hR xRt bt rit ‡ Rt fRt ;D1t ;...;Dnt g

‡

X i2N

‡

i2N

tˆ1

hSi ‰xNSit

where we use ‰xŠ :ˆ maxfx; 0g.

‡ ri;t

ki

i2N ‡

Dit Š ‡

X i2N

vi ‰Dit

xNSit

#) ri;t

‡

ki Š

;

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K. Inderfurth et al. / European Journal of Operational Research 133 (2001) 130±152

We ®nd dynamic restrictions in the form of inventory balance equations for each period t. These restrictions refer to the inventory of returns X xR;t‡1 ˆ xRt bt rit ‡ Rt ; i2N

and to the serviceable options inventories xNSi ;t‡1 ˆ xNSit ‡ ri;t

Dit

ki

8i 2 N :

A static restriction in each period is given by the condition that disposal and remanufacturing must not exceed the number of units in the inventory of returns X bt ‡ rit 6 xRt : i2N

Finally we face non-negativity of all decision variables bt ; rit P 0: The value of any stock at the end of the planning horizon is assumed to be equal to zero. In the sequel the problem formulated above will be called the Multiple Remanufacturing Options Problem (MROP).

3. Policy analysis for the MROP 3.1. The optimal disposal, remanufacturing, and allocation policy For a dynamic stochastic product recovery system as described in Section 2 it is obvious that the state information which is necessary to control such a system in an optimal way consists of the returns stock on hand xRt and the inventory positions of serviceables xSit at the beginning of each period. The latter information is used to describe how the remanufacturing decisions of a period a€ect the expected holding and backorder costs of the next controllable time period, which is ki periods ahead. The balance equations with respect to the serviceables inventory positions are given by xSi;t‡1 ˆ xSit ‡ rit

Dit

8i 2 N :

Thus for the echelon inventory position xEt ˆ xRt ‡ X xE;t‡1 ˆ xEt bt ‡ Rt Dit :

P i

xSit , we ®nd

i2N

As expected holding costs of returns in period t we can write ! ! Z 1 X X hR xRt bt rit ‡ R uR …R† dR ˆ hR xRt bt rit ‡ E‰RŠ : 0

i2N

i2N

Since holding costs hR E‰RŠ cannot be in¯uenced by the decisions to make, this ®xed part of the costs will be omitted in the sequel. The expected holding and backorder costs for a serviceable option i as a function of the inventory position yi after remanufacturing can be described by

K. Inderfurth et al. / European Journal of Operational Research 133 (2001) 130±152

Z Li …yi † ˆ hSi

yi

0

…x

ki ‡1 Di † uDi …Di † dDi

Z ‡ vi

1

yi

…Di

135

ki ‡1 yi † uDi …Di † dDi ;

where ukDii …
† represents the ki -fold convolution of uDi …
†. It can easily be shown and is well known from standard inventory theory that Li …
† are convex functions, (see, e.g., Hillier and Lieberman, 1990, p. 712). Using the above formulations and following Bellman's optimality principle we can express the functional equations of dynamic programming for the MROP in the following way (where for the sake of simplicity we omit time index t for the variables): ( ! X X X ft …xR ; xS1 ; . . . ; xSn † ˆ P min cB b ‡ cRi ri ‡ hR xR b ri ‡ Li …xSi ‡ ri † b‡

ri 6 xR i b;r1 ;...;rn P 0

Z

‡a

0

1

...

i

Z

1 0

i

ft‡1 xR

b

X

i

! ri ‡ R; xS1 ‡ r1

i

D1 ; . . . ; xSn ‡ rn

Dn

)

 uR …R† uD1 …D1 † . . . uDn …Dn † dR dD1 . . . dDn

…1†

for t P 1 and with fT ‡1 …xR ; xS1 ; . . . ; xSn †  0: It is evident that the value functions ft …:; . . . ; :† depend on n ‡ 1 state variables. The way of analyzing the structure of the optimal policy for these kinds of problems is regularly by induction. First, the policy structure is evaluated for a single-period problem. Then it is checked if this kind of policy also holds for a …n ‡ 1†-period problem, given that it is assumed to be valid in a n-period situation. So we start our investigation of the optimal MROP policy by considering a single-period model, i.e., setting T ˆ 1. The respective minimization problem can be written as f1 …xR ; xS1 ; . . . ; xSn † ˆ min g1 …xR ; xS1 ; . . . ; xSn ; b; r1 ; . . . ; rn † b;r1 ;...;rn

w:r:t:

b‡

X

ri 6 x R ;

b; r1 ; . . . ; rn P 0;

i

where g1 …xR ; xS1 ; . . . ; xSn ; b; r1 ; . . . ; rn † ˆ cB b ‡

X i

cRi ri ‡ hR xR

b

X i

! ri

‡

X

Li …xSi ‡ ri †:

…2†

i

Since g1 …:; . . . ; :† can be shown to be convex in the decision variables, we can exploit the Kuhn±Tucker conditions of optimality to ®nd the optimal policy (for proofs and details, see Inderfurth et al., 1998). Before we give a formal description of the optimal policy, we describe and interpret how the optimal disposal and remanufacturing decisions are linked together. Since in the single-period case disposing one unit results in an increase of disposal costs by cB and in a reduction of returns holding costs by hR , we face a simple rule with respect to the disposal decision. If cB 6 hR , stockholding of returns is not economical, and all returned items, which are not used for remanufacturing, will be disposed of. If cB > hR , disposal will never be advantageous yielding b ˆ 0 (where an asterisk denotes the optimal decision). Due to the convexity of the cost functions Li …
†, in principle the remanufacturing decisions follow an order-up-to-Mi policy, where the constant Mi corresponds to the level to which the respective serviceables inventory position should be brought up, if it is below this position. Otherwise no remanufacturing order

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K. Inderfurth et al. / European Journal of Operational Research 133 (2001) 130±152

should be placed. For each option i, the critical level Mi has to be chosen in such a way that the marginal costs of remanufacturing equal the marginal cost advantage by decreasing the number of items in the returns stock and by increasing the serviceables inventory position by one unit cR ˆ minfcB ; hR g

L0i …Mi †;

…3†

where L0i …
† stands for the ®rst-order derivative of function Li …
†. This policy can P only be carried through if a sucient amount of returned items is in the respective stock. ‡ If not, i.e., if xR 6 i2N ‰Mi xSi Š , an optimal allocation of scarce returns to the di€erent options has to be made. This results in a decrease of the original remanufacture-up-to-levels Mi , which in this situation become dependent on all single inventory state variables  i …xR ; xS1 ; . . . ; xSn †: Mi ˆ M  i are determined such that the net marginal cost advantages of increasing all The modi®ed parameters M single serviceables inventory positions by one unit are equal. Additionally, the total number of returns allocated to the di€erent remanufacturing options has to equal the number of available returns:  1 † ‡ cR1 ˆ L0 …M  2 † ‡ cR2 ˆ


ˆ L0 …M  n † ‡ cRn L01 …M 2 n and

X

i ‰M

…4†

xSi Š‡ ˆ xR :

i2N

The optimal disposal quantity is either equal to zero (if cB > hR ), or it equals the number of returned items which are not needed for remanufacturing purposes, i.e., X ‡ b ˆ x R ‰Mi xSi Š if cB 6 hR : i2N

Thus, using U as a critical disposal level, where  1 for cB > hR ;   U ˆ U …xS1 ; . . . ; xSn † ˆ P ‡ ‰M x Š for cB 6 hR ; i Si i we can summarize the combination of optimal disposal and remanufacturing decisions as follows: X  i …xR ; xS1 ; . . . ; xSn † xSi Š‡ ; xR < ‰Mi xSi Š‡ ) b ˆ 0; ri ˆ ‰M X

i

‰Mi

xSi Š 6 xR 6 U …xS1 ; . . . ; xSn † ‡

i

xR > U …xS1 ; . . . ; xSn †

)

b ˆ x R

)

b ˆ 0;

U …xS1 ; . . . ; xSn †;

ri ˆ ‰Mi ri ˆ ‰Mi

‡

xSi Š ;

…5†

xSi Š‡ :

Obviously, this policy has a highly complicated structure, mainly because all decisions are dependent on every single state variable in a complex way b ˆ b …xR ; xS1 ; . . . ; xSn †; ri ˆ ri …xR ; xS1 ; . . . ; xSn †: Unfortunately, such a kind of policy is not suitable to be used in practical decision making because of its structural complexity and its prohibitive amount of numerical computation for optimizing the respective policy parameters.

K. Inderfurth et al. / European Journal of Operational Research 133 (2001) 130±152

137

In the multi-period case the underlying optimization problem becomes even more complex, since an additional …n ‡ 1†-dimensional cost function appears in the functional equations. Thus, the optimal policy also will at best be of the same complexity as shown in the single-period situation. For this reason we will not go into the problem of analyzing the optimal policy for the multi-period problem, but turn to a reasonable approximation, which considerably simpli®es the policy for controlling the recovery management system. Since the main complexity comes into the problem by determining the optimal allocation of returns in case of insucient stock, we make two simplifying assumptions which refer to the way of allocation and to its consequences.

3.2. The optimal policy under linear allocation rules The ®rst assumption we use, concerns the allocation rule which is employed whenever the stock of returned items is not sucient to bring all serviceables inventory positions up to their desired levels Mi . In this case we assume that the scarce amount xR of remanufacturables is distributed among the di€erent options according to a so-called linear allocation rule as follows: " ri0 ˆ ri0 …xE ; xSi † ˆ Mi

qi

X

#‡ Mj

xE

xSi ;

…6†

j2N

P where qi are given allocation fractions which satisfy the conditions qi P 0 8i 2 N and i2N qi ˆ 1. Superindex `0' is used to denote decisions under the conditions of linear allocation. In case of insucient supply this allocated to each option proportional to the P rule divides the amount ‡ overall remanufacturables de®cit ‰ j2N …Mj xSj † xR Š just using the constant fractions qi , which have to be determined in an appropriate way. Such a policy seems promising since it is also successfully employed in multi-echelon divergent inventory systems, where similar allocation problems arise (cf. Diks et al., 1996). A second assumption which is useful to facilitate the evaluation of a tractable policy for the MROP refers to the distribution of serviceables inventory positions. As often used as supposition in related multiechelon inventory systems we assume that the inventory positions of all serviceable options are properly balanced. This so-called balance assumption means that the echelon stock in the system is always distributed in such a way that under application of the remanufacturing policy with linear allocation rule we never face a situation where negative remanufacturing orders occur, i.e., ri0 P 0

8i 2 N :

…7†

Under these assumptions it can be shown that the resulting (only approximately) optimal policy has a very simple structure, and that the associated value function of dynamic programming has nice properties (for proofs and details, see Inderfurth et al., 1998). 3.2.1. The ®nite horizon problem 3.2.1.1. The single-period case. For studying the MROP policy for ®nite planning horizons we start with the single-period case. It turns out that the optimal disposal and remanufacturing decisions in this case, different from the situation in (5), are solely dependent on n ‡ 1 constant inventory levels Mi …i ˆ 1; . . . ; n† and U, if a linear allocation rule is applied and the balance assumption is made.

138

K. Inderfurth et al. / European Journal of Operational Research 133 (2001) 130±152

xE < X

X

Mj

)

b0 ˆ 0;

ri0 ˆ Mi

j

Mj 6 xE 6 U

)

b0 ˆ 0;

j

xE > U

)

b0 ˆ x E

qi

! Mj

xE

xSi ;

j

ri0 ˆ Mi ri0 ˆ Mi

U;

X

…8†

xSi ;

xSi :

This kind of policy is easy to implement in practice. In the sequel it will be called …nM; U †-policy. The remanufacture-up-to-levels Mi are given by (3). For the critical inventory U, which can be interpreted as dispose-down-to level, we have  1 for cB > hR ; Uˆ P for cB 6 hR : i Mi From (8) it can be seen that the way of reacting with disposal and remanufacturing decisions mainly depends on the size of the echelon inventory positions in relation to the dispose-down-to and to the remanufacture-up-to levels. This holds for each feasible combination of allocation fractions. Using the …nM; U †-policy, it can be shown that the resulting minimum single-period costs f10 …xR ; xS1 ; . . . ; xSn † perform a convex function. Furthermore, it can be proved that this value function can be expressed as being dependent on the echelon inventory position xE instead of being a€ected by the stock level xR of returns f10 …xR ; xS1 ; . . . ; xSn † ˆ f1 …xE ; xS1 ; . . . ; xSn †:

…9†

Eventually it turns out that this transformed value function is a sum of separate convex functions in the single state variables X f1i …xSi †: …10† f1 …xE ; xS1 ; . . . ; xSn † ˆ f10 …xE † ‡ i2N

3.2.1.2. The multi-period case. By induction it can be proved that the properties of the policy structure and value function evaluated in the single-period case also hold for each period of a multi-period problem with ®nite horizon T, given the linear allocation policy and the balance assumption. From the functional equations in (1), we can derive the minimization problem in the value function determination for any period t < T to be ft …xE ; xS1 ; . . . ; xSn † ˆ min gt …xE ; xS1 ; . . . ; xSn ; b; r1 ; . . . ; rn † b;r1 ;...;rn

w:r:t:

b‡

X

ri 6 x E

i2N

X

xSi ;

b; r1 ; . . . ; rn P 0:

i2N

The function to be minimized is gt …xE ; xS1 ; . . . ; xSn ; b; r1 ; . . . ; rn † ˆ cB b ‡

X

cRi ri ‡ Gt0 …xE

i

where Gt0 …yE † ˆ hR yE ‡ a

Z

1 1

ft‡1;0 …yE ‡ S† uS …S† dS

b† ‡

X i

Gti …xSi ‡ ri †;

K. Inderfurth et al. / European Journal of Operational Research 133 (2001) 130±152

139

and Z Gti …yi † ˆ

hR yi ‡ Li …yi † ‡ a

1 0

ft‡1;i …yi

Di † uDi …Di † dDi :

Solving this optimization problem under the assumption of Section 3.2, we ®nd a policy to be optimal which has the same …nM; U † structure as in (8) for the single-period case. The policy parameters Ut and Mti follow from cB ˆ G0t0 …Ut † and

cRi ˆ

Gti …Mti †

as long as these conditions result in Ut P following set of equations: cB

8i 2 N ; P i2N

…11†

Mti . If not, the optimal parameters are developed from the

G0t0 …Ut † ˆ cR1 ‡ G0t1 …Mt1 † ˆ


ˆ cRn ‡ G0tn …Mtn †

…12†

and Ut ˆ

X

Mti :

i2N

This result means that in principle the dispose-down-to-level Ut has to be chosen such that the costs of disposing one unit equals the marginal cost advantage of decreasing the echelon stock by one item. The remanufacture-up-to-levels Mti are optimally determined if the cost of recovering one unit with respect to option i equals the marginal cost advantage of increasing the respective inventory positions. This holds as long as the cumulated Mti -values do not exceed the disposal level Ut . Otherwise these levels have to be equalized, and the single parameters have to be determined in such a way that the net marginal cost e€ects are the same for increasing or decreasing by one unit, respectively. 3.2.2. The in®nite horizon problem In stationary situations it can be useful to consider problems with an unlimited planning horizon. The in®nite horizon is a good approximation for products that can be used for many years, as among others holds for products whose technical speci®cations do not rapidly change and which have been designed in a modular way. Examples are washing machines, refrigerators, big medical systems and copiers. For the product recovery problem under consideration this can only make sense, if it is assured that the expected returns per period exceed the sum of expected demands for the serviceable options, i.e., X E‰RŠ > E‰Di Š; …13† i2N

as usually holds during the end of the saturation and the complete decline phase of the business economic life-cycle of products, as well as for the time hereafter. Otherwise, total costs would tend to in®nity since expected backorders would go on accumulating over time. 3.2.2.1. The discounted cost case. In case of strictly discounted costs, i.e., a < 1, it can be shown under the returns excess condition in (13) that in the in®nite horizon case (i.e., T ! 1), the value function ft …xE ; xS1 ; . . . ; xSn † converges to a stationary function (for a proof and details see Inderfurth et al., 1998). So the functional equations in the limiting case can be written as

140

K. Inderfurth et al. / European Journal of Operational Research 133 (2001) 130±152

( f…xE ; xS1 ; . . . ; xSn † ˆ

b‡

P min P ri 6 xE

xSi

cB b ‡

X

cRi ri ‡ hR …xE

i

b;r1 ;...;rn P 0

Z ‡a ‡ rn

1 1

b† ‡

X

Li …xSi ‡ ri †

i

Z

1 0

Z ...

1 0

f…xE

b ‡ S; xS1 ‡ ri

D1 ; . . . ; xSn )

Dn † uS …S† uD1 …D1 † . . . uDn …Dn † dS dD1 . . . dDn :

…14†

For f…xE1 ; xS1 ; . . . ; xSn †, the properties of separability and convexity hold. So in the in®nite horizon case under application of a linear allocation rule, the optimal disposal and remanufacturing decisions still follow a …nM; U †-policy as de®ned in (8). However, the policy parameters U and Mi are no longer time-dependent, but are the same for each period. They will depend on all cost parameters and additional problem data. In order to compute these n ‡ 1 parameters numerically, given a prespeci®ed set of allocation fractions qi , we can transform the problem into a discrete one and apply the policy improvement technique or the method of successive approximations of dynamic programming (see, e.g., Hillier and Lieberman, 1990, pp. 778±792). Because of the multi-dimensionality of the state space numerical solution procedures of dynamic programming are connected with a prohibitive amount of computation. Therefore it is desirable to ®nd a practically acceptable way of computing the optimal values of the policy parameters U and Mi . Such a procedure is available in the case of average cost minimization. 3.2.2.2. The average cost case. The average cost case is equivalent to a problem setting where a discount factor equal to one is chosen, i.e. a ˆ 1. Unfortunately, for this case we were not able to prove formally the convergence property of the respective multi-period value functions of the costs per period if T ! 1. However, we have the strong conjecture that for the average cost criterion also this property holds (as it does in many stochastic inventory models), so that in this case also a stationary …nM; U †-policy would be optimal under the linear allocation rule and balance assumption. By all means, this is valid for any a-value arbitrarily near to one. In the next section we will present a very e€ective procedure to determine the policy parameters U and Mi for an average cost situation. This approach additionally includes an optimization of the allocation fractions qi which are used for the linear allocation rules. It shall be pointed out that di€erent from the discounted cost case under the average cost criterion the disposal cost cB and remanufacturing costs cRi will not a€ect the optimal policy parameter values. This is simply due to the fact that, because each demand ®nally has to be ful®lled, the average remanufacturing quantity of each option equals the respective average demand, whereas the average disposal amount is equal to the expected excess of return over cumulated demands. So the average remanufacturing and disposal costs per period are ®xed and cannot be a€ected by choosing the policy parameters, which only will depend on the di€erent holding and backorder costs. However, if holding costs for returns …hR † and for serviceables …hSi † also re¯ect the interest on capital tied up in the inventories, remanufacturing and disposal costs may in¯uence these values so that they implicitly may a€ect the policy parameters.

3.3. Discussion of the …nM; U †-policy The …nM; U †-policy is highly attractive for practical application, because it is simply structured, direct to understand, and easy to calculate. Nevertheless, in Section 3.1 we have shown that the optimal policy is

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far more complex than this simple rule. For assessing the value of the …nM; U †-policy we have to discuss the signi®cance of both the linear allocation rule and balance assumption, because given these conditions the …nM; U †-policy is optimal. As mentioned above the decision problem under consideration is similar to a two-echelon inventory control problem where goods are ordered at a central stockpoint from an outside supplier and shipped to various decentral stockpoints from where stochastic demands have to be satis®ed. In the MROP the supply side of the problem is signi®cantly di€erent, because the in¯ow of goods is stochastic. But the demand side is almost identical, so that the procedure of providing downstream stockpoints with goods is comparable. For the divergent inventory control case with strictly proportional costs it is well known that order-up-to-policies comparable to the remanufacture-up-to-rules are nearoptimal (cf. Van Houtum et al., 1996). Additionally, it is known that a linear allocation rule with optimized allocation fractions is very well performing for the task of allocating insucient central stock (cf. Van der Heijden et al., 1997). In fact, Diks (1997) shows for a two-echelon two-product model that the optimal allocation function and the optimal linear allocation function are almost identical for all relevant values of the inventory positions of the stockpoints. Besides this, also the balance assumption turns out to be non-critical under a wide range of circumstances which also include lot-for-lot-ordering which is equivalent to an order-up-to-policy (cf. Diks et al., 1996). So we can expect the remanufacturing and allocation part of the …nM; U †-policy to be very reasonable for tackling the product recovery problem under consideration. On the other hand, the disposal part of the …nM; U †-policy is equally well justi®ed by the results from analyzing remanufacturing problems with a single option for product recovery. For these kinds of problems it is shown that under proportional costs the optimal disposal policy follows a dispose-down-to-rule as included in the …nM; U †-policy (cf. Inderfurth, 1997). Altogether, this makes us very con®dent, that the proposed …nM; U †-policy is an ideal candidate for being applied in making coordinated disposal, remanufacturing and allocation decisions in the MROP. Additionally, it turns out that in the special case of only one single remanufacturing option an …nM; U †-policy is optimal, because in this situation the resulting …1M; U †-policy is a degenerated version of an …L; M; U †-type policy which can be shown to be optimal when an additional regular production option is available for procurement (cf. Inderfurth, 1997). For the case of exceptionally high procurement costs, which is equivalent to non-existence of a procurement option, the order-up-to-level L tends to minus in®nity …L ! 1† and both policies coincide. Thus the combination of a remanufacture-up-to and a dispose-down-to policy can be shown to be optimal, if an allocation problem cannot occur.

4. Parameter optimization for the …nM; U†-policy In Section 3 we proved the optimality of the …nM; U †-policy for the discounted cost case for both ®nite and in®nite horizon. In this section we concentrate on the average cost case under the assumption of stationary demand for serviceable options and stationary returns of used products. We propose a computational scheme to derive near-optimal policies within the class of …nM; U †-policies with linear allocation rules.

4.1. The stationary analysis For the steady-state analysis in the sequel we will use the following random variables which are n needed to determine the dynamics of relevant stocks when policy parameters U and fMi giˆ1 are applied.

142

Rt Dit Ditki ‡1 D0t Bt Yt Zt xNSit xARt xASit

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returns of reusable products during …t 1; tŠ demand for serviceable option i during …t 1; tŠ; i 2 N demand for serviceable option i during …t 1; t ‡ ki Š; i 2 N total demand for serviceable options during …t 1; tŠ amount disposed of at time t echelon system stock at time t immediately after disposal U Yt net stock of serviceable option i at time t immediately after satisfying all demands in …t 1; tŠ and immediately before input of newly available serviceable options, i 2 N net stock of products available for remanufacturing at time t immediately after the disposal decision and immediately after the allocation decision echelon inventory position of serviceable option i at time t immediately after the allocation decision, i2N

In the sequel we omit the index t for the above random variables to indicate their stationary equivalent. It readily follows that the long-term average costs under a given …nM; U † policy, which are in¯uenced by the policy parameters, are given by X X C…U ; M† ˆ vi E‰xNSi Š ‡ hSi E‰xNSi‡ Š ‡ hR E‰xAR Š; …15† i2N

i2N

where M stands for the vector of parameters fMi gniˆ1 . As mentioned above, in the average cost case the expected remanufacturing and disposal costs per period are not a€ected by the choice of policy parameters since on the long run remanufacturing has to be equal to demand (due to the backorder assumption) and the disposal quantity per period has to be equal in expectation to the excess of returns over total demands in a period. Note that in (15) we suppressed the dependence of the above random variables on the policy parameters n U and fMi giˆ1 . To compute an expression for C…U ; M† we need to ®nd expressions for E‰xAR Š; E‰xNSi Š 8i 2 N N‡ and E‰xSi Š 8i 2 N . These random variables depend on the dynamics of the echelon stock of the system and on the dynamics of the echelon stocks of the serviceable options. Let us ®rst consider the dynamics of the echelon stock of the system. It is easy to see that the following equations hold: Yt‡1 ˆ min…U ; Yt

D0t ‡ Rt †;

Bt‡1 ˆ max…0; Yt

D0t ‡ Rt

U †:

Substitution of the de®nition of Zt in the above equations yields: Zt‡1 ˆ max…0; Zt ‡ D0t

Rt †;

…16†

Bt‡1 ˆ max…0; Rt

Zt †:

…17†

D0t

Now we note that Eq. (16) is identical to Lindley's integral equation for the G=G=1 queue, where the interarrival times are Rt and the service times are D0t , respectively. The waiting times are given by Zt . In De Kok (1989) a moment-iteration method is developed to compute the ®rst two moments of the waiting time distribution. This method can be applied here as well to compute the ®rst two moments of Z. The essence of the method is to compute E‰Ztk Š for k ˆ 1; 2 from k Š ˆ E‰max…0; Zt ‡ D0t E‰Zt‡1

k

Rt † Š;

…18†

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143

where we ®t mixtures of Erlang distributions to the ®rst two moments of Zt ‡ D0t and Rt , respectively (cf. Tijms, 1994). The expressions resulting from Eq. (18) under this ®t assumption are easy to compute. By subsequent substitution of the newly found E‰Ztk Š in the right-hand side of (18), we obtain a series E‰Ztk Š, which converges fast, provided E‰D0 Š=E‰RŠ < 0:95. From now on we assume that we have accurate approximations for E‰ZŠ and E‰Z 2 Š. The next relevant operating characteristic is E‰xAR Š. It follows P from the allocation policy that products are left at the stock point i€ the P system echelon stock exceeds i2N Mi . The amount left is the di€erence between the echelon stock and i2N Mi . De®ning d as X dˆU Mi ; i2N

we thus ®nd that ‡

E‰xAR Š ˆ E‰…d

Z† Š:

Under the linear allocation policies de®ned in Section 3, we have that in the stationary situation xASi ˆ Mi

qi …Z

d†‡ :

From standard inventory theory arguments, we then ®nd the following expression for xNSi : xNSi ˆ xASi

Diki ‡1 ˆ Mi

qi …Z

d†

‡

Diki ‡1 :

Hence we ®nd for i ˆ 1; 2; . . . ; n E‰xNSi‡ Š ˆ E‰…Mi E‰xNSi Š ˆ E‰…qi …Z

d†‡

qi …Z

Diki ‡1 †‡ Š;

‡

d† ‡ Diki ‡1

‡

Mi † Š:

The above expected values for E‰xAR Š; E‰xNSi‡ Š 8i 2 N and E‰XSiN Š 8i 2 N , can be easily computed when we subsequently ®t mixtures of Erlang distribution to the ®rst two moments of Z and ®t a mixture of Erlang ‡ distributions to the ®rst two moments of qi …Z d† ‡ Diki ‡1 . 4.2. Algorithm for parameter determination Now we want to solve the following problem: min

U ;fMi gniˆ1 ;fqi gniˆ1

n

n

C…U ; fMi giˆ1 ; fqi giˆ1 †:

To solve this problem we compare it with the problem of ®nding cost-optimal policies for twoechelon divergent inventory system as studied in Diks (1997). A close look at the cost equations derived in Diks (1997) shows that these equations are identical when Z is replaced by a random variable that denotes the total demand for options during a ®xed lead time preceding the allocation moment. In Diks (1997) it is proven that the optimal order-up-to-policy with linear allocation rules can be found by recursively solving Newsboy equations. A detailed study of his proof shows that it applies also to our multiple options model. The key condition to be satis®ed is that Z, which replaces the total demand during a ®xed lead time preceding time t, is independent of the demand for each option during another ®xed lead time following time t. This clearly holds. In Dong et al. (1994) a similar approach based on

144

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substitution of random variables shows the equivalence of a multi-stage planned lead times model with a serial N-echelon inventory model studied by Clark and Scarf (1960). Thus the following theorem holds. Theorem 4.1. The optimal policy …U  ; fMi gniˆ1 ; fqi gniˆ1 † satisfies the following Newsboy equations: P fDiki ‡1 6 Mi g ˆ

v i ‡ hR vi ‡ hR ‡ hSi

P fDiki ‡1 ‡ qi …Z

d † 6 Mi g ˆ

X i2N

‡

8i 2 N ; vi vi ‡ hR ‡ hSi

8i 2 N ;

qi ˆ 1:

In order to solve these equations we apply the algorithm proposed in De Kok (1990) according to the re®nements in Diks (1997) (see also Dong et al., 1994). Note that the costs C…U ; M† in (15) among others depend on the shortage costs vi …i 2 N † which will re¯ect the amount of the pro®t margin of option i possibly a€ected by a shortage. Thus, minimizing C…U ; M† under linear allocation rules just following Theorem 4.1, results in optimal allocation fractions qi which will also incorporate the levels of contribution margins of the di€erent options. Algorithm 4.1. Step 1. (a) Fit a mixture of Erlang distributions F^i to the ®rst two moments of Diki ‡1 and use a bisection procedure to solve F^i …Mi † ˆ

vi ‡ h R vi ‡ hR ‡ hSi

8i 2 N :

(b) Set d ˆ 0. Step 2. ‡ (a) Fit a mixture of Erlang distributions F^Z to Z and compute the ®rst two moments of E‰…Z d† Š under this distribution ®t. For i ˆ 1; 2; . . . ; n set qi;min ˆ 0 and qi;max ˆ 1: (b) qi ˆ …qi;min ‡ qi;max †=2: ‡ Fit a mixture of Erlang distributions F~i …
; qi † to the ®rst two moments of Diki ‡1 ‡ qi …Z d† . If F~i …Mi ; qi † 6

vi ; vi ‡ hR ‡ hSi

then qi;min :ˆ qi , otherwise qi;max :ˆ qi (c) If vi F~i …M  ; qi † 6 ; i vi ‡ hR ‡ hSi then goto step 3 else goto step 2(b).

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145

Step P 3. If Pi2N qi P 1 ‡  then decrease d, goto step 2. If i2N qi 6 1 ‡  then increase d, goto step 2. Otherwise goto step 4. Step 4. X Uˆ Mi ‡ d: i2N

If d < 0 then Mi :ˆ Mi ‡ qi d 8i 2 N ; d :ˆ 0 Stop. In the implementation of the above procedure the main diculty occurs when the di€erence in holding costs hR and hSi is small. In that case it may happen that the optimal policy does not maintain any stock of n input material. This results P in a non-positive value of d. To detect this situation one should solve for fqi giˆ1 with d ˆ 0 and check if i2N qi > 1. If so, one should search for negative d, otherwise one should search for positive d. Step 4 of Algorithm 4.1 updates the order-up-to-levels Mi so that the ®nal policy satis®es d > 0. The validity of the update equations follows from Lemma 4.1. P Lemma 4.1. Given a policy …U ; fMi g; fqi g†. Let d ˆ U i2N Mi . If d < 0, then the sample paths of the inventory positions fxSi g under this policy are identical to the sample paths under a …U ; fMi ‡ qi dg; fqi g† policy. Proof. First note that Zt is independent of d. At time t, xSit is given by xASit ˆ Mi

qi …Zt

d†‡ :

Since d is negative this is equivalent to xASit ˆ Mi

qi Zt ‡ qi d:

…19†

Under the …U ; fMi ‡ qi dg; fqi g†-policy we ®nd similarly !!‡ X …Mi ‡ qi d† : xSit ˆ Mi ‡ qi d qi Zt U i2N

Since

X

…Mi ‡ qi d† ˆ

i2N

X

Mi ‡

i2N

we ®nd xASit ˆ Mi ‡ qi d

qi Zt ;

which is identical to (19).

X i2N

qi d ˆ

X i2N

Mi ‡ d ˆ U ;

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In Diks (1997) and De Kok (1989), the accuracy of the two-moment ®ts is demonstrated for the two major building blocks of our approach. From De Kok (1989), we may conclude that our approximation for the ®rst two moments of Z is accurate. From Diks (1997), we may conclude that Algorithm 4.1 yields excellent approximations for the optimal policies within the class of …nM; U †-policies with linear allocation rules. In Section 5 we apply Algorithm 4.1 to gain insight into a number of issues concerning the Multiple Remanufacturing Options Problem.

4.3. A case example In this section we illustrate the application of Algorithm 4.1 to a case example. Suppose a returned copier can be re-used in three di€erent ways: 1. Copier is cleaned and sold at a second-hand third-world market. 2. Copier is cleaned, some parts are substituted, a software-upgrade is made and the copier is sold as new at a second-hand market in Europe. 3. Copier is taken apart completely and parts are used for the assembly of new copiers. This is in fact the dispose option. Since the two markets involved can be seen as separate markets it is reasonable to assume that demand for option 1 is independent of option 2. It is assumed that any disposed item can be absorbed into the manufacturing of new copiers. We assume the following about the inputs of the model:

Reusable products E‰RŠ r…R† hR cB

200 200 40 100

Serviceable options i E‰Di Š r…Di † ki vi hSi

1 80 120 2 1200 200

2 100 50 4 2000 300

In this case cB should be interpreted as the di€erence between the lost opportunity to re-use the copier for options 1 and 2 and the savings realized for not having to buy new components. The values for vi should be interpreted as the cost of not being able to sell the copier immediately and potential future costs for loosing customers because of backorders. From the optimality condition we immediately derive that the non-stockout probabilities for options 1 and 2 under the optimal policy pi are given by 83% and 85%, respectively. Given these data we ®nd the following solution and associated costs:

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Reusable products U E‰xAR Š

7333 5032 (28 periods)

Serviceable options i Mi qi E‰xNSi Š ‡ E‰xNSi Š

1 455 0.8 44 (0.55 periods) 248 (3.1 periods)

147

2 628 0.2 11 (0.11 periods) 136 (1.4 periods)

From the results we ®nd that substantial stocks of reusable products are held, because they are compared to the serviceable options almost for free. In practice we may expect that capacity imposes a constraint on the stock of reusable products. Furthermore we ®nd that option 2 has both lower stocks and backlogs than option 1. This follows from the fact that for option 2 both holding costs and shortage costs are higher. We also ®nd that if a shortage occurs at the stock of reusable products (which is very unlikely), then only 20% of this shortage is allocated to option 2, again safeguarding option 2 against shortages.

5. Managerial insights In this section we provide some insight into the impact of di€erent input characteristics on the average system costs consisting of disposal costs, holding costs and penalty costs. Due to the assumption of backordering demand for serviceable options we do not consider the remanufacturing costs. Although the disposal costs are independent of the policy chosen, we do consider disposal costs in some comparisons since the disposal costs depend on the demand/return ratio q, with qˆ

E‰D0 Š ; E‰RŠ

…20†

which under certain circumstances can be considered to be depending on managerial activities. Under longterm conditions ®nite costs can only be assured if q < 1. For a ®xed ratio q, the optimization procedure under a …nM; U †-policy is restricted to determine the amount of the overall echelon stock in the system and to balance this stock between the remanufacturable products and the di€erent serviceable options in order to keep expected holding and backorder costs at the lowest possible level. In the numerical experiments discussed below we depart from a base case: we assume that n ˆ 3, E‰Di Š ˆ 100 …i ˆ 1; 2; 3†, r…Di † ˆ 100 …i ˆ 1; 2; 3†, hR ˆ 1, hSi ˆ 1:5 …i ˆ 1; 2; 3†. The backorder costs are determined indirectly as follows. From the Newsboy equation in Theorem 4.1 it follows that the optimal policy satis®es a particular non-stockout probability for each option determined by holding and shortage costs. Since stockout probabilities are more intuitive than shortage costs we conversely choose a nonstockout probability and determine the backorder costs from the Newsboy equations. In all cases considered we assume a non-stockout probability of 0.98 for all serviceable options. In Fig. 2 we show the impact of c2R , i.e., the squared coecient of variation of the number of returns per period, on the minimum of the relevant costs in (15). It follows that the costs increase almost linearly with the squared coecient of variation of the number of returns for both values of q considered. This behavior

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Fig. 2. Impact of variability of returns c2R on minimum costs.

Fig. 3. Impact of demand/return ratio q on minimum costs.

is similar to the behavior of the average waiting time in an M/G/1 queue as a function of the squared coecient of variation of the service times, expressed in the famous Pollaczek±Khintchine formula (e.g., Tijms, 1994). In Fig. 3, we show the impact of q on the sum of relevant costs. Similar to the standard behavior of waiting times in queues we ®nd that the costs explode when q approaches 1. For low values of q the costs seem independent of c2R . The results shown in Fig. 3 motivate us to consider the impact of the disposal costs on the total costs. Disposal cost e€ects are particularly considered for two reasons. First, total disposal costs re¯ect the level of disposal activities which is a critical issue from an environmental perspective and is usually desired to be as low as possible from a global point of view. Second, although the latter may not be valid under managerial considerations, disposal costs per unit often determined by governmental price setting are relevant cost inputs which together with other cost parameters a€ect the product recovery policy of a ®rm. In our context managerial reaction on disposal cost parameters variations is only reasonable, if the demand/returns ratio q can be in¯uenced. By increasing q the fraction of returns disposed of can be decreased. In principle this can be done by two di€erent types of managerial actions. On one hand, the amount of products returned per period may be reduced, e.g., by extending the life cycle of the products. On the other, it may be possible to increase demand for serviceable options, e.g., by additional marketing activities in traditional or by penetration of new markets. In the sequel we analyze the in¯uence of the disposal costs in combination with a possible variation of the ratio q. Towards this end we de®ne a base case with the demand and cost parameters as above. Furthermore we assume that q0 ˆ 0:75 and c2R ˆ 0:5. For this base case, we determine the disposal cost

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149

Fig. 4. Impact of demand/return ratio q on total costs including disposal.

per unit cB , such that the disposal costs cB …E‰RŠ E‰D0 Š† equal 20% of the overall holding and backordering costs in (15). For this parameter constellation (where we have cB ˆ 13:36) total costs including expenditures for disposal (but excluding remanufacturing costs) can be shown to follow an U-shaped function as depicted in Fig. 4. There exists an optimal value for q at a medium demand/returns level. This value characterizes the optimal trade-o€ between holding and backorder costs which are monotonically increasing and disposal costs which are monotonically decreasing as q increases. Fig. 4 shows that it makes sense to in¯uence the ¯ow of returns to prevent that the system is in a suboptimal situation. To pursue this point somewhat further we determine the optimal q as a function of /, which is de®ned as the average disposal costs as a percentage of total costs (excluding remanufacturing costs). It follows from the above that we must ®rst de®ne a base case from which we derive the disposal cost per unit that is associated with the given value of /. In Fig. 5 we consider three base cases de®ned by q0 ˆ 0:5; 0:75; 0:9. Next for each value of / we determine the optimal value of q. It follows from the results in Fig. 5 that the optimal q increases with /. This is intuitively clear, since higher the /, the higher is the contribution of disposal cost to the total costs, and therefore we want to reduce the disposal costs by increasing q. The fact that the optimal values of q increase with q0 follows from a similar argument. To gain insight into the sensitivity of the total cost for changes in cB we again start from a particular base case and determine cB for a given value of /. Next we determine for q ˆ 0:1; 0:2; . . . ; 0:9, the relative change in cB that would result in exactly the same cost as in the base case. As base cases we assume q0 ˆ 0:5; 0:8. From Table 1 we ®nd that in case q0 ˆ 0:5, the value of cB may increase substantially for high values of q in case / is high, i.e., disposal costs are a large fraction of the total cost in the base case. Hence decreasing the fraction of input disposed gives a high reduction in cost when cB remains unchanged or gets only slightly

Fig. 5. Impact of disposal cost fraction / on optimal demand/return ratio.

150 Table 1 /

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q (%) 0.1

0.2

0.3

0.4

0.5

q0 ˆ 0:5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

inf. inf. )99 )95 )93 )92 )91 )90 )89

)59 )68 )71 )72 )73 )74 )74 )75 )75

)34 )47 )51 )53 )55 )55 )56 )57 )57

)14 )25 )28 )30 )31 )32 )32 )33 )33

0 0 0 0 0 0 0 0 0

q0 ˆ 0:8 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

inf. inf. )99 )98 )98 )98 )98 )97 )97

)32 )66 )78 )83 )87 )89 )91 )92 )93

)13 )44 )63 )72 )78 )82 )84 )86 )88

64 )18 )45 )59 )67 )72 )76 )79 )82

123 13 )24 )42 )53 )60 )66 )69 )73

0.6

0.7

0.8

0.9

)9 24 35 40 43 46 47 48 49

inf. 23 69 92 106 115 121 126 130

inf. inf. 36 130 187 225 252 272 287

inf. inf. inf. inf. 9 273 461 602 712

189 49 3 )21 )35 )44 )51 )56 )59

227 78 28 3 )12 )22 )29 )34 )38

0 0 0 0 0 0 0 0 0

inf. inf. inf. inf. inf. inf. )54 20 79

Fig. 6. Impact of demand/return ratio q on structure of stocks.

higher. In case q0 ˆ 0:8, Table 1 shows that an increase in disposal costs per unit increases costs in almost all cases. In fact, a substantial decrease in disposal costs per unit is required if q gets lower, i.e., if the amount disposed increases. Finally we study the impact of q on the division of system stocks and holding costs between the stock of returns and the stocks of serviceable options. Again we assume the base case data given above for demand and cost parameters. It follows from Fig. 6 that the fraction of system stock in serviceable options reduces as q increases. This is caused by the fact that with higher q, i.e., average returns closer to average demand, we need a higher system stock to prevent high penalty costs. Since the cost of holding returns is lower than the cost of holding serviceable options, we need to keep relatively higher return stocks. This concludes our discussion of the impact of model parameters on costs and control aspects.

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6. Conclusions The purpose of this paper is two-fold. First, it aims to give an analytical insight into the complexity of optimal control for stochastic remanufacturing problems, if more than a single recovery option is available as often found in practice. Second, it shows that under a reasonable restriction of allocation procedures for reusables a simple policy is optimal, the parameter of which can easily be calculated numerically. Thus this policy is a good candidate for practical application. In more detail, this paper refers to developing optimal and near-optimal control rules for a stochastic product recovery problem where multiple remanufacturing options exist which yield di€erent serviceable products satisfying speci®c demands. It is demonstrated that the structure of the optimal policy in this case is extremely complicated, due to the inherent allocation problem in case of scarce remanufacturables. However, employing the so-called linear allocation rules it turned out that a fairly simple near-optimal control policy exists which is characterized by a single disposedown-to level and a speci®c remanufacture-up-to level for each reuse option. For this policy which can easily be used in practical decision making, an ecient algorithm is developed for numerical evaluation of the policy parameters. Using this algorithm a numerical study was carried out, which helps to gain an insight into the in¯uence of disposal costs and the demand/returns ratio on the performance of such a remanufacturing system. The results presented in this paper are developed under some restrictive assumptions which partly can be relaxed without diculty. Due to the similarity of the serviceable options, correlations of demands across these options may be present. For this case of correlated demands a straightforward extension of the policy analysis demonstrated above is possible. However, it is far more dicult to incorporate the possible e€ect that certain demands can be ful®lled by several options, e.g., by upgraded product versions when downgraded ones that are originally demanded are out of stock. It is easy again, to take into account stochastic remanufacturing leadtimes for the di€erent reuse options. But including ®xed remanufacturing costs or capacity constraints for the remanufacturing facilities makes an analysis of the optimal policy extremely dicult. From problems with a single reuse option we know that in these cases simple policies cannot be expected to be optimal. Under these circumstances plausible, but not necessarily optimal control rules which are known from the single-option case (cf. Van der Laan and Salomon, 1997) should be adapted. Necessary extensions of our model which deserve further research should include the following additional aspects. In some instances demands for remanufactured products can also be ful®lled by newly manufactured items, usually at a considerably higher cost. Then manufacturing of serviceables, possibly in a speci®c mode for each di€erent option, can be an additional source of satisfying demands. In this situation a combined optimization of disposal, remanufacturing and manufacturing is necessary. A further possible extension incorporates di€erent quality levels of the returned products which may lead to di€erent remanufacturing costs and leadtimes. Under these conditions, multiple stocks of returned items have to be taken into account. In this paper we included up- and down-grading for items in the RP inventory. It seems worthwhile to study as well the consequences of up- and down-grading di€erent SP items into each other. Forthcoming research is devoted to investigating these extended multiple remanufacturing options problems based on the results developed in this paper. Acknowledgements The research presented in this paper makes up part of the research on re-use in the context of the EU sponsored TMR project REVerse LOGistics (ERB 4061 PL 97-5650) in which apart from Eindhoven University of Technology and the Otto-von-Guericke Universitaet Magdeburg take part the Aristoteles University of Thessaloniki (GR), the Erasmus University Rotterdam (NL), INSEAD (F), and the University of Piraeus (GR).

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