remanufacturing planning for the reverse wagner/whitin models

Remanufacturing planning for the reverse Wagner/Whitin models

Knut Richter *, Mirko Sombrutzki

Wirtschaftswissenschaftliche Fakult�at, Europa-Universit�at Viadrina Frankfurt (Oder), Groûe Scharrnstr. 59, D-15230 Frankfurt (Oder),

Germany

Abstract

In this paper the reverse Wagner/WhitinÕs dynamic production planning and inventory control model and some of its

extensions are studied. In such reverse (product recovery) models, used products arrive to be stored and to be re-

manufactured at minimum cost. For the reverse model with given demand the zero-inventory-property of optimal

solutions is proved, the corresponding Wagner/Whitin algorithm is presented and the stability of optimal solutions is

discussed for the case of a large quantity of low cost used products. Furthermore, the model of the alternate application

of remanufacturing and manufacturing processes is analysed. Again, for the case of a large quantity of low cost used

products the zero-inventory-property is proved, the Wagner/Whitin algorithm is applied to determine the periods in

which used products are remanufactured or new products are produced, and the stability of Silver/Meal solutions for

this model is sketched. Ó 2000 Elsevier Science B.V. All rights reserved.

Keywords: Inventory; Remanufacturing; Stability

1. Introduction

The traditional models of production planningand inventory control (see [10]) usually do not takeinto account the multiple use of products. Onlyrecently research has been started to integrate theso called product recovery management into pro-duction planning systems (see [19,8,7,13±15]) asone of the approaches of an ecologically orientedproduction management. These approaches try tomodel the product recovery management activityunder several assumptions. While Thierry et al.

[19] describe the management philosophy behindthe di�erent product recovery options like repair,refurbishing, remanufacturing, cannibalisation andrecycling, Laan et al. [8] and also Inderfurth [7]study review policies for production planning andinventory control in stochastic manufacturing/re-manufacturing/disposal systems, and Richter [13±15] applies the usual Economic Order Quantitymodel to a similar deterministic manufacturing/remanufacturing/disposal system.

Having in mind the greening of industry bydeveloping product recovery systems it is an in-teresting task to reconsider all models designed sofar for traditional planning systems. Here an at-tempt will be made to study further the modelsintroduced in [16] as reverse dynamic deterministic

European Journal of Operational Research 121 (2000) 304±315www.elsevier.com/locate/orms

* Corresponding author.

E-mail address: [email protected] (K. Richter).

0377-2217/00/$ - see front matter Ó 2000 Elsevier Science B.V. All rights reserved.

PII: S 0 3 7 7 - 2 2 1 7 ( 9 9 ) 0 0 2 1 9 - 2

models of production planning and inventorycontrol which can be solved by simple modi®ca-tions of the well-known Wagner/Whitin algorithm.

The traditional models, beginning with the pi-oneer work of Wagner and Whitin [21], havecovered many cases of single-item, multi-item, hi-erarchical situations etc. and they are constantlyattracting attention of many researchers (see[5,11,17]). In all these models the demand for oneor more products has to be satis®ed by ordering orproducing the appropriate quantities. Neither thereuse of the products, nor the return of theseproducts from the customer, are considered ex-plicitly. The aim of this paper is therefore to start®lling this gap, and the problem of productionplanning and inventory control is viewed from theopposite side: used products come back to theproducer, have to be stored and perhaps to beremanufactured, recycled, disposed of etc. For thesake of simplicity here manufacturing and re-manufacturing will always be used as synonymsfor the di�erent options of producing new prod-ucts and the product recovery, respectively. Themodels are designed and analysed, starting withthe simplest single item model, up to the rathercomplicated case of a model of alternate remanu-facturing and remanufacturing activities. Multi-item and hierarchical models are to be studied inanother paper.

In Section 2 the original Wagner/Whitin modelis brie¯y characterised as a traditional model formanufacturing, or more generally, producing newproducts to satisfy a given demand for severalperiods. In Section 3 this model is reversed, i.e., theformer demand parameters are treated now asin¯ow information for used products to be re-manufactured. The classical Wagner/Whitin algo-rithm and their planning horizon property will beadopted to this situation. In Section 4, the zero-inventory-property of optimal solutions for thereverse model with given demand are proved, thecorresponding Wagner/Whitin algorithm is pre-sented and the stability of optimal solutions([12,3]) is discussed for the case of a large quantityof low cost used products, i.e. for the case, that asu�ciently large quality of used goods with lowinventory cost is available already at the beginningof the planning process. In Section 5, the model of

the alternate application of remanufacturing andmanufacturing processes is analysed. Again, forthe case of a large quantity of low cost usedproducts the zero-inventory-property is proved,the Wagner/Whitin algorithm is applied to deter-mine the periods in which used products are re-manufactured or new products are produced, andthe stability of Silver/Meal solutions [18] for thismodel is sketched. The presentation concentrateson such models which make it possible to applythe zero-inventory-property. For more complicat-ed models algorithms might be developed easily.However, they probably do not give such clearinsight into the structure of an optimal behaviourin the remanufacturing/manufacturing environ-ment.

2. The original Wagner/Whitin model

The original Wagner/Whitin model describesthe planning process of ordering (or manufactur-ing, producing . . .) and stocking a certain productover some time intervals. The deterministic de-mand Dt > 0 for all periods t is to be satis®ed, andthe total sum of order cost and holding cost is tobe minimised. The cost inputs st > 0 and Ht > 0cover the set-up cost arising when a certainquantity of the product is ordered (or produced) inthe tth period and when some unit of the productis stored for the same period. The manufacturingversion of the model is described here, since laterits reverse process ± the remanufacturing (or moregenerally the product recovery) process will bediscussed.

If zt and It denote the decisions on the quanti-ties to be produced or to be stocked, respectively,and the number of periods regarded is ®xed by thedecision horizon T, the Wagner/Whitin model forthe T-period planning problem is usually providedby the following restrictions and objective func-tion:

I0 � IT � 0; It � Itÿ1 � zt ÿ Dt; zt; It P 0;

t � 1; 2; . . . ; T ;XT

t�1

st � sign zt� � Ht � It� ! min :

�1�

K. Richter, M. Sombrutzki / European Journal of Operational Research 121 (2000) 304±315 305

The model is based on the assumption that all theactivities are carried out just at the beginning ofevery period, i.e. the products are manufacturedand also delivered to the customer at the beginningof a period, and the inventory of a product ap-pears only if it is still on stock at the end of aperiod. The starting inventory level is usually set to0, and the cost minimisation makes the end in-ventory level equal to zero as well. When discuss-ing the reverse models, however, this level is setequal to zero explicitly.

For the optimal solution the zero-inventory-property Itÿ1 � zt � 0 is well known, which meansthat there is no production after a period withpositive inventory. Using this property, the origi-nally published Wagner/Whitin algorithm worksin the following way, although now, of course,much faster methods for solving problem (1) havebeen published [6,20]:

f0 � 0; ft � min06 i<t

citf � si�1 � fig;

cit �Xtÿ1

j�i�1

Hj � Dj�1;t;

i � 0; 1; . . . ; t ÿ 1; t � 1; 2; . . . ;

�2�

where Dj;t �Pt

i�j Di covers the cumulative de-mand for the periods j; j� 1; . . . ; t.

The values ft � ci�t�;t � si�t��1 � fi�t� express theminimal total cost for the periods 1; 2; . . . ; t andi(t) the regeneration point, the period before thelast order/production period. The holding costwhich arises if the demand for the periodsi� 1; i� 2; . . . ; t is produced in the period i� 1 isgiven by the parameters cit. The correspondingoptimal solutions can also be found easily by abackward recursion.

Another important property observed for themodel is the so called planning horizon[21,1,2,4,9,22]. In Wagner/WhitinÕs version a peri-od pÿ 1 is called the planning horizon, if the op-timal solutions of the ®rst pÿ 1 periods for aproblem with a p-period decision horizon do notchange in the case of decision horizon expansion.For the case of constant cost inputs, for instance, aplanning horizon can be found, if fp � sp � fpÿ1

holds, i.e. if it is optimal to start another manu-facturing and inventory holding cycle in the lastperiod. In this paper only the simply version of theplanning horizon for the remanufacturing modelin the sense of Wagner/Whitin will be studied.

The modelled situation can be illustrated inFig. 1.

If only demand of one period is manufactured,then no inventory stock occurs in the model.

3. The purely reverse Wagner/Whitin model

The purely (and somehow academic) reversesituation to that modelled by the original Wagner/Whitin model appears if there is a return ¯ow ofused products which are to be remanufactured,and no demand is known. In the original modelthe case of pull production is covered, whereproduction is induced by demand. Now the re-manufacturing process is pushed by the arrivinggoods and the case of push production is consid-ered. Fig. 2 illustrates this situation.

The used goods come (back) in quantities dt perperiod and are remanufactured after possiblestocking. The remanufacturing activity with set-upcost rt as well as the arrival of the used productswill happen at the beginning of every period, and

Fig. 1. Goods ¯ow in the original Wagner/Whitin model.

Fig. 2. Goods ¯ow in the reverse Wagner/Whitin model.

306 K. Richter, M. Sombrutzki / European Journal of Operational Research 121 (2000) 304±315

the quantity dt of used goods is assumed not toa�ect the inventory level, if it is immediately re-manufactured (see Fig. 3). In this ®gure the returnof the quantities dt at four periods is shown, andthe return quantities of the ®rst three periods arestocked until the remanufacturing activity (boldline) is started at the beginning of the fourth pe-riod. The quantity d4 is remanufactured just whenit arrives, and therefore it is not stocked at all.

In the original Wagner/Whitin model withpositive demand values the manufacturing processalways has to start in the ®rst period. One wouldexpect that in the reverse model the remanufac-turing will be accomplished in the last period ofthe decision horizon. In the latter case no demandhas to be satis®ed. Therefore the last remanufac-turing decision is only determined cost consider-ations, and it can even be optimal not toremanufacture any quantity of the used goods.

Thus the appropriate model will not have arestriction yT � 0 (cf. [16]). If xt presents thequantity remanufactured at the beginning of pe-riod t and yt is the inventory stock at that periodthe reverse model is then given by

y0 � 0; yt � ytÿ1 ÿ xt � dt; xt; yt P 0;

t � 1; 2; . . . ; T ;XT

t�1

rt � signxt� � ht � yt� ! min :

�3�

This model allows the used goods not to be re-manufactured within the decision horizon at all.Since so far no demand on the remanufacturedgoods has been taken into consideration, only thecost minimisation aspect will force the remanu-facturing process to start. For this model the zero-inventory-property yt � xt � 0 holds as well.

Let the following simple example be used toillustrate the di�erent models (1) and (3): For T� 3the demand Dt (or the quantity of arrived usedgoods dt) for the periods is given as 3, 2, 1, re-spectively, with time constant cost inputs s�r� 3.5 and h�H� 1. Then the di�erent solutionsand cost values are provided in Table 1 and Fig. 4.

While the data used for the original Wagner/Whitin model produces a solution with one lot ofsix units in the ®rst period, for the reverse model®ve units are remanufactured, and one unit re-mains as inventory stock.

If for an optimal solution of the model (3)xT > 0 is ful®led, i.e. the last remanufacturing ac-tivity is carried out in the period T, then this pe-riod remains to be remanufacturing periodwhatever extension of the decision horizon T 0 > Tfor the model is considered. The period T can beagain called planning horizon for model (3).

Theorem 1. The period T with xT > 0 in the optimalsolution is a planning horizon for model (3) in thesense, that the period T remains to be a remanu-facturing period, no matter which decision horizonT 0 > T is regarded.

Proof. Let T 0 > T and xT > 0. If T is not a plan-ning horizon, then there is another, optimal solu-tion for the T 0-period problem (3) with yT > 0. Letthe two decision horizons 1; 2; . . . ; T and

Fig. 3. The inventory stock for the reverse Wagner/Whitin model.

Table 1

Optimal solutions for the di�erent models

Model Optimal solution Minimal cost

(1) z1� 6 z2� 0 z3� 0 3.5 + 4� 7.5

(3) x1� 0 x2� 5 x3� 0 3 + 3.5 + 1� 7.5


T � 1; . . . ; T 0 be considered separately. Since theobjective function is additive, the minimal totalcost FT 0 might be also separated into two compo-nents C1 � C2. Due to the optimality FT 6C1

holds. Since in the supposed optimal solutionyT > 0 the cost FT 0;T 0 occurring in the case of yt � 0is also not higher than C2. Hence the solution withyT � 0 is optimal and T is a planning horizon. �

It follows from this theorem that an optimalsolution for the model (3) separates into a numberof locally optimal decisions which cover exactlythe remanufacturing cycle, and only at the end ofthe decision horizon some used products might becollected but not remanufactured. In the examplegiven in Table 2 the decision horizon T changesfrom 3 to 5, and, it can be seen, that the optimalsolution of the three-period problem leaves twounits not remanufactured and that the fourth pe-riod is a planning horizon.

4. The reverse model with given demand for

remanufactured products

This model combines the models (1) and (3) insuch a way that, on one hand, a given demand Dt

arising at the beginning of every period for theremanufactured products is to be satis®ed, and onthe other hand, the quantity dt of used productsarrive at the same time. Inventory levels It andinventory cost Ht, respectively, have to be consid-ered for the remanufactured products, while in-ventory levels yt and inventory cost ht appear forthe returned used products. As in the originalWagner/Whitin model, the equations It � Itÿ1�xt ÿ Dt describe the remanufactured product in-ventory process, and yt � ytÿ1 ÿ xt � dt the usedproduct inventory process. The situation modelledis illustrated by Fig. 5.

For the set-up cost rt the model of minimisingtotal cost is then given by the relations

y0 � I0 � 0; yt � ytÿ1 ÿ xt � dt;

It � Itÿ1 � xt ÿ Dt; It; xt; yt P 0;

t � 1; 2; . . . ; T ;XT

t�1

rt � sign xt� � ht � yt � Ht � It� ! min :

�4�

The model might be treated also as the originalWagner/Whitin model of type (1) with restrictionon the material dt delivered to the production unit.

Table 2

Example of a planning horizon

t� 1 2 3 4 5 Cost

dt 5 3 2 5 3

rt 6 6 6 6 6

ht 1 1 1 1 1

xt 0 8 0 ± ± 13

xt 0 8 0 5 ± 19

xt 0 8 0 5 0 22

Fig. 5. Goods ¯ow in the modelled situation.

Fig. 4. Graphical presentation of the optimal solutions for the two di�erent models.


Then the inventory stock for the material is givenby yt, and the ®nal product inventory by It. Here,however, the focus will be on the environmentalinterpretation.

d1j ÿ D1j P 0; j � 1; 2; . . . ; T : �5�

This model (4) has obviously feasible solutionsif and only if (5) is ful®lled, where di;t �

Ptj�i dj

represents the cumulative quantity of arrived usedgoods.

By adding the two inventory equations for anyfeasible solution of the model (4) the relation

yt � It � St with St � d1t ÿ D1t;

t � 1; 2; . . . ; T ;�6�

is obtained. Now the equivalent simpler model (7)will developed which will solve the original model(4).

First, the relation It6 St; t � 1; 2; . . . ; T holdsdue to (6) if and only if the relations yt P 0 areful®lled. If It6 St and (6) hold the equationsyt � ytÿ1 ÿ xt � dt are ful®lled automatically andcan be dropped.

Secondly, let ht6Ht; t � 1; 2; . . . ; T , be ful-®lled, i.e. the holding cost for remanufacturedproducts is not less than this cost for used prod-ucts, or, in other words, ``low cost used products''have to be remanufactured, then this cost inthe objective function can be expressed ashtyt � HtIt � ht�yt � It� � �Ht ÿ ht�It � htSt � H 0t t,where H 0t � Ht ÿ ht.

Thirdly, let the constant term be C �PTt�1 ht � St.Then the model (4) appears as

I0 � 0; It � Itÿ1 � xt ÿ Dt;

It6 St; t � 1; 2; . . . ; T ; It; xt P 0; t � 1; 2; . . . ; T ;XT

t�1

rt � signxt

ÿ � H �t It

�� C ! min : �7�

This model is really equivalent to the model (4)since an optimal solution which has the compo-nents xt and It can be extended to a feasible solu-tion for the original model by adding yt � St ÿ It.It is optimal, since the objective functions of bothmodels coincide.

This is a typical Wagner/Whitin model withupper bounds and it is known that an optimalsolution can be found among the solutions satis-fying the zero-inventory property Itÿ1 � xt � 0;t � 1; 2; . . . ; T (see [11]). The Wagner/Whitin al-gorithm (2) can be therefore applied by using asimply modi®cation.

Lemma 1. For any feasible solution of model (7)with Ii � It � 0; Ij > 0; j � i� 1; . . . ; t ÿ 1 therelation It6 St; t � 1; 2; . . . ; T is fulfilled if and onlyif the period i is not smaller than Jt, whereJt � minfj : Sj�1 P Dj�2;tg.Proof. Because of the monotonicity of

SJ�1 ÿ IJ�1 � SJ � DJ�1 ÿ DJ�1 ÿ �IJ ÿ DJ�1�� SJ ÿ IJ � DJ�1 > SJ ÿ IJ

the relation Ij � Dj�1;t6 Sj holds for j � i�1; . . . ; t ÿ 1 if and only if Ii�1 � Di�2;t6 Si�1, i.e. ifand only i P Jt is ful®lled. �

Thus the Wagner/Whitin algorithm (2) can bemodi®ed as

f0 � 0; ft � minJt 6 i<t

citf � ri�1 � fig;

cit �Xtÿ1

j�i�1

H �jDj�1;t; i � 0; 1; . . . ; t ÿ 1;

t � 1; 2; . . .

�8�

Example. Let r� 10, h� 1, H� 2 andd � �4; 3; 2; 1�, D � �2; 3; 1; 4�, i.e. S � �2; 2; 3; 0�.Then there is an optimal solution x � �2; 4; 0; 4�with I � �0; 1; 0; 0�, y � �2; 1; 3; 0� and the cost3 � 10� 1� 7 � 30� 2� 6 � 38. The parametersJt equal 0; 1; 1; 3 for t � 1; 2; 3; 4. Therefore the onlydecision to be made is to determine

f3 � minff1 � 10� 1; f2 � 10g� minf10� 11; 30g � 21;

i.e. i�3� � 1:

Remarks.(i) If now

d � d1 P D1t; �9�


i.e. at least the quantity of used products neededto satisfy the total demand arrives just beforethe remanufacturing process starts, then theabove restriction i P Jt can be dropped and theoriginal Wagner/Whitin algorithm solves thatmodel with large quantity of low cost usedproducts.(ii) It can be seen that the values of the holdingcost parameters are not in¯uencing the deci-sions but the di�erence of the values.(iii) For the model (6) satisfying (9) and havingconstant di�erence of the holding cost parame-ters H � Ht ÿ ht; r � rt; t � 1; 2; . . . ; T the sta-bility results from [3,12] can be applied. Due tothese studies the stability region of an optimalsolution with n setups, i.e. the set SRr;H 0 of costparameters r and H 0 for which this solution re-mains optimal, is provided by the following in-equality: Iÿ6 r

H 6 I� where the two bounds arede®ned by the following rules: Let n be the num-ber of setups in the optimal solution and letD(n) the minimal total inventory stock

PTt�1 It

for the case of exactly n setups, i.e.PTt�1 sign xtf g � n. Furthermore let D(0)�+1

and D(T + 1)� 0. This function is proved to beconvex and monotonously decreasing in n (cf.[3]). Then Iÿ � D�n� ÿ D�n� 1� and I� �D�nÿ 1� ÿ D�n� provide the boundaries of thestability region which shows now how the setupcost and the holding cost di�erence mightchange for a solution to remain optimal. Ifthe di�erence I� ÿ Iÿ � D�nÿ 1� ÿ 2 � D�n��D�n� 1� is regarded as a measure of stability,then the optimal solutions with a few setups ap-pearing if the setup cost/inventory cost ratio ishigh, are generally more stable than other solu-tions. An optimal solution with n� 1 is there-fore the most stable, or most robust solution,since then I� ÿ Iÿ � �1 holds.

Example. Let r � 5; H 0 � H ÿ h � 3ÿ 1 � 2;D � �3; 2; 1�. Then the inequalities Iÿ � 16 r

H 63 � I� cover the stability region SR for the opti-mal solution x � �3; 3; 0�, I � �0; 1; 0�, with n� 2,D(2)� 1, D(1)� 4, D(3)� 0 (see Fig. 6). Let r re-main unchanged, then H 0 might change within5=36H 06 5, if additionally also H is ®xed, then

the inventory holding cost h might change withinthe region of 06 h6 4=3, if instead of H the pa-rameter h remains ®xed, then H has the followingstability region: 2=36H 6 6.

The above-mentioned stability measure isI� ÿ Iÿ � 2 for the example.

5. The model of alternate application of manufac-

turing and remanufacturing

5.1. Modelling the case of alternate application ofmanufacturing and remanufacturing

Now the situation is regarded, in which theproblem of the previous section is extended by thealternate option to apply the manufacturing pro-cess instead of the remanufacturing activity.Roughly this case can be illustrated by Fig. 7.

The demand can be satis®ed by using ®nalproducts which have been either remanufacturedor newly manufactured (or bought). The inputs st

express the setup cost of manufacturing newproducts, and the variables zt denote theirquantity. Now, the assumption from [16], that atevery period only one of the options, manufac-turing or remanufacturing, is allowed, will bedropped. For the sake of simplicity, however, theinventory cost of remanufactured and new itemsis assumed to be equal. Then the following modelarises:

y0 � I0 � 0; yt � ytÿ1 ÿ xt � dt;

It � Itÿ1 � xt � zt ÿ Dt; It; xt; yt P 0;

t � 1; 2; . . . ; T ;

Fig. 6. The stability region SR5 ;2 for the given problem.


XT

t�1

rt � signxt� � st � sign zt � ht � yt � Ht � It� ! min :

�10�In that model, in the inventory of ®nal products

no distinction is made between remanufacturedand newly produced items is not distinguished, i.e.they are considered to have the same value for thecustomer. Let again the inventory holding cost fornew and remanufactured products be higher thanthat for used products.

The model (10) can be also treated as a modelof the alternate use of material from outside, or ofusing certain own resources in the productionprocess with setup cost st. This interpretation,however, will not play a major role here.

Since yt � It � St �Pt

j�1 zj the objective func-tion of the model (10) can be expressed as

XT

t�1

rt � signxt

� st � sign zt � ht �

Xt

j�1

zj � H 0t It

!� C

�XT

t�1

rt � signxt

ÿ � st � sign zt � htT � zt � H 0t It

�� C;

where as before C �XT

t�1

ht � St;

H 0t � Ht ÿ ht, and where the total inventory costfor one item of the used product over the periodst; t � 1; . . . ; T is denoted by htT �

PTj�t hj.

Hence, the objective function now showsclearly, that every manufacturing decision bringsadditional inventory cost for used products. Thereare probably no simple algorithms to solve thatgeneral model, but the practically important casepresented in the next section brings us back to theWagner/Whitin algorithm.

5.2. The case of large quantity of low cost usedproducts

Let as in Section 4 the quantity of used prod-ucts at the beginning of the decision period besu�ciently large, i.e. d � d1 P D1T and ht6Ht;t � 1; 2; . . . ; T . Then the model becomes muchmore simple:

I0 � 0; It � Itÿ1 � xt � zt ÿ Dt;

It; xt; zt P 0; t � 1; 2; . . . ; T ;�11�

XT

t�1

rt � sign xt

ÿ � st � sign zt � htT � zt � H 0t � It

�� C

! min :

Although the used products balance from themodel (10) has been dropped, the feasible (opti-mal) solutions xt; Itf g generated by the model (11)and extended by yt � St ÿ It �

Ptj�1 zj are feasible

(optimal) for the original model:

Lemma 2. The extended optimal solution of themodel (11) is also optimal for the model (10).

Proof. For proving the optimality only the feasi-bility of the newly generated solution is to beshown. But

yt � St ÿ It �Xt

j�1

zj

�Xt

j�1

dj

ÿ ÿ Dj

�ÿXt

j�1

xj

ÿ � zj ÿ Dj

��Xt

j�1

zj

�Xt

j�1

dj

ÿ ÿ xj

�P d ÿ

Xt

j�1

xj PXT

j�1

Dj ÿXt

j�1

xj P 0;

Fig. 7. Situation of the alternate application of manufacturing and remanufacturing.


since obviously the quantity of remanufacturedgoods will not exceed the total demand. Thesevariables ful®l also the inventory balance, since

yt � St ÿ It �Xt

j�1

zj

� Stÿ1 � dt ÿ Itÿ1 ÿ xt �Xtÿ1

j�1

zj

� ytÿ1 ÿ xt � dt: �

Hence the model describes correctly the prob-lem of alternate application of manufacturing andremanufacturing. This formulation of the model(10) suggests that some modi®cation of the zero-inventory-property holds again. The objectivefunction is obviously concave and the set of fea-sible solutions is a convex polyhedral set. Thisproperty will be satis®ed, if the extreme points ofthe polyhedral set, or the basis solutions of themodel, satisfy such a property.

Lemma 3. The basis solutions of the model (11) aregiven by

xt � zt � 0; t � 1; 2; . . . ; T ; and

Itÿ1 � zt � xt� � � 0; t � 1; 2; . . . ; T : �12�

Proof. (i) Let a basis solution of the model begiven, i.e. a solution that cannot be expressed asconvex combination of two other solutions. Thenit ful®ls the ®rst property, since in the oppositecase of two positive variables xt; zt, it is the com-bination of the two other feasible solutionsxÿ;�t � xt � D; zÿ;�t � zt � D; xÿ;�i � xi; zÿ;�i � zi;i 6� t, with D � min xt; ztf g. For a basic solutionalso the second property holds, since both vari-ables can be replaced by their sum and then thewell-known structure of the restrictions of theWagner/Whitin model appears.

(ii) Let a feasible solution satisfying (12) begiven. Then it is a basic solution, since any repre-sentation by a convex combination requires neg-ative values of It and xt � zt� �. �

It can be seen that for optimal solutions either amanufacturing or a remanufacturing activity willoccur only if the ®nal product inventory in theprevious period is empty.

Next some condition will be provided when theapplication of the manufacturing process can beexcluded:

Lemma 4. Let st � Dt � htT P rt, t � 1; 2; . . . ; T , befulfilled. Then there is an optimal solution withzt � 0; t � 1; 2; . . . ; T .

Proof. It can be easily seen that if zt > 0 thentransformation xt :� zt; zt :� 0 reduces the cost byst ÿ xt � htT ÿ rt.

Remark. If ± in the case of a su�cient quantity ofavailable used products ± the setup cost for themanufacturing process is higher than that for theremanufacturing process, no production activitywill be carried out. Every time, when new productsare made, the additional holding cost for usedproducts have to be compensated as well. There-fore, to have the manufacturing process to beswitched on at least one of the two conditions mustbe ful®lled: the manufacturing setup cost is lowerthan the remanufacturing setup cost or the quan-tity of used goods is not large enough.

5.3. The Wagner/Whitin algorithm for the largequantity of low cost used products model

The properties presented so far for the case oflarge quantity of low cost used products suggest arather simple algorithm of Wagner/Whitin type:

Let fit � min ri�1; si�1 � Di�1;t � hi�1;Tf g theminimal cost for deciding whether to remanufac-ture used products to satisfy the demand for theperiods i� 1; i� 2; . . . ; t or to produce new items.This cost can be incorporated into the Wagner/Whitin algorithm instead of the setup cost:

f0 � 0;

ft � min06 i<t

citf � fit � fig � ci�t�;t � fi�t�;t � fi�t�;

cit �Xtÿ1

j�i�1

H 0j � Dj�1;t;

i � 0; 1; . . . ; t ÿ 1; t � 1; 2; . . . �13�

Example. Let d� 11, r� 50, s� 40, H� 11, h� 1,H 0 � 10, D � �3; 4; 3; 1�. Then S � �8; 4; 1; 0�,


cost� 13, and the values fit, cit, ft are given below(see Table 3). The generated solution is presentedin detail in Table 4.

The solution consists in remanufacturing sevenunits according to the demand of the ®rst twoperiods, while the demand of the other periods issatis®ed by newly made products. The total costcan be determined directly, or by adding the termC to f4.

It is worth mentioning that the optimal solutionswitches to the manufacturing option only due tothe distinct cost parameters. The probability toswitch to this option increases with increasing t,since at the same time the used products inventorycost decreases.

5.4. Stability results

Let now the conditions

H � Ht > h � ht � 1 and rt � r > s � st �14�

be ful®lled. The monotonicity of the regenerationpoints used to prove some stability properties in[12] probably will not hold for that model. Thestability region SR � r; s;Hf g of the cost parame-ters for which a found optimal solution remainsvalid can be hardly determined explicitly.

In order to obtain more practical results lettherefore now one of the well-known heuristics,the Silver/Meal Heuristic, be applied to the model(11). This methods works in the following way:Starting from a ®rst period the demand of the nextperiods is added to the order (or production)quantity as long as the following ``Silver/Meal''-optimality criterion holds. This rule can be easilyadopted to the model (11):

x1 :� D1; x1 :� x1 � Dt; if t 2 C;

C � t�

> 1 :Ct

t6 Ctÿ1

t ÿ 1

�;

where Ct � f1t � HPt

i�1 Di�iÿ 1�.It should be noted that here this heuristic

applies to a model with a ®xed decision horizonT.

In Table 5, the application of the heuristic tothe last example is illustrated. This time the solu-tion generated by the heuristic, the Silver/Mealsolution, coincides with the optimal solution. Be-low for every period also the range of the setupcost parameters is given which is derived from theinequalities of the Silver/Meal Heuristic as well as

Table 3

Wagner/Whitin algorithm for the case of large quantity of low cost used products

t�i 1 2 3 4

fit 0 50 < 40 + 12 50 50 50

1 50 < 40 + 12 50 50

2 50 > 40 + 6 50 > 40 + 8

3 50 > 40 + 1

cit 0 0 40 100 130

1 0 30 50

2 0 10

3 0

ft 50 90 130 148

i(t) 0 0 1 2

Table 4

The solution of the example

T 1 2 3 4 Cost

xt 7 0 0 0 50

zt 0 0 4 0 40

It 4 0 1 0 55

yt 4 4 4 4 16

Total 161


from the inequalities of the production and re-manufacturing preference.

The �r; s;H 0�-stability region of this Silver/Mealsolution is given by 86 sÿ r < 28; s6H 0 ÿ 8;4H 06 r < 8H 0. If H 0 � 10 is ®xed, the (r, s)-sta-

bility region is 86 sÿ r < 28; s6 2; 406 r < 80(cf. Fig. 8).

6. Conclusion

In this paper the Wagner/Whitin model as oneof the basic models of production planning andinventory control has been studied in a reversefashion. By considering the planning and controlproblems from this position, attention is paid tothe increasing importance of recycling strategies inproduction and marketing systems. So far hereonly the simplest models, which preserve the fa-mous zero-inventory-property and allow the ap-plication of understandable modi®cations of theWagner/Whitin algorithm, have been considered.It appears that for the reverse problems similaralgorithms and planning horizon properties can bederived, and it is still even possible to solve e�-

ciently several combinations of reverse and origi-nal models. The restrictions in the sections show,however, that the complexity of the models solv-able by e�cient algorithms is bounded. A lot ofresearch can be done to expand the scope of thesemodels and algorithms. One of the directions ofbringing the models nearer to practice is to includevariable manufacturing and remanufacturing costparameters into the models. Another point is thatthe models do not estimate properly the costarising after the decision horizon T. The case of alarge quantity of low cost used products studied inthis paper is probably practically important. If,however, the quantity of used products does notmatch the demand of remanufactured goods, themethods proposed here, fail. Therefore the designof appropriate algorithms seems to be anotherimportant research direction.

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remanufacturing planning for the reverse wagner/whitin models

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