Lot-sizing for inventory systems with product recovery

Transcription

1 Lot-sizing fo inventoy systems with oduct ecovey Ruud Teunte August 29, 2003 Econometic Institute Reot EI Abstact We study inventoy systems with oduct ecovey. Recoveed items ae as-good-as-new and satisfy the same demands as new items. The demand ate and etun faction ae deteministic. The elevant costs ae those fo odeing ecovey lots, fo odeing oduction lots, fo holding ecoveable items in stock, and fo holding new/ecoveed items in stock. We deive simle fomulae that detemine the otimal lot-sizes fo the oduction/ocuement of new items and fo the ecovey of etuned items. These fomulae ae valid fo finite and infinite oduction ates as well as finite and infinite ecovey ates, and theefoe moe geneal than those in the liteatue. Moeove, the method of deivation is easy and insightful. Key wods: Poduct etuns, ecovey, lot sizing, EOQ/EPQ. Easmus Univesity Rottedam, Econometic Institute, PO Box 738, 3000 DR Rottedam, The Nethelands.

2 Intoduction Poduct ecovey (eai, efubishing, emanufactuing) is eceiving inceasing attention. In the ast, engagement in ecovey activities was often diven by legislation o by the associated envionmentally fiendly image. But nowadays, the main eason fo comanies to become involved with oduct ecovey is economical. Being active in oduct ecovey educes the need fo vigin mateials and thus leads to educed costs. Ou attention is focused on Oiginal Equiment Manufactues (OEM s) that ae involved with oduct ecovey. These diffe fom secialized ecovey comanies in that they also oduce/ocue new items. Moeove, we assume that ecoveed items ae as-good-as-new and sold on the same maket as new items. See Figue. INSERT FIGURE ABOUT HERE We addess the oblem of detemining otimal lot-sizes fo oduction/ocuement and ecovey. This oblem was fist studied by Schady (967). He analyzes the oblem in the taditional Economic Ode Quantity (EOQ) setting: deteministic and continuous demand and etun, infinite oduction and ecovey ates. The objective is to minimize the total cost e time unit fo lacing odes and fo holding inventoy (diffeent holding cost ates fo ecoveable items and seviceable items). He consides olicies that altenate oduction lot with a fixed numbe R of ecovey lots ((, R) olicies fo shot), and deives a ai of simle EOQ fomulae. Mabini, Pintelon & Geldes (992) discuss an extension of Schady s model to a multi-item case. Teunte (200) genealizes the EOQ fomulae fo (, R) olicies fom Schady (967) by including a disosal otion fo non-seviceable items, and by using diffeent holding cost ates fo oduced and ecoveed seviceable items. Futhemoe, he also deives EOQ fomulae fo olicies that altenate a fixed numbe P of oduction lots and ecovey lot ((P, ) olicies fo shot). Richte (996a,996b) also includes a disosal otion (though a sub-otimal constant disosal ate is equied) and studies both (, R) and (P, ) olicies. Howeve, his model diffes fom that of Schady (967) and Teunte (200) in the existence of a collection oint, whee used items ae collected and fom thee etuned in batches. Richte deives a fomula fo the total aveage cost, but simle fomulae fo the otimal lot sizes ae not obtained. Nahmias & Rivea (979) study an EPQ vaiant of Schady s model with a finite ecovey ate. The oduction ate is still infinite. They assume that the ecovey ate is lage than the demand ate, and deive lot-sizing fomulae fo (, R) olicies. Koh, Hwang, Sohn & Ko (2002) also assume that the oduction ate is infinite and that the ecovey ate is finite. Thei study is moe geneal that that of Nahmias & Rivea (979), since they allow the ecovey ate to be both smalle and lage than the demand ate, and they conside both (P, ) and (, R) olicies. Fo all 4 combinations, they deive a closed-fom exession fo the aveage total cost which can be used to detemine the otimal lot-sizes numeically. We emak that the (, R) olicies oosed by Koh et al. (2002) diffe fom those oosed by Nahmias & Rivea (979) (which ae genealizations of Schady s olicies) in the timing of the ecovey lots. As is exlained in the aendix, the (, R) olicies in Nahmias & Rivea (979) ae bette if the holding cost ate fo seviceable items is lage than that fo ecoveable items, which is usually the case since ecovey adds value to an item. 2

3 To summaize the above mentioned findings, lot-sizing fomulae have been deived fo both (P, ) and (, R) olicies, fo infinite and finite ecovey ates, but only fo infinite oduction ates. In this ae, we deive moe geneal lot-sizing fomulae fo both olicies. (The (, R) olicies ae genealizations fom those in Nahmias & Rivea (979).) These fomulae ae also valid fo finite oduction ates. Futhemoe, the fomulae ae deived in an insightful gahical way, avoiding the tedious calculations in evious wok. The emainde of this ae is oganized as follows. The model and the assumtions ae descibed, and notations ae intoduced in Section 2. Lot-sizing fomulae fo (P, ) and (, R) olicies ae deived in Sections 3 and 4, esectively. Thei use is illustated in Section 5. We end with conclusions and discuss limitations in Section 6. 2 Model and notation The lanning hoizon is infinite and time is continuous. Demand and etun ae continuous and deteministic. The demand ate is d, d > 0. The etun ate is fd, whee 0 < f < is the etun faction. The oduction ate and ecovey ate can be finite as well as infinite (, ). It is assumed that the oduction ate and the ecovey ate ae always lage than the demand ate. Realizing that comanies often oduce/ecove many diffeent items, this seems to be moe ealistic than to assume that the finite oduction/ecovey ate fo a secific item is smalle than its demand ate. We futhe assume that all etuned items ae ecoveed. Results by Teunte (200) and Teunte & Vlachos (2002) show that including a disosal otion fo ecoveable items seldom leads to a consideable cost eduction. Relevant costs ae: the oduction odeing cost K (e lot), the ecovey odeing cost K (e lot), the holding cost fo ecoveable items h (e item e time unit), and the holding cost fo seviceable items h s (e item e time unit). The objective is to find the lot-sizing olicy that minimizes the aveage total cost e time unit. We do not conside all ossible olicies, but estict ou attention to two classes with a simle stuctue. The (, R) olicies altenate oduction lot with a fixed numbe R of ecovey lots (in a cycle ). The (P, ) olicies altenate a fixed numbe P oduction lots with ecovey lot. Both tyes of olicies use fixed lot-sizes Q fo oduction and Q fo ecovey. These two classes of olicies wee also analyzed by evious authos (see Section ). Neithe we no them claim that the otimal olicy is always contained in these classes. The otimal olicy might vay the numbe of lots in a cycle and the lot-sizes. Howeve, such a comlex olicy will be difficult to aly in actice. Futhemoe, as agued by Teunte (200), thee is always a nea-otimal (, R) o (P, ) olicy. So we estict ou attention to olicies in the two classes (, R) o (P, ). In Section 3, (, R) olicies ae consideed. An exession fo the total cost T C (,R) (Q, Q ) is deived. Ignoing the disceteness of R, simle fomulae ae deived fo the otimal lot sizes Q (,R) fo ecovey. Futhemoe, a modified fomulae Q (,R) fo oduction and Q (,R) fo oduction is oosed, which ensues that R is discete. In Section 4, (P, ) olicies ae consideed. An exession fo the total cost T C (Q, Q ) is deived. Ignoing the disceteness of P, simle fomulae ae deived fo the otimal lot sizes Q fo oduction and Q fo ecovey. Futhemoe, a modified fomulae Q fo ecovey is oosed, which ensues that P is discete. The notations ae listed in Table. 3

4 INSERT TABLE ABOUT HERE 3 Policy (, R) Figue 2 shows the stocks of ecoveable and seviceable items associated with the (, R) olicy. INSERT FIGURE 2 ABOUT HERE The dotted line is a helful fo detemining the aveage stock of ecoveables. It cosses all (solid) ecoveable stock line-ats exactly in the middle. It is easy to see that the aveage stock of ecoveables duing a dotted eiod is equal to the aveage height of the dotted line (equal aeas of tiangles above and below the dotted line), which is clealy half of the maximum stock of ecoveables. Since the same holds outside the dotted eiods, it follows that the aveage stock of ecoveables (duing an entie oduction/ecovey cycle ) is half of the maximum stock of ecoveables. Figue 2 also shows that the maximum stock of ecoveables is equal to the numbe of etuns duing a eiod with ( d/)q +Q demands. So, the maximum stock of ecoveables is f(( d/)q +Q ), and hence the aveage stock of ecoveables is 2 f(( d/)q + Q ). Theefoe, the aveage cost e time unit fo holding ecoveable items is C (,R) := h 2 f(( d/)q + Q ). Simila to the taditional EOQ analysis, it can easily be shown that the aveage costs e time unit fo odeing and fo holding seviceable items is duing oduction eiods and C (,R) 2 := K d Q + h s 2 ( d/)q C (,R) 3 := K d Q + h s 2 ( d/)q duing ecovey eiods. Since the etun faction is f, we get the following cost exession fo the total cost e time unit: T C (,R) (Q, Q ) = C (,R) + ( f)c (,R) 2 + fc (,R) 3 = K d( f) Q + K df Q + h s 2 (( f)( d/)q + f( d/)q ) + h 2 f (( d/)q + Q ). () Note that the total cost only deends on the lot-sizes. This is because R is fully detemined by the lot-sizes via the elation RQ R ( f) = Q f, (2) since the oduction faction is ( f) and the ecovey faction is f. Since R has to be discete, this elation esticts the choice of Q and Q. We will ignoe this estiction fo deiving the lot-sizing 4

5 fomulae. At the end of this section, howeve, we discuss how the lot sizes esulting fom the fomulae can easily be modified so that R is discete. Diffeentiating the above cost exession with esect to Q and Q and setting the deivatives to zeo gives and K d( f) Q 2 K df Q 2 esectively. So, the otimal lot-sizes ae Q (,R) = + h s 2 ( f)( d/) + h 2 f = 0 + h s 2 f( d/) + h 2 f( d/)q = 0, 2K d( f) h s ( f)( d/) + h f and Q (,R) = 2K d (h s + h )( d/). (3) We emak that the lot-size fomulae deived by Nahmias & Rivea (979) and Schady (967) ae secial cases of the above fomulae with = = and =, esectively. Using (2), it follows that the value fo R coesonding with Q (,R) and Q (,R) is R (,R) := Q (,R) f Q (,R) ( f). (4) As discussed above, this value geneally is not discete and theefoe Q (,R) imlemented. Howeve, using (2), it is easy to modify Q (,R) and Q (,R) can not be so that the coesonding and/o Q (,R) only, since the numbe of oduction lots is smalle than R is discete. We oose to modify Q (,R) o equal to the numbe of ecovey lots fo (, R) olicies. We define the modified oduction lot size as whee Q (,R) is the ositive intege neaest to R (,R). 4 Policy (P, ) := R (,R) Q (,R) ( f), (5) f { [ R (,R) := max, R (,R)]} (6) Figue 3 shows the stocks of ecoveable and seviceable items associated with the (P, ) olicy. INSERT FIGURE 3 ABOUT HERE Using that figue, it is easy to see that the aveage stock of ecoveables is 2 ( fd/)q. Simila to the analysis of the evious section, it follows that the total cost e time unit is T C (Q, Q ) = K d( f) + K df +h s Q Q 2 (( f)( d/)q + f( d/)q )+h 2 ( fd/)q. (7) 5

6 Note, as in the evious section, that the total cost only deends on the lot-sizes. This is because P is fully detemined by the lot-sizes via the elation Q R ( f) = P Q f. (8) Since P has to be discete, this elation esticts the choice of Q and Q. As in the evious section, we will ignoe this estiction fo deiving the lot-sizing fomulae. At the end of this section, howeve, we discuss how the lot sizes esulting fom the fomulae can easily be modified so that P is discete. Diffeentiating the cost exession with esect to Q and Q and setting the deivatives to zeo gives and K df Q 2 K d( f) Q 2 esectively. So, the otimal lot-sizes ae Q 2K d( f) = h s ( f)( d/) + h s ( f)( d/) = h s 2 f( d/)q + h ( fd/) = 0, 2 and Q = 2K df h s f( d/) + h ( fd/). (9) We emak that the lot-size fomulae deived by Koh et al. (2002) (fo the (P, ) olicy if > d, > d) ae secial cases of the above fomulae with =. Using (8), it follows that the value fo P coesonding with Q and Q P := Q ( f) Q. (0) f As discussed above, this value geneally is not discete and theefoe Q imlemented. Howeve, using (8), it is easy to modify Q is and Q can not be so that the coesonding and/o Q only, since the numbe of ecovey lots is smalle than o P is discete. We oose to modify Q equal to the numbe of oduction lots fo (P, ) olicies. We define the modified ecovey lot size as whee Q := P Q f f, () { [ P := max, P ]} (2) is the ositive intege neaest to P. 5 Examle In this section, we illustate the use of the deived fomulae fo a secific examle. aametes ae given in Table 2. The model 6

7 INSERT TABLE 2 ABOUT HERE Fist, we conside (, R) olicies. Using (3) and (4) we get: Q (,R) = 50.0, Q (,R) = 35.4, and R (,R) = 5.7. The coesonding total cost, detemined using (), is Howeve, this solution can not be imlemented since R (,R) is not intege. Following (5) and (6), we theefoe ound R (,R) to R (,R) (,R) = 6 and modify the oduction lot size to Q = The coesonding total cost is Next, we conside (P, ) olicies. Using (9) and (0) we get: Q = 70.7, Q = 34.3, and P = 0.. Howeve, this solution can not be imlemented since P is not intege. Following () and (2), we theefoe modify P to the neaest ositive intege P = and the ecovey lot size to Q = 8.6. The coesonding total cost is Since the modified otimal (, R) olicy has much lowe cost, it is efeable. This is easily exlained, since (i) f = 0.8 and hence 4 items ae ecoveed fo evey item that is oduced, and (ii) the set-u cost fo oduction is fou times highe than the set-u cost fo ecovey. Indeed, it was obvious befoehand that (, R) olicies ae efeable fo this case, but we also consideed (P, ) olicies fo comleteness. We end this examle with an small sensitivity study. In actical situations, thee may be uncetainty about some of the model aametes, e.g. about the demand ate. Sensitivity analysis is a useful tool in such situations. Since we deived simle fomulae fo all aametes of the (modified) lot sizing olicies, a sensitivity analysis is easy to efom (in any seadsheet softwae ackage). The effects of vaiations in the demand ate on the modified otimal (, R) olicy ae esented gahically in Figue 4. INSERT FIGURE 4 ABOUT HERE The main conclusion fom Figue 4 is that, in geneal, both lot sizes as well as the total cost ae inceasing in the demand ate. The only excetion is the do in the modified oduction lot size when the demand ate inceases fom 00 to 200. This is caused by the change in R (,R) fom 6 (fo demand ates 00 to 00) to 5 (fo demand ates 200 to 500). Indeed, the unmodified oduction lot size Q (,R) is stictly inceasing in the demand ate. 6 Conclusion and limitation We deived squae-oot fomulae that detemine the otimal oduction and ecovey lot-sizes fo two classes of olicies: (, R) and (P, ). Moeove, we oosed simle modifications of esulting lot sizes that ensue disceteness of R o P. These esults ae quite geneal, since they ae valid fo finite and infinite oduction ates as well as finite and infinite ecovey ates. Futhemoe, they ae easy to aly in actice due to thei simle fom and due to the simle stuctue of the olicies. A limitation of the consideed olicies is that they seem too igid fo cases with high vaiations in the demand ate and in the etun faction. In such cases, it is bette to use a olicy that allows vaiations in the numbe of oduction o ecovey lots e cycle. One easonable olicy is as follows: if the stock dos to 0, lace a ecovey ode of size Q if enough ecoveable items ae available, and lace a oduction ode of size Q othewise. Fo such a olicy, the fomulae that we esented can still be used to as aoximations fo the otimal lot-sizes. See also Teunte (2002). 7

8 Acknowledgements: The eseach of D. Ruud H. Teunte has been made ossible by a fellowshi of the Royal Nethelands Academy of Ats and Sciences. The eseach esented in this ae is at of the eseach on e-use in the context of the EU sonsoed TMR oject REVesed LOGistics (ERB 406 PL ) in which take at the Otto-von-Gueicke Univesitaet Magdebug (D), the Easmus Univesity Rottedam (NL), the Eindhoven Univesity of Technology (NL), INSEAD (F), the Aistoteles Univesity of Thessaloniki (GR), and the Univesity of Piaeus (GR). A Sub-otimality of (, R) olicies in Koh et al. (2002) In this aendix, we comae the (, R) lot sizing olicies oosed by Koh et al. (2002) to the (, R) lot sizing olicies oosed by Nahmias & Rivea (979) fo systems with an infinite oduction ate and a finite ecovey ate. Fo ease of notation, these will be efeed to as K-olicies and N-olicies, esectively. We futhe use the notations in Table. We assume that the holding cost ate fo seviceable items is lage than that fo ecoveable items. This assumtion usually holds in actice, since ecovey adds value to a etuned item. We show that unde the assumtion, the total cost fo a K-olicy is lage than that fo an N-olicy with the same numbe of ecovey lots e oduction lot R, and with the same lot sizes Q and Q. A K-olicy stats a ecovey lot of size Q as soon as the stock of ecoveable items eaches level ( /)Q. The ecovey lot is comlete afte Q / time units, at which moment the stock of ecoveable items do to ( /)Q (Q /)( ) = 0. A oduction lot of size Q m is stated as soon as the stock of seviceable items dos to 0. Since thee is an infinite oduction ate, oduction lots ae eady immediately. An N-olicy is a secial case, fo an infinite oduction ate, of the (, R) lot sizing olicies studied in this ae (see Section 3). It diffes fom a K-olicy in only one asect: ecovey lots ae ostoned until the stock of seviceable items dos to 0, i.e. until they ae needed. Figue 5 comaes, fo a secific examle, the stocks of ecoveable and seviceable items fo the K-olicy and the N-olicy with Q = 20, Q = 40, and R = 3. INSERT FIGURE 5 ABOUT HERE As Figue 5 clealy shows, both olicies have the same cycle time. Let us define a cycle as the time eiod between two successive times at which a oduction lot is odeed. Figue 5 shows that just befoe a cycle stats, both olicies have a zeo stock of seviceable items and the same stock of ecoveable items. Since both olicies face the same demand ate and etun faction, this imlies that both olicies have the same total (ecoveable and seviceable) stock at any time duing a cycle, and hence the same aveage total stock. But, contay to a K-olicy, an N-olicy ostones all ecovey lots until the stock of seviceable items dos to 0. Clealy, this educes the aveage stock of seviceable items. And since the aveage total stock is unchanged, it inceases the aveage stock of ecoveable items by the same amount. This esults in a cost eduction if the holding cost ate fo seviceable items is lage than that fo ecoveable items, as was assumed. 8

9 Refeences Koh, S.-G., Hwang, H., Sohn, K.-I., & Ko, C.-S. (2002). An otimal odeing and ecovey olicy fo eusable items. Comutes and Industial Engineeing, 43, Mabini, M.C., Pintelon, L.M., & Geldes, L.F. (998). EOQ tye fomulations fo contolling eaiable inventoies. Intenational Jounal of Poduction Economics, 54, Nahmias, N., & Rivea, H. (979). A deteministic model fo a eaiable item inventoy system with a finite eai ate. Intenational Jounal of Poduction Reseach, 7 (3), Richte, K. (996a). The EOQ and waste disosal model with vaiable setu numbes. Euoean Jounal of Oeational Reseach, 95, Richte, K. (996b). The extended EOQ eai and waste disosal model. Intenational Jounal of Poduction Economics, 45 (-3), Schady, D.A. (967). A deteministic inventoy model fo eaiable items. Naval Reseach Logistics Quately, 4, Teunte, R.H. (200). Economic odeing quantities fo ecoveable item inventoy systems. Naval Reseach Logistics, 48, Teunte, R.H. (2002). Economic Odeing Quantities fo stochastic discounted cost inventoy systems with emanufactuing. Intenational Jounal of Logistics, 5 (2), Teunte, R.H., & Vlachos, D. (2002). On the necessity of a disosal otion fo etuned oducts that can be emanufactued. Intenational Jounal of Poduction economics, 75,

10 d demand ate f etun faction (etun ate f d) oduction ate ecovey ate K odeing (setu) cost e oduction lot K odeing (setu) cost e ecovey lot h holding cost e ecoveable item e time unit h s holding cost e seviceable item e time unit Q oduction lot-size Q ecovey lot-size T C (,R) (Q, Q ) total cost fo (, R) olicy with lot sizes Q and Q otimal oduction lot-size fo (, R) olicy Q (,R) Q (,R) R (,R) Q (,R) R (,R) T C (Q, Q ) Q Q P Q P otimal ecovey lot-size fo (, R) olicy value fo R coesonding with Q (,R) and Q (,R) modified otimal oduction lot-size fo (, R) olicy Q (,R) discete value fo R coesonding with total cost fo (P, ) olicy with lot sizes Q and Q otimal oduction lot-size fo (P, ) olicy otimal ecovey lot-size fo (P, ) olicy value fo P coesonding with Q and Q modified otimal ecovey lot-size fo (P, ) olicy discete value fo P coesonding with Q and and Q (,R) Q Table : Notations. 0

11 Model aamete d f K K h h s Value Table 2: Model aamete settings fo the examle.

12 oduction stock of ecoveables ecovey stock of seviceables s s s s customes Figue : Inventoy system with oduct ecovey. 2

13 sevicable stock ( d/)q d d d d ( d/)q 0 0 time ecoveable stock 0 time Figue 2: Seviceable and ecoveable stock coesonding to the (, R) olicy (R = 5 fo this examle) duing two cycles. The dotted line is a hel-line fo detemining the aveage stock of ecoveables (see text). 3

14 sevicable stock ( d/)q ( d/)q d d d d 0 0 time ecoveable stock ( fd) ( fd)(q /) 0 (Q /) time Figue 3: Seviceable and ecoveable stock coesonding to the (P, ) olicy (P = 3 fo this examle) duing two cycles. 4

15 Q (,R) Q (,R) Q (,R) T C (,R) ( 400, Q (,R) ) demand ate d Figue 4: Effects of vaiations (fom the aamete settings in Table 2) in d on T C (,R) (,R) ( Q, Q (,R) ). Q (,R), Q (,R), and 5

16 seviceable stock d = 0 d = 0 K-olicy N-olicy time ecoveable stock K-olicy N-olicy fd = 5 fd = time Figue 5: Comaison of stock levels between the K-olicy (Koh. et al., 2002) and the N-olicy (Nahmias & Rivea, 979), both with oduction lot size Q = 20, ecovey lot size Q = 40, and R = 3 ecovey lots e oduction lot (demand ate d = 5, etun faction f = 0.5, ecovey ate = 20, oduction ate = ). 6