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Disassembly Scheduling with Multiple Product Types
Disassembly Scheduling with Multiple Product Types Kim'. D,H. Lee'. P.Xirwchakisl. and R. ZOst3(1) 'Laboratmy for Corrputer Aided Design and Production (LICP). Institute of Produdion Research Swiss Federal Institute of T e c h n o l w at Lausanne (EPFL). Switzerland b p a r t m e n t of Industrial Engineering Hanyang Universlty. Seoul. Korea %TH ags - Center for Sustainabillty at ETH Swiss Federal Institute of TechnoQy at Zurich (ETHZ). Switzerland
Abstract We mnsider the problem of determining the disassembly schedules of used produds while satisfyng the demand of their partslcorrponents over a planning hornon. The case of rmltiple produd types with parts mmrxxlallty is mnsidered for the objective of minimizing the sum of setup, disassembly operation. and it ventory holding msts. A heuristic algorithm is proposed using the linear p r q r a m n g relaxation approach. Test results of the case study on thedata obtained from a disassembly shop show that the heuristic algorithm suggested in this paper works well for pradical p r o b l e m . Keywords: Disassembly. Produdion Planning and Scheduling. Optimization
1 INTRODUCTION Disassembly is a process in which used or e n b t l i f e products are separated into parts and mrrponents with sorting operations [ I ] . In general, disassembly is an m w a n t step in material and produd remvery processes since used or endotlife produds are usually disassem bled before recycled. remanufadured. or even disposed. Amxlg various disassembly problem. this paper focuses on disessernblyscheduling, which is the problem of d& termining the disassembly scheddes of used produds while satisfying the demand of their parts and corrponents over a planning horizon. Disassembly scheduling is one of the irrportant planning problems in disassembly system. In other words. from its solution. we Mn deternine the quantrty and timng of disassembly.
There are several previous research artides on disassembly scheduling. Gupta and Taleb [2] mnsider the b s sic case. i.e.. single produd type without parts m a C lty. and suggest an MW-like algorithm. For the case of single produd type with parts m m m a l l t y . Taleb eta/. [3] suggest another MW-Cke algorithm for the objedive of minimizing the number of products to be disassembled. and Neuendorf et al. [4] suggest a Petri-net based solw tion algorithm. Later, Taleb and Gupta [5] m i d e r the case of rmltiple produd types with parts m m a l i t y . and suggest a -phase heuristic. Recently, Lee eta/.[6] extend the three cases of the problem with the m n s i d e r s tion of various cost fadors in the objedive fundion. and suggest an integer prqrammng approach to solve the suggest another problem @mally. Also. Lee d a!. integer p r q r a m n g &el for the proMem with resource capacity mstraints. This paper m s i d e r s the case of rmltiple produd types with parts mmnonallty for the objedive of minimizing the sum of setup, disassembly operation. and inventmy hol& ing msts. A fast heuristic based on the linear p r q r a m mng relaxation is suggested in order to give near optiml solutions up to pradiml sized problem. In fad. the prob lem m s d e r e d here is simlar to the one treated by Lee et a/. However. their integer p r o g r a m i n g approach i s
m.
not adequate to solve pradical sRed p r o b l e m due to e z cessive mrnputation times and the failure to obtain feas C ble solutions for certain test problem. Moreover. even for the small sized p r o b l e m . there is rmch variation in the mrnputation times. depending on produd strudure. mst values. etc. 2 PROBLEM DESCRIPTION This sedion starts with explaining the disassembly p r d ud structure w m rmltiple produd types and parts mm m a l i t y . In the disassembly strudure. the r d item represents the produd itself to be ordered and a leaf item is an item not to be disassembled further. Also, a child item denotes an item that has at least one parent and a parent item is an item that has two or m e child i t e m . Note that in the strudure mnsidered here, there may be two or m e r m t item (rmltiple produd types) and each item may have two or m e parents (parts mrrmonality). which makes the problem m r e mmplex. An example of the disassembly produd structure m n s a ered in this paper is s b w n in Figure 1. which is obtained from Taleb and Gupta p]. The exa@e has three r d item. i.e.. i t e m 1 , 2. and 3. The number in parenthesis represents the yield of the item when its parent is d i s a sembled. e.g.. item 3 is disassembled into in one unit of item 6 and three units of item 7. Also,the d i s a s s m l y lead-time (DLT) is the time required to disassemble a certain parent item. Note that the shaded boxes represent m m item. For example. item 6 is a c m m m item
Figure 1: Disassembly product strudure: an example
since it has two parent items. i.e.. i t e m 2 and 3
Now.the problem mnsidered in this paper can be defned as follows : for a gken disassembly product structure with multiple product types and parts commonality, the prab/em is to determine the disassemblyschedule ofallparent items? while satiswing the demand of leaf items over a planniq homon with the objective of minimhing the sum o f setup, disassembly operation, and inventory holding costs. The disassembly strudure of each produd type can be obtained from the mrrespnding &assembly plan that specifies necessary disassembly operations and their S B quence. See 0’Shea et el. [8]. Lee e t a / . [9]. and SantDchi eta!.[IOIfor literature reviews on disassembly planning. It is assumed that there is rm stwtage of the r m t item. i.e.. produds can be obtained wheneverthey are ordered. F u r t h e m e . the following additional assurrptrons are made: (a) he planning horizon is dvided into disoete planning periods; (b) demand of leaf item is given and is not allowed; (d) disasdeterministic; (c) ba-ing sembly operations are always successful; (e) disassembly lead times with discrete time scales are given and deterministic; and (r) inventory holding msts are mnprted d the e n b r p e r i o d . The problem mnsidered here can be forrmlated as an iw teger p r q r a m . In the formulation. without loss of genera C lty. all Rem are numbered with integers 1, 2 .... ir .... ir .... I . Here, L denotes the number of r m t item. and ir denotes the index for the first leaf Rem. and hence the indices thd are larger than or equal to ir represent leaf i t e m . The n& tations used are summarized below.
Bi
&(r) l,
inventory holding mst of item i demand of item i in period t number of units (yield) of itemjobtained by disassembly of one unitof its parent item i
set o f parents of item i disassembly lead-time (DLT) of parent i t e m / = 1 if there is a setup for item i in period t. and 0
otherwise Xr l,r
disassembly quantity of item i in period t hventory o f Rem i at the end of period t
Now.the integer program is given below
-1
+
z&
&)
ah-XkfA - Xf.
f o r a l l i = L + l ,... i - l a n d t = l
la = /,,f-l
+
Xas MY4 Y,~{0.1}
Deflnltlon 1. For an arbitrary solution (Xa. l,r) of problem p ] . the balance (&) of item i in period t is defined as X k d,)au-Xkrn+ Xr, for i = L + 1... . ir
-
1
/,,r-~-Xkd,] ah-XkfA+da,fori=ir. ... /.
The balance defined above can be used to check the f e a sibility of the roundsolutron More speatimlly. the inventory Row of Rem r in penod t is feasible fi and only fi BR = 0. where the roundeddown sdutron is used to calcw late the balance 4r C & f 0.the m r r e s p d i n g rounddown solutron should be modmed The basic ldea of the modficatron is that the d e a s m vanables. I e , Xr and l,r. are inaeased or deaeased in such a way that the COTTB sponding inventory flow and non-negatrvlty mnstraints are sahsfied. while mnsidenng the cost changes The solutron modmcatron is d m e dlfferenUy for each of the f w r m i n a t r o n s of two dem types (parent and leaf dem) and the sgn of balance ( p i h e and n e g a h e ] In this paper. we explain the method only for the cases wrth posltrve signs. I e , & > 0 This is bemuse the a s e s wdh n e g a h e sgns can be easily dealt wdh by reversing the methods for pos ltrve cases
Case 1 (Parent rtems)
subjed to / a = /,,f
T o explain the method of the rounde&down sohtion modification. w e first define balance as fdlows.
&f=/#-
Decision variables Ylr
3 SOLUTION ALGORITHM The heuristic suggested in this paper. based on the linear p r q r a m n g (LP) relaxation approach. consists of two main steps. First, the integer p r q r a m [PI is sdved directly dter the integrallty m s t r a i n t s (4). (5). and (6) are relaxed. and then the LP solution (real values) is rounded down. Note that the relaxed problem can be solved easily us@ m r r m e r a a l LP software. Semnd. the rwndsdw tion obtained in the first step is rmdified so that all the original mnstraints of [PI are satisfied. The m s t changes are considered during the solution modification.
&r = / a - /,,r.l-
Parametem s, setup cost of item i p, disassembly operation cost o f Rem i h, da
tion that defines the inventory level of m - r o o t i t e m at the end of each period. The ordering quanhty of each root item can be obtained from the disassenbly quantrtyof the mrresponding r m t item and hence is not included in the model. Also. the inventory Row m s t r a i n t s of r m t item are not needed because it is not necessary to have suplus inventories for r m t i t e m . Constraint (3).where Mis an arbitrary large number. guarantees that a setup mst in a period i s incurred whenever there is at least one disassembly operation at that period. Finally. the other m w straints (4). ( 5 ) and (6) represent the mnditions on the deasion variables. In particular. mnstraint (6) ensures that b a d d q i n g is not allowed.
ah-&fA - da. f o r a l l i = i / ,...! a n d t = l ,
,...
T (1)
xh
...
T
(2)
for all i = 1.... ir - 1 and t = 1,... T (3) f o r a l l i = l ,... 6 - 1 a n d t = l ,... T (41
Xa2 0 and integers,
f o r a l l i = l , ... C - I a n d t = l , ... T l a 2 0 and integers, f o r a l l i = i , + l , ... l a n d t = l , ... T
(5)
In this case. the inventory flow m s t r a i n t (1) can b e satrstied b y decreasing the inventory level or the disassembly quanhty. I e , hor Xr. or by increasing the disassembly quantrtyofoneofRemr’sparents.ie.~,rI].] E & ( r ) Note that the case wRh negatwo sgn of balance. I e , &r < 0. can be dealt wrth by inaeasing 6ror Xa. OT by deaeasing 4f //.IE & ( r ) We first m s l d e r the case of deaeasing la or Xr h this case. seleded is the one that gwes the maxlrmm m s t deaease. I e , m a x { L . 0,).where
L=h, BR,andC3=pBR+x~Hu)htat& (6)
The objedive fundion denotes the sum of setup, disassembly operation. and inventory holding msts. Cow straints (1) and (2) represent the inventory flow m s e m
Here, L (0,) denotes the amount o f mst deaease when l,r (XR) is deaeased by the a m n t of balance B.and H[r) denotes the set of children of dem I If both cases desmbed above result in infeasible solubons.
@(o.
2) Calculate balance & lsing the rwndedsolution obtained in (2) of Step 1. (!%me of the current rwndedsolution may have been changed during the earlier modifimtions.) 3) r BR = 0.go to (4). Otherwise. modlry the solk tion using tw moditimtion methods described variables. earlier. and update the mrresponHere. i f the disassembly quantity of a parent of item i . & q . j ~ @(/].is moditied.and j > l a n d ,/ > 0 ,set i = j + 1 and t = t - 4. i.e.. restart the procedure from item j and period f - 4.
we mnsider the m s e of increasing Xg - g. j E Far a parent j E d(i).the amwnt 4 of inaease in disassembly quantty is r4dajl. where r.1 is the smallest integer value that is greater than or equal to m. Now. a formal d e s c r i p tion of mt inaease 4 when disassembly quantity of parent j E @(/] is increased by 4 mn be expressed as
4= Pj.4 +
+
xhH(i], h#i
hh-qh-4
5-4&,r-/d hi-(q,-4- &),
where 40)= 1. if > 0. and 0 otherwise. The first term represents the increase of disassembly operation mst of parent j . and the second term represents the inaease of hventory holding msts of child item (except for item 0 d parent j . Also. the third term represents the setup mst i r curred if &m~.tj = 0. Finally, the last term is the inventory holding mst incurred b y s u ~ u inventory s remained after balance BR is satisfied. Then, a m x l g the candidate parents of item i. we seled the one that results in the m n C m m cost increase. i.e.,
r
1” = argmn,. HI1 { 4 IW .we mn see that the inventory flow mservation of the seleded parentj’ is broken if its disassembly quantty. &-,M,-,is inaeased. and hence its solution should be modt fied. However. if the seleded parent j’ is a root item. i.e.. j’ = i,. it is not necessary to modify its solution since there are no inventory mnservation m s t r a i n k for root item. r h e detailed method i s summarized in the procedure.) Case 2 (Leafitem)
This case is the same a s that of parent item (Case 1) except that it is not necessary to m s i d e r the case of d& creasing Xa bemuse the inventory flow mnstraint (2) has the demand requirement da instead o f Xr.
Now. the Overall procedure for the LP relaxation heuristic suggested in this paper can be s u m r i z e d as follows. In the s e m d step of the procedure. the rwnd&down s d w tion is moditied from the first period to the last period. and for each period. from the leaf items to the rmt item. Procedure (LP relaxation heuristic)
Sep 1. (LPRelaxation and Solution lnifialization) 1) W e the linear p r q r a m obtained after relaxing the integralty m s t r a i n t s (4). (5). and (6)
of the integer prqram [PI and obtain the LP solution Xa and 6r. Here, Xa and Ilr may be real wlues. 2) Rwnd down the LP solution. i.e., set / I ’ = Llld. and xI’ = L x ~ .where L1 . is the largest irteger less than or equal to 0. step 2. (Solution Modification) 1) S e t t = 1 a n d i = l .
4) % t i = i-1. If i > h. go to (2). Otherwise, s e t f = t + 1. If;= b a n d t > T . stopandsave the modt fied solution. Otherwise, go to (2). 4
CASESTUDY
T o show the effediveness of our LP relaxation heuristic, a case study on two types of endd-life inkjet printers with several m m m partslmrrponents was performed and the results are reported in this sedion. The required data for the case study were obtained from a swiss d i s a s s m bly mrrpany. which eams protits from dkassembling various endof-life eledronic produds instead of disposing t h e m without disassembly. Figure 2 shows the disassemty product strudure of the two types of inkjet printers, in which the shaded boxes represent (XXTIITID~partslmrrponents. There are 29 items in total. i.e.. 9 parent item and 20 leaf item. Other a quired data for this case study are summed in Table 1. The disassembly operation mst of a parent item was mC culated by mltiplying the unit direct labor mst and the disassembly operation time for that item; the set up mst was calculated by multiplying the unit d i r e d mst and the required preparation time for that item. Finally, the invert tory holding mst of an item was set to be proportional to the mrresponding endoflife value o f that item. Now. the remaining data are initial inventory. demands of leaf item and disassembly lea&time. In this test, we generated these data r a m since we muld not obtain themfromthe mrrpany. First. the a m w n t s o f initial invert torywere generated from DU(0. 50) and DU(0. 10)for leaf and non-leaf items. sspedively Here. DU(a. b ) is the discrete uniformdistribution with range [a. b ] . Demands of each leaf itemwere set to 0 or DU(50.200)with probabilC ties 0.1 and 0.9.respedively Finally. disassembly lead times were set to 0. 1. and 2 with probabilities 0.2.0.7. and 0.1, respedivw.
In the test, 10 problem were generated randonly for three levels of the number of periods (10. 15, and 20). Performance measures used in the test are percerttage deviations from optimal solution values, i.e.. the gap b e tween a heuristic solution value and an optrmal solution value. and CPU semnds. In this test, we used C P L M 6.5.
Figure 2: Disassembly p r o d u d structure of the two types of inkjet printers
5
Table 1 : Data for two inkjet printers
CONCLUDING REMARKS
We m s i d e r e d the disassembly scheduling proMem with rmltiple produd types and parts m m m l i t y . which is the problem of determining the disassembly schedule of used products whle satisfying the demands of their i n d C vidual parts or m m p e n t s over a given planning horizon. To sdve the problem. a heuristic based on linear prtr grarming relaxation. is suggested in this paper. To show the effediveness of the heuristic, a case study on eWoP life inkjet printers was performed and the results showthat that the heuristic can give near optrmal solutions within very short mq~utationtimes. This research can be extended in several ways. First, it is necessary to m s i d e r the problem with r e w r c e capady mnstramts. Semnd. like other disassembly problem. UIF certainties. such as stochastic demands. stochastic disassembly times. and stochastic i n m r i n g rate of used pro& uds. are irqmrtant mnsiderations.
6 ACKNOWEDGEMENT This research was supported by the swiss National ScC ence Foundation (SNF) under grant rm. 200~066640.01. This support is gratefully acknowledged. Also.the authors wwld like to appreciate Mr. JuwGyu Kang at EPFL-LICP and Mr. Adrien B e s m at BIRD in Switzerland fortheir kind help to gather the necessary information on the caw Study. a m m r c i a l software package. to obtain optrmal s d k tions. However. to avoid eroessive mrnputation times. CPLEX 6.5 was terminated when CPU time reached 3600 s e m n d s . and in this case. the percentage deviationsfrom lower bounds. i.e., the gap between a heuristic solution value and a lower b w n d value, were used as the performance measure. Note that the lower bounds can be obtained diredyfrom CPLEX 6.5. Test results on the case study are s u m r i z e d in Table 2. which shows the percentage deviations from optrmal s d k tions or lower bounds [for the problem that CPLEX 6.5 muld rmt give the optrmal sdutions within 3600 semnds) and CPU semnds. The CPU seconds of the integer prtl g r a m i n g approach are also surrmarized in the table. lt can be seen from the table that the heuristic suggested in this paper can give near optimal solutions b the test problem. i.e.. 1.29%. 1.03%. and 1.33% in average for the cases with 10, 15, and 20 p e r i d s . respectively. Also. the heuristic required quite short mrrputation times. i.e.. within 0.04 semnds for all problem. However. in the case of the mteger prcgrarming approach. the CPU s e m n d s ranged from 0.45 to3603.57 semnds. Note that in the latter case. we muld mt obtain optimal solutions using the integer prcgramming approach. Table 2: Test results on the case study bhrnbnr o f n n n d r ; I0
Rhhm 7 2
3 4 6 6 7 8 g I0 Average
OW' 7 06
177 126 12l 106 I 27 o BI 0 90 141 214
7 29
CPU~ 002( f 7 6 ) 002(217) OOl(O46) 002(210) OOl(0Sa) 0 M(69 38) o 02(1746) 0 0 2 ( 3 61) 002(437) OOl(O71) 0 82)
16
CPU DWJ 0 0 3 ( 7 d f s 7 3 ) 7 39 003(28243) 1 7 8 003( 3 1 6 2 ) I 3 3 003(1MB36) 0 7 4 002(17690) 1 7 9 003(360093) 1 2 6 0 0 3 p 6 0 1 23) I06 003(360083) 138 o w ow(3mm) 1 4 6 OB9 O m ( 3 7 2 9 9 ) I 1 6 T O 3 0 0 3 ( 7 m 2 8 ) 7 33
20
CPU
DWJ
OTB 107 143 I011 106 I011 I00 109
0
OM(360367) OM(36Ml6) OM(3807M) OM(36M13) 0 M(36M 13) o M(36rn~ 7 ) 0 M(36M 19)
ou(36rnm) OM(36M27) 0 M(%-
Percantage deviation from opimal solution value or lower bwnd CPU seconds d the heunstic suggested in t h s paper and the r t q e r progamming approach (in pareflhases)
7
REFEENCES Jovane. F.. A M g . L..Armllotta.A., Eversheirm. W.. Feldmann. K.. Seliger. G. and Roth. N.. 1993, A Key Issue in Produd Life Cyde: Diszsembly. Annals of the C IRP 42: 65 1-658. Gupta. S. M. and Taleb. K. N., 1994, Scheduling Disassembly. International Journal of Produdion Research 32: 1857-1 886. Taleb. K. N.. Gupta. S. M.. and Brennan. L.. 1997. Disassembly of Complex Produd Structures with Parts and Materials Corrmonallty. Produdion plaw ning and Control 8: 255-269. Neuendorf. K.-P.. Lee. D.-H.. Kiritsis. D. and X rwchakis. P.. 2001. Disassembly Scheduling with Parts Comnonallty using Petri-Nets with T i m e starrps. Fundamenta Informatime 47: 295306. Taleb. K. N . and Gupta. S.M.. 1997. Disassembly of Mumple Product Strudures. C m p u t e r s and Industrial Engineering 32: 949-961. Lee, D.-H.. Kim, H.-J.. and Xirwchakis. P. 2002. Disassembly Scheduling: An Integer Prqramning Approach. Technical Report. Department of W chanical Engineering. swiss Federal Institute of T e c h n o l w - Lausanne [EPFL). Switzerland. Lee. D . 4 . Xirwchakis. P.. and ZOst. R.. 2002. Disassembly Scheduling with Capacity Constraints. h nals of the ClRP 51: 387-390. OShea. B., Grewal. S . S., and Kaebemick. H., 1998, State of the Art Literature Survey on Disassembly Planning. Concurrent Engineering: Research and Application 6 : 345357. Lee. D . 4 . Kang. J.-G.. and Xirwchakis, P.. 2001. Disassembly Planning and Scheduling: Review and Further Research. Proceedings of the Institution of Mechanical Engineers: Journal of Engineering Manufadure - Part 215: 695-71 0. Santochi. M.. Dini. G.. and Failli. F.. 2002. Computer Aided Disassembly Planning: State of the Art and Perspedives.Annalsof the ClRP 51 : 1-23.
Abstract We mnsider the problem of determining the disassembly schedules of used produds while satisfyng the demand of their partslcorrponents over a planning hornon. The case of rmltiple produd types with parts mmrxxlallty is mnsidered for the objective of minimizing the sum of setup, disassembly operation. and it ventory holding msts. A heuristic algorithm is proposed using the linear p r q r a m n g relaxation approach. Test results of the case study on thedata obtained from a disassembly shop show that the heuristic algorithm suggested in this paper works well for pradical p r o b l e m . Keywords: Disassembly. Produdion Planning and Scheduling. Optimization
1 INTRODUCTION Disassembly is a process in which used or e n b t l i f e products are separated into parts and mrrponents with sorting operations [ I ] . In general, disassembly is an m w a n t step in material and produd remvery processes since used or endotlife produds are usually disassem bled before recycled. remanufadured. or even disposed. Amxlg various disassembly problem. this paper focuses on disessernblyscheduling, which is the problem of d& termining the disassembly scheddes of used produds while satisfying the demand of their parts and corrponents over a planning horizon. Disassembly scheduling is one of the irrportant planning problems in disassembly system. In other words. from its solution. we Mn deternine the quantrty and timng of disassembly.
There are several previous research artides on disassembly scheduling. Gupta and Taleb [2] mnsider the b s sic case. i.e.. single produd type without parts m a C lty. and suggest an MW-like algorithm. For the case of single produd type with parts m m m a l l t y . Taleb eta/. [3] suggest another MW-Cke algorithm for the objedive of minimizing the number of products to be disassembled. and Neuendorf et al. [4] suggest a Petri-net based solw tion algorithm. Later, Taleb and Gupta [5] m i d e r the case of rmltiple produd types with parts m m a l i t y . and suggest a -phase heuristic. Recently, Lee eta/.[6] extend the three cases of the problem with the m n s i d e r s tion of various cost fadors in the objedive fundion. and suggest an integer prqrammng approach to solve the suggest another problem @mally. Also. Lee d a!. integer p r q r a m n g &el for the proMem with resource capacity mstraints. This paper m s i d e r s the case of rmltiple produd types with parts mmnonallty for the objedive of minimizing the sum of setup, disassembly operation. and inventmy hol& ing msts. A fast heuristic based on the linear p r q r a m mng relaxation is suggested in order to give near optiml solutions up to pradiml sized problem. In fad. the prob lem m s d e r e d here is simlar to the one treated by Lee et a/. However. their integer p r o g r a m i n g approach i s
m.
not adequate to solve pradical sRed p r o b l e m due to e z cessive mrnputation times and the failure to obtain feas C ble solutions for certain test problem. Moreover. even for the small sized p r o b l e m . there is rmch variation in the mrnputation times. depending on produd strudure. mst values. etc. 2 PROBLEM DESCRIPTION This sedion starts with explaining the disassembly p r d ud structure w m rmltiple produd types and parts mm m a l i t y . In the disassembly strudure. the r d item represents the produd itself to be ordered and a leaf item is an item not to be disassembled further. Also, a child item denotes an item that has at least one parent and a parent item is an item that has two or m e child i t e m . Note that in the strudure mnsidered here, there may be two or m e r m t item (rmltiple produd types) and each item may have two or m e parents (parts mrrmonality). which makes the problem m r e mmplex. An example of the disassembly produd structure m n s a ered in this paper is s b w n in Figure 1. which is obtained from Taleb and Gupta p]. The exa@e has three r d item. i.e.. i t e m 1 , 2. and 3. The number in parenthesis represents the yield of the item when its parent is d i s a sembled. e.g.. item 3 is disassembled into in one unit of item 6 and three units of item 7. Also,the d i s a s s m l y lead-time (DLT) is the time required to disassemble a certain parent item. Note that the shaded boxes represent m m item. For example. item 6 is a c m m m item
Figure 1: Disassembly product strudure: an example
since it has two parent items. i.e.. i t e m 2 and 3
Now.the problem mnsidered in this paper can be defned as follows : for a gken disassembly product structure with multiple product types and parts commonality, the prab/em is to determine the disassemblyschedule ofallparent items? while satiswing the demand of leaf items over a planniq homon with the objective of minimhing the sum o f setup, disassembly operation, and inventory holding costs. The disassembly strudure of each produd type can be obtained from the mrrespnding &assembly plan that specifies necessary disassembly operations and their S B quence. See 0’Shea et el. [8]. Lee e t a / . [9]. and SantDchi eta!.[IOIfor literature reviews on disassembly planning. It is assumed that there is rm stwtage of the r m t item. i.e.. produds can be obtained wheneverthey are ordered. F u r t h e m e . the following additional assurrptrons are made: (a) he planning horizon is dvided into disoete planning periods; (b) demand of leaf item is given and is not allowed; (d) disasdeterministic; (c) ba-ing sembly operations are always successful; (e) disassembly lead times with discrete time scales are given and deterministic; and (r) inventory holding msts are mnprted d the e n b r p e r i o d . The problem mnsidered here can be forrmlated as an iw teger p r q r a m . In the formulation. without loss of genera C lty. all Rem are numbered with integers 1, 2 .... ir .... ir .... I . Here, L denotes the number of r m t item. and ir denotes the index for the first leaf Rem. and hence the indices thd are larger than or equal to ir represent leaf i t e m . The n& tations used are summarized below.
Bi
&(r) l,
inventory holding mst of item i demand of item i in period t number of units (yield) of itemjobtained by disassembly of one unitof its parent item i
set o f parents of item i disassembly lead-time (DLT) of parent i t e m / = 1 if there is a setup for item i in period t. and 0
otherwise Xr l,r
disassembly quantity of item i in period t hventory o f Rem i at the end of period t
Now.the integer program is given below
-1
+
z&
&)
ah-XkfA - Xf.
f o r a l l i = L + l ,... i - l a n d t = l
la = /,,f-l
+
Xas MY4 Y,~{0.1}
Deflnltlon 1. For an arbitrary solution (Xa. l,r) of problem p ] . the balance (&) of item i in period t is defined as X k d,)au-Xkrn+ Xr, for i = L + 1... . ir
-
1
/,,r-~-Xkd,] ah-XkfA+da,fori=ir. ... /.
The balance defined above can be used to check the f e a sibility of the roundsolutron More speatimlly. the inventory Row of Rem r in penod t is feasible fi and only fi BR = 0. where the roundeddown sdutron is used to calcw late the balance 4r C & f 0.the m r r e s p d i n g rounddown solutron should be modmed The basic ldea of the modficatron is that the d e a s m vanables. I e , Xr and l,r. are inaeased or deaeased in such a way that the COTTB sponding inventory flow and non-negatrvlty mnstraints are sahsfied. while mnsidenng the cost changes The solutron modmcatron is d m e dlfferenUy for each of the f w r m i n a t r o n s of two dem types (parent and leaf dem) and the sgn of balance ( p i h e and n e g a h e ] In this paper. we explain the method only for the cases wrth posltrve signs. I e , & > 0 This is bemuse the a s e s wdh n e g a h e sgns can be easily dealt wdh by reversing the methods for pos ltrve cases
Case 1 (Parent rtems)
subjed to / a = /,,f
T o explain the method of the rounde&down sohtion modification. w e first define balance as fdlows.
&f=/#-
Decision variables Ylr
3 SOLUTION ALGORITHM The heuristic suggested in this paper. based on the linear p r q r a m n g (LP) relaxation approach. consists of two main steps. First, the integer p r q r a m [PI is sdved directly dter the integrallty m s t r a i n t s (4). (5). and (6) are relaxed. and then the LP solution (real values) is rounded down. Note that the relaxed problem can be solved easily us@ m r r m e r a a l LP software. Semnd. the rwndsdw tion obtained in the first step is rmdified so that all the original mnstraints of [PI are satisfied. The m s t changes are considered during the solution modification.
&r = / a - /,,r.l-
Parametem s, setup cost of item i p, disassembly operation cost o f Rem i h, da
tion that defines the inventory level of m - r o o t i t e m at the end of each period. The ordering quanhty of each root item can be obtained from the disassenbly quantrtyof the mrresponding r m t item and hence is not included in the model. Also. the inventory Row m s t r a i n t s of r m t item are not needed because it is not necessary to have suplus inventories for r m t i t e m . Constraint (3).where Mis an arbitrary large number. guarantees that a setup mst in a period i s incurred whenever there is at least one disassembly operation at that period. Finally. the other m w straints (4). ( 5 ) and (6) represent the mnditions on the deasion variables. In particular. mnstraint (6) ensures that b a d d q i n g is not allowed.
ah-&fA - da. f o r a l l i = i / ,...! a n d t = l ,
,...
T (1)
xh
...
T
(2)
for all i = 1.... ir - 1 and t = 1,... T (3) f o r a l l i = l ,... 6 - 1 a n d t = l ,... T (41
Xa2 0 and integers,
f o r a l l i = l , ... C - I a n d t = l , ... T l a 2 0 and integers, f o r a l l i = i , + l , ... l a n d t = l , ... T
(5)
In this case. the inventory flow m s t r a i n t (1) can b e satrstied b y decreasing the inventory level or the disassembly quanhty. I e , hor Xr. or by increasing the disassembly quantrtyofoneofRemr’sparents.ie.~,rI].] E & ( r ) Note that the case wRh negatwo sgn of balance. I e , &r < 0. can be dealt wrth by inaeasing 6ror Xa. OT by deaeasing 4f //.IE & ( r ) We first m s l d e r the case of deaeasing la or Xr h this case. seleded is the one that gwes the maxlrmm m s t deaease. I e , m a x { L . 0,).where
L=h, BR,andC3=pBR+x~Hu)htat& (6)
The objedive fundion denotes the sum of setup, disassembly operation. and inventory holding msts. Cow straints (1) and (2) represent the inventory flow m s e m
Here, L (0,) denotes the amount o f mst deaease when l,r (XR) is deaeased by the a m n t of balance B.and H[r) denotes the set of children of dem I If both cases desmbed above result in infeasible solubons.
@(o.
2) Calculate balance & lsing the rwndedsolution obtained in (2) of Step 1. (!%me of the current rwndedsolution may have been changed during the earlier modifimtions.) 3) r BR = 0.go to (4). Otherwise. modlry the solk tion using tw moditimtion methods described variables. earlier. and update the mrresponHere. i f the disassembly quantity of a parent of item i . & q . j ~ @(/].is moditied.and j > l a n d ,/ > 0 ,set i = j + 1 and t = t - 4. i.e.. restart the procedure from item j and period f - 4.
we mnsider the m s e of increasing Xg - g. j E Far a parent j E d(i).the amwnt 4 of inaease in disassembly quantty is r4dajl. where r.1 is the smallest integer value that is greater than or equal to m. Now. a formal d e s c r i p tion of mt inaease 4 when disassembly quantity of parent j E @(/] is increased by 4 mn be expressed as
4= Pj.4 +
+
xhH(i], h#i
hh-qh-4
5-4&,r-/d hi-(q,-4- &),
where 40)= 1. if > 0. and 0 otherwise. The first term represents the increase of disassembly operation mst of parent j . and the second term represents the inaease of hventory holding msts of child item (except for item 0 d parent j . Also. the third term represents the setup mst i r curred if &m~.tj = 0. Finally, the last term is the inventory holding mst incurred b y s u ~ u inventory s remained after balance BR is satisfied. Then, a m x l g the candidate parents of item i. we seled the one that results in the m n C m m cost increase. i.e.,
r
1” = argmn,. HI1 { 4 IW .we mn see that the inventory flow mservation of the seleded parentj’ is broken if its disassembly quantty. &-,M,-,is inaeased. and hence its solution should be modt fied. However. if the seleded parent j’ is a root item. i.e.. j’ = i,. it is not necessary to modify its solution since there are no inventory mnservation m s t r a i n k for root item. r h e detailed method i s summarized in the procedure.) Case 2 (Leafitem)
This case is the same a s that of parent item (Case 1) except that it is not necessary to m s i d e r the case of d& creasing Xa bemuse the inventory flow mnstraint (2) has the demand requirement da instead o f Xr.
Now. the Overall procedure for the LP relaxation heuristic suggested in this paper can be s u m r i z e d as follows. In the s e m d step of the procedure. the rwnd&down s d w tion is moditied from the first period to the last period. and for each period. from the leaf items to the rmt item. Procedure (LP relaxation heuristic)
Sep 1. (LPRelaxation and Solution lnifialization) 1) W e the linear p r q r a m obtained after relaxing the integralty m s t r a i n t s (4). (5). and (6)
of the integer prqram [PI and obtain the LP solution Xa and 6r. Here, Xa and Ilr may be real wlues. 2) Rwnd down the LP solution. i.e., set / I ’ = Llld. and xI’ = L x ~ .where L1 . is the largest irteger less than or equal to 0. step 2. (Solution Modification) 1) S e t t = 1 a n d i = l .
4) % t i = i-1. If i > h. go to (2). Otherwise, s e t f = t + 1. If;= b a n d t > T . stopandsave the modt fied solution. Otherwise, go to (2). 4
CASESTUDY
T o show the effediveness of our LP relaxation heuristic, a case study on two types of endd-life inkjet printers with several m m m partslmrrponents was performed and the results are reported in this sedion. The required data for the case study were obtained from a swiss d i s a s s m bly mrrpany. which eams protits from dkassembling various endof-life eledronic produds instead of disposing t h e m without disassembly. Figure 2 shows the disassemty product strudure of the two types of inkjet printers, in which the shaded boxes represent (XXTIITID~partslmrrponents. There are 29 items in total. i.e.. 9 parent item and 20 leaf item. Other a quired data for this case study are summed in Table 1. The disassembly operation mst of a parent item was mC culated by mltiplying the unit direct labor mst and the disassembly operation time for that item; the set up mst was calculated by multiplying the unit d i r e d mst and the required preparation time for that item. Finally, the invert tory holding mst of an item was set to be proportional to the mrresponding endoflife value o f that item. Now. the remaining data are initial inventory. demands of leaf item and disassembly lea&time. In this test, we generated these data r a m since we muld not obtain themfromthe mrrpany. First. the a m w n t s o f initial invert torywere generated from DU(0. 50) and DU(0. 10)for leaf and non-leaf items. sspedively Here. DU(a. b ) is the discrete uniformdistribution with range [a. b ] . Demands of each leaf itemwere set to 0 or DU(50.200)with probabilC ties 0.1 and 0.9.respedively Finally. disassembly lead times were set to 0. 1. and 2 with probabilities 0.2.0.7. and 0.1, respedivw.
In the test, 10 problem were generated randonly for three levels of the number of periods (10. 15, and 20). Performance measures used in the test are percerttage deviations from optimal solution values, i.e.. the gap b e tween a heuristic solution value and an optrmal solution value. and CPU semnds. In this test, we used C P L M 6.5.
Figure 2: Disassembly p r o d u d structure of the two types of inkjet printers
5
Table 1 : Data for two inkjet printers
CONCLUDING REMARKS
We m s i d e r e d the disassembly scheduling proMem with rmltiple produd types and parts m m m l i t y . which is the problem of determining the disassembly schedule of used products whle satisfying the demands of their i n d C vidual parts or m m p e n t s over a given planning horizon. To sdve the problem. a heuristic based on linear prtr grarming relaxation. is suggested in this paper. To show the effediveness of the heuristic, a case study on eWoP life inkjet printers was performed and the results showthat that the heuristic can give near optrmal solutions within very short mq~utationtimes. This research can be extended in several ways. First, it is necessary to m s i d e r the problem with r e w r c e capady mnstramts. Semnd. like other disassembly problem. UIF certainties. such as stochastic demands. stochastic disassembly times. and stochastic i n m r i n g rate of used pro& uds. are irqmrtant mnsiderations.
6 ACKNOWEDGEMENT This research was supported by the swiss National ScC ence Foundation (SNF) under grant rm. 200~066640.01. This support is gratefully acknowledged. Also.the authors wwld like to appreciate Mr. JuwGyu Kang at EPFL-LICP and Mr. Adrien B e s m at BIRD in Switzerland fortheir kind help to gather the necessary information on the caw Study. a m m r c i a l software package. to obtain optrmal s d k tions. However. to avoid eroessive mrnputation times. CPLEX 6.5 was terminated when CPU time reached 3600 s e m n d s . and in this case. the percentage deviationsfrom lower bounds. i.e., the gap between a heuristic solution value and a lower b w n d value, were used as the performance measure. Note that the lower bounds can be obtained diredyfrom CPLEX 6.5. Test results on the case study are s u m r i z e d in Table 2. which shows the percentage deviations from optrmal s d k tions or lower bounds [for the problem that CPLEX 6.5 muld rmt give the optrmal sdutions within 3600 semnds) and CPU semnds. The CPU seconds of the integer prtl g r a m i n g approach are also surrmarized in the table. lt can be seen from the table that the heuristic suggested in this paper can give near optimal solutions b the test problem. i.e.. 1.29%. 1.03%. and 1.33% in average for the cases with 10, 15, and 20 p e r i d s . respectively. Also. the heuristic required quite short mrrputation times. i.e.. within 0.04 semnds for all problem. However. in the case of the mteger prcgrarming approach. the CPU s e m n d s ranged from 0.45 to3603.57 semnds. Note that in the latter case. we muld mt obtain optimal solutions using the integer prcgramming approach. Table 2: Test results on the case study bhrnbnr o f n n n d r ; I0
Rhhm 7 2
3 4 6 6 7 8 g I0 Average
OW' 7 06
177 126 12l 106 I 27 o BI 0 90 141 214
7 29
CPU~ 002( f 7 6 ) 002(217) OOl(O46) 002(210) OOl(0Sa) 0 M(69 38) o 02(1746) 0 0 2 ( 3 61) 002(437) OOl(O71) 0 82)
16
CPU DWJ 0 0 3 ( 7 d f s 7 3 ) 7 39 003(28243) 1 7 8 003( 3 1 6 2 ) I 3 3 003(1MB36) 0 7 4 002(17690) 1 7 9 003(360093) 1 2 6 0 0 3 p 6 0 1 23) I06 003(360083) 138 o w ow(3mm) 1 4 6 OB9 O m ( 3 7 2 9 9 ) I 1 6 T O 3 0 0 3 ( 7 m 2 8 ) 7 33
20
CPU
DWJ
OTB 107 143 I011 106 I011 I00 109
0
OM(360367) OM(36Ml6) OM(3807M) OM(36M13) 0 M(36M 13) o M(36rn~ 7 ) 0 M(36M 19)
ou(36rnm) OM(36M27) 0 M(%-
Percantage deviation from opimal solution value or lower bwnd CPU seconds d the heunstic suggested in t h s paper and the r t q e r progamming approach (in pareflhases)
7
REFEENCES Jovane. F.. A M g . L..Armllotta.A., Eversheirm. W.. Feldmann. K.. Seliger. G. and Roth. N.. 1993, A Key Issue in Produd Life Cyde: Diszsembly. Annals of the C IRP 42: 65 1-658. Gupta. S. M. and Taleb. K. N., 1994, Scheduling Disassembly. International Journal of Produdion Research 32: 1857-1 886. Taleb. K. N.. Gupta. S. M.. and Brennan. L.. 1997. Disassembly of Complex Produd Structures with Parts and Materials Corrmonallty. Produdion plaw ning and Control 8: 255-269. Neuendorf. K.-P.. Lee. D.-H.. Kiritsis. D. and X rwchakis. P.. 2001. Disassembly Scheduling with Parts Comnonallty using Petri-Nets with T i m e starrps. Fundamenta Informatime 47: 295306. Taleb. K. N . and Gupta. S.M.. 1997. Disassembly of Mumple Product Strudures. C m p u t e r s and Industrial Engineering 32: 949-961. Lee, D.-H.. Kim, H.-J.. and Xirwchakis. P. 2002. Disassembly Scheduling: An Integer Prqramning Approach. Technical Report. Department of W chanical Engineering. swiss Federal Institute of T e c h n o l w - Lausanne [EPFL). Switzerland. Lee. D . 4 . Xirwchakis. P.. and ZOst. R.. 2002. Disassembly Scheduling with Capacity Constraints. h nals of the ClRP 51: 387-390. OShea. B., Grewal. S . S., and Kaebemick. H., 1998, State of the Art Literature Survey on Disassembly Planning. Concurrent Engineering: Research and Application 6 : 345357. Lee. D . 4 . Kang. J.-G.. and Xirwchakis, P.. 2001. Disassembly Planning and Scheduling: Review and Further Research. Proceedings of the Institution of Mechanical Engineers: Journal of Engineering Manufadure - Part 215: 695-71 0. Santochi. M.. Dini. G.. and Failli. F.. 2002. Computer Aided Disassembly Planning: State of the Art and Perspedives.Annalsof the ClRP 51 : 1-23.