Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs Gaean Belvaux Laurence A. Wolsey Cockerll-Sambre, Flémalle Work carred ou a CORE, Unversé Caholque de Louvan CORE and INMA, Unversé Caholque de Louvan, Voe du Roman Pays 34, 1348 Louvan-la-Neuve, Belgum gaean.belvaux@cockerll-sambre.com wolsey@core.ucl.ac.be In spe of he remarkable mprovemens n he qualy of general purpose mxed-neger programmng sofware, he effecve soluon of a varey of lo-szng problems depends crucally on he developmen of gh formulaons for he specal problem feaures occurrng n pracce. Afer revewng some of he basc preprocessng echnques for handlng safey socks and mullevel problems, we dscuss a varey of aspecs arsng parcularly n small and large bucke (me perod) models such as sar-ups, changeovers, mnmum bach szes, choce of one or wo se-ups per perod, ec. A se of applcaons s descrbed ha conans one or more of hese specal feaures, and some ndcave compuaonal resuls are presened. Fnally, o show anoher echnque ha s useful, a slghly dfferen (supply chan) applcaon s presened, for whch he a pror addon of some smple mxed-neger nequales, based on aggregaon, leads o mporan mprovemens n he resuls. (Lo-Szng; Producon Plannng; Mxed-Ineger Programmng; Vald Inequales; Reformulaon) 1. Inroducon Much work has been done n acklng academc lo-szng problems by usng Lagrangan relaxaon combned wh branch-and-bound (Afenaks and Gavsh 1986, Daby e al. 1992, Tempelmeer and Dersroff 1996, Thzy and van Wassenhove 1985) or by heurscs usng boh one-pass greedy-ype approaches (Dxon and Slver 1981), meaheurscs such as abu search (Smpson and Erenguc 1998a, b), and column generaon (Carysse e al. (1990, 1993) and Vanderbeck (1998)). Relavely lle compuaon s repored based on mxed-neger programmng reformulaons and/or cung planes, wh he excepon of (Barany e al. 1984, Consanno 1996, Eppen and Marn 1987, Poche and Wolsey 1991). The hess here, and n he companon paper (Belvaux and Wolsey 2000), s ha many praccal lo-szng problems can be effecvely solved va mxed-neger programmng. Two mporan elemens of such an approach are () gh a pror formulaons of he specal problem feaures ncurred n pracce, and () general or specal purpose mxed-neger programmng (MIP) sofware ncorporang cung planes. The second elemen can be found n recen commercal MIP sysems such as XPRESS and CPLEX, or research codes such as MINTO (Savelsbergh and Nemhauser 1993) and BC-OPT (Corder e al. 1999), or n BC- PROD (a modellng and opmzaon sysem specfcally desgned for lo-szng problems, descrbed n a recen paper (Belvaux and Wolsey 2000)). Here we concenrae on he frs elemen, descrbng and formulang varous aspecs of lo-szng encounered n pracce. Specfcally n 2, we remnd he reader of some of he basc nequales ha can be added a pror or as cung planes n lo-szng models, as well as 0025-1909/01/4707/0993$5.00 1526-5501 elecronc ISSN Managemen Scence 2001 INFORMS Vol. 47, No. 7, July 2001 pp. 993 1007
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs wo basc modellng choces, ways of reang safey socks, and he possble use of echelon socks n mullevel problems. These choces ofen nfluence he compuaonal performance of commercal and specalzed mxed-neger programmng sysems. In 3 we consder how o bes model sar-ups, changeovers, and swch-offs n boh small bucke models wh one or wo se-ups per perod, and n bg bucke models n whch numerous ems can be produced per perod. In 4 we consder how o model aspecs such as mnmum bach szes, fullcapacy producon, and mnmum down-mes n small bucke models. We hen descrbe brefly a se of praccal applcaons conanng one or more of he above feaures, and ndcae how he formulaons can be mproved/modfed usng he observaons from he earler secons. Some lmed compuaonal resuls are presened. In 6 we consder a dfferen, bu sgnfcan (supply chan) applcaon nvolvng a large number of producs, ses, and sales areas, where he dffculy les n he requremen o produce n mulples of a fxed bach sze or o produce mnmum amouns. Here, based on aggregaon, some smple vald nequales are added o he formulaon a pror. We close wh some bref remarks. 2. Some Basc Modellng Opons 2.1. A Basc Model The noaon used resembles as closely as possble ha n Belvaux and Wolsey (2000). We consder frs lo-szng problems wh NI ems/producs ha can be produced n parallel on NK machnes wh a horzon of NT me perods. The basc daa (apar from coss) are s he prespecfed demand for em n perod. d k s he rae of producon of em on machne k n perod. We wll assume ha k = 1 unless oherwse saed. C k s he maxmum amoun of em ha can be produced on machne k n perod. L k s he oal capacy of machne k n perod. a k s he capacy requred per un of em on machne k n perod. b k s he fxed amoun of capacy requred f machne k s se up o produce em n perod. The basc varables are defned as follows: s s he sock of a he end of. r s he backlog of a he end of. x k s he producon of on k n. y k = 1 f machne k s made avalable o produce em n, and y k = 0 oherwse. y k s he se-up varable. The mnmal consrans of a mul-em, mulmachne sngle-level problem n whch one or several ems are produced on each machne durng a perod are s 1 r 1 + k a k xk k xk = d + s r for all (1) x k C k y k for all k (2) + b k yk L k for all k (3) x s r 0 y 0 1 (4) When only one or wo ems can be produced per perod, s ofen mporan o ake no accoun he cos and/or me of preparng he machne for he producon of a new em or for cleanng he machne aferwards. The addonal varables needed follow. z k = 1 f machne k s se up o produce em n perod, bu was no se up for em mmedaely beforehand; z k s called he sar-up varable. w k s he analogous swch-off varable. w k = 1f machne k s se up o produce em n perod, bu s no se up mmedaely aferwards. The basc model exended o nclude sar-ups has he addonal consrans z k +1 wk = y k +1 yk for all k (5) z k y k for all k (6) z y 0 1 (7) Consrans (5) are ofen wren n he alernave form or z k +1 yk +1 yk for all k (8) w k y k y k +1 for all k (9) 994 Managemen Scence/Vol. 47, No. 7, July 2001
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs when here s no need for he swch-off or he sarup varables, respecvely. Also, wh sar-ups presen he machne capacy consran (3) ofen akes he smple form y k 1 for all k (10) wh L k = 1 for all k, and ak = 0 b k = 1 for all k. In he nex subsecon we presen some of he nequales ha can be added a pror o he basc model; n he followng subsecon we consder how hey can also be used as cung planes. 2.2. Vald Inequales: A Pror Reformulaon Consder frs a sngle em wh a sngle machne and no backloggng. We drop ndces when possble so as o smplfy he noaon. For each em he flow balance consran (1), he capacy consran (2), and he bound consrans (4) gve a se of he form s 1 + x = d + s for all (11) x C y for all (12) x s 0 y 0 1 (13) Ignorng he values of he C, and reang he problem as uncapacaed, he nequales s k 1 d j 1 y k y j (14) are vald, because he sock a he end of perod k 1 mus be used o sasfy he demand n perod j f no producon akes place n he nerval k k + 1 j y k + +y j = 0. As here are only O n 2 such nequales, some or all of hem can be added a pror o he formulaon. Smple expermenaon shows ha even jus addng he n nequales s k 1 d k 1 y k for k = 1 n (15) obaned by akng he nequales (14) wh = k ypcally produces an mporan mprovemen n he lower bound value of he LP relaxaon. When C = max C s resrcve, gher nequales can be obaned. Specfcally by aggregang he balance consrans over he nerval k and hen usng he varable upper bound consrans, one obans he surrogae nequaly s k 1 + C y j d k wh s k 1 0 and y j neger. For hs se, he basc mxed-neger roundng nequaly (Nemhauser and Wolsey 1988) s s k 1 k ( k y j ) (16) where k = d k /C, and k = d k C k 1. Ths agan gves O n 2 nequales ha can possbly be added a pror. We now ndcae how he above nequales can be easly adaped n he presence of sar-up or backloggng varables. Wh sar-ups he nequales (14) can be replaced by s k 1 d j 1 y k z k+1 z j (17) because y k +z k+1 + +z j = 0 also mples ha here s no producon n he nerval k j. Wh backloggng, s no dffcul o see ha (14) can be replaced by s k 1 + r j d j 1 y k y j (18) and (17) can be exended n he same way, whereas (16) can be replaced by s k 1 + r k ( k y j ) (19) 2.3. Vald Inequales: Cus When usng one of he branch-and-cu sysems CPLEX, XPRESS, MINTO, and BC-OPT s possble o add some or all of he above vald nequales a pror. However, hese sysems are also able o generae some of hem auomacally as cung planes. These sysems ypcally generae cus by recognzng and combnng consrans much as above. For nsance, akng (11) wh = k, combnng wh he varable upper bound consran (12), and usng he nonnegavy of s k leads o he surrogae s k 1 + C k y k d k s k 1 0 y k 0 1 Managemen Scence/Vol. 47, No. 7, July 2001 995
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs from whch each of he sysems can generae he nequaly (15). By combnng several consecuve flow balance consrans wh he varable upper bound consrans hey can hen generae nequales such as (16), and possbly (14). Boh nequales (14) and (16) are members of larger exponenal famles of nequales. For example, he nequales (14) form par of he exponenal famly s k 1 + ( x j d j 1 y ) j k \T j T for all k T wh k and T k. See Barany e al. (1984), and Poche and Wolsey (1993) for a generalzaon of he nequales (16) for he capacaed case. However, should be poned ou ha he way he nal lo-szng model s formulaed may be crucal o wheher he above general mxed-neger programmng sysems succeed n generang cus or no. For nsance, f he flow balance consrans (11) were wren as x j d 1 for all j=1 none of hese sysems s lkely o recognze he exsence of mplc flow conservaon consrans, and so he nequaly (15) wll no be generaed. 2.4. Safey Socks Calculang Ne Demands Suppose now ha here s an nal sock S 0, and ha here are nonzero lower bounds S (safey socks) on he socks a he end of each perod. S 0 + x 1 = d 1 + s 1 s 1 + x = d + s for = 2 NT s S for = 1 NT Now he nequaly (15) s sll vald, bu s unlkely o be volaed because s 1 S 1. Here, he alernave s o rewre he sysem of equaons based on ne demands. Specfcally, frs se S max { } S S 1 d for = 1 NT Then, defne new varables s = s S 0, and ne demands d = d + S S 1. The resulng sysem s s 1 + x = d + s s 0 Wh hs change, he nequaly (15) ha now akes he form s 1 d 1 y may be volaed, even hough he orgnal nequaly s no. A second modfcaon may be approprae f d >C for some. Workng backwards from = NT 2, we see ha s 1 ŝ 1 = d + ŝ C + wh ŝ NT = 0. We hen reapply he same procedure as above o elmnae he lower bounds ŝ on he modfed sock varables s. 2.5. Mullevel Problems Echelon Socks We now consder mullevel problems. For smplcy of noaon we consder an assembly sysem where each em requres exacly one un of each of s predecessors, bu he ransformaon below apples jus as well o problems wh arbrary produc srucure. For an assembly sysem n whch s he successor em of em, and em 1 s he sngle-end em, he flow conservaon equaons ake he form s 1 + x = x + s for = 2 NI all (20) s 1 1 + x1 = d 1 + s1 for all (21) Here, by combnng hese equaons wh he varable upper bound consran x C y, nequaly (14) can be generaed for produc = 1, bu here s no apparen sngle-em lo-szng problem for he oher producs on whch o generae vald nequales. To overcome hs dffculy, remember ha he echelon sock of an em a an nsan n me s he oal sock of ha em whn he sysem, counng s own sock plus he number of uns of he em conaned n he socks of all s successors (Clark and Scarf 1960). So, we nroduce echelon sock varables defned by e 1 = s1 and e = s + e for 2 for each. Now he sysem (20), (21) can be rewren as e 1 + x = d1 + e for all (22) e e for all (23) 996 Managemen Scence/Vol. 47, No. 7, July 2001
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs where he addonal consrans (23) come from he nonnegavy of s. Now vald nequaly (15), akng he form e 1 d1 1 y can be generaed for each em and no jus for he end Produc 1. In erms of he orgnal varables, hs nequaly s of he form s j 1 d1 1 y j P where P s he se of all ems conanng Iem. 3. Sar-ups and Changeovers In he nex wo secons we explcly examne ways o formulae or reformulae pars of a parcular loszng problem a pror. Frs, we consder sar-ups, swch-offs, and changeovers. We say ha here s a changeover from o j f here s a swch-off of em followed mmedaely be a sar-up of em j. Here, an mporan decson o be made n buldng a model s he sze of he me nerval. If he me nerval s large, leadng o a bg bucke model, a relavely small number of perods may be needed, bu many evens can hen occur whn a me nerval, whch may lead o dffcules n modellng. Alernavely, f he me nervals are shor, leadng o a small bucke model, reang n deal wha happens n a me nerval or beween consecuve me nervals may be easer, bu he number of nervals may become large. 3.1. Small Bucke Models Frs, we consder small bucke models n whch we are lmed o producng one or wo ems per perod. The rade-off beween hese wo modellng opons s obvously problem-dependen. I s possble ha by allowng wo ems per perod, one can somewha ncrease he sze of he perods, and hus decrease he number of perods and he sze of he LP formulaon. On he oher hand, fndng a gh formulaon may be more complcaed. The specfc goal n hs subsecon s o fnd gh formulaons for sar-ups, swch-offs, and changeovers n he one and wo se-up models. The smples modellng soluon s jus o allow one em o be produced per perod on a sngle machne. Here, a producon sequence s very smple. We jus ndcae whch em, f any, s se up n each perod. Thus, f NT = 5, he sequence 1 2 2 4 3 ndcaes ha he machne s se up o produce Iem 1 n Perod 1, Iem 2 n Perods 2 and 3, Iem 4 n Perod 4, ec. Also, a sar-up of Iem 2 occurs n Perod 2, of Iem 3 n 5, ec. Ths suaon s easly modelled usng consrans (5), (6), and (10) resrced o a sngle machne and he addonal consrans z 1 y 1 for all. When we allow wo ems o be produced per perod, we allow only one produc change per perod. Now a feasble sequence such as 12 2 24 43 3 ndcaes ha Iems 1 and 2 are se up n Perod 1 n ha order, Iem 2 s se-up n Perod 2, Iems 2 and 4 are se up n Perod 3 n ha order, and so on. Now Iem 2 s sared up n Perod 1 because he machne s se up for Iem 2 a he end of Perod 1, and was se up for anoher em earler n he perod. Smlarly, Iem 4 s sared up n Perod 3. If we rewre he sequence as 12 22 24 43 33, we see ha he second em se up n 1 mus always be he frs em se up n. Noe ha hs makes possble for one em o be produced hroughou perod 1 and for wo oher ems and o be produced n perod by usng he se up sequence, bu s mpossble for four dfferen ems o be produced whn wo me perods. I can be shown by nducon ha he 0 1 pons sasfyng (5), and he consrans y 1 + z for all z 1 for all z + z 1 y for all precsely descrbe such wo se-up sequences. To undersand wheher hese formulaons can be mproved, we sudy a model nvolvng sequence dependen changeovers (Karmarkar and Schrage 1985), whch s mporan n s own rgh. Suppose ha here s a dummy em/produc correspondng o he dle sae. We consder he polyhedron Q represenng he flow of a sngle un from Managemen Scence/Vol. 47, No. 7, July 2001 997
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs Fgure 1 Small Buckes: One and Two Se-ups = 0 = 1 = 2 = 3 = 4 = 5 = 1 = 2 = 3 = 4 = 1 = 2 = 3 = 4 = 3 = 5 = 1 = 2 = 3 = 4 = 3 = 4 1 2 2 4 3 12 2 24 43 em o em over me, j = q j for j = 1 NI = 1 NT j = q 1 for = 1 NI = 1 NT j q 0 = 1 j 0 for j = 1 NI = 1 NT shown n Fgure 1, where q s he flow hrough node, and j s he flow from node 1 o node j. Ths formulaon represenng a flow of one un (or a pah) s known o be as gh as possble n he sense ha, f no oher consrans are presen, he lnear programmng relaxaon leads o an neger soluon. The correspondng gh formulaon whou he changeover varables j wh j s gven by he followng resul. Proposon 1 (Consanno 1995). proj q Q s he polyhedron = 1 for = 0 NT (24) q + q 1 + j j q q q 1 jj 1 + for = 1 NI = 1 NT (25) for = 1 NI = 1 NT (26) for = 1 NI = 1 NT (27) 0 for = 1 NI = 1 NT (28) We now nerpre hs proposon n he cases of one and wo se-ups per perod. 3.2. One Se-up per Perod To use he polyhedron Q o represen hs suaon, we need o defne he se-up and sar-up varables appropraely. Specfcally, we nerpre a flow hrough node j hrough he frs nework n Fgure 1 o mean ha here s a se-up of em j n perod. Thus, we se y j = q j. Now j = 1fy 1 = yj = 1, namely, s se up n 1 and j n. Wh hese defnons, z = j j j y y +1 s he sar-up varable, and w = j j j s he swch-off varable. = +1 = Usng hese equaons and proj y Q, we oban he y z w gh formulaon for se-ups, sar-ups, and swch-offs: y = 1 for = 0 NT (29) z y for = 1 NI = 1 NT (30) y z = y 1 w 1 for = 1 NI = 1 NT (31) 998 Managemen Scence/Vol. 47, No. 7, July 2001
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs y 1 + z + y j z j 1 j j z w 0 for = 1 NI = 1 NT (32) for = 1 NI = 1 NT (33) y + ( j y +1 zj +1 ) zj 1 for =1 NI j j z w 0 =1 NT (37) for =1 NI =1 NT (38) where (29) follows from (24); (33) because z and w 1 are he slack varables n (25) and (26), respecvely; (31) by elmnang n he equaons defnng z and w ; (30) from (28) and he defnon of z ; and (32) from (27) afer subsuon. Noe ha nequaly (32) s a cu srenghenng he formulaon gven a he begnnng of hs secon; see Consanno (1995). I says ha he hree possbles em s se up n perod 1, em s sared up n perod, and some em oher han s produced n boh perods 1 and are muually exclusve. where (34) follows from (24), and he defnons of y and z, (35) from (28) and he equaons defnng z and y, (38) as z and w are he slack varables n (25) and (26), respecvely, (36) from usng he defnons of z and w+1 o elmnae q, and hen he defnon of y o elmnae, and (37) from (27) afer approprae subsuons. Noe ha here (37) s a cu srenghenng he formulaon gven a he begnnng of hs secon. I says ha em s se up durng perod and some em oher han s se up hroughou perod are muually exclusve. 3.3. Two Se-ups per Perod We agan use formulaon Q bu wh he second nework n Fgure 1, so he nerpreaon s dfferen. Iems and j are se up n perod n ha order f and only f q 1 = qj = j = 1. So, q j = 1 means ha he machne s se up o produce em j a he end of perod. Wh hese defnons, z = j j j sar-up varable of em n perod, = q s he w = j j j = q 1 s he swch-off varable of em n perod, and y = q 1 +q = q 1 + z = w +z + s he seup varable of em n perod. Usng hese equaons and proj y Q, we oban he followng y z w formulaon: y z = 1 for = 1 NT (34) z + z 1 y for = 1 NI = 1 NT (35) y z = y 1 w 1 for = 1 NI = 1 NT (36) 3.4. Bg Bucke Models wh Changeovers Consder now a dfferen suaon n whch many ems can be se up n each perod, bu changeover coss and/or mes mus be modelled. Suppose for smplcy ha he problem only nvolves one me perod, and he frs and las ems produced n he perod, denoed by = 0 and = n + 1, respecvely, are known and dsnc. Le y be he number of seups of em n he perod, and j be he number of mes producon s changed from o j, wh q an a pror upper bound on he number of mes a gven em s produced n he perod. Thus, he seup sequence 1, 2, 2, 3, 4, 2, 3 s represened by y 1 = y 4 = 1 y 2 = 3 y 3 = 2, and 01 = 12 = 22 = 34 = 42 = 35 = 1 23 = 2. Ths can be modelled n much he same way as he prze collecng ravellng salesman or he vehcle roung problem (Balas 1989) wh 0j = 1 (39) j 0 j = y for 0 n+ 1 (40) j j = y j for j 0 n+ 1 (41) Managemen Scence/Vol. 47, No. 7, July 2001 999
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs n+1 n+1 = 1 (42) y j 0 1 q for all j (43) wh addonal consrans o elmnae subours. The dfference here s ha we mus deal wh neger raher han 0 1 varables. The followng resul s easly verfed. Proposon 2. () A vecor y sasfyng he sysem (39) (43) provdes a feasble producon sequence f and only f he nduced dreced mulgraph does no conan a dsconneced dreced Euleran subgraph. () The vald nequaly j E S j y 1 q yk for k S (44) S elmnaes such a mulgraph on he node se S 1 n, where E S s he se of pars j wh j S. 4. Mnmum Producon Runs and Full-Capacy Producon Here we show how o model varous ypcal consrans arsng n small bucke models. Some of hese reformulaons were frs presened n Poche and Wolsey (1996); see also PAMIPS (1995). Mnmum On and Off Tmes. We consder sngleem consrans hroughou hs subsecon, so he produc superscrp s agan dropped. If a machne mus reman se up for an em for a mnmum of perods, we have z y A gher formulaon s for = + 1 z +1 + +z y (45) or equvalenly w + +w + 1 y. If afer producng an em, a machne canno produce agan whn perods, we have z 1 y for = + 1 + and agan a gher formulaon s z +1 + +z + 1 y (46) or equvalenly w 1 + +w 1 y. Full-Capacy Producon. If an em mus be produced a full capacy n all bu he frs and las perods of a se-up sequence (nonpreemped producon bach), we have ha producon s a full capacy n perod f he machne s se up n perods 1 and, and neher s a swch-off perod. Ths gves he consran x C y 1 + y w 1 w 1 A gher formulaon s x C y 1 w 1 w (47) correspondng o he observaon ha full-capacy producon s enforced n f he machne s se up n 1 and s no swched off n perods 1or. Full-Capacy and Mnmum Producon Runs. Suppose ha he mnmum producon quany for some em s P, full capacy s C, and he maxmum capacy n a changeover perod s C, wh C < C<P. Le a = P C /C and b = P/C. Clearly, he leas number of perods requred o produce P s max a + 1 b, and he mos s b + 1. The basc consran x Pw = b can be ghened. Specfcally f a = b, he nequaly x a + x P a 1 C w (48) s vald for all >a, and f a = b 1, he nequaly x b + x a + x P a 1 C w (49) s vald for all >b. Noe also ha f he mnmum producon quany s fully produced durng some nerval + 1 l and exceeds he oal demand d l l = d for he nerval, hen he excess mus be used o sasfy demands n perods ousde he nerval. Ths shows he valdy of he nequales and s l + r 1 P d l + l =+b w (50) l b s l + r 1 P d l + z (51) = 1000 Managemen Scence/Vol. 47, No. 7, July 2001
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs 5. A Selecon of Problems Here we descrbe some praccal lo-szng nsances, gvng he orgnal formulaon, and some mprovemens ha can be made based on 2 4. 5.1. b4 Ths s a mul-em, mulmachne problem wh backloggng, cleanng mes CLT, and lower and upper bounds on socks. I s a small bucke model wh wo se-ups per perod a mos. The specal feaure of he problem s he requremen of lower bounds MB (n number of days) on producon runs for end ems, denoed EP, and full-capacy producon n all bu he frs and las perods of a producon run. There s a small number of nermedae producs, denoed IP, wh a few machnes dedcaed o hem. The coeffcens RML j represen he number of uns of nermedae produc IP requred o produce one un of fnal produc j EP. Noe ha he producon srucure s no of assembly ype, bu he se of end ems s paroned accordng o he unque nermedae em ha each uses. An nal formulaon of hs problem s mn k s 1 + k s 1 r 1 + k ( p k x k + c w k k x k = k x k j k j ) ( + h s + ) e r RML j j k x j k +s for IP for all (52) = d +s r for EP for all (53) x k +CLT w k C k y k for all k (54) x k + CLT w k C k for all k (55) w k 1 y k 1 y k for all k (56) w k +w k +1 y k for all k (57) y k w k 1 for all k (58) w k 1 for all k (59) w k l k x k l l= k k x k y k for EP all k l + k (60) MB w k C k( y k for EP all k (61) 1 +yk w k w k 1 1) for all k (62) s r x 0 y w 0 1 s 0 = S 0 r 0 =0 s s s where k = MB /C k, and k = MB + CLT C k /C k. Noe ha (57) (59) represen wo se-ups per perod, (60) a mnmum run lengh consran, (61) he mnmum bach consran, and (62) he full-capacy producon consran. A reformulaon of hs problem based on 3 and 4s mn ( ) ( p k x k + c w k + h s + ) e r s 1 + k s 1 r 1 + k k x k = k x k j k j RML j j k x j k +s for IP for all (63) = d +s r for EP for all (64) x k +CLT w k C k y k for all k (65) x k + CLT w k C k for all k (66) y k k + k l= x k l l= k w k 1 y k 1 y k for all k (67) w k 1 for all k (68) w k l +x k x k y k for EP all k (69) ( MB / k ( k 1 ) C k) w k for EP all k (70) C k( ) y k 1 w k w k 1 Managemen Scence/Vol. 47, No. 7, July 2001 1001
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs for all k (71) r 1 +s l ( ) MB d + l l w k k =+b for k l wh +b l (72) ( l l s 1 + r d 1 1 ) w k u y k = = k u= k s r x 0 y w 0 1 for k l (73) s 0 = S 0 r 0 =0 s s s Noe ha (69) s from he nequaly (45); (70) from 48 49 ; (71) from (47); (72) from (50); and (73) from a combnaon of (17) and (18). Resuls for nsances wh beween 10 and 20 perods, 22 ems and 7 machnes are presened n Belvaux and Wolsey (2000). Daa and model fles for hs problem are avalable a Belvaux and Wolsey (1999). 5.2. Chesapeake The CHES problems are a se of fve ndusral problems nvolvng he allocaon and sequencng of producon operaons wh sequence dependen se-up coss on connuous parallel facles. They are based on real problems (CHES 1989). We use he formulaon from he orgnal paper (CHES 1989) wh he addon of he generalzed subour elmnaon consrans proposed n 3. Here, y k s modfed o denoe he number of se-ups of em on machne k n perod wh upper bound q. jk s he number of ransons from producon of o producon of j n on machne k, whereas jk = 1 f he las producon on machne k n s and he frs producon n + 1sj. k = 1 f he las em produced on machne k s. Y jk 0 s daa akng value 1 f machne k s nally se up for em j, and value 0 oherwse. The formulaon proposed s mn h s + k s 1 + k p xk + f y + j k q jk( jk x k = d + s for all k x k C k y k for all k + jk ) jk + jk + j j j x k LB k yk for all a k x k L k for all k s x 0 y 0 1 q jk + Y jk 0 = y jk jk 1 = yjk jk = y k jk + k = y k j E S jk for = 1 for >1 for <NT for = NT y k 1 q yl k for l S S S 1 NI all Compuaonal resuls have been obaned by runnng BC-PROD n branch-and-bound mode wh he formulaon excludng he subour consrans. When an neger soluon s found, he sysem checks f represens a feasble machne sequence, and f no, a generalzed subour elmnaon consran s added, and he ree search connues. Such an approach s easly mplemenable wh sysems such as MINTO, or wh he subroune lbrares of CPLEX or XPRESS. A beer approach would be o also generae consrans o cu off fraconal soluons. Resuls on he unsolved problems of he orgnal daa se are presened n Table 1 (he resuls repored on n Belvaux and Wolsey (2000) are for slghly modfed daa). The frs seven columns descrbe he nsance: s name, he number of Iems NI, he number of Machnes NK, he number of Perods NT, he number of rows r, columns (c), and neger varables n n he MIP formulaon. The las four columns presen resuls: LB denoes he value of he bes lower bound. BIP he value of he bes feasble soluon, Secs he runnng me n seconds (wh a maxmum of 3,600), and GAP = 100 BIP LB / LB he dualy gap on ermnaon. All runs have been carred ou on a Penum 200 Mhz runnng under Wndows NT. Noe ha, apar from he bes feasble soluon for ches5, hese mprove on he resuls n Kang e al. (1999) obaned usng an orgnal approach, conssng 1002 Managemen Scence/Vol. 47, No. 7, July 2001
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs Table 1 Resuls for Unsolved CHES Problems Insance NI NK NT r c n LB BIP Secs Gap ches2 21 8 1 517 1168 881 2865 7 2852 5 3600 0 46% ches3 11 1 3 235 726 363 1284114 1284106 3600 0 00% ches4 11 2 1 146 330 242 646862 646860 8 5 0 ches5 12 2 3 367 972 486 7246 96 6894 51 3600 4 86% of breakng he producon sequence durng a perod no a lmed number of spl sequences each conanng 3 5 ems. By enumeraon, an opmal orderng can be found for each spl sequence. An MIP s hen solved n whch hese spl sequences are joned o produce feasble producon schedules, whle column generaon and heurscs are used o generae neresng spl sequences. 5.3. SmpEreng In Smpson and Erenguc (1998b), a general model for mullevel problems wh assembly produc srucure s proposed, nvolvng produc famles conssng of one or more ems, where each famly can n urn have a fxed cos, a se-up me, or a resource consran assocaed wh. The formulaon proposed s F f a f x + g V f mn h s + c f f f s 1 + x = x + s + d for all x My for all y f y f 0 1 x s 0 for all f wh F f gf g C f for all f for all f where he new varable f akes he value 1 f some em n famly f s se up n perod, s he mmedae successor of produc. F f s he se of ems n famly f. M s a large posve value. C f s he resource avalable for famly f n perod. a f s he rae a whch produc uses he resource assocaed o famly f. V f s he se of famles occurrng n he famly f budge consran. gf s he fxed amoun of he resource of famly f used f famly g s se up. Two observaons can be used o modfy hs formulaon. The frs s o use echelon socks as suggesed n 2. The second s ha he se-up varables y are unnecessary as here are no se-up coss. Ths leads o he followng revsed formulaon wh many fewer neger varables: F f mn h s + c f f f e 1 + x = dq + e for all e e for all x M f for all f wh F f a f x + gf g C f f for all f g V f f 0 1 x s 0 for all f where q s he fnal produc conanng em. Insances s NI v are versons of wo praccal problems wh NI ems, a sngle machne, and NT = 16 me perods. Problem s 78 v s a sorage rack producon problem wh 6 end producs, whereas problem s 80 v s an anmal feed packng problem wh 8 feeds, each packed n 2 conaner ypes, makng 16 end producs. All he daa are provded excep for he demand daa for he end producs for whch dscree probably dsrbuons are proposed. We gve resuls for wo dfferen approaches n Table 2. The frs s o run he reformulaed problem wh he general purpose sysem BC-OPT. The second s o run he orgnal formulaon wh he specalzed lo-szng sysem BC-PROD. We are no able o run he reformulaed problem wh BC-PROD as he sysem does no a presen recognze produc Managemen Scence/Vol. 47, No. 7, July 2001 1003
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs Table 2 Resuls for Smpson and Erenguc Problems Insance r c n LP XLP BIP Secs Gap s-78-lec 3505 2688 192 3238 9 10579 9 11672 9 1836 8 08 s-78-2ec 3505 2688 192 4548 8 10329 0 10995 3 1837 4 92 s-80-1ec 3905 2848 288 3533 3 8587 5 26274 0 1839 55 94 s-80-2ec 3905 2848 288 4696 8 9478 5 27353 2 1849 57 78 s-78-1 3313 3648 1151 2192 9 9892 8 11937 9 1892 15 71 s-78-2 3313 3648 1151 1649 8 9965 0 11004 2 1903 8 46 s-80-1 3905 3872 1312 3378 0 21527 6 25251 8 1977 13 48 s-80-2 3905 3872 1312 4533 5 22148 0 26052 0 2005 13 81 famles. In Table 2, LP denoes he nal LP value, XLP he value afer he addon of cus, BIP he value of he bes feasble soluon found, and GAP = BIP BLB /BIP 100, where BLB s he value of he bes lower bound on ermnaon. The runs have been carred ou on a Penum 350 Mhz runnng under Wndows NT. For he s NI v models he gaps of 10% 20% may appear large. However, n Smpson and Erenguc (1998b), where s suggesed ha local search heursc mehods should replace relaxaon-based mehods, only average resuls are repored and he gaps are much larger. 5.4. L1 Ths problem nvolves capacy consrans, desrable sock levels, backloggng, mnmum run lenghs, and producon of ems from only one famly durng a gven perod on each machne. The coss are for backloggng, socks and for socks fallng below he safey sock level. S s he safey sock level for Iem n Perod, and LB k s a lower bound on he amoun of produced on k n f s produced. The varable bs measures he amoun by whch he sock when posve falls below he safey sock level. s 1 r 1 + k x k = d + s r for all (74) bs + s S for all (75) k x k B k for all k (76) LB k yk x k C k y k f y k for all k (77) fk for all k wh F f (78) fk 1 for all k (79) y k fk 0 1 s r bs xk 0 for all k f Two observaons can be used o modfy hs formulaon. The frs s o change he base sock level, much as n 2.2. Specfcally, f we se he base sock level a S raher han 0 by defnng s = bs + s S 0, and r = bs + r 0, hen s s now he amoun by whch he sock exceeds he safey sock level S, and r s he amoun by whch he sock s below S. Now seng d = d S 1 + S, he flow balance equaon (74) becomes s 1 r 1 + k x k = d + s r and he nequaly ( s 1 + r d 1 ) y k k whch s a generalzaon of (15), may be volaed. The second s algorhmc. Even hough he famly se-up varables do no need o be se o be negral because of he negraly of he se-up varables y k and he forcng consrans (78), s perhaps neresng o make hem neger, and hen use prores o branch on he famly se-up varables before branchng on he se-up varables for he ndvdual ems. 1004 Managemen Scence/Vol. 47, No. 7, July 2001
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs Fgure 2 Se and Sales Area v l v l s l x k = l L l l LL l sh l l + bb l for all l (82) d l for all l (83) sh ll +1 for all l (84) = B k y k for all k (85) x 0 y 0 and neger (86) 6. A Producon-Dsrbuon Bg Bucke Problem Ths s a mulse, mulproduc problem n whch producon shpmen and sales mus be deermned over a 3-monh (perod) plannng horzon. The sysem consss of 4 producon ses comprsng 15 producon uns and 150 producs, and 30 sales areas. The flows a he producon ses and sales areas are shown n Fgure 2. Parculares of he problem are producs are socked only a he producon ses, he nvenory a he end of a me perod mus cover a ceran fracon of he sales n he succeedng perod (because long me perods are used and producon s no nsananeous), here s ransporaon beween ceran producon ses and from producon ses o sales areas, bu no beween sales areas, some producs are manufacured n mulples of a fxed bach sze, whereas for ohers here are mnmum producon quanes, lower bounds are gven on he sales per em a each sales area for each perod. A formulaon of a slghly smplfed problem s s l 1 + bl + x k + sh l l k K l l l l = s l + l l l sh ll for all l (80) x k C k for all k (81) wh he followng daa: s he fracon of sales ha mus be n sock n he prevous perod. d l s he lower bound on sales of n sales area l n perod. B k s he bach sze for em n un k. K l s he se of uns k such ha em s produceable on un k a producon se l. L l s he se of producon ses. LL l s he se of sales areas. The varables are s l s he sock of n producon se l a he end of perod. b l s he amoun of bough a producon se l n perod. sh ll s he amoun of shpped from l o l n perod. bb l s he amoun of bough a sales area l n perod. v l s he amoun of sold a sales area l n perod. y k s he number of baches of produced on un k n. Consrans (80) and (82) represen flow conservaon a he producon ses and sales areas, respecvely, (83) represens he lower bound on sales, (84) he sock avalably condon, and (81) he capacy of each producon un. To oban effecve vald nequales, we aggregae ems over ses and sales areas. Summng (80) over l, and nroducng aggregaed varables s sh b, ec., leads o s 1 + x k + b = s + sh (87) k Summng (82) and (83) over l gves where u = l bb l. sh + u d (88) Managemen Scence/Vol. 47, No. 7, July 2001 1005
Modellng Praccal Lo-Szng Problems as Mxed-Ineger Programs Aggregang (84) gves s sh +1 (89) Now combnng 87 88, and (89) gves s 1 + b + u + u +1 + k x k d + d +1 (90) Fnally, usng (85) and defnng a varable = s 1 + b +u + u +1 0 and leng D = d + d +1, we oban + B y k D k For hs we oban he mxed-neger roundng nequaly (Nemhauser and Wolsey 1988) ( ) y k k where = D/B, and = D B 1. Smlar nequales are obaned for ems wh mnmum producon runs. In Table 3 we presen compuaonal resuls for four ypcal nsances. We compare resuls usng he nal formulaon (mod1), mod1 plus all he aggregaed MIR nequales (mod2), and mod1 wh he aggregaed MIR nequales acve a he op node (mod3). Resuls were obaned on a Penum Pro 200 Mhz runnng under Wndows NT wh he XPRESS opmser mp-op 10.24. A me lm of 900 seconds s used for each nsance, and he opmsaon sraegy consss of a bes-bound node selecon sraegy for 127 nodes, followed by he mp-op defaul based on a choce among he wo descendan nodes. Ths lm s mposed n pracce as he model s run durng plannng meengs every monh. 7. Remarks The resuls n Belvaux and Wolsey (2000) and here sugges ha reformulaon and cung planes combned wh mxed-neger programmng provde an effecve ool for acklng a varey of praccal loszng problems. The undoubed srengh of he approach les n he qualy of he dual (lower) bounds provded by he ghened relaxaons. To fnd good feasble soluons gvng prmal (upper) bounds, here are many more possble approaches, and a general MIP sysem does no properly make use of problem srucure eher o branch or o search for feasble soluons. I does, however, have as sarng pon he lnear programmng soluon from he srenghened formulaon, so heurscs ncorporang hs soluon and problem srucure deserve furher exploraon. For dscree lo-szng problems whou realvalued producon varables nvolvng sar-up or changeover coss and mes (Fleschmann 1994), he reformulaons presened here reman mporan. However, he large number of varables needed o model changeovers s an mporan drawback. For such problems research no a combnaon of consran programmng and mxed-neger programmng appears an neresng drecon. Table 3 Resuls for Four Typcal Insances Insance r c LP IP Gap bm1mod1 9738 11539 120329241 116907408 2 56 bm1mod2 10354 11782 120084537 117090143 2 37 bm1mod3 10085 11782 120084537 117297001 2 19 bm2mod1 9892 11352 124572624 111504469 11 65 bm2mod2 10532 11592 123272639 121533925 1 37 bm2mod3 10261 11592 123272639 121496301 1 41 bm3mod1 9412 10838 104455434 100118515 4 14 bm3mod2 10026 11066 103269512 100856470 2 15 bm3mod3 9768 11066 103269512 100738832 2 40 bm4mod1 9778 11168 122958911 117865067 4 14 bm4mod2 10409 11393 121147329 118519707 2 03 bm4mod3 10149 11393 121147329 118519707 2 03 1006 Managemen Scence/Vol. 47, No. 7, July 2001
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