European Journal of Operaonal Research 267 2018 86 95 Conens lss avalable a ScenceDrec European Journal of Operaonal Research journal homepage: www.elsever.com/locae/ejor Producon, Manufacurng and Logscs Vald nequales for wo-perod relaxaons of bg-bucke lo-szng problems: Zero seup case Mahd Doosmohammad a, Kerem Akarunalı b, a School of Mahemacs, Unvers of Ednburgh, Ednburgh EH9 3FD, UK b Deparmen of Managemen Scence, Unvers of Srahclde, Glasgow G4 0GE, UK a r c l e n f o a b s r a c Arcle hsor: Receved 27 Sepember 2016 Acceped 8 November 2017 Avalable onlne 16 November 2017 Kewords: Producon Lo-szng Ineger programmng Polhedral analss Vald nequales In hs paper, we nvesgae wo-perod subproblems for bg-bucke lo-szng problems, whch have shown a grea poenal for obanng srong bounds. In parcular, we nvesgae he specal case of zero seup mes and denf wo mporan mxed neger ses represenng relaxaons of hese subproblems. We analze he polhedral srucure of hese ses, dervng several famles of vald nequales and presenng her face-defnng condons. We hen exend hese nequales n a novel fashon o he orgnal space of wo-perod subproblems, and also propose a new faml of vald nequales n he orgnal space. In order o nvesgae he rue srengh of he proposed nequales, we propose and mplemen exac separaon algorhms, whch are compuaonall esed over a broad range of es problems. In addon, we develop a heursc framework for separaon, n order o exend compuaonal ess o larger nsances. These compuaonal expermens ndcae he proposed nequales can be ndeed ver effecve m provng lower bounds subsanall. 2017 The Auhors. Publshed b Elsever B.V. Ths s an open access arcle under he CC BY lcense. hp://creavecommons.org/lcenses/b/4.0/ 1. Inroducon The lo-szng problem ams o deermne an opmal producon plan dealng how much o produce and sock n each me perod of he plannng horzon, gven manufacurng ssem lmaons such as machne capaces and cusomer orders/forecased demand. Due o s srong mpac on manufacurng companes performance n erms of cusomer servce qual and operang coss, lo-szng has been a ver acve research area for man decades wh sgnfcan aenon from researchers as well as praconers. Due o s praccal mporance and lmed knowledge n he leraure, we focus n hs paper on he mul-em lo-szng problem wh bg-bucke capaces, where each resource s shared b mulple ems and more han one pe of em can be produced n an me perod. We sud hs problem from a heorecal perspecve, where we analze a wo-perod relaxaon of hs problem and characerze s mporan properes. Our man conrbuons are several famles of new vald nequales and her The research presened n hs paper s suppored b he EPSRC gran EP/L0 0 0911/1, enled Mul-Iem Producon Plannng: Theor, Compuaon and Pracce. Correspondng auhor. E-mal addresses: mahd.doosmohammad@ed.ac.uk M. Doosmohammad, kerem.akarunal@srah.ac.uk K. Akarunalı. face-defnng properes for he relaxaons of he wo-perod subproblem, novel exensons of hese nequales no he orgnal space of he wo-perod relaxaon, as well as new vald nequales for he orgnal space, and exac separaon algorhms desgned o es he praccal srengh of he proposed nequales. We also develop a smple bu effecve heursc approach n order o exend compuaonal expermens. Our compuaonal resuls show ha he proposed nequales have grea poenal o srenghen he lower bounds sgnfcanl. 1.1. Leraure revew Mos lo-szng problems are nherenl dffcul problems: from he heorecal complex perspecve, even he mul-em problem wh a sngle jon capac and whou seup mes s srongl N P -hard Chen & Thz, 1990. From a compuaonal perspecve, problems wh mulple ems and capaces, n parcular of ndusral scale, reman noorousl dffcul o solve o opmal, ofen resulng n hgh dual gaps Buschkühl, Sahlng, Helber, & Tempelmeer, 2010. Therefore, here s a wde specrum of research on lo-szng problems, rangng from praccall effcen heurscs e.g., Federgruen, Messner, & Tzur, 2007 and mea-heurscs e.g., Jans & Degraeve, 2007 o mahemacal programmng echnques, whch we dscuss nex n more deal due o her relevance o our sud. hps://do.org/10.1016/j.ejor.2017.11.014 0377-2217/ 2017 The Auhors. Publshed b Elsever B.V. Ths s an open access arcle under he CC BY lcense. hp://creavecommons.org/lcenses/b/4.0/
M. Doosmohammad, K. Akarunalı / European Journal of Operaonal Research 267 2018 86 95 87 Because of her complex, mos researchers n he mahemacal programmng commun suded specal cases of lo-szng problems, whch sll provde valuable nsghs on some nheren srucures of more general problems and hence suppor he soluon mehodologes proposed. The exac approaches mos ofen emploed eher defnng vald nequales e.g., Baran, Van Ro, & Wolse, 1984; Küçükavuz & Poche, 2009; Poche & Wolse, 1994 or exended reformulaons e.g., Eppen & Marn, 1987; Krarup & Blde, 1977; Poche & Wolse, 2010 for varans of sngle-em problem, some of whch were also exended o mul-em problems e.g., Belvaux & Wolse, 20 0 0. There are also few sudes usng oher echnques such as Lagrangan relaxaon e.g., Bllngon, McClan, & Thomas, 1986 and Danzg Wolfe decomposon e.g., Degraeve & Jans, 2007. Poche and Wolse 2006 provde a horough revew of dfferen varans of lo-szng problems, her complexes and soluon mehods used. Mos recenl, here have been nsghful polhedral resuls on mul-level problems, such as he vald nequales of Zhang, Küçükavuz, and Yaman 2012, and he compac formulaons of Van Vve, Wolse, and Yaman 2014 for small-bucke capaces,.e., ems do no share resources. Despe hs exensve leraure, he research explcl nvesgang complcaons arsng from mulple ems compeng for he lmed capaces nheren n bg-bucke problems s raher lmed, and onl few excepons exs o he bes of our knowledge. The polhedral analss of a sngle-perod relaxaon b Mller, Nemhauser, and Savelsbergh 20 0 0, 20 03 provded us some nsghful properes of hs polhedron ncludng new vald nequales. The sud of Jans and Degraeve 2004 presened varous decomposons and ndcaed ha perod decomposons provde sronger bounds, whch s recenl nvesgaed furher b he branch-and-cu framework of de Araujo, Reck, Degraeve, Fragkos, and Jans 2015 resulng n promsng compuaonal resuls wh regards o gaps. The work of Van Vve and Wolse 2005 obaned srong lower bounds, mos ofen sronger han an prevous resuls, b applng approxmae exended reformulaons onl for a small number of perods. The exensve compuaonal sud of Akarunalı and Mller 2012 noed he boleneck n bg-bucke problems as he lack of a good undersandng of he convex hull of sngle-machne, mul-perod problems. Ths movaed he novel framework of Akarunalı, Fragkos, Mller, and Wu 2016, where he smalles such problem, a wo-perod relaxaon, s used o separae all volaed nequales b generang he exreme pons of s convex hull, whou pre-defnng famles of nequales. The compuaonal resuls of hs sud have shown grea promse o sgnfcanl close dual gaps for bg-bucke problems n general, whch movaed us o sud such a wo-perod relaxaon from a polhedral perspecve. In hs paper, we presen our work nvesgang he specal case of zero seup mes. Ths does no onl enable us o analcall sud nheren srucures and hence provde useful nsghs ha can be poenall exended o more complcaed problems, bu also mprove our undersandng abou hs mul-em problem wh zero seup mes ha has been acvel suded for man ears n he lo-szng leraure, e.g., he earler work of Dxon and Slver 1981 proposed heursc approaches for hs problem, and essenall he movaon for he semnal work of Padberg, van Ro, and Wolse 1985 semmed from hs problem. From a praccal perspecve, s worh nong ha seup mes are no necessarl zero, however, usng zero seup mes has been a ver effecve modellng approach n case of neglgble seup mes,.e., ver low seup mes compared o processng mes, n order o reduce he problem complex, see, e.g., Kuk, Salomon, & van Wassenhove, 1994 for a dscusson. Neglgble seup mes can be observed n varous manufacurng sengs, e.g., assembl operaons n auomoble ndusr and packng operaons n food ndusr. Moreover, as noed b Olaan and Geragh 2013 echnologcal advances such as agle oolng and maeral handlng make possble o produce dfferen producs on he same se of machnes and herefore enable producon lne desgners reduce seup coss sgnfcanl. We fnall refer he neresed reader o he revew of Jans and Degraeve 2008 for a horough dscusson abou hs specal case, achevemens and open challenges. In he nex secon, we presen he problem formulaon and he wo-perod relaxaon X 2 PL, orgnall proposed b Akarunalı e al. 2016, and sud some of s polhedral properes, ncludng he specal case wh no seup mes. In Secon 3, we presen wo relaxaons of X 2 PL, propose a number of vald nequales for hese relaxaons and dscuss her face-defnng properes. Secon 4 presens novel exensons of hese nequales no he orgnal space of X 2 PL, as well as a new faml of vald nequales. We presen exac separaon algorhms n Secon 5, and compuaonall es he srengh of he nequales n Secon 6, whch show promsng resuls for her effecveness. We also develop a smple bu effecve heursc separaon approach n Secon 7 enablng expermenaon wh larger nsances, where furher encouragng resuls are obaned. We conclude he paper wh a dscusson of possble exensons and generalzaons. We noe ha all essenal proofs are provded n he Onlne Supplemen due o her lengh and nvolved naure. 2. A wo-perod relaxaon for bg-bucke lo-szng problem Before we defne and sud he wo-perod relaxaon of neres, we frs provde he mahemacal formulaon of he mulem lo-szng problem wh bg-bucke capaces. We le T, I and R denoe he ses of me perods, ems, and machne resource pes, respecvel. We represen he producon, seup, and nvenor varables for em n perod b x,, and s, respecvel. mn T I f + T I h s 1 s.. x + s 1 s = d T, I 2 I a r x + ST r C r T, r R 3 x M T, I 4 { 0, 1 } T x I ; x, s 0 5 The objecve funcon 1 mnmzes oal cos, where f and h ndcae he seup and nvenor cos coeffcens, respecvel. The flow balance consrans 2 ensure ha he demand for each em n perod, denoed b d, s sasfed. We noe ha he model can be generalzed o nvolve mulple levels see, e.g., Akarunalı & Mller, 2012, however, we om hs for he sake of smplc. The bg-bucke capac consrans 3 ensure ha he capac C r of machne r s no exceeded n me perod, where a r and ST r represen he per un producon me and seup me for em, respecvel. Consrans 4 guaranee ha he seup varable s equal o 1 f producon occurs, where M represens he maxmum number of em ha can be produced n perod, whch s he mnmum of eher he remanng cumulave demand or he capac avalable. Fnall, he negral and non-negav consrans are gven b 5. 2.1. A wo-perod relaxaon: X 2 PL Nex, we presen he feasble regon of a wo-perod, sngle-machne relaxaon, as orgnall proposed b
88 M. Doosmohammad, K. Akarunalı / European Journal of Operaonal Research 267 2018 86 95 Akarunalı e al. 2016, of he mul-em producon plannng problem wh bg-bucke capaces, denoed b X 2 PL. x M I, = 1, 2 x d + s I, = 1, 2 x 1 + x 2 d 1 1 + d 2 2 + s I x 1 + x 2 d 1 + s I a x + ST C = 1, 2 I 2 x, s 0, { 0, 1 } I As we consder a sngle machne r R n hs relaxaon, we dropped he ndex r. We noe ha, for a gven me perod, he choce for he horzon of hs wo-perod subproblem wll be + α wh α = 1,..., NT. An obvous choce for α would be 1,.e., = 1, 2 relae o he perods of, + 1. The parameers can be assocaed wh he orgnal problem parameers usng he relaons M = M + 1 α, C = C r + 1 α, and d = d + 1 α, + α for all and = 1, 2. We refer he neresed reader o Akarunalı e al. 2016 for a dealed dscusson of srucurng wo-perod subproblems. Nex, we noe some polhedral properes of X 2 PL. Proposon 1 Akarunalı e al. 2016. W.l.o.g., we assume 0 < M and ST < C hold I, = 1, 2. Then, con v X 2 PL s fulldmensonal. Proposon 2. The rval face-defnng nequales for con v X 2 PL and her face-defnng condons f an are: 1. x 0, I, = 1, 2. 2. 1, I, = 1, 2. 3. s 0, I. 4. x M, I, = 1, 2. 5. x d + s, I, = 1, 2 f d < M. 6. x 1 + x 2 d 1 1 + d 2 2 + s, I f d < M, { 1, 2 }. 7. I a x + ST C, = 1, 2 f for {1, 2}, I a M + ST C + a k M k + ST k, k I. We om he proof for he sake of smplc of he presenaon. Nex, we presen some non-rval faces of con v X 2 PL. W.l.o.g., we assume a = 1, I n he remander of he paper snce varables can be scaled as needed. Proposon 3. For I, 1. The followng nequal s vald for X 2 PL : x 1 + x 2 d 1 1 + d 2 + s If d + ST C, { 1, 2 }, defnes a face of con v X 2 PL. 2. If ST C 2, hen he followng nequal s vald for X 2 PL : x 1 + x 2 d 1 C 2 ST 1 + C 2 ST + s If d 1 + ST C 1 and d 2 + ST > C 2, s face-defnng for con v X 2 PL. 3. If d 1 + ST C 1, hen he followng nequal s vald for X 2 PL : x 1 + x 2 d 1 1 + d 1 C 1 ST 2 + C 1 ST d 1 + s If d 1 + ST > C 1 and d 2 + ST C 2, defnes a face of con v X 2 PL. 4. If d 1 + 2 ST C 1 + C 2, hen he followng nequal s vald for X 2 PL : x 1 + x 2 d 1 C 2 ST 1 + d 1 C 1 ST 2 + C 1 ST + C 2 ST d + s If d 1 + ST > C 1 and d 2 + ST > C 2, defnes a face of con v X 2 PL. 1 The proof s omed for he sake of brev. In hs paper, we nvesgae he specal case of zero seups,.e., ST = 0, I. We noe ha a companon paper Akarunalı, Doosmohammad, & Fragkos, 2017 sudes he polhedral properes of he general case of non-zero seups as well as srucurng an effecve compuaonal framework. In he nex secon, we esablsh wo relaxaons of X 2 PL and sud her polhedral srucures. We frs presen he known face-defnng nequales, and hen derve several classes of vald nequales and esablsh her face-defnng condons. 3. Polhedral analss of he relaxaons of X 2 PL Frs, we make necessar defnons for he remander of he paper. Defnon 1. For a gven : A cover of I for perod s a se S such ha λ = S d C > 0. For gven non-emp ses S I and T I\ S, we defne he paron slack as ξ = T M + S d C. We defne he se S + of srcl posve cover/paron elemens as follows: {{ S S + = d > λ } f S s a cover. { S d > ξ } f S s par of a paron. The posve maxmum funcon as b + = max { b, 0 }. Frs, for a gven, we defne he followng relaxaon, denoed b PIR 0, for X 2 PL, snce s suded n he leraure b varous researchers. x M, x C I I x 0, { 0, 1 } I We dropped here all he ndces as well as for he sake of smplc. We noe ha Defnon 1 remans vald n he same fashon ha we use same defnons for hs relaxaon wh all he ndces as well as dropped. Nex, we presen known face-defnng nequales for PIR 0. Proposon 4 Flow cover nequales Padberg e al., 1985. Le S be a cover, and S M = C + λ. Assume ha M = max S M > λ. Then, x M C M 6 S S S s vald and defnes a face of con v P IR 0. Moreover, for L I \ S and M = max M, M, he nequal x M M λ C M S L S L s vald and defnes a face of con v P IR 0 f 0 < M λ < M M, L. In addon o hs known relaxaon and s face-defnng nequales, we presen a second relaxaon of X 2 PL for a gven. We call hs as PIR 1 and sud mporan properes of n he remander of hs secon: x M, x d + s, x C I I I S 7
M. Doosmohammad, K. Akarunalı / European Journal of Operaonal Research 267 2018 86 95 89 x, s 0, { 0, 1 } I Smlar o he prevous relaxaon, we dropped here all he ndces as well as, and Defnon 1 remans vald n he same fashon. Snce he srucure of he se X 2 PL s que complex from a polhedral perspecve, hs relaxaon of X 2 PL wh a smpler polhedral srucure enables us o poenall derve valuable nsghs on he nheren srucure of X 2 PL. Nex, we noe some obvous properes of hs polhedron, ncludng he full dmensonal and rval faces of con v P IR 1. These proposons can be easl proven and herefore, we om dealed proofs here for he sake of brev. Proposon 5. dm con v P IR 1 = 3 I. Proposon 6. The followng nequales are he rval faces of PIR 1 : 1. x 0, I. 2. 1, I. 3. s 0, I. 4. x M, I. 5. x d + s f d < M, I 6. I x C f I M C + M k, k I. Nex, we dscuss famles of vald nequales and esablsh her face-defnng condons for PIR 1. Proposon 7. Le S be a cover of I. Then he followng nequal called cover nequal s vald for PIR 1 : x d s + C d 8 S S S Moreover, f K I \ S such ha M d holds K, where d = max S d λ and d = max d, d, K, hen he followng nequal called em-exended cover nequal s vald for PIR 1 : x d d λ s + C d S K S K The vald of 8 and 9 can be shown b consderng he mxed-neger se { I x s C, x s d, { 0, 1 } I }, whch s a sngle-node flow se whou he nonnegav consrans as well as a specal relaxaon of PIR 1, and hen dervng flow covers and exended flow covers from. Inuvel speakng, snce he ems n he cover have a oal demand ha s srcl exceedng he capac b λ, he cover nequal consders onl ems from hs se ha have an ndvdual demand srcl hgher han hs excess of λ, as s no possble o produce a leas one such em o s full demand. Proposon 8. If d < M S and S + 2 hold, 8 defnes a face of con v P IR 1. If, n addon, 0 < d λ < d d holds K, hen 9 defnes a face of con v P IR 1. We provde a dealed proof n he Onlne Supplemen. Proposon 9. Le S = be a subse of I, T = I\ S, and T, T be a paron of T such ha T = and ξ 0. Then he followng nequal called paron nequal s vald for PIR 1 : x + d ξ + 1 + M ξ + 1 s + C S T S T S 10 We noe ha f eher S = or T =, hen hs reduces o eher flow cover nequales of Proposon 4 or cover nequales of Proposon 7, respecvel. The vald can be shown b consderng he mxed-neger se { S x s + I\ S x C, x s S S S 9 d, x M, { 0, 1 } I }, whch s a sngle-node flow se whou he non-negav consrans as well as a specal relaxaon of PIR 1, and hen dervng flow covers from. Proposon 10. Le ξ > 0, and assume ha for T + = { T M > ξ}, T + 1 holds. Moreover, assume ha d < M holds S. Then, 10 defnes a face of con v P IR 1. For he proof, we refer he reader o he Onlne Supplemen. Proposon 11. Le S be a non-emp subse of I, T = I\ S, T, T be a paron of T, K T and ξ 0. We defne { d p = : S M : T and p = max S T p ξ. We also defne p = max M, p, K. Then he followng nequal called em-exended paron nequal s vald for PIR 1 : x d ξ + M ξ + p ξ S T K S T K s + C d ξ + M ξ + 11 S S T The vald follows b dervng generalzed flow covers for he mxed-neger se { S x s + I\ S x C, x s d, x M, { 0, 1 } I }, whch s a sngle-node flow se whou he nonnegav consrans as well as a specal relaxaon of PIR 1. Proposon 12. Assume ha he condons presened n Proposon 10 hold. Moreover, le 0 < p ξ < M p hold K. Then, 11 defnes a face of con v P IR 1. For he proof, we refer he reader o he Onlne Supplemen. Example. Le I = { 1, 2, 3 }, and PIR 1 defned b: x 1 14 1, x 2 10 2, x 3 11 3 x 1 10 1 + s 1, x 2 6 2 + s 2, x 3 8 3 + s 3 x 1 + x 2 + x 3 14 Consder S = { 1 } and T = { 3 }. Hence ξ = 10 + 11 14 = 7. Then, we can generae a face-defnng paron nequal as follows: x 1 + x 3 10 7 1 11 7 3 s 1 + 14 3 4 x 1 + x 3 3 1 4 3 s 1 + 7 Wh S = { 1 } and T = { 3 }, we noe p 1 = 10, p 3 = 11. Hence, p = max S T p = 11 > ξ = 7. Le K = { 2 } and hence p 2 = max 11, 10 = 11. Then, we can derve he face-defnng emexended paron nequal: x 1 + x 2 + x 3 3 1 11 7 2 4 3 s 1 + 7 where bold elemens ndcae all erms ha are addonal compared o he prevous nequal. Usng PORTA Chrsof & Lobel, 2009, we can denf 6 face-defnng paron nequales and 3 face-defnng em-exended paron nequales for hs se. In Secon 5, we descrbe separaon algorhms for hese nequales. Nex, we dscuss how we can exend he resuls of hs secon o he space of he wo-perod relaxaon of X 2 PL. 4. Vald nequales n he orgnal space of X 2 PL We recall he orgnal space defned earler as X 2 PL. We frs map he nequales developed for PIR 1 n he prevous secon o he orgnal space of X 2 PL, and also defne new vald nequales for X 2 PL. Noe ha all he ndces as well as are nroduced here agan, whch were dropped n he prevous secon, snce he wll be par of he dscusson here.
90 M. Doosmohammad, K. Akarunalı / European Journal of Operaonal Research 267 2018 86 95 Corollar 1. Le {1, 2}, and S be a cover of I n perod. Then he nequal mapped from he cover nequal n PIR 1 s vald for X 2 PL : S x + S d + λ 1 S We can also exend hs nequal as follows. s + C 12 Proposon 13. Le, {1, 2}, =, and S be a cover of I for perod. In addon, assume L S. Then he followng nequal called perod-exended cover nequal s vald for X 2 PL : S x + L x + S d + λ 1 L d S s + C 13 We om he proof here, as Proposon 14 covers a more general case and hs s he specal case when K =. Nex, we dscuss em-exended cover nequales and how he can be represened n he orgnal space. Corollar 2. Le {1, 2} and S be a cover of I for perod. Le K I \ S such ha M d holds K, where d = max S d λ and d = max d, d. Then he followng nequal mapped from he em-exended cover nequal n PIR 1 s vald for X 2 PL : S K x + S d + λ 1 K d λ S s + C 14 The proof s sraghforward as follows he same logc as Proposon 7. We can also exend hs nequal as follows. Proposon 14. In addon o he assumpons and defnons of Corollar 2, le, {1, 2}, =, and L S. Then he followng nequal called em-and-perod-exended cover nequal s vald for X 2 PL : S K x L + L x + S S d + λ 1 K d λ d s + C 15 For he proof, we refer he reader o he Onlne Supplemen. Nex, we dscuss he exenson of paron nequales o orgnal space. Corollar 3. Le {1, 2} and S be a non-emp subse of I n perod. Le T = I\ S, and T, T be a paron of T such ha ξ 0. Then he followng nequal mapped from he paron nequal n PIR 1 s vald for X 2 PL : S T x + S d + ξ 1 S + T M + ξ 1 s + C 16 The proof s sraghforward as follows a smlar logc o he proof of Proposon 9. We can also exend hs nequal as follows. Proposon 15. In addon o he assumpons and defnons of Corollar 3, le L S, where, {1, 2} and =. Then he followng nequal called perod-exended paron nequal s vald for X 2 PL : S T L x + L x d + S S d + ξ 1 + T M + ξ 1 s + C 17 We om he proof as follows a smlar logc o he proof of Proposons 9 and 14. Nex, we dscuss em-exended paron nequales. Corollar 4. Le {1, 2}, S = be a subse of I for perod, T = I\ S and T, T be a paron of T such ha ξ 0 and K T. Le { d = : S M : T p p = max S T p ξ, and p = max M, p, K. Then he nequal mapped from he em-exended paron nequal n PIR 1 s vald for X 2 PL : + d + ξ 1 + M + ξ 1 x S T K K S p ξ S T s + C 18 The proof s omed as follows a smlar logc o he proof of Proposon 11. We can also exend hs nequal furher as follows. Proposon 16. In addon o he assumpons and defnons of Corollar 4, le L S,, {1, 2} and =. Then, he followng nequal called em-and-perod-exended paron nequal s vald for X 2 PL : x S T K + T + L x + M + ξ 1 S d + ξ 1 K p ξ L d S s + C 19 We om he proof here snce s ver smlar o he proof of Proposon 15. Fnall, we nroduce n he nex proposon a new faml of vald nequales ha s no derved from an of he nequales based on he sngle-perod relaxaon. Proposon 17. For, {1, 2} and =, le S I and T I\ S such ha θ = S d + T M C 0, and S Iand T I\ S such ha θ = S d + T M C 0. Then, he followng nequal called wo-perod paron nequal s vald for X 2 PL : S T x + S T x + S d + θ 1 + M θ + 1 + T + T S M θ + 1 C + C + d θ + 1 S S s 20 We provde a dealed proof n Onlne Supplemen. We noe ha hs nequal s non-domnaed f S S =.
M. Doosmohammad, K. Akarunalı / European Journal of Operaonal Research 267 2018 86 95 91 5. Separaon algorhms for relaxaons and orgnal space The purpose of hs secon s o descrbe n deal he separaon problems assocaed wh all he famles of nequales defned n he prevous secons. Snce he man purpose of hs paper s o nvesgae he rue srengh of he cus generaed, we focus on defnng exac separaon algorhms raher han her compuaonal effcenc. Here, we follow he same srucure and order of he prevous wo secons: we frsl defne separaon problems assocaed wh famles of nequales defned for he relaxaons of he problems, and hen for hose assocaed wh he orgnal space. In he remander of hs secon, we le x, ȳ, s o represen a fraconal soluon vecor n he assocaed space ha s o be cu off. W.l.o.g., we assume all problem parameers o be neger valued. 5.1. Separaon n he relaxaon space We sar frs wh he faml of cover nequales as defned b Eq. 8. Frs, we noe ha we can rewre hese nequales as follows: x + d λ + 1 s C S Snce S mus be a cover, for a gven value of λ > 0, we can fnd he mos volaed nequal f an b solvng he followng knapsack problem: { } f λ = max τ j λ w j d w = C + λ, w { 0, 1 } I, j I I where τ j λ = x j + d j λ + 1 ȳ j s j. If f λ > C, hen a volaed cover nequal s denfed for he specfed λ. We noe ha snce λ Z +, one can solve hs separaon problem for an value of λ [1, I d C]. Nex, we dscuss he separaon procedure for he faml of em-exended cover nequales 9. We frs rewre hese nequales as follows: S x + d λ + 1 s + K x d λ C For a gven cover S, f d λ, hen we can defne he se K as follows: { K = I\ S x d λ ȳ > 0, M d } Therefore, one can generae covers usng he procedure defned for he cover nequales, and hen heurscall generae he se K. We noe ha hs s a smlar approach o he one proposed b Padberg e al. 1985 p.854 for flow cover nequales. Fnall, we noe ha such a procedure s appled for he separaon of he known nequales 6 and 7. Nex, we dscuss he separaon procedure for he faml of paron nequales 10. Frs noe ha we can rewre hese nequales as follows: S x + d ξ + 1 s + T x + M ξ + 1 C For a gven value of ξ, we defne he followng IP for he separaon: f ξ = max x + d ξ + 1 ȳ s u I I + x + M ξ + 1 ȳ s.. d u + M v = C + ξ I I v C > d u I u + v 1, I u, v { 0, 1 }, I Here, u and v varables ndcae wheher an em belongs o se S or T, respecvel. The frs wo consrans ensure ha I v 1,.e., T =. A volaed nequal s found f f ξ > C. Sm- lar o he process for cover nequales, snce ξ Z +, one can solve hs separaon problem for an value of ξ [1, I max { d, M } C]. The separaon procedure for em-exended paron nequales 11 s smlar o he procedure descrbed for em-exended cover nequales 9. 5.2. Separaon n he orgnal space We sar hs secon wh he separaon of he perodexended cover nequales n he orgnal space as defned b Eq. 13. Frs, we rewre hese nequales as follows, where S I and L S, = : S x + d λ + 1 s + L x d C For a gven and fxed λ > 0, we can solve he separaon problem: max x + d λ + 1 ȳ s u + x d ȳ v I d I s.. u = C + λ ; v u I; u, v { 0, 1 } I If he opmal value of he problem s srcl greaer han C, hen a volaed nequal s denfed. Nex, we noe ha he separaon procedures n he orgnal space for he em-and-perod-exended cover nequales 15, for he perod-exended paron nequales 17, and for he emand-perod-exended paron nequales 19 follow a ver smlar logc o he separaon procedures of he perod-exended cover nequales n he orgnal space 13. Therefore, we om a dealed descrpon here for he sake of brev. Fnall, we nroduce he separaon procedure for wo-perod paron nequales defned b Eq. 20. Frs, we rewre as follows: x + d + ξ 1 s S + x + M + ξ 1 T + S T x + d ξ + 1 s + x + M ξ + 1 + S S I s C + C For a gven par of, as well as fxed ξ > 0 and ξ > 0 values, we can defne he followng separaon problem 2PPI: 2PPI f ξ,ξ = max x + d + ξ 1 ȳ s u I I + x + M + ξ 1 ȳ v I + x + d ξ + 1 ȳ s I + x + M ξ + 1 ȳ v u + s z I
92 M. Doosmohammad, K. Akarunalı / European Journal of Operaonal Research 267 2018 86 95 s.. d u + M v = C + ξ I d u I I + M I v u + v 1, I u + v 1, I z u, I z u, I = C + ξ u, v, z, u, v { 0, 1 }, I If f ξ,ξ > C + C holds, hen he nequal s volaed and hence he cung plane s added o he formulaon. 6. Compuaonal resuls In hs secon, we presen numercal resuls ndcang he srengh of he varous cus proposed earler. We noe ha our prmar am here s no necessarl o buld a praccall effcen compuaonal framework, whch s addressed n a companon paper Akarunalı e al., 2017, bu nsead o exhausvel generae all volaed nequales b exac separaon o measure her praccal srengh and effecveness. All he separaon algorhms and mahemacal models are mplemened and execued usng he Mosel language of FICO Xpress Opmzaon Sue Mosel verson 3.6.0, Xpress-MP v7.7 on a PC wh Inel Core 5 3.10 ggaherz processor and 4 ggabe RAM, where all possble wo-perod relaxaons, boh consecuve and non-consecuve, were consdered. In order o es he effecveness of he cus proposed, we have generaed 240 random es nsances n oal, whch we descrbe n deal nex. Frs of all, we noe ha exac separaon s compuaonall expensve, causng ssues wh avalable memor or prohbvel long mes when he problem sze became bgger han T = 12 and I = 10, so ha we se he maxmum sze o hese values. We also noe ha even wh hs maxmum sze, compuaonal mes can be exensve. On he oher hand, we have se he mnmum sze o T = 2 and I = 3, n order o capure he smples problem wh he wo-perod, mul-em srucure. We have vared T and I values wh nervals growng exponenall, n order o capure he vare creaed b he fac ha he problem complex grows exponenall raher han usng equal lengh nervals, resulng n 16 dfferen T, I combnaons. On he oher hand, we have consdered low, medum and hgh demand varabl for a good mx of problems, randoml generang d parame- ers n he nervals of [10, 20], [10, 40] and [10, 60], respecvel. Ths resuls n 48 combnaons, where for each combnaon, we have generaed 5 es nsances. The capaces n each perod are generaed as a random varable from he nerval [0.8 I md, 1.2 I md ], where md ndcaes he medan demand n ha nerval. Fnall, we noe ha he holdng coss h are randoml generaed from he nerval [0.1, 1] and he seup coss f akes a value of {1, 10, 50}, each wh probabl of 1 3, n order o generae a good mx of low and hgh seup cos ems and n beween. Nex, we presen he compuaonal resuls for low, medum and hgh demand varabl, n Tables 1, 2 and 3, respecvel. Snce he exac separaon procedures are sgnfcanl meconsumng for our proposed nequales, we generaed onl one round of volaed cus and added o he formulaon. In each able, he frs column ndcaes he combnaon T, I, followed b he columns ndcang average I nal G ap, G ap C losed b F low C over nequales onl.e., onl exended flow cover nequales are generaed, and he percenage G ap C losed for 5 nsances wh all he volaed cus generaed. Noe ha he nal gap s based on he srenghened LP relaxaon wh all volaed l, S nequales added a pror, whch are known o be ver effecve n pracce for mul-em problems, see, e.g., Akarunalı & Mller, 2012. In he remander of he ables, he columns ndcae he oal number of cus generaed of each pe for 5 nsances, n he followng order: C over 8, I em-exended C over 9, P erod-exended C over 13, I em-and- P erod-exended C over 15, P aron 10, I em-exended P aron 11, P erodexended P aron 17, I em-and- P erod-exended P aron 19, T wo- P erod P aron 20, and F low C over 7. We noe ha for bgges nsances of d [10, 40], [10, 60], he separaon procedure for wo-perod paron nequales ook prohbve mes and herefore he were removed from he framework, whch are ndcaed b n he ables. As he resuls n Tables 1 3 ndcae, he cus can close on average more han 25% of he nal gap. As one could naurall expec, he average gap closed b he cus deeroraes when eher he number of ems or he number of perods s ncreased, where hs deeroraon seems more sensve o he ncrease n he numbers of ems han o he ncrease n he numbers of perods. When he number of ems ncreases, he problem resembles more he srucure of an uncapacaed problem, he convex hull of whch can be effecvel descrbed b he l, S nequales and hence here s lle room for mprovemen b oher cus. Ths can be ndeed conssenl observed from he average nal gaps for all he problems wh 10 ems. On he oher hand, as he number of perods ncreases, he problem becomes furher awa from he deal wo-perod problem, for whch hese cus are orgnall derved. However, we noe ha when all nsances wh 10 ems are aken ou, even he average gap closed for he nsances wh 12 perods s 23.05%, whch s a subsanal mprovemen wh onl one round of added cus, and also onl slghl lower han 24.48%, he average gap closed for he nsances wh 6 perods and 3/4/6 ems. As we wll dscuss n Secon 7, he expermenaon wh 24 perod problems, albe usng a heursc approach, ndcae smlar gap mprovemens compared o nsances wh 12 perods. The resuls also ndcae whch pes of nequales are more nheren for dfferen szes of problems. A small number of cover nequales seem o close subsanal gaps for he 2-perod problems almos onl on her own, and he number of hese nequales do no var much as he problem sze ges bgger. On he oher hand, he number of paron nequales and wo-perod paron nequales generaed ncreases sgnfcanl as he number of ems and perods ncrease, makng boh pes of nequales mos ofen generaed nequales n our framework, hence also ponng o where an compuaonall effcen framework can focus on. We also noe our observaon from he compuaonal ess ha addng wo-perod paron nequales on op of cover and paron nequales and her varans s no ver effecve n closng he negral gap, alhough hs mgh also be he consequence of a sngle round of separaon. Anoher neresng aspec he resuls pon a s he fac ha he number of em- and perod-exended versons of he cus reman small compared o he smple versons of hese cus. Fnall, we make a remark on he effec of he proposed cus when he demand varabl changes. As he ables clearl ndcae, he cus n parcular paron nequales and s varans are more ofen volaed when he demand varabl ncreases: hese are also he nsances when our cus make more of an mpac for he amoun of he gap closed. Ths makes nuve sense ha paron nequales are more flexble han covers and hence a hgher demand varabl wll be able o generae more volaed nequales of hs pe. 7. A Heursc separaon approach As he exac separaon procedures dscussed n Secon 5 are sgnfcanl me-consumng and even prohbve wh respec
M. Doosmohammad, K. Akarunalı / European Journal of Operaonal Research 267 2018 86 95 93 Table 1 Average closed gaps and number of cus generaed of each pe for es problems wh d [10, 20]. # Cus generaed T, I IG GCFC GC C IC PC IPC P IP PP IPP TPP FC 2,3 18.01 52.88 52.88 5 0 0 0 0 0 0 0 0 5 2,4 18.03 45.11 50.52 5 0 0 0 0 0 0 0 0 5 2,6 9.85 48.30 48.30 7 0 0 0 0 0 0 0 0 7 2,10 6.01 34.52 34.52 2 0 0 0 0 0 0 0 0 2 3,3 17.40 10.34 38.79 5 0 0 0 10 0 0 0 0 4 3,4 14.24 0 28.99 1 0 0 0 9 0 0 0 0 0 3,6 12.65 10.27 27.68 5 2 0 0 12 1 0 0 0 5 3,10 10.06 0 15.60 0 0 0 0 20 4 0 0 0 2 6,3 19.39 8.53 25.90 8 3 4 0 31 8 3 0 22 7 6,4 15.35 0.82 20.37 6 0 1 0 26 4 1 1 16 2 6,6 8.93 1.63 16.62 6 4 0 0 16 13 0 0 14 7 6,10 8.40 4.41 7.87 4 2 0 0 22 11 0 0 8 6 12,3 16.47 3.05 18.25 13 6 14 0 51 13 23 7 99 11 12,4 13.81 0.73 17.21 8 4 6 2 64 23 12 7 98 6 12,6 15.22 0 18.28 10 0 1 0 46 13 0 0 108 0 12,10 7.46 0 5.88 5 1 0 0 144 20 0 0 204 2 Ave = 13.08 13.79 26.84 5.63 1.38 1.63 0.13 28.19 6.88 2.44 0.94 35.56 4.44 Table 2 Average closed gaps and number of cus generaed of each pe for es problems wh d [10, 40]. # Cus generaed T, I IG GCFC GC C IC PC IPC P IP PP IPP TPP FC 2,3 11.15 59.49 59.49 5 0 0 0 0 0 0 0 0 5 2,4 16.63 41.02 42.27 7 2 0 0 1 0 0 0 0 9 2,6 10.19 46.62 46.62 7 0 0 0 0 0 0 0 0 7 2,10 5.88 8.39 8.39 3 0 0 0 0 0 0 0 0 3 3,3 20.20 11.50 40.74 5 0 0 0 10 0 0 0 7 3 3,4 10.15 0 30.43 4 0 0 0 12 0 0 0 2 0 3,6 10.67 14.65 25.30 7 1 0 0 14 4 0 0 0 7 3,10 4.79 0 29.51 5 0 0 0 41 11 0 0 0 0 6,3 11.23 4.12 29.55 12 2 3 0 28 6 4 0 42 5 6,4 12.28 3.01 21.11 7 2 4 0 34 13 1 0 46 5 6,6 9.09 0 17.81 3 0 2 0 36 20 1 2 41 0 6,10 7.06 1.58 17.35 7 1 0 0 50 12 3 0 47 4 12,3 10.52 4.79 38.95 25 1 20 3 72 10 38 14 119 5 12,4 13.82 0.41 25.39 16 1 7 0 45 10 10 2 60 5 12,6 6.30 0.09 17.65 10 0 7 0 77 12 10 3 2 12,10 5.21 0 2.32 4 0 0 1 366 22 2 1 0 Ave = 10.32 12.23 28.31 7.94 0.63 2.69 0.25 49.13 7.5 4.31 1.38 22.75 3.25 o wo-perod paron nequales for larges nsances esed n Secon 6, we canno expec o generae all cus whn accepable compuaonal mes, n parcular for large nsances. Ths movaed us o furher analze he naure of he volaed nequales denfed b he exac separaon framework, n order o develop a smple bu effecve heursc approach for subsanall reducng he compuaonal mes requred b he separaon process. Frs of all, we conduced an exensve compuaonal expermen focusng prmarl on he cover nequales 8 and paron nequales 10, as her numbers and effecveness plaed a sgnfcan role n he gaps closed for he nsances esed n he prevous secon. In order o acheve unbased resuls, we generaed 80 new random nsances wh d [10, 60] and analzed he volaed nequales denfed. Ths analss ndcaed ha major of he volaed nequales of cover and paron nequales are generaed from he wo-perod subproblems conssng of wo consecuve perods raher han wo non-consecuve perods. Ths provdes a sgnfcan poenal for reducon n he number of separaon problems solved e.g., n a 12 perod problem, here are 66 possble combnaons of wo perods, whereas he consecuve wo perods are onl lmed o 11 combnaons, and unsurprsngl, for hese 80 nsances, we observed on average 70% mprovemen n compuaonal mes b generang consecuve wo-perod subproblems compared o all wo-perod subproblems. Mos mporanl, usng consecuve wo-perod subproblems, he closed gaps obaned were on average 95% of he closed gaps obaned b generang cus from all wo-perod subproblems. These observaons movaed us o generae he volaed nequales from he consecuve wo-perod subproblems onl. In addon, we performed an analss on he wo-perod paron nequales denfed for a randoml seleced se of 12 perod nsances from Secon 6. As dscussed earler, he exac separaon of hese nequales requres subsanall more compuaonal me compared o oher pes of nequales due o he sheer number of possble combnaons of ξ and ξ values n he separaon problem and hence he were no generaed for he larges nsances esed n Secon 6, and n addon, her effecveness for closng he dual gaps were observed o be raher lmed. Therefore, we lmed our analss here o prevousl used nsances raher han newl generaed nsances. Frs of all, smlar o our observaon for oher pes of nequales, we observed ha mos of he volaed wo-perod paron nequales were generaed from consecuve wo-perod subproblems. We also observed ha mos of he volaed wo-perod paron nequales occurred n he cenre of he plannng horzon, raher han earler or laer perods. Our prelmnar analss suggesed o focus on he woperod subproblems from he cenral 1/3 of he horzon, so, e.g., for a 12-perod problem, perods 5 o 8 o be used for he n he heursc separaon. Alhough we could no observe an parcular paern for he poenall more effecve value ranges of ξ,
94 M. Doosmohammad, K. Akarunalı / European Journal of Operaonal Research 267 2018 86 95 Table 3 Average closed gaps and number of cus generaed of each pe for es problems wh d [10, 60]. # Cus generaed T, I IG GCFC GC C IC PC IPC P IP PP IPP TPP FC 2,3 14.61 44.77 44.77 5 0 0 0 1 0 0 0 0 5 2,4 16.73 26.01 37.21 7 2 0 0 2 0 0 0 0 9 2,6 9.42 33.50 33.50 8 0 0 0 0 0 0 0 0 8 2,10 7.01 33.81 33.81 3 0 0 0 0 0 0 0 0 3 3,3 20.26 16.65 41.64 8 0 0 0 16 0 0 0 13 5 3,4 8.80 14.67 38.37 6 0 0 0 11 3 0 0 4 3 3,6 10.92 11.34 31.77 6 0 0 0 19 5 0 0 0 4 3,10 6.82 2.01 27.28 10 0 0 0 53 7 0 0 0 1 6,3 11.98 12.43 33.30 15 1 7 0 31 3 10 1 45 7 6,4 7.79 1.42 31.72 15 1 2 0 38 3 4 0 50 5 6,6 6.16 0 23.98 4 0 2 0 32 19 6 4 59 0 6,10 2.03 0 16.33 1 0 1 0 55 28 0 0 76 2 12,3 11.20 1.11 35.84 19 1 22 4 79 6 22 5 114 3 12,4 12.43 0.02 20.05 15 3 11 1 73 14 19 2 7 1 12,6 8.69 0.06 14.13 9 0 4 1 60 21 9 4 1 12,10 5.99 0 7.18 9 0 1 0 614 16 2 0 2 Ave = 10.03 12.36 29.43 8.75 0.5 3.13 0.38 67.75 7.81 4.5 1 27.44 3.69 we made he observaon ha ξ values ofen concde wh he ξ values of he paron nequales 10 from perod. Ths movaed us for he smple bu sgnfcanl me savng approach as presened n Algorhm 1. Algorhm 1: Heursc separaon algorhm for wo-perod paron nequales 20. Inpu : A fraconal soluon x, ȳ, s ; a wo-perod subproblem X 2 PL for, n he cenre of he plannng horzon Oupu : A volaed wo-perod paron nequal for ξ > 0 do f ξ { ξ here exss volaed paron nequal for ξ } hen for ξ > 0 do Solve he maxmzaon problem 2PPI f f ξ,ξ > C + C and S S hen Add he volaed cu 20 end end end end I s noeworh o remark ha even hough we are sll solvng he exac separaon problems of Secon 5, hs heursc algorhm sgnfcanl reduces he number of problems o be solved, and herefore, a subsanal reducon n compuaonal mes requred can be acheved. Thanks o hs advanage and based on our prelmnar ess, we decded o run 3 rounds of separaon wh hs heursc approach, as opposed o he sngle round we performed wh he exac separaon approach. The resuls are presened n Table 4, whch follows he same noaon and smlar srucure used n Secon 6, wh he addon of specfng he gaps closed wh he EXAc and HEUrsc approaches for he sake of comparson. Frs of all, when we compare he average performance of he heursc approach wh he performance of he exac approach for he nsances wh 12 perods, we noe ha s ver compeve, and even ofen superor, hanks o he advanage of 3 rounds of separaon and herefore beng able o denf furher nequales whch could no be denfed b he sngle round of he exac separaon approach. As expeced, compuaonal mes are sgnfcanl mproved, for example, nsances wh N T = 12, N I = 10 and d [10, 60] ook on average 7 das usng he exac approach whereas he heursc approach ook on average onl 3.5 hours for he same nsances. In order o furher es our heursc approach on larger nsances, we have also generaed nsances wh 24 perods, usng he same random generaon procedure descrbed n Secon 6. As compuaonal mes sgnfcanl ncrease even n case of our heursc, we noe ha he larges and mos challengng nsances n hs se.e., nsances wh N T = 24, N I = 10 and d [10, 60] ook on average 1.5 das, whch s sll superor o he mes aken b he exac separaon for nsances wh 12 perods. We presen our resuls n Table 4, where, for a gven number of ems, he average gap closure of he heursc for hese nsances s no sgnfcanl dfferen from s average performance on he nsances wh 12 perods. I s also noeworh o remark ha on average over all hese 120 nsances wh 12 and 24 perods, gaps are closed around 18%. We conclude ha n addon o he subsanal gans n compuaonal mes and abl o expermen wh larger nsances, he smple heursc framework acheves successful closure of gaps. 8. Conclusons In hs paper, we nvesgaed a wo-perod subproblem of he bg-bucke lo-szng problem from a heorecal perspecve. In parcular, we have denfed varous famles of vald nequales for a relaxaon of hs subproblem n he specal case of zero seup mes, descrbed her face-defnng properes, and we have also mapped and exended hese nequales o he orgnal space of he wo-perod subproblem. The compuaonal resuls ndcaed sgnfcan poenal for mprovng lower bounds, and we are currenl nvesgang hs horoughl n a companon sud n wo mmedae drecons: undersandng polhedral characerscs of he general case wh non-zero seup mes and denfng furher vald nequales, and desgnng a branch-and-cu framework wh rounes generang cung planes of boh zero and non-zero seup me sengs n realsc mes for mul-em loszng problems. The heorecal resuls we presened n hs paper can be exended o oher MIP problems hanks o he commonal of he mxed neger ses nheren n dfferen problems. We have alread noed varous sudes on he polhedron of he PIR 0, he sngle node fxed charge se, whch s a common mxed neger se n varous MIP problems. On he oher hand, he srucure of PIR 1 poses dfferen challenges and opporunes, and s worh nvesgang furher s lnk o oher mxed neger ses. Fnall, here s also mmedae neres n nvesgang f and how our undersand-
M. Doosmohammad, K. Akarunalı / European Journal of Operaonal Research 267 2018 86 95 95 Table 4 Average closed gaps usng he heursc procedure HEU vs. usng he exac separaon EXA. d [10, 20] d [10, 40] d [10, 60] GC GC GC T, I IG EXA HEU IG EXA HEU IG EXA HEU 12,3 16.47 18.25 25.12 10.52 38.95 37.34 11.20 35.84 36.84 12,4 13.81 17.21 17.87 13.82 25.39 25.03 12.43 20.05 20.61 12,6 15.22 18.28 17.74 6.30 17.65 19.23 8.69 14.13 16.51 12,10 7.46 5.88 7.30 5.21 2.32 4.39 5.99 7.18 7.38 24,3 21.51-31.94 17.15-25.77 9.60-44.33 24,4 16.13-22.58 13.59-16.53 11.39-30.01 24,6 12.84-13.09 8.64-8.27 7.86-5.58 24,10 7.84-2.08 6.65-3.99 5.49-4.32 Ave = 13.91-17.22 10.24-17.56 9.08-20.69 ng of he wo-perod subproblems can be furher exended o more sophscaed subproblems, e.g., a k -perod subproblem. As Van Vve and Wolse 2005 observed n her framework, even lmng o he values of k = 3 and k = 4, here s subsanal poenal o develop a horough undersandng of he complex loszng problems, whch we plan o sud n he fuure. 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