A CONSTRUCTIVE HEURISTIC FOR THE INTEGRATED INVENTORY-DISTRIBUTION PROBLEM

Transcription

1 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9), Caro, Egyp, Jan. 8-, 28. A COSTRUCTIVE HEURISTIC FOR THE ITEGRATED IVETORY-DISTRIBUTIO PROBLEM Tamer F. Abdelmagud *, Maged M. Dessouky ** and Fernando Ordonez *** * Asssan Professor, Mechancal Desgn and Producon Deparmen Faculy of Engneerng, Caro Unersy, Gza Egyp. ** Professor, *** Asssan Professor, Danel J. Epsen Dep. of Indusral and Sysems Engneerng, Unersy of Souhern Calforna, Calforna, USA. abdelmagud@eng.cu.edu.eg ABSTRACT We sudy he negraed nenory dsrbuon problem whch s concerned wh mulperod nenory holdng, backloggng, and ehcle roung decsons for a se of cusomers who recee uns of a sngle em from a depo wh nfne supply. We consder an enronmen n whch he demand a each cusomer s deermnsc and relaely small compared o he ehcle capacy, and he cusomers are locaed closely such ha a consoldaed shppng sraegy s approprae. We deelop a consruce heursc o oban an approxmae soluon for hs P-hard problem and demonsrae s effeceness hrough compuaonal expermens. KEYWORDS Vehcle roung, nenory managemen, heurscs.. ITRODUCTIO Recen decades hae seen ferce compeon n local and global markes, forcng manufacurng enerprses o sreamlne her logsc sysems, as hey consue oer 3% of he cos of goods sold for many producs []. The major componens of logsc coss are ransporaon coss, represenng approxmaely one hrd, and nenory coss, represenng one ffh [2]. The ransporaon and nenory cos reducon problems hae been horoughly suded separaely; whle, he negraed problem has recenly araced more neres n he research communy as new deas of cenralzed supply chan managemen sysems, such as endor managed nenory (VMI), hae ganed accepance n many supply chan enronmens. The negraon of ransporaon and nenory decsons s represened n he leraure by a general class of problems referred o as dynamc roung and nenory (DRAI) problems. As defned by Baa e al. [3], hs class of problems s characerzed by he smulaneous ehcle roung and nenory decsons ha are presen n a dynamc framework such ha earler decsons nfluence laer decsons. Baa e al. [3] classfy he approaches used for DRAI problems no wo caegores. The frs caegory operaes n he frequency doman where he decson arables are replenshmen frequences, or headways beween shpmens. Examples n he leraure nclude [4-8]. 587

2 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) A COSTRUCTIVE HEURISTIC FOR THE ITEGRATED IVETORY-DISTRIBUTIO PROBLEM The second caegory, referred o as he me doman approach, deermnes he schedule of shpmens. Wh dscree me models, quanes and roues are decded a fxed me nerals. Whn hs caegory he mos famous problem s he nenory roung problem (IRP), whch arses n he applcaon of he dsrbuon of ndusral gases. The man concern for hs knd of applcaon s o manan an adequae leel of nenory for all cusomers and o aod any sockou. In he IRP, s assumed ha each cusomer has a fxed demand rae and he focus s on mnmzng he oal ransporaon cos; whle nenory coss are mosly no of concern. Examples of hs applcaon n he leraure nclude [9-4]. In hs paper, we consder a DRAI problem ha addresses he negraed nenory and ehcle roung decsons n he me doman a he operaonal plannng leel. Ths problem, referred o as he Inegraed Inenory Dsrbuon Problem (IIDP), consders mulple plannng perods, boh nenory and ransporaon coss, and a suaon n whch backorders are permed. The knd of applcaon ha perms backorders s, of course, dfferen from he dsrbuon of ndusral gases, where no shorage s allowed. The proposed model s suable o ndusral applcaons n whch a manufacurer dsrbues s produc o geographcally dsbursed facores/realers whch are locaed n ces close o s warehouse. A he operaonal plannng leel, backorder decsons are generally jusfed n wo cases. The frs s when here s a ransporaon cos sang ha s hgher han he ncurred shorage cos by a cusomer. The second case s when here s nsuffcen ehcle capacy o deler o a cusomer gen ha renng addonal ehcles s no an opon due o echnologcal or economc consrans. In he leraure, he negraon of ehcle roung and nenory decsons wh he consderaon of nenory coss n he me doman approaches of he DRAI problems has aken dfferen forms. In a few cases a sngle perod plannng problem has been addressed as found n [5] and [6]. In he mul-perod problem, he decsons are conduced for a specfc number of plannng perods, or he problem s reduced o a sngle perod problem by consderng he effec of he long erm decsons on he shor erm ones. Examples nclude [7-9]. Oher researchers ake no consderaon arous forms such as dsrbung pershable producs [2], and he consderaon of he me alue of money for long-erm plannng [2]. Some work focused on dfferen srucures of he dsrbuon nework such as he case of saelle facles [22], he case where warehouses ac as ransshpmen pons n a 3-leel dsrbuon nework [23], and he case of a mul-depo problem [24-25]. Soluon heurscs ha hae been proposed n he leraure for he dfferen araons of he negraed nenory-dsrbuon problem, parcularly he nenory roung problem, are eher based on subgraden opmzaon of a Lagrangan relaxaon as n [9] and [6] or consruce and mproemen heurscs. The consruce heurscs are broadly classfed no heurscs ha allocae cusomers o serce days and hen sole a VRP o generae ehcle roues for each day [2]; and heurscs ha allocae cusomers o days and ehcles and hen sole a raelng salesman problem for eery assgnmen []. Improemen heurscs found n he leraure [5 and 26] are generally consdered as exensons o he arcexchange and node-exchange heurscs as found n he ehcle roung leraure. In he leraure of he me doman approaches of he DRAI problems, some models n he case of mul-perod plannng may nclude shorage or sockou coss; howeer, backorder 588

3 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) Abdelmagud, Dessouky and Ordonez decsons are generally no explcly consdered. Insead, he shorage or sock-ou cos s reaed as he penaly cos ha s ncurred due o makng drec deleres o cusomers whose demand s no fulflled n he regular delery roue n a gen perod. Examples of such models n he leraure nclude [9] and [27]. In hs paper, we consder a suaon n whch backorder decsons are eher unaodable or more economcal, and hey hae o be coordnaed wh oher nenory holdng and ehcle roung decsons oer a specfc plannng horzon. We nroduce a consruce heursc for solng hs P-hard problem, and benchmark agans lower and upper bounds obaned by a commercal sofware package, CEX. The consruce heursc nroduced n hs paper s an enhanced erson of a preously deeloped one n [28]. The res of hs paper s organzed as follows. In secon 2 we formulae he problem as a mxed neger lnear program. The moang deas and search plan for he deeloped heursc s presened n secon 3. Secon 4 prodes descrpon of he deeloped consruce heursc. In secon 5, he expermenal resuls are presened followed by he concluson and drecons for fuure research n secon PROBLEM DESCRIPTIO AD MIXED ITEGER PROGRAMMIG MODEL In he IIDP, we sudy a dsrbuon sysem conssng of a depo, denoed, and geographcally dspersed cusomers, ndexed,,. Each cusomer faces a dfferen demand d for a sngle em per me perod (day/week). As radonally consdered, a sngle em does no resrc he problem o he case of a sngle produc dsrbuon, as he word em can refer o a un wegh or olume of he dsrbued producs and each cusomer can be ewed as a consumpon cener for packages of un wegh or olume [8]. Accordngly, he proposed model can be appled o he case of mulple producs gen ha he alues of he nenory holdng and shorage coss per un olume/wegh hae small arance among he dfferen producs. We consder he case n whch he demand of each cusomer s relaely small compared o he ehcle capacy, and he cusomers are locaed closely such ha a consoldaed shppng sraegy s approprae. Deleres o cusomers,, are o be made by a capacaed heerogeneous flee of V ehcles, each wh capacy q sarng from he depo a he begnnng of each perod. Vehcles mus reurn o he depo a he end of he perod, and no furher delery assgnmens should be made n he same perod. In hs model, we consder he case n whch renng addonal ehcles s no an opon due o echnologcal or economc consrans, and s assumed ha he flee of ehcles remans unchanged hroughou he plannng horzon. Each cusomer manans s own nenory up o capacy C and ncurs nenory carryng cos of h per perod per un and a backorder penaly (shorage cos) of per perod per un on he end of perod nenory poson. We assume ha he depo has suffcen supply of ems ha can coer all cusomers demands hroughou he plannng horzon. The plannng horzon consders T perods. Transporaon coss nclude f a fxed usage cos per ehcle, whch depends on he perod, and c j a arable ransporaon cos beween and j, whch sasfes he rangular nequaly. The objece s o mnmze he oerall ransporaon, nenory carryng and backloggng coss ncurred oer a specfc plannng horzon. We consder an neger arable x j, whch equals f ehcle raels from o j n perod, and f does no. The amoun ranspored on ha rp s represened by y j. A cusomer, he 589

4 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) A COSTRUCTIVE HEURISTIC FOR THE ITEGRATED IVETORY-DISTRIBUTIO PROBLEM nenory and backorder a he end of me s I and B respecely. The followng s a mxed neger programmng formulaon for he problem. [IIDP] Inegraed nenory dsrbuon problem T Ø ø V œ p () œ œß Mn Œ f x + c x + ( h I + B ) Œ Œº V j j j = j= = = j= = = j subjec o: j= j x =,,, =,, T and =,, V () j x k - x k = l= k l l = =,,, =,, T and =,, V (2) y - q x =,,, j =,,, j, =,, T and =,, V (3) j k= k j y - y =,,, =,, T and =,, V (4) k l= l l V - - B - - I + B + yl - yk d =,, and =,, T (5) = l = k = l k I = Ł ł I C =,, and =,, T (6) I =,, and =,, T (7) B =,, and =,, T (8) y j =,,, j =,,, j, =,, T and =,, V (9) x j = or, =,,, j =,,, j, =,, T and =,, V () The objece funcon () ncludes ransporaon coss and nenory carryng and shorage coss on he end of perod nenory poson. Consrans () make sure ha a ehcle wll s a locaon no more han once n a me perod, and consrans (2) ensure roue connuy. Consrans (3) sere for wo purposes. The frs one s o ensure ha he amoun ranspored beween wo locaons wll always be zero wheneer here s no ehcle mong beween hese locaons, and he second s o ensure ha he amoun ranspored s less han or equal o he ehcle s capacy. Consrans (4) are necessary o elmnae sub-ours. Consrans (5) are he nenory balance equaons for he cusomers. Consrans (6) lm he nenory leel of he cusomers o he correspondng sorage capacy. I s assumed ha he amoun consumed 59

5 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) Abdelmagud, Dessouky and Ordonez by each cusomer n a gen perod s no kep n he cusomer s sorage locaon; accordngly, s no accouned for n consrans (6). Consrans (7) o () are he doman consrans. 3. MOTIVATIG IDEAS AD HEURISTIC DESIG The IIDP s P-hard snce ncludes he capacaed ehcle roung problem (VRP) as a subproblem. In hs secon we presen he key deas behnd he proposed consruce heursc for hs problem. A key decson n solng he IIDP s he amoun delered o cusomer n perod, as hs quany, le us defne by w = V y l - y = l = k = l k k, effecely separaes he roung and Ł ł nenory problems. In fac, gen delery alues w for all cusomers and perods, he nenory and backorder problem s decded by mnmzng he las erm n he objece funcon of he MIP formulaon subjec o consrans (5) o (8). A he same me, he bes roung soluon for hese w s obaned by solng T separae capacaed ehcle roung problems. Each VRP compues he opmal ransporaon coss o deler ( w : = ) W =,..., n perod by solng he followng feasble problem f he delery amouns sasfy = V w q = : TC (W ) = mn V j= = Subjec o: j= j f x j + V = j= = j c j x j x =, and =, V ( ) j x k - x k = l= k l l = =, and =, V (2 ) y - q x, j =,, j and =, V (3 ) j k = k j y - y =,, and =, V (4 ) k l= l l V l - = l = k = l k Ł y y = w y j and j k =, () ł x = or, j =,, j and =, V () Therefore, he key n solng he IIDP s o be able o denfy he opmal delery amouns w snce for a gen alues of he w arables, he IIDP can be separaed no nenory and 59

6 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) A COSTRUCTIVE HEURISTIC FOR THE ITEGRATED IVETORY-DISTRIBUTIO PROBLEM roung problems for whch here exs seeral effcen algorhms. Our proposed heursc bulds on hs obseraon by focusng on how o deermne he w arables effcenly. The procedure used o deermne he w alues mus ake no consderaon he radeoff exsng beween nenory and ransporaon coss. In secon 4 we propose a consruce heursc ha ses he delery amouns by balancng hs radeoff. The dea of he heursc s o esmae a ransporaon cos alue for each cusomer n each perod from an approxmae roung soluon. Acual delery amouns, w, are hen decded by comparng hese ransporaon cos esmaes wh he correspondng nenory coss. Ths process s done sequenally from he frs perod onward and n each perod he comparson of ransporaon and nenory coss s done n wo phases. The frs phase looks no backorder decsons ha are eher mposed by nsuffcen ehcle capacy or preferred due o sangs n ransporaon coss ha are hgher han backorderng coss. The second phase nesgaes nenory decsons ha would coer demand requremens n fuure perods n he case ha excess ehcle capacy s aalable a he curren perod. The heursc looks no nenory decsons ha prode sangs n fuure ransporaon coss ha are hgher han nenory carryng coss. A key sep n hs heursc s o be able o effecely esmae he ransporaon cos of each cusomer. Below we presen a resul ha prodes nsgh no he srucure of he oal ransporaon cos n perod as a funcon of he delery amoun W. Proposon. TC (W ) s a mul-dmensonal monoone ncreasng sep funcon. Proof. Gen ha he defnon of TC (W ) s based on an MIP model for he capacaed ehcle roung problem (VRP) n whch rangular nequaly holds. Sarng from an opmal soluon of a specfc VRP a an nal W = ( w : =,..., ), and by addng ( w : w, = ) D W + =,..., o W (.e. ncreasng he demand alues for a subse of he cusomers) such ha ( w + w ) = V = q, one of wo possble consequences wll occur: ) new arc or arcs wll be added o he curren soluon o sasfy he ehcle capacy consrans (3 ), whch wll ncrease TC (W ) by he correspondng c j and/or f amouns as needed, or 2) he curren VRP soluon remans opmal. Thus TC (W +DW + ) TC (W ) when W +DW + > W. Snce he changes of TC (W ) occur a dscree pons accordng o he ehcle capaces, TC (W ) akes he form of a muldmensonal sep funcon. As a resul of proposon, he soluon scheme can focus only on hose alues of he connuous arables, w, a whch changes o he ransporaon cos occur. We can look a hs resul from anoher perspece. Gen planned delery amouns o cusomers n a perod, by reducng he delery quany of a specfc cusomer, he ransporaon coss wll be reduced a dscree pons and he maxmum possble reducon wll occur when he delery o ha cusomer s dropped o zero. Alhough proposon s proen for opmal soluons o he VRP, hs resul can sll be used for soluons generaed by effcen heurscs as an approxmaon, such as he sangs algorhm [29]. 592

7 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) Abdelmagud, Dessouky and Ordonez 4. THE COSTRUCTIVE HEURISTIC As menoned earler, he consruce heursc s based on he dea of esmang a ransporaon cos alue for each cusomer n each perod, whch s necessary o faclae he comparson beween ransporaon and nenory carryng and shorage coss. We herefore refer o he consruce heursc as he Esmaed Transporaon Coss Heursc (ETCH). In subsecon 4., we descrbe how he ransporaon cos esmaes are ealuaed and connuously updaed hroughou he course of he heursc. Usng hese esmaes, we show n subsecon 4.2 how he nenory problem n IIDP can be decomposed no wo subproblems ha are soled by he heursc n wo phases. The soluon echnques for hese subproblems are llusraed n subsecon Esmang ransporaon coss Le j = W ( ) w be he planned delery amoun for cusomer n perod. For perod n whch w j q V, le W = ( w j : w j = w j, j =,..., ) = ( w : w =, w = w, j =,..., j ) =, j j j. For cusomer whose w >, le. Then, he ransporaon cos reducon ha would resul from reducng cusomer s delery n perod o zero can be calculaed as () TC ( W )- TC ( W ). Snce he ransporaon cos funcon noles he soluon of a VRP, whch s known o be P-hard, may no be possble o calculae s exac alue, especally for large problem szes; nsead, an effcen heursc can be used o approxmae. In our mplemenaon, he sangs algorhm s used for hs purpose. Le ATC ( W ) be an approxmaon for TC (W ) when he sangs algorhm s used o sole he assocaed VRP. The ransporaon cos esmae for cusomer n perod s calculaed as ( ( ) ( ) ( ) ) ETC W = ATC W - ATC W. Howeer, resolng a VRP eery me he ransporaon cos esmae for each cusomer s calculaed may be compuaonally neffcen. Insead, a faser approxmaon scheme can be consruced by ealuang he ransporaon cos sang ha wll resul when a cusomer s remoed from s delery our assgned o n a gen VRP soluon. Ths means ha for gen delery amouns, W, he assocaed VRP wll be soled only once and he resulng ehcle ours wll be used for generang ransporaon cos esmaes. ATC ( W ) and ( W ) ETC are funcons of he planned delery amouns w whch are deermned based on he cusomers ne demand requremens n perod. Howeer, he alues of w mus be defned such ha he ehcle capacy consran, j= w V j q =, s sasfed. Gen he nenory poson a he begnnng of perod, I,- - B,-, and he demand requremens d for all perods, ETCH ealuaes he ne demand requremen for each cusomer, and based on ha esmaes. If he ehcle capacy consran s no sasfed n a gen perod, he w w alues are adjused such ha cusomers wh he lowes un 593

8 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) A COSTRUCTIVE HEURISTIC FOR THE ITEGRATED IVETORY-DISTRIBUTIO PROBLEM shorage coss, p, wll hae par of her demand requremens posponed o fuure perods. The followng ls descrbes he seps of hs approach. Procedure DLV(). Le OC = ordered se of all cusomers n whch cusomers are sored n a non-ncreasng order of her p alues; 2. For eery cusomer OC, le n = I,- B,- ; 3. For perod = o T do 3.. Le Q max = V q = ; 3.2. For eery cusomer OC usng he order n se OC do w = mn(q max, max(d n, )); End-Loop; End-Loop; Q max = max(q max n = n + w, ); w d ; The resulan ( ) W ETC w alues can be safely used n ealuang boh funcons ( ) ATC and W. Durng he course of he algorhm, f a change n he planned delery amouns occurs, a VRP for he perod n whch he change occurred s nsanaed and soled o updae he alues of he ransporaon cos esmaes Problem decomposon and soluon scheme In he ETCH procedure, he comparson beween he ransporaon cos esmaes and nenory carryng and shorage coss s separaed no wo subproblems ha are soled sequenally. Ths comparson s conduced for eery perod sarng from he frs perod onward. The frs subproblem s concerned wh decdng wheher o hae backorders on perod and he second subproblem s concerned wh decdng wheher o use remanng ehcle capacy n perod, f any, o coer fuure cusomer demand. Backorders can be profable for wo reasons; s eher cheaper o pay he backorder cos han he ransporaon cos, or here s nsuffcen capacy n he ehcles o sasfy demand. Le d, = max(d, - I,- + B,-, ) be he ousandng demand a cusomer a he begnnng of perod, and CD be he se of cusomers ha hae d, >. The followng subproblem decdes wheher o deler o cusomer n perod or no (z = or respecely) and he quany r o deler such ha he sum of backorder cos and esmaed ransporaon cos s mnmzed and ehcle capacy consrans are sasfed. [SUB] - Backorder decsons subproblem Mn ATC ( W ) + p ( d, - r ) Subjec o: CD 594

9 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) Abdelmagud, Dessouky and Ordonez r CD r d, V q = (2) = z " CD (3) W = ( w : z = or. w = r, CD) (4) " CD (5) In SUB, he objece funcon s composed of wo pars, an approxmaon of he ransporaon coss n perod and backorder penaly coss. Boh pars are funcons of he decson arables r. Consran (2) ensures ha we do no exceed he oal ehcle capacy, and consrans (3) enforces ha we deler he exac amoun of he ousandng demand only o cusomers ncluded n he delery n perod. Consran (4) defnes he ecor of delery amouns used n approxmang he ransporaon cos funcon. The man oucome from solng SUB s he backorder decsons ealuaed as B = d, for eery cusomer CD ha has z =, and accordngly w =, n he soluon of SUB. The delery amouns, w, for cusomers n se CD ha hae z = n he soluon of SUB are no decded ye as fuure demand requremens may be added. These decsons are nesgaed hrough subproblem SUB2. For eery oher cusomer j ˇ CD, w j =, B j = and I j = I j- d j. Le FD be he se of cusomers ha hae z = n he soluon of SUB. Consder he neger arable u o decde wheher o deler cusomer s demand for perod n he curren perod, where >. Le Q r denoe he oal remanng ehcle capacy,.e. Q r = V = q - r, and CD le T max be he laes perod where cusomer s demand can be consdered whou olang s sorage capacy consran,.e. T max = mn T max max = max( T ). arg max L Ł L d = + C, T ł. We also defne Le w be he planned delery amoun for cusomer n a fuure perod >. The alues of w are nally calculaed usng he DLV(+) procedure as descrbed n subsecon 4. wh a small modfcaon o make sure ha for eery cusomer j FD, nal alues of w j = max d j. If s no possble o achee hs condon n a fuure perod for cusomer j FD, T j s se o -. The w alues for cusomers ha do no belong o se FD are fxed; howeer, he alues of w for cusomers n se FD change wh he change of he u decson arables. The followng problem decdes wheher o nclude fuure demand for any cusomer n he curren delery by mnmzng he oal ransporaon and nenory coss and sasfyng capacy lms. Ths par s formulaed as follows: [SUB2] - Inenory decsons subproblem max T Mn ATC( W ) + [( - ) h d ] = + max T FD = + u 595

10 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) A COSTRUCTIVE HEURISTIC FOR THE ITEGRATED IVETORY-DISTRIBUTIO PROBLEM Subjec o: max T FD = + d u Q r (6) u - u = +,, T max, " FD (7) w = d (-u ) = +,, T max, " FD (8) W = ( w : w = w, =,,) = +,, T max (9) u = or. = +,, T max, " FD (2) Consran (6) represens he aalable ehcle capacy lm. For smplfcaon, he cusomers sorage lms are represened by he me ndex (T max ), whch s compued n adance as descrbed earler. The precedence consrans (7) are added o represen he fac ha fuure demand n a ceran perod s o be consdered only f he cusomer s precedng perod demand s fulflled. Consrans (8) defne he relaonshp beween he fuure planned delery amouns for cusomers n se FD and he decson arables u. SUB2 neglecs he effec of changes n ransporaon cos n perod ha may resul from changng delery amouns n ha perod. Sar se day ndex = Insanae and sole subproblem SUB o decde he delery amouns for cusomers n day and decde wheher backorder decsons wll be made. Is here remanng ehcle capacy n day? o Yes If < T, nsanae and sole subproblem SUB2 o decde wheher o use he remanng ehcle capacy o ncrease he deleres decded n day such ha fuure demand requremens are coered. Usng cusomers decded delery amouns for day, w, calculae he nenory and backorder arables, I and B, and sole a VRP o generae feasble ehcle ours. = + Yes T Sop o Fg.. An oulne of ETCH 596

11 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) Abdelmagud, Dessouky and Ordonez By solng SUB2, he delery amouns for cusomers n se FD can be calculaed as w = r + max T = + d u. Accordngly, he nenory and backorder decson arables n perod can be easly calculaed. Fnally, delery roues n perod are decded by solng a VRP usng he resulng delery amouns. The flow char n Fg. summarzes he major seps of he proposed heursc. The followng subsecon prodes he algorhmc soluons for boh subproblems and her relaed analyses Solng subproblems The wo subproblems are resource allocaon problems n whch he scarce resource s he assocaed aalable ehcle capacy and he man decson arables, z and u, are bnary arables. Accordngly, boh of hem can be soled opmally usng dynamc programmng (DP) as descrbed n [3]. Howeer, wh he ncrease of he problem sze, manly due o he number of cusomers and he plannng horzon, he DP mplemenaons suffer from he curse of dmensonaly. In hs secon, we presen effcen heurscs ha can be used nsead. Frs, we presen he followng resul ha characerzes opmal soluons o subproblem SUB. Proposon 2. There s an opmal soluon o SUB ha only makes deleres o cusomer f he quany delered sasfes r > ETC (W ) / p. Also, eery opmal soluon o SUB only makes deleres f r ETC (W ) / p. Proof. Assume ha n he opmal soluon o SUB, some cusomer s delered r ha ( ) p d - r + ATR W p d + ATR W. If sasfes r ETC (W ) / p, or equalenly ( ) ( ) ( ) we consder he modfed soluon obaned by seng z = r =, hen he preous nequaly shows ha he modfed soluon, whch s feasble, s a leas as good as he opmal soluon. In he case when r < ETC (W ) / p, hen he modfed soluon s srcly beer. Thus, he orgnal soluon canno be opmal. Proposon 2 ges a necessary condon for he opmaly of he delery decson made for a specfc cusomer; howeer, sasfyng hs condon for all cusomers ha hae planned deleres does no guaranee opmaly for he soluon of SUB. Ye, snce backorder decsons are generally no preferable, we wll consder soluons ha hae hs characersc suffcenly good. We desgn he followng algorhm ha ulzes hs rule. Le DL k = {dl: dl CD and dl = CD - k}, where. denoes he sze of a se. We defne f SUB (dl) as he objece funcon alue of subproblem SUB when z = for eery cusomer dl and z j = for eery cusomer j CD - dl, where dl DL k for some k. If he ehcle capacy consran of SUB assocaed wh seng z = for all cusomers n a se dl s no sasfed, we defne f SUB (dl) =. The followng ls descrbes he seps of a breadh-frsbased heursc approach ha searches for effcen soluons o SUB. Procedure SUBALG. Le k = and dl mn = CD;,, 597

12 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) A COSTRUCTIVE HEURISTIC FOR THE ITEGRATED IVETORY-DISTRIBUTIO PROBLEM 2. If f SUB (dl mn ) and r ETC (W ) / p " dl mn hen go o 9; 3. For eery dl DL k ealuae f SUB (dl); 4. Fnd dl s from se DL k ha has he mnmum f SUB (dl) seleced from he members of DL k ha sasfy he followng condons: f SUB (dl) and r ETC (W ) / p " dl; 5. If dl s Ø hen le dl mn = dl s go o 9; 6. Fnd dl * from se DL k ha has he mnmum f SUB (dl); 7. If f SUB (dl * ) < f SUB (dl mn ) hen le dl mn = dl * ; 8. If k < CD hen le k = k +, go o 3; 9. Generae a soluon for SUB n whch deleres are only made o cusomers n se dl mn ; SUBALG ealuaes he f SUB (dl) alue for eery se dl DL k a alues of k=,, CD. If a some leel of k, he condon ha r ETC (W ) / p s sasfed for all dl, we fnd an approxmae soluon and he algorhm ermnaes. Howeer, f seps 2, 4 and 5 are remoed, he algorhm guaranees ha an opmal soluon for SUB has been denfed. perod Cusomers n se FD S 2 S S 2 S 3 S S 22 S 23 S 32 4 S 42 S 43 S 44 Fg. 2. Graphcal llusraon of subproblem SUB2 for a sample case Subproblem SUB2 can be llusraed graphcally. Consder he sample case for SUB2 llusraed n Fg. 2. The decson arables u are represened by dreced arcs, where he cos sang assocaed wh each arc S - = ETC ( W ) - ( - ) h d. A sold ercal lne s drawn o represen he me lm T max for cusomer. Sarng from node, arcs are o be seleced usng he order gen by her drecons, such ha he oal cos sang s maxmzed and he ehcle capacy consran s sasfed. We noe here ha f one or more arcs n a gen perod are seleced, he sang alues S - of he unseleced arcs n he same perod wll be changed due o changes n he ransporaon cos esmaes and herefore hae o be recalculaed. 598

13 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) Abdelmagud, Dessouky and Ordonez Inspred by hs graphcal represenaon, subproblem SUB2 can be deal wh as a precedence consraned knapsack problem (PCKP) n whch he coeffcens of he objece funcon, S -, are dependen on he decson arables. The PCKP s known o be P-hard [3]; howeer, Johnson and em [32] prode a dynamc programmng algorhm for he PCKP ha can sole he problem n a pseudo-polynomal me, gen ha he underlyng precedence graph s a ree, whch s forunaely a propery of SUB2 as can be seen n Fg. 2. We presen here a smpler algorhm based on a greedy search ha selecs he nex possble arc (see Fg. 2) ha has he maxmum pose sang. Ths algorhm does no guaranee opmaly o he soluon of SUB2; howeer, can produce relaely good soluons n polynomal me. The followng seps descrbe he algorhm. Procedure SUBALG2. Le D max = Q r and TD = FD; 2. For eery cusomer n se TD, Le D = ; 3. Fnd cusomer j n se TD ha has he larges pose alue of ETC ( ) - h d ; If none found hen ermnae; ( ) 4. If D max d j, + D hen j Le D max = D max - Add d j j W + D D j j j j, + D j + D j d j, + D ; j, o cusomer j s delery amoun and updae ransporaon cos esmaes n perod +D j ; Le D j = D j +; If D j > T j max hen remoe cusomer j from se TD; End-If Else remoe cusomer j from se TD; 5. If TD = Ø hen ermnae; Else go o sep EXPERIMETATIO AD RESULTS Two ersons of ETCH hae been mplemened. In he frs one, opmal soluons for he wo subproblems are generaed usng a complee breadh-frs search for SUB and a dynamc programmng algorhm for SUB2. We refer o hs mplemenaon as ETCH-O. The second erson uses he proded breadh-frs heursc for SUB and he greedy-search algorhm for SUB2, and s referred o as ETCH-H. These heurscs are programmed and compled usng Borland C++ Bulder erson 3 and benchmarked agans he lower and upper bounds obaned by AM-CEX 8. wh a specfed execuon me lm under an Inel Penum 4 processor runnng wh a clock speed of 2.4 GHz wh GB RAM. 5.. Expermenal desgn We consder wo dfferen scenaros o examne he effeceness of he deeloped heurscs under dfferen crcumsances. These scenaros smulae he negraed nenory-dsrbuon decsons faced by manufacurng companes ha deal wh small number of cusomers, each 599

14 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) A COSTRUCTIVE HEURISTIC FOR THE ITEGRATED IVETORY-DISTRIBUTIO PROBLEM locaed n a dfferen major cy. An example for smlar cases n he leraure can be found n [33]. The frs scenaro s desgned o es he qualy of he nenory holdng decsons of ETCH; whle, n he second one, some parameers are uned o prode condons n whch backorder decsons are economcal, so ha he backorder decsons of ETCH are assessed. The man facors ha are conrolled o produce such cases are he rao of he aalable ehcle capacy o he aerage daly demand by cusomers, he aerage un shorage cos and he ransporaon cos per un dsance. In boh scenaros, cusomers are allocaed n a square of 2 2 dsance uns and her coordnaes are generaed usng a unform dsrbuon whn hese lms. The depo s locaed n he mddle of he square. Cusomers un holdng coss are generaed usng a normal dsrbuon wh a mean of. and a sandard deaon of.2, and each cusomer has a sorage capacy of 2 ems. In he frs scenaro, he ransporaon cos per un dsance s se o, he cusomers un shorage coss are generaed usng a normal dsrbuon wh a mean of 5 and a sandard deaon of.5, and he cusomers demands are generaed usng a unform dsrbuon from 25 o 5 ems per day. In he second scenaro, we se he parameer alues so s opmal o carry backorders. In hs scenaro, he ransporaon cos per un dsance s se o 2, he cusomers un shorage coss are generaed usng a normal dsrbuon wh a mean of 3 and a sandard deaon of.5, and he cusomers demands are generaed usng a unform dsrbuon from 5 o 5 ems per day. For each scenaro, sxy problems hae been generaed by aryng he number of cusomers (), he number of plannng perods (T) and he number of homogenous ehcles (V). We generae hree leels of (5, and 5), wo leels of T (5 and 7), and wo leels of V ( and 2). For each problem seng defned by a combnaon of, T, and V, we randomly generae fe problems. The oal ehcle capacy n he frs scenaro s seleced o be fxed a 5,, and 5 for each leel of, respecely. In he second scenaro, he seleced oal ehcle capaces are 5, 3, and 45. The namng conenon used for he es problems sars wh a number ha refers o he scenaro. Afer a hyphen, wo dgs are assgned for he number of cusomers, followed by a dg represenng he lengh of he plannng horzon. The nex dg represens he number of ehcles. Fnally, he replcae number s gen a he las dg afer a hyphen. Thus, he problem -55- represens he frs run of he frs scenaro wh 5 cusomers, a plannng horzon of 5 perods and ehcle Resuls and dscusson The dealed expermenal resuls are no ncluded n hs paper for he sake of brey. The percenage dfferences beween he oal cos obaned by each heursc and he lower bound are used as performance ndcaors. The percenage dfference, also referred o as opmaly gap, s calculaed by akng he rao of he dfference beween he heursc s oal cos and he lower bound o he lower bound. A comparson agans he lower bound prodes a measure of deaon from opmaly. The CEX upper bound n a maxmum of one-hour 6

15 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) Abdelmagud, Dessouky and Ordonez runnng me s used as an alernae heursc and s percenage dfference agans he lower bound s smlarly calculaed. The resuls of he frs scenaro problems, as shown n Fg. 3, ndcae ha he oal cos obaned by ETCH-O s less han ha of ETCH-H n mos of he cases. ETCH-O generaes he lower cos by generally reducng he ransporaon coss wh a slgh ncrease n nenory holdng coss as compared o ETCH-H. Ths dscrepancy n he cos elemens of boh ersons of he consruce heursc s arbued o he search echnque used for solng he second subproblem. The dynamc programmng par of ETCH-O allows for nesgang larger number of combnaons for allocang he aalable ehcle capacy o fuure demand coerage. Consequenly, ETCH-O nesgaes addonal soluon alernaes ha may hae lower oal cos for he second subproblem. Howeer, solng he second subproblem opmally does no guaranee generang lower oal cos o he whole problem due o he myopc naure of he decsons made for he second subproblem. Ths explans why ETCH-H can generae soluons n a few nsances wh lower oal cos. Wh a smlar argumen, he resuls obaned for he second scenaro problems show ha solng he frs subproblem opmally does no guaranee generang lower cos for he whole problem. Howeer, on aerage, ETCH-O s capable of generang soluons wh lower oal cos. We noe ha hese general fndngs do no change wh aryng problem parameers of, T, or V for boh scenaros. An AOVA analyss showed no effec of, T, or V on he opmaly gaps for he wo heurscs (resuls of he AOVA analyss no ncluded n he paper for brey). We nex nesgae he mpac of he soluon qualy as a funcon of he problem sze n erms of he number of bnary arables. Tha s, for each problem seng, he aerage of he percenage dfferences of he 5 replcaes s calculaed and ploed agans he number of bnary arables of ha seng as shown n Fgs. 3 and 4. Table summarzes he aerage compuaonal me of he deeloped heurscs for each problem se n boh scenaros. As shown n Fgs. 3 and 4, he consruce heursc ouperforms he CEX upper bound for nsances wh more han 5 cusomers n boh scenaros. Whle he growh of he CEX opmaly gap s seady wh he ncrease of he number of bnary arables, he opmaly gap for he deeloped heurscs s below 3% on aerage and remans almos leel wh he ncrease of he number of bnary arables. The ETCH-O erson of he consruce heursc s on aerage 2% closer o he lower bound han ETCH-H. The compuaonal me for ETCH-H s found o be less han one second n all he esed cases. For ETCH-O, due o he dynamc programmng par of he algorhm, he compuaonal me ncreases wh he ncrease of he problem sze; howeer on aerage, has no reached he 9 seconds lm n all problem ses. The ncrease of compuaonal me of ETCH-O s mosly arbued o he ncrease n boh and T; whle, he number of ehcles, V, does no seem o hae a sgnfcan effec on compuaonal me. 6

16 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) A COSTRUCTIVE HEURISTIC FOR THE ITEGRATED IVETORY-DISTRIBUTIO PROBLEM UB LB dff % ETCH-O LB dff % ETCH-H LB dff % o. of bnary arables Fg. 3. Aerage percenage dfferences agans lower bounds for he frs scenaro problems UB LB dff % ETCH-O LB dff % ETCH-H LB dff % o. of bnary arables Fg. 4. Aerage percenage dfferences agans lower bounds for he second scenaro problems 62

17 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) Abdelmagud, Dessouky and Ordonez Table. Aerage compuaonal mes (n mnues) for he deeloped heurscs # bnary Frs scenaro Second scenaro T V arables ETCH-O ETCH-H ETCH-O ETCH-H From he preous resuls we conclude he followng. The ETCH-O erson of he consruce heursc s capable of generang slghly beer soluons compared o ETCH-H wh up o 2% dfference on aerage n he opmaly gap. Howeer, wh he ncrease of he problem sze, especally he number of cusomers and he number of plannng perods, he compuaonal me of ETCH-O wll be sgnfcanly hgher han he compuaonal me of ETCH-H. To nesgae he performance of he deeloped heurscs wh larger problem szes, we consruc an addonal expermenal se based on a hrd scenaro. In hs scenaro, medum ehcle capacy o aerage daly demand rao s used such ha a suaon n he mddle of he frs wo exreme scenaros s addressed. Ths scenaro consders smlar parameers as n he second one wh some modfcaons o reduce he frequency n whch backorder decsons are needed. The man dfference beween he parameers used n he hrd scenaro as compared o he second one s ha he rael cos per un dsance s se o and he cusomers daly demand s generaed usng a unform dsrbuon beween and 25. We consder hree dfferen leels of he number of cusomers, : 2, 25 and 3, and a oal ehcle capacy of 3, 35 and 4 a each leel of respecely. We only consder one leel for boh T and V a 7 and 2 respecely. Fe random replcaes are generaed a each leel of. We use he preously defned namng conenon for he hrd scenaro problems. For he hrd scenaro problems, CEX lower and upper bounds are obaned afer a runnng me of hree hours. Due o he nably of he opmzaon rounes o fnd soluons for he wo subproblems for hese large problem nsances, we only ran he heursc ETCH-H. The aerage cos and me resuls for he ETCH-H erson are shown n able 2. We can see ha he rae of ncrease of he heursc's opmaly gap s almos consan wh he ncrease of he number of cusomers. When we compare hs wh he rapd rae of ncrease for he CEX upper bound percenage dfference, we can see he poenal benef of he deeloped consruce heursc for larger problem szes. In erms of compuaonal me, he ETCH-H erson of he consruce heursc remans below one second for larger problems wh up o 3 cusomers. 63

18 Proceedngs of he nnh Caro Unersy Inernaonal Conference on Mechancal Desgn and Producon (MDP-9) A COSTRUCTIVE HEURISTIC FOR THE ITEGRATED IVETORY-DISTRIBUTIO PROBLEM Table 2. Aerage resuls for he hrd scenaro problems # bnary CEX ETCH-H T V arables UB LB dff % LB dff % Tme (mn) I s mporan o noe ha he aboe expermenal resuls are represenng he specfc algorhmc mplemenaons ha are based on he Sangs algorhm for prodng soluons o he VRP subproblem. More effcen algorhms for he VRP may prode beer soluons as a resul of generang lower ransporaon coss. Inesgang hs ssue s beyond he scope of hs paper. Regardng he CEX runs, was found ha he "presole" heurscs ha are appled a he preparaon sage o generae effcen cus before applyng he branch and bound algorhm play an mporan role n generang effcen lower and upper bounds. Howeer, hese bounds mproe wh a ery slow rae as he branch and bound algorhm proceeds. I was found ha runs longer han he used me lms do no mproe he bounds sgnfcanly. 6. COCLUSIO AD FUTURE WORK Ths arcle addressed he negraed nenory dsrbuon problem n whch mulperod ehcle roung and nenory holdng and backloggng decsons for a se of cusomers are o be made. We consdered an enronmen n whch he demand a each cusomer s relaely small compared o he ehcle capacy, and he cusomers are closely locaed such ha a consoldaed shppng sraegy s approprae. We presened a consruce heursc based on he dea of allocang sngle ransporaon cos esmaes for each cusomer. Two subproblems, comparng nenory holdng and backloggng decsons wh hese ransporaon cos esmaes, are formulaed and her soluon mehods are ncorporaed n he deeloped heursc. The man dea behnd he consruce heursc as seen n he formulaon of he wo subproblems s o consder only delery plans n whch fulfllmen of par of he curren or he fuure demand requremens n a currenly suded perod s no allowed. A mxed neger programmng formulaon s proded and used o oban lower and upper bounds usng AM-CEX o assess he performance of he deeloped heurscs. For small szed problems wh up o 5 cusomers, he expermenal resuls show ha he deeloped consruce heursc can achee soluons ha are on aerage no farher han 3% from he opmal n a few mnues. Wh he ncrease of problem sze, he opmaly gap of he deeloped heurscs ncreases wh almos a consan rae and resuls can be obaned n a few mnues. Ths shows he poenal benef of he deeloped heurscs for larger problem szes. Fuure research may consder buldng upon he algorhmc deas deeloped n hs paper o sole oher problems n he manufacurng ndusry ha hae wder scope n he supply chan. The negraon of producon and nenory-dsrbuon decsons a he operaonal plannng leel s a good example of one such problem. 64

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