THE SINGLE-NODE DYNAMIC SERVICE SCHEDULING AND DISPATCHING PROBLEM

Transcription

1 THE SINGLE-NODE DYNAMIC SERVICE SCHEDULING AND DISPATCHING PROBLEM Leonardo Campo Dall Oro Vale do Rio Doce and Deparmen of Indusrial Engineering Ponifícia Universidade Caólica do Rio de Janeiro Teodor Gabriel Crainic Deparmen of Managemen and Technology École des sciences de la gesion Universié du Québec à Monréal and Cenre for Research on Transporaion, Monréal José Eugenio Leal Deparmen of Indusrial Engineering Ponifícia Universidade Caólica do Rio de Janeiro Warren B. Powell Deparmen of Operaions Research and Financial Engineering Princeon Universiy June 10, 2004

2 Absrac In his paper, we focus on a paricular version of he Dynamic Service Nework Design (DSND) problem, namely he case of a single-erminal ha dispaches services o a number of cusomers and oher erminals. We presen a ime-dependen, sochasic formulaion ha aims o opimize he problem over a given planning horizon, and propose a soluion approach based on dynamic programming principles. We also presen a saic, single-period, formulaion of he single-node problem ha appears as a subproblem when addressing he ime-dependen version and general service nework design cases. Despie is apparen simpliciy, i is sill a nework design problem and eac soluion mehods are no sufficienly fas. We herefore propose wo abu search mea-heurisics based on he ejecion-chain concep. We also inroduce a learning mechanism ha akes advanage of eperience gahered in repeaed eecuions. Eperimens wih problem insances derived from real cases indicae ha he proposed soluion mehods are efficien and yield good soluions. Keywords: dynamic service nework design, sochasic formulaion, approimae dynamic programming, abu search, ejecion chains. Résumé Dans ce aricle, nous eaminons un cas pariculier du problème de concepion dynamique du réseau de service : le cas d un erminal qui envoie des services de ranspor vers des cliens ou d aures erminau. Nous présenons une formulaion dynamique e sochasique qui vise à rouver une soluion opimale au problème pour l ensemble de l horizon de planificaion, ainsi qu une approche de résoluion basée sur des principes de programmaion dynamique. Nous eaminons égalemen la formulaion saique du problème, définie pour une seule période, qui apparaî comme sousproblème dans le modèle dynamique, ainsi que lors de la résoluion de problèmes générau de concepion de réseau dynamiques. En dépi de son apparene simplicié, ce dernier problème apparien à la classe de formulaions de concepion de réseau e les méhodes eaces de résoluion n offren pas de performances saisfaisanes. Nous proposons donc des méa-heurisiques de recherche avec abous, basées sur des chaînes d éjecion. Nous proposons égalemen un mécanisme d apprenissage qui accumule l informaion au cours d eécuions répéées. Les résulas d epérimenaions à l aide de cas dérivés de données réelles indiquen que les méhodes proposées son efficaces e fournissen des soluions de bonne qualié. Mos clefs : concepion de réseau de service dynamique, formulaion sochasique, programmaion dynamique, recherches avec abous, chaînes d éjecion. 2

3 INTRODUCTION Less-han-ruckload (LTL) moor carriers, railways, and ocean conainer lines handle he flows of small shipmens by consolidaing hem on common vehicles ha mus be dispached over ime. In he case of LTL carriers (and o a lesser een rail), i is necessary o dynamically dispach vehicles o move shipmens ha arrive according o a poenially random, and nonsaionary, process. The decision o dispach a vehicle mus rade off he cos of holding freigh ha has already arrived agains he cos of dispaching a vehicle ha is no quie full. When hese dispaches occur in he cone of larger neworks, we mus also consider he cos of moving shipmens from he desinaion of he vehicle o he final desinaion of each shipmen. Service Nework Design (SND) is ofen associaed wih he class of problems ha address he issue of deermining he se of services and characerisics ha opimize hese goals. Dynamic, or ime-dependen, Service Nework Design (DSND) arises when he schedule is a funcion of ime. SND problems are NP-Hard and are generally represened as mied-ineger muli-commodiy, capaciaed, nework opimizaion formulaions (Magnani and Wong 1986; Minou 1986; Balachrishnan, Magnani, and Mirchandani 1997; Crainic 2000). Crainic (2000, 2003; see also Crainic and Lapore 1997 and Crainic and Kim 2004) provides a general view of hese problems and surveys he main mehodological conribuions o he area. Cordeau, Toh, and Vigo (1998) presen a review of he mos recen conribuions dealing wih rain rouing and scheduling wih regard o boh freigh and passenger ransporaion. Chrisiansen e al. (2004) address he same issues for mariime ransporaion. These problems ehibi very weak LP relaaions, which cripple any algorihms ha depend on branch and bound. As a resul, he field is dominaed by various heurisics (e.g., Armacos, Barnhar, and Ware 2002, Barnhar, Jin, and Vance 2000; Barnhar and Schneur 1996; Kim e al. 1999; Crainic, Ferland, and Rousseau 1984; Crainic and Rousseau 1986; Crainic and Roy 1988; Grüner and Sebasian 2000; Powell 1986; Powell and Sheffi, 1989). Dynamic nework design problems are even harder, and as a resul have received even less aenion (e.g., Haghani 1989; Farvolden and Powell 1994; Equi e al. 1997; Gorman 1998). To our knowledge, sochasic versions of he nework design problem have no been addressed in he lieraure. There have been separae lines of invesigaion ino special cases of he nework design problem. There is a hisory of research ino he single-link problem, which involves modeling he conrol of a bach dispach process of a vehicle from one node o anoher. Kosen (1967, 1973) was he firs o consider he conrol of a single server sochasic, bach service problem in seady sae. Ignall and Kolesar (1972), Deb and Serfozo (1973), and Powell and Humble (1986) made furher conribuions o his problem class. Speranza and Ukovich (1992, 1994, and 1996) presen deerminisic formulaions of he single link problem. Burns e al. (1984), Blumenfeld e al. (1985), and Daganzo (1991) invesigae properies of he single link problem in he cone of sraegies in shipper logisics. Berazzi and Speranza (1999) use approimae dynamic programming mehods o solve a discree sae version of he sochasic problem wih a small number of produc ypes. Berazzi and Speranza (2002) presen shipping sraegies for several producs on a capaciaed link, such ha he sum of ransporaion and invenory coss is minimized; The discree and coninuous ranspor frequencies cases are derived from he general analysis framework. Papadaki and Powell (2002) eploi he monoone srucure of he value funcion o esimae a discree version of he value funcion for a single commodiy. Papadaki and Powell (2003) inroduce linear approimaions and nonlinear, concave approimaions and show ha hey produce much faser convergence and can be easily applied o problems wih a large number of shipmen ypes. 3

4 We propose o eend he basic sraegy used in Papadaki and Powell (2003) o he problem of a single node, wih vehicles deparing over a se of oubound links o differen erminals. We assume ha here is a se of shipmens waiing o be dispached o various desinaions ha migh serve as inermediae ransfer poins. Thus, he desinaions of he vehicles will generally be a small subse of he se of desinaions of shipmens. This also inroduces he dimension ha we have o decide which oubound vehicle a shipmen should be assigned o, recognizing ha here are boh immediae and downsream coss o be considered in such a decision. Our sraegy is o formulae a precise version of he problem faced a a single node, using approimaions of downsream coss. This view is a realisic model of he dispaching process of large less-han-ruckload neworks, where erminal managers make decisions using a high level of informaion abou heir own erminal (a he poin in ime ha a decision is made), wih approimaions of downsream impacs (he cos of sending a shipmen wih final desinaion k on a ruck whose desinaion is j). This paper direcly eends his line of research o single-node problems, and represens, we believe, a poenial sepping sone o a sraegy ha can be applied o large-scale neworks. A realisic dispach problem from a single node migh involve dispaching rucks o several dozen desinaions. Given he relaive ineffeciveness of LP relaaions in branch and bound algorihms for his problem class, i is compuaionally inracable o formulae even deerminisic versions of his problem over muliple ime periods. Our sraegy is o solve he single-node, muliple ime period problem using approimae dynamic programming mehods, where he impac of decisions now on he fuure is approimaed using linear approimaions. The use of coninuous approimaions for value funcions has been sudied for years (see, for eample, Tsisiklis and van Roy 1997; Pflug 1996) bu only recenly has been recognized for is abiliy o allow dynamic programming echniques o be applied o high dimensional problems (Powell and Carvalho 1997, 1998; Powell and Topaloglu 2003; Powell and van Roy, 2004), mos recenly in his specific problem class (Papadaki and Powell 2003). Even wih hese echniques, he single-period problem (which mus be solved very quickly) remains compuaionally challenging. A single node a a single poin in ime may face several dozen oubound links, creaing an ineger programming problem wih several dozen ineger variables. Wihou a reasonable bound, branch and bound would virually enumerae he enire ree. Our approimaion sraegy requires ha many of hese problems be solved, so speed is essenial. We herefore propose a abu search mea-heurisic (Glover 1989; Glover and Laguna 1997) based on he concep of ejecion chains. We also inroduce a learning mechanism ha goes beyond he usual wihin-he-search-rajecory memories o ake advanage of eperiences gahered in repeaed eecuions. Eperimens wih problem insances derived from real cases indicae ha he proposed soluion mehods are efficien and yield good soluions. The paper is organized as follows: Secion 1 presens he problem and inroduces he imedependen formulaions and he soluion algorihm. Secion 2 describes he single-period formulaions, he abu search mea-heurisics, and he long-erm learning mechanism. Secion 3 presens he eperimenal seing and he resul analysis. We conclude in Secion 4. 1 THE SINGLE-NODE MULTIPERIOD SERVICE NETWORK DESIGN PROBLEM In he single-node-service design problem, shipmens mus be sen from a erminal o cusomers using he services available a he erminal. Services could be direc or no. Direc services link he origin direcly o he desinaion, while he ohers use inermediae erminals o consolidae cargo. A fied cos mus be paid o operae a given service (o dispach a vehicle), independenly of he acual load. Each direc service has a fied capaciy. Transporaion coss proporional o he ship- 4

5 mens carried may also be involved. Shipmens can be kep a he erminal, incurring a holding cos ha reflecs he service requiremens of each individual shipmen. We mus make he decision a ime o deermine wheher or no o dispach a vehicle o each of a se of inermediae ransfer poins, and which shipmens should go on each vehicle. A shipmen ha is pu on a vehicle ha akes i o an inermediae ransfer erminal also incurs a cos o move he shipmen from he inermediae ransfer erminal o he final desinaion. For his paper, his downsream cos is assumed known. Our goal is o minimize he sum of dispach coss, holding coss a he origin erminal, and he downsream coss for shipmens arriving a an inermediae erminal. Coss are minimized over a finie planning horizon. When he scheduling (or he dispaching) of services is conemplaed, a ime dimension mus be inroduced in he DSND formulaions. This is usually achieved by represening he operaions over a cerain number of ime periods by using a space-ime nework. This increases he size of he problem and also makes i more compuaionally comple, since he model mus incorporae he impac of decisions aken in one period on decisions aken in laer periods. The general approach is as follows. The represenaion of he physical nework, of erminals in paricular, is replicaed in each period. Services are generaed on emporal links. Saring from is origin in a given period, a service arrives (and leaves, in he case of inermediary sops) laer a oher erminals. Each service leg hus represens a service link beween wo differen erminals a differen ime periods. Links beween nodes represening he same erminal a wo consecuive periods are used o model holding decisions. Figure 1 illusraes his concep hrough a nework wih four erminals and several services offered in each erminal. Services S1, S2, and S3 originae a erminal A. Services S1 and S2 are direc, while S3 uses erminal C as an inermediae erminal for consolidaion operaions. Similarly, service S4 iniiaes a erminal B, S5 from C, and S6 from D. Since erminals and services are replicaed over ime, he deparing period is indicaed (e.g., service S1,1 sands for S1 leaving erminal A a ime 1 and arriving a erminal D a period 2). The opion of holding freigh a he erminal is illusraed by he links ha connec he same erminal in subsequen periods. Two ypes of decision variables are usually defined. Ineger design variables are associaed wih each service. Resriced o {0, 1} values, hese variables indicae wheher or no he service will be offered a he specified ime. When several deparures may ake place in he same period, eiher general (nonnegaive) ineger variables are used or he lengh of he period is redefined. Coninuous variables are used o represen he disribuion of freigh flows hrough his service nework. Addiional decision variables may be inroduced o represen he flow of empy vehicles in order o ensure he balance of vehicle flows a erminals. The resuling mied-ineger formulaion minimizes he sum of service fied coss, holding coss, and downsream ransporaion coss. I considers consrains such as flow conservaion a nodes, capaciy on service links, and general rouing and frequency consrains. The formulaions we presen in his paper follow hese general principles. The single-node muliperiod service nework design problem may be viewed as a special case of he general dynamic case wih only one source node. Alhough his seing appears o simplify he formulaion and he soluion mehods, he problem sill ehibis srong combinaorial feaures and mos difficulies relaed o solving nework design problems are also encounered in his case. To simplify he presenaion, we describe he problem and models in erms of less-han-ruckload moor carrier ransporaion. The developmens are general, however, and may be applied o any consolidaion ransporaion case. 5

6 A1 S2,1 A2 A3 A4 A5 S1,1 S4,1 B1 B2 B3 B4 B5 S5,1 C 1 C2 C3 C4 C5 D1 S6,1 D2 D3 D 4 D5 S3,1 Figure 1 A Dynamic Service Nework The problem represens he decision of a dispacher of a ruck erminal a a given momen. The dispacher has o deermine he bes sraegy o move freigh from he erminal o is desinaion, selecing beween i) sending in he curren period, eiher direcly or by using some inermediae erminals, or ii) holding he freigh in invenory for shipmen a a laer period. A fied cos is incurred when a service is seleced (following a dispaching decision). Each service has a finie capaciy represening he maimum load he service can carry. Holding coss apply o freigh kep for fuure periods. We presen boh a deerminisic and a sochasic formulaion of he problem. An algorihm is also described for he laer. The wo formulaions are eperimenally conrased in Secion The deerminisic model The problem is illusraed in Figure 2 for periods = 1,, T. All shipmens are waiing a a single source node (which is no indeed o simplify noaion). Le: K = The se of desinaion nodes for shipmens waiing a ime. Rˆ = The number of new arrivals of shipmens wih desinaion k arriving o he Rˆ k = sysem a ime. ( Rˆ k) k K We define our sae variable using: R k = The number of shipmens wih desinaion k remaining in he sysem a he end of period, afer he vehicles have depared. ( k) R = R k K 6

7 f j1,u j1 {R ik1 } {R ik,+1 } = T Figure 2 - The Single-Node Muliperiod Service Nework Design Diagram The vecor R is known as he pos-decision sae variable (he sae of he sysem afer decisions have been made, hence he superscrip), which plays an imporan role in he developmen of approimaions for sochasic versions of he problem. R is also he vecor of shipmens available a he beginning of ime +1 before new arrivals are added in. We inde he variable by (insead of +1) because, in he sochasic version, i indicaes he informaion conen of he variable. We adop his version of he sae variable here o mainain consisency. The oal number of shipmens ha need o be moved in period is given by he pre-decision sae variable, which we wrie R ˆ = R 1 + R. Shipmens may be moved eiher direcly o heir final desinaions or o an inermediary erminal, which will hen be responsible for he ne dispach. A service is a direc movemen of a vehicle from he source o anoher erminal. Le: J = The se of desinaions o which vehicles may be sen a ime. The decision variables are given by: 1 If a vehicle is dispached from he source node o node j, deparing a ime ; y j = 0 Oherwise. jk = The number of shipmens wih final desinaion k o be moved on a vehicle going o erminal j a ime. Once a shipmen has arrived a he ransfer erminal j, we hen approimae he cos of he remainder of he rip by a linear funcion. Shipmens ha are no dispached o a erminal in J are held. The coss we wish o minimize are given by: f j = The fied cos of sending a vehicle o desinaion j a ime ; g jk = The uni cos of moving a shipmen from inermediae erminal j o final desinaion k when i is dispached o j a ime ; h = The uni cos of holding a origin a ime a shipmen wih ulimae desinaion k. k 7

8 h k The holding coss can be used o capure differences in he value of he shipmens. The invenory equaion can now be wrien as. R = R + Rˆ = R k k, 1 k, jk k jk j J j J Le u j represen he service capaciy o desinaion j deparing a ime. The deerminisic muliperiod node service nework design formulaion may hen be wrien as follows: T Minimize Z(, y) = ( f j yj + g jk jk ) + hk ( Rk jk ) (1) = 1 j J k K k K j J Subjec o: jk k K y j u j j J, = 1,, T (2) ˆ jk Rk, 1+ Rk k K, = 1,, T j J R R Rˆ = +, 2,, k k, 1 k jk j J jk 0 (3) k K = T (4) j J, k K, = 1,, T (5) y = {0,1} j J, = 1,, T (6) j Equaion (1) capures he cos of moving vehicles and shipmens, as well as keeping shipmens in sorage, over he planning horizon. Equaion (2) is he usual linking (or knapsack) resricions on using only open services ha also double as capaciy consrains. Equaion (3) is he demand consrains indicaing he maimum flow possible ou o a given erminal in any period, while equaions (4) capure he dynamics of he sysem. Equaions (5) and (6) are he usual non-negaiviy and inegraliy (on design decisions) consrains on decision variables. The muli-period problem belongs o he family of fied cos, capaciaed, mulicommodiy nework design problems ha is known o be NP-hard and have proved difficul o solve for realisic problem insances. Consequenly, heurisics are used in mos cases and, as illusraed in Secions 2 and 3, his soluion sraegy is also used here. Load arrivals are generally no deerminisic, however. Consequenly, we reformulae he problem o reflec he sochasiciy of he demand and derive a soluion sraegy based on approimae dynamic programming echniques. The resuling procedure requires solving several insances of he deerminisic single-period varian of formulaion (1) (6). Secion 1.2 is dedicaed o he presenaion of his model and he corresponding soluion mehod. 1.2 The sochasic problem Compared o he previous deerminisic model, he new formulaion considers eplicily he arrival over ime of new informaion. The mos common form of new informaion is he cusomers arriving o be served, bu we can also allow for new informaion abou ransporaion coss and holding coss. In ransporaion, holding coss ypically represen a penaly ha measures he qualiy of service. For eample, he carrier could ge a reques o rush a shipmen, or learn ha a shipmen is acually 8

9 arriving early (which can be a problem in some siuaions). We solve he resuling sochasic model by formulaing a sochasic opimizaion problem over ime, which we can approach using dynamic programming. Our developmen follows and generalizes he work of Papadaki and Powell (2002, 2003). We le be he vecor of random variables represening all he informaion arriving during W W ime inerval. includes informaion abou new cusomer arrivals and all coss, and can be wrien as W ( ˆ = R, f, g, h), which means ha he arrivals Rˆ and coss ( f, g, h ) firs become known W during ime inerval. represens eogenous informaion. We now adop he convenion ha any variable indeed by ime is a funcion of he informaion up hrough ime. This means ha a ime, any variable indeed by is deerminisic, while any variable indeed by > is random. Le Ω be he se of elemenary oucomes of he informaion process, wih ( W ( )) T ω = 1 represening a sample realizaion of he informaion. To define a formal probabiliy space, we le F be he se of oucomes of W (more formally, he σ algebra generaed by W ) and le P be a probabiliy measure on ( Ω, F ). Then, our probabiliy space is ( Ω, FP, ). We would hen le be he σ algebra generaed by he hisory( W s ) s = 1. By consrucion, all funcions indeed by are - measurable. Our problem consiss of deermining he dispach policy ha minimizes he oal epeced coss over ime, ha is, he policy ha achieves he bes rade-off beween he dispach and he invenory coss. Le C(, y R ) be he one-period cos funcion given he pre-decision sae R : C (, y R ) = f j yj + g jk jk + (7) hkrk j J k K k K Noe ha he cos funcion depends on he pre-decision sae variable R, bu he holding coss are assessed only on he shipmens lef over a he end of he ime period, given by R. Decisions (,y) are made according o a policy π. We represen he decision funcions using: π X ( R ) = The flow decision funcion, which reurns a ime given sae R; π Y ( R ) = The dispach decision funcion, which reurns y a ime given sae R. Le Π be our family of decision funcions. The objecive funcion can now be formulaed as: T π π π F ( R) = E C' ( X', Y' R' ) ' = The opimizaion problem, now, is o find he bes policy, which is o say ha we wish o solve: * π F0 = inf F0 ( R 0 ). π Π { } Given he sheer compleiy of solving his problem eacly, we resor o a search for good policies. We sar by wriing he problem in erms of he Bellman opimaliy equaions. Given a pol- F F 9

10 . We ne de- icyπ, each oucome ω Ω produces a specific sequence of sae variables( R ( ) T ω = fine: π V ( R ) = The fuure epeced coss given ha we sar in period in sae R and follow policy π unil he end of he horizon. Since we are using he pos-decision sae variable, he value funcions V using a nonsandard form of he Bellman equaions: π { ( ) y, } π π ) 0 ( R ) are defined V ( R ) = E min C ( y, R ) + V R ( y, ) R (8) The value funcion V π is oo comple, however, and canno be compued eacly. The challenge is o find an approimaion V( R ) ha is compuaionally racable and sill represens in a saisfacory way he impac of he decisions aken in he period of decision over ime. If we are in sae R 1, we can sample ω (which deermines R ˆ as well as he coss in period ). Then, using an approimaion V ( R ), we can find he decisions in period using: π ( ˆ π ˆ Y ( R 1, R( ω)), X ( R 1, R( ω)) ) = argmin C( y, R ( ω) ) + V ( R ( y,, ω) ) R is a deermi- A criical feaure of (9) is ha given ( R ˆ 1, R( ω) ), he pos-decision sae variable nisic funcion of and y. y, (9) We see, hen, ha an approimaion V ( R ) deermines a policy. Papadaki and Powell (2002, 2003) evaluae hree ypes of approimaions when R is a scalar: discree, linear, and nonlinear (concave). If R is a scalar, equaion (8) may be solved opimally. The resuls show ha he linear and nonlinear approimaions converge much more quickly han he discree approimaion, wih resuls wihin one o wo percen of opimaliy. The nonlinear approimaion ouperforms he linear, bu only afer hundreds of ieraions. The linear approimaion works well because he funcion V( R ) is monoone. No surprisingly, he linear approimaion works especially well when he underlying problem is sochasic. For our applicaion, we will generally be able o run only a few dozen ieraions, and hence he linear approimaion offers no only simpliciy, bu also works he bes among he approimaions ha have been esed. Equaion (9) also illusraes he imporance of using he pos-decision sae variable for sochasic problems. When When R is a scalar, he epecaion in equaion (8) is generally easy o compue. R is a vecor, he epecaion is compuaionally inracable. In equaion (9), we ook a sam- ple realizaion and compued( ˆ Y ( R 1, R( )), X ( R 1, Rˆ ω ( ω)), where he decisions are allowed o π π ) see R ˆ = R 1 +R. Had we used a pre-decision sae variable, he same rick would have required finding he decisions in period allowing he decisions o see R + 1, which is a violaion of he informaion consrain on he decision funcion. This rick of solving approimaions of he Bellman equaions by formulaing he opimaliy equaions around he pos-decision sae variable, combined wih he use of appropriae funcional forms for he value funcion, has been he basis of work on large-scale flee managemen problems (Godfrey and Powell 2002a, 2002b) and muliproduc bach 10

11 Le { } k k service problems (Papadaki and Powell 2003). The general idea is summarized in Powell and Topaloglu 2003 and Powell and van Roy v = v be he vecor of smoohed esimaes of he uni value of shipmens for desinaion k a ime before new arrivals. A linear approimaion of he value funcion can hen be wrien ˆ ˆ V( R 1, R( ω)) = vr = vkrk = vk Rk, 1+ Rk, ( ω) jk (10) k K k K j J Subsiuing (10) ino (9) gives π ( ˆ π Y ( ˆ R 1, R( ω)), X ( R 1, R( ω)) ) = arg min C ( y, R ( ω) ) + v R + Rˆ ( ω) k 1 k, jk y, k K j J = arg min C y, R ( ) vk jk + vk R + Rk ( ) y, k K j J k K ( ω ) ( ˆ 1, ω ) (11) The las erm on he righ hand side of (11) is a consan wih respec o and y, and hence can be ignored. The middle erm is linear in. Combining he definiion of C y, R ( ω ) in equaion (7) wih equaion (11), and dropping he consan erm, gives: ( ) Vˆ ( R, Rˆ ( ω)) = min f y + ( g v ) + h R (12) 1 1 j j jk k jk k k y, j J k K k K where we le Vˆ ˆ 1( R 1, R ( ω) ) represen a sample esimae of he value funcion. The challenge now is o devise a sraegy o updae v in such a way ha he formulaion for he dynamic problem can be solved saisfacorily. Le vˆ be a sample esimae of he slope of V in a paricular ieraion. If he value funcion were differeniable, we could wrie: vˆ V k = (13) Rk We use he noaion ha v is a smoohed esimae of he slope, which combines esimaes over muliple ieraions, while vˆ is a sample esimae obained a a paricular ieraion. For our seing, we use a finie difference approimaion of he derivaive: vˆk ˆ ˆ [ ( ) ˆ v V R + R V ( R )]/ R (14) k k k The slope capures he impac on he value of he objecive funcion of having in invenory a he beginning of period (before he arrival of he new shipmens) one era uni of flow wih desinaion k. By using he definiion of he value funcionv, one obains 11

12 where vˆ k f j yj + ( g jk vk ) jk 1 j J k K = Rk ˆ h 1 ( ) ˆ k K k R Rk Rk jk jk R 1 Rk jk j J j J (15) and y = y ( R + R ) y ( R j j k k j k = ( R + R ) ( R ). jk jk k k jk k R k The soluion algorihm is schemaically displayed in Figure 3. The iniializaion phase ses slope esimaes and indices o 0 and akes a single sample of demands (shipmens) a all periods of he planning horizon. The algorihm hen proceeds ieraively in a series of alernaing forward and backward passes unil some sopping crieria are saisfied. Each ieraion is composed of a forward pass and a backward pass. In he forward pass, he deerminisic muliperiod service selecion and flow assignmen problem is solved from ime = 0 o he end of he planning horizon ( = T), using he evaluaions of he shipmen values a nodes updaed during he previous backward pass. The problem is solved using he mehod presened in Secion 3, which is based on he mea-heurisic sraegy deailed in he ne secion. The backward pass proceeds from he end of he planning horizon o is beginning. I compues new shipmen value approimaions a nodes (slope values) using epression (15) for a perurbaion of one uni of he invenory available a each node and ime period for each desinaion ( = 1) and he updaed dispaching and rouing decisions yielded by he previous forward sep. New ses of flows (y j, jk ) and shipmen value approimaions ) are hus obained a each ieraion n. Since he values vˆk are sample esimaes, we approimae heir epecaion by performing he sandard smoohing operaion: v = (1 γ ) v + γ n v ˆ n, 0 γ n 1. (16) n n n 1 k k k vˆk 2 DETERMINISTIC SINGLE-PERIOD, SINGLE-NODE PROBLEM The general sraegy used o address he muliperiod formulaions of he previous secion decomposes he problem by period and solves a sequence of deerminisic single-period, single-erminal nework design problems. The problem belongs o he class of capaciaed, mulicommodiy nework design problem and is known o be NP-Hard. Moreover, i has o be solved repeaedly and a very efficien soluion mehod is required. In fac, good soluions obained very quickly are of greaer ineres han opimal ones obained a greaer ime cos. Opimaliy in his sense is a bonus, no an objecive per se. We propose herefore an efficien abu search mea-heurisic procedure. We firs sae he mahemaical formulaion, followed by he descripion of he ejecion-chain neighborhood we propose, and, finally, we inroduce he complee abu search mea-heurisics. 12

13 Sep 0 Iniializaion Se v = 0, for all k and. Le n = 1, = 0. 0 k Sep 1 Forward Pass Draw a sample realizaion ω ha deermines he oucome of he informaion process a each ime period. For 1 o T Solve he single-node, single-period deerminisic service selecion and demand rouing problem using he mehod of Secion 2.3. This yields new values for (y j, jk ). Sep 2 Backward Pass For T o 1 For k 1 o K Compue a new sample gradien ˆn (using (14) and (15)). Use (16) o smooh he vˆn k v k values o obain updaed esimaes of v. Sep 3 Sop If he sopping crierion is reached STOP, oherwise, se n n + 1 and reurn o 1. k Figure 3 Algorihm for he Sochasic Muliperiod Problem 2.1 The single-period formulaion The nework of he saic, single-period problem is rooed a node (erminal) i, in period, and ieraion n. I is denoed by G = N, A and represens he decision of a carrier, given a momen n n n in ( ) i i i ime and he availabiliy of a cerain ype of informaion. Figure 4 illusraes he concep. To simplify he noaion in his secion, we drop he indices n,, and i ecep when necessary o avoid ambiguiies. Following he noaion of he previous secion, demand is defined as he volume (number of shipmens) R k o be sen from he origin node o a desinaion k K. Nodes in K may be erminals or concepual nodes creaed by aggregaion. Le R = k K R k be he oal demand, in number of shipmens, a node i. To reach a desinaion, a firs movemen (dispach) is performed o one of a se of adjacen nodes j J N. The roo erminal is conneced o he erminals in J by a se of service design arcs (i,j) ha represen he possible services a he erminal. From each j J, one can reach a desinaion k K by using a ransporaion arc (j,k) ha represens a direc link or a pah linking he erminals. Transporaion links capure he fuure periods effec of he muliperiod formulaion. Holding arcs (i,k) direcly link he roo erminal o each desinaion k and capure he decisions o hold shipmens for dispach in laer periods. Thus, N = {, i J, K} and A = {( i, j),( j, k),( i, k) j J, k K}. 13

14 A fied cos f j and a capaciy u j are associaed o each design link. A cos g jk is associaed wih ransporaion links, which represens he cos of sending one uni of flow o a desinaion k hrough an inermediae erminal j. When he inermediae node is he final desinaion, g jj = 0. There is always he opion of no moving he freigh a his momen and his opion implies a holding cos h k. The decision variables represen wheher or no a ruck is o be dispached o j (or, equivalenly, he uilizaion of he inermediae erminals), and he rouing of freigh o each desinaion k: n Design (dispaching) variables: y j ( ), y j = 1 if node j is used (a vehicle is dispached for i o j) y ij and 0 oherwise. We assume in his paper ha he lengh of he ime period is se in such a way ha only one deparure owards each erminal j is possible a each period. n Freigh flows: volumes sen from i o k hrough j: jk ( ). ijk Design Link J Transporaion Link K f j,u j {R k } hg j k Holding Arc h k Figure 4 - Saic, Single-Period Single-Node Service Nework Design Problem The general formulaion of Secion 1 hen becomes Minimize Z(,y) = ( f y + g ) + h ( R ) (17) j j jk jk k k jk j J k K k K j J Subjec o: jk yu j j j J (18) k K jk Rk k K j J (19) 14

15 0 j Jk, K (20) jk y = {0,1} j J (21) j p pahs are all made ou of wo arcs: { } ijk where consrains (18) are he usual linking (or knapsack) resricions on using only available services, while equaions (19) are flow (or demand) consrains. Noice ha for each desinaion k, one can eplicily enumerae all he pahs from i o k hrough nodes j. These p ijk = ( i, j),( j, k). The cos of sending a full ruck on each of he pahs is given bycp = f ijk j + g jkuj. I is hen clear ha for each desinaion k, here is a preferred roue ha ensures he smaller cos for moving a full ruck. Therefore, when R k u j, R k / u j rucks will be dispached on he preferred roue and only roues for he re- R R / u ruck-equivalens will have o be deermined. In he res of his paper, we maining k k j assume herefore ha R k represens his remaining quaniy (and hus R k < u j ). Noice also ha he objecive funcion can be recas as: Cy (, ) = cy+ ( g h) + hr. j j jk k jk k k j J k K j J k K I is hen clear ha for all k such ha g jk > h k, C(,y) will be minimized for jk = 0. Thus, one has only o consider pahs for which he ransporaion cos is less han he cos of do nohing (hold) opion for he paricular desinaion. I is also ineresing o noe ha he pah-based and he arc-based formulaions of he nework design model are he same, and ha he problem may also be saed as a capaciaed, mulicommodiy node design (locaion) model (Daskin 1995; Labbé and Louveau 1997). Three heurisics have been developed o address his problem and are compared in Secion 3. All hree are ieraive improvemen procedures based on moving hrough a search space defined by he coninuous ransporaion variables. The firs heurisic acceps only changes ha improve he objecive funcion. The oher wo are abu search mea-heurisics. These sraegies are described ne, following he presenaion of he ejecion-chain neighborhood used by all procedures. 2.2 Ejecion chain neighborhood and greedy descen procedure The search sraegies we eamine proceed in he space of he coninuous flow variables according o a neighborhood defined by ejecion chains (Glover and Laguna 1997; Rego and Roucairol 1996). An ejecion chain combines simple moves o produce a composed movemen. The simple move generally modifies he aribues of a single elemen of a soluion. If he resuling soluion is feasible, he procedure sops. Oherwise, he aribues of anoher soluion are modified o accommodae he previous modificaion. And so on, unil a cerain sopping crierion is me. Iniial applicaions were dedicaed o vehicle rouing problems where he simple move consised in moving one cusomer o a differen roue and, evenually, ejecing anoher cusomer from he receiving roue o make room for i. The applicaion of he ejecion-chain idea o he single-node service nework design problem is o consider he shipmens as he individual soluion elemens and heir assignmen o services as he aribue o modify. Thus, he chain is iniiaed by selecing a load and moving i o anoher service. If he capaciy resricion of he new service is sill verified, he procedure sops. Oherwise, a load previously on he receiving service is seleced and moved (i is ejeced) o anoher service. And so on, in a sequence of move a load from one service o anoher and, evenually, ejec anoher load 15

16 o make room for i. A limi is imposed on he maimum size of an ejecion chain. When his limi is reached wihou reaching feasibiliy, he load is moved o he holding arc. Figure 5 illusraes an ejecion chain. The upper lef figure represens a feasible soluion. The figures in he sequence show he reacion caused by a move of a load from one service o anoher. The values wihin square parenheses represen he size of he shipmens, while values wihin parenheses represen arc capaciies. In his eample, he chain sops by moving a load o he holding arc. [10] ( ) [10] ( ) [10,15] (30) [15] ( ) [10] (30) [12] (15) [12] ( ) 1 [12,15] (15)* *Unfeasible [12,15] ( ) [8] (15) [8] (15) [8] ( ) [8] ( ) [0] ( ) [0] ( ) 2 [10] ( ) [10] ( ) [10] (30) [10] (30) [15] (15) [15] ( ) 3 [15] (15) [15] ( ) [8,12] (15) *Unfeasible [8,12] ( ) [12] (15) [12] ( ) [0] ( ) [8] ( ) Figure 5 - Illusraion of an Ejecion Chain A cos-based greedy procedure is used o deermine an ejecion chain. The firs move in he chain corresponds o a load on he pah wih he highes uni cos. The load is moved o is minimum uni cos pah. Because one of he goals of he procedure is o reduce he number of services used (empy some and load as much as possible he ohers), he iniial load has he highes volume among hose on he service, while he loads ejeced in subsequen moves have he lowes volume among hose on he ejecing service. Figure 6 displays he main seps of his greedy sraegy. 16

17 Sep 1 Sor pahs in decreasing order of uni cos Sep 2 Selec he maimum volume load on he pah wih he highes uni cos Ejec his shipmen and idenify i as he firs of he chain Sep 3 While (chain no compleed) or (maimum chain size no reached)) Do Find he minimum uni cos pah for his shipmen If (residual capaciy of he service size of shipmen) Then The chain is compleed Else Idenify he lowes volume shipmen on he receiving service Ejec his load and add i o he chain Sep 4 If (chain reached he maimum size) Then Pu he las ejeced shipmen on is invenory arc End Figure 6 Greedy Algorihm o Deermine an Ejecion Chain A Simple Heurisic (SH) ha greedily acceps an ejecion chain ha improves he curren soluion can be described as follows. Given a feasible soluion, deermine an ejecion chain using he procedure described above. If he new soluion is beer han he curren soluion, implemen i; oherwise, deermine a new ejecion chain (saring wih he ne load in decreasing order of volumes on he pahs ordered by decreasing uni coss). Coninue unil an improving soluion is found (or a ime limi is reached). I is clear ha he procedure will sop wih he firs local opimum encounered, irrespecive of he qualiy of he soluion. The abu search mea-heurisics described ne aemp o overcome his shorcoming. 2.3 Tabu Search Mea-heurisics Tabu search is a mea-heurisic ha has been successfully used o address many hard, combinaorial opimizaion problems, including nework design (Glover 1986; Glover and Laguna 1997; Crainic, Farvolden, and Gendreau 2000; Ghamelouche, Crainic, and Gendreau 2003). Tabu search is an ieraive procedure ha uses memories of soluions (or aribues hereof) already visied o learn abou he soluion space and o guide he search ou of local opima and owards promising regions. One or several neighborhoods are defined, ogeher wih he corresponding moves. Then, a each ieraion, soluions neighboring he curren soluion are idenified and he bes, no necessarily improving, one is seleced as he new curren soluion. A shor-erm memory, usually idenified as he abu lis, is used o avoid cycling by forbidding moves o soluions recenly visied or sharing aribues wih such soluions. More advanced memory and neighborhood srucures may be advanageously used o aemp o reach higher qualiy soluions. In he presen case, however, speed is of essence and we do no implemen hese feaures. Eperimenal resuls show, however, ha he simple abu search sraegy we propose sill achieves good qualiy soluions. 17

18 The abu search procedures we propose are based on he ejecion-chain neighborhood defined previously (see Dall Oro 2001 for deailed descripions of hese procedures). Soluions are evaluaed using he objecive funcion of he design problem, which is also used o define an aspiraion crierion: a move o a abu soluion is acceped if he new soluion has a lower objecive funcion value han he curren bes. Once an ejecion chain is implemened, i becomes abu (one canno use i again) for a given number of ieraions. The induced neighborhood is huge, however, since in heory, one should consruc all ejecion chains before selecing he soluion o move o. This is impracical in any circumsance. We hus adop a well-known echnique in mea-heurisics in general and abu search in paricular, and use a candidae lis. A candidae lis is a collecion of neighbors among which he selecion of he ne soluion is performed. We define wo abu search procedures ha differ in how he candidae lis is buil: Search Firs Feasible Chain (SFFC) and Search Firs Improving Chain (SFIC). The SFFC sraegy builds ejecion chains saring from he curren soluion and acceps he firs feasible soluion ha eiher is no abu or fulfills he aspiraion crierion. Noice ha non improving soluions are acceped. Therefore, if afer a cerain number of ieraions, neiher he bes soluion nor he curren soluion are improved, he procedure reurns o he curren bes soluion and proceeds wih he search. If he search reurns o he same soluion several consecuive imes, he search is sopped. A broader par of he neighborhood is eplored by he SFIC procedure. Here, ejecion chains are buil unil a feasible soluion is found ha improves he curren soluion. If no chain improves he curren soluion, he one ha leas degrades he soluion is implemened. If, afer a cerain number of ieraions he curren soluion is sill no improved, he procedure reurns o he bes curren soluion and he search coninues on he second bes chain for ha soluion. The process coninues unil i reurns a cerain number of imes o he same bes soluion. 2.3 Iniial soluion Irrespecive of he sraegy seleced, an iniial feasible soluion has o be idenified. The heurisic we use sars by soring all pahs in increasing order of cos. I hen proceeds ieraively hrough he shipmens and aemps o send every load on is shores (cheapes) pah. The ne pah in he lis is used when he capaciy of he pah is reached. This heurisic is easy o implemen and fas. Moreover, as illusraed by he compuaional resuls of he ne secion, he abu search procedures obain good soluions in shor compuaional imes. Shipmens are assigned o and moved among services based on he pah valuaions ha reflec boh he fied and he variable coss of he sysem. The fied service cos has o be paid as soon as he service is seleced. In order no o penalize he shipmens ha are assigned firs o a previously unseleced service, approimaions are used. Thus, o compue he iniial soluion, we associae he fied cos o he capaciy of he service, which yields he following pah cos: f j c p ijk = + g jk u j Once he iniial soluion is found, he uni cos of a pah using seleced services is compued in he usual way: f j c p ijk = + g jk j 18

19 On he oher hand, unused arcs are closed (y ij = 0) during he search. Then, o evaluae he cos of sending a load on a pah ha makes use of a no ye seleced service, he corresponding arc is given a cos equal o he maimum cos value compued for ha arc during he previous ieraions, including hose of he iniial soluion. 2.4 Long-erm memory As already noiced, he single-period, single-erminal problem has o be solved repeaedly when DSND problems are conemplaed: a each ime period and forward-backward ieraion. This may be eremely ime-consuming as illusraed in Secion 3. Moreover, i is fel ha he behavior of a erminal, he number of poenial service deparures, as well as he composiion and ampliude of demand and, hus, he acual dispach decisions, will no vary widely from period o period or from one ieraion o he ne. Some order eiss in he randomness of demand arrivals and he operaions of he sysem. Observaions of operaions of acual ransporaion sysems end o suppor his hypohesis. Then, if such a behavioral paern could be idenified for a erminal, i could be used o accelerae he search by, for eample, idenifying he preferred service for shipmens deparing o cerain final desinaions. This should help o reduce significanly he compuaional ime. Our sraegy is based on long-erm memories ha record, in a frequency mari, he frequencies of services used o move each demand o is final desinaion. This mari is buil during he eecuion of he algorihm. Once he saic problem is solved for he curren period, he mari is updaed by increasing he frequencies of he services used. Afer a few ieraions, he dispach paern for each demand should sar o emerge as he services mos heavily used. We will use his paern o bias he selecion of services when iniial soluions need o be compued. This approach is somewha similar o he arge analysis mehodology proposed iniially by Glover (see Glover and Laguna 1997). 3 EXPERIMENTAL RESULTS The eperimenal phase of our work was planned o eamine several issues: 1. Wha is he performance in erms of compuing efficiency and soluion qualiy of he heurisics for he saic, single-period problem? Which one is beer? Recall ha his problem has o be solved repeaedly when he dynamic problem is addressed, and ha he number of repeiions is epeced o ge huge when full dynamic service nework design problems are conemplaed. 2. How efficienly can dynamic problems be solved? Is he long-erm memory mechanism (he mari of frequencies) helpful? 3. Wha is gained by eplicily considering he dynamics of he sysem compared o simply solving a series of single-period saic insances? 3.1 Eperimenal Design We esed he models on problem insances based on daa from a large U.S. LTL moor carrier, operaing numerous breakbulk and end-of-line erminals, ha serve he coninenal Unied Saes. We used his informaion o se he general parameers of our problem, bu used randomly generaed daa o conrol he characerisics of individual problem insances. Three classes of problems were generaed, each wih a differen demand-o-capaciy raio. Class I problems have more demand han he capaciy of he nework, while he conrary is rue for problems of Class III. Problem insances in Class II display a balance beween demand and capaciy. Problem insances were generaed randomly, by using uniform disribuions on he inervals presened in Table 1 for capaciy and demand, as well as for service (fied), ransporaion, and holding 19

20 coss. Problems of four differen sizes, small, medium, large, and era large, were generaed as defined in Table 2. Dynamic problem insances were generaed following a similar process bu generaing a differen vecor of demand for each period of a 7-day planning horizon. Problem Class Table 1 General Characerisics of Tes Problem Insances Capaciy Demand Fied Cos Holding Cos Transporaion Cos Class I [50,150] [150,300] [50,300] [100,300] [20,120] Class II [150,400] [130,260] [50,300] [100,300] [20,120] Class III [150,450] [30,60] [50,300] [100,300] [20,120] Problem Size Table 2 Tes Problem Insances Dimensions # of Terminals # of Inermediae Terminals # of Inermediae for each Desinaion Small [10,20] [3,8] [2,5] Medium [21,100] [10,50] [4,12] Large [101,250] [50,100] [8,20] XLarge [260,450] [100,200] [16,50] For he sake of conciseness and clariy, only aggregaed resuls are included in he paper. Deailed resuls are presened in Dall Oro (2001). 3.2 Saic Eperimens As indicaed previously, he soluion procedure for he single-period, single-erminal service design problem mus be very efficien. Of course, he qualiy of he soluion is also of major imporance. A number of parameers characerize he heurisics and impac heir performance. These parameers mus be calibraed prior o underaking he performance analysis. The calibraion phase has been conduced using 250 problem insances (small, medium, and large of all ypes), on a PC equipped wih a Penium III processor wih an 800 MHz clock. Lindo 6.1 was used o obain he opimal soluions. Five parameers characerize he heurisics. The values esed for each one of hem were: 1. Maimum number of ieraions he procedure was allowed o run (if no sopped oherwise). We esed values equal o 100, 200, 300, and 500 for HS, and 500, 1000, 1500, 2000, 2500, and 3500 for he SFFC and SFIC abu search procedures. 2. Maimum size of he ejecion chain. This parameer limis he number of load echanges allowed when ejecion chains are buil. Values equal o 4, 6, 8, 10, and 12 were esed. 3. Tabu lis size ha indicaes he number of ieraions a soluion is agged abu, e.g., one canno reurn o i unless i saisfies he aspiraion crieria (beer han he curren bes). This parameer aemps o avoid cycling and o foser a broader eploraion of he soluion space. Tabu lis lengh of 5, 10, 15, and 20 were esed. 20

21 4. Maimum number of reurns o he same soluion before sopping he procedure. When his limi is aained, i is an indicaion ha, given he seings of he oher parameers, he search has reached a local opimum from which i is incapable o escape. We esed 4, 6, 8, 10, and Maimum number of successive ieraions wihou improvemen. When he search reaches his limi, i reurns o he curren bes soluion o iniiae a new search hread. We esed 3, 5, and 10. The procedures and parameer seings were compared according o he percenage of deviaion from he opimal soluion and he compuaion ime required. The general conclusion of hese eperimens is ha he procedures are robus wih respec o he parameer seings. For each heurisic, he resuls for differen parameer seings are very similar, he calibraion appearing more as fine uning of parameers, raher han a discriminaion procedure. As epeced, abu search ouperforms greedy descen in soluion qualiy, he procedure using Firs Improvemen (SFIC) reaching higher qualiy soluions han he one (SFFC) ha moves o he Firs Feasible soluion. Again, wihou surprise, he relaion on compuing imes displays eacly he opposie characerisics. In fac, he number of ieraions has no impac on he performance of he greedy HS heurisic, provided i is sufficienly long o allow he procedure o reach a local opimum. This is no rue for he abu search procedures ha generally obain beer soluions when allowed o search longer. Ye, he rae of improvemen slows down afer a cerain poin ha we fied empirically o 2500 ieraions. We somehow epeced o find a relaionship beween he lengh of he ejecion chains and he problem dimensions. We did no find such a relaionship in our eperimens, however. We raher idenified a relaionship beween he ejecion chain lengh and he ype of soluion mehod. The oher hree parameers do no apply o he HS procedure. For abu search, he duraion of abu ags is generally an imporan parameer. This urned ou o be he case here as well, he abu lis lengh impacing boh he soluion qualiy and he compuaional imes. The las 2 parameers, he maimum number of reurns o he same soluion and he maimum number of successive ieraions wihou improvemen, did no impac he soluion qualiy in any significan way. They did influence compuing imes, hough. The final parameer seings appear in Table 3 (an X indicaes ha he parameer is no used by he procedure) and were used o perform he comparison analysis of he hree heurisics presened ne. A new se of 100 problems of all ypes and dimensions were generaed for he comparaive analysis of he proposed procedures. We generaed a larger number of Class II problems, where demand and capaciy are balanced, because hey appear more difficul o solve han Class I (he soluion consiss in using all services and holding he ecess demand) and Class III (where he preferred service is used for almos all demands) problem insances. Table 3 The Parameers for he Soluion Sraegies Parameer SH SFFC SFIC Ma. ieraions Ma. chain size Tabu lis size X Ma. reurns o he same soluion X 8 4 Ma. ieraions wihou improvemen X Opimal soluions have been obained by using CPLEX 7.1. All eperimens have been performed on a SUN Worksaion. The resuls are summarized in Tables 4, 5, and 6. We indicae average resuls for he percenage of opimaliy reached (a value of 100 indicaes ha opimal soluions 21