An Integrated Matching and Partitioning Problem with Applications in Intermodal Transport

Transcription

1 2015 IEEE Symposium Series o Computatioal Itelligece A Itegrated Matchig ad Partitioig Problem with Applicatios i Itermodal Trasport Erwi Pesch, Domiik Kress, Sebastia Meiswikel Uiversity of Siege Deptartmet of Maagemet Iformatio Sciece Kohlbettstr. 15, Siege, Germay {erwi.pesch, domiik.kress, sebastia.meiswikel}@ui-siege.de Abstract We itroduce a combiatio of the problem of partitioig a set of vertices of a bipartite graph ito disjoit subsets of restricted size ad the Mi-Max Weighted Matchig Problem. The resultig problem has applicatios i itermodal trasport. We propose a mathematical model ad prove the problem to be NP-hard i the strog sese. Two heuristic frameworks that decompose the problem ito its partitioig ad matchig compoets are preseted. Additioally, we aalyze a basic implemetatio of tabu search ad a geetic algorithm for the itegrated problem. All algorithms outperform stadard optimizatio software. Moreover, the decompositio heuristics outperform the classical metaheuristic approaches. I. INTRODUCTION I this paper we cosider a itegrated matchig ad partitioig problem that is motivated by real world problems arisig i itermodal trasport. Cosider, for example, a railroad termial, that maily serves as a iterface i itermodal trasport. Here, gatry craes trasship cotaiers betwee trais ad trucks. A typical layout of a rail-road termial (yard) is schematically depicted i Fig. 1. This yard was preseted i details i [1]. Freight trais are parked o parallel trashipmet tracks of the yard. Trucks arrive o parallel truck laes, which are usually separated ito a drivig lae ad a parkig lae. I the most moder trasshipmet yards or MegaHubs there additioally is a fully automated sortig system. Such a sorter cosists of shuttle cars that ca receive cotaiers close to their iitial positios from iboud trais ad move them alogside the yard to their target positios. Usually, the yard area is subdivided horizotally ito slots of equal size measured i uits of the legth of a stadard railcar (or ay other uit). The resultig grid is used to idetify the coordiates of ay give cotaier i the yard ad the coordiates of all parkig slots of trucks. The area of the rail-road termial is divided ito several subareas accordig to the umber of craes (see [2]). A crae is assiged to each area ad each crae operates oly i the assiged area. Ay cotaier trasport betwee subareas is performed usig the sorter. We will make the followig simplificatios. First, we oly cosider loaded moves, i.e., moves of craes carryig cotaiers, which is a assumptio that proved to be realistic i previous research. Secod, a cotaier travellig through the sorter ivolves two craes where the first crae delivers 1st crae area 2d crae area 3rd crae area Fig. 1. Schematic layout of a rail-road yard Sorter Trais with cars carryig cotaiers Road with parkig slots the cotaier to the shuttle car while the secod oe picks it up agai. We drop this assumptio ad cosider oly oe crae operatig i the area from which the cotaier is delivered to the sorter. We will assume the umber of cotaiers to be less or equal to the umber of parkig slots. Let the completio time of trai operatios be the momet whe the last cotaier of the trai is trashipped o the truck. Each trai has a specified due date. The problem is to schedule cotaier trashipmet operatios so that the maximum lateess is miimized. The, give the above simplificatios, the described situatio withi a rail-road termial ca be modelled as a DEDICATED VARIATIVE JOBS Problem, which is defied as follows (see [3]). Set J of dedicated jobs is to be processed o m parallel machies. The job set is partitioed i advace ito mk subsets J lz,l =1,...,m,z =1,...,k, ad all jobs from the set k z=1j lz are assiged to machie l. Each job j has a due date d j ad for all j m l=1 J lz the due date is the same, i.e., d j = D z. Processig times of jobs are itercoected ad they are the subject of the decisio makig. The choice of the particular processig times is as follows. We are give a complete bipartite graph G = G(J, V, A) with bipartitios J ad V, = J V, ad edge set A. For each j J ad each v V, edge (j, v) A has a weight c(j, v). Each c(j, v), v V, is a possible processig time for job j. If we have chose c(j, v 0 ) as the processig time for some job j the oe of c(j,v 0 ) may be chose as the processig time for job j j. Thus, ay feasible choice of the processig times for the jobs should provide a maximum (by iclusio) matchig for graph G. The goal is to choose a processig time for each job i the feasible way ad to /15 $ IEEE DOI /SSCI

2 costruct a schedule S that miimizes maximum lateess L max (S) =max j J L j (S) =max j J (C j (S) d j ), where C j (S) is the completio time of the job j i the schedule S. To defie a particular schedule, we should choose the processig time for each job ad costruct a sequece of jobs for each machie. I terms of the Dedicated Variative Jobs Problem, the described above situatio withi a rail-road termial ca be modelled as follows. We have m craes ad k trais. Set J is the set of all cotaiers to be trashipped. Subset J lz,l = 1,...,m,z =1,...,k,is the subset of all cotaiers that are situated o the z-th trai i the l-th crae zoe. Set V is the set of parkig slots of the trucks. The positios of cotaiers ad parkig slots are predefied ad kow. Moreover, the duratio of trasportig the particular cotaier to the defied parkig slot is kow. These trasportatio times defie the weights of the edges. Due date D z, 1 z k, correspods to the cotaiers of the z-th trai. Whe we assume the trais i the rail-road termial to arrive (ad leave) i budles, ad thus aim at workload balacig of the gatry craes, the above metioed realworld problem ca be modelled as the so called MIN- MAX WEIGHTED MATCHING Problem, that has recetly bee itroduced i [4]. This problem ca be described as follows. There is a complete weighted bipartite graph G = G(U, V, E) where U ad V are the bipartitios ad E is the set of edges. Each edge e = (u, v) E,u U, v V, has a ratioal weight c(e). Set U is partitioed ito m subsets U 1,U 2,...,U m, the so-called compoets. For ay maximum matchig Π, we defie the weight of compoet U k as the sum of the weights of those edges of the matchig with a vertex belogig to this compoet: c(u k, Π) = u U k,(u,v) Π c(u, v). The value of Π is the maximum of the weights of the compoets: c(π) = max k {1,...,m} c(u k, Π). The problem is to fid a maximum matchig of miimum value. I terms of the Mi-Max Weighted Matchig Problem, the rail-road termial example is modelled as follows. Set U deotes the set of all cotaiers to be trasshipped. Subset U k is the set of all cotaiers that are located i the k-th crae area. The set of parkig slots is deoted by V. The trasportatio times defie the weights of the correspodig edges. The problem is to balace the workload over the craes. Give the strogly NP-hard Mi-Max Weighted Matchig Problem, the mai focus of this paper lies o relaxig the assumptio of the compoets beig fixed i advace. I terms of the cosidered problem at rail-road termials, this correspods to cosiderig the decisio o the crae areas (i.e. the partitioig decisio) as part of the problem (which becomes more adequate whe cosiderig crossover craes that are allowed to pass oe aother). We will refer to this problem as the PARTITIONING MIN-MAX WEIGHTED MATCHING Problem. It has first bee itroduced i [5]. Besides the rail-road termial case, aother applicatio of Partitioig Mi-Max Weighted Matchig arises at small to medium sized sea ports where cotaiers are hadled vessel quay crae reach stacker temporary storage area log-term storage area Fig. 2. Schematic layout of a reach stacker based termial (see [5]) by reach stackers as represeted i Fig. 2. These termials ca iclude large log-term storage areas ad additioal temporary storage areas (or marshalig-areas, [6], [7]). The latter areas aim at improvig the performace of the termials by iducig short turaroud times of vessels whe distaces to log-term storage areas are relatively large. Whe a vessel arrives at a berth at the termial, cotaiers are uloaded by quay craes ad are the stored i a temporary storage area that is located ext to the berth. Cotaiers that leave the termial by ship are moved to the temporary storage area durig previous idle times. A applicatio of Partitioig Mi-Max Weighted Matchig arises, whe cosiderig the process of emptyig or refillig the temporary storage area usig reach stackers durig idle times. We assume the vehicles to be fast if they are uloaded ad slow if they are loaded. This is a commo assumptio whe cosiderig cotaier movemets [2] ad is reasoable whe otig that reach stackers [i compariso to straddle carriers] are less stable i the forward directio as the machies will fall forward whe breakig i a emergecy, particularly if the load is beig carried high for visibility reasos [8]. We ca the restrict ourselves to cosiderig lade movemets of the vehicles oly. We are faced with the problem of assigig each cotaier i the temporary storage area to a empty slot i the log-term storage area (or vice versa) ad assigig the correspodig cotaier movemets to a limited umber of available vehicles. Naturally, the objective is to perform all movemets as fast as possible. This paper proceeds as follows. I Sectios II ad III (see also [5]), we preset a detailed problem descriptio alog with a mathematical model of Partitioig Mi-Max Weighted Matchig ad a proof of the problem s NPhardess, respectively. Sectio IV is cocered with describig heuristic approaches for solvig Partitioig Mi-Max Weighted Matchig. O the oe had, we are cocered with heuristics that decompose the problem ito its matchig ad partitioig compoets. O the other had, we cosider classical metaheuristic approaches for the itegrated problem. 1759

3 The heuristics are aalyzed i a computatioal study i Sectio V. The paper closes with a summary i Sectio VI. II. PROBLEM DEFINITION Let G(U, V, E) be a weighted bipartite graph with bipartitios U ad V ad edge set E. The elemets of U ad V are idexed i =1,..., 1 ad j =1,..., 2, respectively. Assume 1 2. A weight c(e) =c uv Q + 0 is associated with each edge e =(u, v) E of G. Defie a matchig as a set M E of pairwise oadjacet edges ad a maximum matchig as a matchig havig the largest possible size M amogst all matchigs o G. Throughout the paper, we will assume that, for ay give bipartite graph, there exists a maximum matchig Π with Π = 1. As i [4], give a partitioig of U ito m disjoit subsets, U 1,U 2,...,U m, the value of a maximum matchig Π is defied to be c(π) := max k {1,...,m} { u U c k,(u,v) Π uv}. The Partitioig Mi- Max Weighted Matchig ca formally be defied as follows: Fid a partitioig of the vertex set U ito m (potetially empty) disjoit subsets, U 1,U 2,...,U m, with at most ū elemets i each subset, ad a maximum matchig Π o G, such that the value of Π is miimum amogst all maximum matchigs over all possible partitioigs of U. We defie the followig biary variables: { 1 if (i, j) Π, x ij := (i, j) E (1) 0 else, ad y ik := { 1 if i U k, 0 else, i U, k {1,...,m}. (2) The a oliear mathematical model for Partitioig Mi- Max Weighted Matchig is as follows: mi max c ij y ik x ij (3) x,y k {1,...,m} s.t. i U j V x ij =1 i U, (4) j V x ij 1 j V, (5) i U m y ik =1 i U, (6) k=1 y ik ū k {1,...,m}, (7) i U x ij {0, 1} (i, j) E, (8) y ik {0, 1} i U, k {1,...,m}. (9) The objective fuctio (3) miimizes the value of the maximum matchig over all possible partitioigs ad all maximum matchigs. Costraits (4) (5) are well kow maximum matchig costraits (recall that 1 2 ). Costraits (6) eforce every vertex u U to be a elemet of exactly oe partitio U k, k 1,...,m. Costraits (7) restrict the umber of vertices i each partitio to be at most ū. The domais of the variables are defied by (8) (9). Model (3) (9) ca easily be liearized by itroducig additioal variables z ijk 0 for all i U, j V ad k {1,...,m}: s.t. mi x,y c (10) costraits (4) (9), c c ij z ijk k {1,...,m}, (11) i U j V z ijk y ik + x ij 1 i U, j V, k {1,...,m},(12) z ijk 0 i U, j V, k {1,...,m}.(13) III. COMPUTATIONAL COMPLEXITY I [3] the authors proved that problem Dedicated Variative Jobs is NP-Hard eve if m = 1 ad fidig a approximate solutio to problem Dedicated Variative Jobs with a approximatio ratio less tha 2 is strogly NP-hard problem. Moreover Mi-Max Weighted Matchig is NP-hard i the strog sese ad fidig a approximate solutio to problem Mi-Max Weighted Matchig with a approximatio ratio less tha 2 is strogly NP-hard, see [4] Note that Partitioig Mi-Max Weighted Matchig is ot a straight forward geeralizatio of Mi-Max Weighted Matchig i the sese that, give a istace I of Mi-Max Weighted Matchig, we eed oly copy the correspodig bipartite graph, fix a set of Partitioig Mi-Max Weighted Matchig parameters to specific values, ad solve the resultig Partitioig Mi-Max Weighted Matchig istace to receive a optimal solutio to I. More geerally, it is ot trivial to costruct a istace of Partitioig Mi-Max Weighted Matchig that (whe solved to optimality) is guarateed to result i the partitioig give i I ad thus to provide a optimal solutio to I. Thus we caot simply coclude that Partitioig Mi-Max Weighted Matchig is NP-hard from the NP-hardess of Mi-Max Weighted Matchig. A formal proof of strog NP-hardess of Partitioig Mi- Max Weighted Matchig, which we will ow provide, is more aturally based o a reductio of a classical partitioig problem tha o a matchig problem. We will cosider the decisio problem related to Partitioig Mi-Max Weighted Matchig: Defiitio 1 (PARTITIONING MIN-MAX WEIGHTED MATCHING - D). Give a weighted bipartite graph as defied i Sectio II. Does there exist a partitioig of the vertex set U ito m (potetially empty) disjoit subsets, U 1,U 2,...,U m, with at most ū elemets i each subset, ad a maximum matchig Π o G, such that the value of Π is o larger tha a give ω Q +? The proof is based o a reductio of the strogly NPcomplete 3-PARTITION Problem [9]: Defiitio 2 (3-PARTITION). Give a set A of 3k elemets, a boud B Z +, ad sizes s(a) Z + for all a A such 1760

4 that B/4 <s(a) <B/2 ad such that a A s(a) =kb. Is there a partitioig of A ito k disjoit subsets A 1,...,A k such that, for 1 i k, a A i s(a) =B? Theorem 1. Partitioig Mi-Max Weighted Matchig-D is NP-complete i the strog sese. Proof: Obviously Partitioig Mi-Max Weighted Matchig-D NP because for ay partitioig ad maximum matchig it ca be checked i polyomial time if the correspodig value is o larger tha ω. Now cosider a arbitrary istace I of 3-Partitio ad costruct a complete bipartite graph G as a istace J of Partitioig Mi-Max Weighted Matchig-D with U = A ad vertex set V such that V = U. Furthermore, for each u U, defie the weights of all edges icidet to u i G to be equal to the correspodig elemet s size s(a), a A. Set m = k, ū =3, ad ω = B. Now ote that, give a arbitrary partitioig U 1,...,U m of U, every maximum matchig has the same value. We are left with the task of showig that there exists a partitioig U 1,...,U m of U ad a maximum matchig Π with a value of o more tha ω if ad oly if the aswer to I is yes (we will say that I has a solutio if its aswer is yes ad call a correspodig partitioig of A a solutio ). Suppose we are give a solutio A 1,...,A k to I. Costruct a partitioig U 1,...,U m of U such that subset U i cotais all vertices that correspod to elemets of A i for all i =1,...,m, ad cosider ay maximum matchig o G. It is immediately implied that the value is ω. Suppose we are give a partitioig U 1,...,U m of U ad a maximum matchig Π such that its value is o more tha ω. Asū =3ad U =3m, we ecessarily have U i =3 for all i =1,...,m. Now ote that, as the sum of the edge weights of ay maximum matchig o G equals mω ad the value of Π is at most ω, wehave u U i,(u,v) Π c uv = ω for all i =1,...,m. Hece, the subsets of A that correspod to the give partitioig of U establish a solutio to I. We coclude: Corollary 1. Partitioig Mi-Max Weighted Matchig is NP-hard i the strog sese. IV. HEURISTIC AND METAHEURISTIC APPROACHES FOR SOLVING PARTITIONING MIN-MAX WEIGHTED MATCHING Partitioig Mi-Max Weighted Matchig ca be cosidered as a combiatio of a matchig ad a partitioig problem. Hece, a atural way of costructig heuristics for Partitioig Mi-Max Weighted Matchig is to decompose the problem ito its matchig ad partitioig compoets, solve the resultig problems separately (either exact or heuristically), ad combie the resultig solutios to a solutio of Partitioig Mi-Max Weighted Matchig. Obviously, we ca costruct differet heuristics by chagig the order of solvig the separate stages. We will cosider such decompositio approaches i Sectio IV-A. For details, see [5]. O the other had, oe ca try to solve Partitioig Mi- Max Weighted Matchig directly ( itegrated approaches ), for example by applyig metaheuristic approaches such as tabu search [10], [11] or geetic algorithms [12]. I Sectio IV-B we will briefly discuss how to adapt these approaches to the Partitioig Mi-Max Weighted Matchig Problem. A. Decompositio Approaches Partitio-Match heuristics assume that there is a vertex weight associated with each elemet of vertex set U ad, i the first stage, partitio U ito o more tha m compoets with at most ū elemets i each compoet, such that the maximum weight of the compoets is as small as possible. Here, the weight of a compoet is defied as the sum of the weights of the vertices of the compoet. Whe cosiderig the objective of miimizig the maximum weight of the compoets, we refer to this problem as the RESTRICTED PARTITIONING (RP) Problem. Based o the proof of Theorem 1, it is easy to see that RP is NP-hard i the strog sese. I the ext stage of a Partitio-Match heuristic, we drop the weights of the vertices. Give the compoets resultig from the first stage, we ow eed to determie a maximum matchig of small value. Hece, we are faced with a istace of Mi-Max Weighted Matchig. Solvig this problem results i a maximum matchig, that we may use for restartig the overall procedure by solvig RP with each vertex weight beig equal to the weight of the edge of the maximum matchig that is icidet to the very vertex. Summig up, Partitio-Match heuristics proceed as follows: 1) Geerate or modify vertex weights. 2) Partitioig stage: Solve the resultig Restricted Partitioig Problem. 3) Drop vertex weights. 4) Matchig stage: Solve the resultig Mi-Max Weighted Matchig Problem. 5) If there has bee o improvemet of the best kow solutio for 20 iteratios, the stop. Otherwise go to step 1. We iitialize a Partitio-Match heuristic by geeratig vertex weights that are used for determiig the first partitioig of the vertex set U. We tested several strategies of geeratig these weights. As they all performed similarly i computatioal tests, we decided o applyig the average weight of all edges icidet to vertex u for each vertex u U. Match-Partitio heuristics first cosider the classical problem of fidig a maximum matchig of miimum weight, i.e. miimum sum of the weights of all edges of the matchig, which we refer to as the MIN-SUM WEIGHTED MATCHING (MSWM) Problem. Hereafter, we proceed i aalogy to our approach i Partitio-Match heuristics, i.e. by defiig vertex weights based o the maximum matchig ad solvig RP. Fially, we modify the edge weights of the bipartite graph to restart the procedure by solvig MSWM o the modified graph: 1) Matchig stage: Solve the (resultig) Mi-Sum Weighted Matchig Problem. 2) Geerate or modify vertex weights. 1761

5 3) Partitioig stage: Solve the resultig Restricted Partitioig Problem. 4) If there has bee o improvemet of the best kow solutio for 20 iteratios, the stop. 5) Modify edge weights ad go to step 1. Cocerig the modificatio of edge weights i step 5, we apply a simple trasformatio strategy, that icreases the weight of exactly oe edge e E i each iteratio. The selectio of edge e depeds o the curret solutio of the Partitioig Mi-Max Weighted Matchig iput-istace: Select the compoet with the largest weight. Amog the curret edges of the matchig that are icidet to vertices of this compoet, select the oe with largest weight. The, set the weight of this edge to a large value (we set the weight to 10 2 c max, where c max is the largest edge weight of the iput graph, i our computatioal tests) to reduce the probability that it will be part of the solutio of the altered MSWM istace. After λ iteratios of the overall procedure, restore the origial edge weight of edge e. I prelimiary computatioal tests, we have see that λ = is a reasoable value. Note that the perturbed edge weights are oly used for the matchig problem i stage oe. The vertex weights assiged i stage two are based o the origial weights of the iput graph. We costruct multiple variats of Partitio-Match ad Match-Partitio heuristics by combiig differet methods of solvig the correspodig subproblems: A heuristic algorithm for solvig RP, deoted by RPH. As RP is similar to the well kow MULTIPROCES- SOR SCHEDULING Problem, that has bee extesively studied i the literature (see, for example, the literature review [13]), our approach adapts ideas preseted i [14] i its first stage that costructs a feasible solutio to RP. I a secod stage, we apply a local search o the obtaied solutio. The well kow O( 3 ) Hugaria algorithm [15], [16] for MSWM (where := max{ 1, 2 }). A approximate algorithm by [4], deoted by BPS, ad a simple regret heuristic, deoted by REG, for solvig Mi-Max Weighted Matchig. Both methods apply a additioal local search stage, deoted by LS. While [4] show BPS to be a adequate method for solvig Mi-Max Weighted Matchig aloe, we implemeted the rather simple REG heuristic motivated by the fact that, whe callig a Partitio-Match heuristic, we have to solve multiple istaces of Mi-Max Weighted Matchig. Hece, i order to reduce ruig times of the overall procedure, a simple heuristic seems to be promisig. Its ame derives from the fact that it is based o the regret priciple (cf., for istace, [17]). This results i two variats of the Partitio-Match framework, PM BPS (applyig BPS ad LS for solvig Mi-Max Weighted Matchig) ad PM REG (applyig REG ad LS for solvig Mi-Max Weighted Matchig), ad two variats of the Match-Partitio framework, combiig the Hugaria algorithm ad RPH with (MP) ad without (MP LS ) a additioal local search stage. B. Itegrated Approaches 1) Tabu Search: Tabu search is a well kow metaheuristic due to Fred W. Glover [11]. We assume the reader to be familiar with its basic cocepts ad refer to [10] for a detailed itroductio. I the cotext of Partitioig Mi-Max Weighted Matchig, we apply tabu search i its basic form. I order to do so, we defie the eighborhood N(s) of a give solutio s of Partitioig Mi-Max Weighted Matchig to cosist of three disjoit sets N(s) =M(s) S(s) V (s). Each set cotais all feasible solutios that ca be costructed based o s by performig oe specific type of trasformatio: M(s) cotais all feasible solutios that ca be costructed by applyig a MATCH trasformatio. A MATCH trasformatio alters a solutio s by replacig two edges, (u i,v j ) ad (u k,v l ), that are part of the maximum matchig of s, with (u i,v l ) ad (u k,v j ). The edges are chose such that u i is part of a compoet of the partitioig of s with largest weight. S(s) cotais all feasible solutios that ca be costructed by applyig a SWAP trasformatio. A SWAP trasformatio alters the partitioig of vertex set U i solutio s by swappig two vertices from differet compoets. More precisely, we determie a partitio U max of maximal weight i s ad a partitio U mi (with U mi U max ) of miimal weight i s. Afterwards we move oe vertex u i U max from U max to U mi ad oe vertex u j U mi from U mi to U max. V (s) cotais all feasible solutios that ca be costructed by applyig a MOVE trasformatio. Here, we determie U mi ad U max as before ad move oe vertex u i U max from U max to U mi. 2) A Geetic Algorithm: Geetic algorithms are well kow evolutioary search algorithms due to Joh H. Hollad [18]. As i Sectio IV-B1, we assume the reader to be familiar with the basic cocepts of geetic algorithms ad refer to [12] for a itroductio. I this sectio, we will defie a appropriate represetatio of a solutio of Partitioig Mi-Max Weighted Matchig, as well as a mutatio operator, a selectio operator, ad a crossover operator that our implemetatio of a basic geetic algorithm for Partitioig Mi-Max Weighted Matchig is based upo. Our represetatio of a solutio s of Partitioig Mi- Max Weighted Matchig is based o fixed sizes U k of the compoets k =1,...,m of the partitioig ad two lists, L U ad L V, cotaiig a permutatio of U ad V, respectively. Let L U (i) deote the i-th elemet of L U. The coectio of the partitioig of s ad the list L U is i 1 U i = {L U (j) k=1 U k < j i k=1 U k,j N}, i =1,...,k. The maximum matchig M of s is represeted by M = {(L U (i),l V (i)) i {1,..., 1 }}. To overcome the drawback of the sizes of the compoets of the partitioig beig fixed, oe may restart the geetic algorithm several times with differet sizes that are determied radomly. The iitial populatio is geerated radomly. The selectio operator radomly selects two idividuals from the curret geeratio ad adds the oe with the smaller objective fuctio value to the subsequet geeratio. Afterwards this 1762

6 solutio is mutated or crossed with a other solutio with a give probability util the desired populatio size is reached. Give two solutios s 1 ad s 2, the crossover operator costructs two ew solutios, s 1 ad s 2. This is realized by geeratig two radom umbers 1 p 1 < p 2 1 ad proceedig as follows. Let L U1 ad L V1 deote the lists represetig solutio s 1 ad L U2 ad L V2 the lists represetig s 2. The the list L U 1 represetig the ew solutio s 1 is { L L U2 (i) if p 1 <i p 2, U 1 (i) = (14) L U1 (i) else. The lists L V 1, L U 2 ad L V 2 are geerated i the same way. If ecessary, a simple repair operatio is performed to obtai feasible solutios after havig called the crossover operator. The mutatio operator radomly selects a pair of elemets i L U ad iterchages these elemets. Similarly, a iterchage operatio is performed i L V. V. COMPUTATIONAL RESULTS I order to assess the performace of the algorithms itroduced i Sectio IV, we ra extesive computatioal tests. All computatioal tests were performed o a Itel Core i7 mobile CPU at 2.8GHz ad 8GB of RAM, ruig Widows 7 64bit. All algorithms were implemeted i C ++ (Microsoft Visual Studio 2010). We used IBM ILOG CPLEX i versio 12.5 with 64bit. I order to get a comprehesive overview of the performace of our algorithms, we use differet strategies of geeratig istaces. All istaces are defied o bipartite graphs with 1 = 2, sice the case 1 2 trivially reduces to the case 1 = 2 by addig edges of zero weight. We defie := 1 = 2. The first strategy of geeratig istaces is ispired by [4]. We refer to the istace sets geerated by this strategy as BPS70 ad BPS80. We geerate 2 ratioal umbers uiformly distributed i the iterval [1, 1000] ad a complete bipartite graph with vertex sets U = {u 1,...,u } ad V = {v 1,...,v }. We sort the umbers i o-decreasig order ad deote the resultig list by L. Hereafter, we cosecutively assig these umbers (as edge weights) to the edges i the followig maer: We start with vertex v 1 ad assig the first 0.8 (i case of BPS80) or 0.7 (i case of BPS70) elemets of L to the edges (u 1,v 1 ), (u 2,v 1 ),...,(u 0.8,v 1 ) (BPS80) or (u 1,v 1 ), (u 2,v 1 ),...,(u 0.7,v 1 ) (BPS70) ad remove them from L. For the remaiig edges beig icidet to vertex v 1, we radomly choose ad assig elemets of L. Oce a elemet of L has bee assiged, we remove it from L. The procedure is repeated for all remaiig vertices v V, v v 1. The secod strategy of geeratig istaces costructs complete bipartite graphs ad radomly determies ad assigs edge weights uiformly distributed i the iterval [1, 1000]. We refer to istaces geerated by this strategy as RAND. Besides BPS70, BPS80, ad RAND, all of which are based o complete bipartite graphs, we geerate istaces defied o o-complete (or sparse) bipartite graphs with E = or E = We refer to them as SPARSE70 ad SPARSE80, respectively. Whe geeratig the edges, we guaratee that there exists at least oe feasible solutio for the resultig Partitioig Mi-Max Weighted Matchig istace (cf. restrictios (4)). Agai, edge weights are determied radomly ad uiformly distributed i the iterval [1, 1000]. For each strategy of geeratig istaces, BPS70, BPS80, RAND, SPARSE70, ad SPARSE80, we costruct istaces based o differet parameter settigs. We vary the size of the bipartitios from 10 to 170, i.e. = 10, 30,...,170. Furthermore, we set m to 2, ad (if larger tha 2) to 0.04, 0.08 or Similarly, we set ū to m, ( m ) m,or. For each combiatio of parameters ad each geeratio strategy, we costruct five Partitioig Mi-Max Weighted Matchig istaces. Let F be the value of the objective fuctio (10) of the best solutio foud by a specific heuristic. The, we rate the quality of this heuristic with the ratio F, where F best is F best the best objective fuctio value amog the objective fuctio values of the solutios obtaied by ay of the heuristics ad the best solutio foud by CPLEX (model (10) (13); prelimiary tests proved CPLEX to perform better for the liearized model tha for directly feedig it with model (3) (9)) withi a time limit of 180 secods. Cocerig the performace of CPLEX, oly istaces with a size of =10 could be solved to optimality withi this time limit. I geeral we foud the objective fuctio values to be ot competitive whe compared with the oes determied by the heuristics, which was especially obvious for RAND, SPARSE70, ad SPARSE80 istaces. The results for BPS70 ad BPS80 istaces were better. However, the objective fuctio values determied by CPLEX were (o average) still up to 600% larger tha the oes determied by the best heuristic. A. Decompositio Approaches As described i Sectio IV-A, we implemeted REG to solve Mi-Max Weighted Matchig maily i order to speed up Partitio-Match heuristics i compariso to usig BPS. This motivatio is illustrated i Fig. 3, which compares the average rutimes of REG ad BPS o the RAND istaces where the partitioig has bee fixed radomly with a additioal lower boud o the size of each partitio. For each, the figure depicts the average rutimes over the results of all parameter settigs. As to be expected, the correspodig solutio qualities are better for the BPS heuristic, see Fig. 4. For this figure, F best is set to the best objective fuctio value amog both heuristics. Ufortuately, the same effects arise whe embeddig REG ad BPS ito the Partitio-Match framework, i.e. PM REG is ot competitive i terms of solutio quality. Except for PM REG, however, all heuristics perform very well, eve for large values of ; for details see [5]. We fid that both, MP ad MP LS, outperform PM BPS i terms of solutio quality. However, PM BPS still performs at a high level. While, for all heuristics, rutimes icrease roughly quadratically i, we fid that PM BPS requires by far the most computatioal effort (resultig from the effect 1763

7 1.5 BPS REG 8 6 TABU5 TABU1 GA5 GA1 Rutime i s 1 Solutio quality Fig. 3. Rutime compariso of REG ad BPS o RAND istaces Fig. 5. Solutio quality of metaheuristic approaches o RAND istaces 1.6 BPS REG 10 8 MPLS TABU5 TABU1 Solutio quality 1.4 Solutio quality Fig. 4. Solutio quality compariso of REG ad BPS o RAND istaces Fig. 6. Tabu search solutio quality o RAND istaces i compariso to best decompositio approach MP LS see i Fig. 3). This fact stads agaist the gap i qualities of PM REG ad PM BPS. PM REG, MP, ad MP LS are quite similar i terms of rutimes. Based o these isights, it seems advisable to choose the Match-Partitio framework whe facig a istace of Partitioig Mi-Max Weighted Matchig. While MP ad MP LS perform equally well o the RAND, SPARSE70, ad SPARSE80 istaces, MP seems to be less effective for BPS70 ad BPS80 istaces. MP LS, however, does ot have this hadicap. Eve i the case of compariso with optimal solutios, the quality ratio of MP LS is less tha For all cosidered values of, the rutime of MP LS is smaller tha 6 secods. The average rutime for = 170 is aroud 1.5 secods. The ifluece of icludig LS o rutimes is (o average) less tha a secod for all ad all types of istaces. Thus it is advisable to always apply MP LS whe facig a radom istace of Partitioig Mi- Max Weighted Matchig. B. Itegrated Approaches As idicated i Sectio IV-B, we have implemeted differet versios of the cosidered metaheuristics. TABU1 ad GA1 refer to the basic tabu search procedure ad the geetic algorithm as described i Sectio IV-B1 ad IV-B2, respectively. TABU5 correspods to callig the tabu search procedure startig from five differet, radomly chose solutios, ad returig the best solutio. Similarly, GA5 returs the best solutio out of five rus of the geetic algorithm, startig with differet, radomly chose iitial populatios ad sizes of compoets of the partitioig (see Sectio IV-B2). We ra comprehesive prelimiary computatioal tests to determie the remaiig parameters for the tabu search algorithm ad the geetic algorithm. As a result of these tests we set the legth of the tabu list to max(5, 1 10 ) ad the stoppig criterio to 500 iteratios without improvemet for the tabu search procedure. For the geetic algorithm we set the populatio size to 10, the crossover probability to 0.6, 1 the mutatio probability to 2 1, ad the stoppig criterio to 500 iteratios without improvemet. Fig. 5 gives a overview of the performace of the cosidered metaheuristics solely based o the RAND istaces. Note that, i order to measure the quality of the algorithms, F best i this case is set to the best objective fuctio value amog the objective fuctio values of the solutios obtaied by ay of the itegrated approaches. Clearly, TABU1 ad TABU5 outperform the geetic algorithms. Furthermore, cosiderig the best solutio out of five rus seems to have a similar impact, i.e. a sigificat improvemet of solutio quality, i both metaheuristic frameworks. It is a iterestig questio, whether the metaheuristic approaches ca provide high quality results whe comparig them with the best decompositio approach. As metioed i Sectio V-A, MP LS has prove to be most effective out of these approaches whe it comes to solvig radom problem istaces. Hece, i Fig. 6 we compare the solutio quality of TABU1 ad TABU5 with MP LS o the RAND istaces. Ufortuately, tabu search seems to be iferior 1764

8 Solutio quality BPS70 BPS80 Fig. 7. TABU5 solutio quality o BPS70 ad BPS80 istaces Solutio quality CPLEX TABU5 Fig. 8. TABU5 solutio quality o RAND istaces i compariso to CPLEX Rutime i s MPLS TABU5 TABU1 GA5 GA1 Fig. 9. Rutime compariso for RAND istaces whe compared with MP LS. However, whe focussig o BPS70 ad BPS80 istaces, TABU5 proves to be competitive, as ca be see i Fig. 7. Additioally, it is worth metioig that tabu search will outperform IBM ILOG CPLEX Fig. 8 illustrates this fact by comparig the performace of CPLEX ad TABU5 o the RAND istaces. Fially, i Fig. 9 we compare the rutimes of our metaheuristic approaches ad MP LS o the RAND istaces. While rutimes icrease roughly quadratically i for all heuristics, both tabu search ad the geetic algorithm require more computatioal effort tha MP LS. VI. SUMMARY I this paper we have itroduced a itegrated matchig ad partitioig problem, the Partitioig Mi-Max Weighted Matchig Problem, with applicatios i itermodal trasport. We have preseted a mathematical formulatio of the problem ad prove its strog NP-hardess. Moreover, we have preseted differet heuristic ad metaheuristic approaches for solvig Partitioig Mi-Max Weighted Matchig. More specifically, we have implemeted a tabu search ad a geetic algorithm for solvig Partitioig Mi-Max Weighted Matchig ad we have foud that tabu search outperforms the geetic algorithm i terms of solutio quality. However, heuristic approaches that make use of the structure of the problem at had by decomposig it ito its matchig ad partitioig subproblems, perform better tha the metaheuristic approaches uder cosideratio. Nevertheless, all heuristic approaches outperform a stadard commercial solver (CPLEX), which has show to ot perform competitive i computatioal tests. REFERENCES [1] N. Boyse, M. Flieder, F. Jaeh, ad E. Pesch, A survey o cotaier processig i railway yards, Trasport. Sci., vol. 47, o. 3, pp , [2] N. Boyse ad M. Flieder, Determiig crae areas i itermodal trasshipmet yards: The yard partitio problem, Eur. J. Oper. Res., vol. 204, o. 2, pp , [3] M. Barketau, E. Pesch, ad Y. Shafrasky, Schedulig dedicated jobs with variative processig times, J. Comb. Optim., vol. i press, [4], Miimizig maximum weight of subsets of a maximum matchig i a bipartite graph, Discrete Appl. Math., vol. i press, [5] D. Kress, S. Meiswikel, ad E. Pesch, The partitioig mi-max weighted matchig problem, Eur. J. Oper. Res., vol. 247, o. 3, pp , [6] P. Presto ad E. Koza, A approach to determie storage locatios of cotaiers at seaport termials, Comput. Oper. Res., vol. 28, o. 10, pp , [7] E. Koza ad P. Presto, Geetic algorithms to schedule cotaier trasfers at multimodal termials, It. T. Oper. Res., vol. 6, o. 3, pp , [8] Isoloader. (2012) Whe to choose a straddle ad whe to choose a forklift. [Olie]. Available: [9] M. R. Garey ad D. S. Johso, Computers ad Itractability A Guide to the Theory of NP-Completeess. New York: Freema, [10] F. Glover ad M. Lagua, Tabu Search. Dordrecht: Kluwer, [11] F. Glover, Future paths for iteger programmig ad liks to artificial itelligece, Comput. Oper. Res., vol. 13, o. 5, pp , [12] M. Mitchell, A Itroductio to Geetic Algorithms. Cambridge: MIT Press, [13] B. Che, C. N. Potts, ad G. J. Woegiger, A review of machie schedulig: Complexity, algorithms ad approximability, i Hadbook of Combiatorial Optimizatio, Vol. 1, D.-Z. Du ad P. M. Pardalos, Eds. Bosto: Kluwer, 1999, pp [14] C.-Y. Lee ad J. D. Massey, Multiprocessor schedulig: Combiig LPT ad MULTIFIT, Discrete Appl. Math., vol. 20, o. 3, pp , [15] H. W. Kuh, The Hugaria method for the assigmet problem, Nav. Res. Logist. Quat., vol. 2, o. 1-2, pp , [16] R. Burkard, M. Dell Amico, ad S. Martello, Assigmet Problems. Philadelphia: Siam, [17] W. Domschke ad A. Scholl, Grudlage der Betriebswirtschaftslehre, 3rd ed. Berli: Spriger, [18] J. Hollad, Adaptatio i Natural ad Artificial Systems. A Arbor: Uiversity of Michiga Press,