research article modeling the multicommodity multimodal...

Research ArticleModeling the Multicommodity Multimodal RoutingProblem with Schedule-Based Services and Carbon DioxideEmission Costs

Yan Sun12 and Maoxiang Lang12

1School of Traffic and Transportation Beijing Jiaotong University Beijing 100044 China2MOE Key Laboratory for Urban Transportation Complex Systems Theory and TechnologyBeijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Maoxiang Lang mxlangbjtueducn

Received 7 September 2015 Revised 22 November 2015 Accepted 23 November 2015

Academic Editor Young Hae Lee

Copyright copy 2015 Y Sun and M LangThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We explore a freight routing problem wherein the aim is to assign optimal routes to move commodities through a multimodaltransportation network This problem belongs to the operational level of service network planning The following formulationcharacteristics will be comprehensively considered (1) multicommodity flow routing (2) a capacitated multimodal transportationnetwork with schedule-based rail services and time-flexible road services (3) carbon dioxide emissions consideration and (4) ageneralized costs optimum oriented to customer demands The specific planning of freight routing is thus defined as a capacitatedtime-sensitive multicommodity multimodal generalized shortest path problem To solve this problem systematically we firstestablish a node-arc-based mixed integer nonlinear programming model that combines the above formulation characteristics ina comprehensive manner Then we develop a linearization method to transform the proposed model into a linear one Finally acomputational experiment from the Chinese inland container export business is presented to demonstrate the feasibility of themodel and linearization method The computational results indicate that implementing the proposed model and linearizationmethod in the mathematical programming software Lingo can effectively solve the large-scale practical multicommoditymultimodal transportation routing problem

1 Introduction

11 Multimodal Transportation Multimodal transportationutilizes more than one transportation service (rail road airand maritime transportation) on the routes that serve tomove commodities from their origins to their destinations[1ndash3] Multimodal transportation differs significantly fromtraditional unimodal transportation which employs singletransportation service provided by one transportation carrierDue to the integrative combination of different advantages ofvarious transportation services multimodal transportationshows superiorities in lowering transportation costs andabating environmental pollution when compared with uni-modal transportation in a long-haul transportation settingMoreover containers arewidely used to carry commodities inmultimodal transportation They can simplify the packaging

of product economize on the package materials and ensuretransportation safety [4] The standardization of containersin the physical structure of the transportation system alsobenefits the mechanization of the loading and unloadingoperations and can help improve the efficiency of theseoperations at terminals [5]

Several empirical comparative studies have demonstratedthe superiorities of multimodal transportation in costs andenvironmental protection Janic [6] formulated the full costs(linear summation of the internal costs and external costs)of a multimodal and road freight transportation networkBy using the simplified European multimodal transportationnetwork (truck-rail combination) and its equivalent roadtransportation network as a case that study summarized thetrends of variation of the full costs of multimodal transporta-tion and road transportation in relation to the transportation

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 406218 21 pageshttpdxdoiorg1011552015406218

2 Mathematical Problems in Engineering

distances These trends showed that compared with roadtransportation multimodal transportation decreases theaverage full costs more rapidly as the distance increasesand shows fewer full costs if the transportation distanceexceeds approximately 1000 km (the critical distance whenthe rail service frequency is 5 trainsweek) Moreover thecritical distance decreases significantly as the rail servicefrequency increases Liao et al [7] compared the carbondioxide emissions of the long-haul truck transportation andthemultimodal transportation (truck-ocean combination) ina case study of container transportation between Kaohsiungand Keelung in Taiwan The activity-based method wasadopted to estimate the carbon dioxide emissions in thetransportation process and the computational results ofthe case indicated that the carbon dioxide emissions fellfrom 566525 tonnes to 163830 tonnes corresponding to a71 decrease when truck transportation was replaced bymultimodal transportation These results supported the viewthat the use of multimodal transportation as an alternative tolong-haul truck transportation canmarkedly decrease carbondioxide emissions from the transportation sector

Many researchers have studied the empirical applicationsof multimodal transportation in specific cases for exampleBookbinder and Fox [8] discussed multimodal transporta-tion shortest time and minimal costs route selection for fivecommodity flows that need to be routed from Canada toMexico in the North American Free Trade Agreement areaBanomyong and Beresford [9] explored multimodal trans-portation route selection for exporting commodities fromLaos to the European Union and proposed a cost model formultimodal transportation to help exporters make decisionsCho et al [10] presented a weighted constrained shortestpath problem model and a label setting algorithm to selectmultimodal transportation routes from Busan to RotterdamMeethom and Kengpol [11] combined the AHP (AnalyticHierarchy Process) method with 0-1 goal programming todesign a decision support system for selecting an optimalmultimodal transportation route from Bangkok to Danang

Currently with the rapid development of economic glob-alization international trade and the accompanying globalcirculation of commodities have grown remarkably Conse-quently transportation networks have expanded widely andcommodity distribution routes have extended significantlyThese changes represent challenges to transportation perfor-mance in terms of for example costs timeliness and envi-ronmental concerns [2 12] In response to these challengesincreasing numbers of companies have adopted multimodaltransportation schemes to transport their products or rawmaterials According to the relevant statistics logistics costsrepresent 30ndash50 of the product cost [2] For this reasonlogistics cost is one of themost effective targets for companiesto lower product cost raise profits and maintain competi-tiveness in the international trade market Therefore freightrouting for commodity transportation has been given greatimportance by both the management decision makers andmultimodal transport operators

12 Formulation Characteristics of the Study The practi-cal demand for lowering logistics costs motivates research

interests in the freight routing problem in multimodaltransportation networks The essence of this problem is toselect optimal routes tomove commodities from their originsto their destinations through a multimodal transportationnetwork In our study we consider following formulationcharacteristics to make the formulation of this problemcorrespond more closely to actual practice

(1) Multicommodity Flow Routing Freight routing aims atsatisfying customer demands at minimal costs Each trans-portation demand can be represented by a commodity flowthat is characterized by five attributes origin destinationvolume release time and due date From the perspective ofthe entiremultimodal transportation networkmore than onetransportation demand within the planning horizon needs tobe satisfied Various transportation demands differ from eachother in at least one of the five above-mentioned attributionsThus multiple commodity flows must be routed to satisfythe various transportation demands Therefore the freightrouting problem explored in this study will be developed as amulticommodity problem

(2) Schedule-Based andTime-Flexible Transportation ServicesIn a multimodal transportation network we can combinedifferent transportation services to move commodities alongroutes The most obvious distinction among the transporta-tion services is in terms of their operating modes In thisstudy we define two categories for classifying transportationservices The first category is that of schedule-based servicesthat is operations controlled by schedules The secondcategory is that of time-flexible services that is operationsnot controlled by schedulesThis distinction is similar to thatdrawn in Moccia et alrsquos study [13] Schedule-based services(eg rail services) are operated based on schedules that arespecified in advance Conversely time-flexible services (egroad services) are not constrained by time and can travelthrough the network flexiblyThe operations of the two typesof transportation services and the transshipments amongthem will be emphasized in Section 2

(3) Consideration of Carbon Dioxide Emissions With theevolution of society and improvement in the awareness ofsustainable development environmental issues have drawnincreasing attention from both the government and thepublic especially in developing countries for exampleChina a rapidly industrializing country One of the mostpressing environmental issues is global warming causedby greenhouse gas emissions Carbon dioxide representsapproximately 80 of the total greenhouse gas emissions [7]and the reduction of these emissions is acknowledged to be ahighly challenging problem worldwide In the transportationsector various activities have been shown to represent up to19 of the global energy consumption [14] and to producelarge amounts of greenhouse gases (eg carbon dioxidecarbon monoxide and methane) and air pollutants (egsulfur dioxide and oxynitride) [15] For example accordingto Li et alrsquos evaluation [16] carbon dioxide emissions fromthe Chinese transportation sector represent 837 (5396)of the total carbon dioxide emissions from fossil fuel (liquid

Mathematical Problems in Engineering 3

fuel) consumption in 2007 Therefore transportation is con-sidered a key target in reducing carbon dioxide emissions Forthis reason it is necessary for decision makers to seriouslyconsider the carbon dioxide emissions when planning multi-modal transportation routes

(4) Generalized Costs Optimum Oriented to CustomerDemands Transportation is a service industry and trans-portation operation oriented to customer demands has beenpromoted vigorously with the development of traditionaltransportation practice into modern logistics In freightrouting the goal of the shippers is to payminimal costs for thetransportation of their commodities As service providers themultimodal transport operators and the third party logisticscompanies should base their planning on customer demandswhich means that lowering costs should be chosen as theirplanning objective for the transportation process In thisstudy we use generalized costs to evaluate how much moneythe shippers should pay for transportation The generalizedcosts cover not only the transportation costs but also theinventory costs operation costs at terminals carbon dioxideemissions costs and additional service costs

13 Literature Review Because multimodal transportationrouting has attracted substantial interest [17 18] manyrelevant studies have been conducted in previous decadesAmong the early studies (studies conducted before 2005)Min [2] developed a chance-constrained goal programmodelto provide the distribution manager with decision supportin choosing international multimodal transportation routesMinimizing the transportation costs and risk and satisfyingthe requirements of on-time service are formulated in theproposed model Barnhart and Ratliff [19] discussed theminimal costs multimodal transportation routing problemand introduced two solution approaches that separatelyinvolved a shortest path algorithm and amatching algorithmThe two approaches were adopted to solve the routingproblems with rail service costs expressed per trailer andper flatcar respectively Boardman et al [20] designeda 119896-shortest path double-swap method to solve a multi-modal transportation routing problem and incorporated thismethod with the a database and user interface to builda decision support system for the real-time routing ofshipments through a multimodal transportation networkZiliaskopoulos and Wardell [21] systematically presented atime-dependent multimodal optimum path algorithm formultimodal transportation networks In their algorithmmany time parameters including schedules dynamic arctravel times and transshipment delays were comprehensivelyconsidered This approach enhanced the feasibility of thealgorithm in addressing the practical problem Lozano andStorchi [22] defined the shortest viable path within an origin-destination pair in the multimodal transportation networkand proposed an ad hoc modification of the chronologicalalgorithm to solve the multimodal shortest path problem byobtaining a solution set of the problem Lam and Srikanthan[18] improved the computational efficiency of the 119896-shortestalgorithm by using clustering technique and applied thisalgorithm to multimodal transportation routing Boussedjra

et al [23] addressed the multimodal shortest path problemand proposed a shortest path algorithm involving labelcorrecting method to solve the single origin-destination pairproblem The efficiency of the algorithm was verified bycomparing its performance with that of the branch-and-bound method

Essentially in these previous studies with the exceptionof Minrsquos study researchers have emphasized the use ofalgorithms for selecting optimal multimodal transportationroutes for commodities and the development of optimizationmodels attracted limited attention However it is necessaryto construct optimization models because it is difficultto find a universal optimization model to solve all typesof multimodal transportation routing problems consideringdifferent formulation characteristics Moreover optimizationmodels can provide an exact benchmark for systematicallytesting various solution algorithms

In recent years with the constant improvement of thedesign of algorithms for solving problems in this area ofstudy increasing numbers of multimodal transportationrouting models have been developed In these studies afew researchers have concentrated on the development ofGIS-basedmodels for the multimodal transportation routingproblem For example Winebrake et al [24] constructeda geospatial multimodal transportation routing model toselect minimal costs minimal time and minimal carbondioxide emissions routes for an origin-destination pair Theconstruction and solution of them model were conductedwith ArcGIS software Other researchers have continued topursue studies on goal programming models and solutionalgorithms Zhang and Guo [25] and Zhang et al [26]separately presented the foundational frameworks of integerprogramming models for the multimodal transportationrouting problem Sun et al [27] studied the basic uncapaci-tated single-commodity multimodal transportation routingproblem without a demanded delivery time constraint andused the label correcting algorithm to solve it A similar studywas also conducted by Sun and Chen [28] whereas this studyaddressed an uncertain transportation case and consideredbiobjective optimization including the minimization of totaltransportation costs and total carbon dioxide emissionsUsing these studies as a springboard many studies havehighlighted transportation due date constraints and capacityconstraints In the formulation of transportation due dateconstraints several studies (eg Liu et al [29]) have treatedthe transportation due date as a hard constraint which thetotal transportation time should not exceed Furthermoreothers have considered the transportation due date as anindex for charging penalty costs which means that if thetransportation due date is violated penalty costsmust be paidto compensate for the loss as in for example Zeng et al[30] Wang et al [31] Fan and Le [32] Li et al [33] andTang andHuo [34] Regarding the capacity constraint vehiclecarrying capacity has been widely considered for exampleby Kang et al [35] Wang et al [31] Liu et al [29] Li etal [33] Cakır [36] Verma et al [37] Cai et al [38] Tangand Huo [34] and Lei et al [39] A few studies (eg Changet al [40]) have also defined terminal operating capacityand vehicle carrying capacity as capacity constraints As for


the scheduling issues current studies for example Cai et al[38] Xiong and Wang [41] and Lei et al [39] have viewedschedules as ldquoterminal schedule-based service timewindowsrdquoto formulate a multimodal transportation routing problemwith time windows In these studies the terminal scheduledservice time windows are only related to the terminals anddifferent transportation services share one schedule-basedservice time window at the same terminal A terminal cannotbe covered in the multimodal transportation route if thearrival time of the commodity at this terminal is out of therange of the schedule-based service time window and thisconsideration is also formulated as a hard constraint in theoptimization models [13 38 39 41 42]

In terms of comparisons with our study first apart fromCakır [36] Verma et al [37] and Chang et al [40] all otherprevious studies cited above preferred to optimize the single-commodity routing problem in a multimodal transportationnetwork [27ndash35 38 39 41] and this type of optimizationcannot guarantee an optimum result for the overall per-formance of the entire multimodal transportation networkSecond in many studies all types of transportation serviceswere assumed to adopt a time-flexible service mode andthe operations at terminals were simplified as a continuousldquoarrival rarr transshipment rarr departurerdquo process [27ndash41]which cannot be expected to match the practical reality thatsome transportation services for example rail ocean andair services are operated according to schedules The actualtransshipments among the schedule-based and time-flexibletransportation services are much more complicated than theldquoarrival rarr transshipment rarr departurerdquo process as willbe explained in Section 2 Although some researchers usedtime windows to address the scheduling issues [38 39 41]the consideration of terminal schedule-based service timewindows is not expected to represent the schedules in themodel formulation exactly because schedules regulate notonly the transportation service times but also the transporta-tion service routes Additionally different schedules regulatedifferent service times even at the same terminal Moreoverformulation of terminal schedule-based service time win-dows as a hard constraint itself [13 38 39 41 42] does notreflect the actual practice because if their arrival times arenot within the time windows the commodities can wait untilthe lower bound of the current time window or that of thenext time window and then be transshipped Consequentlythe transportation schemes designed by the studies above areless supportive of decision-making Additionally the networkdeformation method is widely used by researchers to convertactual multimodal transportation networks into a standardgraph with one link between two conjoint nodes [27ndash29 3438] This method can simplify model formulation but willresult in a substantial expansion of the network scale Thusthis procedure may be feasible for a small-scale multimodaltransportation networks but it will be unfeasible for largescale ones in practice

14 Similar Works and Problem Among existing studiesthose most similar to ours are those of Chang [42] Mocciaet al [13] and Ayar and Yaman [5] Chang [42] addressedthe problem of selecting the best routes tomove commodities

through international multimodal transportation networksIn his study he considered multiobjective optimizationand schedule-based transportation services and demandeddelivery times and transportation economies of scale andhe formulated the route selection problem as a multiobjec-tive multimodal multicommodity flow problem with timewindows and concave costs In his study time windowconstraints were used to represent the restriction of schedulesand demanded delivery times relative to best route selectionin empirical cases A concave piecewise linear functionwas adopted to measure the transportation costs of allcommodity flows Each commodity flow was considered tobe splittable In the proposed model minimizing the totaltransportation costs and total transportation time were setas the objectives and the weighted summation method wasused to address the multicriteria-based optimization Theproblem was decomposed into a series of more easily solvedsingle-commodity flow subproblems by using the Lagrangianrelaxation technique to relax the capacity constraint of themodel The subgradient optimization algorithm was thenused to obtain the solutions to the subproblems Finallya reoptimization technique was designed to modify thesolutions of subproblems to construct a feasible solution ofthe initial problem

Moccia et al [13] explored the multimodal multicom-modity flowproblemwith pickup and delivery timewindowsand their study was similar to Changrsquos [42] The problemaddressed in their study was oriented to a multimodaltransportation network with time-flexible road services andschedule-based rail services To address this problem theauthors first presented a virtual network representation toconvert the physical multimodal transportation network intoa detailed representation of operations by adding nodesto represent the pickup and delivery time windows andthe scheduled departure times Then two mixed integerprogramming models including an arc-node-based modeland a path-based model were formulated The two modelswere both single-objective ones wherein the aim was tominimize the total transportation costs of all commodityflows Nonconvex piecewise linear costs time windowsand side constraints were all considered and formulated inthe optimization models and each commodity flow wasunsplittable Finally a column generation algorithm wasdeveloped to achieve the lower bound of the problem and itwas embedded within heuristics to obtain feasible solutionsof the problem

Ayar andYamanrsquos study [5] was a special case of themulti-modal multicommodity flow problem previously formulatedby Moccia et al [13] In their study the release times and duedates of commodities replaced the pickup time and deliverytime windows respectively that is the transportation of acertain commodity was defined to begin at or after its releasetime at its origin and was to be achieved before its due date atits destination Additionally in contrast to Changrsquos [42] andMoccia et alrsquos [13] approaches to make the costs calculationsmore accessible and to make them correspond better tothe real-world cases the total transportation costs wereevaluated by generalized costs covering transportation costsen route operation costs and inventory costs at terminals


Then the authors proposed two mixed integer programmingmodels and valid inequalities to solve the multicommodityrouting problem in a truck-ocean transportation networkwhere truck services were considered to be time flexible anduncapacitated whereas maritime services were capacitatedand operated according to schedules The solution algorithmdesigned in this study was somewhat similar to Changrsquos [42]in that a Lagrangian relaxation technique was used to relaxthe capacity constraint

Note that the freight routing problem that this studyfocuses on is extremely similar to the multicommoditymultimodal transportation network design problem Both ofthese problems relate to planning optimal routes to movemultiple commodities through the transportation network byusing multiple transportation services rationally They alsoshow obvious similarities in model formulation and solutionalgorithm design The multicommodity multimodal trans-portation network design problem has always been a focusof transportation planning and many studies have alreadybeen conducted to address this problem Some representativestudies are reviewed here Crainic and Rousseau [43] pre-sented a foundational modeling and algorithmic frameworkto solve the multicommodity multimodal transportationnetwork design problem An integer nonlinear programmingmodel was constructed and a solution algorithm involvingdecomposition and column generation was developed in thisstudy Their work provided a solid foundation for futurerelated studies Daeki et al [44] explored the truck-aircraftservice network design problem for express package deliveryThe authors addressed this problem as a multimodal trans-portation network design problem with time windows sep-arately formulated an approximate model and an exact oneto describe this problem and designed a linear programmingrelaxation method optimized by valid inequalities to attainthe optimal solution of the large-scale problem Zhang et al[45] developed a bilevel programming model to optimize themulticommodity multimodal transportation design problemwith carbon dioxide emissions and economies of terminalscale The upper level of the model adopted a genetic algo-rithm to design the optimal topology of the terminal networkwhile the lower level served to distribute themulticommodityflow through the multimodal transportation network Quet al [46] built an integer nonlinear programming modelconsidering transfer costs and carbon dioxide emission costsfor the multicommodity multimodal transportation networkdesign problem Using linearization technique the proposedmodel was transformed into a linear one that could be solvedeasily by mathematical programming software

Additionally Crainic [47] Southworth and Peterson [48]Jansen et al [49] and Yaghini and Akhavan [50] separatelyconducted systematic reviews of the service network designproblem All these reviews introduced this problem compre-hensively from the perspectives of research content currentprogress model formulation algorithm development andresearch prospects thus contributing to making the generalresearch architecture more mature and ideal Furthermoreto enhance the effectiveness of solving the network designproblem many studies discussed the solution approaches forthe network design problem in detail and a large number of

solution approaches with high feasibility have been proposedfor example a dual-ascentmethod by Balakrishnan et al [51]a Lagrangian heuristic-based branch-and-bound approachby Holmberg and Yuan [52] a first multilevel cooperativetabu search algorithm by Crainic et al [53] and a multi-ple choice 0-1 reformation and column-and-row generationmethod by Frangioni and Gendron [54] All the studiesdiscussed above significantly advanced the knowledge of themulticommodity multimodal transportation network designproblem

Although similarities between the two types of problemsare obvious there are still three remarkable differencesbetween them Considering these differences helps to defineobvious distinctions between the two types of problems interms of model formulation

(1) In addition to the question of route selection thenetwork design problem also needs to solve the allocationof limited transportation resources to the network thatis to determine the number of facilities or capacities oftransportation services that should be installed on the routesor the levels of transportation services that should be offeredon the routes [47] By contrast the freight routing problemis based on an existing multimodal transportation networkwhose transportation resources have already been allocatedand it only focuses on planning origin-to-destination routesby selecting proper terminals and transportation servicesconnecting them

(2) The network design problem is a form of tacticalplanning The due dates of the commodities are usually notconsidered in this problem By contrast the freight routingproblem is a part of the operational planning andmust assigngreat importance to customer demands The due dates of thecommodities must be formulated as a constraint to satisfy thecustomer demands regarding timeliness

(3) In the network design problem the detailed sched-ules of transportation services are rarely formulated by theresearchers However in the freight routing problem whenutilizing schedule-based services to move commodities theschedulesmust be followed strictly to guarantee the feasibilityof routing in empirical cases Consideration of service sched-ules improves the complexity of the freight routing problemin the multimodal transportation network

15 Organization of the Rest of the Sections All the studiesreviewed above regardless of the problem (freight routingproblem or freight network design problem) they addressedlaid a solid foundation for our study with regard to the modelformulation and algorithm designThe remaining sections ofour study are organized as follows

In Section 2 we first give a detailed introduction tomultiple transportation services in the multimodal trans-portation network including schedule-based rail servicesand the time-flexible road services We then analyze thevarious transshipment manners that are available in thenetwork All this background information is considered informulating the model In Section 3 we define themulticom-modity multimodal transportation routing problem fromthe perspectives of capacity constraint and time sensitivityand we present an example containing a single-commodity


Time axis

h

i

j

k

Term

inal

sequ

ence

A rail servicefrom terminal h

to terminal k

Scheduleddeparture

time

Scheduledservice time

windowScheduled

arrival time

tm tnmiddot middot middot middot middot middotmiddot middot middot

Figure 1 Diagram of rail services between two terminals

flow to illustrate the selection of time-sensitive multimodaltransportation routes In Section 4 we establish an arc-node-based mixed integer nonlinear programming modelto formulate the specific freight routing problem that weexplore in this study The proposed optimization modelintegrates the various formulation characteristics mentionedin the abstract Then we develop a linearization methodto linearize the proposed model to make it easily solvableby mathematical programming software In Section 5 acomputational example from the Chinese inland containerexport business is presented to demonstrate the feasibility ofthe proposed model and the use of the linearization methodto address the practical problem Finally the conclusions ofthis study are presented in Section 6

2 Multiple Transportation ServiceModes and Transshipments

21 Schedule-Based and Time-Flexible Services Rail servicesin the transportation network are controlled by a centralauthority in ChinaThe rail services are scheduled in advanceunder macroscopic control The freight trains are oper-ated through the multimodal transportation network strictlyaccording to the rail service schedulestimetables A typicalrail service diagram is shown in Figure 1 Some rail servicesshow periodicity in their operating frequencies within theplanning horizon For the convenience of modeling weconsider the same rail services viewed in different operatingperiods as different services

The following components of a rail service are regulatedby its schedule to control the operations of a train

(1) Route including the operation direction of the trainand its terminal sequence

(2) Arrival times of the train at the terminals on the routeand its departure times from them

(3) Service time windows of the train at the terminalson the route Each service time window is a closedinterval from operation (loadingunloading) starttime to operation cutoff time

(4) Carrying capacity of the train

Accordingly the operations of commodities at terminals area process with the following characteristics

(1) The commodity cannot be unloaded from the train ata terminal until the rail service reaches its operationstart time

(2) Once the operation cutoff time is reached the railservice will stop operating immediately

(3) After being loaded on board the commodity willdepart from the terminal at the scheduled departuretime of the rail service

(4) If the commodity arrives at the terminal and com-pletes the unloading operation earlier than the oper-ation start time of the rail service it must wait untilthe operation start time In this case the commodityneeds to occupy a freight yard or warehouse to bestored Hence its inventory costs are created accord-ing to the charged inventory time

The commodity must observe the following two rules to beable to use a rail service at a terminal

(1) The capacity of the service at the terminal must admitits volume

(2) Its loading completion time on the train should not belater than the operation cutoff time of the rail serviceat the terminal

In this study we specifically focus on freight routing inthe multimodal truck-rail transportation network where allthe commodities are carried in containers In China toallow a rapid development of container transportation blockcontainer trains are extensively operated in the rail servicenetwork and have gradually become the backbone of railcontainer transportation For this reason the rail serviceaddressed in this study is the block container train serviceAlthough block container train service is a component ofschedule-based rail service its operation differs from thatof the common rail service shown in Figure 1 Comparedwith the common rail service the block container train is apoint-to-point rail service (see in Figure 2) and it conductsloading and unloading operations separately at its loadingorganization terminals and unloading organization terminalson the route Intermediate terminals along the route whereloading and unloading operations are not undertaken canbe regarded as invalid terminals in the routing and canbe removed from the multimodal transportation networktopology Apart from the difference cited above the operationof block container train is identical to the common railserviceTheutilization of block container trains in the routingshould also observe the rules above Many block containertrains have multiple loadingunloading organization termi-nals For convenience we further classify the same blockcontainer train into different train units according to itsloadingunloading organization terminals that is if a blockcontainer train has 119899 origin terminals and 119898 destinationterminals it will be considered as (119899 times 119898) a different blockcontainer train


Loading organizationterminal

Intermediate terminals(containers stay on board)

Loadingoperation

Unloadingoperation

Shippercarrier Receivercarrier

Unloading organizationterminal

Figure 2 Operation of a block container train

OriginDestinationRail terminalBlock container train route

Pickup service from origins to rail terminalsDelivery service form rail terminals to destinationsRail terminal connecting serviceDirect origin-destination road service

1

1

1

1 12

2

2 2

2

4

4

Shippersrsquo area

Receiversrsquo area

3

3

2

1

2

Figure 3 Road services in the multimodal transportation network

The outstanding advantage of road services is their highflexibility due to the rapid travel speeds of the trucks andwidedistribution and high density of the road service networkCompared with rail services road services have much lesscontrol over their operations and the trucks can travelfeasibly over the road service network Moreover it is easyto assign adequate numbers of trucks to the terminals tocarry the commodities Hence we consider that road servicesare uncapacitated in the multimodal transportation networkRoad services are not scheduled and their unloading andloading operations are not restricted in time A commoditycan start to be loaded on trucks immediately upon arrival at aterminal and trucks can depart from the terminals once theloading is completed There are mainly three types of roadservices in a multimodal transportation network includingpickup anddelivery services rail terminal connecting serviceand direct origin-destination road service as indicated byFigure 3

When moving a commodity from one terminal (egterminal ℎ in Figure 4) to another one (terminal 119896) by

road service one or more intermediate terminals (terminal119894 and terminal 119895) may be visited along the road serviceroute There is no need to transship commodities at theintermediate terminals because the extra transshipments notonly delay the delivery of the commodities but also increasethe transportation costs Therefore commodities can bemoved directly among the intermediate terminal To facilitatemodeling we split every road service along a route fromorigin to destination into several segments (eg in Figure 4road service (ℎ 119896) is split into 6 segments including (ℎ 119894)(ℎ 119895) (ℎ 119896) (119894 119895) (119894 119896) and (119895 119896)) and consider that twoconjoint segments (eg (119894 119895) and (119895 119896) in Figure 4) cannot becovered in a commodity flow route simultaneously to avoidan extra transshipment

22 Transshipments betweenMultiple Transportation ServicesThe transshipments that consider schedules in the multi-modal transportation network can be divided into three cate-gories ldquoroad service rarr rail servicerdquo ldquorail service rarr anotherrail servicerdquo and ldquorail service rarr road servicerdquo However


i Terminal

Road service

kjih

Figure 4 An example of road services along a route

h jiRoad service Rail service s

Rail service routei Terminal

Road service

Tim

e

Location

Step 3 waiting for loading at terminal i

Step 5 waiting for departure

Step 2 unloading the commodity from the trucks

Step 6 moving commodity from terminal ito terminal j by rail service s

Step 1 moving commodity from terminal hto terminal i by road service

t7

t6t5

t4

t3

t2

t1

Step 4 loading commodity on the train (rail service s)Δt34

Figure 5 Transshipment from road service to rail service

h jiRail service sRail service r


Tim

e

Location



Step 1 moving commodity from terminal hto terminal i by rail service r

t7

t6

t5

t4

t3

t2

t1

Step 5 loading commodity on the train (rail service s)

Δt34

t998400


Step 3 unloading the commodity from the train(rail service r)

Step 2 waiting for unloading from the train(rail service r)

Figure 6 Transshipment between different rail services

there is no ldquoroad service rarr road servicerdquo transshipmentbecause we view road service as a type of time-flexible servicemode and trucks can travel feasibly trough the road servicenetwork the ldquoroad service rarr road servicerdquo transshipment isextra

The transshipment from road service to rail service andthat between different rail services have similar processeswhich are illustrated by Figures 5 and 6 respectively For

Table 1 Symbols in Figures 5 and 6 and their representations

Symbol Representation

1199051

Departure time of road service from terminal ℎ(Figure 5) scheduled departure time of rail service 119904

from terminal ℎ (Figure 6)

1199052

Arrival time of road service at terminal 119894 (Figure 5)scheduled arrival time of rail service 119903 at terminal 119894(Figure 6)

1199051015840 Scheduled operation start time of rail service 119903

(Figure 6)

1199053

Completion time of unloading commodity from thetrucks (road service) at terminal 119894 (Figure 5)completion time of unloading commodity from thetrain (rail service 119903) at terminal 119894 (Figure 6)

1199054

Scheduled operation start time of rail service 119904 atterminal 119894

1199055

Completion time of loading commodity on the train(rail service 119904) at terminal 119894

1199056

Scheduled departure time of rail service 119904 fromterminal 119894

1199057 Scheduled arrival time of rail service 119904 at terminal 119895

Δ11990534

Waiting time for loading at terminal 119894Δ11990534

= max0 1199054minus 1199053

the two figures the representations of time parameters areshown in Table 1

As shown in Figures 5 and 6 the main difference betweenthe two transshipments is that the transshipment betweendifferent rail services has an additional unloading waitingprocess the commodity should wait until the operation starttime of the predecessor rail service before being unloadedCompared with the two transshipments above the processof transshipping a commodity from a rail service to a roadservice is much simpler as can be observed in Figure 7Representations of the time parameters in Figure 7 are givenin Table 2

Note that due to the highly mechanized operation ofthe block container trains at the terminals the times ofloadingunloading commodities onfrom the trains can benegligible as can be the times of loadingunloading com-modities onfrom the trucks For the sake of generalizationwe still consider these times in Figures 5 6 and 7 but wewill neglect these loadingunloading times in the followingproblem description and model formulation

3 Problem Description

As asserted in Section 1 the multicommodity multimodaltransportation routing problem aims at selecting minimalgeneralized costs routes to move multiple commodities fromtheir origins to their destinations via capacitated schedule-based rail services and uncapacitated time-flexible roadservices A route can be created by a single road servicea single rail service with necessary additional pickup anddelivery services or various truck-rail combinations Becausethe carrying capacities of the block container trains arelimited the routing problem in this study is a capacitated


Table 2 Symbols in Figure 7 and their representations


1199051

Scheduled departure time of rail service 119904 fromterminal ℎ


1199053


1199054

Completion time of unloading commodity from thetrain (rail service s) at terminal 119894

1199055

Completion time of loading commodity on the trucks(road service) at terminal 119894

1199056 Arrival time of road service at terminal 119895

h jiRoad serviceRail service s


Road service

Tim

e

Location

Step 5 moving commodity from terminal ito terminal j by road service

Step 1 moving commodity from terminal hto terminal i by rail service s

t6

t5

t4

t3

t2

t1

Step 4 loading commodity on the trucks (road service)Step 3 unloading commodity from the train

(rail service s)Step 2 waiting for unloading from the train

(rail service s)

Figure 7 Transshipment from rail service to road service

one The commodities that are loaded on a block containertrain at its loading organization terminal should not exceedits carrying capacity

The routing problem in this study is a time-sensitive onethat is constrained by the transportation due dates of themultiple commodities and the schedules of the rail servicesThe selected routes therefore must be time feasible from twoaspects The first time feasibility is that the arrival time ofeach commodity at the destination along the route shouldnot be later than its transportation due date Second if thecommodity needs to be moved from the current terminal tothe successor terminal by rail service the loading completiontime of the commodity on the train at the current terminalshould not be later than the operation cutoff time of the railservice

Here we give an example of a single-commodity flowin Figure 8 to illustrate the time-sensitive routing problemFigure 8 shows a directed multimodal transportation net-work containing seven terminals (1 2 7 where 1 and 7represent the origin and destination resp) and seven blockcontainer train services (119877

1 119877

7) The release time of the

commodity at the origin is 10 orsquoclock on the first day (shownas 10) and its due date at the destination is 22 orsquoclock onthe second day (shown as 22 + 24 = 46) In Figure 8the rail services are represented by solid lines and the roadservices are represented by dashed lines The travel times ofroad services on different arcs are given beside the dashed

lines (unit h) For the rail services we sequentially give theoperation start time operation cutoff time and departuretime of each block container train at and from the loadingorganization terminal and the arrival time and operation starttime at the unloading organization terminal In Figure 8all the time parameters are converted into natural numbers(eg 1030 is converted into 105) and a time that exceeds 24corresponds to the second day (2400 to 4800)

In the example in Figure 8 there are overall 18 routesto route the commodity flow from terminal 1 to terminal 7However only the 9 following routes are time feasible

(1) Depart from terminal 1 by road service at 10 andarrive at terminal 2 at 14 complete unloading fromtrucks at 14 (loadingunloading times at the terminalsare neglected here and below) wait until 25 andcomplete loading on 119877

4at 25 depart from terminal 2

by 1198774at 275 and arrive at terminal 5 at 355 complete

unloading from 1198774and loading on 119877

5at 36 depart

from terminal 5 by 1198775at 385 and finally arrive at

terminal 7 at 445(2) Complete loading on 119877



unloading from1198771at 185 wait until 25 and complete

loading on 1198774at 25 depart from terminal 2 by 119877

4

at 275 and arrive at terminal 5 at 355 completeunloading from 119877

4and loading on 119877

5at 36 depart


terminal 7 at 445(3) Depart from terminal 1 by road service at 10 and

arrive at terminal 2 at 14 complete unloading fromtrucks and loading on 119877

3at 14 depart from terminal

2 by 1198773at 16 and arrive at terminal 4 at 24 complete

unloading from1198773and loading on trucks at 25 depart

from terminal 4 by road service at 25 visit terminal 6without transshipment and finally arrive at terminal7 at 40

(4) Depart from terminal 1 by road service at 10 and arriveat terminal 2 at 14 complete unloading from trucksat 14 wait until 25 and complete loading on 119877

4at

25 depart from terminal 2 by 1198774at 275 and arrive at

terminal 5 at 355 complete unloading from 1198774and


4at

385 and finally arrive at terminal 7 at 445(5) Depart from terminal 1 by road service at 10 visit

terminals 4 and 6 without transshipment and finallyarrive at terminal 7 at 35


6at 27

depart from terminal 4 by 1198776at 32 and finally arrive

at terminal 7 at 43(7) Depart from terminal 1 by road service at 10 visit ter-

minal 4 without transshipment and arrive at terminal5 at 25 complete unloading from trucks at 25 waituntil 33 and complete loading on 119877

5at 33 depart


terminal 7 at 445


5

6

8

4

8

10

7

[19 215 2225] [28 285]

[27 31] 32 43 [44 47]

[10 15 16] [24 25]

[25 27 275] [355 36]

[11 13 1325] [18 185]

[33 38 385] [445 45][35 39 40] [47 475]

1

2

4

56

7

3

Originrelease time = 10

Destinationdue date = 46

R1

R4

R5

R7

R6

R3

R2

Figure 8 A multimodal transportation network example

(8) Depart from terminal 1 by road service at 10 and arriveat terminal 3 at 16 complete unloading from trucks at16 wait until 19 and then complete loading on 119877

2at

19 depart from terminal 3 by 1198772at 2225 and arrive

at terminal 6 at 28 complete unloading from 1198772and

loading on trucks at 285 depart from terminal 6 byroad service at 185 and finally arrive at terminal 7 at355

(9) Depart from terminal 1 by road service at 10 visitterminals 3 and 6 without transshipment and finallyarrive at terminal 7 at 31

4 Model Formulation and Linearization

41 Estimation of Carbon Dioxide Emissions In 2012 to con-trol greenhouse gas emissions theNationalDevelopment andReform Commission of China proposed the development ofrelevant pilot projects on trading carbon emissions rightsThus the transportation industry may be charged for carbondioxide emissions in the future to reduce their emissions andpromote the development of ecofriendly freight transporta-tion [55] To price the carbon dioxide emissions accuratelythe first step is to estimate the volume of carbon dioxideemissions when multimodal transportation is adopted tomove commodities

Carbon dioxide emitted en route during the transporta-tion represents themajority of total carbondioxide emissionsBy contrast the carbon dioxide emitted by the operationsat the terminals is negligible In this study we considerthe former emissions when estimating the carbon dioxideemitted by the transportation of commodities Generallythere are two primary methods that can be used to estimatethe carbon dioxide emitted in the transportation process that

is energy-basedmethod and activity-basedmethod [46]Theactivity-based method has been developed more extensivelyand has been widely used in many studies for example Liaoet al [7] Chang et al [40] and Qu et al [46] which inturn demonstrate its feasibility For this reason we adopt theactivity-based method to estimate carbon dioxide emissionsin this study

The activity-based method calculates the carbon dioxideemissions of a transportation service 119904 as (120578

119904times 119876 times 119871) where

120578119904is the emission factor for carbon dioxide of service 119904 (unit

gTEU-km) 119876 is the volume of the commodity that must bemoved (unit TEU) and119871 is the transportation distance (unitkm) Different types of transportation services have differentcarbon dioxide emission factor According to the data fromWinebrake et alrsquos study [24] (unit gTEU-mi) the carbondioxide emissions factors of rail services and road services areset to 125 gTEU-km and 626 gTEU-km respectively

42 Notations In this study we use 119866 = (119873119860 119878) torepresent a multimodal transportation network where 119873119860 and 119878 are the terminal set the directed arc set andtransportation service set in the network respectively Let 119870denotes the set of the commodities that need to be movedthrough the network For a certain commodity 119896 isin 119870 itsorigin 119900

119896 destination119889

119896 release time at origin 119905

119896

release due date119879119896 and the volume 119902

119896(unit TEU) demanded by the shipper

are all known The rest of the symbols in the optimizationmodel and their representations are listed shown in Table 3

43 Node-Arc-Based Mixed Integer Nonlinear ProgrammingModel Using the symbols above we propose the follow-ing node-arc-based mixed integer nonlinear programmingmodel to describe the capacitated time-sensitive multicom-modity multimodal generalized shortest path problem


(i) Mixed Integer Nonlinear Programming ModelObjective FunctionWe have the following

minimize sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895

119862119896

119894119895119904sdot 119883119896

119894119895119904 (1)

+ sum

119896isin119870

sum

119894isin119873

( sum

ℎisin120575minus(119894)

sum

119903isin119878ℎ119894

119888119903sdot 119902119896sdot 119883119896

ℎ119894119903+ sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119888119904sdot 119902119896sdot 119883119896

119894119895119904) (2)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isinΓ119894119895

119888store sdot 119902119896 sdot 119885119896

119894119895119904 (3)

+ sum

119896isin119870

120593119896sdot sum

119895isin120575+(119900119896)

sum

119904isinΓ119900119896119895

119888pick sdot 119902119896sdot 119883119896

119900119896119895119904+ sum

119896isin119870

120583119896sdot sum

119894isin120575minus(119889119896)

sum

119904isinΓ119894119889119896

119888deliver sdot 119902119896 sdot 119883119896

119894119889119896119904 (4)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895

119888co2 sdot 120578119904 sdot 119902119896 sdot 119889119894119895119904 sdot 119883119896

119894119895119904 (5)

In this formulation the objective function is to minimizethe total generalized costs of moving multiple commoditiesthrough the multimodal transportation network The gener-alized costs are the linear summation of Components (1)ndash(5) where Components (1)ndash(5) are the transportation costsen route loading and unloading operation costs at terminalsinventory costs at terminals additional origin picking upservice costs and destination delivery costs and carbondioxide emissions costs respectively

Subject to sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119883119896

119894119895119904minus sum


sum

119903isin119878ℎ119894

119883119896

ℎ119894119903

=

1 119894 = 119900119896

0 forall119894 isin 119873 119900119896 119889119896

minus1 119894 = 119889119896

forall119896 isin 119870 forall119894 isin 119873

(6)

sum

119904isin119878119894119895

119883119896

119894119895119904le 1 forall119896 isin 119870 forall (119894 119895) isin 119860 (7)

Constraint (6) is the general commodity flow conservationconstraint Constraint (7) ensures that only one transporta-tion service can be adopted to move a specific commodity onone arc The combination of the two constraints ensures thateach commodity is unsplittablenonbifurcated and followsexactly one route from its origin terminal to its destinationterminal through the multimodal transportation network[54 56]

119883119896

ℎ119894119903+ 119883119896

119894119895119904le 1

forall119896 isin 119870 forall119894 isin 119873 forallℎ isin 120575minus(119894) forall119895 isin 120575

+(119894) forall119904 isin Ω

119894119895 forall119903 isin Ω

ℎ119894

(8)

Constraint (8) ensures that the two conjoint segments of aroad service route cannot be covered in a route simultane-ously to avoid an extra transshipment at the terminals that

is if commodity 119896 plans to be moved from terminal ℎ toterminal 119895 via terminal 119894 by road service it should directlyuse the equivalent segment (ℎ 119895) that is generated in themultimodal transportation network based on the road serviceroute deformation illustrated by Figure 4 For more detailswe can refer to the statement in Section 21

sum

119896isin119870

119902119896sdot 119883119896

119894119895119904le 119876119904

119894119895forall (119894 119895) isin 119860 forall119904 isin Γ

119894119895 (9)

Constraint (9) is the rail service carrying capacity constraintIt ensures that the total volume of commodities loaded on theblock container trains at the origin terminals will not exceedtheir available carrying capacities at the same terminals

119884119896

119900119896= 119905119896

release forall119896 isin 119870 (10)

Constraint (10) means that the arrival time of each commod-ity at the origin equals its release time at the origin

(max 119884119896

119894 SD119904119894 + 119905119894119895119904

minus 119884119896

119895) sdot 119883119896

119894119895119904= 0

forall119896 isin 119870 forall (119894 119895) isin 119860 forall119904 isin 119878119894119895

(11)

Constraint (11) reflects the relationship between the arrivaltime variables and the commodity flow variables It alsosuccessively computes the arrival times of each commodityat all the terminals that are covered in their transportationroutes For 119904 isin Ω

119894119895 there exists no scheduled departure time

and SD119904119894is therefore set to 0 in this study For 119904 isin Γ

119894119895 we are far

more concerned about its operation start time at terminal 119895than about its arrival time at the same terminal becausewhenthe commodity 119896 arrives at terminal 119895 it will not be unloadeduntil 119897119904

119895 Therefore we consider 119897119904

119895to be the valid arrival time

of commodity 119896 at terminal 119895whenmoved by rail service 119904 on


Table 3 Symbols in the optimization model and their representa-tions

Indices Representationℎ 119894 119895 Index of the terminals and ℎ 119894 119895 isin 119873119904 119903 Index of the transportation service and 119904 119903 isin 119878

(119894 119895) Directed arc from terminal 119894 to terminal 119895 and(119894 119895) isin 119860

Sets RepresentationΓ119894119895

Set of rail services on arc(119894 119895) and Γ119894119895sube 119878

Ω119894119895

Set of road services on arc(119894 119895) and Ω119894119895sube 119878

119878119894119895

Set of the transportation services on arc(119894 119895)119878119894119895= Γ119894119895cup Ω119894119895 and 119878

119894119895sube 119878

120575minus(119894)

Set of the predecessor terminals to terminal 119894 and120575minus(119894) sube 119873

120575+(119894)

Set of the successor terminals to terminal 119894 and120575+(119894) sube 119873

Parameters Representation119897119904

119894Operation start time of rail service 119904 at terminal 119894

119906119904

119894Operation cutoff time of rail service 119904 at terminal 119894

SA119904119894

Scheduled arrival time of rail service 119904 at terminal 119894

SD119904119894

Scheduled departure time of service 119904 from terminal119894

119876119904

119894119895

Available capacity of rail service 119904 at terminal 119894 whenbeing operated on arc(119894 119895) unit TEU

119889119894119895119904

Transportation distance of service 119904 on arc(119894 119895) unitkm

119905119894119895119904

Transportation time of service 119904 on arc(119894 119895) unit h

119862119896

119894119895119904

Transportation costs of moving commodity 119896 onarc(119894 119895) by service 119904 unit yen

119888119904

Unit costs of loadingunloading operation of service119904 unit yenTEU

119888store Unit inventory costs of rail service unit yenTEU-h

119888pick

Additional charges for picking up the unitcommodity from shipper at a rail terminal by railservice at origin unit yenTEU

119888deliver

Additional charges for delivering the unitcommodity from rail terminal to receiver by railservice at destination unit yenTEU

120593119896

A parameter indicating whether origin pickupservice is needed for commodity 119896 If the service isneeded 120593

119896= 1 otherwise 120593

119896= 0

120583119896

A parameter indicating whether destination deliveryservice is needed for commodity 119896 If the service isneeded 120583

119896= 1 otherwise 120583

119896= 0

119888co2 Unit carbon dioxide emissions costs unit yeng

120578119904

Emission factor for carbon dioxide of service 119904 unitgTEU-km

119872 A large positive numberDecisionvariables Representation

119883119896

119894119895119904

0-1 decision variable If commodity 119896 is moved fromterminal 119894 to terminal 119895 by service 119904 119883119896

119894119895119904= 1

otherwise 119883119896119894119895119904

= 0119884119896

119894Arrival time of commodity 119896 at terminal 119894

119885119896

119894119895119904

Charged inventory time of commodity 119896 at terminal119894 before being moved on arc(119894 119895) by rail service 119904unit h

arc (119894 119895) Hence119884119896119895equals 119897119904

119895instead of SA119904

119894and consequently

in this case 119905119894119895119904

= (119897119904

119895minus SD119904119894)

119884119896

119894le 119906119904

119894sdot 119883119896

119894119895119904+ 119872 sdot (1 minus 119883

119896

119894119895119904)

forall119896 isin 119870 forall (119894 119895) isin 119860 forall119904 isin Γ119894119895

(12)

Constraint (12) is the operation service time constraint Itmeans that when adopting rail service the arrival time ofthe commodity at the terminal should not be later than theoperation cutoff time of the service For this equation if119883119896

119894119895119904= 0 then 119884

119896

119894le 119872 This inequality is always satisfied

that is the arrival time of the commodity at the terminal isnot constrained by the operation cutoff time of a rail servicethat is not adopted in the routing

119884119896

119889119896le 119879119896

forall119896 isin 119870 (13)

Constraint (13) is the due date constraint It ensures that thearrival time of each commodity at the destination does notexceed its claimed due date

(max 0 119897119904

119894minus 119884119896

119894minus 120587 minus 119885

119896

119894119895119904) sdot 119883119896

119894119895119904= 0


(14)

Constraint (14) indicates the relationship between thecharged inventory time variables and the commodity flowvariables It also computes the charged inventory times ofeach commodity at all the origin terminals of the rail servicesthat are adopted in their transportation routesThe inventoryis needed only if the arrival times of the commodity at theterminals are earlier than the operation start times of the railservices at the same terminals There also exists a period ofinventory free of charge denoted by 120587 (unit h) such that ifthe inventory time of the commodity at a terminal is shorterthan 120587 no inventory costs will be charged For road servicesthere is no inventory at the terminals

119862119896

119894119895119904=

(1198881

rail + 1198882

rail sdot 119889119894119895119904) sdot 119902119896

119904 isin Γ119894119895

1198882

road sdot 119902119896sdot 119889119894119895119904

119904 isin Ω119894119895


(15)

Constraint (15) computes the costs of operating services onthe arcs according to the regulations proposed by the ChinaMinistry of Railways and Ministry of Transport In thisequation 1198881rail are the cost parameters related to the volumeof the commodities (unit yenTEU) while 119888

2

rail and 1198882

road are thecost parameters related to the turnover of commodities (unityenTEU-km)The values of the above parameters are regulatedby the two ministries

119883119896

119894119895119904isin 0 1 forall119896 isin 119870 forall (119894 119895) isin 119860 forall119904 isin 119878

119894119895 (16)

119884119896

119894ge 0 forall119896 isin 119870 forall119894 isin 119873 (17)

119885119896

119894119895119904ge 0 forall119896 isin 119870 forall (119894 119895) isin 119860 forall119904 isin Γ

119894119895 (18)

Constraints (16)ndash(18) are the variable domain constraints


44 Model Linearization The proposed model above is atypical nonlinear programming formulationWhen using thestandard mathematical programming software to solve thisproblem directly the computational results will representlocal optimum and much computational time will also beconsumed that is the quality of the solutions and the solvingefficiency will both be lowered due to the nonlinearity of theproposed model

The nonlinearity of the model is caused by Constraints(11) and (14) which include a nonlinear function (maxsdotfunction) and multiplications of different variables Thus tomake the proposed model easily solvable by mathematicalprogramming software we linearize the two nonlinear con-straints to transform the proposedmodel into amixed integerlinear programming formulation according to the followingtwo propositions

Proposition 1 Nonlinear constraint (max119884119896119894 119878119863119904

119894+119905119894119895119904minus119884119896

119895)sdot

119883119896

119894119895119904= 0 forall119896 isin 119870 forall(119894 119895) isin 119860 and forall119904 isin 119878

119894119895can be linearized by

using the following four constraints

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(19)

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(20)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)

forall119896 isin 119870 forall (119894 119895) isin 119860 forall119904 isin Ω119894119895

(21)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(22)

Proof Constraints (19) and (20) and Constraints (21) and(22) correspond separately to two independent but similarscenarios Here we only address Scenario 1 Scenario 1contains two independent subscenarios Scenarios 11 and 12

Scenario 1 For rail service 119904 on arc (119894 119895) we have the following

Scenario 11 If it is used to move commodity 119896 then119883119896

119894119895119904= 1

and according to Constraint (11) SD119904119894+ 119905119894119895119904

minus 119884119896

119895= 0 In this

scenario Constraints (19) and (20) equal (SD119904119894+ 119905119894119895119904

minus119884119896

119895) ge 0

and (SD119904119894+ 119905119894119895119904

minus 119884119896

119895) le 0 respectively Hence the following

deductive process exists The deductive result ensures theexisting equation relationship between SD119904

119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge 0

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0

997904rArr 0 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0 997904rArr 119884

119896

119895= SD119904119894+ 119905119894119895119904 (23)

Scenario 12 Otherwise119883119896119894119895119904

= 0 and according toConstraint(11) SD119904

119894and119884

119896

119895are not related to each other In this scenario

Constraints (19) and (20) equal (SD119904119894+ 119905119894119895119904

minus 119884119896

119895) ge minus119872 and

(SD119904119894+ 119905119894119895119904

minus 119884119896

119895) le 119872 respectively Therefore the following

deductive process exists The deductive result ensures thatthere is no relationship between SD119904

119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge minus119872

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872

997904rArr minus119872 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872 (24)

Proposition 2 Nonlinear constraint (max0 119897119904119894minus 119884119896

119894minus 120587 minus

119885119896

119894119895119904) sdot 119883119896

119894119895119904= 0 forall119896 isin 119870 forall(119894 119895) isin 119860 and forall119904 isin Γ

119894119895can be

linearized by using the following two constraints

119885119896

119894119895119904ge 119872 sdot (119883

119896

119894119895119904minus 1) + (119897

119904

119894minus 119884119896

119894minus 120587)


(25)

119885119896

119894119895119904le 119872 sdot 119883

119896

119894119895119904forall119896 isin 119870 forall (119894 119895) isin 119860 forall119904 isin Γ

119894119895 (26)

Proof We consider two independent Scenarios 1 and 2

Scenario 1 When the commodity 119896 is not moved fromterminal 119894 to terminal 119895 by rail service 119904 then 119883

119896

119894119895119904= 0 and

according to Constraint (14) there is no charged inventorytime for commodity 119896 towards rail service 119904 on arc(119894 119895) Inthis scenario because 119872 is a large positive number minus119872 +

(119897119904

119894minus 119884119896

119894minus 120587) is equivalent to minus119872 Consequently Constraints

(25) and (26) separately equal 119885119896119894119895119904

ge minus119872 and 119885119896

119894119895119904le 0 When

combinedwith the nonnegative constraint of119885119896119894119895119904(Constraint

(17)) the following deductive process exists The deductiveresult ensures that there is no charged inventory time forcommodity 119896 toward rail service 119904 on arc(119894 119895) in this scenario

119885119896

119894119895119904ge minus119872

119885119896

119894119895119904le 0

119885119896

119894119895119904ge 0

997904rArr 0 le 119885119896

119894119895119904le 0 997904rArr 119885

119896

119894119895119904= 0 (27)

Scenario 2 When commodity 119896 is moved from terminal 119894to terminal 119895 by rail service 119904 then 119883

119896

119894119895119904= 1 This scenario

contains two following independent subscenarios Scenarios21 and 22

Scenario 21 If (119897119904119894minus119884119896

119894minus120587) le 0 there is no charged inventory

time for commodity 119896 towards rail service 119904 on arc(119894 119895)based on Constraint (14) In this scenario Constraints (21)and (22) equal 119885

119896

119894119895119904ge (119897119904

119894minus 119884119896

119894minus 120587) and 119885

119896

119894119895119904le 119872

respectively Considering the nonnegative constraint of 119885119896119894119895119904


(Constraint (17)) the following deductive process exists inwhich the control of the minimization of Component (3) inthe objective function towards the deductive result in the firststepwill lead to the final deductive resultThedeductive resultensures that when commodity 119896 is moved on arc(119894 119895) by railservice 119904 as long as (119897119904

119894minus 119884119896

119894minus 120587) le 0 the charged inventory

time will not be generated

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)

Scenario 22 If (119897119904119894minus 119884119896

119894minus 120587) gt 0 the charged inventory time

(119897119904

119894minus 119884119896

119894minus 120587) exists for commodity 119896 toward rail service 119904 on

arc(119894 119895) based on Constraint (14) In this scenario consider-ing the nonnegative constraint of 119885119896

119894119895119904(Constraint (17)) the

following deductive process exists in which the control ofthe minimization of Component (3) in the objective functiontowards the deductive result in the first step will lead tothe final deductive result The deductive result ensures thatwhen commodity 119896 is moved on arc(119894 119895) by rail service 119904 and(119897119904

119894minus119884119896

119894minus120587) gt 0 the charged inventory timewill be (119897119904

119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)

The final model is now a mixed integer linear programmingmodel as follows It can be easily solved by any mathematicalprogramming software for example Lingo Cplex or GAMSto obtain the optimal solution to the capacitated time-sensitive multicommodity multimodal generalized shortestpath problem in a specific case

(ii) Mixed Integer Linear Programming ModelObjective FunctionWe have the following

minimize (1) + (2) + (3) + (4) + (5) (30)

Subject to Constraints (6)ndash(10) (12) (13) and (15)ndash(22) (25)and (26)

5 Computational Experiment

In this section we design a large-scale empirical example todemonstrate the feasibility of the proposed model and thelinearization method in addressing the practical problem byusing the mathematical programming software Lingo Thevalues of the cost parameters in the proposedmodel are givenin Table 4 The unit loadingunloading costs of the rail androad services and the rail servicersquos unit inventory costs areshown in Table 5 Moreover the period of inventory free ofcharge of rail services is 48 h and the unit carbon dioxideemissions cost is 100 yenton

In this empirical example which is based on a Chi-nese scenario we study multimodal transportation routingfor moving multiple commodities carried in containersfrom inland cities (eg Lanzhou Hohhot Guiyang andChangsha) to the eastern sea ports (eg Qingdao Shang-hai Guangzhou and Shenzhen) Multimodal transportationrouting for this example starts from a deterministic and

known date and the data used in this example should firstbe discretized into real numbers

The rail terminals of the block container trains arethe backbone of the multimodal transportation network Atotal of 40 terminals and 118 arcs were used to build theentire multimodal transportation network topology shownin Figure 9 There are 42 periodic operated rail services and78 road services in the multimodal transportation networkthat can be adopted to move containers from inland cities tothe sea ports In the large-scale multimodal transportationnetwork below the transportation distances and times ofthe road services are given in Table 6 The data for thistable were collected by using the online digital map andthe relevant operating information for the rail services isgiven in Table 7 In China the average time interval betweenthe arrival time and the operation start time of a blockcontainer train at the unloading organization terminals isapproximately 40min and the average time interval betweenthe departure time and the operation cutoff time at theloading organization terminals is approximately 30min Ifthere aremarshaling or classifying operations at the two typesof terminals the two time intervals will increase to 70minand 60min respectively

The multiple commodities for this example are presentedin Table 8These commodities are all carried in 20 ft contain-ersThe due dates of the commodities are usually determinedby the loading cutoff times of the container vessels at thesea ports If the arrival times of the commodities at the seaport exceed the container vesselsrsquo loading cutoff time thecommodities cannot be carried by the container vessels tobe further moved overseas Considering these commoditiesas the optimization object we then used the mathematicalprogramming software Lingo 12 on a Lenovo Laptop withIntel Core i5 3235M 260GHz CPU and 4GB RAM to solvelarge-scale empirical example


1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang

13 Lianyungang11Putianxi 12Putian

Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang

Beijiao19 Hejiawan15

Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi

Road serviceRail serviceTerminal

X954039 Train number

Figure 9 A large-scale multimodal transportation network for China inland container export

Table 4 Values of the cost parameters

Service Parameter Container feet Unit20 ft 40 ft

Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882

rail 2025 1377 yenTEU-km

The computational results for the large-scale empiricalexample are presented in Table 9 Moreover we simulatedthe routing in the road service network that is containedin the large-scale empirical example The correspondingcomputational result for the single road service routing is5806742 yen The computational results for this example indi-cate that the multimodal transportation routing can lower


Table 5 Values of the loadingunloading and inventory costs

Service Container feet Loadingunloading costs Pickupdelivery costs Inventory costs(unit yenTEU) (unit yenTEU) (unit yenTEU-h)

Road 20 ft 25 mdash mdash40ft 19 mdash mdash

Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (

Figure 10 Cost structure for the large-scale empirical example

the generalized costs of moving commodities by approxi-mately 10 (|5251563 minus 5806742|5806742) compared withsingle road service routing which indicates the superiorityof multimodal transportation in the terms of economicefficiency

The cost structure of the two types of servicemodes in thelarge-scale empirical example is shown in Figure 10 whereC1 to C5 successively represent the costs incurred duringthe routes the loadingunloading costs the inventory coststhe additional pickupdelivery costs and the carbon dioxideemissions costs which separately correspond to Components(1)ndash(5) in the objective function of the proposed model

Finally the corresponding optimal multimodal trans-portation routing framework is given in Table 10 where theitems such as X95143 and X95187 are the train numbers ofthe utilized block container trains and their operation routesare all presented in Figure 9

6 Conclusions

In this study we explore the multimodal transportationrouting problem To address this combinatorial optimizationproblem a node-arc-based mixed integer nonlinear pro-gramming model is first established A linearization methodis then developed to transform the proposed nonlinearmodelinto a linear one by linearizing its nonlinear constraintsFinally a computational example is presented to demon-strate the feasibility of the proposed model and linearizationmethod in addressing the large-scale empirical example

The main contribution of this study is that it compre-hensively considers (1)multicommodity flow (2) capacitatedschedule-based rail service and uncapacitated time-flexibleroad service (3) carbon dioxide emissions estimation and

Table 6 Transportation distances and times for road services in thelarge-scale empirical example

Arc Distance Time Arc Distance Time(unit km) (unit h) (unit km) (unit h)

(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35


Table 7 Operating information for rail services in the large-scale empirical example

Rail service number 1 2 3 4 5 6 7Origin 1 2 5 6 7 8 8Operation start time mdash 207 58 26 9 mdash mdashOperation cutoff time 08 222 68 41 11 mdash mdashDeparture time 18 227 73 46 12 01 01Destination 3 3 3 3 3 9 10Arrival time 465 465 429 429 429 452 459Operation start time 477 477 436 436 436 464 466Distance (unit km) 686 847 1415 1358 1272 1281 1326Capacity (unit TEU) 44 61 50 30 20 50 55Operation period (unit daytrain) 2 2 2 2 2 35 35Rail service number 8 9 10 11 12 13 14Origin 11 11 11 12 12 12 14Operation start time mdash mdash 9 mdash mdash mdash 18Operation cutoff time mdash mdash 119 mdash mdash mdash 224Departure time 04 04 124 08 08 08 229Destination 9 10 13 9 10 13 15Arrival time 259 265 303 259 265 303 111Operation start time 271 272 31 271 272 31 1117Distance (unit km) 763 821 585 767 812 576 2210Capacity (unit TEU) 22 38 40 20 20 60 16Operation period (unit daytrain) 2 2 2 2 2 2 2Rail service number 15 16 17 18 19 20 21Origin 14 14 14 14 14 14 14Operation start time 18 18 18 18 18 145 145Operation cutoff time 224 224 224 224 224 173 173Departure time 229 229 229 229 229 178 178Destination 16 17 18 19 20 23 24Arrival time 1048 106 109 1054 1014 1027 109Operation start time 1055 1067 1077 1061 1021 1034 1097Distance (unit km) 2240 2265 2334 2249 2255 2357 2254Capacity (unit TEU) 10 22 42 10 5 100 28Operation period (unit daytrain) 2 2 2 2 2 2 2Rail service number 22 23 24 25 26 27 28Origin 14 14 14 21 21 21 21Operation start time 145 145 145 19 1 1 1Operation cutoff time 173 173 173 207 35 35 35Departure time 178 178 178 212 4 4 4Destination 26 27 28 23 24 26 27Arrival time 1022 1021 1088 787 109 1022 1021Operation start time 1029 1028 1095 799 1097 1029 1028Distance (unit km) 2213 2223 2280 1848 1745 1704 1714Capacity (unit TEU) 6 10 4 100 14 8 10Operation period (unit daytrain) 2 2 2 2 2 2 2


Table 7 Continued

Rail service number 29 30 31 32 33 34 35Origin 21 22 22 22 22 22 22Operation start time 1 7 7 7 7 7 7Operation cutoff time 35 95 95 95 95 95 95Departure time 4 10 10 10 10 10 10Destination 28 24 25 26 28 29 30Arrival time 1088 935 875 822 746 781 852Operation start time 1095 942 882 829 753 788 859Distance (unit km) 1771 1661 1644 1640 1573 1590 1603Capacity (unit TEU) 20 8 10 6 8 4 12Operation period (unit daytrain) 2 2 2 2 2 2 2Rail service number 36 37 38 39 40 41 42Origin 22 22 34 35 36 37 38Operation start time 7 7 17 205 10 16 155Operation cutoff time 95 95 179 218 115 184 179Departure time 10 10 184 223 12 189 184Destination 31 32 40 40 40 40 40Arrival time 808 895 566 566 566 566 566Operation start time 815 902 573 573 573 573 573Distance (unit km) 1610 1660 789 826 671 553 861Capacity (unit TEU) 16 36 14 16 10 12 48Operation period (unit daytrain) 2 2 2 2 2 2 2

Table 8 Information on the multiple commodity flows in the large-scale empirical example

Number Pickup service Delivery service Origin Destination Volume (TEU) Release time Due date1 Y N 1 3 21 8 402 Y Y 1 10 30 36 1093 N Y 1 10 10 39 1204 Y N 2 3 43 16 1115 Y Y 2 3 14 74 1616 N Y 2 9 45 10 1387 Y Y 4 13 32 28 1838 N Y 4 13 16 111 1989 Y N 4 18 19 15 10510 Y N 4 18 37 90 20111 Y N 8 13 48 22 10612 Y N 8 13 28 137 21013 Y Y 14 32 44 18 7014 Y Y 14 32 18 68 19615 N Y 22 40 52 4 6016 Y N 22 40 8 15 9817 Y N 22 40 40 67 20518 Y N 33 15 39 5 10619 Y Y 33 15 10 81 17320 Y Y 33 15 14 125 20821 N Y 33 40 55 12 5822 Y Y 33 40 19 77 19923 Y N 39 23 11 124 21024 Y Y 39 23 15 85 19225 N Y 39 32 37 98 226


Table 9 Computational results of Lingo 12 towards the large-scale empirical example

Service mode Solver type Solution Computational time StateMultimodal service B-and-B 5251563 yen 1min 35 sec Global opt

Table 10 Optimal multimodal transportation routing framework for the large-scale empirical example

Number Multimodal transportation route Arrival time at destination1 1

997888997888997888997888997888997888997888997888997888997888rarrroad service 3 18

2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3


7 4997888997888997888997888997888997888997888997888997888997888rarrroad service 12

997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175

13 14997888997888997888997888997888997888997888997888997888997888rarrroad service 32 44


997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759


(4) generalized costs optimum to satisfy customer demandsMoreover it formulates the multimodal transportation rout-ing problem as a capacitated time-sensitive multicommoditymultimodal generalized shortest path problem Specificallythe time-sensitive routing better matches the operationalcharacteristics of the schedule-based services correspondsto transportation practice and can hence provide betterdecision support for decision makers Additionally from theperspectives of solvability the linearization method that wehave developed can easily linearize the model which makesit effectively solvable by any mathematical programmingsoftware

Although several advances have been made by this studyweaknesses still exist The most significant one is that theproblem addressed by this study is oriented toward thedemands of a deterministic transportation environmentwhich means that all the transportation demands are knownand determined In reality many studies and practical datahave indicated that transportation demands show remarkable

spatial and temporal uncertainty For this reason satisfyingfluctuating transportation demands presents great challengesfor both decision makers and researchers [57 58] Thereforefurther study is needed to address the stochastic multimodaltransportation routing problem

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This study was supported by the National Natural ScienceFoundation Project (no 71390332-3) of China The authorswould like to thank Dr Xuesong Zhou at School of Sus-tainable Engineering and the Built Environment ArizonaState University USA for his constructive suggestions and


comments that lead to a significant improvement of thispaper

References

[1] Y Sun M X Lang and D Z Wang ldquoOptimization models andsolution algorithms for freight routing problem in the multi-modal transportation networks a review of the state-of-the-artrdquo The Open Civil Engineering Journal vol 9 no 1 pp 714ndash723 2015

[2] H Min ldquoInternational intermodal choices via chance-constrained goal programmingrdquo Transportation Research PartA General vol 25 no 6 pp 351ndash362 1991

[3] S Atalay M Canci G Kaya C Oguz and M Turkay ldquoInter-modal transportation in Istanbul via Marmarayrdquo IBM Journalof Research and Development vol 54 pp 1ndash9 2010

[4] K Liu Modern Logistics Technology Foundation TsinghuaUniversity Press amp Beijing Jiaotong University Press BeijingChina 2004

[5] B Ayar and H Yaman ldquoAn intermodal multicommodity rout-ing problemwith scheduled servicesrdquoComputational Optimiza-tion and Applications vol 53 no 1 pp 131ndash153 2012

[6] M Janic ldquoModelling the full costs of an intermodal androad freight transport networkrdquo Transportation Research DTransport and Environment vol 12 no 1 pp 33ndash44 2007

[7] C-H Liao P-H Tseng and C-S Lu ldquoComparing carbondioxide emissions of trucking and intermodal container trans-port in Taiwanrdquo Transportation Research Part D Transport andEnvironment vol 14 no 7 pp 493ndash496 2009

[8] J H Bookbinder andN S Fox ldquoIntermodal routing of Canada-Mexico shipments under NAFTArdquo Transportation Research ELogistics and Transportation Review vol 34 no 4 pp 289ndash3031998

[9] R Banomyong and A K C Beresford ldquoMultimodal transportthe case of Laotian garment exportersrdquo International Journal ofPhysical Distribution amp Logistics Management vol 31 no 9 pp663ndash685 2001

[10] J H Cho H S Kim and H R Choi ldquoAn intermodal transportnetwork planning algorithm using dynamic programming-acase study from Busan to Rotterdam in intermodal freightroutingrdquo Applied Intelligence vol 36 no 3 pp 529ndash541 2012

[11] W Meethom and A Kengpol ldquoDesign of a decision supportsystem for selecting multimodal transportation route an inte-grated model using AHP and ZOGP case study Thailand-Vietnamrdquo in Proceedings of the 2009 IEEE International Confer-ence on Industrial Engineering andEngineeringManagement pp2304ndash2308 December 2009

[12] A Caris C Macharis and G K Janssens ldquoPlanning prob-lems in intermodal freight transport accomplishments andprospectsrdquo Transportation Planning and Technology vol 31 no3 pp 277ndash302 2008

[13] L Moccia J-F Cordeau G Laporte S Ropke and M PValentini ldquoModeling and solving a multimodal transportationproblem with flexible-time and scheduled servicesrdquo Networksvol 57 no 1 pp 53ndash68 2011

[14] A J Craig E E Blanco and Y Sheffi ldquoEstimating the CO2

intensity of intermodal freight transportationrdquo TransportationResearch Part D Transport and Environment vol 22 pp 49ndash532013

[15] E Van Ierland C Graveland and R Huiberts ldquoAn environ-mental economic analysis of the new rail link to Europeanmain

port RotterdamrdquoTransportation Research Part D Transport andEnvironment vol 5 no 3 pp 197ndash209 2000

[16] H Li Y Lu J Zhang and T Wang ldquoTrends in road freighttransportation carbon dioxide emissions and policies in ChinardquoEnergy Policy vol 57 pp 99ndash106 2013

[17] Y M Bontekoning C Macharis and J J Trip ldquoIs a newapplied transportation research field emergingmdasha review ofintermodal rail-truck freight transport literaturerdquo Transporta-tion Research Part A Policy and Practice vol 38 no 1 pp 1ndash342004

[18] S K Lam and T Srikanthan ldquoAccelerating the k-shortestpaths computation in multimodal transportation networksrdquoin Proceedings of the IEEE 5th International Conference onIntelligent Transportation Systems pp 491ndash495 Singapore 2002

[19] C Barnhart and H D Ratliff ldquoModeling intermodal routingrdquoJournal of Business Logistics vol 14 no 1 pp 205ndash223 1993

[20] B S Boardman E M Malstrom D P Butler and M HCole ldquoComputer assisted routing of intermodal shipmentsrdquoComputers and Industrial Engineering vol 33 no 1-2 pp 311ndash314 1997

[21] A Ziliaskopoulos and W Wardell ldquoAn intermodal optimumpath algorithm for multimodal networks with dynamic arctravel times and switching delaysrdquo European Journal of Oper-ational Research vol 125 no 3 pp 486ndash502 2000

[22] A Lozano and G Storchi ldquoShortest viable path algorithm inmultimodal networksrdquo Transportation Research A Policy andPractice vol 35 no 3 pp 225ndash241 2001

[23] M Boussedjra C Bloch and A El Moudni ldquoAn exact methodto find the intermodal shortest path (ISP)rdquo in Proceedings ofthe IEEE International Conference on Networking Sensing andControl (ICNSC rsquo04) vol 2 pp 1075ndash1080 IEEE Taipei TaiwanMarch 2004

[24] J J Winebrake J J Corbett A Falzarano et al ldquoAssessingenergy environmental and economic tradeoffs in intermodalfreight transportationrdquo Journal of the Air ampWaste ManagementAssociation vol 58 no 8 pp 1004ndash1013 2008

[25] J Y Zhang andYHGuo ldquoAmultimode transportationnetworkassignmentmodelrdquo Journal of the China Railway Society vol 24no 4 pp 114ndash116 2002

[26] Y-H Zhang B-L Lin D Liang and H-Y Gao ldquoResearch ona generalized shortest path method of optimizing intermodaltransportation problemsrdquo Journal of the China Railway Societyvol 28 no 4 pp 22ndash26 2006

[27] H C Sun X H Li and D W Chen ldquoModeling and solutionmethods for viable routes in multimodal networksrdquo in Proceed-ings of the 4th IEEE International Conference on Managementof Innovation and Technology (ICMIT rsquo08) pp 1384ndash1388Bangkok Thailand September 2008

[28] B Sun and Q S Chen ldquoThe routing optimization for multi-modal transport with carbon emission consideration underuncertaintyrdquo in Proceedings of the 32nd Chinese Control Confer-ence (CCC rsquo13) pp 8135ndash8140 Xirsquoan China July 2013

[29] J Liu S-WHe R Song andH-D Li ldquoStudy on optimization ofdynamic paths of intermodal transportation network based onalternative set of transport modesrdquo Journal of the China RailwaySociety vol 33 no 10 pp 1ndash6 2011

[30] Y-C Zeng Y Wang and Z-Z Lai ldquoResearch on model andalgorithm of multimodal transportation with time windowsrdquoIndustrial Engineering Journal vol 12 no 2 pp 24ndash28 2009

[31] X Wang Z B Chi and X L Ge ldquoResearch and analysis fortime-limited multimodal transport model of vehiclerdquo Applica-tion Research of Computers vol 28 no 2 pp 563ndash565 2011


[32] Z Q Fan and M L Le ldquoResearch on multimodal transportrouting problem with soft time windows under stochasticenvironmentrdquo Industrial Engineering and Management vol 16no 5 pp 68ndash72 2011

[33] Y Li J Zhao G Wu and J Chen ldquoSolving the mode selectionproblem with fixed transportation cost in intermodal trans-portationrdquo Journal of Southwest Jiaotong University vol 47 no5 pp 881ndash887 2012

[34] L S Tang and J Z Huo ldquoMultimodal transport distributionnetwork design with time windowrdquo in Proceedings of theInternational Conference on Management and Service Science(MASS rsquo11) pp 1ndash4 Wuhan China August 2011

[35] K Kang H J Niu Y J Zhu and W C Zhang ldquoResearchon improved integrated optimization model for mode andRoute in multimodal transportation basing on the PSO-ACOrdquoin Proceedings of the International Conference on LogisticsSystems and Intelligent Management (ICLSIM rsquo10) pp 1445ndash1449 Harbin China January 2010

[36] O Cakır ldquoBenders decomposition applied tomulti-commoditymulti-mode distribution planningrdquo Expert Systems with Appli-cations vol 36 no 4 pp 8212ndash8217 2009

[37] M Verma V Verter and N Zufferey ldquoA bi-objective model forplanning andmanaging rail-truck intermodal transportation ofhazardous materialsrdquo Transportation Research Part E Logisticsand Transportation Review vol 48 no 1 pp 132ndash149 2012

[38] Y P Cai L Zhang and L N Shao ldquoOptimalmulti-modal trans-port model for full loads with time windowsrdquo in Proceedings ofthe International Conference on Logistics Systems and IntelligentManagement (ICLSIM rsquo10) pp 147ndash151 IEEE Harbin ChinaJanuary 2010

[39] K Lei X Zhu J Hou andW Huang ldquoDecision of multimodaltransportation scheme based on swarm intelligencerdquo Mathe-matical Problems in Engineering vol 2014 Article ID 93283210 pages 2014

[40] Y-T Chang P T-W Lee H-J Kim and S-H Shin ldquoOptimiza-tion model for transportation of container cargoes consideringshort sea shipping and external cost South Korean caserdquoTransportation Research Record vol 2166 pp 99ndash108 2010

[41] G W Xiong and Y Wang ldquoBest routes selection in multimodalnetworks using multi-objective genetic algorithmrdquo Journal ofCombinatorial Optimization vol 28 no 3 pp 655ndash673 2014

[42] T-S Chang ldquoBest routes selection in international intermodalnetworksrdquo Computers amp Operations Research vol 35 no 9 pp2877ndash2891 2008

[43] T G Crainic and J-M Rousseau ldquoMulticommodity multi-mode freight transportation a general modeling and algo-rithmic framework for the service network design problemrdquoTransportation Research B vol 20 no 3 pp 225ndash242 1986

[44] K Daeki C Barnhart KWare and G Reinhardt ldquoMultimodalexpress package delivery a service network design applicationrdquoTransportation Science vol 33 no 4 pp 391ndash407 1999

[45] M Zhang B Wiegmans and L Tavasszy ldquoOptimization ofmultimodal networks including environmental costs a modeland findings for transport policyrdquo Computers in Industry vol64 no 2 pp 136ndash145 2013

[46] Y Qu T Bektas and J Bennell ldquoSustainability SI multimodemulticommodity network design model for intermodal freighttransportation with transfer and emission costsrdquo Networks andSpatial Economics 2014

[47] T G Crainic ldquoService network design in freight transporta-tionrdquo European Journal of Operational Research vol 122 no 2pp 272ndash288 2000

[48] F Southworth andB E Peterson ldquoIntermodal and internationalfreight network modelingrdquo Transportation Research Part CEmerging Technologies vol 8 no 1ndash6 pp 147ndash166 2000

[49] B Jansen P C J Swinkels G J A Teeuwen B van Antwerpende Fluiter and H A Fleuren ldquoOperational planning of a large-scale multi-modal transportation systemrdquo European Journal ofOperational Research vol 156 no 1 pp 41ndash53 2004

[50] M Yaghini and R Akhavan ldquoMulticommodity network designproblem in rail freight transportation planningrdquo ProcediamdashSocial and Behavioral Sciences vol 43 pp 728ndash739 2012 8thInternational Conference on Traffic and Transportation Studies(ICTTS 2012)

[51] A Balakrishnan T L Magnanti and R T Wong ldquoA dual-ascent procedure for large-scale uncapacitated network designrdquoOperations Research vol 37 no 5 pp 716ndash740 1989

[52] K Holmberg and D Yuan ldquoA Lagrangian heuristic basedbranch-and-bound approach for the capacitated networkdesign problemrdquo Operations Research vol 48 no 3 pp 461ndash481 2000

[53] T G Crainic Y Li and M Toulouse ldquoA first multilevel coop-erative algorithm for capacitated multicommodity networkdesignrdquo Computers and Operations Research vol 33 no 9 pp2602ndash2622 2006

[54] A Frangioni and B Gendron ldquo0-1 reformulations of themulticommodity capacitated network design problemrdquoDiscreteApplied Mathematics vol 157 no 6 pp 1229ndash1241 2009

[55] X H Wen B L Lin S W He J Zhang and J Li ldquoTheoptimization model for flow assignment and routing plan inintegrated transportation based on carbon emissionsrdquo Transac-tions of Beijing Institute of Technology vol 33 supplement 1 pp5ndash8 2013

[56] Y Sun andM Lang ldquoBi-objective optimization formulti-modaltransportation routing planning problem based on Paretooptimalityrdquo Journal of Industrial Engineering and Managementvol 8 no 4 pp 1195ndash1217 2015

[57] R A Garrido and H S Mahmassani ldquoForecasting freighttransportation demandwith the spacendashtimemultinomial probitmodelrdquo Transportation Research B Methodological vol 34 no5 pp 403ndash418 2000

[58] R Bai S W Wallace J Li and A Y-L Chong ldquoStochasticservice network designwith reroutingrdquoTransportation ResearchPart B Methodological vol 60 pp 50ndash65 2014

Submit your manuscripts athttpwwwhindawicom

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Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


distances These trends showed that compared with roadtransportation multimodal transportation decreases theaverage full costs more rapidly as the distance increasesand shows fewer full costs if the transportation distanceexceeds approximately 1000 km (the critical distance whenthe rail service frequency is 5 trainsweek) Moreover thecritical distance decreases significantly as the rail servicefrequency increases Liao et al [7] compared the carbondioxide emissions of the long-haul truck transportation andthemultimodal transportation (truck-ocean combination) ina case study of container transportation between Kaohsiungand Keelung in Taiwan The activity-based method wasadopted to estimate the carbon dioxide emissions in thetransportation process and the computational results ofthe case indicated that the carbon dioxide emissions fellfrom 566525 tonnes to 163830 tonnes corresponding to a71 decrease when truck transportation was replaced bymultimodal transportation These results supported the viewthat the use of multimodal transportation as an alternative tolong-haul truck transportation canmarkedly decrease carbondioxide emissions from the transportation sector

Many researchers have studied the empirical applicationsof multimodal transportation in specific cases for exampleBookbinder and Fox [8] discussed multimodal transporta-tion shortest time and minimal costs route selection for fivecommodity flows that need to be routed from Canada toMexico in the North American Free Trade Agreement areaBanomyong and Beresford [9] explored multimodal trans-portation route selection for exporting commodities fromLaos to the European Union and proposed a cost model formultimodal transportation to help exporters make decisionsCho et al [10] presented a weighted constrained shortestpath problem model and a label setting algorithm to selectmultimodal transportation routes from Busan to RotterdamMeethom and Kengpol [11] combined the AHP (AnalyticHierarchy Process) method with 0-1 goal programming todesign a decision support system for selecting an optimalmultimodal transportation route from Bangkok to Danang

Currently with the rapid development of economic glob-alization international trade and the accompanying globalcirculation of commodities have grown remarkably Conse-quently transportation networks have expanded widely andcommodity distribution routes have extended significantlyThese changes represent challenges to transportation perfor-mance in terms of for example costs timeliness and envi-ronmental concerns [2 12] In response to these challengesincreasing numbers of companies have adopted multimodaltransportation schemes to transport their products or rawmaterials According to the relevant statistics logistics costsrepresent 30ndash50 of the product cost [2] For this reasonlogistics cost is one of themost effective targets for companiesto lower product cost raise profits and maintain competi-tiveness in the international trade market Therefore freightrouting for commodity transportation has been given greatimportance by both the management decision makers andmultimodal transport operators

12 Formulation Characteristics of the Study The practi-cal demand for lowering logistics costs motivates research

interests in the freight routing problem in multimodaltransportation networks The essence of this problem is toselect optimal routes tomove commodities from their originsto their destinations through a multimodal transportationnetwork In our study we consider following formulationcharacteristics to make the formulation of this problemcorrespond more closely to actual practice

(1) Multicommodity Flow Routing Freight routing aims atsatisfying customer demands at minimal costs Each trans-portation demand can be represented by a commodity flowthat is characterized by five attributes origin destinationvolume release time and due date From the perspective ofthe entiremultimodal transportation networkmore than onetransportation demand within the planning horizon needs tobe satisfied Various transportation demands differ from eachother in at least one of the five above-mentioned attributionsThus multiple commodity flows must be routed to satisfythe various transportation demands Therefore the freightrouting problem explored in this study will be developed as amulticommodity problem

(2) Schedule-Based andTime-Flexible Transportation ServicesIn a multimodal transportation network we can combinedifferent transportation services to move commodities alongroutes The most obvious distinction among the transporta-tion services is in terms of their operating modes In thisstudy we define two categories for classifying transportationservices The first category is that of schedule-based servicesthat is operations controlled by schedules The secondcategory is that of time-flexible services that is operationsnot controlled by schedulesThis distinction is similar to thatdrawn in Moccia et alrsquos study [13] Schedule-based services(eg rail services) are operated based on schedules that arespecified in advance Conversely time-flexible services (egroad services) are not constrained by time and can travelthrough the network flexiblyThe operations of the two typesof transportation services and the transshipments amongthem will be emphasized in Section 2

(3) Consideration of Carbon Dioxide Emissions With theevolution of society and improvement in the awareness ofsustainable development environmental issues have drawnincreasing attention from both the government and thepublic especially in developing countries for exampleChina a rapidly industrializing country One of the mostpressing environmental issues is global warming causedby greenhouse gas emissions Carbon dioxide representsapproximately 80 of the total greenhouse gas emissions [7]and the reduction of these emissions is acknowledged to be ahighly challenging problem worldwide In the transportationsector various activities have been shown to represent up to19 of the global energy consumption [14] and to producelarge amounts of greenhouse gases (eg carbon dioxidecarbon monoxide and methane) and air pollutants (egsulfur dioxide and oxynitride) [15] For example accordingto Li et alrsquos evaluation [16] carbon dioxide emissions fromthe Chinese transportation sector represent 837 (5396)of the total carbon dioxide emissions from fossil fuel (liquid



























Time axis

h

i

j

k

Term

inal

sequ

ence


to terminal k

Scheduleddeparture

time


windowScheduled

arrival time























Loadingoperation

Unloadingoperation






1

1

1

1 12

2

2 2

2

4

4

Shippersrsquo area

Receiversrsquo area

3

3

2

1

2







i Terminal

Road service

kjih




Road service

Tim

e

Location






t7

t6t5

t4

t3

t2

t1





Tim

e

Location




t7

t6

t5

t4

t3

t2

t1


Δt34

t998400









1199051



1199052



(Figure 6)

1199053


1199054


1199055


1199056



Δ11990534


= max0 1199054minus 1199053









1199051



1199053


1199054


1199055





Road service

Tim

e

Location



t6

t5

t4

t3

t2

t1



(rail service s)





1 119877









5at 36 depart







4



5at 36 depart









4at




4at




6at 27




5at 33 depart


terminal 7 at 445


5

6

8

4

8

10

7

[19 215 2225] [28 285]

[27 31] 32 43 [44 47]

[10 15 16] [24 25]

[25 27 275] [355 36]

[11 13 1325] [18 185]

[33 38 385] [445 45][35 39 40] [47 475]

1

2

4

56

7

3



R1

R4

R5

R7

R6

R3

R2



2at
















119896







minimize sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895

119862119896

119894119895119904sdot 119883119896

119894119895119904 (1)

+ sum

119896isin119870

sum

119894isin119873

( sum


sum

119903isin119878ℎ119894

119888119903sdot 119902119896sdot 119883119896

ℎ119894119903+ sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119888119904sdot 119902119896sdot 119883119896

119894119895119904) (2)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isinΓ119894119895

119888store sdot 119902119896 sdot 119885119896

119894119895119904 (3)

+ sum

119896isin119870

120593119896sdot sum

119895isin120575+(119900119896)

sum

119904isinΓ119900119896119895

119888pick sdot 119902119896sdot 119883119896

119900119896119895119904+ sum

119896isin119870

120583119896sdot sum

119894isin120575minus(119889119896)

sum

119904isinΓ119894119889119896


119894119889119896119904 (4)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895


119894119895119904 (5)


Subject to sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119883119896

119894119895119904minus sum


sum

119903isin119878ℎ119894

119883119896

ℎ119894119903

=

1 119894 = 119900119896

0 forall119894 isin 119873 119900119896 119889119896

minus1 119894 = 119889119896


(6)

sum

119904isin119878119894119895

119883119896



119883119896

ℎ119894119903+ 119883119896

119894119895119904le 1


+(119894) forall119904 isin Ω

119894119895 forall119903 isin Ω

ℎ119894

(8)



sum

119896isin119870

119902119896sdot 119883119896

119894119895119904le 119876119904


119894119895 (9)


119884119896

119900119896= 119905119896



(max 119884119896

119894 SD119904119894 + 119905119894119895119904

minus 119884119896

119895) sdot 119883119896

119894119895119904= 0


(11)




119894119895 we are far











Ω119894119895


119878119894119895


119894119895sube 119878

120575minus(119894)


120575+(119894)




119906119904


SA119904119894


SD119904119894


119876119904

119894119895


119889119894119895119904


119905119894119895119904


119862119896

119894119895119904


119888119904



119888pick


119888deliver


120593119896


119896= 1 otherwise 120593

119896= 0

120583119896


119896= 1 otherwise 120583

119896= 0


120578119904



119883119896

119894119895119904


119894119895119904= 1

otherwise 119883119896119894119895119904

= 0119884119896


119885119896

119894119895119904


arc (119894 119895) Hence119884119896119895equals 119897119904



in this case 119905119894119895119904

= (119897119904

119895minus SD119904119894)

119884119896

119894le 119906119904

119894sdot 119883119896

119894119895119904+ 119872 sdot (1 minus 119883

119896

119894119895119904)


(12)


119894119895119904= 0 then 119884

119896



119884119896

119889119896le 119879119896

forall119896 isin 119870 (13)


(max 0 119897119904

119894minus 119884119896

119894minus 120587 minus 119885

119896

119894119895119904) sdot 119883119896

119894119895119904= 0


(14)


119862119896

119894119895119904=

(1198881

rail + 1198882

rail sdot 119889119894119895119904) sdot 119902119896

119904 isin Γ119894119895

1198882

road sdot 119902119896sdot 119889119894119895119904

119904 isin Ω119894119895


(15)


2

rail and 1198882


119883119896


119894119895 (16)

119884119896


119885119896


119894119895 (18)






119894+119905119894119895119904minus119884119896

119895)sdot

119883119896




119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(19)

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(20)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(21)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(22)




119894119895119904= 1


minus 119884119896

119895= 0 In this


minus119884119896

119895) ge 0

and (SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge 0

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0

997904rArr 0 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0 997904rArr 119884

119896

119895= SD119904119894+ 119905119894119895119904 (23)



119894and119884

119896



minus 119884119896

119895) ge minus119872 and

(SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge minus119872

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872

997904rArr minus119872 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872 (24)



119885119896

119894119895119904) sdot 119883119896


119894119895can be


119885119896

119894119895119904ge 119872 sdot (119883

119896

119894119895119904minus 1) + (119897

119904

119894minus 119884119896

119894minus 120587)


(25)

119885119896

119894119895119904le 119872 sdot 119883

119896


119894119895 (26)



119896

119894119895119904= 0 and


(119897119904

119894minus 119884119896



ge minus119872 and 119885119896

119894119895119904le 0 When



119885119896

119894119895119904ge minus119872

119885119896

119894119895119904le 0

119885119896

119894119895119904ge 0

997904rArr 0 le 119885119896

119894119895119904le 0 997904rArr 119885

119896

119894119895119904= 0 (27)


119896

119894119895119904= 1 This scenario





119896

119894119895119904ge (119897119904

119894minus 119884119896

119894minus 120587) and 119885

119896

119894119895119904le 119872




119894minus 119884119896



119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)



(119897119904

119894minus 119884119896



119894119895119904(Constraint (17)) the


119894minus119884119896


119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)



minimize (1) + (2) + (3) + (4) + (5) (30)









1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



































Time axis

h

i

j

k

Term

inal

sequ

ence


to terminal k

Scheduleddeparture

time


windowScheduled

arrival time























Loadingoperation

Unloadingoperation






1

1

1

1 12

2

2 2

2

4

4

Shippersrsquo area

Receiversrsquo area

3

3

2

1

2







i Terminal

Road service

kjih




Road service

Tim

e

Location






t7

t6t5

t4

t3

t2

t1





Tim

e

Location




t7

t6

t5

t4

t3

t2

t1


Δt34

t998400









1199051



1199052



(Figure 6)

1199053


1199054


1199055


1199056



Δ11990534


= max0 1199054minus 1199053









1199051



1199053


1199054


1199055





Road service

Tim

e

Location



t6

t5

t4

t3

t2

t1



(rail service s)





1 119877









5at 36 depart







4



5at 36 depart









4at




4at




6at 27




5at 33 depart


terminal 7 at 445


5

6

8

4

8

10

7

[19 215 2225] [28 285]

[27 31] 32 43 [44 47]

[10 15 16] [24 25]

[25 27 275] [355 36]

[11 13 1325] [18 185]

[33 38 385] [445 45][35 39 40] [47 475]

1

2

4

56

7

3



R1

R4

R5

R7

R6

R3

R2



2at
















119896







minimize sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895

119862119896

119894119895119904sdot 119883119896

119894119895119904 (1)

+ sum

119896isin119870

sum

119894isin119873

( sum


sum

119903isin119878ℎ119894

119888119903sdot 119902119896sdot 119883119896

ℎ119894119903+ sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119888119904sdot 119902119896sdot 119883119896

119894119895119904) (2)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isinΓ119894119895

119888store sdot 119902119896 sdot 119885119896

119894119895119904 (3)

+ sum

119896isin119870

120593119896sdot sum

119895isin120575+(119900119896)

sum

119904isinΓ119900119896119895

119888pick sdot 119902119896sdot 119883119896

119900119896119895119904+ sum

119896isin119870

120583119896sdot sum

119894isin120575minus(119889119896)

sum

119904isinΓ119894119889119896


119894119889119896119904 (4)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895


119894119895119904 (5)


Subject to sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119883119896

119894119895119904minus sum


sum

119903isin119878ℎ119894

119883119896

ℎ119894119903

=

1 119894 = 119900119896

0 forall119894 isin 119873 119900119896 119889119896

minus1 119894 = 119889119896


(6)

sum

119904isin119878119894119895

119883119896



119883119896

ℎ119894119903+ 119883119896

119894119895119904le 1


+(119894) forall119904 isin Ω

119894119895 forall119903 isin Ω

ℎ119894

(8)



sum

119896isin119870

119902119896sdot 119883119896

119894119895119904le 119876119904


119894119895 (9)


119884119896

119900119896= 119905119896



(max 119884119896

119894 SD119904119894 + 119905119894119895119904

minus 119884119896

119895) sdot 119883119896

119894119895119904= 0


(11)




119894119895 we are far











Ω119894119895


119878119894119895


119894119895sube 119878

120575minus(119894)


120575+(119894)




119906119904


SA119904119894


SD119904119894


119876119904

119894119895


119889119894119895119904


119905119894119895119904


119862119896

119894119895119904


119888119904



119888pick


119888deliver


120593119896


119896= 1 otherwise 120593

119896= 0

120583119896


119896= 1 otherwise 120583

119896= 0


120578119904



119883119896

119894119895119904


119894119895119904= 1

otherwise 119883119896119894119895119904

= 0119884119896


119885119896

119894119895119904


arc (119894 119895) Hence119884119896119895equals 119897119904



in this case 119905119894119895119904

= (119897119904

119895minus SD119904119894)

119884119896

119894le 119906119904

119894sdot 119883119896

119894119895119904+ 119872 sdot (1 minus 119883

119896

119894119895119904)


(12)


119894119895119904= 0 then 119884

119896



119884119896

119889119896le 119879119896

forall119896 isin 119870 (13)


(max 0 119897119904

119894minus 119884119896

119894minus 120587 minus 119885

119896

119894119895119904) sdot 119883119896

119894119895119904= 0


(14)


119862119896

119894119895119904=

(1198881

rail + 1198882

rail sdot 119889119894119895119904) sdot 119902119896

119904 isin Γ119894119895

1198882

road sdot 119902119896sdot 119889119894119895119904

119904 isin Ω119894119895


(15)


2

rail and 1198882


119883119896


119894119895 (16)

119884119896


119885119896


119894119895 (18)






119894+119905119894119895119904minus119884119896

119895)sdot

119883119896




119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(19)

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(20)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(21)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(22)




119894119895119904= 1


minus 119884119896

119895= 0 In this


minus119884119896

119895) ge 0

and (SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge 0

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0

997904rArr 0 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0 997904rArr 119884

119896

119895= SD119904119894+ 119905119894119895119904 (23)



119894and119884

119896



minus 119884119896

119895) ge minus119872 and

(SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge minus119872

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872

997904rArr minus119872 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872 (24)



119885119896

119894119895119904) sdot 119883119896


119894119895can be


119885119896

119894119895119904ge 119872 sdot (119883

119896

119894119895119904minus 1) + (119897

119904

119894minus 119884119896

119894minus 120587)


(25)

119885119896

119894119895119904le 119872 sdot 119883

119896


119894119895 (26)



119896

119894119895119904= 0 and


(119897119904

119894minus 119884119896



ge minus119872 and 119885119896

119894119895119904le 0 When



119885119896

119894119895119904ge minus119872

119885119896

119894119895119904le 0

119885119896

119894119895119904ge 0

997904rArr 0 le 119885119896

119894119895119904le 0 997904rArr 119885

119896

119894119895119904= 0 (27)


119896

119894119895119904= 1 This scenario





119896

119894119895119904ge (119897119904

119894minus 119884119896

119894minus 120587) and 119885

119896

119894119895119904le 119872




119894minus 119884119896



119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)



(119897119904

119894minus 119884119896



119894119895119904(Constraint (17)) the


119894minus119884119896


119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)



minimize (1) + (2) + (3) + (4) + (5) (30)









1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Loadingoperation

Unloadingoperation






1

1

1

1 12

2

2 2

2

4

4

Shippersrsquo area

Receiversrsquo area

3

3

2

1

2







i Terminal

Road service

kjih




Road service

Tim

e

Location






t7

t6t5

t4

t3

t2

t1





Tim

e

Location




t7

t6

t5

t4

t3

t2

t1


Δt34

t998400









1199051



1199052



(Figure 6)

1199053


1199054


1199055


1199056



Δ11990534


= max0 1199054minus 1199053









1199051



1199053


1199054


1199055





Road service

Tim

e

Location



t6

t5

t4

t3

t2

t1



(rail service s)





1 119877









5at 36 depart







4



5at 36 depart









4at




4at




6at 27




5at 33 depart


terminal 7 at 445


5

6

8

4

8

10

7

[19 215 2225] [28 285]

[27 31] 32 43 [44 47]

[10 15 16] [24 25]

[25 27 275] [355 36]

[11 13 1325] [18 185]

[33 38 385] [445 45][35 39 40] [47 475]

1

2

4

56

7

3



R1

R4

R5

R7

R6

R3

R2



2at
















119896







minimize sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895

119862119896

119894119895119904sdot 119883119896

119894119895119904 (1)

+ sum

119896isin119870

sum

119894isin119873

( sum


sum

119903isin119878ℎ119894

119888119903sdot 119902119896sdot 119883119896

ℎ119894119903+ sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119888119904sdot 119902119896sdot 119883119896

119894119895119904) (2)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isinΓ119894119895

119888store sdot 119902119896 sdot 119885119896

119894119895119904 (3)

+ sum

119896isin119870

120593119896sdot sum

119895isin120575+(119900119896)

sum

119904isinΓ119900119896119895

119888pick sdot 119902119896sdot 119883119896

119900119896119895119904+ sum

119896isin119870

120583119896sdot sum

119894isin120575minus(119889119896)

sum

119904isinΓ119894119889119896


119894119889119896119904 (4)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895


119894119895119904 (5)


Subject to sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119883119896

119894119895119904minus sum


sum

119903isin119878ℎ119894

119883119896

ℎ119894119903

=

1 119894 = 119900119896

0 forall119894 isin 119873 119900119896 119889119896

minus1 119894 = 119889119896


(6)

sum

119904isin119878119894119895

119883119896



119883119896

ℎ119894119903+ 119883119896

119894119895119904le 1


+(119894) forall119904 isin Ω

119894119895 forall119903 isin Ω

ℎ119894

(8)



sum

119896isin119870

119902119896sdot 119883119896

119894119895119904le 119876119904


119894119895 (9)


119884119896

119900119896= 119905119896



(max 119884119896

119894 SD119904119894 + 119905119894119895119904

minus 119884119896

119895) sdot 119883119896

119894119895119904= 0


(11)




119894119895 we are far











Ω119894119895


119878119894119895


119894119895sube 119878

120575minus(119894)


120575+(119894)




119906119904


SA119904119894


SD119904119894


119876119904

119894119895


119889119894119895119904


119905119894119895119904


119862119896

119894119895119904


119888119904



119888pick


119888deliver


120593119896


119896= 1 otherwise 120593

119896= 0

120583119896


119896= 1 otherwise 120583

119896= 0


120578119904



119883119896

119894119895119904


119894119895119904= 1

otherwise 119883119896119894119895119904

= 0119884119896


119885119896

119894119895119904


arc (119894 119895) Hence119884119896119895equals 119897119904



in this case 119905119894119895119904

= (119897119904

119895minus SD119904119894)

119884119896

119894le 119906119904

119894sdot 119883119896

119894119895119904+ 119872 sdot (1 minus 119883

119896

119894119895119904)


(12)


119894119895119904= 0 then 119884

119896



119884119896

119889119896le 119879119896

forall119896 isin 119870 (13)


(max 0 119897119904

119894minus 119884119896

119894minus 120587 minus 119885

119896

119894119895119904) sdot 119883119896

119894119895119904= 0


(14)


119862119896

119894119895119904=

(1198881

rail + 1198882

rail sdot 119889119894119895119904) sdot 119902119896

119904 isin Γ119894119895

1198882

road sdot 119902119896sdot 119889119894119895119904

119904 isin Ω119894119895


(15)


2

rail and 1198882


119883119896


119894119895 (16)

119884119896


119885119896


119894119895 (18)






119894+119905119894119895119904minus119884119896

119895)sdot

119883119896




119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(19)

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(20)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(21)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(22)




119894119895119904= 1


minus 119884119896

119895= 0 In this


minus119884119896

119895) ge 0

and (SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge 0

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0

997904rArr 0 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0 997904rArr 119884

119896

119895= SD119904119894+ 119905119894119895119904 (23)



119894and119884

119896



minus 119884119896

119895) ge minus119872 and

(SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge minus119872

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872

997904rArr minus119872 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872 (24)



119885119896

119894119895119904) sdot 119883119896


119894119895can be


119885119896

119894119895119904ge 119872 sdot (119883

119896

119894119895119904minus 1) + (119897

119904

119894minus 119884119896

119894minus 120587)


(25)

119885119896

119894119895119904le 119872 sdot 119883

119896


119894119895 (26)



119896

119894119895119904= 0 and


(119897119904

119894minus 119884119896



ge minus119872 and 119885119896

119894119895119904le 0 When



119885119896

119894119895119904ge minus119872

119885119896

119894119895119904le 0

119885119896

119894119895119904ge 0

997904rArr 0 le 119885119896

119894119895119904le 0 997904rArr 119885

119896

119894119895119904= 0 (27)


119896

119894119895119904= 1 This scenario





119896

119894119895119904ge (119897119904

119894minus 119884119896

119894minus 120587) and 119885

119896

119894119895119904le 119872




119894minus 119884119896



119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)



(119897119904

119894minus 119884119896



119894119895119904(Constraint (17)) the


119894minus119884119896


119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)



minimize (1) + (2) + (3) + (4) + (5) (30)









1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










i Terminal

Road service

kjih




Road service

Tim

e

Location






t7

t6t5

t4

t3

t2

t1





Tim

e

Location




t7

t6

t5

t4

t3

t2

t1


Δt34

t998400









1199051



1199052



(Figure 6)

1199053


1199054


1199055


1199056



Δ11990534


= max0 1199054minus 1199053









1199051



1199053


1199054


1199055





Road service

Tim

e

Location



t6

t5

t4

t3

t2

t1



(rail service s)





1 119877









5at 36 depart







4



5at 36 depart









4at




4at




6at 27




5at 33 depart


terminal 7 at 445


5

6

8

4

8

10

7

[19 215 2225] [28 285]

[27 31] 32 43 [44 47]

[10 15 16] [24 25]

[25 27 275] [355 36]

[11 13 1325] [18 185]

[33 38 385] [445 45][35 39 40] [47 475]

1

2

4

56

7

3



R1

R4

R5

R7

R6

R3

R2



2at
















119896







minimize sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895

119862119896

119894119895119904sdot 119883119896

119894119895119904 (1)

+ sum

119896isin119870

sum

119894isin119873

( sum


sum

119903isin119878ℎ119894

119888119903sdot 119902119896sdot 119883119896

ℎ119894119903+ sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119888119904sdot 119902119896sdot 119883119896

119894119895119904) (2)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isinΓ119894119895

119888store sdot 119902119896 sdot 119885119896

119894119895119904 (3)

+ sum

119896isin119870

120593119896sdot sum

119895isin120575+(119900119896)

sum

119904isinΓ119900119896119895

119888pick sdot 119902119896sdot 119883119896

119900119896119895119904+ sum

119896isin119870

120583119896sdot sum

119894isin120575minus(119889119896)

sum

119904isinΓ119894119889119896


119894119889119896119904 (4)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895


119894119895119904 (5)


Subject to sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119883119896

119894119895119904minus sum


sum

119903isin119878ℎ119894

119883119896

ℎ119894119903

=

1 119894 = 119900119896

0 forall119894 isin 119873 119900119896 119889119896

minus1 119894 = 119889119896


(6)

sum

119904isin119878119894119895

119883119896



119883119896

ℎ119894119903+ 119883119896

119894119895119904le 1


+(119894) forall119904 isin Ω

119894119895 forall119903 isin Ω

ℎ119894

(8)



sum

119896isin119870

119902119896sdot 119883119896

119894119895119904le 119876119904


119894119895 (9)


119884119896

119900119896= 119905119896



(max 119884119896

119894 SD119904119894 + 119905119894119895119904

minus 119884119896

119895) sdot 119883119896

119894119895119904= 0


(11)




119894119895 we are far











Ω119894119895


119878119894119895


119894119895sube 119878

120575minus(119894)


120575+(119894)




119906119904


SA119904119894


SD119904119894


119876119904

119894119895


119889119894119895119904


119905119894119895119904


119862119896

119894119895119904


119888119904



119888pick


119888deliver


120593119896


119896= 1 otherwise 120593

119896= 0

120583119896


119896= 1 otherwise 120583

119896= 0


120578119904



119883119896

119894119895119904


119894119895119904= 1

otherwise 119883119896119894119895119904

= 0119884119896


119885119896

119894119895119904


arc (119894 119895) Hence119884119896119895equals 119897119904



in this case 119905119894119895119904

= (119897119904

119895minus SD119904119894)

119884119896

119894le 119906119904

119894sdot 119883119896

119894119895119904+ 119872 sdot (1 minus 119883

119896

119894119895119904)


(12)


119894119895119904= 0 then 119884

119896



119884119896

119889119896le 119879119896

forall119896 isin 119870 (13)


(max 0 119897119904

119894minus 119884119896

119894minus 120587 minus 119885

119896

119894119895119904) sdot 119883119896

119894119895119904= 0


(14)


119862119896

119894119895119904=

(1198881

rail + 1198882

rail sdot 119889119894119895119904) sdot 119902119896

119904 isin Γ119894119895

1198882

road sdot 119902119896sdot 119889119894119895119904

119904 isin Ω119894119895


(15)


2

rail and 1198882


119883119896


119894119895 (16)

119884119896


119885119896


119894119895 (18)






119894+119905119894119895119904minus119884119896

119895)sdot

119883119896




119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(19)

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(20)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(21)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(22)




119894119895119904= 1


minus 119884119896

119895= 0 In this


minus119884119896

119895) ge 0

and (SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge 0

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0

997904rArr 0 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0 997904rArr 119884

119896

119895= SD119904119894+ 119905119894119895119904 (23)



119894and119884

119896



minus 119884119896

119895) ge minus119872 and

(SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge minus119872

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872

997904rArr minus119872 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872 (24)



119885119896

119894119895119904) sdot 119883119896


119894119895can be


119885119896

119894119895119904ge 119872 sdot (119883

119896

119894119895119904minus 1) + (119897

119904

119894minus 119884119896

119894minus 120587)


(25)

119885119896

119894119895119904le 119872 sdot 119883

119896


119894119895 (26)



119896

119894119895119904= 0 and


(119897119904

119894minus 119884119896



ge minus119872 and 119885119896

119894119895119904le 0 When



119885119896

119894119895119904ge minus119872

119885119896

119894119895119904le 0

119885119896

119894119895119904ge 0

997904rArr 0 le 119885119896

119894119895119904le 0 997904rArr 119885

119896

119894119895119904= 0 (27)


119896

119894119895119904= 1 This scenario





119896

119894119895119904ge (119897119904

119894minus 119884119896

119894minus 120587) and 119885

119896

119894119895119904le 119872




119894minus 119884119896



119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)



(119897119904

119894minus 119884119896



119894119895119904(Constraint (17)) the


119894minus119884119896


119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)



minimize (1) + (2) + (3) + (4) + (5) (30)









1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












1199051



1199053


1199054


1199055





Road service

Tim

e

Location



t6

t5

t4

t3

t2

t1



(rail service s)





1 119877









5at 36 depart







4



5at 36 depart









4at




4at




6at 27




5at 33 depart


terminal 7 at 445


5

6

8

4

8

10

7

[19 215 2225] [28 285]

[27 31] 32 43 [44 47]

[10 15 16] [24 25]

[25 27 275] [355 36]

[11 13 1325] [18 185]

[33 38 385] [445 45][35 39 40] [47 475]

1

2

4

56

7

3



R1

R4

R5

R7

R6

R3

R2



2at
















119896







minimize sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895

119862119896

119894119895119904sdot 119883119896

119894119895119904 (1)

+ sum

119896isin119870

sum

119894isin119873

( sum


sum

119903isin119878ℎ119894

119888119903sdot 119902119896sdot 119883119896

ℎ119894119903+ sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119888119904sdot 119902119896sdot 119883119896

119894119895119904) (2)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isinΓ119894119895

119888store sdot 119902119896 sdot 119885119896

119894119895119904 (3)

+ sum

119896isin119870

120593119896sdot sum

119895isin120575+(119900119896)

sum

119904isinΓ119900119896119895

119888pick sdot 119902119896sdot 119883119896

119900119896119895119904+ sum

119896isin119870

120583119896sdot sum

119894isin120575minus(119889119896)

sum

119904isinΓ119894119889119896


119894119889119896119904 (4)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895


119894119895119904 (5)


Subject to sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119883119896

119894119895119904minus sum


sum

119903isin119878ℎ119894

119883119896

ℎ119894119903

=

1 119894 = 119900119896

0 forall119894 isin 119873 119900119896 119889119896

minus1 119894 = 119889119896


(6)

sum

119904isin119878119894119895

119883119896



119883119896

ℎ119894119903+ 119883119896

119894119895119904le 1


+(119894) forall119904 isin Ω

119894119895 forall119903 isin Ω

ℎ119894

(8)



sum

119896isin119870

119902119896sdot 119883119896

119894119895119904le 119876119904


119894119895 (9)


119884119896

119900119896= 119905119896



(max 119884119896

119894 SD119904119894 + 119905119894119895119904

minus 119884119896

119895) sdot 119883119896

119894119895119904= 0


(11)




119894119895 we are far











Ω119894119895


119878119894119895


119894119895sube 119878

120575minus(119894)


120575+(119894)




119906119904


SA119904119894


SD119904119894


119876119904

119894119895


119889119894119895119904


119905119894119895119904


119862119896

119894119895119904


119888119904



119888pick


119888deliver


120593119896


119896= 1 otherwise 120593

119896= 0

120583119896


119896= 1 otherwise 120583

119896= 0


120578119904



119883119896

119894119895119904


119894119895119904= 1

otherwise 119883119896119894119895119904

= 0119884119896


119885119896

119894119895119904


arc (119894 119895) Hence119884119896119895equals 119897119904



in this case 119905119894119895119904

= (119897119904

119895minus SD119904119894)

119884119896

119894le 119906119904

119894sdot 119883119896

119894119895119904+ 119872 sdot (1 minus 119883

119896

119894119895119904)


(12)


119894119895119904= 0 then 119884

119896



119884119896

119889119896le 119879119896

forall119896 isin 119870 (13)


(max 0 119897119904

119894minus 119884119896

119894minus 120587 minus 119885

119896

119894119895119904) sdot 119883119896

119894119895119904= 0


(14)


119862119896

119894119895119904=

(1198881

rail + 1198882

rail sdot 119889119894119895119904) sdot 119902119896

119904 isin Γ119894119895

1198882

road sdot 119902119896sdot 119889119894119895119904

119904 isin Ω119894119895


(15)


2

rail and 1198882


119883119896


119894119895 (16)

119884119896


119885119896


119894119895 (18)






119894+119905119894119895119904minus119884119896

119895)sdot

119883119896




119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(19)

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(20)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(21)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(22)




119894119895119904= 1


minus 119884119896

119895= 0 In this


minus119884119896

119895) ge 0

and (SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge 0

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0

997904rArr 0 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0 997904rArr 119884

119896

119895= SD119904119894+ 119905119894119895119904 (23)



119894and119884

119896



minus 119884119896

119895) ge minus119872 and

(SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge minus119872

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872

997904rArr minus119872 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872 (24)



119885119896

119894119895119904) sdot 119883119896


119894119895can be


119885119896

119894119895119904ge 119872 sdot (119883

119896

119894119895119904minus 1) + (119897

119904

119894minus 119884119896

119894minus 120587)


(25)

119885119896

119894119895119904le 119872 sdot 119883

119896


119894119895 (26)



119896

119894119895119904= 0 and


(119897119904

119894minus 119884119896



ge minus119872 and 119885119896

119894119895119904le 0 When



119885119896

119894119895119904ge minus119872

119885119896

119894119895119904le 0

119885119896

119894119895119904ge 0

997904rArr 0 le 119885119896

119894119895119904le 0 997904rArr 119885

119896

119894119895119904= 0 (27)


119896

119894119895119904= 1 This scenario





119896

119894119895119904ge (119897119904

119894minus 119884119896

119894minus 120587) and 119885

119896

119894119895119904le 119872




119894minus 119884119896



119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)



(119897119904

119894minus 119884119896



119894119895119904(Constraint (17)) the


119894minus119884119896


119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)



minimize (1) + (2) + (3) + (4) + (5) (30)









1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










5

6

8

4

8

10

7

[19 215 2225] [28 285]

[27 31] 32 43 [44 47]

[10 15 16] [24 25]

[25 27 275] [355 36]

[11 13 1325] [18 185]

[33 38 385] [445 45][35 39 40] [47 475]

1

2

4

56

7

3



R1

R4

R5

R7

R6

R3

R2



2at
















119896







minimize sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895

119862119896

119894119895119904sdot 119883119896

119894119895119904 (1)

+ sum

119896isin119870

sum

119894isin119873

( sum


sum

119903isin119878ℎ119894

119888119903sdot 119902119896sdot 119883119896

ℎ119894119903+ sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119888119904sdot 119902119896sdot 119883119896

119894119895119904) (2)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isinΓ119894119895

119888store sdot 119902119896 sdot 119885119896

119894119895119904 (3)

+ sum

119896isin119870

120593119896sdot sum

119895isin120575+(119900119896)

sum

119904isinΓ119900119896119895

119888pick sdot 119902119896sdot 119883119896

119900119896119895119904+ sum

119896isin119870

120583119896sdot sum

119894isin120575minus(119889119896)

sum

119904isinΓ119894119889119896


119894119889119896119904 (4)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895


119894119895119904 (5)


Subject to sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119883119896

119894119895119904minus sum


sum

119903isin119878ℎ119894

119883119896

ℎ119894119903

=

1 119894 = 119900119896

0 forall119894 isin 119873 119900119896 119889119896

minus1 119894 = 119889119896


(6)

sum

119904isin119878119894119895

119883119896



119883119896

ℎ119894119903+ 119883119896

119894119895119904le 1


+(119894) forall119904 isin Ω

119894119895 forall119903 isin Ω

ℎ119894

(8)



sum

119896isin119870

119902119896sdot 119883119896

119894119895119904le 119876119904


119894119895 (9)


119884119896

119900119896= 119905119896



(max 119884119896

119894 SD119904119894 + 119905119894119895119904

minus 119884119896

119895) sdot 119883119896

119894119895119904= 0


(11)




119894119895 we are far











Ω119894119895


119878119894119895


119894119895sube 119878

120575minus(119894)


120575+(119894)




119906119904


SA119904119894


SD119904119894


119876119904

119894119895


119889119894119895119904


119905119894119895119904


119862119896

119894119895119904


119888119904



119888pick


119888deliver


120593119896


119896= 1 otherwise 120593

119896= 0

120583119896


119896= 1 otherwise 120583

119896= 0


120578119904



119883119896

119894119895119904


119894119895119904= 1

otherwise 119883119896119894119895119904

= 0119884119896


119885119896

119894119895119904


arc (119894 119895) Hence119884119896119895equals 119897119904



in this case 119905119894119895119904

= (119897119904

119895minus SD119904119894)

119884119896

119894le 119906119904

119894sdot 119883119896

119894119895119904+ 119872 sdot (1 minus 119883

119896

119894119895119904)


(12)


119894119895119904= 0 then 119884

119896



119884119896

119889119896le 119879119896

forall119896 isin 119870 (13)


(max 0 119897119904

119894minus 119884119896

119894minus 120587 minus 119885

119896

119894119895119904) sdot 119883119896

119894119895119904= 0


(14)


119862119896

119894119895119904=

(1198881

rail + 1198882

rail sdot 119889119894119895119904) sdot 119902119896

119904 isin Γ119894119895

1198882

road sdot 119902119896sdot 119889119894119895119904

119904 isin Ω119894119895


(15)


2

rail and 1198882


119883119896


119894119895 (16)

119884119896


119885119896


119894119895 (18)






119894+119905119894119895119904minus119884119896

119895)sdot

119883119896




119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(19)

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(20)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(21)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(22)




119894119895119904= 1


minus 119884119896

119895= 0 In this


minus119884119896

119895) ge 0

and (SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge 0

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0

997904rArr 0 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0 997904rArr 119884

119896

119895= SD119904119894+ 119905119894119895119904 (23)



119894and119884

119896



minus 119884119896

119895) ge minus119872 and

(SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge minus119872

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872

997904rArr minus119872 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872 (24)



119885119896

119894119895119904) sdot 119883119896


119894119895can be


119885119896

119894119895119904ge 119872 sdot (119883

119896

119894119895119904minus 1) + (119897

119904

119894minus 119884119896

119894minus 120587)


(25)

119885119896

119894119895119904le 119872 sdot 119883

119896


119894119895 (26)



119896

119894119895119904= 0 and


(119897119904

119894minus 119884119896



ge minus119872 and 119885119896

119894119895119904le 0 When



119885119896

119894119895119904ge minus119872

119885119896

119894119895119904le 0

119885119896

119894119895119904ge 0

997904rArr 0 le 119885119896

119894119895119904le 0 997904rArr 119885

119896

119894119895119904= 0 (27)


119896

119894119895119904= 1 This scenario





119896

119894119895119904ge (119897119904

119894minus 119884119896

119894minus 120587) and 119885

119896

119894119895119904le 119872




119894minus 119884119896



119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)



(119897119904

119894minus 119884119896



119894119895119904(Constraint (17)) the


119894minus119884119896


119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)



minimize (1) + (2) + (3) + (4) + (5) (30)









1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











minimize sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895

119862119896

119894119895119904sdot 119883119896

119894119895119904 (1)

+ sum

119896isin119870

sum

119894isin119873

( sum


sum

119903isin119878ℎ119894

119888119903sdot 119902119896sdot 119883119896

ℎ119894119903+ sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119888119904sdot 119902119896sdot 119883119896

119894119895119904) (2)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isinΓ119894119895

119888store sdot 119902119896 sdot 119885119896

119894119895119904 (3)

+ sum

119896isin119870

120593119896sdot sum

119895isin120575+(119900119896)

sum

119904isinΓ119900119896119895

119888pick sdot 119902119896sdot 119883119896

119900119896119895119904+ sum

119896isin119870

120583119896sdot sum

119894isin120575minus(119889119896)

sum

119904isinΓ119894119889119896


119894119889119896119904 (4)

+ sum

119896isin119870

sum

(119894119895)isin119860

sum

119904isin119878119894119895


119894119895119904 (5)


Subject to sum

119895isin120575+(119894)

sum

119904isin119878119894119895

119883119896

119894119895119904minus sum


sum

119903isin119878ℎ119894

119883119896

ℎ119894119903

=

1 119894 = 119900119896

0 forall119894 isin 119873 119900119896 119889119896

minus1 119894 = 119889119896


(6)

sum

119904isin119878119894119895

119883119896



119883119896

ℎ119894119903+ 119883119896

119894119895119904le 1


+(119894) forall119904 isin Ω

119894119895 forall119903 isin Ω

ℎ119894

(8)



sum

119896isin119870

119902119896sdot 119883119896

119894119895119904le 119876119904


119894119895 (9)


119884119896

119900119896= 119905119896



(max 119884119896

119894 SD119904119894 + 119905119894119895119904

minus 119884119896

119895) sdot 119883119896

119894119895119904= 0


(11)




119894119895 we are far











Ω119894119895


119878119894119895


119894119895sube 119878

120575minus(119894)


120575+(119894)




119906119904


SA119904119894


SD119904119894


119876119904

119894119895


119889119894119895119904


119905119894119895119904


119862119896

119894119895119904


119888119904



119888pick


119888deliver


120593119896


119896= 1 otherwise 120593

119896= 0

120583119896


119896= 1 otherwise 120583

119896= 0


120578119904



119883119896

119894119895119904


119894119895119904= 1

otherwise 119883119896119894119895119904

= 0119884119896


119885119896

119894119895119904


arc (119894 119895) Hence119884119896119895equals 119897119904



in this case 119905119894119895119904

= (119897119904

119895minus SD119904119894)

119884119896

119894le 119906119904

119894sdot 119883119896

119894119895119904+ 119872 sdot (1 minus 119883

119896

119894119895119904)


(12)


119894119895119904= 0 then 119884

119896



119884119896

119889119896le 119879119896

forall119896 isin 119870 (13)


(max 0 119897119904

119894minus 119884119896

119894minus 120587 minus 119885

119896

119894119895119904) sdot 119883119896

119894119895119904= 0


(14)


119862119896

119894119895119904=

(1198881

rail + 1198882

rail sdot 119889119894119895119904) sdot 119902119896

119904 isin Γ119894119895

1198882

road sdot 119902119896sdot 119889119894119895119904

119904 isin Ω119894119895


(15)


2

rail and 1198882


119883119896


119894119895 (16)

119884119896


119885119896


119894119895 (18)






119894+119905119894119895119904minus119884119896

119895)sdot

119883119896




119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(19)

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(20)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(21)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(22)




119894119895119904= 1


minus 119884119896

119895= 0 In this


minus119884119896

119895) ge 0

and (SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge 0

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0

997904rArr 0 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0 997904rArr 119884

119896

119895= SD119904119894+ 119905119894119895119904 (23)



119894and119884

119896



minus 119884119896

119895) ge minus119872 and

(SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge minus119872

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872

997904rArr minus119872 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872 (24)



119885119896

119894119895119904) sdot 119883119896


119894119895can be


119885119896

119894119895119904ge 119872 sdot (119883

119896

119894119895119904minus 1) + (119897

119904

119894minus 119884119896

119894minus 120587)


(25)

119885119896

119894119895119904le 119872 sdot 119883

119896


119894119895 (26)



119896

119894119895119904= 0 and


(119897119904

119894minus 119884119896



ge minus119872 and 119885119896

119894119895119904le 0 When



119885119896

119894119895119904ge minus119872

119885119896

119894119895119904le 0

119885119896

119894119895119904ge 0

997904rArr 0 le 119885119896

119894119895119904le 0 997904rArr 119885

119896

119894119895119904= 0 (27)


119896

119894119895119904= 1 This scenario





119896

119894119895119904ge (119897119904

119894minus 119884119896

119894minus 120587) and 119885

119896

119894119895119904le 119872




119894minus 119884119896



119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)



(119897119904

119894minus 119884119896



119894119895119904(Constraint (17)) the


119894minus119884119896


119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)



minimize (1) + (2) + (3) + (4) + (5) (30)









1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Ω119894119895


119878119894119895


119894119895sube 119878

120575minus(119894)


120575+(119894)




119906119904


SA119904119894


SD119904119894


119876119904

119894119895


119889119894119895119904


119905119894119895119904


119862119896

119894119895119904


119888119904



119888pick


119888deliver


120593119896


119896= 1 otherwise 120593

119896= 0

120583119896


119896= 1 otherwise 120583

119896= 0


120578119904



119883119896

119894119895119904


119894119895119904= 1

otherwise 119883119896119894119895119904

= 0119884119896


119885119896

119894119895119904


arc (119894 119895) Hence119884119896119895equals 119897119904



in this case 119905119894119895119904

= (119897119904

119895minus SD119904119894)

119884119896

119894le 119906119904

119894sdot 119883119896

119894119895119904+ 119872 sdot (1 minus 119883

119896

119894119895119904)


(12)


119894119895119904= 0 then 119884

119896



119884119896

119889119896le 119879119896

forall119896 isin 119870 (13)


(max 0 119897119904

119894minus 119884119896

119894minus 120587 minus 119885

119896

119894119895119904) sdot 119883119896

119894119895119904= 0


(14)


119862119896

119894119895119904=

(1198881

rail + 1198882

rail sdot 119889119894119895119904) sdot 119902119896

119904 isin Γ119894119895

1198882

road sdot 119902119896sdot 119889119894119895119904

119904 isin Ω119894119895


(15)


2

rail and 1198882


119883119896


119894119895 (16)

119884119896


119885119896


119894119895 (18)






119894+119905119894119895119904minus119884119896

119895)sdot

119883119896




119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(19)

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(20)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(21)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(22)




119894119895119904= 1


minus 119884119896

119895= 0 In this


minus119884119896

119895) ge 0

and (SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge 0

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0

997904rArr 0 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0 997904rArr 119884

119896

119895= SD119904119894+ 119905119894119895119904 (23)



119894and119884

119896



minus 119884119896

119895) ge minus119872 and

(SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge minus119872

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872

997904rArr minus119872 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872 (24)



119885119896

119894119895119904) sdot 119883119896


119894119895can be


119885119896

119894119895119904ge 119872 sdot (119883

119896

119894119895119904minus 1) + (119897

119904

119894minus 119884119896

119894minus 120587)


(25)

119885119896

119894119895119904le 119872 sdot 119883

119896


119894119895 (26)



119896

119894119895119904= 0 and


(119897119904

119894minus 119884119896



ge minus119872 and 119885119896

119894119895119904le 0 When



119885119896

119894119895119904ge minus119872

119885119896

119894119895119904le 0

119885119896

119894119895119904ge 0

997904rArr 0 le 119885119896

119894119895119904le 0 997904rArr 119885

119896

119894119895119904= 0 (27)


119896

119894119895119904= 1 This scenario





119896

119894119895119904ge (119897119904

119894minus 119884119896

119894minus 120587) and 119885

119896

119894119895119904le 119872




119894minus 119884119896



119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)



(119897119904

119894minus 119884119896



119894119895119904(Constraint (17)) the


119894minus119884119896


119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)



minimize (1) + (2) + (3) + (4) + (5) (30)









1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













119894+119905119894119895119904minus119884119896

119895)sdot

119883119896




119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(19)

119878119863119904

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(20)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895ge 119872 sdot (119883

119896

119894119895119904minus 1)


(21)

119884119896

119894+ 119905119894119895119904

minus 119884119896

119895le 119872 sdot (1 minus 119883

119896

119894119895119904)


(22)




119894119895119904= 1


minus 119884119896

119895= 0 In this


minus119884119896

119895) ge 0

and (SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge 0

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0

997904rArr 0 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 0 997904rArr 119884

119896

119895= SD119904119894+ 119905119894119895119904 (23)



119894and119884

119896



minus 119884119896

119895) ge minus119872 and

(SD119904119894+ 119905119894119895119904

minus 119884119896



119894and 119884

119896

119895

SD119904119894+ 119905119894119895119904

minus 119884119896

119895ge minus119872

SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872

997904rArr minus119872 le SD119904119894+ 119905119894119895119904

minus 119884119896

119895le 119872 (24)



119885119896

119894119895119904) sdot 119883119896


119894119895can be


119885119896

119894119895119904ge 119872 sdot (119883

119896

119894119895119904minus 1) + (119897

119904

119894minus 119884119896

119894minus 120587)


(25)

119885119896

119894119895119904le 119872 sdot 119883

119896


119894119895 (26)



119896

119894119895119904= 0 and


(119897119904

119894minus 119884119896



ge minus119872 and 119885119896

119894119895119904le 0 When



119885119896

119894119895119904ge minus119872

119885119896

119894119895119904le 0

119885119896

119894119895119904ge 0

997904rArr 0 le 119885119896

119894119895119904le 0 997904rArr 119885

119896

119894119895119904= 0 (27)


119896

119894119895119904= 1 This scenario





119896

119894119895119904ge (119897119904

119894minus 119884119896

119894minus 120587) and 119885

119896

119894119895119904le 119872




119894minus 119884119896



119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)



(119897119904

119894minus 119884119896



119894119895119904(Constraint (17)) the


119894minus119884119896


119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)



minimize (1) + (2) + (3) + (4) + (5) (30)









1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119894minus 119884119896



119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 le 0

997904rArr 0 le 119885119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 0 (28)



(119897119904

119894minus 119884119896



119894119895119904(Constraint (17)) the


119894minus119884119896


119894minus119884119896

119894minus120587)

119885119896

119894119895119904ge 119897119904

119894minus 119884119896

119894minus 120587

119885119896

119894119895119904le 119872

119885119896

119894119895119904ge 0

119897119904

119894minus 119884119896

119894minus 120587 gt 0

997904rArr 119897119904

119894minus 119884119896

119894minus 120587 le 119885

119896

119894119895119904le 119872 997904rArr 119885

119896

119894119895119904= 119897119904

119894minus 119884119896

119894minus 120587 (29)



minimize (1) + (2) + (3) + (4) + (5) (30)









1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1Hohhot

2Baotoudong

3Xingang

10 Huangdaogang

9

Xinzhu

21Tuanjiecun

Chengxiang


Huinong

Yinchuannan

Qingtongxia

Yingtannan

Nanchangbei

Shangrao

34Jingdezhendong

Bajing

40 Beilungang

17

18

20

16

Yangpu

Luchaogang


Anting

36

35

37

38

8

6

5

23 Pinghunan

Wangjiayingxi

24

2627

3231

25DalangTangxi

Xiayuan

Huangpu

Jianggaozhen

3028

7

Lanzhou 4

14

29

33Changsha

22

Jiaozhou

X95143

X954039

X95187

X95165

X95621

X95565

X95587

X95521

X956059

X95543

Minhang

39Guiyang

X95221

N

S

W E

Foshandong

Jiebian

Xiaotangxi

Sanshuixi






Road 1198882

road 6 45 yenTEU-km

Rail 1198881

rail 500 340 yenTEU1198882







Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Rail 20 ft 195 225 312540 ft 14625 3375 3125

5000

4000

3000

2000

1000

0C1 C2 C3 C4 C5

Cost types

49575

2253 45 196 33

10

3yen)

Cos

ts (





6 Conclusions





(1 2) 147 45 (20 40) 225 6(1 3) 656 10 (22 14) 868 14(1 10) 1081 245 (22 40) 2273 34(2 1) 147 45 (22 34) 1923 23(2 3) 801 125 (22 35) 1645 295(2 9) 1161 295 (22 36) 1748 275(3 9) 449 105 (22 37) 1846 28(3 10) 489 12 (22 38) 1578 235(3 13) 558 145 (23 32) 96 2(4 5) 362 5 (24 23) 103 3(4 6) 419 7 (24 32) 9 05(4 7) 525 95 (25 23) 133 3(4 8) 618 115 (25 32) 41 15(4 11) 1083 18 (26 23) 130 3(4 12) 1082 195 (26 32) 37 15(4 13) 1613 29 (27 23) 124 4(4 14) 821 16 (27 32) 29 1(4 18) 2016 345 (28 23) 168 5(8 11) 474 65 (28 32) 72 2(8 12) 473 8 (29 23) 153 45(8 13) 1004 175 (29 32) 59 2(9 13) 225 45 (30 23) 138 45(9 18) 788 9 (30 32) 47 15(10 13) 218 3 (31 23) 135 45(10 18) 774 9 (31 32) 45 15(13 18) 566 7 (32 23) 100 2(14 21) 271 35 (33 15) 1016 215(14 22) 869 14 (33 40) 989 16(14 32) 1613 26 (33 34) 519 95(15 18) 93 35 (33 35) 335 6(15 40) 207 35 (33 36) 467 12(16 18) 71 25 (33 37) 566 125(16 40) 223 6 (33 38) 292 8(17 18) 59 15 (39 14) 659 85(17 40) 183 45 (39 21) 319 155(18 40) 235 3 (39 22) 518 85(19 18) 71 25 (39 23) 1106 20(19 40) 229 6 (39 32) 1005 185(20 18) 61 2 (40 15) 209 35





Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













Table 7 Continued










2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















2 1997888997888997888997888997888997888997888rarrX95143 3


3 1997888997888997888997888997888997888997888rarrX95143 3


4 2997888997888997888997888997888997888997888rarrX95143 3 957

5 2997888997888997888997888997888997888997888rarrX95143 3 1437

6 2997888997888997888997888997888997888997888rarrX95143 3



997888997888997888997888997888997888997888rarrX95187 13 127


997888997888997888997888997888997888997888rarrX95187 13 175


997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13



997888997888997888997888997888997888997888rarrX95187 13 79


997888997888997888997888997888997888997888rarrX95187 13 175



997888997888997888997888997888997888997888rarrX95587 23





997888997888997888997888997888997888997888997888rarrX954039 40 2013


997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40



997888997888997888997888997888997888997888997888rarrX954039 40




997888997888997888997888997888997888997888997888rarrX954039 40 1533



997888997888997888997888997888997888997888rarrX95587 23 1759







Acknowledgments




References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











References





































































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article modeling the multicommodity multimodal...

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