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Research ArticleGreen and Reliable Freight Routing Problem in the Road-RailIntermodal Transportation Network with Uncertain ParametersA Fuzzy Goal Programming Approach

Yan Sun

School of Management Science and Engineering Shandong University of Finance and Economics No 7366Second Ring East Road Jinan Shandong Province 250014 China

Correspondence should be addressed to Yan Sun sunyanbjtu163com

Received 9 September 2019 Accepted 8 January 2020 Published 19 February 2020

Academic Editor Yuchuan Du

Copyright copy 2020 Yan Sun )is is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this study the author focuses on modeling and optimizing a freight routing problem in a road-rail intermodal transportationnetwork that combines the hub-and-spoke and point-to-point structures )e operations of road transportation are time flexiblewhile rail transportation has fixed departure times)e reliability of the routing is improved bymodeling the uncertainty of the road-rail intermodal transportation network Parameters that are influenced by the real-time status of the network including capacitiestravel times loading and unloading times and container trainsrsquo fixed departure times are considered uncertain in the routingdecision-making Based on fuzzy set theory triangular fuzzy numbers are employed to formulate the uncertain parameters as well asresulting uncertain variables Green routing is also discussed by treating the minimization of carbon dioxide emissions as anobjective First of all a multiobjective fuzzy mixed integer nonlinear programming model is established for the specific reliable andgreen routing problem )en defuzzification linearization and weighted sum method are implemented to present a crisp linearmodel whose global optimum solutions can be effectively obtained by the exact solution algorithm run by mathematical pro-gramming software Finally a numerical case is given to demonstrate how the proposed methods work In the case sensitivityanalysis is adopted to reveal the effects of uncertainty on the routing optimization Fuzzy simulation is then performed to helpdecision makers to select the best crisp route plan by determining the best confidence level shown in the fuzzy chance constraints

1 Introduction

Rail transportation has been acknowledged to be a cost-effective means of long-distribution transportation It yieldsmassive capacity and can effectively realize the economies ofscale in bulk transportation [1] As another representativetransportation mode road transportation shows excellentmobility and flexibility in short- and medium-distancecollection and delivery activities [2] Coordination of the twotransportation modes in one distribution chain in which theloading units are ISO standard 20 ft containers [3] leads to anadvanced transportation system named road-rail intermodaltransportation

Road-rail transportation combines the above twotransportation modes and integrates their respective ad-vantages to provide seamless door-to-door services for

containerized freight delivery that are difficult to achieve bythe unimodal transportation [4] In recent years road-railintermodal transportation has become a promising means ofinland freight transportation in the Euro-China Expresswayto support the growing international trade motivated by thedevelopment of the ldquoOne Belt One Roadrdquo initiatives [5])eroad-rail intermodal transportation system has been therebywidely established and promoted not only in transportationpractice but also in logistics [6] and supply chain [7]

To realize effective resource allocation and utilizationas well as successful operations of the road-rail intermodaltransportation system it is necessary to use modelingoptimization and simulation methods and techniques tohelp decision makers (eg transportation planners in-termodal transportation operators policy makers andtransportation providers and customers) to establish an

HindawiJournal of Advanced TransportationVolume 2020 Article ID 7570686 21 pageshttpsdoiorg10115520207570686

optimal transportation network in strategic tactical andoperational levels [8] As summarized by Caris et al [9]recent research topics in the road-rail intermodal trans-portation planning field include policy support terminalnetwork design intermodal service network design in-termodal routing and drayage operations Among thesetopics road-rail intermodal routing that belongs to op-erational-level planning is a spotlight highlighted by re-searchers in theory and practitioners in transportationpractice [10] and is the issue targeted by this study

Road-rail intermodal routing involves the optimal uti-lization of existing transportation resources that are limitedin the road-rail intermodal transportation network It aimsat the selection of the best routes to move containers oftransportation orders demanded by customers from originsto destinations through the road-rail intermodal trans-portation network [11] )e best routes can either achieveoptimum on any one of the objectives eg minimal cost[3 12 13] minimal time [14 15] lowest risk [16 17] andminimal greenhouse gas emissions [18] or make tradeoffsamong them in multiobjective optimizations

Since logistics cost accounts for as much as 30 to50 of the total production cost of companies [14 19]reducing costs created in the road-rail intermodaltransportation process is the main task of its routingoptimization which also motivates that majority of therelative literature takes minimization of costs as a prioroptimization objective [2] Simultaneously as a greatcontributor to greenhouse gas emissions that causeglobal warming the transportation industry has beenurgently demanded for an environmental-friendly de-velopment [20ndash22] In this case the transportation in-dustry is challenged by reducing environmental impactwhile keeping profitable [23] As stated by Winebrakeet al [20] planning routes with environmental con-siderations through the intermodal transportation sys-tem is an effective way to improve the environmentalsustainability of transportation Carbon dioxide ac-counts for sim80 of the total greenhouse gas emissions[24] )is study investigates a green road-rail intermodalrouting problem considering the reduction of carbondioxide emissions

Reliability is a crucial issue that is involved in the in-termodal transportation system that needs to coordinatevarious transportation modes large number of facilitiesand different actors in an extensive network that is under amore complicated environment than unimodal trans-portation [18 25] In the road-rail intermodal routingproblem reliability means the successful accomplishmentof transportation orders by using planned road-rail in-termodal routes [26] Uncertainty is the primary factor thatinfluences the reliability of intermodal transportation[27ndash29] Although a short-term planning [8] the routingoptimization should be undertaken before the actualtransportation starts [30] However the operations of thetransportation network are easily disrupted by variousfactors such as bad weather traffic congestion accidentsbreakdown or shortage of transportation facilities and soon [26] As a result network parameters regarding

operational times and service capacities that are exceed-ingly sensitive to the real-time status of the transportationnetwork cannot keep stable and are also challenging to beforecasted precisely in the advanced routing decision-making which results in the uncertainty of the intermodaltransportation network )erefore this study discusses thereliable road-rail intermodal routing problem under un-certainty in this study

Above all in this study the author systematically ex-plores a green and reliable road-rail intermodal routingproblem by making the following contributions

(1) A road-rail intermodal routing problem is extendedby enhancing its environmental sustainability withcarbon dioxide emission optimization and improv-ing transportation reliability with multiple sources ofuncertainty

(2) Based on the fuzzy set theory triangular fuzzy numbersare adopted to model the uncertainty of the road-railintermodal transportation network including capac-ities travel times loading and unloading times andfixed departure times Multiobjective optimization isemployed to deal with the carbon dioxide emissions bygenerating Pareto solutions to make tradeoffs betweenthe economic and environmental objectives

(3) Fuzzy goal programming approach is used to for-mulate the green and reliable freight routing prob-lem in the road-rail intermodal transportationnetwork with hub-and-spoke and point-to-pointstructures A multiobjective fuzzy mixed integernonlinear programming model is established andassociated exact solution approach is developed

(4) )e proposed methods are demonstrated in a nu-merical case in which sensitivity analysis and fuzzysimulation are utilized to analyze the effects oftransportation network uncertainty on the routingoptimization quantitatively and finally determinethe optimum confidence level in the fuzzy chanceconstraints

)e remaining sections of this study are organized asfollows Section 2 reviews relative literature in order tofind the research gap and identify the contributions ofthis study Section 3 provides the modeling foundationfor the road-rail intermodal routing problem includingmodeling network uncertainty by fuzzy set theory andformulating the transportation system Section 4 presentsa multiobjective fuzzy mixed integer nonlinear pro-gramming model for the specific green and reliablerouting problem )en an exact solution approach thatcombines defuzzification linearization and the weightedsum method is designed for the sake of obtaining theglobal optimum solutions to the problem A numericalcase is given in Section 6 to demonstrate the feasibility ofthe proposed fuzzy goal programming approach )isstudy also uses sensitivity analysis and fuzzy simulationto quantify the effects of transportation network uncer-tainty on the routing optimization in this section Finallyconclusions and insights are drawn in Section 7

2 Journal of Advanced Transportation

2 Literature Review

Although Caris et al [9] pointed out in 2013 that few studiespay attention to the intermodal routing problem manyrelative types of research on this topic published in recentyears can be found

21 Review on Green Intermodal Routing Problem As aneffective way to improve the environmental sustainability ofthe intermodal transportation system the green routingproblem with carbon dioxide emission consideration is paidgrowing attention by existing literature Carbon tax regu-lation might be the most widely used method in the greenintermodal routing problem to reduce carbon dioxideemissions [31] Under carbon tax regulation transportationorders are charged for each unit of emission with a tax [32])e tax is then integrated into the economic objective Byoptimizing the economic objective the carbon dioxideemission tax has opportunities to be reduced which can helpto lower carbon dioxide emissions in the routing

As a simple and straightforward method carbon taxregulation has been used by Chang et al [33] in modeling asea-land intermodal routing problem with external cost Sunand Lang [34] and Zhang et al [35] also adopt this regulationin a multicommodity multimodal routing problem and atransportation mode selection problem in intermodaltransportation respectively Hrusovsky et al [36] explore agreen intermodal routing problem with travel time uncer-tainty in which the carbon dioxide emission tax is a part ofthe economic objective and is also attached to weight as-sociated with decision makersrsquo preference to the environ-mental sustainability of transportation It is also employedby Zhang et al [37] in a green multimodal routing problemwith time windows In this study sensitivity of the greenrouting optimization on the unit carbon tax is analyzedwhich shows that the performance of carbon tax regulationis sensitive to the setting of time windows Sun et al [11]discuss a green intermodal routing problem with capacityuncertainty and road traffic congestion )e empirical casestudy presented in this study indicates that the green routingoptimization is not sensitive to the unit emission tax and theroutes are not changed unless the unit emission tax reach asubstantial value that is infeasible in practice

Since the performance of carbon dioxide emissionregulation depends on the setting of routing cases [36] and isnot always stable [11] finding an alternative for the greenintermodal routing problem is necessary Currently a fewarticles try multiobjective optimization to reduce the carbondioxide emissions in the intermodal routing Vale andRibeiro [38] propose a multiobjective model for a multi-modal routing problem in which minimization of carbondioxide emissions is set as an objective that is independent ofthe objective of minimizing the transportation time Demiret al [18] also employ multiobjective optimization for re-ducing carbon dioxide-equivalent emissions in the inter-modal routing in which several methods that can be used togenerate the Pareto solutions to the green routing optimi-zation are tested Moreover in the case study given by Sun

et al [11] multiobjective optimization that can providedecision makers with Pareto solutions to the green routingproblem shows better feasibility than carbon tax regulationin making tradeoffs between improving environmentalsustainability and lowering logistics cost Besides the in-termodal routing problem limited researches on the in-termodal transportation network design problem also utilizemultiobjective optimization to realize the design of greenintermodal transportation network eg Qu et al [39] andResat and Turkay [40]

Additionally Sun et al [31] establish a carbon dioxideemission constraint in the hazardous materials road-railmultimodal routing problem )is constraint ensures thecarbon dioxide emissions generated by the hazardous ma-terials road-rail multimodal transportation using the plan-ned routes kept lower than a prescribed thresholdSensitivity analysis on the routing optimization concerningthe allowable carbon dioxide emissions in this study indi-cates that the carbon dioxide emission constraint has nearlythe same performance to the multiobjective optimization

22 Review on Reliable Intermodal Routing Problem underUncertainty Transportation process optimization shouldtake uncertainty into account [41 42] As indicated inSection 1 parameters that are sensitive to the real-timestatus of the road-rail intermodal transportation network areuncertainty in the routing decision-making However thedeterministic intermodal routing problem is still the currentdominance Large numbers of deterministic routing opti-mizations can be found from early studies eg Barnhart andRatliff [43] Boardman et al [44] Bookbinder and Fox [45]and Ziliaskopoulos and Wardell [46] to recently publishedarticles eg Chang et al [14] Xiong andWang [15] Sun andLang [34] Ayar and Yaman [47] Moccia et al [48] Heggenet al [49] andWolfinger et al [50] A few studies contributeto the intermodal routing problem under uncertainty [3]

In the intermodal routing problem under uncertaintytravel time uncertainty gains the most extensive attentionMajority of the relative studies focuses on the intermodalrouting problem under travel time uncertainty An earlystudy conducted by Min [51] developed a stochasticchance-constrained programming model to solve the in-termodal routing problem with travel time uncertainty Asimilar approach is employed by Zhao et al [3] to find thebest sea-rail intermodal routes with stochastic travel timesin a space-time network and by Uddin and Huynh [13 27]in the real-world road-rail intermodal transportationsystem Hrusovsky et al [36] also take advantage of sto-chastic programming and design a hybrid simulation andoptimization approach for the green intermodal routingproblem with travel time uncertainty Conducted by thesame group of authors to Hrusovsky et al [36] there is astudy that uses stochastic programming to address a greenintermodal transportation network design problem thatconsiders travel times as uncertain parameters [22] Basedon the fuzzy set theory Sun and Li [2] establish a fuzzyprogramming model to formulate a road-rail intermodalrouting problem with road travel time fuzziness Similarly

Journal of Advanced Transportation 3

fuzzy programming method is also utilized by Wang et al[1] in optimizing a road-rail intermodal hub-and-spokenetwork design problem in which travel times are one ofseveral uncertain network parameters

Other intermodal transportation network time param-eters receive limited interests from routing optimizationLoading and unloading time uncertainty (fuzziness) isstudied by Sun and Li [2] together with road travel timeuncertainty in the road-rail intermodal routing problemTransfer times between different transportation modes areconsidered as stochastic parameters by Zhao et al [3] in theirrelative study As stressed by various studies eg Demiret al [18] Hrusovsky et al [36] Moccia et al [48] andHeinold and Meisel [52] there are some transportationmodes eg rail and vessel that should be operated by fixeddeparture times Fixed departure times are sensitive to thereal-time status of operations of associated transportationmodes at terminals under disruptions claimed in Section 1and detailly explained in Section 3)erefore the departuresof container trains from terminals are not always punctualand thus are also uncertain However to the best of ourknowledge there is no existing literature that considers thisissue in the intermodal routing problem

Besides time parameters the capacity of the intermodaltransportation network is also uncertain [11 30] while fewstudies can be found Sun et al [30] systematically inves-tigate the effects of capacity uncertainty on the intermodalrouting optimization from a fuzzy programming perspec-tive )ere is also a study [30] that considers capacityfuzziness in a green intermodal routing problem Althoughthe intermodal routing problem pays limited attentioncapacity uncertainty is highlighted by the solid trans-portation problem [53ndash55] and supply chain planningproblem [56 57] A combination of multiple sources ofuncertainty can remarkably improve the planning reliability[2] As a result this study will comprehensively model theroad-rail intermodal transportation network uncertainty byconsidering capacities fixed departure times travel timesand loading and unloading times as uncertain parameters

Stochastic programming is widely used to deal withuncertain optimization problems in the transportationplanning field eg Uddin and Huynh [13] Demir et al [18]andHrusovsky et al [36] Large numbers of reliable previousdata must be attained to fit the possibility distributions foruncertain parameters [58 59] However in most practicalcases the previous data are missing or vague [60] Conse-quently there is not enough data to carry out stochasticprogramming Its feasibility is hence reduced As claimed byZarandi et al [58] fuzzy set theory and fuzzy programmingare effective alternatives to stochastic programming whendata availability is limited )e previous data of an uncertainparameter usually fall into a particular range Based on thefuzzy set theory decision makers can define uncertain pa-rameters as fuzzy numbers eg triangular fuzzy numbersby referring to the expert experience and limited previousdata After that fuzzy programming approaches eg fuzzychance-constrained programming can be used to establishoptimization models for the uncertain optimization prob-lems As a result in this study fuzzy programming is

selected to address the intermodal routing problem asso-ciated with uncertain parameters

23 Review Overview Above all an overall comparisonbetween this study and relevant existing literature is pre-sented in Table 1 With the help of Table 1 improvementsmade by this study can be clearly identified

3 Modeling Foundation

In this section the author systematically introduces how thisstudy models the uncertainty and the road-rail intermodaltransportation system which provides a modeling foun-dation for building the optimization model in the nextsection

31 Modeling Road-Rail Intermodal Transportation NetworkUncertainty As claimed in Section 2 in this study theauthor uses the fuzzy set theory to model the uncertainparameters and resulting uncertain decision variables Tri-angular fuzzy numbers are used to describe the fuzzy pa-rameters and decision variables due to its better simplicityand flexibility in fuzzy arithmetic operations than intervaland trapezoidal fuzzy numbers that are other widely usedforms of fuzzy numbers [57] Triangular fuzzy numbersadopt three prominent points including minimum mostlikely and maximum estimations to represent the fuzziness[2] and comprehensively show decision makersrsquo pessimisticoptimistic and most likely opinions on fuzzy events whichcan be seen in Figure 1 Specifically as for the uncertainparameters in the road-rail intermodal transportation sys-tem the three prominent points are defined as follows

For minimum estimation on the network parameters itreflects the road-rail intermodal transportation system underfollowing situations that are slightly possible in practice (i)the road and rail traffic is in a quite good status and movingconstrainers from one node to another one can be rapidlyrealized (ii) the operations of trucks and trains at the railterminal are of high efficiency ie loading and unloadingoperations can be finished rapidly by using enough me-chanical equipment handled by enough skilled staff and thework that need to be undertaken before the departure of thecontainer trains can be finished in a short period (iii) thetransportation network however is faced with lousy capacityavailability For example large numbers of container trucksand trains might be occupied by other tasks [30]

For most likely estimation it represents the usual statusof the road-rail intermodal transportation system )e roadand rail traffic the operations of trucks and trains at railterminals and the network capacity are in common situa-tions that are neither too good nor too bad Under suchstatus the parameters of the network yield values appear inmost cases

For maximum estimation it is contrary to minimumestimation It represents the following situations that are alsoslightly possible in practice (i) the road and rail traffic is in aterrible status that might be caused by congestion badweather or accidents which result in the slow movement of


Table 1 Comparison between the existing literature and this study

Authors Year Methods for greenrouting

Uncertain parameters in reliable routing under uncertainty

Modeling approachCapacities

Fixeddeparturetimes

Traveltimes

Loading andunloading times

Min [51] 1991 Stochasticprogramming

Barnhart and Ratliff[43] 1993 Deterministic

programming

Boardman et al [44] 1997 Deterministicprogramming

Bookbinder and Fox[45] 1998 Deterministic

programmingZiliaskopoulos andWardell [46] 2000 Deterministic

programming

Chang [14] 2008 Deterministicprogramming

Chang et al [33] 2010 Carbon taxregulation Deterministic

programming

Zhang et al [35] 2011 Carbon taxregulation Deterministic

programming

Moccia et al [48] 2011 Deterministicprogramming

Verma et al [17] 2012 Deterministicprogramming

Ayar and Yaman [47] 2012 Deterministicprogramming

Xiong and Wang [15] 2014 Deterministicprogramming

Sun and Lang [12] 2015 Carbon taxregulation Deterministic

programming

Sun and Lang [34] 2015 Deterministicprogramming

Uddin and Huynh[27] 2016 Stochastic

programming


programming

Zhao et al [3] 2018 (transfer timeuncertainty)

Stochasticprogramming

Sun et al [11] 2018 Carbon taxregulation Time-dependent fuzzy

programmingSun et al [30] 2018 Fuzzy programming

Hrusovsky et al [36] 2018 Carbon taxregulation Stochastic

programming

Vale and Ribeiro [38] 2018 Multiobjectiveoptimization Deterministic

programming

Wolfinger et al [50] 2018 Deterministicprogramming

Sun and Li [2] 2019 Fuzzy programmingUddin and Huynh[13] 2019 Stochastic

programming

Demir et al [18] 2019 Multiobjectiveoptimization Deterministic

programming

Rivera et al [23] 2019 Deterministicprogramming

Sun et al [31] 2019 Carbon emissionconstraint Deterministic

programming

Heggen et al [49] 2019 Deterministicprogramming

1is study Multiobjectiveoptimization Fuzzy goal

programming


containers on transportation lines (ii) )e operations oftrucks and trains are delayed by information errors me-chanical breakdown shortage of equipment or staff orunskilled handling of inexperienced workers (iii) )etransportation network has adequate equipment and facilityresources for carrying containers

Above all triangular fuzzy numbers are feasible torepresent the different status of the road-rail intermodaltransportation system and can thus fully reflect the uncer-tainty of the network parameters

Additionally fuzzy chance-constrained programmingmodeling and associated defuzzification involve some fuzzyarithmetic operations of triangular fuzzy numbers )eseoperations are presented as follows where 1113957m (m1 m2 m3)

and 1113957n (n1 n2 n3) are the triangular fuzzy numbers and a isa non-negative deterministic number

1113957m + 1113957n m1 m2 m3( 1113857 + n1 n2 n3( 1113857

m1 + n1 m2 + n2 m3 + n3( 1113857(1)

1113957m minus 1113957n m1 m2 m3( 1113857 minus n1 n2 n3( 1113857

m1 minus n3 m2 minus n2 m3 minus n1( 1113857(2)

a middot 1113957n a middot n1 n2 n3( 1113857 a middot n1 a middot n2 a middot n3( 1113857 (3)

32 Modeling Road-Rail Intermodal Transportation SystemRoad-rail intermodal routing problem is more compli-cated than the classical vehicle routing problem It shoulddetermine the consolidation network of the road-railtransportation system that is more complex than the less-than-truck-load system of vehicle routing and model thedifferent transportation modes instead of only consid-ering one transportation mode ie trucks in vehiclerouting )e question of how to coordinate the twodifferent transportation modes also improves the com-plexity of the road-rail intermodal routing )e road-rail

intermodal transportation system in this study shows thefollowing characteristics

(1) Consolidation network of the system contains hub-and-spoke structures (prehaul by trucks long haulby trains and end haul by trucks) that is consideredas the most suitable physical construction for road-rail intermodal transportation [1] and point-to-pointstructures (all-road transportation) that might besuitable for transportation orders with tight duedates [11 13 61] Such a network of the road-railintermodal transportation system is getting moreand more attention from transportation planningfield eg Sun et al [11] Uddin and Huynh [13] andSun et al [61]

(2) Road transportation is considered time flexible [12]which means that loading containers on trucks canbe immediately conducted when the containers getunloaded and unloading containers can be imme-diately started once the trucks arrive at the nodesHowever rail transportation ie container trainsshould follow fixed departure times [18 36 48 52])at is to say when the containers get loaded ontrains the trains should wait until the fixed departuretimes and then depart from the current nodes As aresult if containers are planned to be moved by acontainer train the time when they are loaded on theselected train should not be later than the fixeddeparture time of the train

(3) )e full-truck-load strategy is utilized by the road-rail intermodal transportation system that is asso-ciated with bulk transportation Moreover in theprehaul end haul and all-road transportation linesthere might exist more than one truck fleet Asexplained by Sun et al [31] all the truck fleets in oneroad transportation line can be combined into onetruck fleet group under the coordination of the in-termodal transportation operators As a result under

Estimation value

EV1

EV2

EV3

EV1Most likelyestimation(opinion)

Maximumestimation

Minimumestimation

EV2 EV3

Fuzzy membership degree

0

1

(i) Optimistic opinion on uncertain time(ii) Pessimistic opinion on uncertain capacity

(i) Pessimistic opinion on uncertain time(ii) Optimistic opinion on uncertain capacity

Figure 1 A triangular fuzzy number


the full-truck-load strategy one road transportationline has one truck fleet group in which the trucks canbe flexibly assigned into several fleets to carrycontainers from various transportation ordersContainer trains are operated periodically [12] )esame container trains in different periods areindexed as different transportation in the modelingso that the dimensions of associated parameters anddecision variables can be reduced [12 31]

4 Fuzzy Goal Programming Model

In this section a multiobjective fuzzy mixed integernonlinear programming model is established to formulatethe green and reliable freight routing problem in the road-rail intermodal transportation network with multiplesources of uncertainty that are addressed by fuzzy settheory

41 Assumptions )e following assumptions should befollowed by the modeling in this study in order to make theoptimization model rigorous

Assumption 1 All the transportation orders that are servedby the road-rail intermodal routing are known and deter-ministic especially the demand for the containers of atransportation order is deterministic since this study onlyconsiders uncertainty from the road-rail intermodal trans-portation network

Assumption 2 Containers of a transportation order start toget loaded at the corresponding release time at the origin

After loading the containers at the destination is accom-plished the transportation of containers by prehaul or all-road transportation carried out by trucks immediately starts

Assumption 3 A transportation order is considered ac-complished when its containers arrive and then get unloadedat the destination

Assumption 4 It is not allowed that containers of a trans-portation order are moved through the road-rail intermodaltransportation network in a splittable way so that the in-tegrality of each transportation order is ensured and thecustomers can receive all the containers that they need at thesame time

Assumption 5 Early and late deliveries of the containerslead to a penalty in order to improve the timeliness of thedeliveries for the sake of on-time transportation

42 Notations )e symbols used in the optimization modeland their representations are presented in Tables 2 and 3 Itshould be noted that the two kinds of decision variables arealso fuzzy since they are related to the fuzzy parametersshown in Table 2 in computation

43 Multiobjective Fuzzy Mixed Integer Nonlinear Pro-grammingModel )e proposed optimization model for thespecific road-rail intermodal routing problem is shown asfollows

minimize 1113944pisinP

1113944(i j)isinA

1113944nisinTij

cn middot qp middot dijn middot xpijn + 1113944

pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi

em middot qp middot xp

him + 1113944jisinN+

i

1113944nisinTij

en middot qp middot xpijn

⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij

g middot qp middot 1113957zp

ijn + 1113944pisinP

pen middot qp middot max duep minus 1113957yp

dp 01113882 1113883 + max 1113957y

p

dpminus duep 01113882 11138831113874 1113875

(4)


1113944(i j)isinA

1113944nisinTij

emn middot qp middot dijn middot xp

ijn(5)

Equation (4) is the economic objective that aims atminimizing the total costs paid for accomplishing all thetransportation orders served by the routing It contains fourformulas that are defined as follows

Formula 1113936pisinP1113936(i j)isinA1113936nisinTijcn middot qp middot dijn middot x

pijn is the

travel costs for moving containers from one node toanother on transportation arcsFormula 1113936pisinP1113936iisinN(1113936hisinNminus

i1113936misinThi


him + 1113936jisinN+i

1113936nisinTijen middot qp middot x

pijn) represents the operational costs for

loading and unloading operations at origins and des-tinations as well as rail terminals where transshipmentbetween rail and road services is realized

Formula 1113936pisinP1113936(i j)isinA1113936nisinTRijg middot qp middot 1113957z

pijn denotes the

storage costs for storing containers at rail terminalsbefore they are further moved by successive containertrainsFormula 1113936pisinPpen middot qp middot (max1113864duep minus 1113957y

p

dp 01113865 +

max11138641113957yp

dpminus duep 01113865) is the penalty costs caused by

deliveries of containers at destinations that are earlieror later than the due dates claimed by customers

Equation (5) is the environmental objective that considersthe minimization of all the carbon dioxide emissions foraccomplishing all the transportation orders )e calculation ofthe emissions is based on the activity-based method [11 31]


1113944hisinNminus

i

1113944misinThi

xp

him minus 1113944jisinN+

i

1113944nisinTij

xpijn 0 foralli isin N op dp1113966 1113967forallp isin P

(6)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij

xpijn minus 1 i isin op1113966 1113967forallp isin P

(7)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij

xpijn 1 i isin dp1113966 1113967forallp isin P

(8)

1113944nisinTij

xp

ijn le 1 forall(i j) isin A forallp isin P(9)

ypπop

t0p forallp isin Pforallπ isin 1 2 3 (10)

ypπi + otnπi middot qp + ttπijn + otnπj middot qp minus y

pπj1113872 1113873 middot x

pijn 0 forallp isin Pforall(i j) isin Aforalln isin TKijforallπ isin 1 2 3 (11)

Table 3 Decision variables

xp

ijn

0-1 decision variable if transportation service n is used to move containers of transportation order p on arc(i j) xpijn 1 otherwise

xpijn 0

1113957ypi

Non-negative fuzzy decision variable that represents the time when the containers of transportation order p arrive at node i and getunloaded and 1113957y

p

i (yp

1i yp

2i yp

3i)

1113957zpijn

Non-negative fuzzy decision variable that represents the storage time in hour of container of transportation order p at node i beforebeing moved on arc(i j) by train service n and 1113957z

pijn (z

1pijn z

2pijn z

3pijn)

Table 2 Sets indexes and parameters

P Transportation order set served by the road-rail intermodal routingp Index of a transportation order and p isin Pop Index of the origin of the containers of transportation order pdp Index of the destination of the containers of transportation order pqp Demand in TEU for the containers of transportation order pt0p Release time of the containers of transportation order p at origin opduep Due date of accomplishing transportation order pN Node set of the road-rail intermodal transportation networkA Directed arc set of the road-rail intermodal transportation networkT Transportation service set of the road-rail intermodal transportation networkh i j Indexes of the nodes in the road-rail intermodal transportation network and h i j isin NNminus

i Predecessor node set to node i and Nminusi subeN

N+i Successor node set to node i and N+

i subeN(i j) Directed arc from node i to node j and (i j) isin ATij Transportation service set on arc (i j) and Tij subeTTKij Truck service set on arc (i j) and TKij subeTijTNij Train service set (i j) and TNij subeTijm n Indexes of transportation services in the road-rail intermodal transportation network and m n isin Tdijn Distance in km of transportation service n on arc (i j)1113957ttijn Travel time in hour of transportation service n on arc(i j) and 1113957ttijn (tt1ijn tt2ijn tt3ijn)1113957rijn Fuzzy capacity in TEU of transportation service n on arc(i j) and 1113957rijn (r1ijn r2ijn r3ijn)1113957otni

Separate fuzzy loading and unloading operation time in hour per TEU of transportation service n at node i and1113957otni (otn1i otn2i otn3i)

1113957fn

i Fuzzy fixed departure time of train service n from node i and 1113957fn

i (fn1i fn

2i fn3i)

cn Travel costs in CNY per TEU per km of transportation service nen Separate loading and unloading costs in CNY per TEU of transportation service ng Storage costs in CNY per TEU per hour of railway serviceemn Carbon dioxide emission factor in kg per TEU per km of transportation service npen Penalty costs in CNY per TEU per hour caused by early or late deliveryM A large enough positive numberπ An index used in the calculation of the fuzzy decision variables and π isin 1 2 3


fnπi + ttπijn + otnπj middot qp minus y

p

πj1113872 1113873 middot xp

ijn 0 forallp isin Pforall(i j) isin Aforalln isin TNijforallπ isin 1 2 3 (12)

1113957yp

i yp

1i yp

2i yp

3i1113872 1113873 foralli isin Nforallp isin P (13)

fnπi minus y

p

(4minus π)i minus otn(4minus π)i middot qp minus zπpijn1113872 1113873 middot x

pijn 0 forallp isin Pforall(i j) isin A foralln isin TNijforallπ isin 1 2 3 (14)

1113957zp

ijn z1p

ijn z2p

ijn z3p

ijn1113872 1113873 forallp isin Pforall(i j) isin Aforalln isin TNij (15)

1113957yp

i + 1113957otni middot qp le 1113957fn

i middot xp

ijn + M middot 1 minus xp

ijn1113872 1113873 forallp isin Pforall(i j) isin A foralln isin TNij (16)

1113944pisinP

qp middot xp

ijn le 1113957rijn forall(i j) isin A foralln isin Tij (17)

xpijn isin 0 1 forallp isin Pforall(i j) isin A foralln isin Tij (18)

yp

1i geyp

2i geyp

3i ge 0 foralli isin Nforallp isin P (19)

z1pijn ge z

2pijn ge z

3pijn ge 0 forallp isin Pforall(i j) isin Aforalln isin TNij (20)

)e constraint set of the optimization model includesequations (6)ndash(20) )eir representations are explained asfollows

Equations (6) to (8) is the flow conservation constraintsthat ensure that an origin-to-destination route can begenerated for each transportation orderEquation (9) is the transportation order integrityconstraint that ensures that the containers of a trans-portation order should not be splittable which matchesAssumption 5Equation (10) is the constraint that ensures that con-tainers of a transportation order can start to get loadedat their release time which represents Assumption 2Equations (11) to (13) are the constraints that are usedto calculate fuzzy decision variable 1113957y

p

i by firstlyobtaining its three prominent pointsEquations (14) and (15) are the constraints that cal-culate fuzzy decision variable 1113957z

pijn in the way that is the

same to the determination of 1113957yp

i Equation (16) is the fixed departure time constraint thatensures that the loading operation of containers on theselected container train should not be accomplishedlater than its fixed departure timeEquation (17) is the capacity constraint that ensuresthat all the containers loaded on one transportationservice should not exceed its capacityEquations (18) to (20) are the variable domain con-straints that ensure that the values of the decisionvariables should strictly follow their definitions

44 Characteristics of the Proposed Model )e proposedmodel is a multiobjective fuzzy mixed integer nonlinearprogramming model It cannot be straightforwardly solved

to provide decision makers with crisp solutions that cansupport decision-making since its economic objective andconstraints contain both fuzzy parameters and fuzzy deci-sion variables )us the model is not clearly definedDefuzzification should be first of all realized to obtain a crispreformulation that is equivalent to the initial fuzzy pro-gramming model

)e reformulation is nonlinear since its economic ob-jective has maximum function and constraints are involvedwith multiplications of decision variables It is widely ac-knowledged that using mathematical programming softwareto run an exact solution algorithm is an effective way to solvethe routing problem by getting its global optimum solutions[62] Furthermore the linear programming model is mostsuitable for this method )erefore after the defuzzificationlinearization should be undertaken to generate an equivalentmodel that is both crisp and linear

Moreover the model yields two different objectives andusually has Pareto solutions (also known as nondominatedsolutions) One way to obtain the Pareto solutions to therouting problem is to transfer the two objectives into oneobjective Based on the characteristics above this studydesigns a solution approach given in Section 5

5 Exact Solution Approach

In this section based on the analysis of the characteristics ofthe model presented in Section 44 an exact solution ap-proach is developed as follows

51 Defuzzification )e fuzzy nonlinear programmingmodel proposed in Section 43 cannot be solved straightfor-wardly to provide the decision maker with a crisp road-railintermodal transportation plan )erefore defuzzificationshould be first of all undertaken to generate a crisp nonlinear


programming model that can be further processed to make itmore solvable

511 Defuzzification of the Fuzzy Objective )e proposedfuzzy programming model yields a fuzzy objective thatcontains fuzzy decision variables 1113957z

pijn and 1113957y

p

dp In this study

the author adoptsthe widely used fuzzy expected valuemodel to realize the defuzzification of the fuzzy objective

)e fuzzy expected value model aims to minimize ormaximize the expected value of the fuzzy objective [1]

)e fuzzy expected value of the fuzzy objective shown asequation (4) is as equation (21) where E[1113957z

pijn] and E[1113957y

p

dp] are

the fuzzy expected values of the associated fuzzy decisionvariables For a given triangular fuzzy number 1113957a there isE[1113957a] (a1 + 2 middot a2 + a3)4 [2] Consequently equation (21)can be written as equation (22) that is a crisp linear function


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij

g middot qp middot E 1113957zpijn1113960 1113961 + 1113944

pisinPpen middot qp middot max duep minus E 1113957y

p

dp1113876 1113877 01113882 1113883 + max E 1113957y

p

dp1113876 1113877 minus duep 01113882 11138831113874 1113875

(21)


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij

g middot qp middotz1p

ijn + 2 middot z2p

ijn + z3p

ijn

4+ 1113944

pisinPpen middot qp middot max duep minus

yp

1dp+ 2 middot y

p

2dp+ y

p

3dp

4 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝

+ maxy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4minus duep 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎞⎟⎠

(22)

512 Defuzzification of the Fuzzy Constraints )e fuzzyprogrammingmodel contains three fuzzy constraints includingequation (16) (fixed departure time constraint) and equation(17) (capacity constraint) since they involve either fuzzy pa-rameters or fuzzy decision variables )eir existence also makesthemodel impossible to be solved As a result after undertakingthe defuzzification of the fuzzy objective it is necessary togenerate the crisp reformulations of these fuzzy constraints

Currently fuzzy chance-constrained programming iswidely acknowledged to be an effective way to deal with thefuzzy constraint and shows good feasibility [1 2] )us thisstudy takes advantage of this method to address theabovementioned fuzzy constraints

)ree fuzzy measures can be used to establish the fuzzychance constraint ie fuzzy possibility necessity and credi-bility measures [59] Among the three measures only the fuzzycredibilitymeasure is self-dual whichmeans that suchmeasurecan ensure a fuzzy event must hold when its credibility reachesup to 1 while must fail when 0 [30 59] )e other measureshowever lack such property A fuzzy event might still fail whenits possibility is 1 and hold when its necessity equals 0

Considering the self-duality property of the fuzzy credi-bility measure this study utilizes fuzzy credibilistic chance-constrained programming to take care of the fuzzy constraints

)e fuzzy credibilistic chance constraints of equations(16) and (17) are as following equations (23) and (24)respectively where Cr is the credibility that a fuzzy eventin happens and λ isin [0 10] is the credibility level that is

set subjectively by decision makers based on theirpreference

Cr 1113957yp


i middot xp


ijn1113872 11138731113966 1113967ge λ

forallp isin Pforall(i j) isin Aforalln isin TNij(23)

Cr 1113944pisinP

qp middot xpijn le 1113957rijn

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭ge λ forall(i j) isin Aforalln isin Tij (24)

Although the fuzzy credibility chance constraints havebeen constructed they still cannot be programmed andsolved directly by mathematical programming software)erefore further crisp reformulations on them areneeded A deterministic number a and a triangular fuzzynumber 1113957n have a relationship shown as the followingequation [2 59]

Cr 1113957nge a

1 if ale n1

2n2 minus n1 minus a

2 n2 minus n1( 1113857 if n1 le ale n2

n3 minus a


0 if age n3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)


Based on equation (25) Cr 1113957nge a ge λ can be furtherrewritten as equations (26) and (27) [1 2]

2λ middot n2 minus (2λ minus 1) middot n3 ge a if λ isin [0 05] (26)

2(1 minus λ) middot n2 +(2λ minus 1) middot n1 ge a if λ isin (05 10] (27)

In equation (23) the left-hand formulaCr[1113957y

pi + 1113957otni middot qp le 1113957f

n

i middot xpijn + M middot (1 minus x

pijn)]ge λ can be

modified into Cr[1113957fn

i middot xpijn minus 1113957y

pi minus 1113957otni middot qp geM middot (x

pijn minus 1)]

that corresponds to Cr 1113957nge a Accordingly equation (23)can be reformulated as equations (28) and (29) Equation(24) can be directly converted into equations (30) and (31)

2λ middot fn2i middot x

pijn minus y

p2i minus otn2i middot qp1113872 1113873 minus (2λ minus 1) middot f

n3i middot x

pijn minus y

p1i minus otn1i middot qp1113872 1113873geM middot x

pijn minus 11113872 1113873

forallp isin Pforall(i j) isin A foralln isin TNij if λ isin [0 05](28)

2(1 minus λ) middot fn2i middot x

p

ijn minus yp

2i minus otn2i middot qp1113872 1113873 +(2λ minus 1) middot fn1i middot x

p

ijn minus yp

3i minus otn3i middot qp1113872 1113873geM middot xp

ijn minus 11113872 1113873

forallp isin Pforall(i j) isin A foralln isin TNij if λ isin (05 10] (29)

2λ middot r2ijn minus (2λ minus 1) middot r3ijn ge 1113944pisinP

qp middot xpijn forall(i j) isin Aforalln isin Tij if λ isin [0 05]

(30)

2(1 minus λ) middot r2ijn +(2λ minus 1) middot r1ijn ge 1113944pisinP

qp middot xp

ijn forall(i j) isin A foralln isin Tij if λ isin (05 10] (31)

52 Linearization After the defuzzification we can obtain acrisp mixed integer nonlinear programming model for thespecific routing problem discussed in this study )e ob-jective functions of this model are equations (22) and (5) Itsconstraint set contains equations (6) to (12) and (14)equations (18) to (20) and equations (28) to (31) Amongthese equations equations (11) (14) and (16) are nonlinearconstraints and equation (22) is a nonlinear objective

Exact solution algorithms eg branch-and-bound al-gorithm are useful tools to solve the optimization problemsby providing global optimum solutions Furthermore exactsolution algorithms can be easily programmed and run bysophisticated standard mathematical programming soft-ware eg LINGO However the exact solution algorithm ismost suitable for solving the linear programming model[63] If a nonlinear programming model describes theproblem solutions to the problem generated by an exactsolution algorithmmight fall into local optimum Moreover

the computation would consume a significant amount oftime which significantly reduces the computationalefficiency

Consequently after defuzzification linear reformulationof the crisp mixed integer nonlinear programming modelshould be conducted in order to enable the routing problemto be effectively solved by using exact solution algorithms toobtain its global optimum solution with the help of math-ematical programming software

Sun and Lang [12] proposed the linearization techniquesthat can generate equivalent linear functions to nonlinearconstraints of the mixed integer nonlinear programmingmodel indicated at the beginning of this section Based onthe linearization techniques proposed by Sun and Lang [12]equations (11) (14) and (16) can be replaced by theirequivalent linear representations shown as equations (32) to(37)


pπj geM middot x

pijn minus 11113872 1113873 forallp isin Pforall(i j) isin A foralln isin TKijforallπ isin 1 2 3

(32)

yp

πi + otnπi middot qp + ttπijn + otnπj middot qp minus yp

πj leM middot 1 minus xp

ijn1113872 1113873 forallp isin Pforall(i j) isin A foralln isin TKijforallπ isin 1 2 3

(33)


pπj geM middot x

pijn minus 11113872 1113873 forallp isin Pforall(i j) isin A foralln isin TNijforallπ isin 1 2 3

(34)


p


ijn1113872 1113873 forallp isin Pforall(i j) isin A foralln isin TNijforallπ isin 1 2 3 (35)


zπp

ijn gefnπi minus y

p

(4minus π)i minus otn(4minus π)i middot qp + M middot xp

ijn minus 11113872 1113873

forallp isin Pforall(i j) isin Aforalln isin TNijforallπ isin 1 2 3 (36)

zπp

ijn leM middot xp

ijn forallp isin Pforall(i j) isin Aforalln isin TNijforallπ isin 1 2 3 (37)

Sun et al [11] design a linearization technique that canbe used to linearize the nonlinear max function in theoptimization objectives As for the nonlinear objectiveequation (22) it can be reformulated as equation (38) by

using two non-negative auxiliary decision variables and twoauxiliary linear constraints including equations (39) and(41)


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij

gn middot qp middotz1pijn + 2 middot z

2pijn + z

3pijn

4+ 1113944

pisinPpen middot qp middot φp + τp1113872 1113873

(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4 forallp isin P (39)

φp ge 0 forallp isin P (40)

τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4minus duep forallp isin P (41)

τp ge 0 forallp isin P (42)

53 Weighted Sum Approach )e weighted sum approachhas been widely utilized by current studies to deal with themultiobjective optimizations [64] It has been acknowledgedto be the most classical approach to make tradeoffs betweenconflict objectives

As a result in this study the weighted sum approach isemployed by us to address the economic objective andenvironmental objective of the routing optimization Letobjcost and objemission separately denote the economic ob-jective ie equation (38) and environmental objective ieequation (5) of the road-rail intermodal routing problemand non-negative parametersω1 andω2 represent the weightdistributed to the economic objective and environmentalobjective respectively By using the weighted sum approachthe new objective is as the following equation

minimize ω1 middot objcost + ω2 middot objemission( 1113857 (43)

Non-negative parameters ω1 and ω2 are manually set bythe decision makers according to their preference to get arouting decision For example decision makers can set ω1 as1000 while ω2 as 1 which means that they attach moreimportance to the economic objective However setting ω1as 1 while ω2 as 1000 shows that decision makers give morepriority to the environmental objective By changing thevalues of ω1 and ω2 Pareto solutions to the routing problem

with different economic objective values and environmentalobjective values might be obtained

Finally this study can obtain a single-objective mixedinteger linear programming model for the road-rail inter-modal routing problem investigated by this study Such amodel consists of equation (43) as the objective andequations (6) to (10) equations (18) to (20) equations (28) to(37) and equations (39) to (42) as the constraints Since themodel is linear and yields only one optimization objective itcan be solved by the exact solution algorithm implementedby the mathematical programming software

6 Computational Experiment

In this section a numerical case is given to demonstrate thefeasibility of the proposed method in dealing with thespecific road-rail intermodal routing problem Some helpfulinsights are also revealed based on the analysis of the nu-merical case

61 Case Design )e road-rail intermodal transportationnetwork in this case is shown in Figure 2 that is derived fromthe authorrsquos previous study [61] In this network there are sixrail transportation lines where container trains with fixeddeparture times are operated periodically Twelve road


transportation lines undertake prehaul and end haul as well asall-road services for transportation orders with tight due dates

In the road-rail transportation network the fuzzy ca-pacities in TEU fuzzy travel times in hour and traveldistances in km of the truck fleet groups in the roadtransportation lines are listed in Table 4 )e information onthe container trains operated in the road-rail intermodaltransportation network is available in Table 5 )e values ofthe parameters shown in Table 2 are given in Table 6 byreferring to Sun et al [11] and Sun and Li [2] Finally theinformation on the transportation orders served by the road-rail intermodal routing is provided in Table 7

62 Experimental Platform In this study the author usesmathematical programming software LINGO version 120developed by LINDO Systems Inc to run the standardbranch-and-bound algorithm to solve the specific road-railintermodal routing problem formulated by the mixed in-teger linear programming model All the computation andsimulation are performed on a )inkPad Laptop with IntelCore i5-5200U 220GHz CPU 8GB RAM

63 Computational Results First of all this study setscredibility level λ as 10 ie the decision makers would notlike to bear any risk that might lead to transportation failureon capacity constraint and fixed departure time constraint)en this study conducts a serial of calculations usingdifferent values of ω1 and ω2 As for the case presented inSection 61 this study sets ω1 as 1000 and ω2 as 1 to get theoptimum value of the economic objective and obtain thevalue of the environmental objective Meanwhile the op-timum value of the environmental objective and corre-sponding value of the economic objective can be alsogenerated by setting ω1 as 1 and ω2 as 1000 )e compu-tational results are shown in Table 8

It can be observed from Table 8 that the two objectivescannot reach their optimum simultaneously Consequentlythere exist Pareto solutions to the numerical case By varying the

values of ω1 and ω2 according to the magnitudes of the twoobjectives this study can generate the Pareto solutions illustratedin Figure 3 All the Pareto solutions can be obtained within20 seconds using the solution approach proposed in Section 5

As we can see from Figure 3 the economic objective andenvironmental objective of the road-rail intermodal routingproblem are at conflict with each other Improving eitherone of the objectives will worsen the other Since the twoobjectives have a conflicted relationship decision makersmust make tradeoffs between them in the practical decision-making )e Pareto solutions illustrated in Figure 3 can helpdecision makers to determine the road-rail intermodal routeplan that is suitable for a specific decision-making situationAs indicated by Sun and Li [2] andWang et al [1] multiple-criteria decision-making methods eg AHPmethod can beused to help decision makers to select the most suitablePareto solutions under a given decision-making situation

64 Sensitivity Analysis on Routing Optimization with respectto Network Uncertainty After the decision makers deter-mine the weights distributed to the two objectives what theyneed to do is to determine the value of the credibility levelCredibility level reflects the decision makersrsquo preference tothe reliability of the routing optimization concerning thefixed departure time constraint ie equation (23) and thecapacity constraint ie equation (24) In this section theauthor analyzes whether and how the network uncertaintyinfluences the road-rail intermodal routing optimizationthrough credibility level

)e analysis in this section and the following sections alltakes the road-rail intermodal routing optimization underω1 1 and ω2 150 as an example )e abovementionedsetting of weights corresponds to the sixth Pareto solutionfrom the left side in Figure 3 when credibility level is set as10 and means that decision makers give priority to theeconomic objective while still paying some attention to theenvironmental objective )e analysis presented as followsremain the same if decision makers hold different weights

)is study varies confidence level λ from 01 to 10 with astep size of 01 and calculate the optimization results cor-responding to each credibility level )e sensitivity is shownin Figure 4

Figure 4 demonstrates that transportation networkuncertainty has a significant effect on the road-rail inter-modal routing optimization Different credibility level mightlead to different values of both economic objective andenvironment objective )e routing optimization is verysensitive to the credibility level when it changes from 04 to05 and from 06 to 07 )e increase in credibility level doesnot always result in constantly improved reliability of therouting optimization which has been demonstrated by Sunet al [2 11 30] Sometimes when increasing the credibilitythe reliability is not enhanced while the objectives get worseFor example in Figure 4 when changing the confidence levelfrom 06 to 07 it is not sure if the reliability of thetransportation concerning the capacity constraint and fixeddeparture time constraint is getting better but the economicobjective that decision makers attach more importance

3

4

5

6

7

8

92

1

Rail terminal

Origindestination

Rail transportation line(long haul by container train)

All-road transportation lineRoad transportation line(prehaul by container truck fleets)Road transportation line(end haul by container truck fleets)

Figure 2 Road-rail intermodal transportation network in thenumerical case


becomes worse As a result it cannot be determined for surethat 07 as the credibility level is better than 06

65 Fuzzy Simulation for Identifying the Best Credibility LevelUnder the above circumstance shown in Figure 4 it isnecessary to quantify the reliability of the routing

optimization under different credibility levels However theactual reliability can be only quantified when the plannedroad-rail intermodal routes are used to accomplish thetransportation orders in practice However the fact is thatthe routing is an advanced task that should be undertakenbefore the actual transportation starts [30] We can onlysimulate the actual transportation by simulating the

Table 4 Information on the truck fleet groups in the road-rail intermodal transportation network

Road transportation lines (truck fleet groups) Fuzzy capacities in TEU Fuzzy travel times in hour Travel distances in km(1 3) 45 50 65 32 40 48 90(1 4) 60 67 73 42 50 63 110(1 5) 55 64 70 85 100 142 200(1 8) 55 67 71 145 180 234 600(1 9) 50 54 63 185 244 280 680(2 3) 43 48 58 85 122 155 285(2 4) 48 55 60 60 104 135 210(2 5) 45 54 60 45 53 68 140(2 8) 40 55 64 203 255 300 700(2 9) 55 67 72 185 220 253 650(6 8) 55 58 64 55 67 75 140(6 9) 48 54 68 36 45 52 200(7 8) 50 60 65 76 88 93 320(7 9) 60 67 71 25 30 45 100

Table 5 Information on the container trains in the road-rail intermodal transportation network

Rail transportation lines(container trains)

Fuzzy fixeddeparture times

Fuzzy capacitiesin TEU

Fuzzy travel timesin hour

Travel distancesin km

Periods in number oftrains per day

(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1

Table 6 Values of the parameters indicated by Table 1 in the numerical case

Parameters ValuesPenalty costs in CNY per TEU per hour 1000

Rail transportation Road transportationTravel costs in CNY per TEU per km 2025 6Separate loading and unloading costs in CNY per TEU 195 25Storage costs in CNY per TEU per hour 3125 mdashCarbon dioxide emission factor in kg per TEU per km 0262 1064Separate fuzzy loading and unloading time in hour per TEU 005 010 015 010 020 025

Table 7 Transportation orders served by the road-rail intermodal routing optimization in the numerical case

Transportation order no Origins Destinations Release times Due dates Demands in TEU1 1 8 8 27 242 1 8 11 45 283 1 9 4 23 214 1 9 13 50 235 2 8 2 30 186 2 8 10 38 157 2 9 3 27 258 2 9 6 31 21


deterministic values of fuzzy parameters that can be onlyknown when the actual transportation starts

Fuzzy simulation developed by the previous studies of Sunet al [2 11 30] can be used to simulate the actual trans-portation by generating the deterministic values of fuzzyparameters of the transportation network based on their fuzzymembership function Fuzzy simulation can be carried outseveral times to obtain a serial of actual transportation sce-narios )en using these deterministic scenarios this studycan test the reliability of the planned road-rail intermodalroutes with respect to the capacity constraint and fixed de-parture time constraint under different credibility levels in aquantitative way )e process of fuzzy simulation is shown inFigure 5

651 Quantifying Reliability of Planned Routes under Dif-ferent Credibility Levels In this section the author definesadditional notations as follows for deterministic trans-portation scenarios

ypi non-negative deterministic decision variable that

represents the time when the containers of trans-portation order p arrive at node i and get unloadedz

pijn non-negative deterministic decision variable that

represents the storage time in hour of the containers oftransportation order p at node i before being moved onarc (i j) by train service nK total times of fuzzy simulation in the numerical casek index of a certain fuzzy simulation that correspondsto a deterministic transportation scenario and k= 1 23 Kttkijn deterministic travel time in hour of transportationservice n on arc(i j) generated in the kth fuzzysimulationrk

ijn deterministic capacity in TEU of transportationservice n on arc(i j) generated in the kth fuzzysimulationotnik separate deterministic fuzzy loading and unload-ing operation time in hour per TEU of transportationservice n at node in the kth fuzzy simulationfn

ik deterministic fixed departure time of train service n

from node i in the kth fuzzy simulation

In the kth fuzzy simulation this study tests if the plannedroutes under different credibility levels satisfy the followingtwo constraints

ypi + otnik middot qp lef

nik middot x

pijn + M middot 1 minus x

pijn1113872 1113873

forallp isin Pforall(i j) isin A foralln isin TNij(44)

1113944pisinP

qp middot xpijn le rk

ijn forall(i j) isin Aforalln isin Tij (45)

If equations (44) and (45) are both satisfied the plannedroutes are considered feasible for kth deterministic trans-portation scenario otherwise failed In this study the au-thor runs fuzzy simulation 20 times ie K 20 and thusgenerate 20 deterministic transportation scenarios )eauthor can then obtain the times that the planned routesunder a certain credibility level are feasible or failed in thesimulated deterministic scenarios Furthermore the authorcan get the ratios that the planned routes are feasible or failedin these scenarios )e results are indicated in Figure 6

As we can see from Figure 6 with the increase incredibility level the ratio that the planned routes are feasibleimproves )e improvement is stepwise When the credi-bility level is set as 05 or 06 the corresponding ratio is only15 which is very low and means that the planned routesunder such credibility level yield extensive reliability riskthat the capacity constraint or fixed departure time con-straint is violated )erefore in practical transportation theplanned routes under the credibility level of 05 and 06 willnot be considered by the decision makers

When the credibility level is changed to 07 08 or 09the ratio that the planned routes are feasible is significantlyenhanced from 15 to 75 by 3 times In practice thedecision makers might accept such reliability to a certain

Table 8 Optimum values of the two objectives

Economicobjective

Environmentalobjective

ω1 1000 and ω2 1 1787112 CNY 112814 kgω1 1 and ω2 1000 4637615 CNY 83342 kg

150

200

250

300

350

400

450

500

Environmental objective in ton

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580

Figure 3 Pareto solutions to the road-rail intermodal routingproblem when credibility level is 10

01 02 03 04 05 06 07 08 09 10Credibility level

250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton

Economic objectiveEnvironmental objective

Figure 4 Sensitivity of the road-rail intermodal routing optimi-zation with respect to the confidence level


degree on condition that they would like to bear some riskWhen the credibility level reaches up to 10 such ratioincreases to 100 which means that the planned routesunder such credibility level are considerably feasible inpractical transportation

652 Quantifying Gaps between Planned Routes and ActualBest Routes In this section this study assumes that decisionmakers accept the reliability of the planned routes under thecredibility level of 07 08 09 or 10 Since the plannedroutes obtained by the optimization under the credibility

u(a) = (a ndash n1)(n2 ndash n1)

Set a as the simulated deterministic value of ntilde

Calculate the fuzzy membership u(a) of a withrespect to ntilde = (n1 n2 n3)

For a triangular fuzzy number n = (n1 n2 n3)randomly generate a deterministic number a that

falls into range [n1 n3]

Randomly generate a real number π that falls intorange [0 1]

Start fuzzy simulation

u(a) = (n3 ndash a)(n3 ndash n2)

u(a) = 0

Yes

No

End fuzzy simulation

If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π

Figure 5 Fuzzy simulation for generating deterministic values of triangular fuzzy numbers [2 11 30]


level of 07 08 or 09 differ from these under the credibilitylevel of 10 it is necessary to compare them to help decisionmakers to determine a crisp route plan

Using the deterministic forms defined in Section 651 toreplace the corresponding fuzzy parameters and fuzzy de-cision variables this study can get a deterministic model forthe routing problem in the deterministic road-rail inter-modal transportation network

For the kth deterministic transportation scenario thisstudy can employ the deterministic model to generate itsactual best routes )is study can also calculate the values ofthe two objectives when using the planned routes under thecredibility level of 07 08 09 or 10 to move containers inthe kth deterministic transportation scenario Finally thisstudy can compare the objectives between the actual bestroutes and the planned routes )e comparisons are indi-cated in Figures 7 and 8 It should be noted that the plannedroutes under the credibility level of 07 08 or 09 is notfeasible in the 8th 10th 11th 14th and 18th deterministictransportation scenarios )us this study cannot obtain thevalues of the objectives in them

Based on Figures 7 and 8 this study can quantify the gapsof the two objectives between the planned routes and theactual best routes )is study defines the following symbols

ecodeterminsitck (envdeterminsitc

k ) optimum value of theeconomic (environmental) objective given by the de-terministic model in the kth deterministic trans-portation scenarioecofuzzyminus λ

k (envfuzzyminus λk ) value of the economic (envi-

ronmental) objective when moving containers alongthe planned routes designed by the fuzzy programmingmodel under credibility level λ in the kth deterministictransportation scenarioRMSEλ

eco(RMSEλenv) root mean square error of the

economic (environmental) objective values whenmoving containers along the planned routes undercredibility level λ with respect to the correspondingoptimum values given by the deterministic model in thetotal K deterministic transportation scenariosRMSEλ

eco and RMSEλenv can be calculated by equations

(46) and (47)

RMSEλeco

111393620k1 ecofuzzyminus λ

k minus ecodeterminsitck1113872 1113873

2

K

11139741113972

(46)

RMSEλenv

111393620k1 envfuzzyminus λ

k minus evndeterminsitck1113872 1113873

2

K

11139741113972

(47)

Using the two kinds of root mean square error definedabove this study can effectively quantify the gaps betweenthe planned routes given by the fuzzy programming modeland the actual best routes given by the deterministic model)e computational results of the root mean square error areshown in Table 9 It should be noted that the calculation ofthe root mean square error does not include the deter-ministic transportation scenarios that the planned routesunder the credibility level of 07 08 or 09 are infeasible iefor these routes K equals 15 instead of 20

As shown in Table 9 when using the planned routesunder the credibility level of 10 the root mean square errorsof the economic objective and environmental objectivevalues can decrease by sim605 and sim102 Consequently

05 06 07 08 09 10Credibility level

010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()

Ratio that the planned routes are feasibleRatio that the planned routes are failed

Figure 6 Ratio of the planned routes are feasible or failed in thesimulated deterministic scenarios

1 2 3 4 5 6 8 9 10 11 12 13 14 15 16 17 18 19 207Deterministic transportation scenario no

Actual best routesPlanned routes under credibility level of 07 08 or 09Planned routes under credibility level of 10

1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y

Figure 7 Comparison of the economic objective between theplanned routes and the actual best routes

Deterministic transportation scenario no1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n


Figure 8 Comparison of the environmental objective between theplanned routes and the actual best routes


this study can draw the conclusion that the planned routesgiven by the fuzzy programming model under the credibilitylevel of 10 are closer to the actual optimum situationTogether with the higher reliability over the planned routesunder the credibility level of 07 08 or 09 illustrated inFigure 6 the planned routes under the credibility level of 10are recommended to the decision makers to consider as thecrisp road-rail intermodal route plan that can be used in thepractical transportation

66 Comparing Proposed Fuzzy Programming Approach withExisting Deterministic Modeling In the existing literaturethat focuses on the freight routing problem in the deter-ministic road-rail intermodal transportation network pa-rameters including capacities fixed departure times traveltimes and loadingunloading times are estimated in adeterministic way by using their most likely values )eirmodeling can be easily realized by using r2ijn fn

2i tt2ijn and

otn2i to take the place of 1113957rijn 1113957fn

i 1113957ttijn and 1113957otni in the fuzzyprogramming model established in this study respectivelyAccordingly 1113957y

pi and 1113957z

pijn will be separately replaced by y

p2i

and z2p

ijnOn the condition that ω1 1 and ω2 150 this study

first uses the standard branch-and-bound algorithm to solvethe deterministic road-rail intermodal routing problem onLINGO version 120 )en this study also tests if theplanned routes under certainty are feasible or failed in the 20deterministic transportation scenarios simulated by usingfuzzy simulation)is study can also obtain the ratio that theplanned routes under road-rail intermodal transportationnetwork certainty are feasible in all the deterministic sce-narios )e ratio is only 15 which is the same as comparedto the planned routes given by the fuzzy programmingmodel under the credibility level of 05 or 06 but is con-siderably lower than results of the fuzzy programmingmodel when the credibility level is larger than 06

Additionally the planned routes under the road-railintermodal transportation network certainty have smallerroot mean square errors of the economic objectives (ie7135 thousand CNY that decreases by sim187 comparedwith the planned routes under a credibility level of 10)while yielding larger root mean square errors of the envi-ronmental objectives (ie 456 ton that increases by sim329compared with the planned routes under a credibility level of10) in the three deterministic transportation scenarios thatare feasible

Above all although the planned routes under networkcertainty can slightly improve the economic objective itworsens the environmental objective and also lead to an ex-tensive risk that the routes violate the capacity constraint andthe fixed departure time constraint As a result this study canconclude that considering the parameters of the road-rail

intermodal transportation network as uncertainty can re-markably improve reliability and thereby reduce failure risk ofthe associated freight routing optimization when comparedwith the deterministic road-rail intermodal routingoptimization

7 Conclusions

In this study the author aims at modeling and solving agreen and reliable road-rail intermodal routing problem)e objectives of the road-rail intermodal routing modelinginclude (1) minimizing total costs for accomplishing all thetransportation orders served by the routing optimization (2)minimizing total carbon dioxide emissions created in thetransportation process to realize green transportation )ereliability of the routing optimization is enhanced by fullyconsidering the uncertainty of the road-rail intermodaltransportation network Parameters that are considerablysensitive to the real-time status of the network includingcapacities fixed departure times of container trains traveltimes and loadingunloading times are modeled as trian-gular fuzzy numbers based on fuzzy set theory

To deal with the green and reliable road-rail intermodalrouting problem under network uncertainty a fuzzy goalprogramming approach is developed by this study )especific routing problem is initially formulated by a mul-tiobjective fuzzy mixed integer nonlinear programmingmodel )en an exact solution approach that combinesdefuzzification linearization and weighted sum method isproposed to address the initial model to enable that theproblem can be solved by the exact solution algorithm runby the mathematical programming software In the casestudy besides demonstrating the feasibility of the proposedmethods this study uses sensitivity analysis and fuzzysimulation to draw some helpful insights as follows

(1) )e economic objective is in conflict with the envi-ronmental objective ie improving one objective willworsen the other one By using the weighted summethod the Pareto solutions to the problem can beobtainedwhich provides candidates for decisionmakersto make effective tradeoffs between the objectives andfurther select the road-rail intermodal routes plan thatmatches a particular decision-making situation

(2) Transportation network uncertainty yields signifi-cant effects on the two objectives of the road-railintermodal routing optimization as well as its reli-ability Using fuzzy set theory to model the uncertainparameters and modeling the problem by fuzzy goalprogramming show good feasibility

(3) )e quality of credibility levels can be quantified byusing fuzzy simulation demonstrated in our study)e best credibility level can be identified to provide

Table 9 Root mean square error of the two objectives in the fuzzy simulation

Credibility level Economic objective in thousand CNY Environmental objective in ton07 08 and 09 22245 38210 8777 343


the decision makers with the best crisp routes planthat is most suitable for practical transportation

In the future work on this study the author will focus onthe following three aspects

(1) Uncertainty from the side of customers eg demanduncertainty [61] is not considered in this study Inthe future work a comprehensive consideration ofuncertainty from both demanders (shippers andreceivers) and providers (transportation network)will be investigated

(2) Some other defuzzification methods eg fuzzyranking method [60] can be also adopted to obtainthe crisp reformulation of the fuzzy programmingmodel )e comparison among different defuzzifi-cation methods can be discussed

(3) )e potential utilization of some emerging fuzzynumbers eg type-II fuzzy numbers [65] also isworth a try to check if the forms of fuzzy numberssignificantly influence the optimization results of therouting problem under uncertainty

Data Availability

)e fuzzy simulation data used to support the findings of thestudy are included within the supplementary informationfile

Conflicts of Interest

)e author declares that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

)is research was funded by the Shandong ProvincialNatural Science Foundation of China under Grant noZR2019BG006 the Project for Humanities and Social Sci-ences Research of Ministry of Education of China underGrant no 19YJC630149 and the Shandong ProvincialHigher Educational Social Science Program of China underGrant no J18RA053 )e author would like to thank DrCevin Zhang from KTH Royal Institute of Technology forhis contribution to improving the writing of this paper

Supplementary Materials

)e supplementary material contains an appendix thatpresents the data of the fuzzy simulation results on the fuzzyparameters )ese data are used for the analysis in Sections65 and 66 (Supplementary Materials)

References

[1] R Wang K Yang L Yang and Z Gao ldquoModeling andoptimization of a road-rail intermodal transport system underuncertain informationrdquo Engineering Applications of ArtificialIntelligence vol 72 pp 423ndash436 2018

[2] Y Sun and X Li ldquoFuzzy programming approaches formodeling a customer-centred freight routing problem in the

road-rail intermodal hub-and-spoke network with fuzzy softtime windows and multiple sources of time uncertaintyrdquoMathematics vol 7 no 8 p 739 2019

[3] Y Zhao R Liu X Zhang and A Whiteing ldquoA chance-constrained stochastic approach to intermodal containerrouting problemsrdquo PLoS One vol 13 no 2 Article IDe0192275 2018

[4] B Slack ldquoIntermodal transportationrdquo in Handbook of Lo-gistics and Supply-Chain Management pp 141ndash154 EmeraldGroup Publishing Limited Bingley UK 2008

[5] S Zhang X Ruan Y Xia and X Feng ldquoFoldable container inempty container repositioning in intermodal transportationnetwork of Belt and Road Initiative strengths and limita-tionsrdquo Maritime Policy amp Management vol 45 no 3pp 351ndash369 2018

[6] Y Kayikci ldquoA conceptual model for intermodal freight lo-gistics centre location decisionsrdquo Procedia-Social and Be-havioral Sciences vol 2 no 3 pp 6297ndash6311 2010

[7] D M Z Islam J Dinwoodie and M Roe ldquoTowards supplychain integration through multimodal transport in devel-oping economies the case of Bangladeshrdquo Maritime Eco-nomics amp Logistics vol 7 no 4 pp 382ndash399 2005

[8] T G Crainic ldquoService network design in freight trans-portationrdquo European Journal of Operational Researchvol 122 no 2 pp 272ndash288 2000

[9] A Caris C Macharis and G K Janssens ldquoDecision supportin intermodal transport a new research agendardquo Computersin Industry vol 64 no 2 pp 105ndash112 2013

[10] Y Sun M Lang and D Wang ldquoOptimization models andsolution algorithms for freight routing planning problem inthe multi-modal transportation networks a review of thestatus-of-the-artrdquo 1e Open Civil Engineering Journal vol 9no 1 2015

[11] Y Sun M Hrusovsky C Zhang and M Lang ldquoA time-dependent Fuzzy programming approach for the greenmultimodal routing problem with rail service capacity un-certainty and road traffic congestionrdquo Complexity vol 2018Article ID 8645793 22 pages 2018

[12] Y Sun and M Lang ldquoModeling the multicommodity mul-timodal routing problem with schedule-based services andcarbon dioxide emission costsrdquo Mathematical Problems inEngineering vol 2015 Article ID 406218 21 pages 2015

[13] M Uddin and N Huynh ldquoReliable routing of road-rail in-termodal freight under uncertaintyrdquo Networks and SpatialEconomics vol 19 no 3 pp 929ndash952 2019

[14] T-S Chang ldquoBest routes selection in international inter-modal networksrdquo Computers amp Operations Research vol 35no 9 pp 2877ndash2891 2008

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

[16] A Ghaderi and R L Burdett ldquoAn integrated location androuting approach for transporting hazardousmaterials in a bi-modal transportation networkrdquo Transportation Research PartE Logistics and Transportation Review vol 127 pp 49ndash652019

[17] M Verma V Verter and N Zufferey ldquoA bi-objective modelfor planning and managing rail-truck intermodal trans-portation of hazardous materialsrdquo Transportation ResearchPart E Logistics and Transportation Review vol 4C8 no 1pp 132ndash149 2012

[18] E Demir M Hrusovsky W Jammernegg andT Van Woensel ldquoGreen intermodal freight transportation


bi-objective modelling and analysisrdquo International Journal ofProduction Research vol 57 no 19 pp 6162ndash6180 2019

[19] J H Cho H S Kim and H R Choi ldquoAn intermodaltransport network planning algorithm using dynamic pro-gramming-A case study from Busan to Rotterdam in in-termodal freight routingrdquo Applied Intelligence vol 36 no 3pp 529ndash541 2012

[20] J J Winebrake J J Corbett A Falzarano et al ldquoAssessingenergy environmental and economic tradeoffs in intermodalfreight transportationrdquo Journal of the Air amp Waste Man-agement Association vol 58 no 8 pp 1004ndash1013 2008

[21] J Jiang D Zhang S Li and Y Liu ldquoMultimodal green lo-gistics network design of urban agglomeration with stochasticdemandrdquo Journal of Advanced Transportation vol 2019Article ID 4165942 19 pages 2019

[22] E Demir W Burgholzer M Hrusovsky E ArıkanW Jammernegg and T V Woensel ldquoA green intermodalservice network design problem with travel time uncertaintyrdquoTransportation Research Part B Methodological vol 93pp 789ndash807 2016

[23] A E P Rivera and M R Mes ldquoIntegrated scheduling ofdrayage and long-haul operations in synchromodal trans-portrdquo Flexible Services and Manufacturing Journal vol 31no 3 pp 763ndash806 2019

[24] C-H Liao P-H Tseng and C-S Lu ldquoComparing carbondioxide emissions of trucking and intermodal containertransport in Taiwanrdquo Transportation Research Part DTransport and Environment vol 14 no 7 pp 493ndash496 2009

[25] F Fotuhi and N Huynh ldquoA reliable multi-period intermodalfreight network expansion problemrdquo Computers amp IndustrialEngineering vol 115 pp 138ndash150 2018

[26] D Ambrosino A Sciomachen and C Surace ldquoEvaluation offlow dependent external costs in freight logistics networksrdquoNetworks vol 74 no 2 pp 111ndash123 2019

[27] M M Uddin and N Huynh ldquoRouting model for multi-commodity freight in an intermodal network under disrup-tionsrdquo Transportation Research Record Journal of theTransportation Research Board vol 2548 no 1 pp 71ndash802016

[28] M Mohammadi P Jula and R Tavakkoli-MoghaddamldquoDesign of a reliable multi-modal multi-commodity model forhazardous materials transportation under uncertaintyrdquo Eu-ropean Journal of Operational Research vol 257 no 3pp 792ndash809 2017

[29] B Y Chen W H K Lam A Sumalee Q Li H Shao andZ Fang ldquoFinding reliable shortest paths in road networksunder uncertaintyrdquo Networks and Spatial Economics vol 13no 2 pp 123ndash148 2013

[30] Y Sun G Zhang Z Hong and K Dong ldquoHow uncertaininformation on service capacity influences the intermodalrouting decision a fuzzy programming perspectiverdquo Infor-mation vol 9 no 1 p 24 2018

[31] Y Sun X Li X Liang and C Zhang ldquoA Bi-objective fuzzycredibilistic chance-constrained Programming approach forthe hazardous materials road-rail multimodal routing Prob-lem under uncertainty and sustainabilityrdquo Sustainabilityvol 11 no 9 p 2577 2019

[32] Q Bai and M Chen ldquo)e distributionally robust newsvendorproblem with dual sourcing under carbon tax and cap-and-trade regulationsrdquo Computers amp Industrial Engineeringvol 98 pp 260ndash274 2016

[33] Y-T Chang P T-W Lee H-J Kim and S-H Shin ldquoOp-timization model for transportation of container cargoesconsidering short sea shipping and external costrdquo

Transportation Research Record Journal of the TransportationResearch Board vol 2166 no 1 pp 99ndash108 2010

[34] Y Sun and M Lang ldquoBi-objective optimization for multi-modal transportation routing planning problem based onPareto optimalityrdquo Journal of Industrial Engineering andManagement vol 8 no 4 pp 1195ndash1217 2015

[35] J Zhang H W Ding X Q Wang W J Yin T Z Zhao andJ Dong ldquoMode choice for the intermodal transportationconsidering carbon emissionsrdquo in Proceedings of 2011 IEEEInternational Conference on Service Operations Logistics andInformatics pp 297ndash301 IEEE Piscataway NJ USA 2011July

[36] M Hrusovsky E Demir W Jammernegg andT Van Woensel ldquoHybrid simulation and optimization ap-proach for green intermodal transportation problem withtravel time uncertaintyrdquo Flexible Services and ManufacturingJournal vol 30 no 3 pp 486ndash516 2018

[37] D Zhang R He S Li and Z Wang ldquoA multimodal logisticsservice network design with time windows and environmentalconcernsrdquo PLoS One vol 12 no 9 Article ID e0185001 2017

[38] C Vale and I M Ribeiro ldquoIntermodal routing model forsustainable transport through multi-objective optimizationrdquoin First International Conference on Intelligent TransportSystems pp 144ndash154 Springer New York NY USA 2018

[39] Y Qu T Bektas and J Bennell ldquoSustainability SI multimodemulticommodity network design model for intermodalfreight transportation with transfer and emission costsrdquoNetworks and Spatial Economics vol 16 no 1 pp 303ndash3292016

[40] H G Resat and M Turkay ldquoA bi-objective model for designand analysis of sustainable intermodal transportation systemsa case study of Turkeyrdquo International Journal of ProductionResearch vol 57 no 19 pp 6146ndash6161 2019

[41] Y Chen Z Yuan and B Chen ldquoProcess optimization withconsideration of uncertainties-an overviewrdquo Chinese Journalof Chemical Engineering vol 26 no 8 pp 1700ndash1706 2018

[42] I E Grossmann R M Apap B A Calfa P Garcıa-Herrerosand Q Zhang ldquoRecent advances in mathematical program-ming techniques for the optimization of process systemsunder uncertaintyrdquo Computers amp Chemical Engineeringvol 91 pp 3ndash14 2016

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

[44] B S Boardman E M Malstrom D P Butler andM H ColeldquoComputer assisted routing of intermodal shipmentsrdquoComputers amp Industrial Engineering vol 33 no 1-2pp 311ndash314 1997

[45] J H Bookbinder and N S Fox ldquoIntermodal routing ofCanada-Mexico shipments under NAFTArdquo TransportationResearch Part E Logistics and Transportation Review vol 34no 4 pp 289ndash303 1998

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

[47] B Ayar and H Yaman ldquoAn intermodal multicommodity routingproblem with scheduled servicesrdquo Computational Optimizationand Applications vol 53 no 1 pp 131ndash153 2012

[48] L Moccia J-F Cordeau G Laporte S Ropke andM P Valentini ldquoModeling and solving a multimodaltransportation problem with flexible-time and scheduledservicesrdquo Networks vol 57 no 1 pp 53ndash68 2011

[49] H Heggen Y Molenbruch A Caris and K Braekers ldquoIn-termodal container routing integrating long-haul routing and


local drayage decisionsrdquo Sustainability vol 11 no 6 p 16342019

[50] DWolfinger F Tricoire and K F Doerner ldquoAmatheuristic fora multimodal long haul routing problemrdquo EURO Journal onTransportation and Logistics vol 8 no 4 pp 397ndash433 2019

[51] HMin ldquoInternational intermodal choices via chance-constrainedgoal programmingrdquo Transportation Research Part A Generalvol 25 no 6 pp 351ndash362 1991

[52] A Heinold and F Meisel ldquoEmission oriented vs Time ori-ented routing in the European intermodal railroad freighttransportation networkrdquo in Logistics Management pp 188ndash202 Springer New York NY USA 2019

[53] S-T Liu ldquoFuzzy total transportation cost measures for fuzzysolid transportation problemrdquo Applied Mathematics andComputation vol 174 no 2 pp 927ndash941 2006

[54] M Kaur and A Kumar ldquoMethod for solving unbalanced fullyfuzzy multi-objective solid minimal cost flow problemsrdquoApplied Intelligence vol 38 no 2 pp 239ndash254 2013

[55] P Kundu S Kar and M Maiti ldquoMulti-objective multi-itemsolid transportation problem in fuzzy environmentrdquo AppliedMathematical Modelling vol 37 no 4 pp 2028ndash2038 2013

[56] B Vahdani R Tavakkoli-Moghaddam F Jolai and A BabolildquoReliable design of a closed loop supply chain network underuncertainty an interval fuzzy possibilistic chance-constrainedmodelrdquo Engineering Optimization vol 45 no 6 pp 745ndash7652013

[57] E Ozceylan and T Paksoy ldquoInteractive fuzzy programmingapproaches to the strategic and tactical planning of a closed-loop supply chain under uncertaintyrdquo International Journal ofProduction Research vol 52 no 8 pp 2363ndash2387 2014

[58] M H F Zarandi A Hemmati and S Davari ldquo)e multi-depot capacitated location-routing problem with fuzzy traveltimesrdquo Expert Systems with Applications vol 38 no 8pp 10075ndash10084 2011

[59] Y Zheng and B Liu ldquoFuzzy vehicle routing model withcredibility measure and its hybrid intelligent algorithmrdquoApplied Mathematics and Computation vol 176 no 2pp 673ndash683 2006

[60] S Fazayeli A Eydi and I N Kamalabadi ldquoLocation-routingproblem in multimodal transportation network with timewindows and fuzzy demands presenting a two-part geneticalgorithmrdquo Computers amp Industrial Engineering vol 119pp 233ndash246 2018

[61] Y Sun X Liang X Li and C Zhang ldquoA fuzzy programmingmethod for modeling demand uncertainty in the capacitatedroad-rail multimodal routing problem with time windowsrdquoSymmetry vol 11 no 1 p 91 2019

[62] Y Xie W Lu W Wang and L Quadrifoglio ldquoA multimodallocation and routing model for hazardous materials trans-portationrdquo Journal of Hazardous Materials vol 227-228pp 135ndash141 2012

[63] S Chakraborty K Bhattacharjee and S P Sarmah ldquoAn ef-fective MILPmodel for food grain inventory transportation inIndiamdasha heuristic approachrdquo in Optimization and InventoryManagement pp 361ndash376 Springer New York NY USA2020

[64] B Sawik ldquoWeighted-sum approach for bi-objective optimi-zation of fleet size with environmental aspectsrdquo in Applica-tions of Management Science pp 101ndash116 PublishingLimited Bingley UK 2018

[65] S J Sharahi A-R Abtahi A Rashidi-Komijan andK K Damghani ldquoType-II fuzzy multi-product multi-levelmulti-period location-allocation Production-distributionproblem in supply chains modelling and optimisation

approachrdquo Fuzzy Information and Engineering vol 10 no 2pp 260ndash283 2018


optimal transportation network in strategic tactical andoperational levels [8] As summarized by Caris et al [9]recent research topics in the road-rail intermodal trans-portation planning field include policy support terminalnetwork design intermodal service network design in-termodal routing and drayage operations Among thesetopics road-rail intermodal routing that belongs to op-erational-level planning is a spotlight highlighted by re-searchers in theory and practitioners in transportationpractice [10] and is the issue targeted by this study

Road-rail intermodal routing involves the optimal uti-lization of existing transportation resources that are limitedin the road-rail intermodal transportation network It aimsat the selection of the best routes to move containers oftransportation orders demanded by customers from originsto destinations through the road-rail intermodal trans-portation network [11] )e best routes can either achieveoptimum on any one of the objectives eg minimal cost[3 12 13] minimal time [14 15] lowest risk [16 17] andminimal greenhouse gas emissions [18] or make tradeoffsamong them in multiobjective optimizations

Since logistics cost accounts for as much as 30 to50 of the total production cost of companies [14 19]reducing costs created in the road-rail intermodaltransportation process is the main task of its routingoptimization which also motivates that majority of therelative literature takes minimization of costs as a prioroptimization objective [2] Simultaneously as a greatcontributor to greenhouse gas emissions that causeglobal warming the transportation industry has beenurgently demanded for an environmental-friendly de-velopment [20ndash22] In this case the transportation in-dustry is challenged by reducing environmental impactwhile keeping profitable [23] As stated by Winebrakeet al [20] planning routes with environmental con-siderations through the intermodal transportation sys-tem is an effective way to improve the environmentalsustainability of transportation Carbon dioxide ac-counts for sim80 of the total greenhouse gas emissions[24] )is study investigates a green road-rail intermodalrouting problem considering the reduction of carbondioxide emissions

Reliability is a crucial issue that is involved in the in-termodal transportation system that needs to coordinatevarious transportation modes large number of facilitiesand different actors in an extensive network that is under amore complicated environment than unimodal trans-portation [18 25] In the road-rail intermodal routingproblem reliability means the successful accomplishmentof transportation orders by using planned road-rail in-termodal routes [26] Uncertainty is the primary factor thatinfluences the reliability of intermodal transportation[27ndash29] Although a short-term planning [8] the routingoptimization should be undertaken before the actualtransportation starts [30] However the operations of thetransportation network are easily disrupted by variousfactors such as bad weather traffic congestion accidentsbreakdown or shortage of transportation facilities and soon [26] As a result network parameters regarding

operational times and service capacities that are exceed-ingly sensitive to the real-time status of the transportationnetwork cannot keep stable and are also challenging to beforecasted precisely in the advanced routing decision-making which results in the uncertainty of the intermodaltransportation network )erefore this study discusses thereliable road-rail intermodal routing problem under un-certainty in this study

Above all in this study the author systematically ex-plores a green and reliable road-rail intermodal routingproblem by making the following contributions

(1) A road-rail intermodal routing problem is extendedby enhancing its environmental sustainability withcarbon dioxide emission optimization and improv-ing transportation reliability with multiple sources ofuncertainty

(2) Based on the fuzzy set theory triangular fuzzy numbersare adopted to model the uncertainty of the road-railintermodal transportation network including capac-ities travel times loading and unloading times andfixed departure times Multiobjective optimization isemployed to deal with the carbon dioxide emissions bygenerating Pareto solutions to make tradeoffs betweenthe economic and environmental objectives

(3) Fuzzy goal programming approach is used to for-mulate the green and reliable freight routing prob-lem in the road-rail intermodal transportationnetwork with hub-and-spoke and point-to-pointstructures A multiobjective fuzzy mixed integernonlinear programming model is established andassociated exact solution approach is developed

(4) )e proposed methods are demonstrated in a nu-merical case in which sensitivity analysis and fuzzysimulation are utilized to analyze the effects oftransportation network uncertainty on the routingoptimization quantitatively and finally determinethe optimum confidence level in the fuzzy chanceconstraints

)e remaining sections of this study are organized asfollows Section 2 reviews relative literature in order tofind the research gap and identify the contributions ofthis study Section 3 provides the modeling foundationfor the road-rail intermodal routing problem includingmodeling network uncertainty by fuzzy set theory andformulating the transportation system Section 4 presentsa multiobjective fuzzy mixed integer nonlinear pro-gramming model for the specific green and reliablerouting problem )en an exact solution approach thatcombines defuzzification linearization and the weightedsum method is designed for the sake of obtaining theglobal optimum solutions to the problem A numericalcase is given in Section 6 to demonstrate the feasibility ofthe proposed fuzzy goal programming approach )isstudy also uses sensitivity analysis and fuzzy simulationto quantify the effects of transportation network uncer-tainty on the routing optimization in this section Finallyconclusions and insights are drawn in Section 7


2 Literature Review



























Fixeddeparturetimes

Traveltimes




programming




programming



programming


programming






programming



programming


programming






programming


programming



programming


programming



programming



programming






1113957m + 1113957n m1 m2 m3( 1113857 + n1 n2 n3( 1113857

m1 + n1 m2 + n2 m3 + n3( 1113857(1)









Estimation value

EV1

EV2

EV3


Maximumestimation

Minimumestimation

EV2 EV3


0

1


















1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 1113944pisinP


dp 01113882 1113883 + max 1113957y

p

dpminus duep 01113882 11138831113874 1113875

(4)


1113944(i j)isinA

1113944nisinTij


ijn(5)



pijn is the


i1113936misinThi







pijn denotes the


p

dp 01113865 +

max11138641113957yp





1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(6)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(7)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(8)

1113944nisinTij

xp


ypπop



pπj1113872 1113873 middot x



xp

ijn


xpijn 0

1113957ypi


p

i (yp

1i yp

2i yp

3i)

1113957zpijn


pijn (z

1pijn z

2pijn z

3pijn)







1113957fn


i (fn1i fn

2i fn3i)




p

πj1113872 1113873 middot xp


1113957yp

i yp

1i yp

2i yp


fnπi minus y

p



1113957zp

ijn z1p

ijn z2p

ijn z3p


1113957yp


i middot xp



1113944pisinP

qp middot xp



yp

1i geyp

2i geyp


z1pijn ge z

2pijn ge z




p















pijn and 1113957y

p

dp In this study





p

dp] are



1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij



p

dp1113876 1113877 01113882 1113883 + max E 1113957y

p

dp1113876 1113877 minus duep 01113882 11138831113874 1113875

(21)


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 2 middot z2p

ijn + z3p

ijn

4+ 1113944


yp

1dp+ 2 middot y

p

2dp+ y

p

3dp

4 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝

+ maxy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4minus duep 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎞⎟⎠

(22)







Cr 1113957yp


i middot xp


ijn1113872 11138731113966 1113967ge λ


Cr 1113944pisinP


⎧⎪⎨

⎪⎩

⎫⎪⎬



Cr 1113957nge a

1 if ale n1



n3 minus a


0 if age n3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)







n


pijn)]ge λ can be




pijn minus 1)]



pijn minus y


n3i middot x

pijn minus y


pijn minus 11113872 1113873



p

ijn minus yp


p

ijn minus yp


ijn minus 11113872 1113873




(30)


qp middot xp








pπj geM middot x


(32)

yp




(33)


pπj geM middot x


(34)


p




zπp

ijn gefnπi minus y

p


ijn minus 11113872 1113873


zπp

ijn leM middot xp





1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


2pijn + z

3pijn

4+ 1113944


(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp



τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp
























3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References










































































2 Literature Review



























Fixeddeparturetimes

Traveltimes




programming




programming



programming


programming






programming



programming


programming






programming


programming



programming


programming



programming



programming






1113957m + 1113957n m1 m2 m3( 1113857 + n1 n2 n3( 1113857

m1 + n1 m2 + n2 m3 + n3( 1113857(1)









Estimation value

EV1

EV2

EV3


Maximumestimation

Minimumestimation

EV2 EV3


0

1


















1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 1113944pisinP


dp 01113882 1113883 + max 1113957y

p

dpminus duep 01113882 11138831113874 1113875

(4)


1113944(i j)isinA

1113944nisinTij


ijn(5)



pijn is the


i1113936misinThi







pijn denotes the


p

dp 01113865 +

max11138641113957yp





1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(6)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(7)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(8)

1113944nisinTij

xp


ypπop



pπj1113872 1113873 middot x



xp

ijn


xpijn 0

1113957ypi


p

i (yp

1i yp

2i yp

3i)

1113957zpijn


pijn (z

1pijn z

2pijn z

3pijn)







1113957fn


i (fn1i fn

2i fn3i)




p

πj1113872 1113873 middot xp


1113957yp

i yp

1i yp

2i yp


fnπi minus y

p



1113957zp

ijn z1p

ijn z2p

ijn z3p


1113957yp


i middot xp



1113944pisinP

qp middot xp



yp

1i geyp

2i geyp


z1pijn ge z

2pijn ge z




p















pijn and 1113957y

p

dp In this study





p

dp] are



1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij



p

dp1113876 1113877 01113882 1113883 + max E 1113957y

p

dp1113876 1113877 minus duep 01113882 11138831113874 1113875

(21)


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 2 middot z2p

ijn + z3p

ijn

4+ 1113944


yp

1dp+ 2 middot y

p

2dp+ y

p

3dp

4 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝

+ maxy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4minus duep 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎞⎟⎠

(22)







Cr 1113957yp


i middot xp


ijn1113872 11138731113966 1113967ge λ


Cr 1113944pisinP


⎧⎪⎨

⎪⎩

⎫⎪⎬



Cr 1113957nge a

1 if ale n1



n3 minus a


0 if age n3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)







n


pijn)]ge λ can be




pijn minus 1)]



pijn minus y


n3i middot x

pijn minus y


pijn minus 11113872 1113873



p

ijn minus yp


p

ijn minus yp


ijn minus 11113872 1113873




(30)


qp middot xp








pπj geM middot x


(32)

yp




(33)


pπj geM middot x


(34)


p




zπp

ijn gefnπi minus y

p


ijn minus 11113872 1113873


zπp

ijn leM middot xp





1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


2pijn + z

3pijn

4+ 1113944


(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp



τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp
























3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References



























































































Fixeddeparturetimes

Traveltimes




programming




programming



programming


programming






programming



programming


programming






programming


programming



programming


programming



programming



programming






1113957m + 1113957n m1 m2 m3( 1113857 + n1 n2 n3( 1113857

m1 + n1 m2 + n2 m3 + n3( 1113857(1)









Estimation value

EV1

EV2

EV3


Maximumestimation

Minimumestimation

EV2 EV3


0

1


















1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 1113944pisinP


dp 01113882 1113883 + max 1113957y

p

dpminus duep 01113882 11138831113874 1113875

(4)


1113944(i j)isinA

1113944nisinTij


ijn(5)



pijn is the


i1113936misinThi







pijn denotes the


p

dp 01113865 +

max11138641113957yp





1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(6)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(7)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(8)

1113944nisinTij

xp


ypπop



pπj1113872 1113873 middot x



xp

ijn


xpijn 0

1113957ypi


p

i (yp

1i yp

2i yp

3i)

1113957zpijn


pijn (z

1pijn z

2pijn z

3pijn)







1113957fn


i (fn1i fn

2i fn3i)




p

πj1113872 1113873 middot xp


1113957yp

i yp

1i yp

2i yp


fnπi minus y

p



1113957zp

ijn z1p

ijn z2p

ijn z3p


1113957yp


i middot xp



1113944pisinP

qp middot xp



yp

1i geyp

2i geyp


z1pijn ge z

2pijn ge z




p















pijn and 1113957y

p

dp In this study





p

dp] are



1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij



p

dp1113876 1113877 01113882 1113883 + max E 1113957y

p

dp1113876 1113877 minus duep 01113882 11138831113874 1113875

(21)


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 2 middot z2p

ijn + z3p

ijn

4+ 1113944


yp

1dp+ 2 middot y

p

2dp+ y

p

3dp

4 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝

+ maxy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4minus duep 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎞⎟⎠

(22)







Cr 1113957yp


i middot xp


ijn1113872 11138731113966 1113967ge λ


Cr 1113944pisinP


⎧⎪⎨

⎪⎩

⎫⎪⎬



Cr 1113957nge a

1 if ale n1



n3 minus a


0 if age n3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)







n


pijn)]ge λ can be




pijn minus 1)]



pijn minus y


n3i middot x

pijn minus y


pijn minus 11113872 1113873



p

ijn minus yp


p

ijn minus yp


ijn minus 11113872 1113873




(30)


qp middot xp








pπj geM middot x


(32)

yp




(33)


pπj geM middot x


(34)


p




zπp

ijn gefnπi minus y

p


ijn minus 11113872 1113873


zπp

ijn leM middot xp





1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


2pijn + z

3pijn

4+ 1113944


(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp



τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp
























3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References














































































1113957m + 1113957n m1 m2 m3( 1113857 + n1 n2 n3( 1113857

m1 + n1 m2 + n2 m3 + n3( 1113857(1)









Estimation value

EV1

EV2

EV3


Maximumestimation

Minimumestimation

EV2 EV3


0

1


















1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 1113944pisinP


dp 01113882 1113883 + max 1113957y

p

dpminus duep 01113882 11138831113874 1113875

(4)


1113944(i j)isinA

1113944nisinTij


ijn(5)



pijn is the


i1113936misinThi







pijn denotes the


p

dp 01113865 +

max11138641113957yp





1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(6)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(7)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(8)

1113944nisinTij

xp


ypπop



pπj1113872 1113873 middot x



xp

ijn


xpijn 0

1113957ypi


p

i (yp

1i yp

2i yp

3i)

1113957zpijn


pijn (z

1pijn z

2pijn z

3pijn)







1113957fn


i (fn1i fn

2i fn3i)




p

πj1113872 1113873 middot xp


1113957yp

i yp

1i yp

2i yp


fnπi minus y

p



1113957zp

ijn z1p

ijn z2p

ijn z3p


1113957yp


i middot xp



1113944pisinP

qp middot xp



yp

1i geyp

2i geyp


z1pijn ge z

2pijn ge z




p















pijn and 1113957y

p

dp In this study





p

dp] are



1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij



p

dp1113876 1113877 01113882 1113883 + max E 1113957y

p

dp1113876 1113877 minus duep 01113882 11138831113874 1113875

(21)


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 2 middot z2p

ijn + z3p

ijn

4+ 1113944


yp

1dp+ 2 middot y

p

2dp+ y

p

3dp

4 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝

+ maxy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4minus duep 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎞⎟⎠

(22)







Cr 1113957yp


i middot xp


ijn1113872 11138731113966 1113967ge λ


Cr 1113944pisinP


⎧⎪⎨

⎪⎩

⎫⎪⎬



Cr 1113957nge a

1 if ale n1



n3 minus a


0 if age n3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)







n


pijn)]ge λ can be




pijn minus 1)]



pijn minus y


n3i middot x

pijn minus y


pijn minus 11113872 1113873



p

ijn minus yp


p

ijn minus yp


ijn minus 11113872 1113873




(30)


qp middot xp








pπj geM middot x


(32)

yp




(33)


pπj geM middot x


(34)


p




zπp

ijn gefnπi minus y

p


ijn minus 11113872 1113873


zπp

ijn leM middot xp





1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


2pijn + z

3pijn

4+ 1113944


(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp



τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp
























3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References























































































1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 1113944pisinP


dp 01113882 1113883 + max 1113957y

p

dpminus duep 01113882 11138831113874 1113875

(4)


1113944(i j)isinA

1113944nisinTij


ijn(5)



pijn is the


i1113936misinThi







pijn denotes the


p

dp 01113865 +

max11138641113957yp





1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(6)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(7)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(8)

1113944nisinTij

xp


ypπop



pπj1113872 1113873 middot x



xp

ijn


xpijn 0

1113957ypi


p

i (yp

1i yp

2i yp

3i)

1113957zpijn


pijn (z

1pijn z

2pijn z

3pijn)







1113957fn


i (fn1i fn

2i fn3i)




p

πj1113872 1113873 middot xp


1113957yp

i yp

1i yp

2i yp


fnπi minus y

p



1113957zp

ijn z1p

ijn z2p

ijn z3p


1113957yp


i middot xp



1113944pisinP

qp middot xp



yp

1i geyp

2i geyp


z1pijn ge z

2pijn ge z




p















pijn and 1113957y

p

dp In this study





p

dp] are



1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij



p

dp1113876 1113877 01113882 1113883 + max E 1113957y

p

dp1113876 1113877 minus duep 01113882 11138831113874 1113875

(21)


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 2 middot z2p

ijn + z3p

ijn

4+ 1113944


yp

1dp+ 2 middot y

p

2dp+ y

p

3dp

4 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝

+ maxy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4minus duep 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎞⎟⎠

(22)







Cr 1113957yp


i middot xp


ijn1113872 11138731113966 1113967ge λ


Cr 1113944pisinP


⎧⎪⎨

⎪⎩

⎫⎪⎬



Cr 1113957nge a

1 if ale n1



n3 minus a


0 if age n3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)







n


pijn)]ge λ can be




pijn minus 1)]



pijn minus y


n3i middot x

pijn minus y


pijn minus 11113872 1113873



p

ijn minus yp


p

ijn minus yp


ijn minus 11113872 1113873




(30)


qp middot xp








pπj geM middot x


(32)

yp




(33)


pπj geM middot x


(34)


p




zπp

ijn gefnπi minus y

p


ijn minus 11113872 1113873


zπp

ijn leM middot xp





1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


2pijn + z

3pijn

4+ 1113944


(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp



τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp
























3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References










































































1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(6)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(7)

1113944hisinNminus

i

1113944misinThi

xp


i

1113944nisinTij


(8)

1113944nisinTij

xp


ypπop



pπj1113872 1113873 middot x



xp

ijn


xpijn 0

1113957ypi


p

i (yp

1i yp

2i yp

3i)

1113957zpijn


pijn (z

1pijn z

2pijn z

3pijn)







1113957fn


i (fn1i fn

2i fn3i)




p

πj1113872 1113873 middot xp


1113957yp

i yp

1i yp

2i yp


fnπi minus y

p



1113957zp

ijn z1p

ijn z2p

ijn z3p


1113957yp


i middot xp



1113944pisinP

qp middot xp



yp

1i geyp

2i geyp


z1pijn ge z

2pijn ge z




p















pijn and 1113957y

p

dp In this study





p

dp] are



1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij



p

dp1113876 1113877 01113882 1113883 + max E 1113957y

p

dp1113876 1113877 minus duep 01113882 11138831113874 1113875

(21)


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 2 middot z2p

ijn + z3p

ijn

4+ 1113944


yp

1dp+ 2 middot y

p

2dp+ y

p

3dp

4 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝

+ maxy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4minus duep 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎞⎟⎠

(22)







Cr 1113957yp


i middot xp


ijn1113872 11138731113966 1113967ge λ


Cr 1113944pisinP


⎧⎪⎨

⎪⎩

⎫⎪⎬



Cr 1113957nge a

1 if ale n1



n3 minus a


0 if age n3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)







n


pijn)]ge λ can be




pijn minus 1)]



pijn minus y


n3i middot x

pijn minus y


pijn minus 11113872 1113873



p

ijn minus yp


p

ijn minus yp


ijn minus 11113872 1113873




(30)


qp middot xp








pπj geM middot x


(32)

yp




(33)


pπj geM middot x


(34)


p




zπp

ijn gefnπi minus y

p


ijn minus 11113872 1113873


zπp

ijn leM middot xp





1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


2pijn + z

3pijn

4+ 1113944


(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp



τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp
























3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References











































































p

πj1113872 1113873 middot xp


1113957yp

i yp

1i yp

2i yp


fnπi minus y

p



1113957zp

ijn z1p

ijn z2p

ijn z3p


1113957yp


i middot xp



1113944pisinP

qp middot xp



yp

1i geyp

2i geyp


z1pijn ge z

2pijn ge z




p















pijn and 1113957y

p

dp In this study





p

dp] are



1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij



p

dp1113876 1113877 01113882 1113883 + max E 1113957y

p

dp1113876 1113877 minus duep 01113882 11138831113874 1113875

(21)


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 2 middot z2p

ijn + z3p

ijn

4+ 1113944


yp

1dp+ 2 middot y

p

2dp+ y

p

3dp

4 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝

+ maxy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4minus duep 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎞⎟⎠

(22)







Cr 1113957yp


i middot xp


ijn1113872 11138731113966 1113967ge λ


Cr 1113944pisinP


⎧⎪⎨

⎪⎩

⎫⎪⎬



Cr 1113957nge a

1 if ale n1



n3 minus a


0 if age n3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)







n


pijn)]ge λ can be




pijn minus 1)]



pijn minus y


n3i middot x

pijn minus y


pijn minus 11113872 1113873



p

ijn minus yp


p

ijn minus yp


ijn minus 11113872 1113873




(30)


qp middot xp








pπj geM middot x


(32)

yp




(33)


pπj geM middot x


(34)


p




zπp

ijn gefnπi minus y

p


ijn minus 11113872 1113873


zπp

ijn leM middot xp





1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


2pijn + z

3pijn

4+ 1113944


(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp



τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp
























3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References












































































pijn and 1113957y

p

dp In this study





p

dp] are



1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij



p

dp1113876 1113877 01113882 1113883 + max E 1113957y

p

dp1113876 1113877 minus duep 01113882 11138831113874 1113875

(21)


1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


ijn + 2 middot z2p

ijn + z3p

ijn

4+ 1113944


yp

1dp+ 2 middot y

p

2dp+ y

p

3dp

4 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎛⎜⎝

+ maxy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp

4minus duep 0

⎧⎪⎨

⎪⎩

⎫⎪⎬

⎪⎭⎞⎟⎠

(22)







Cr 1113957yp


i middot xp


ijn1113872 11138731113966 1113967ge λ


Cr 1113944pisinP


⎧⎪⎨

⎪⎩

⎫⎪⎬



Cr 1113957nge a

1 if ale n1



n3 minus a


0 if age n3

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(25)







n


pijn)]ge λ can be




pijn minus 1)]



pijn minus y


n3i middot x

pijn minus y


pijn minus 11113872 1113873



p

ijn minus yp


p

ijn minus yp


ijn minus 11113872 1113873




(30)


qp middot xp








pπj geM middot x


(32)

yp




(33)


pπj geM middot x


(34)


p




zπp

ijn gefnπi minus y

p


ijn minus 11113872 1113873


zπp

ijn leM middot xp





1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


2pijn + z

3pijn

4+ 1113944


(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp



τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp
























3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References















































































n


pijn)]ge λ can be




pijn minus 1)]



pijn minus y


n3i middot x

pijn minus y


pijn minus 11113872 1113873



p

ijn minus yp


p

ijn minus yp


ijn minus 11113872 1113873




(30)


qp middot xp








pπj geM middot x


(32)

yp




(33)


pπj geM middot x


(34)


p




zπp

ijn gefnπi minus y

p


ijn minus 11113872 1113873


zπp

ijn leM middot xp





1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


2pijn + z

3pijn

4+ 1113944


(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp



τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp
























3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References










































































zπp

ijn gefnπi minus y

p


ijn minus 11113872 1113873


zπp

ijn leM middot xp





1113944(i j)isinA

1113944nisinTij


pisinP1113944iisinN

1113944hisinNminus

i

1113944misinThi



i

1113944nisinTij


⎛⎜⎝ ⎞⎟⎠

+ 1113944pisinP

1113944(i j)isinA

1113944nisinTRij


2pijn + z

3pijn

4+ 1113944


(38)

φp ge duep minusy

p

1dp+ 2 middot y

p

2dp+ y

p

3dp



τp gey

p

1dp+ 2 middot y

p

2dp+ y

p

3dp
























3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References





















































































3

4

5

6

7

8

92

1

Rail terminal

Origindestination

















(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References






















































































(3 6) 5 65 7 86 93 100 45 52 60 520 1(3 7) 72 84 9 78 85 94 62 68 75 600 1(4 6) 14 146 153 82 89 95 65 74 82 650 1(4 7) 11 116 125 90 94 102 46 50 56 480 1(5 6) 15 153 158 90 98 110 58 64 68 610 1(5 7) 96 103 105 89 95 98 68 72 78 660 1



















nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References






















































































nik middot x


pijn1113872 1113873


1113944pisinP







Economicobjective



150

200

250

300

350

400

450

500


Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

85 90 95 100 105 110 11580



250270290310330350370390410

Econ

omic

obj

ectiv

e in

10 th

ousa

nd C

NY

65

70

75

80

85

90

Envi

ronm

enta

l obj

ectiv

e in

ton














u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References




















































































u(a) = 0

Yes

No


If n1 le a le n2Yes

No

No

Yes

If n2 le a le n3

If u(a) ge π














(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References





















































































(46) and (47)

RMSEλeco



2

K

11139741113972

(46)

RMSEλenv



2

K

11139741113972

(47)




010203040

60708090

50

100

Ratio

of t

he p

lann

ed ro

utes

are f

easib

le o

r fai

led

()





1000150020002500300035004000450050005500

Econ

omic

obj

ectiv

e in

thou

sand

CN

Y



70

75

80

85

90

95

Envi

ronm

enta

lob

ject

ive i

n to

n






2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References












































































2i tt2ijn and



pi and 1113957z


p2i

and z2p






7 Conclusions














Data Availability




Acknowledgments




References















































































Data Availability




Acknowledgments




References










































































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