Research ArticlePath Planning and Vehicle Scheduling Optimization for LogisticDistribution of Hazardous Materials in Full Container Load

Huo Chai12 Ruichun He2 Changxi Ma2 Cunjie Dai13 and Kun Zhou1

1Mechatronics Technology and Research Institute Lanzhou Jiaotong University Lanzhou 730070 China2School of Traffic and Transportation Lanzhou Jiaotong University Lanzhou 730070 China3Engineering Technology Center for Information of Logistics amp Transport Equipment Lanzhou 730070 China

Correspondence should be addressed to Ruichun He tranman163com

Received 4 July 2017 Revised 30 September 2017 Accepted 26 October 2017 Published 16 November 2017

Academic Editor Luisa Di Paola

Copyright copy 2017 Huo Chai et alThis is an open access article distributed under theCreativeCommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Mathematical models for path planning and vehicle scheduling for logistic distribution of hazardous materials in full containerload (FCL) are established with their problem-solvingmethods proposed First a two-stage multiobjective optimization algorithmis designed for path planning In the first stage pulse algorithm is used to obtain the Pareto paths from the distribution center toeach destination In the second stage a multiobjective optimizationmethod based on Nondominated Sorting Genetic Algorithm II(NSGA-II) is designed to obtain candidate transport paths Secondwith analysis on the operating process of vehicleswith hazardousmaterials in FCL the vehicle scheduling problem is converted to Vehicle Routing Problem with Time Windows (VRPTW) Aproblem-solving method based on estimation of distribution is adopted A transport timetable for all vehicles based on theirtransport paths is calculated with participation of the decision-makers A visual vehicle scheduling plan is presented for thedecision-makers Last two examples are used to test the method proposed in this study distribution of hazardous materials ina small-scale test network and distribution of oil products for sixteen gas stations in the main districts of Lanzhou city In bothexamples our method is used to obtain the path selection and vehicle scheduling plan proving that validity of our method isverified

1 Introduction

Similar to that of ordinary goods logistic distribution ofhazardous materials can be classified as full container load(FCL) [1 2] and less than container load (LCL) [3] Ifthe transport volume required by a destination is no lessthan the capacity of one transportation vehicle there willbe no less than one vehicle involved in a logistic job Toefficiently finish the job operation of vehicles in FCL isnecessary Distribution of goods in LCL can be attributedto the commonly seen Vehicle Routing Problem (VRP)To plan for the logistic distribution of ordinary goods inFCL the generalized shortest paths between the distributioncenter and multiple destinations can be adopted Howeveroptimization of distribution of hazardous materials involvesat least two objectives transportation cost and risk Insuch scenario multiple nondominated paths between the

distribution center and the destinations exist As a result avehicle transport path has to be chosen according to riskpreferences in a specific situation

The problem of path planning and vehicle scheduling forlogistic distribution of hazardous materials in LCL is funda-mentally the multiobjective VRP [4] Zografos and Androut-sopoulos established a Vehicle Routing Problem with TimeWindows (VRPTW) model for distribution of hazardousmaterials and designed an insertion-based biobjective heuris-tic algorithm On such basis they introduced a model for theissue of emergency site selection and designed a Lagrangianrelaxation heuristic algorithm [5]Their contributions consti-tuted a full set of decision support system for transport pathplanning of hazardous materials and coordinated emergencyresponse arrangement after accidents In a case study ofnatural gas transportation in Thailand Pradhananga et alproposed a genetic algorithm converting the HAZMAT

HindawiDiscrete Dynamics in Nature and SocietyVolume 2017 Article ID 9685125 13 pageshttpsdoiorg10115520179685125

2 Discrete Dynamics in Nature and Society

Vehicle Routing Problem with Time Windows (HVRPTW)withmultiple objectives namely transportation cost and riskinto a single objective problem through linear weighting [6]Later they proposed a multiobjective ant colony algorithmtargeting at solving this problem the validity of which wasverified in a case of distribution of liquefied petroleum gas inOsaka Japan [7]

Existing studies on path planning and vehicle schedulingfor logistic distribution of hazardous materials in FCL focuson transportation of hazardous materials between an originand a destination (OD) while those between a distributioncenter andmultiple destinations are rarely seen In this studythe problem-solving of logistic distribution of hazardousmaterials in FCL is divided into two stages path planningand vehicle scheduling First a multiobjective optimizationmodel targeting at transportation cost and risk is establishedfor path planning However if existing evolutionary multi-objective optimization algorithms are adopted all the nodesalong the transportation network have to be encoded Bydoing so the extended length of code in the case of large-scale transportation networks will result in large volumes ofcalculation which leads to difficulty in finding the optimalsolution under limited number of iterations To avoid suchsituation we design the two-stage method mentioned Inthe first stage a pulse algorithm is used to obtain thePareto-optimal paths from the distribution center to eachdestination so that only the Pareto-optimal paths obtainedare allowed as candidate solutions subsequently In the secondstage the Pareto-optimal paths obtained in the first stage areencoded to calculate the final results using algorithms basedon Nondominated Sorting Genetic Algorithm II (NSGA-II)By segmenting the problem into subproblems in two stagesthe volume of calculation is significantly reduced which leadsto an increased efficiency in calculation and correspondinglyraises probability of obtaining the optimal solution Thenbased on the path plan selected by decision-makers eachdistribution job of a destination is treated as a node in VRPwith time window of the node being the time window ofthe destination By doing so vehicle scheduling problem isconverted to a single objective VRPTW An algorithm basedon Univariate Marginal Distribution Algorithm (UMDA) isdesigned to solve the VRPTW to obtain a transport schedulefor all vehicles

2 Mathematical Model

To solve the distribution problem of vehicles with hazardousmaterials in FCL optimizations are carried out in both pathplanning and vehicle scheduling stages with a purpose ofminimizing transportation cost and total risk of all trans-portation jobs To simplify the risk evaluation process oftransportation of hazardous materials two assumptions areproposed on the premise of not violating practical conditions

(1) Risks produced for vehicles with the same type in thesame road section are equivalent The risks are relevant onlyto conditions of a specific road section without consideringother factors such as whether loaded volume of a vehiclereaches its capacity and vehicle conditions

(2)The risk value of a vehicle transportingwith no loads iszeroThat is when a vehicle returns to the distribution centerafter unloading hazardous materials in a destination site itchooses a path with the minimum transportation cost

In a transport network 119866 = (119873 119864) 119873 represents a setof 119899 nodes and 119864 represents a set of road sections betweenthese nodes The potential risk for a vehicle of a certain typetransporting hazardous materials in a road section (119894 119895) is 119903119894119895The transportation cost induced can be classified into twocases cost of 1198881119894119895 for transport with FCL when the travel timeis Δ1199051119894119895 and cost of 1198882119894119895 for transport with no load when thetravel time is Δ1199052119894119895 Assume there are119898 destinations (119863 = 11988911198892 119889119898 119863 sub 119873) to distribute hazardous materialsto them Each destination requires a transport volume of1198921 1198922 119892119898 with a time window of [119864119879119889 119871119879119889] for desti-nation V There are multiple paths 119875119900119889 from the distributioncenter 119900 to the destination 119889 All vehicles start from thedistribution center 119900 and return to the distribution centerafter finishing its job The capacity of each vehicle is 119902The average loading time and unloading time for a vehicleare Δ1199051 and Δ1199052 respectively Variable 120575119900119889119894119895 = 1 indicates(119894 119895) isin 119864 namely the road section is in the path from thedistribution center 119900 to the destination 119889 otherwise 120575119900119889119894119895 = 0Subsequently we analyze the distribution time number ofvehicles as well as the total traveling distance of vehicles andthe total risk

21 Number of Vehicles Distribution of vehicles with FCLnamely 119892119889 ge 119902 is mainly concerned in this section In thiscondition a vehicle can either finish the whole job or commita portion of a job The number of vehicles required by adestination is 119886119889 119886119889 can be determined in the following rules

(1) When 119892119889119902 is an integer 119886119889 = 119892119889119902(2) When 119892119889119902 is not an integer 119886119889 = lceil119892119889119902rceilThe total number of vehicles required to finish distri-

bution jobs of all destinations is sum119889 119886119889 The number ofvehicles required on the condition of given required transportvolume of a destination is constant Although the totalnumber of vehicles is constant the same vehicle is allowed toparticipate in distribution jobs of multiple destinations whenthe condition of time window is satisfied Issues concerningthe number of vehicles dispatched and schedules for vehiclesleavingarriving will be discussed in the section of vehiclescheduling optimization

22 Total Cost and Risk of Vehicle Travel To finish the distri-bution job of destination 119889 119886119889 vehicles have to be involvedThe transportation cost of all vehicles from the distributioncenter to the destinations is the sum of transportation cost ofall road sections traveled namely

1198881119900119889 = 119886119889sum(119894119895)

1198881119894119895120575119900119889119894119895 (1)

Based on assumption (2) the returning path of a vehicle withno load from the destination to the distribution center is the

Discrete Dynamics in Nature and Society 3

path with minimum cost represented as 1198882119900119889 with its valuecalculated using the shortest path algorithm According tothe analysis above the transportation cost involved in thedistribution job for the destination 119889 is 1198881119900119889 + 1198882119900119889

Similarly the total risk involved in the distribution job ofdestination 119889 is

119903119900119889 = 119886119889sum(119894119895)

119903119894119895120575119900119889119894119895 (2)

23 Vehicle Path Optimization The following mathematicalmodel (P1) of vehicle path optimization can be established

P1 min 1198911 = sum119889isin119873

119886119889( sum(119894119895)isin119864

1198881119894119895120575119900119889119894119895 + 1198882119900119889) (3)

min 1198912 = sum119889isin119873

119886119889 sum(119894119895)isin119864

119903119894119895120575119900119889119894119895 (4)

st sum119895

120575119900119889119894119895 minussum119895

120575119900119889119895119894 =

1 119894 = 119900minus1 119894 = 1198890 otherwise

forall119895 isin 119873 119900 isin 119873 119889 isin 119873(5)

120575119900119889119894119895 = 0 1 forall119900 119889 isin 119873 (119894 119895) isin 119864 (6)

The objective function (3) is a functional expressionof transportation cost and the objective function (4) is afunctional expression of transportation risk Constraint (5)ensures the formation of a complete transportation path fromthe distribution center to destinations

24 Distribution Time The travel time of a vehicle from thedistribution center to destination 119889 and its correspondingreturning time are respectively

Δ1199051119900119889 = sum(119894119895)

Δ1199051119894119895 (7)

Δ1199052119900119889 = sum(119894119895)

Δ1199052119894119895 (8)

The time used by a vehicle for a round trip from thedistribution center to the destination 119889 and after it finishesdistribution back to the distribution center is in total

Δ119905119900119889 = Δ1199051 + Δ1199051119900119889 + Δ1199052 + Δ1199052119900119889 (9)

Assuming the leaving time of a vehicle from the dis-tribution center is 1199050119900119889 the time moment when the vehiclearrives at the distribution center from destination 119889 shouldbe 1199050119900119889 +Δ119905119900119889 Since the required time window for destination119889 is [119864119879119889 119871119879119889] the time moment when the vehicle arrives atthe destination 119889 should satisfy the following condition

119864119879119889 le 1199050119900119889 + Δ1199051 + Δ1199051119900119889 le 119871119879119889 (10)

1

8

9

5

2

3

67

4

10

Assignment 1

Assignment 3

Assignment 2

Figure 1 A sketch diagram of vehicle scheduling assignmentsconverted to VRPTW

25 Vehicle Scheduling By solving model P1 the vehicletravel paths from the distribution center to various desti-nations and back to the distribution center are obtainedDecision-makers are allowed to choose favorable travel pathsdepending on the specific risk preferences though how todetermine a vehicle dispatching plan is still an issue to beaddressed With limited vehicles available the number ofvehicles for dispatching is an optimization objective to beconsidered

For convenience each distribution job for a destination isdivided into several assignments namely a distribution for avehicle is considered as an assignment Any one assignmentshould be finished within the time range from the earlieststarting time to the latest ending time Each assignmentshould be undertaken by only one vehicle Each vehicleundertakes a new assignment only after it finishes an assign-ment on hand Each assignment can be viewed as a node in atwo-dimensional plane All the destinations can be arrangedaccording to the serial numbers of nodes In this way eachassignment in a distribution job can be thus numberedFor example the assignments for the distribution job todestination 1198891 can be represented by 1198691198891 = 1 2 1198861198891assignments for 1198892 can be represented by 1198691198892 = 1198861198891 + 1 1198861198891 +2 1198861198891 + 1198861198892 assignments for 1198893 can be represented by1198691198893 = 1198861198891 + 1198861198892 + 1 1198861198891 + 1198861198892 + 2 1198861198891 + 1198861198892 + 1198861198893119869 = 1198691198891 cup 1198691198892 cup sdot sdot sdot cup 119869119889119898 represents the set of all assignmentsThe vehicle scheduling problem can be converted to VRPwith time windows (Figure 1) with the objective functionindicating a balance between minimum number of vehiclesand averageminimum loads carried by each vehicleThe timewindow and service time for assignment 119894 (119894 isin 119869119889) are [119864119879119889 minus(Δ1199051 + Δ11990510119889) 119871119879119889 + (Δ1199052 + Δ11990520119889)] and 1199050119889 respectively Sincea node carries only information of assignments transportvolume is not contained and can be regarded as zero and norestriction is posed on the capacity of vehicles119909119894119895V (119909119894119895V isin [0 1]) is used to indicate whether a vehicleV continues with assignment 119895 after it finishes assignment 119894If yes 119909119894119895V = 1 else 119909119894119895V = 0 119910119894V (119910119894V isin [0 1]) is used to

4 Discrete Dynamics in Nature and Society

Distrib

ution

cent

er

Destin

ation

1

Distrib

ution

cent

er

Destin

ation

dDist

ribut

ionce

nter

Destin

ation

mDist

ribut

ionce

nter

(cminusd r

minusd )

middot middot middot middot middot middot

(c+d r

+d )

(cd r

d)

(cd rd)

Figure 2 Total cost and total risk of path formed by connecting the distribution center with each destination

indicate whether assignment 119894 is implemented by vehicle VIf yes 119910119894V = 1 else 119910119894V = 0 The vehicle scheduling problemcan be described by the following model (P2)

P2 min 119891= 119872sum119894isin119869

sumVisin1198811199090119894V + sum

Visin119881(sum119894isin119869

sum119895isin119869

Δ119905119894119895119909119894119895V minus 120583)2 (11)

st sumVisin119881119910119894V = 1 forall119894 isin 119869 (12)

sum119894isin119869

119909119894119895V = 119910119895V forall119895 isin 119869 V isin 119881 (13)

sum119895isin119869

119909119894119895V = 119910119894V forall119894 isin 119869 V isin 119881 (14)

sum119894isin119869

119909119894119895119896 minussum119895isin119869

119909119894119895119896 = 0 forallV isin 119881 (15)

119879119895V = 119879119894V + Δ1199050119889119909119894119895Vforall119889 isin 119873 119894 isin 119869119889 119895 isin 119869 V isin 119881 (16)

119864119879119889 minus (Δ1199051 + Δ11990510119889) le 119879119895Vle 119871119879119889 + (Δ1199052 + Δ11990520119889)

forall119889 isin 119873 119895 isin 119869119889 V isin 119881(17)

119909119894119895V isin [0 1] forall119894 119895 isin 119869 V isin 119881 (18)

119910119894V isin [0 1] forall119894 isin 119869 V isin 119881 (19)

The objective function (11) represents the minimumnumber of vehicles and difference of their travel time where120583 = sumVisin119881sum119894isin119869sum119895isin119869 Δ119905119894119895119909119894119895Vsum119894isin119869sumVisin119881 1199090119894V The first part ofthis function represents the minimum number of vehicles119872 is a sufficiently large integer to ensure the priority level ofthe objective of vehicle number The second part representsminimized standard deviation of travel time of all vehiclesunder the same number of vehicles to ensure minimumdifference in transport jobs for each vehicle Constraints(12)ndash(15) guarantee that each vehicle can carry out only oneassignment at a time That is assignments can be carried outby a vehicle consecutively but not in parallel In additionan assignment can be carried out by only one vehicleConstraint (16) represents the time moment when node 119895 is

reached Constraint (17) ensures satisfaction of time windowrestriction at the moment when node 119895 is reached3 Problem-Solving Method

The optimization objective of model P1 is to obtain thePareto-optimal solution sets of total cost and total risk for dis-tribution jobs from the distribution center to all destinationsThis process can be viewed as a calculation of transportationcost and risk involved in the path formed by connecting thedistribution center to each destination (see Figure 2) (119888minus119889 119903minus119889 )represents the cost and risk of the path before reachingdestination 119889 from the distribution center (119888+119889 119903+119889 ) representsthe cost and risk of the path from the destination 119889 back tothe distribution center Assume (119888minus119889 + 119888119889 + 119888+119889 119903minus119889 + 119903119889 + 119903+119889 ) is asolution among the Pareto solution set If (119888119889 119903119889) ≺ (1198881015840119889 1199031015840119889) (≺indicating domination) then (119888minus119889 + 1198881015840119889 + 119888+119889 119903minus119889 + 1199031015840119889 + 119903+119889 ) willnot emerge in the Pareto solution set This conclusion can besimply proved by contradiction If (119888minus119889 + 1198881015840119889 + 119888+119889 119903minus119889 + 1199031015840119889 + 119903+119889 )is one solution among the Pareto solution set then either119888minus119889 + 1198881015840119889 + 119888+119889 le 119888minus119889 + 119888119889 + 119888+119889 or 119903minus119889 + 1199031015840119889 + 119903+119889 le 119903minus119889 + 119903119889 + 119903+119889has to hold Namely either 1198881015840119889 le 119888119889 or le 119903119889 should be correctwhich contradicts (119888119889 119903119889) ≺ (1198881015840119889 1199031015840119889)

As can be seen in order to obtain the Pareto solution setof model P1 a Pareto-optimal solution is calculated for thepath from the distribution center to each destination so thatother paths are excluded out A two-stage method is adopted

The problem of finding the path from the distributioncenter to each destination is essentially a biobjective shortestpath problem (BSP) Generally there are two types of solu-tions to it dynamic programming (DP) [8 9] and ranking[10] Pulse algorithm [11] is a precise algorithm to solve BSPAlthough an idea of recursively traversing all paths in a net-work is adopted it is still efficient by using a four-step pruningmechanism to exclude dominated paths effectively Otherthan the advantage of high efficiency this algorithm has avirtue of finding a full set of Pareto-optimal solution whichis hardly achievable in heuristic or evolutionary algorithmsThe key procedure to pulse algorithm is a newly added nodeif satisfying one of the following four conditions incomingpartial pathwill be prunedwhichmeans the incoming partialpath containing this node will be eliminated

(1) It contains cycles(2) It exceeds either one or both upper bounds [11]

defined by the nadir point before reaching the endnode

Discrete Dynamics in Nature and Society 5

n1 n2 middot middot middot middot middot middot nm

Figure 3 Encoding format

(3) It is dominated by any solution which is in currentefficient set before reaching the end node

(4) Each node that is passed by a feasible path is markedwith a label storing two objective values of the pathIf the incoming partial path is dominated by eitherobjective value stored in the label then this path isdropped or else the objective value of this path willbe added in the label of the node

In the second stage the Pareto paths obtained in thefirst stage are encoded to calculate the final results usingmultiobjective optimization based on NSGA-II [12] Sinceno paths other than the Pareto paths are involved in thesecond stage the length of code is significantly shrunk whenusing multiobjective optimization Therefore the chance ofobtaining the optimal solution increases prominently underthe same size of population and number of iterations

The following introduces encoding method fitness func-tion and population regeneration strategy for NSGA-II

(1) Individual Coding Natural number is adopted for individ-ual coding the length of coding is119898 and Figure 3 shows theformat of the encoding

Where 119899119894 isin [1max(|119875119900119889119894 |)] 119875119900119889119894 is Pareto-optimal pathset from the distribution center 0 to the destination nodes 119889119894If the 119894th value of the code is 119899119894 then it means 119899119894mod|119875119900119889119894 |thpath (obtained by the first stage) from the distribution center119900 to the destination node 119889119894 is adopted for transportationFor example the numbers of Pareto-optimal paths from thedistribution center 0 to the destination nodes 1198891 sim 1198894 are 2 35 and 4 respectively Then a code of [1435] means the 1st (1mod 2) 1st (4 mod 3) 3rd (3 mod 5) and 1st (5 mod 4) of thePareto paths from o to 1198891 sim 1198894 for the individual are adoptedfor transportation

(2) Fitness Function Assuming 119894119899119889119894 represents an individual[119888 V119886119897119906119890 119903 V119886119897119906119890] = 119891(119894119899119889119894) is used to represent fitness of theindividual 119894119899119889119894 where 119888 V119886119897119906119890 and 119903 V119886119897119906119890 are transportationcost and risk respectively based on the selected path plan ofindividual 119894119899119889119894(3) Population Updating Strategy A population adds in newindividuals by crossover for which integers are used Firsttwo individuals are randomly selected in the population withtwo positions 1199011199001199041 and 1199011199001199042 (1199011199001199041 lt 1199011199001199042) randomlygenerated These two individuals 1199011199001199041 and 1199011199001199042 are crossedover to generate two new individuals

Mutation is also operated with integers to obtain newindividuals An individual is selected randomly while twopositions 1199011199001199041 and 1199011199001199042 (1199011199001199041 lt 1199011199001199042) are randomlygenerated Positions of the selected individual1199011199001199041 and1199011199001199042

Table 1 Lengths of each road section and risks of vehicles travelingin each road section

Road section Lengthkm Risk(0 1) 3656 00456(0 3) 985 00214(0 4) 3029 00154(0 5) 1582 00286(0 6) 926 00059(1 2) 4516 00217(1 3) 3036 00020(1 4) 4547 00083(2 3) 2426 00232(2 5) 2795 00225(2 7) 3918 00252(4 6) 2716 00186(4 8) 4889 00068(5 7) 2126 00546(5 8) 3635 00032(6 8) 2415 00457(7 8) 4265 00022

are subtracted by max(|119875(119899)119900119889|) separately with absolute values

taken to generate a new individualModel P2 is targeted at the objective of balance between

number of vehicles and transport job intensity undertakenby each vehicle which is essentially VRP Except that thetransport distance and number of vehicles in the objectivefunction should be replaced by number of vehicles andfairness of transport assignments other steps can be solvedby referring to the problem-solving method for VRPTW Inthis study the problem-solving method based on UMDA forVRPHTW given by literature [13] based on estimation ofdistribution is adopted

The problem-solving process for path planning and vehi-cle scheduling of hazardousmaterials transportation is shownin Figure 4

4 Numerical Example

41 A Small-Scale Network Figure 5 shows a test transporta-tion network where node 0 represents the distribution centerand nodes 1 2 4 7 and 8 represent destinationsThe distanceand risk of each road section in the network are shown inTable 1 The required transport volume and time windowof each destination are shown in Table 2 The capacity of avehicle is 135m3 The average loading time unloading timeand the average travel speed of a vehicle are 075 h 075 hand 45 kmh respectively The cost for transport in FCL is50 $km the cost for transport with no load is 10 $km Avehicle transport plan aims to minimize the transportationcost and risk

Table 2 shows the required transport volumes of alldestinations as well as the accepted unloading time windowswhich is hard indicating a waiting action if a vehicle arrivesearlier than the earliest starting time specified

6 Discrete Dynamics in Nature and Society

Table 2 Required transport volume and time window of a destination

Destination 1 2 4 7 8Required volumem3 35 18 295 45 60Time window [800 1600] [1200 2000] [800 1230] [800 1800] [800 2000]

The pulse algorithm is used to obtain thePareto solutions of all paths from the

End

Initialize network and

Get all paths selection solution byNSGA-II algorithm (path planning)

Decision-makers chose favorable travel pathsdepending on the specific risk preferences

The vehicle scheduling plan under thecurrent path selection plan is acquiredby UMDA (vehicle scheduling)

Satisfied

Yes

No

Yes

No

Start

requirement d = 1

distribution center destination d d = d + 1

d le m

Figure 4 Flow chart for solving problem of path planning andvehicle scheduling of hazardous materials transportation

After the first stage Pareto shortest path (departingpath) and shortest distance paths (returning path) betweenthe distribution center to each destination can be obtained(Table 3)

When the Pareto paths set is obtained the Pareto solutionset of total cost and risk for transportation between thedistribution center and all destinations can be obtainedTable 4 shows the path selection plans after decoding ofPareto solutions

When a certain transport plan is selected model P2 isused to solve the results for the path adopted In this waya vehicle scheduling plan under the current path selectionplan is acquired For example when path plan 1 is selectedin Table 4 the vehicle scheduling plan of Table 5 can beobtained and when path plan 14 is selected the vehicle

1

8

02

34

5 6

7

Distribution centerOther nodesDestination

Figure 5 A test network for transportation of hazardous materials

scheduling plan of Table 6 can be obtained The columnof node represents the distribution node arrived at by avehicle The timetable shows the time moments of a vehiclein a round trip for loading start (LS) departure from thedistribution center (DD) unloading start at the destination(US) returning after unloading finishes (RU) and arrivingback at the distribution center (AC)

42 Comparison with General NSGA-II In addition a gen-eral NSGA-II algorithm (GNA) is designed for the pathplanning in order to illustrate the effectiveness of the two-stage algorithm (TSA) proposed Different from the two-stage algorithm in the general NSGA-II algorithm thepriority based encoding [14] method is adopted and theencoding length is |119873| lowast 119898 However in the second stage ofthe two-stage algorithm the encoding length is only 119898 Forthe example in Section 41 the parameters of two algorithmsare set as follows the size of population is 100 and theiteration number is 50 generations In Figures 6(a)ndash6(d) isthe Pareto-optimal front obtained by iterative 5 10 20 and50 generations respectively By comparison it can be foundthat the final solution can be obtained by using the TSAwhenthe iteration reaches the 10th generation (14 plans see Table 4)while the GNA still fails to obtain the optimal solution untilthe fiftieth iterationThis is just a network of 9 nodes and thisgap will become even bigger for large-scale networks Thereis no direct relationship between the encoding length and thenetwork size in the two-stage algorithm so for large-scalenetworks the algorithm is still valid

Discrete Dynamics in Nature and Society 7

Table 3 Departing path and returning path between the distribution center and each destination

Destination Departing path Lengthkm Risk Returning path Lengthkm

1 0rarr1 3656 00456 1rarr0 36560rarr3rarr1 4021 00234

2 0rarr3rarr2 3411 00446 2rarr3rarr0 34114 0rarr4 3029 00154 4rarr0 3029

7

0rarr5rarr7 3708 00832

7rarr5rarr0 37080rarr3rarr2rarr7 7329 006980rarr6rarr8rarr7 7606 005380rarr5rarr8rarr7 9482 003400rarr4rarr8rarr7 12183 00244

80rarr6rarr8 3341 00516

8rarr6rarr0 33410rarr5rarr8 5217 003180rarr4rarr8 7918 00222

Table 4 Path selection plans

Solution Path Total cost$ Total risk

1

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

216132 08630

2

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

218541 07964

3

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

236768 07640

4

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

239177 06974

5

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

252843 06788

6

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

266943 06662

7

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr4rarr8rarr6rarr0

268888 06494

8

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

269352 05996

8 Discrete Dynamics in Nature and Society

Table 4 Continued

Solution Path Total cost$ Total risk

9

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr6rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

273479 05798

10

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

287579 05672

11

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

289988 05006

12

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

313757 04622

13

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

319699 04526

14

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

343468 04142

Table 5 Vehicle scheduling plan (for solution 1)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

2

1 626 711 800 845 9344 934 1019 1059 1144 12257 1225 1310 1359 1444 15337 1533 1618 1708 1753 1842

3

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

4

4 635 720 800 845 9254 925 1010 1051 1136 12168 1216 1301 1346 1431 15158 1515 1600 1645 1730 18148 1814 1859 1944 2029 2113

Discrete Dynamics in Nature and Society 9

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

2500 3000 3500 40002000Transportation cost$

TSAGNA

(a) Iteration number = 5

2500 3000 3500 40002000Transportation cost$

TSAGNA

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k(b) Iteration number = 10

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k

2500 3000 3500 40002000Transportation cost$

TSAGNA

(c) Iteration number = 20

2500 3000 3500 40002000Transportation cost$

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

TSAGNA

(d) Iteration number = 50

Figure 6 Pareto solutions obtained in path optimization

5 Case Study

The proposed methodology was implemented for the urbantransportation network in Lanzhou China Daily trafficvolumes of all road sections in realistic road networks ofmain districts are used to estimate vehicle travel time for allroad sections In addition other factors such as distributionof population and facilities as well as probabilities andconsequences of traffic accident along the road sectionsinvolved are also considered to establish a traditional model[15] to estimate the risk value of each road section for vehicletransportation Thus a transport network for hazardous

materials (see Figure 7) in main districts of Lanzhou isformed containing 170 nodes and 272 edges Distribution ofgasoline 92 (permil) for 16 gas stations (e) under PetroChinain this network is taken as an example to calculate vehiclescheduling plans for oil product transportation

Table 7 shows the positions of 16 gas stations in thenetwork indicated by their nearest nodes (destination site119889) as well as their needs of gasoline and the time windowof distribution The position of the distribution center isindicated by node 1 The volume of a transportation vehicle(tank car) is 135m3 Traveling fee of a tank car in FCL inurban highways is 200 RMBh while that with no load is

10 Discrete Dynamics in Nature and Society

1

2

3

4

56

78

910

11

12

13

14

15

1617

181920

21

22

23

24

25

26

27

28

29

30

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49 50

51

5255

53

5456

57

58

5960

6162

63

64

65

66

67

69

70

7172

7374

75

7677 78

79

80

81

8283

84

85

86

8788

899091 92

93

94

9596

979899100101102

103

104105

106

107108

109110

111112 113

114115116

117

118

119

120121

122

123124125

126127128

129

130

131 132133

134

135

136

137139138

140141

142 143

144145

146147

148

149

150 151

152153

154

155

156157

158

159

160

161

162163

164

165

166 167

68

168

169

170

Figure 7 A transport network for hazardous materials in main districts of Lanzhou

Table 6 Vehicle scheduling plan (for solution 14)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

11 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

34 0715 0800 0840 0925 10061 1006 1051 1144 1229 13187 1318 1403 1646 1731 1820

52 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

32 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

41 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

6 8 0729 0814 1000 1045 11307 1130 1215 1457 1542 1631

150 RMBh eliminating cost of loading and unloading Timefor either loading or unloading is 075 h

C is the programming language to implement the algo-rithmThe running environment is PC i5-3470 320GHz 4GParameters setting in the program and CPU times of vehiclesare shown in Table 8 A total of 164 path selection plans areobtained When a certain plan is selected the correspondingtimetable for vehicle operation can be obtained by the vehiclescheduling program

A path plan is a choice based on game between trans-portation cost and risk by a decision-maker with a specific

risk preference For example in plan 1 which has the lowesttransportation cost but the highest transportation risk thetransportation time is 439658 RMB while the transporta-tion risk is 32262 8 vehicles are needed to complete jobsAs another example in plan 164 which has the lowesttransportation risk but the highest transportation cost thetransportation time is 460983 RMB while the transporta-tion risk is 25586 also 8 vehicles are needed to complete jobsThe traveling path and transport timetable for all vehicles inplan 164 are shown in Tables 9 and 10

Discrete Dynamics in Nature and Society 11

Table 7 Nodes representing 16 gas stations as well as their needs of gasoline and the time window of distribution

Number Destination Required volumem3 Time window1 2 15 [1200 1600]2 12 26 [830 1330]3 16 12 [1030 1600]4 25 16 [800 1200]5 43 16 [800 1600]6 44 29 [1000 1600]7 47 32 [900 1500]8 52 25 [900 1200]9 62 24 [800 1000]10 63 10 [800 1500]11 94 18 [1400 1800]12 138 25 [800 1500]13 146 40 [1200 1800]14 150 23 [1000 1600]15 167 9 [1200 1800]16 170 25 [800 1600]

Table 8 Transport parameters setting in the program and CPU times

Stage Parameters setting CPU timessPopulation size Iteration

Path selectionStage 1 the pulse algorithm is used to obtain the Pareto solutions of all pathsfrom the distribution center to each destination mdash mdash 17

Stage 2 get all paths selection solution by NSGA-II algorithm 500 100 32Vehicle scheduling

The vehicle scheduling plan under the current path selection plan is acquired byUMDA 200 100 9

6 Conclusion

Distribution of hazardous materials in FCL is essentially amultistage combinatorial optimization problem In this studya decision approach for distribution of hazardous materialsin FCL with a single distribution center is formed which canprovide a decision support for transportation enterprises andadministrative authorities

In this study mathematical models for path planningand vehicle scheduling problems are established respectivelywith corresponding problem-solving algorithms designedIn order to solve the path planning model a two-stagealgorithm is designed Based on screening of paths in thefirst stage the computational complexity of the second stageinvolved in the global path optimization is significantlyreduced which improves calculation efficiency In obtainingthe vehicle scheduling plan the operating process of atransporting vehicle is converted to single objective VRPTWThen the problem-solvingmethod forVRPTWon the basis ofestimation of distribution is adopted to obtain the timetablefor all vehicles

The proposed approach was applied to a hypotheticalnumerical example as well as an application to a real casein distribution of gasoline of Lanzhou ChinaThe numerical

example showed that for small-scale problems the two-stagealgorithm can obtain Pareto-optimal front faster in the sameiterationHowever for large-scale problems due to the lengthof coding it is difficult to obtain Pareto-optimal path set in alimited iterative step in the second stage the problem size ofVRPTW is only related to the number of destination nodesnot to the scale of the transportation network Therefore thevehicle scheduling method has high efficiency whether it issmall-scale or large-scale transportation network

Our purpose is to design a decision support systemfor path planning and vehicle scheduling of distribution ofhazardous materials in FCL Further research has to be doneat least in two directions distribution of hazardous materialsin FCL with multiple distribution centers and transportationwith different volumes vehicle will be focused on

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (nos 61064012 61364026

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 3

path with minimum cost represented as 1198882119900119889 with its valuecalculated using the shortest path algorithm According tothe analysis above the transportation cost involved in thedistribution job for the destination 119889 is 1198881119900119889 + 1198882119900119889

Similarly the total risk involved in the distribution job ofdestination 119889 is

119903119900119889 = 119886119889sum(119894119895)

119903119894119895120575119900119889119894119895 (2)

23 Vehicle Path Optimization The following mathematicalmodel (P1) of vehicle path optimization can be established

P1 min 1198911 = sum119889isin119873

119886119889( sum(119894119895)isin119864

1198881119894119895120575119900119889119894119895 + 1198882119900119889) (3)

min 1198912 = sum119889isin119873

119886119889 sum(119894119895)isin119864

119903119894119895120575119900119889119894119895 (4)

st sum119895

120575119900119889119894119895 minussum119895

120575119900119889119895119894 =

1 119894 = 119900minus1 119894 = 1198890 otherwise

forall119895 isin 119873 119900 isin 119873 119889 isin 119873(5)

120575119900119889119894119895 = 0 1 forall119900 119889 isin 119873 (119894 119895) isin 119864 (6)

The objective function (3) is a functional expressionof transportation cost and the objective function (4) is afunctional expression of transportation risk Constraint (5)ensures the formation of a complete transportation path fromthe distribution center to destinations

24 Distribution Time The travel time of a vehicle from thedistribution center to destination 119889 and its correspondingreturning time are respectively

Δ1199051119900119889 = sum(119894119895)

Δ1199051119894119895 (7)

Δ1199052119900119889 = sum(119894119895)

Δ1199052119894119895 (8)

The time used by a vehicle for a round trip from thedistribution center to the destination 119889 and after it finishesdistribution back to the distribution center is in total

Δ119905119900119889 = Δ1199051 + Δ1199051119900119889 + Δ1199052 + Δ1199052119900119889 (9)

Assuming the leaving time of a vehicle from the dis-tribution center is 1199050119900119889 the time moment when the vehiclearrives at the distribution center from destination 119889 shouldbe 1199050119900119889 +Δ119905119900119889 Since the required time window for destination119889 is [119864119879119889 119871119879119889] the time moment when the vehicle arrives atthe destination 119889 should satisfy the following condition

119864119879119889 le 1199050119900119889 + Δ1199051 + Δ1199051119900119889 le 119871119879119889 (10)

1

8

9

5

2

3

67

4

10

Assignment 1

Assignment 3

Assignment 2

Figure 1 A sketch diagram of vehicle scheduling assignmentsconverted to VRPTW

25 Vehicle Scheduling By solving model P1 the vehicletravel paths from the distribution center to various desti-nations and back to the distribution center are obtainedDecision-makers are allowed to choose favorable travel pathsdepending on the specific risk preferences though how todetermine a vehicle dispatching plan is still an issue to beaddressed With limited vehicles available the number ofvehicles for dispatching is an optimization objective to beconsidered

For convenience each distribution job for a destination isdivided into several assignments namely a distribution for avehicle is considered as an assignment Any one assignmentshould be finished within the time range from the earlieststarting time to the latest ending time Each assignmentshould be undertaken by only one vehicle Each vehicleundertakes a new assignment only after it finishes an assign-ment on hand Each assignment can be viewed as a node in atwo-dimensional plane All the destinations can be arrangedaccording to the serial numbers of nodes In this way eachassignment in a distribution job can be thus numberedFor example the assignments for the distribution job todestination 1198891 can be represented by 1198691198891 = 1 2 1198861198891assignments for 1198892 can be represented by 1198691198892 = 1198861198891 + 1 1198861198891 +2 1198861198891 + 1198861198892 assignments for 1198893 can be represented by1198691198893 = 1198861198891 + 1198861198892 + 1 1198861198891 + 1198861198892 + 2 1198861198891 + 1198861198892 + 1198861198893119869 = 1198691198891 cup 1198691198892 cup sdot sdot sdot cup 119869119889119898 represents the set of all assignmentsThe vehicle scheduling problem can be converted to VRPwith time windows (Figure 1) with the objective functionindicating a balance between minimum number of vehiclesand averageminimum loads carried by each vehicleThe timewindow and service time for assignment 119894 (119894 isin 119869119889) are [119864119879119889 minus(Δ1199051 + Δ11990510119889) 119871119879119889 + (Δ1199052 + Δ11990520119889)] and 1199050119889 respectively Sincea node carries only information of assignments transportvolume is not contained and can be regarded as zero and norestriction is posed on the capacity of vehicles119909119894119895V (119909119894119895V isin [0 1]) is used to indicate whether a vehicleV continues with assignment 119895 after it finishes assignment 119894If yes 119909119894119895V = 1 else 119909119894119895V = 0 119910119894V (119910119894V isin [0 1]) is used to

4 Discrete Dynamics in Nature and Society

Distrib

ution

cent

er

Destin

ation

1

Distrib

ution

cent

er

Destin

ation

dDist

ribut

ionce

nter

Destin

ation

mDist

ribut

ionce

nter

(cminusd r

minusd )

middot middot middot middot middot middot

(c+d r

+d )

(cd r

d)

(cd rd)

Figure 2 Total cost and total risk of path formed by connecting the distribution center with each destination

indicate whether assignment 119894 is implemented by vehicle VIf yes 119910119894V = 1 else 119910119894V = 0 The vehicle scheduling problemcan be described by the following model (P2)

P2 min 119891= 119872sum119894isin119869

sumVisin1198811199090119894V + sum

Visin119881(sum119894isin119869

sum119895isin119869

Δ119905119894119895119909119894119895V minus 120583)2 (11)

st sumVisin119881119910119894V = 1 forall119894 isin 119869 (12)

sum119894isin119869

119909119894119895V = 119910119895V forall119895 isin 119869 V isin 119881 (13)

sum119895isin119869

119909119894119895V = 119910119894V forall119894 isin 119869 V isin 119881 (14)

sum119894isin119869

119909119894119895119896 minussum119895isin119869

119909119894119895119896 = 0 forallV isin 119881 (15)

119879119895V = 119879119894V + Δ1199050119889119909119894119895Vforall119889 isin 119873 119894 isin 119869119889 119895 isin 119869 V isin 119881 (16)

119864119879119889 minus (Δ1199051 + Δ11990510119889) le 119879119895Vle 119871119879119889 + (Δ1199052 + Δ11990520119889)

forall119889 isin 119873 119895 isin 119869119889 V isin 119881(17)

119909119894119895V isin [0 1] forall119894 119895 isin 119869 V isin 119881 (18)

119910119894V isin [0 1] forall119894 isin 119869 V isin 119881 (19)

The objective function (11) represents the minimumnumber of vehicles and difference of their travel time where120583 = sumVisin119881sum119894isin119869sum119895isin119869 Δ119905119894119895119909119894119895Vsum119894isin119869sumVisin119881 1199090119894V The first part ofthis function represents the minimum number of vehicles119872 is a sufficiently large integer to ensure the priority level ofthe objective of vehicle number The second part representsminimized standard deviation of travel time of all vehiclesunder the same number of vehicles to ensure minimumdifference in transport jobs for each vehicle Constraints(12)ndash(15) guarantee that each vehicle can carry out only oneassignment at a time That is assignments can be carried outby a vehicle consecutively but not in parallel In additionan assignment can be carried out by only one vehicleConstraint (16) represents the time moment when node 119895 is

reached Constraint (17) ensures satisfaction of time windowrestriction at the moment when node 119895 is reached3 Problem-Solving Method

The optimization objective of model P1 is to obtain thePareto-optimal solution sets of total cost and total risk for dis-tribution jobs from the distribution center to all destinationsThis process can be viewed as a calculation of transportationcost and risk involved in the path formed by connecting thedistribution center to each destination (see Figure 2) (119888minus119889 119903minus119889 )represents the cost and risk of the path before reachingdestination 119889 from the distribution center (119888+119889 119903+119889 ) representsthe cost and risk of the path from the destination 119889 back tothe distribution center Assume (119888minus119889 + 119888119889 + 119888+119889 119903minus119889 + 119903119889 + 119903+119889 ) is asolution among the Pareto solution set If (119888119889 119903119889) ≺ (1198881015840119889 1199031015840119889) (≺indicating domination) then (119888minus119889 + 1198881015840119889 + 119888+119889 119903minus119889 + 1199031015840119889 + 119903+119889 ) willnot emerge in the Pareto solution set This conclusion can besimply proved by contradiction If (119888minus119889 + 1198881015840119889 + 119888+119889 119903minus119889 + 1199031015840119889 + 119903+119889 )is one solution among the Pareto solution set then either119888minus119889 + 1198881015840119889 + 119888+119889 le 119888minus119889 + 119888119889 + 119888+119889 or 119903minus119889 + 1199031015840119889 + 119903+119889 le 119903minus119889 + 119903119889 + 119903+119889has to hold Namely either 1198881015840119889 le 119888119889 or le 119903119889 should be correctwhich contradicts (119888119889 119903119889) ≺ (1198881015840119889 1199031015840119889)

As can be seen in order to obtain the Pareto solution setof model P1 a Pareto-optimal solution is calculated for thepath from the distribution center to each destination so thatother paths are excluded out A two-stage method is adopted

The problem of finding the path from the distributioncenter to each destination is essentially a biobjective shortestpath problem (BSP) Generally there are two types of solu-tions to it dynamic programming (DP) [8 9] and ranking[10] Pulse algorithm [11] is a precise algorithm to solve BSPAlthough an idea of recursively traversing all paths in a net-work is adopted it is still efficient by using a four-step pruningmechanism to exclude dominated paths effectively Otherthan the advantage of high efficiency this algorithm has avirtue of finding a full set of Pareto-optimal solution whichis hardly achievable in heuristic or evolutionary algorithmsThe key procedure to pulse algorithm is a newly added nodeif satisfying one of the following four conditions incomingpartial pathwill be prunedwhichmeans the incoming partialpath containing this node will be eliminated

(1) It contains cycles(2) It exceeds either one or both upper bounds [11]

defined by the nadir point before reaching the endnode

Discrete Dynamics in Nature and Society 5

n1 n2 middot middot middot middot middot middot nm

Figure 3 Encoding format

(3) It is dominated by any solution which is in currentefficient set before reaching the end node

(4) Each node that is passed by a feasible path is markedwith a label storing two objective values of the pathIf the incoming partial path is dominated by eitherobjective value stored in the label then this path isdropped or else the objective value of this path willbe added in the label of the node

In the second stage the Pareto paths obtained in thefirst stage are encoded to calculate the final results usingmultiobjective optimization based on NSGA-II [12] Sinceno paths other than the Pareto paths are involved in thesecond stage the length of code is significantly shrunk whenusing multiobjective optimization Therefore the chance ofobtaining the optimal solution increases prominently underthe same size of population and number of iterations

The following introduces encoding method fitness func-tion and population regeneration strategy for NSGA-II

(1) Individual Coding Natural number is adopted for individ-ual coding the length of coding is119898 and Figure 3 shows theformat of the encoding

Where 119899119894 isin [1max(|119875119900119889119894 |)] 119875119900119889119894 is Pareto-optimal pathset from the distribution center 0 to the destination nodes 119889119894If the 119894th value of the code is 119899119894 then it means 119899119894mod|119875119900119889119894 |thpath (obtained by the first stage) from the distribution center119900 to the destination node 119889119894 is adopted for transportationFor example the numbers of Pareto-optimal paths from thedistribution center 0 to the destination nodes 1198891 sim 1198894 are 2 35 and 4 respectively Then a code of [1435] means the 1st (1mod 2) 1st (4 mod 3) 3rd (3 mod 5) and 1st (5 mod 4) of thePareto paths from o to 1198891 sim 1198894 for the individual are adoptedfor transportation

(2) Fitness Function Assuming 119894119899119889119894 represents an individual[119888 V119886119897119906119890 119903 V119886119897119906119890] = 119891(119894119899119889119894) is used to represent fitness of theindividual 119894119899119889119894 where 119888 V119886119897119906119890 and 119903 V119886119897119906119890 are transportationcost and risk respectively based on the selected path plan ofindividual 119894119899119889119894(3) Population Updating Strategy A population adds in newindividuals by crossover for which integers are used Firsttwo individuals are randomly selected in the population withtwo positions 1199011199001199041 and 1199011199001199042 (1199011199001199041 lt 1199011199001199042) randomlygenerated These two individuals 1199011199001199041 and 1199011199001199042 are crossedover to generate two new individuals

Mutation is also operated with integers to obtain newindividuals An individual is selected randomly while twopositions 1199011199001199041 and 1199011199001199042 (1199011199001199041 lt 1199011199001199042) are randomlygenerated Positions of the selected individual1199011199001199041 and1199011199001199042

Table 1 Lengths of each road section and risks of vehicles travelingin each road section

Road section Lengthkm Risk(0 1) 3656 00456(0 3) 985 00214(0 4) 3029 00154(0 5) 1582 00286(0 6) 926 00059(1 2) 4516 00217(1 3) 3036 00020(1 4) 4547 00083(2 3) 2426 00232(2 5) 2795 00225(2 7) 3918 00252(4 6) 2716 00186(4 8) 4889 00068(5 7) 2126 00546(5 8) 3635 00032(6 8) 2415 00457(7 8) 4265 00022

are subtracted by max(|119875(119899)119900119889|) separately with absolute values

taken to generate a new individualModel P2 is targeted at the objective of balance between

number of vehicles and transport job intensity undertakenby each vehicle which is essentially VRP Except that thetransport distance and number of vehicles in the objectivefunction should be replaced by number of vehicles andfairness of transport assignments other steps can be solvedby referring to the problem-solving method for VRPTW Inthis study the problem-solving method based on UMDA forVRPHTW given by literature [13] based on estimation ofdistribution is adopted

The problem-solving process for path planning and vehi-cle scheduling of hazardousmaterials transportation is shownin Figure 4

4 Numerical Example

41 A Small-Scale Network Figure 5 shows a test transporta-tion network where node 0 represents the distribution centerand nodes 1 2 4 7 and 8 represent destinationsThe distanceand risk of each road section in the network are shown inTable 1 The required transport volume and time windowof each destination are shown in Table 2 The capacity of avehicle is 135m3 The average loading time unloading timeand the average travel speed of a vehicle are 075 h 075 hand 45 kmh respectively The cost for transport in FCL is50 $km the cost for transport with no load is 10 $km Avehicle transport plan aims to minimize the transportationcost and risk

Table 2 shows the required transport volumes of alldestinations as well as the accepted unloading time windowswhich is hard indicating a waiting action if a vehicle arrivesearlier than the earliest starting time specified

6 Discrete Dynamics in Nature and Society

Table 2 Required transport volume and time window of a destination

Destination 1 2 4 7 8Required volumem3 35 18 295 45 60Time window [800 1600] [1200 2000] [800 1230] [800 1800] [800 2000]

The pulse algorithm is used to obtain thePareto solutions of all paths from the

End

Initialize network and

Get all paths selection solution byNSGA-II algorithm (path planning)

Decision-makers chose favorable travel pathsdepending on the specific risk preferences

The vehicle scheduling plan under thecurrent path selection plan is acquiredby UMDA (vehicle scheduling)

Satisfied

Yes

No

Yes

No

Start

requirement d = 1

distribution center destination d d = d + 1

d le m

Figure 4 Flow chart for solving problem of path planning andvehicle scheduling of hazardous materials transportation

After the first stage Pareto shortest path (departingpath) and shortest distance paths (returning path) betweenthe distribution center to each destination can be obtained(Table 3)

When the Pareto paths set is obtained the Pareto solutionset of total cost and risk for transportation between thedistribution center and all destinations can be obtainedTable 4 shows the path selection plans after decoding ofPareto solutions

When a certain transport plan is selected model P2 isused to solve the results for the path adopted In this waya vehicle scheduling plan under the current path selectionplan is acquired For example when path plan 1 is selectedin Table 4 the vehicle scheduling plan of Table 5 can beobtained and when path plan 14 is selected the vehicle

1

8

02

34

5 6

7

Distribution centerOther nodesDestination

Figure 5 A test network for transportation of hazardous materials

scheduling plan of Table 6 can be obtained The columnof node represents the distribution node arrived at by avehicle The timetable shows the time moments of a vehiclein a round trip for loading start (LS) departure from thedistribution center (DD) unloading start at the destination(US) returning after unloading finishes (RU) and arrivingback at the distribution center (AC)

42 Comparison with General NSGA-II In addition a gen-eral NSGA-II algorithm (GNA) is designed for the pathplanning in order to illustrate the effectiveness of the two-stage algorithm (TSA) proposed Different from the two-stage algorithm in the general NSGA-II algorithm thepriority based encoding [14] method is adopted and theencoding length is |119873| lowast 119898 However in the second stage ofthe two-stage algorithm the encoding length is only 119898 Forthe example in Section 41 the parameters of two algorithmsare set as follows the size of population is 100 and theiteration number is 50 generations In Figures 6(a)ndash6(d) isthe Pareto-optimal front obtained by iterative 5 10 20 and50 generations respectively By comparison it can be foundthat the final solution can be obtained by using the TSAwhenthe iteration reaches the 10th generation (14 plans see Table 4)while the GNA still fails to obtain the optimal solution untilthe fiftieth iterationThis is just a network of 9 nodes and thisgap will become even bigger for large-scale networks Thereis no direct relationship between the encoding length and thenetwork size in the two-stage algorithm so for large-scalenetworks the algorithm is still valid

Discrete Dynamics in Nature and Society 7

Table 3 Departing path and returning path between the distribution center and each destination

Destination Departing path Lengthkm Risk Returning path Lengthkm

1 0rarr1 3656 00456 1rarr0 36560rarr3rarr1 4021 00234

2 0rarr3rarr2 3411 00446 2rarr3rarr0 34114 0rarr4 3029 00154 4rarr0 3029

7

0rarr5rarr7 3708 00832

7rarr5rarr0 37080rarr3rarr2rarr7 7329 006980rarr6rarr8rarr7 7606 005380rarr5rarr8rarr7 9482 003400rarr4rarr8rarr7 12183 00244

80rarr6rarr8 3341 00516

8rarr6rarr0 33410rarr5rarr8 5217 003180rarr4rarr8 7918 00222

Table 4 Path selection plans

Solution Path Total cost$ Total risk

1

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

216132 08630

2

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

218541 07964

3

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

236768 07640

4

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

239177 06974

5

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

252843 06788

6

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

266943 06662

7

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr4rarr8rarr6rarr0

268888 06494

8

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

269352 05996

8 Discrete Dynamics in Nature and Society

Table 4 Continued

Solution Path Total cost$ Total risk

9

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr6rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

273479 05798

10

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

287579 05672

11

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

289988 05006

12

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

313757 04622

13

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

319699 04526

14

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

343468 04142

Table 5 Vehicle scheduling plan (for solution 1)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

2

1 626 711 800 845 9344 934 1019 1059 1144 12257 1225 1310 1359 1444 15337 1533 1618 1708 1753 1842

3

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

4

4 635 720 800 845 9254 925 1010 1051 1136 12168 1216 1301 1346 1431 15158 1515 1600 1645 1730 18148 1814 1859 1944 2029 2113

Discrete Dynamics in Nature and Society 9

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

2500 3000 3500 40002000Transportation cost$

TSAGNA

(a) Iteration number = 5

2500 3000 3500 40002000Transportation cost$

TSAGNA

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k(b) Iteration number = 10

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k

2500 3000 3500 40002000Transportation cost$

TSAGNA

(c) Iteration number = 20

2500 3000 3500 40002000Transportation cost$

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

TSAGNA

(d) Iteration number = 50

Figure 6 Pareto solutions obtained in path optimization

5 Case Study

The proposed methodology was implemented for the urbantransportation network in Lanzhou China Daily trafficvolumes of all road sections in realistic road networks ofmain districts are used to estimate vehicle travel time for allroad sections In addition other factors such as distributionof population and facilities as well as probabilities andconsequences of traffic accident along the road sectionsinvolved are also considered to establish a traditional model[15] to estimate the risk value of each road section for vehicletransportation Thus a transport network for hazardous

materials (see Figure 7) in main districts of Lanzhou isformed containing 170 nodes and 272 edges Distribution ofgasoline 92 (permil) for 16 gas stations (e) under PetroChinain this network is taken as an example to calculate vehiclescheduling plans for oil product transportation

Table 7 shows the positions of 16 gas stations in thenetwork indicated by their nearest nodes (destination site119889) as well as their needs of gasoline and the time windowof distribution The position of the distribution center isindicated by node 1 The volume of a transportation vehicle(tank car) is 135m3 Traveling fee of a tank car in FCL inurban highways is 200 RMBh while that with no load is

10 Discrete Dynamics in Nature and Society

1

2

3

4

56

78

910

11

12

13

14

15

1617

181920

21

22

23

24

25

26

27

28

29

30

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49 50

51

5255

53

5456

57

58

5960

6162

63

64

65

66

67

69

70

7172

7374

75

7677 78

79

80

81

8283

84

85

86

8788

899091 92

93

94

9596

979899100101102

103

104105

106

107108

109110

111112 113

114115116

117

118

119

120121

122

123124125

126127128

129

130

131 132133

134

135

136

137139138

140141

142 143

144145

146147

148

149

150 151

152153

154

155

156157

158

159

160

161

162163

164

165

166 167

68

168

169

170

Figure 7 A transport network for hazardous materials in main districts of Lanzhou

Table 6 Vehicle scheduling plan (for solution 14)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

11 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

34 0715 0800 0840 0925 10061 1006 1051 1144 1229 13187 1318 1403 1646 1731 1820

52 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

32 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

41 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

6 8 0729 0814 1000 1045 11307 1130 1215 1457 1542 1631

150 RMBh eliminating cost of loading and unloading Timefor either loading or unloading is 075 h

C is the programming language to implement the algo-rithmThe running environment is PC i5-3470 320GHz 4GParameters setting in the program and CPU times of vehiclesare shown in Table 8 A total of 164 path selection plans areobtained When a certain plan is selected the correspondingtimetable for vehicle operation can be obtained by the vehiclescheduling program

A path plan is a choice based on game between trans-portation cost and risk by a decision-maker with a specific

risk preference For example in plan 1 which has the lowesttransportation cost but the highest transportation risk thetransportation time is 439658 RMB while the transporta-tion risk is 32262 8 vehicles are needed to complete jobsAs another example in plan 164 which has the lowesttransportation risk but the highest transportation cost thetransportation time is 460983 RMB while the transporta-tion risk is 25586 also 8 vehicles are needed to complete jobsThe traveling path and transport timetable for all vehicles inplan 164 are shown in Tables 9 and 10

Discrete Dynamics in Nature and Society 11

Table 7 Nodes representing 16 gas stations as well as their needs of gasoline and the time window of distribution

Number Destination Required volumem3 Time window1 2 15 [1200 1600]2 12 26 [830 1330]3 16 12 [1030 1600]4 25 16 [800 1200]5 43 16 [800 1600]6 44 29 [1000 1600]7 47 32 [900 1500]8 52 25 [900 1200]9 62 24 [800 1000]10 63 10 [800 1500]11 94 18 [1400 1800]12 138 25 [800 1500]13 146 40 [1200 1800]14 150 23 [1000 1600]15 167 9 [1200 1800]16 170 25 [800 1600]

Table 8 Transport parameters setting in the program and CPU times

Stage Parameters setting CPU timessPopulation size Iteration

Path selectionStage 1 the pulse algorithm is used to obtain the Pareto solutions of all pathsfrom the distribution center to each destination mdash mdash 17

Stage 2 get all paths selection solution by NSGA-II algorithm 500 100 32Vehicle scheduling

The vehicle scheduling plan under the current path selection plan is acquired byUMDA 200 100 9

6 Conclusion

Distribution of hazardous materials in FCL is essentially amultistage combinatorial optimization problem In this studya decision approach for distribution of hazardous materialsin FCL with a single distribution center is formed which canprovide a decision support for transportation enterprises andadministrative authorities

In this study mathematical models for path planningand vehicle scheduling problems are established respectivelywith corresponding problem-solving algorithms designedIn order to solve the path planning model a two-stagealgorithm is designed Based on screening of paths in thefirst stage the computational complexity of the second stageinvolved in the global path optimization is significantlyreduced which improves calculation efficiency In obtainingthe vehicle scheduling plan the operating process of atransporting vehicle is converted to single objective VRPTWThen the problem-solvingmethod forVRPTWon the basis ofestimation of distribution is adopted to obtain the timetablefor all vehicles

The proposed approach was applied to a hypotheticalnumerical example as well as an application to a real casein distribution of gasoline of Lanzhou ChinaThe numerical

example showed that for small-scale problems the two-stagealgorithm can obtain Pareto-optimal front faster in the sameiterationHowever for large-scale problems due to the lengthof coding it is difficult to obtain Pareto-optimal path set in alimited iterative step in the second stage the problem size ofVRPTW is only related to the number of destination nodesnot to the scale of the transportation network Therefore thevehicle scheduling method has high efficiency whether it issmall-scale or large-scale transportation network

Our purpose is to design a decision support systemfor path planning and vehicle scheduling of distribution ofhazardous materials in FCL Further research has to be doneat least in two directions distribution of hazardous materialsin FCL with multiple distribution centers and transportationwith different volumes vehicle will be focused on

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (nos 61064012 61364026

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Discrete Dynamics in Nature and Society

Distrib

ution

cent

er

Destin

ation

1

Distrib

ution

cent

er

Destin

ation

dDist

ribut

ionce

nter

Destin

ation

mDist

ribut

ionce

nter

(cminusd r

minusd )

middot middot middot middot middot middot

(c+d r

+d )

(cd r

d)

(cd rd)

Figure 2 Total cost and total risk of path formed by connecting the distribution center with each destination

indicate whether assignment 119894 is implemented by vehicle VIf yes 119910119894V = 1 else 119910119894V = 0 The vehicle scheduling problemcan be described by the following model (P2)

P2 min 119891= 119872sum119894isin119869

sumVisin1198811199090119894V + sum

Visin119881(sum119894isin119869

sum119895isin119869

Δ119905119894119895119909119894119895V minus 120583)2 (11)

st sumVisin119881119910119894V = 1 forall119894 isin 119869 (12)

sum119894isin119869

119909119894119895V = 119910119895V forall119895 isin 119869 V isin 119881 (13)

sum119895isin119869

119909119894119895V = 119910119894V forall119894 isin 119869 V isin 119881 (14)

sum119894isin119869

119909119894119895119896 minussum119895isin119869

119909119894119895119896 = 0 forallV isin 119881 (15)

119879119895V = 119879119894V + Δ1199050119889119909119894119895Vforall119889 isin 119873 119894 isin 119869119889 119895 isin 119869 V isin 119881 (16)

119864119879119889 minus (Δ1199051 + Δ11990510119889) le 119879119895Vle 119871119879119889 + (Δ1199052 + Δ11990520119889)

forall119889 isin 119873 119895 isin 119869119889 V isin 119881(17)

119909119894119895V isin [0 1] forall119894 119895 isin 119869 V isin 119881 (18)

119910119894V isin [0 1] forall119894 isin 119869 V isin 119881 (19)

The objective function (11) represents the minimumnumber of vehicles and difference of their travel time where120583 = sumVisin119881sum119894isin119869sum119895isin119869 Δ119905119894119895119909119894119895Vsum119894isin119869sumVisin119881 1199090119894V The first part ofthis function represents the minimum number of vehicles119872 is a sufficiently large integer to ensure the priority level ofthe objective of vehicle number The second part representsminimized standard deviation of travel time of all vehiclesunder the same number of vehicles to ensure minimumdifference in transport jobs for each vehicle Constraints(12)ndash(15) guarantee that each vehicle can carry out only oneassignment at a time That is assignments can be carried outby a vehicle consecutively but not in parallel In additionan assignment can be carried out by only one vehicleConstraint (16) represents the time moment when node 119895 is

reached Constraint (17) ensures satisfaction of time windowrestriction at the moment when node 119895 is reached3 Problem-Solving Method

The optimization objective of model P1 is to obtain thePareto-optimal solution sets of total cost and total risk for dis-tribution jobs from the distribution center to all destinationsThis process can be viewed as a calculation of transportationcost and risk involved in the path formed by connecting thedistribution center to each destination (see Figure 2) (119888minus119889 119903minus119889 )represents the cost and risk of the path before reachingdestination 119889 from the distribution center (119888+119889 119903+119889 ) representsthe cost and risk of the path from the destination 119889 back tothe distribution center Assume (119888minus119889 + 119888119889 + 119888+119889 119903minus119889 + 119903119889 + 119903+119889 ) is asolution among the Pareto solution set If (119888119889 119903119889) ≺ (1198881015840119889 1199031015840119889) (≺indicating domination) then (119888minus119889 + 1198881015840119889 + 119888+119889 119903minus119889 + 1199031015840119889 + 119903+119889 ) willnot emerge in the Pareto solution set This conclusion can besimply proved by contradiction If (119888minus119889 + 1198881015840119889 + 119888+119889 119903minus119889 + 1199031015840119889 + 119903+119889 )is one solution among the Pareto solution set then either119888minus119889 + 1198881015840119889 + 119888+119889 le 119888minus119889 + 119888119889 + 119888+119889 or 119903minus119889 + 1199031015840119889 + 119903+119889 le 119903minus119889 + 119903119889 + 119903+119889has to hold Namely either 1198881015840119889 le 119888119889 or le 119903119889 should be correctwhich contradicts (119888119889 119903119889) ≺ (1198881015840119889 1199031015840119889)

As can be seen in order to obtain the Pareto solution setof model P1 a Pareto-optimal solution is calculated for thepath from the distribution center to each destination so thatother paths are excluded out A two-stage method is adopted

The problem of finding the path from the distributioncenter to each destination is essentially a biobjective shortestpath problem (BSP) Generally there are two types of solu-tions to it dynamic programming (DP) [8 9] and ranking[10] Pulse algorithm [11] is a precise algorithm to solve BSPAlthough an idea of recursively traversing all paths in a net-work is adopted it is still efficient by using a four-step pruningmechanism to exclude dominated paths effectively Otherthan the advantage of high efficiency this algorithm has avirtue of finding a full set of Pareto-optimal solution whichis hardly achievable in heuristic or evolutionary algorithmsThe key procedure to pulse algorithm is a newly added nodeif satisfying one of the following four conditions incomingpartial pathwill be prunedwhichmeans the incoming partialpath containing this node will be eliminated

(1) It contains cycles(2) It exceeds either one or both upper bounds [11]

defined by the nadir point before reaching the endnode

Discrete Dynamics in Nature and Society 5

n1 n2 middot middot middot middot middot middot nm

Figure 3 Encoding format

(3) It is dominated by any solution which is in currentefficient set before reaching the end node

(4) Each node that is passed by a feasible path is markedwith a label storing two objective values of the pathIf the incoming partial path is dominated by eitherobjective value stored in the label then this path isdropped or else the objective value of this path willbe added in the label of the node

In the second stage the Pareto paths obtained in thefirst stage are encoded to calculate the final results usingmultiobjective optimization based on NSGA-II [12] Sinceno paths other than the Pareto paths are involved in thesecond stage the length of code is significantly shrunk whenusing multiobjective optimization Therefore the chance ofobtaining the optimal solution increases prominently underthe same size of population and number of iterations

The following introduces encoding method fitness func-tion and population regeneration strategy for NSGA-II

(1) Individual Coding Natural number is adopted for individ-ual coding the length of coding is119898 and Figure 3 shows theformat of the encoding

Where 119899119894 isin [1max(|119875119900119889119894 |)] 119875119900119889119894 is Pareto-optimal pathset from the distribution center 0 to the destination nodes 119889119894If the 119894th value of the code is 119899119894 then it means 119899119894mod|119875119900119889119894 |thpath (obtained by the first stage) from the distribution center119900 to the destination node 119889119894 is adopted for transportationFor example the numbers of Pareto-optimal paths from thedistribution center 0 to the destination nodes 1198891 sim 1198894 are 2 35 and 4 respectively Then a code of [1435] means the 1st (1mod 2) 1st (4 mod 3) 3rd (3 mod 5) and 1st (5 mod 4) of thePareto paths from o to 1198891 sim 1198894 for the individual are adoptedfor transportation

(2) Fitness Function Assuming 119894119899119889119894 represents an individual[119888 V119886119897119906119890 119903 V119886119897119906119890] = 119891(119894119899119889119894) is used to represent fitness of theindividual 119894119899119889119894 where 119888 V119886119897119906119890 and 119903 V119886119897119906119890 are transportationcost and risk respectively based on the selected path plan ofindividual 119894119899119889119894(3) Population Updating Strategy A population adds in newindividuals by crossover for which integers are used Firsttwo individuals are randomly selected in the population withtwo positions 1199011199001199041 and 1199011199001199042 (1199011199001199041 lt 1199011199001199042) randomlygenerated These two individuals 1199011199001199041 and 1199011199001199042 are crossedover to generate two new individuals

Mutation is also operated with integers to obtain newindividuals An individual is selected randomly while twopositions 1199011199001199041 and 1199011199001199042 (1199011199001199041 lt 1199011199001199042) are randomlygenerated Positions of the selected individual1199011199001199041 and1199011199001199042

Table 1 Lengths of each road section and risks of vehicles travelingin each road section

Road section Lengthkm Risk(0 1) 3656 00456(0 3) 985 00214(0 4) 3029 00154(0 5) 1582 00286(0 6) 926 00059(1 2) 4516 00217(1 3) 3036 00020(1 4) 4547 00083(2 3) 2426 00232(2 5) 2795 00225(2 7) 3918 00252(4 6) 2716 00186(4 8) 4889 00068(5 7) 2126 00546(5 8) 3635 00032(6 8) 2415 00457(7 8) 4265 00022

are subtracted by max(|119875(119899)119900119889|) separately with absolute values

taken to generate a new individualModel P2 is targeted at the objective of balance between

number of vehicles and transport job intensity undertakenby each vehicle which is essentially VRP Except that thetransport distance and number of vehicles in the objectivefunction should be replaced by number of vehicles andfairness of transport assignments other steps can be solvedby referring to the problem-solving method for VRPTW Inthis study the problem-solving method based on UMDA forVRPHTW given by literature [13] based on estimation ofdistribution is adopted

The problem-solving process for path planning and vehi-cle scheduling of hazardousmaterials transportation is shownin Figure 4

4 Numerical Example

41 A Small-Scale Network Figure 5 shows a test transporta-tion network where node 0 represents the distribution centerand nodes 1 2 4 7 and 8 represent destinationsThe distanceand risk of each road section in the network are shown inTable 1 The required transport volume and time windowof each destination are shown in Table 2 The capacity of avehicle is 135m3 The average loading time unloading timeand the average travel speed of a vehicle are 075 h 075 hand 45 kmh respectively The cost for transport in FCL is50 $km the cost for transport with no load is 10 $km Avehicle transport plan aims to minimize the transportationcost and risk

Table 2 shows the required transport volumes of alldestinations as well as the accepted unloading time windowswhich is hard indicating a waiting action if a vehicle arrivesearlier than the earliest starting time specified

6 Discrete Dynamics in Nature and Society

Table 2 Required transport volume and time window of a destination

Destination 1 2 4 7 8Required volumem3 35 18 295 45 60Time window [800 1600] [1200 2000] [800 1230] [800 1800] [800 2000]

The pulse algorithm is used to obtain thePareto solutions of all paths from the

End

Initialize network and

Get all paths selection solution byNSGA-II algorithm (path planning)

Decision-makers chose favorable travel pathsdepending on the specific risk preferences

The vehicle scheduling plan under thecurrent path selection plan is acquiredby UMDA (vehicle scheduling)

Satisfied

Yes

No

Yes

No

Start

requirement d = 1

distribution center destination d d = d + 1

d le m

Figure 4 Flow chart for solving problem of path planning andvehicle scheduling of hazardous materials transportation

After the first stage Pareto shortest path (departingpath) and shortest distance paths (returning path) betweenthe distribution center to each destination can be obtained(Table 3)

When the Pareto paths set is obtained the Pareto solutionset of total cost and risk for transportation between thedistribution center and all destinations can be obtainedTable 4 shows the path selection plans after decoding ofPareto solutions

When a certain transport plan is selected model P2 isused to solve the results for the path adopted In this waya vehicle scheduling plan under the current path selectionplan is acquired For example when path plan 1 is selectedin Table 4 the vehicle scheduling plan of Table 5 can beobtained and when path plan 14 is selected the vehicle

1

8

02

34

5 6

7

Distribution centerOther nodesDestination

Figure 5 A test network for transportation of hazardous materials

scheduling plan of Table 6 can be obtained The columnof node represents the distribution node arrived at by avehicle The timetable shows the time moments of a vehiclein a round trip for loading start (LS) departure from thedistribution center (DD) unloading start at the destination(US) returning after unloading finishes (RU) and arrivingback at the distribution center (AC)

42 Comparison with General NSGA-II In addition a gen-eral NSGA-II algorithm (GNA) is designed for the pathplanning in order to illustrate the effectiveness of the two-stage algorithm (TSA) proposed Different from the two-stage algorithm in the general NSGA-II algorithm thepriority based encoding [14] method is adopted and theencoding length is |119873| lowast 119898 However in the second stage ofthe two-stage algorithm the encoding length is only 119898 Forthe example in Section 41 the parameters of two algorithmsare set as follows the size of population is 100 and theiteration number is 50 generations In Figures 6(a)ndash6(d) isthe Pareto-optimal front obtained by iterative 5 10 20 and50 generations respectively By comparison it can be foundthat the final solution can be obtained by using the TSAwhenthe iteration reaches the 10th generation (14 plans see Table 4)while the GNA still fails to obtain the optimal solution untilthe fiftieth iterationThis is just a network of 9 nodes and thisgap will become even bigger for large-scale networks Thereis no direct relationship between the encoding length and thenetwork size in the two-stage algorithm so for large-scalenetworks the algorithm is still valid

Discrete Dynamics in Nature and Society 7

Table 3 Departing path and returning path between the distribution center and each destination

Destination Departing path Lengthkm Risk Returning path Lengthkm

1 0rarr1 3656 00456 1rarr0 36560rarr3rarr1 4021 00234

2 0rarr3rarr2 3411 00446 2rarr3rarr0 34114 0rarr4 3029 00154 4rarr0 3029

7

0rarr5rarr7 3708 00832

7rarr5rarr0 37080rarr3rarr2rarr7 7329 006980rarr6rarr8rarr7 7606 005380rarr5rarr8rarr7 9482 003400rarr4rarr8rarr7 12183 00244

80rarr6rarr8 3341 00516

8rarr6rarr0 33410rarr5rarr8 5217 003180rarr4rarr8 7918 00222

Table 4 Path selection plans

Solution Path Total cost$ Total risk

1

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

216132 08630

2

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

218541 07964

3

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

236768 07640

4

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

239177 06974

5

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

252843 06788

6

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

266943 06662

7

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr4rarr8rarr6rarr0

268888 06494

8

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

269352 05996

8 Discrete Dynamics in Nature and Society

Table 4 Continued

Solution Path Total cost$ Total risk

9

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr6rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

273479 05798

10

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

287579 05672

11

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

289988 05006

12

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

313757 04622

13

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

319699 04526

14

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

343468 04142

Table 5 Vehicle scheduling plan (for solution 1)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

2

1 626 711 800 845 9344 934 1019 1059 1144 12257 1225 1310 1359 1444 15337 1533 1618 1708 1753 1842

3

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

4

4 635 720 800 845 9254 925 1010 1051 1136 12168 1216 1301 1346 1431 15158 1515 1600 1645 1730 18148 1814 1859 1944 2029 2113

Discrete Dynamics in Nature and Society 9

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

2500 3000 3500 40002000Transportation cost$

TSAGNA

(a) Iteration number = 5

2500 3000 3500 40002000Transportation cost$

TSAGNA

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k(b) Iteration number = 10

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k

2500 3000 3500 40002000Transportation cost$

TSAGNA

(c) Iteration number = 20

2500 3000 3500 40002000Transportation cost$

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

TSAGNA

(d) Iteration number = 50

Figure 6 Pareto solutions obtained in path optimization

5 Case Study

The proposed methodology was implemented for the urbantransportation network in Lanzhou China Daily trafficvolumes of all road sections in realistic road networks ofmain districts are used to estimate vehicle travel time for allroad sections In addition other factors such as distributionof population and facilities as well as probabilities andconsequences of traffic accident along the road sectionsinvolved are also considered to establish a traditional model[15] to estimate the risk value of each road section for vehicletransportation Thus a transport network for hazardous

materials (see Figure 7) in main districts of Lanzhou isformed containing 170 nodes and 272 edges Distribution ofgasoline 92 (permil) for 16 gas stations (e) under PetroChinain this network is taken as an example to calculate vehiclescheduling plans for oil product transportation

Table 7 shows the positions of 16 gas stations in thenetwork indicated by their nearest nodes (destination site119889) as well as their needs of gasoline and the time windowof distribution The position of the distribution center isindicated by node 1 The volume of a transportation vehicle(tank car) is 135m3 Traveling fee of a tank car in FCL inurban highways is 200 RMBh while that with no load is

10 Discrete Dynamics in Nature and Society

1

2

3

4

56

78

910

11

12

13

14

15

1617

181920

21

22

23

24

25

26

27

28

29

30

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49 50

51

5255

53

5456

57

58

5960

6162

63

64

65

66

67

69

70

7172

7374

75

7677 78

79

80

81

8283

84

85

86

8788

899091 92

93

94

9596

979899100101102

103

104105

106

107108

109110

111112 113

114115116

117

118

119

120121

122

123124125

126127128

129

130

131 132133

134

135

136

137139138

140141

142 143

144145

146147

148

149

150 151

152153

154

155

156157

158

159

160

161

162163

164

165

166 167

68

168

169

170

Figure 7 A transport network for hazardous materials in main districts of Lanzhou

Table 6 Vehicle scheduling plan (for solution 14)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

11 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

34 0715 0800 0840 0925 10061 1006 1051 1144 1229 13187 1318 1403 1646 1731 1820

52 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

32 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

41 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

6 8 0729 0814 1000 1045 11307 1130 1215 1457 1542 1631

150 RMBh eliminating cost of loading and unloading Timefor either loading or unloading is 075 h

C is the programming language to implement the algo-rithmThe running environment is PC i5-3470 320GHz 4GParameters setting in the program and CPU times of vehiclesare shown in Table 8 A total of 164 path selection plans areobtained When a certain plan is selected the correspondingtimetable for vehicle operation can be obtained by the vehiclescheduling program

A path plan is a choice based on game between trans-portation cost and risk by a decision-maker with a specific

risk preference For example in plan 1 which has the lowesttransportation cost but the highest transportation risk thetransportation time is 439658 RMB while the transporta-tion risk is 32262 8 vehicles are needed to complete jobsAs another example in plan 164 which has the lowesttransportation risk but the highest transportation cost thetransportation time is 460983 RMB while the transporta-tion risk is 25586 also 8 vehicles are needed to complete jobsThe traveling path and transport timetable for all vehicles inplan 164 are shown in Tables 9 and 10

Discrete Dynamics in Nature and Society 11

Table 7 Nodes representing 16 gas stations as well as their needs of gasoline and the time window of distribution

Number Destination Required volumem3 Time window1 2 15 [1200 1600]2 12 26 [830 1330]3 16 12 [1030 1600]4 25 16 [800 1200]5 43 16 [800 1600]6 44 29 [1000 1600]7 47 32 [900 1500]8 52 25 [900 1200]9 62 24 [800 1000]10 63 10 [800 1500]11 94 18 [1400 1800]12 138 25 [800 1500]13 146 40 [1200 1800]14 150 23 [1000 1600]15 167 9 [1200 1800]16 170 25 [800 1600]

Table 8 Transport parameters setting in the program and CPU times

Stage Parameters setting CPU timessPopulation size Iteration

Path selectionStage 1 the pulse algorithm is used to obtain the Pareto solutions of all pathsfrom the distribution center to each destination mdash mdash 17

Stage 2 get all paths selection solution by NSGA-II algorithm 500 100 32Vehicle scheduling

The vehicle scheduling plan under the current path selection plan is acquired byUMDA 200 100 9

6 Conclusion

Distribution of hazardous materials in FCL is essentially amultistage combinatorial optimization problem In this studya decision approach for distribution of hazardous materialsin FCL with a single distribution center is formed which canprovide a decision support for transportation enterprises andadministrative authorities

In this study mathematical models for path planningand vehicle scheduling problems are established respectivelywith corresponding problem-solving algorithms designedIn order to solve the path planning model a two-stagealgorithm is designed Based on screening of paths in thefirst stage the computational complexity of the second stageinvolved in the global path optimization is significantlyreduced which improves calculation efficiency In obtainingthe vehicle scheduling plan the operating process of atransporting vehicle is converted to single objective VRPTWThen the problem-solvingmethod forVRPTWon the basis ofestimation of distribution is adopted to obtain the timetablefor all vehicles

The proposed approach was applied to a hypotheticalnumerical example as well as an application to a real casein distribution of gasoline of Lanzhou ChinaThe numerical

example showed that for small-scale problems the two-stagealgorithm can obtain Pareto-optimal front faster in the sameiterationHowever for large-scale problems due to the lengthof coding it is difficult to obtain Pareto-optimal path set in alimited iterative step in the second stage the problem size ofVRPTW is only related to the number of destination nodesnot to the scale of the transportation network Therefore thevehicle scheduling method has high efficiency whether it issmall-scale or large-scale transportation network

Our purpose is to design a decision support systemfor path planning and vehicle scheduling of distribution ofhazardous materials in FCL Further research has to be doneat least in two directions distribution of hazardous materialsin FCL with multiple distribution centers and transportationwith different volumes vehicle will be focused on

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (nos 61064012 61364026

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 5

n1 n2 middot middot middot middot middot middot nm

Figure 3 Encoding format

(3) It is dominated by any solution which is in currentefficient set before reaching the end node

(4) Each node that is passed by a feasible path is markedwith a label storing two objective values of the pathIf the incoming partial path is dominated by eitherobjective value stored in the label then this path isdropped or else the objective value of this path willbe added in the label of the node

In the second stage the Pareto paths obtained in thefirst stage are encoded to calculate the final results usingmultiobjective optimization based on NSGA-II [12] Sinceno paths other than the Pareto paths are involved in thesecond stage the length of code is significantly shrunk whenusing multiobjective optimization Therefore the chance ofobtaining the optimal solution increases prominently underthe same size of population and number of iterations

The following introduces encoding method fitness func-tion and population regeneration strategy for NSGA-II

(1) Individual Coding Natural number is adopted for individ-ual coding the length of coding is119898 and Figure 3 shows theformat of the encoding

Where 119899119894 isin [1max(|119875119900119889119894 |)] 119875119900119889119894 is Pareto-optimal pathset from the distribution center 0 to the destination nodes 119889119894If the 119894th value of the code is 119899119894 then it means 119899119894mod|119875119900119889119894 |thpath (obtained by the first stage) from the distribution center119900 to the destination node 119889119894 is adopted for transportationFor example the numbers of Pareto-optimal paths from thedistribution center 0 to the destination nodes 1198891 sim 1198894 are 2 35 and 4 respectively Then a code of [1435] means the 1st (1mod 2) 1st (4 mod 3) 3rd (3 mod 5) and 1st (5 mod 4) of thePareto paths from o to 1198891 sim 1198894 for the individual are adoptedfor transportation

(2) Fitness Function Assuming 119894119899119889119894 represents an individual[119888 V119886119897119906119890 119903 V119886119897119906119890] = 119891(119894119899119889119894) is used to represent fitness of theindividual 119894119899119889119894 where 119888 V119886119897119906119890 and 119903 V119886119897119906119890 are transportationcost and risk respectively based on the selected path plan ofindividual 119894119899119889119894(3) Population Updating Strategy A population adds in newindividuals by crossover for which integers are used Firsttwo individuals are randomly selected in the population withtwo positions 1199011199001199041 and 1199011199001199042 (1199011199001199041 lt 1199011199001199042) randomlygenerated These two individuals 1199011199001199041 and 1199011199001199042 are crossedover to generate two new individuals

Mutation is also operated with integers to obtain newindividuals An individual is selected randomly while twopositions 1199011199001199041 and 1199011199001199042 (1199011199001199041 lt 1199011199001199042) are randomlygenerated Positions of the selected individual1199011199001199041 and1199011199001199042

Table 1 Lengths of each road section and risks of vehicles travelingin each road section

Road section Lengthkm Risk(0 1) 3656 00456(0 3) 985 00214(0 4) 3029 00154(0 5) 1582 00286(0 6) 926 00059(1 2) 4516 00217(1 3) 3036 00020(1 4) 4547 00083(2 3) 2426 00232(2 5) 2795 00225(2 7) 3918 00252(4 6) 2716 00186(4 8) 4889 00068(5 7) 2126 00546(5 8) 3635 00032(6 8) 2415 00457(7 8) 4265 00022

are subtracted by max(|119875(119899)119900119889|) separately with absolute values

taken to generate a new individualModel P2 is targeted at the objective of balance between

number of vehicles and transport job intensity undertakenby each vehicle which is essentially VRP Except that thetransport distance and number of vehicles in the objectivefunction should be replaced by number of vehicles andfairness of transport assignments other steps can be solvedby referring to the problem-solving method for VRPTW Inthis study the problem-solving method based on UMDA forVRPHTW given by literature [13] based on estimation ofdistribution is adopted

The problem-solving process for path planning and vehi-cle scheduling of hazardousmaterials transportation is shownin Figure 4

4 Numerical Example

41 A Small-Scale Network Figure 5 shows a test transporta-tion network where node 0 represents the distribution centerand nodes 1 2 4 7 and 8 represent destinationsThe distanceand risk of each road section in the network are shown inTable 1 The required transport volume and time windowof each destination are shown in Table 2 The capacity of avehicle is 135m3 The average loading time unloading timeand the average travel speed of a vehicle are 075 h 075 hand 45 kmh respectively The cost for transport in FCL is50 $km the cost for transport with no load is 10 $km Avehicle transport plan aims to minimize the transportationcost and risk

Table 2 shows the required transport volumes of alldestinations as well as the accepted unloading time windowswhich is hard indicating a waiting action if a vehicle arrivesearlier than the earliest starting time specified

6 Discrete Dynamics in Nature and Society

Table 2 Required transport volume and time window of a destination

Destination 1 2 4 7 8Required volumem3 35 18 295 45 60Time window [800 1600] [1200 2000] [800 1230] [800 1800] [800 2000]

The pulse algorithm is used to obtain thePareto solutions of all paths from the

End

Initialize network and

Get all paths selection solution byNSGA-II algorithm (path planning)

Decision-makers chose favorable travel pathsdepending on the specific risk preferences

The vehicle scheduling plan under thecurrent path selection plan is acquiredby UMDA (vehicle scheduling)

Satisfied

Yes

No

Yes

No

Start

requirement d = 1

distribution center destination d d = d + 1

d le m

Figure 4 Flow chart for solving problem of path planning andvehicle scheduling of hazardous materials transportation

After the first stage Pareto shortest path (departingpath) and shortest distance paths (returning path) betweenthe distribution center to each destination can be obtained(Table 3)

When the Pareto paths set is obtained the Pareto solutionset of total cost and risk for transportation between thedistribution center and all destinations can be obtainedTable 4 shows the path selection plans after decoding ofPareto solutions

When a certain transport plan is selected model P2 isused to solve the results for the path adopted In this waya vehicle scheduling plan under the current path selectionplan is acquired For example when path plan 1 is selectedin Table 4 the vehicle scheduling plan of Table 5 can beobtained and when path plan 14 is selected the vehicle

1

8

02

34

5 6

7

Distribution centerOther nodesDestination

Figure 5 A test network for transportation of hazardous materials

scheduling plan of Table 6 can be obtained The columnof node represents the distribution node arrived at by avehicle The timetable shows the time moments of a vehiclein a round trip for loading start (LS) departure from thedistribution center (DD) unloading start at the destination(US) returning after unloading finishes (RU) and arrivingback at the distribution center (AC)

42 Comparison with General NSGA-II In addition a gen-eral NSGA-II algorithm (GNA) is designed for the pathplanning in order to illustrate the effectiveness of the two-stage algorithm (TSA) proposed Different from the two-stage algorithm in the general NSGA-II algorithm thepriority based encoding [14] method is adopted and theencoding length is |119873| lowast 119898 However in the second stage ofthe two-stage algorithm the encoding length is only 119898 Forthe example in Section 41 the parameters of two algorithmsare set as follows the size of population is 100 and theiteration number is 50 generations In Figures 6(a)ndash6(d) isthe Pareto-optimal front obtained by iterative 5 10 20 and50 generations respectively By comparison it can be foundthat the final solution can be obtained by using the TSAwhenthe iteration reaches the 10th generation (14 plans see Table 4)while the GNA still fails to obtain the optimal solution untilthe fiftieth iterationThis is just a network of 9 nodes and thisgap will become even bigger for large-scale networks Thereis no direct relationship between the encoding length and thenetwork size in the two-stage algorithm so for large-scalenetworks the algorithm is still valid

Discrete Dynamics in Nature and Society 7

Table 3 Departing path and returning path between the distribution center and each destination

Destination Departing path Lengthkm Risk Returning path Lengthkm

1 0rarr1 3656 00456 1rarr0 36560rarr3rarr1 4021 00234

2 0rarr3rarr2 3411 00446 2rarr3rarr0 34114 0rarr4 3029 00154 4rarr0 3029

7

0rarr5rarr7 3708 00832

7rarr5rarr0 37080rarr3rarr2rarr7 7329 006980rarr6rarr8rarr7 7606 005380rarr5rarr8rarr7 9482 003400rarr4rarr8rarr7 12183 00244

80rarr6rarr8 3341 00516

8rarr6rarr0 33410rarr5rarr8 5217 003180rarr4rarr8 7918 00222

Table 4 Path selection plans

Solution Path Total cost$ Total risk

1

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

216132 08630

2

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

218541 07964

3

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

236768 07640

4

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

239177 06974

5

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

252843 06788

6

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

266943 06662

7

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr4rarr8rarr6rarr0

268888 06494

8

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

269352 05996

8 Discrete Dynamics in Nature and Society

Table 4 Continued

Solution Path Total cost$ Total risk

9

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr6rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

273479 05798

10

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

287579 05672

11

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

289988 05006

12

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

313757 04622

13

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

319699 04526

14

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

343468 04142

Table 5 Vehicle scheduling plan (for solution 1)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

2

1 626 711 800 845 9344 934 1019 1059 1144 12257 1225 1310 1359 1444 15337 1533 1618 1708 1753 1842

3

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

4

4 635 720 800 845 9254 925 1010 1051 1136 12168 1216 1301 1346 1431 15158 1515 1600 1645 1730 18148 1814 1859 1944 2029 2113

Discrete Dynamics in Nature and Society 9

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

2500 3000 3500 40002000Transportation cost$

TSAGNA

(a) Iteration number = 5

2500 3000 3500 40002000Transportation cost$

TSAGNA

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k(b) Iteration number = 10

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k

2500 3000 3500 40002000Transportation cost$

TSAGNA

(c) Iteration number = 20

2500 3000 3500 40002000Transportation cost$

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

TSAGNA

(d) Iteration number = 50

Figure 6 Pareto solutions obtained in path optimization

5 Case Study

The proposed methodology was implemented for the urbantransportation network in Lanzhou China Daily trafficvolumes of all road sections in realistic road networks ofmain districts are used to estimate vehicle travel time for allroad sections In addition other factors such as distributionof population and facilities as well as probabilities andconsequences of traffic accident along the road sectionsinvolved are also considered to establish a traditional model[15] to estimate the risk value of each road section for vehicletransportation Thus a transport network for hazardous

materials (see Figure 7) in main districts of Lanzhou isformed containing 170 nodes and 272 edges Distribution ofgasoline 92 (permil) for 16 gas stations (e) under PetroChinain this network is taken as an example to calculate vehiclescheduling plans for oil product transportation

Table 7 shows the positions of 16 gas stations in thenetwork indicated by their nearest nodes (destination site119889) as well as their needs of gasoline and the time windowof distribution The position of the distribution center isindicated by node 1 The volume of a transportation vehicle(tank car) is 135m3 Traveling fee of a tank car in FCL inurban highways is 200 RMBh while that with no load is

10 Discrete Dynamics in Nature and Society

1

2

3

4

56

78

910

11

12

13

14

15

1617

181920

21

22

23

24

25

26

27

28

29

30

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49 50

51

5255

53

5456

57

58

5960

6162

63

64

65

66

67

69

70

7172

7374

75

7677 78

79

80

81

8283

84

85

86

8788

899091 92

93

94

9596

979899100101102

103

104105

106

107108

109110

111112 113

114115116

117

118

119

120121

122

123124125

126127128

129

130

131 132133

134

135

136

137139138

140141

142 143

144145

146147

148

149

150 151

152153

154

155

156157

158

159

160

161

162163

164

165

166 167

68

168

169

170

Figure 7 A transport network for hazardous materials in main districts of Lanzhou

Table 6 Vehicle scheduling plan (for solution 14)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

11 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

34 0715 0800 0840 0925 10061 1006 1051 1144 1229 13187 1318 1403 1646 1731 1820

52 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

32 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

41 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

6 8 0729 0814 1000 1045 11307 1130 1215 1457 1542 1631

150 RMBh eliminating cost of loading and unloading Timefor either loading or unloading is 075 h

C is the programming language to implement the algo-rithmThe running environment is PC i5-3470 320GHz 4GParameters setting in the program and CPU times of vehiclesare shown in Table 8 A total of 164 path selection plans areobtained When a certain plan is selected the correspondingtimetable for vehicle operation can be obtained by the vehiclescheduling program

A path plan is a choice based on game between trans-portation cost and risk by a decision-maker with a specific

risk preference For example in plan 1 which has the lowesttransportation cost but the highest transportation risk thetransportation time is 439658 RMB while the transporta-tion risk is 32262 8 vehicles are needed to complete jobsAs another example in plan 164 which has the lowesttransportation risk but the highest transportation cost thetransportation time is 460983 RMB while the transporta-tion risk is 25586 also 8 vehicles are needed to complete jobsThe traveling path and transport timetable for all vehicles inplan 164 are shown in Tables 9 and 10

Discrete Dynamics in Nature and Society 11

Table 7 Nodes representing 16 gas stations as well as their needs of gasoline and the time window of distribution

Number Destination Required volumem3 Time window1 2 15 [1200 1600]2 12 26 [830 1330]3 16 12 [1030 1600]4 25 16 [800 1200]5 43 16 [800 1600]6 44 29 [1000 1600]7 47 32 [900 1500]8 52 25 [900 1200]9 62 24 [800 1000]10 63 10 [800 1500]11 94 18 [1400 1800]12 138 25 [800 1500]13 146 40 [1200 1800]14 150 23 [1000 1600]15 167 9 [1200 1800]16 170 25 [800 1600]

Table 8 Transport parameters setting in the program and CPU times

Stage Parameters setting CPU timessPopulation size Iteration

Path selectionStage 1 the pulse algorithm is used to obtain the Pareto solutions of all pathsfrom the distribution center to each destination mdash mdash 17

Stage 2 get all paths selection solution by NSGA-II algorithm 500 100 32Vehicle scheduling

The vehicle scheduling plan under the current path selection plan is acquired byUMDA 200 100 9

6 Conclusion

Distribution of hazardous materials in FCL is essentially amultistage combinatorial optimization problem In this studya decision approach for distribution of hazardous materialsin FCL with a single distribution center is formed which canprovide a decision support for transportation enterprises andadministrative authorities

In this study mathematical models for path planningand vehicle scheduling problems are established respectivelywith corresponding problem-solving algorithms designedIn order to solve the path planning model a two-stagealgorithm is designed Based on screening of paths in thefirst stage the computational complexity of the second stageinvolved in the global path optimization is significantlyreduced which improves calculation efficiency In obtainingthe vehicle scheduling plan the operating process of atransporting vehicle is converted to single objective VRPTWThen the problem-solvingmethod forVRPTWon the basis ofestimation of distribution is adopted to obtain the timetablefor all vehicles

The proposed approach was applied to a hypotheticalnumerical example as well as an application to a real casein distribution of gasoline of Lanzhou ChinaThe numerical

example showed that for small-scale problems the two-stagealgorithm can obtain Pareto-optimal front faster in the sameiterationHowever for large-scale problems due to the lengthof coding it is difficult to obtain Pareto-optimal path set in alimited iterative step in the second stage the problem size ofVRPTW is only related to the number of destination nodesnot to the scale of the transportation network Therefore thevehicle scheduling method has high efficiency whether it issmall-scale or large-scale transportation network

Our purpose is to design a decision support systemfor path planning and vehicle scheduling of distribution ofhazardous materials in FCL Further research has to be doneat least in two directions distribution of hazardous materialsin FCL with multiple distribution centers and transportationwith different volumes vehicle will be focused on

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (nos 61064012 61364026

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Complex AnalysisJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Discrete Dynamics in Nature and Society

Table 2 Required transport volume and time window of a destination

Destination 1 2 4 7 8Required volumem3 35 18 295 45 60Time window [800 1600] [1200 2000] [800 1230] [800 1800] [800 2000]

The pulse algorithm is used to obtain thePareto solutions of all paths from the

End

Initialize network and

Get all paths selection solution byNSGA-II algorithm (path planning)

Decision-makers chose favorable travel pathsdepending on the specific risk preferences

The vehicle scheduling plan under thecurrent path selection plan is acquiredby UMDA (vehicle scheduling)

Satisfied

Yes

No

Yes

No

Start

requirement d = 1

distribution center destination d d = d + 1

d le m

Figure 4 Flow chart for solving problem of path planning andvehicle scheduling of hazardous materials transportation

After the first stage Pareto shortest path (departingpath) and shortest distance paths (returning path) betweenthe distribution center to each destination can be obtained(Table 3)

When the Pareto paths set is obtained the Pareto solutionset of total cost and risk for transportation between thedistribution center and all destinations can be obtainedTable 4 shows the path selection plans after decoding ofPareto solutions

When a certain transport plan is selected model P2 isused to solve the results for the path adopted In this waya vehicle scheduling plan under the current path selectionplan is acquired For example when path plan 1 is selectedin Table 4 the vehicle scheduling plan of Table 5 can beobtained and when path plan 14 is selected the vehicle

1

8

02

34

5 6

7

Distribution centerOther nodesDestination

Figure 5 A test network for transportation of hazardous materials

scheduling plan of Table 6 can be obtained The columnof node represents the distribution node arrived at by avehicle The timetable shows the time moments of a vehiclein a round trip for loading start (LS) departure from thedistribution center (DD) unloading start at the destination(US) returning after unloading finishes (RU) and arrivingback at the distribution center (AC)

42 Comparison with General NSGA-II In addition a gen-eral NSGA-II algorithm (GNA) is designed for the pathplanning in order to illustrate the effectiveness of the two-stage algorithm (TSA) proposed Different from the two-stage algorithm in the general NSGA-II algorithm thepriority based encoding [14] method is adopted and theencoding length is |119873| lowast 119898 However in the second stage ofthe two-stage algorithm the encoding length is only 119898 Forthe example in Section 41 the parameters of two algorithmsare set as follows the size of population is 100 and theiteration number is 50 generations In Figures 6(a)ndash6(d) isthe Pareto-optimal front obtained by iterative 5 10 20 and50 generations respectively By comparison it can be foundthat the final solution can be obtained by using the TSAwhenthe iteration reaches the 10th generation (14 plans see Table 4)while the GNA still fails to obtain the optimal solution untilthe fiftieth iterationThis is just a network of 9 nodes and thisgap will become even bigger for large-scale networks Thereis no direct relationship between the encoding length and thenetwork size in the two-stage algorithm so for large-scalenetworks the algorithm is still valid

Discrete Dynamics in Nature and Society 7

Table 3 Departing path and returning path between the distribution center and each destination

Destination Departing path Lengthkm Risk Returning path Lengthkm

1 0rarr1 3656 00456 1rarr0 36560rarr3rarr1 4021 00234

2 0rarr3rarr2 3411 00446 2rarr3rarr0 34114 0rarr4 3029 00154 4rarr0 3029

7

0rarr5rarr7 3708 00832

7rarr5rarr0 37080rarr3rarr2rarr7 7329 006980rarr6rarr8rarr7 7606 005380rarr5rarr8rarr7 9482 003400rarr4rarr8rarr7 12183 00244

80rarr6rarr8 3341 00516

8rarr6rarr0 33410rarr5rarr8 5217 003180rarr4rarr8 7918 00222

Table 4 Path selection plans

Solution Path Total cost$ Total risk

1

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

216132 08630

2

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

218541 07964

3

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

236768 07640

4

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

239177 06974

5

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

252843 06788

6

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

266943 06662

7

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr4rarr8rarr6rarr0

268888 06494

8

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

269352 05996

8 Discrete Dynamics in Nature and Society

Table 4 Continued

Solution Path Total cost$ Total risk

9

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr6rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

273479 05798

10

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

287579 05672

11

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

289988 05006

12

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

313757 04622

13

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

319699 04526

14

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

343468 04142

Table 5 Vehicle scheduling plan (for solution 1)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

2

1 626 711 800 845 9344 934 1019 1059 1144 12257 1225 1310 1359 1444 15337 1533 1618 1708 1753 1842

3

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

4

4 635 720 800 845 9254 925 1010 1051 1136 12168 1216 1301 1346 1431 15158 1515 1600 1645 1730 18148 1814 1859 1944 2029 2113

Discrete Dynamics in Nature and Society 9

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

2500 3000 3500 40002000Transportation cost$

TSAGNA

(a) Iteration number = 5

2500 3000 3500 40002000Transportation cost$

TSAGNA

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k(b) Iteration number = 10

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k

2500 3000 3500 40002000Transportation cost$

TSAGNA

(c) Iteration number = 20

2500 3000 3500 40002000Transportation cost$

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

TSAGNA

(d) Iteration number = 50

Figure 6 Pareto solutions obtained in path optimization

5 Case Study

The proposed methodology was implemented for the urbantransportation network in Lanzhou China Daily trafficvolumes of all road sections in realistic road networks ofmain districts are used to estimate vehicle travel time for allroad sections In addition other factors such as distributionof population and facilities as well as probabilities andconsequences of traffic accident along the road sectionsinvolved are also considered to establish a traditional model[15] to estimate the risk value of each road section for vehicletransportation Thus a transport network for hazardous

materials (see Figure 7) in main districts of Lanzhou isformed containing 170 nodes and 272 edges Distribution ofgasoline 92 (permil) for 16 gas stations (e) under PetroChinain this network is taken as an example to calculate vehiclescheduling plans for oil product transportation

Table 7 shows the positions of 16 gas stations in thenetwork indicated by their nearest nodes (destination site119889) as well as their needs of gasoline and the time windowof distribution The position of the distribution center isindicated by node 1 The volume of a transportation vehicle(tank car) is 135m3 Traveling fee of a tank car in FCL inurban highways is 200 RMBh while that with no load is

10 Discrete Dynamics in Nature and Society

1

2

3

4

56

78

910

11

12

13

14

15

1617

181920

21

22

23

24

25

26

27

28

29

30

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49 50

51

5255

53

5456

57

58

5960

6162

63

64

65

66

67

69

70

7172

7374

75

7677 78

79

80

81

8283

84

85

86

8788

899091 92

93

94

9596

979899100101102

103

104105

106

107108

109110

111112 113

114115116

117

118

119

120121

122

123124125

126127128

129

130

131 132133

134

135

136

137139138

140141

142 143

144145

146147

148

149

150 151

152153

154

155

156157

158

159

160

161

162163

164

165

166 167

68

168

169

170

Figure 7 A transport network for hazardous materials in main districts of Lanzhou

Table 6 Vehicle scheduling plan (for solution 14)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

11 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

34 0715 0800 0840 0925 10061 1006 1051 1144 1229 13187 1318 1403 1646 1731 1820

52 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

32 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

41 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

6 8 0729 0814 1000 1045 11307 1130 1215 1457 1542 1631

150 RMBh eliminating cost of loading and unloading Timefor either loading or unloading is 075 h

C is the programming language to implement the algo-rithmThe running environment is PC i5-3470 320GHz 4GParameters setting in the program and CPU times of vehiclesare shown in Table 8 A total of 164 path selection plans areobtained When a certain plan is selected the correspondingtimetable for vehicle operation can be obtained by the vehiclescheduling program

A path plan is a choice based on game between trans-portation cost and risk by a decision-maker with a specific

risk preference For example in plan 1 which has the lowesttransportation cost but the highest transportation risk thetransportation time is 439658 RMB while the transporta-tion risk is 32262 8 vehicles are needed to complete jobsAs another example in plan 164 which has the lowesttransportation risk but the highest transportation cost thetransportation time is 460983 RMB while the transporta-tion risk is 25586 also 8 vehicles are needed to complete jobsThe traveling path and transport timetable for all vehicles inplan 164 are shown in Tables 9 and 10

Discrete Dynamics in Nature and Society 11

Table 7 Nodes representing 16 gas stations as well as their needs of gasoline and the time window of distribution

Number Destination Required volumem3 Time window1 2 15 [1200 1600]2 12 26 [830 1330]3 16 12 [1030 1600]4 25 16 [800 1200]5 43 16 [800 1600]6 44 29 [1000 1600]7 47 32 [900 1500]8 52 25 [900 1200]9 62 24 [800 1000]10 63 10 [800 1500]11 94 18 [1400 1800]12 138 25 [800 1500]13 146 40 [1200 1800]14 150 23 [1000 1600]15 167 9 [1200 1800]16 170 25 [800 1600]

Table 8 Transport parameters setting in the program and CPU times

Stage Parameters setting CPU timessPopulation size Iteration

Path selectionStage 1 the pulse algorithm is used to obtain the Pareto solutions of all pathsfrom the distribution center to each destination mdash mdash 17

Stage 2 get all paths selection solution by NSGA-II algorithm 500 100 32Vehicle scheduling

The vehicle scheduling plan under the current path selection plan is acquired byUMDA 200 100 9

6 Conclusion

Distribution of hazardous materials in FCL is essentially amultistage combinatorial optimization problem In this studya decision approach for distribution of hazardous materialsin FCL with a single distribution center is formed which canprovide a decision support for transportation enterprises andadministrative authorities

In this study mathematical models for path planningand vehicle scheduling problems are established respectivelywith corresponding problem-solving algorithms designedIn order to solve the path planning model a two-stagealgorithm is designed Based on screening of paths in thefirst stage the computational complexity of the second stageinvolved in the global path optimization is significantlyreduced which improves calculation efficiency In obtainingthe vehicle scheduling plan the operating process of atransporting vehicle is converted to single objective VRPTWThen the problem-solvingmethod forVRPTWon the basis ofestimation of distribution is adopted to obtain the timetablefor all vehicles

The proposed approach was applied to a hypotheticalnumerical example as well as an application to a real casein distribution of gasoline of Lanzhou ChinaThe numerical

example showed that for small-scale problems the two-stagealgorithm can obtain Pareto-optimal front faster in the sameiterationHowever for large-scale problems due to the lengthof coding it is difficult to obtain Pareto-optimal path set in alimited iterative step in the second stage the problem size ofVRPTW is only related to the number of destination nodesnot to the scale of the transportation network Therefore thevehicle scheduling method has high efficiency whether it issmall-scale or large-scale transportation network

Our purpose is to design a decision support systemfor path planning and vehicle scheduling of distribution ofhazardous materials in FCL Further research has to be doneat least in two directions distribution of hazardous materialsin FCL with multiple distribution centers and transportationwith different volumes vehicle will be focused on

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (nos 61064012 61364026

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 7

Table 3 Departing path and returning path between the distribution center and each destination

Destination Departing path Lengthkm Risk Returning path Lengthkm

1 0rarr1 3656 00456 1rarr0 36560rarr3rarr1 4021 00234

2 0rarr3rarr2 3411 00446 2rarr3rarr0 34114 0rarr4 3029 00154 4rarr0 3029

7

0rarr5rarr7 3708 00832

7rarr5rarr0 37080rarr3rarr2rarr7 7329 006980rarr6rarr8rarr7 7606 005380rarr5rarr8rarr7 9482 003400rarr4rarr8rarr7 12183 00244

80rarr6rarr8 3341 00516

8rarr6rarr0 33410rarr5rarr8 5217 003180rarr4rarr8 7918 00222

Table 4 Path selection plans

Solution Path Total cost$ Total risk

1

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

216132 08630

2

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr6rarr8rarr6rarr0

218541 07964

3

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

236768 07640

4

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr5rarr8rarr6rarr0

239177 06974

5

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

252843 06788

6

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

266943 06662

7

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr7rarr5rarr00rarr4rarr8rarr6rarr0

268888 06494

8

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr6rarr8rarr6rarr0

269352 05996

8 Discrete Dynamics in Nature and Society

Table 4 Continued

Solution Path Total cost$ Total risk

9

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr6rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

273479 05798

10

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

287579 05672

11

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

289988 05006

12

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

313757 04622

13

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

319699 04526

14

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

343468 04142

Table 5 Vehicle scheduling plan (for solution 1)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

2

1 626 711 800 845 9344 934 1019 1059 1144 12257 1225 1310 1359 1444 15337 1533 1618 1708 1753 1842

3

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

4

4 635 720 800 845 9254 925 1010 1051 1136 12168 1216 1301 1346 1431 15158 1515 1600 1645 1730 18148 1814 1859 1944 2029 2113

Discrete Dynamics in Nature and Society 9

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

2500 3000 3500 40002000Transportation cost$

TSAGNA

(a) Iteration number = 5

2500 3000 3500 40002000Transportation cost$

TSAGNA

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k(b) Iteration number = 10

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k

2500 3000 3500 40002000Transportation cost$

TSAGNA

(c) Iteration number = 20

2500 3000 3500 40002000Transportation cost$

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

TSAGNA

(d) Iteration number = 50

Figure 6 Pareto solutions obtained in path optimization

5 Case Study

The proposed methodology was implemented for the urbantransportation network in Lanzhou China Daily trafficvolumes of all road sections in realistic road networks ofmain districts are used to estimate vehicle travel time for allroad sections In addition other factors such as distributionof population and facilities as well as probabilities andconsequences of traffic accident along the road sectionsinvolved are also considered to establish a traditional model[15] to estimate the risk value of each road section for vehicletransportation Thus a transport network for hazardous

materials (see Figure 7) in main districts of Lanzhou isformed containing 170 nodes and 272 edges Distribution ofgasoline 92 (permil) for 16 gas stations (e) under PetroChinain this network is taken as an example to calculate vehiclescheduling plans for oil product transportation

Table 7 shows the positions of 16 gas stations in thenetwork indicated by their nearest nodes (destination site119889) as well as their needs of gasoline and the time windowof distribution The position of the distribution center isindicated by node 1 The volume of a transportation vehicle(tank car) is 135m3 Traveling fee of a tank car in FCL inurban highways is 200 RMBh while that with no load is

10 Discrete Dynamics in Nature and Society

1

2

3

4

56

78

910

11

12

13

14

15

1617

181920

21

22

23

24

25

26

27

28

29

30

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49 50

51

5255

53

5456

57

58

5960

6162

63

64

65

66

67

69

70

7172

7374

75

7677 78

79

80

81

8283

84

85

86

8788

899091 92

93

94

9596

979899100101102

103

104105

106

107108

109110

111112 113

114115116

117

118

119

120121

122

123124125

126127128

129

130

131 132133

134

135

136

137139138

140141

142 143

144145

146147

148

149

150 151

152153

154

155

156157

158

159

160

161

162163

164

165

166 167

68

168

169

170

Figure 7 A transport network for hazardous materials in main districts of Lanzhou

Table 6 Vehicle scheduling plan (for solution 14)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

11 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

34 0715 0800 0840 0925 10061 1006 1051 1144 1229 13187 1318 1403 1646 1731 1820

52 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

32 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

41 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

6 8 0729 0814 1000 1045 11307 1130 1215 1457 1542 1631

150 RMBh eliminating cost of loading and unloading Timefor either loading or unloading is 075 h

C is the programming language to implement the algo-rithmThe running environment is PC i5-3470 320GHz 4GParameters setting in the program and CPU times of vehiclesare shown in Table 8 A total of 164 path selection plans areobtained When a certain plan is selected the correspondingtimetable for vehicle operation can be obtained by the vehiclescheduling program

A path plan is a choice based on game between trans-portation cost and risk by a decision-maker with a specific

risk preference For example in plan 1 which has the lowesttransportation cost but the highest transportation risk thetransportation time is 439658 RMB while the transporta-tion risk is 32262 8 vehicles are needed to complete jobsAs another example in plan 164 which has the lowesttransportation risk but the highest transportation cost thetransportation time is 460983 RMB while the transporta-tion risk is 25586 also 8 vehicles are needed to complete jobsThe traveling path and transport timetable for all vehicles inplan 164 are shown in Tables 9 and 10

Discrete Dynamics in Nature and Society 11

Table 7 Nodes representing 16 gas stations as well as their needs of gasoline and the time window of distribution

Number Destination Required volumem3 Time window1 2 15 [1200 1600]2 12 26 [830 1330]3 16 12 [1030 1600]4 25 16 [800 1200]5 43 16 [800 1600]6 44 29 [1000 1600]7 47 32 [900 1500]8 52 25 [900 1200]9 62 24 [800 1000]10 63 10 [800 1500]11 94 18 [1400 1800]12 138 25 [800 1500]13 146 40 [1200 1800]14 150 23 [1000 1600]15 167 9 [1200 1800]16 170 25 [800 1600]

Table 8 Transport parameters setting in the program and CPU times

Stage Parameters setting CPU timessPopulation size Iteration

Path selectionStage 1 the pulse algorithm is used to obtain the Pareto solutions of all pathsfrom the distribution center to each destination mdash mdash 17

Stage 2 get all paths selection solution by NSGA-II algorithm 500 100 32Vehicle scheduling

The vehicle scheduling plan under the current path selection plan is acquired byUMDA 200 100 9

6 Conclusion

Distribution of hazardous materials in FCL is essentially amultistage combinatorial optimization problem In this studya decision approach for distribution of hazardous materialsin FCL with a single distribution center is formed which canprovide a decision support for transportation enterprises andadministrative authorities

In this study mathematical models for path planningand vehicle scheduling problems are established respectivelywith corresponding problem-solving algorithms designedIn order to solve the path planning model a two-stagealgorithm is designed Based on screening of paths in thefirst stage the computational complexity of the second stageinvolved in the global path optimization is significantlyreduced which improves calculation efficiency In obtainingthe vehicle scheduling plan the operating process of atransporting vehicle is converted to single objective VRPTWThen the problem-solvingmethod forVRPTWon the basis ofestimation of distribution is adopted to obtain the timetablefor all vehicles

The proposed approach was applied to a hypotheticalnumerical example as well as an application to a real casein distribution of gasoline of Lanzhou ChinaThe numerical

example showed that for small-scale problems the two-stagealgorithm can obtain Pareto-optimal front faster in the sameiterationHowever for large-scale problems due to the lengthof coding it is difficult to obtain Pareto-optimal path set in alimited iterative step in the second stage the problem size ofVRPTW is only related to the number of destination nodesnot to the scale of the transportation network Therefore thevehicle scheduling method has high efficiency whether it issmall-scale or large-scale transportation network

Our purpose is to design a decision support systemfor path planning and vehicle scheduling of distribution ofhazardous materials in FCL Further research has to be doneat least in two directions distribution of hazardous materialsin FCL with multiple distribution centers and transportationwith different volumes vehicle will be focused on

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (nos 61064012 61364026

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Discrete Dynamics in Nature and Society

Table 4 Continued

Solution Path Total cost$ Total risk

9

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr6rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

273479 05798

10

0rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

287579 05672

11

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

289988 05006

12

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr5rarr8rarr6rarr0

313757 04622

13

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr5rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

319699 04526

14

0rarr3rarr1rarr00rarr3rarr2rarr3rarr0

0rarr4rarr00rarr4rarr8rarr7rarr5rarr00rarr4rarr8rarr6rarr0

343468 04142

Table 5 Vehicle scheduling plan (for solution 1)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

2

1 626 711 800 845 9344 934 1019 1059 1144 12257 1225 1310 1359 1444 15337 1533 1618 1708 1753 1842

3

7 626 711 800 845 9341 934 1019 1108 1153 12428 1242 1327 1411 1456 15412 1541 1626 1712 1757 1842

4

4 635 720 800 845 9254 925 1010 1051 1136 12168 1216 1301 1346 1431 15158 1515 1600 1645 1730 18148 1814 1859 1944 2029 2113

Discrete Dynamics in Nature and Society 9

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

2500 3000 3500 40002000Transportation cost$

TSAGNA

(a) Iteration number = 5

2500 3000 3500 40002000Transportation cost$

TSAGNA

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k(b) Iteration number = 10

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k

2500 3000 3500 40002000Transportation cost$

TSAGNA

(c) Iteration number = 20

2500 3000 3500 40002000Transportation cost$

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

TSAGNA

(d) Iteration number = 50

Figure 6 Pareto solutions obtained in path optimization

5 Case Study

The proposed methodology was implemented for the urbantransportation network in Lanzhou China Daily trafficvolumes of all road sections in realistic road networks ofmain districts are used to estimate vehicle travel time for allroad sections In addition other factors such as distributionof population and facilities as well as probabilities andconsequences of traffic accident along the road sectionsinvolved are also considered to establish a traditional model[15] to estimate the risk value of each road section for vehicletransportation Thus a transport network for hazardous

materials (see Figure 7) in main districts of Lanzhou isformed containing 170 nodes and 272 edges Distribution ofgasoline 92 (permil) for 16 gas stations (e) under PetroChinain this network is taken as an example to calculate vehiclescheduling plans for oil product transportation

Table 7 shows the positions of 16 gas stations in thenetwork indicated by their nearest nodes (destination site119889) as well as their needs of gasoline and the time windowof distribution The position of the distribution center isindicated by node 1 The volume of a transportation vehicle(tank car) is 135m3 Traveling fee of a tank car in FCL inurban highways is 200 RMBh while that with no load is

10 Discrete Dynamics in Nature and Society

1

2

3

4

56

78

910

11

12

13

14

15

1617

181920

21

22

23

24

25

26

27

28

29

30

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49 50

51

5255

53

5456

57

58

5960

6162

63

64

65

66

67

69

70

7172

7374

75

7677 78

79

80

81

8283

84

85

86

8788

899091 92

93

94

9596

979899100101102

103

104105

106

107108

109110

111112 113

114115116

117

118

119

120121

122

123124125

126127128

129

130

131 132133

134

135

136

137139138

140141

142 143

144145

146147

148

149

150 151

152153

154

155

156157

158

159

160

161

162163

164

165

166 167

68

168

169

170

Figure 7 A transport network for hazardous materials in main districts of Lanzhou

Table 6 Vehicle scheduling plan (for solution 14)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

11 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

34 0715 0800 0840 0925 10061 1006 1051 1144 1229 13187 1318 1403 1646 1731 1820

52 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

32 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

41 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

6 8 0729 0814 1000 1045 11307 1130 1215 1457 1542 1631

150 RMBh eliminating cost of loading and unloading Timefor either loading or unloading is 075 h

C is the programming language to implement the algo-rithmThe running environment is PC i5-3470 320GHz 4GParameters setting in the program and CPU times of vehiclesare shown in Table 8 A total of 164 path selection plans areobtained When a certain plan is selected the correspondingtimetable for vehicle operation can be obtained by the vehiclescheduling program

A path plan is a choice based on game between trans-portation cost and risk by a decision-maker with a specific

risk preference For example in plan 1 which has the lowesttransportation cost but the highest transportation risk thetransportation time is 439658 RMB while the transporta-tion risk is 32262 8 vehicles are needed to complete jobsAs another example in plan 164 which has the lowesttransportation risk but the highest transportation cost thetransportation time is 460983 RMB while the transporta-tion risk is 25586 also 8 vehicles are needed to complete jobsThe traveling path and transport timetable for all vehicles inplan 164 are shown in Tables 9 and 10

Discrete Dynamics in Nature and Society 11

Table 7 Nodes representing 16 gas stations as well as their needs of gasoline and the time window of distribution

Number Destination Required volumem3 Time window1 2 15 [1200 1600]2 12 26 [830 1330]3 16 12 [1030 1600]4 25 16 [800 1200]5 43 16 [800 1600]6 44 29 [1000 1600]7 47 32 [900 1500]8 52 25 [900 1200]9 62 24 [800 1000]10 63 10 [800 1500]11 94 18 [1400 1800]12 138 25 [800 1500]13 146 40 [1200 1800]14 150 23 [1000 1600]15 167 9 [1200 1800]16 170 25 [800 1600]

Table 8 Transport parameters setting in the program and CPU times

Stage Parameters setting CPU timessPopulation size Iteration

Path selectionStage 1 the pulse algorithm is used to obtain the Pareto solutions of all pathsfrom the distribution center to each destination mdash mdash 17

Stage 2 get all paths selection solution by NSGA-II algorithm 500 100 32Vehicle scheduling

The vehicle scheduling plan under the current path selection plan is acquired byUMDA 200 100 9

6 Conclusion

Distribution of hazardous materials in FCL is essentially amultistage combinatorial optimization problem In this studya decision approach for distribution of hazardous materialsin FCL with a single distribution center is formed which canprovide a decision support for transportation enterprises andadministrative authorities

In this study mathematical models for path planningand vehicle scheduling problems are established respectivelywith corresponding problem-solving algorithms designedIn order to solve the path planning model a two-stagealgorithm is designed Based on screening of paths in thefirst stage the computational complexity of the second stageinvolved in the global path optimization is significantlyreduced which improves calculation efficiency In obtainingthe vehicle scheduling plan the operating process of atransporting vehicle is converted to single objective VRPTWThen the problem-solvingmethod forVRPTWon the basis ofestimation of distribution is adopted to obtain the timetablefor all vehicles

The proposed approach was applied to a hypotheticalnumerical example as well as an application to a real casein distribution of gasoline of Lanzhou ChinaThe numerical

example showed that for small-scale problems the two-stagealgorithm can obtain Pareto-optimal front faster in the sameiterationHowever for large-scale problems due to the lengthof coding it is difficult to obtain Pareto-optimal path set in alimited iterative step in the second stage the problem size ofVRPTW is only related to the number of destination nodesnot to the scale of the transportation network Therefore thevehicle scheduling method has high efficiency whether it issmall-scale or large-scale transportation network

Our purpose is to design a decision support systemfor path planning and vehicle scheduling of distribution ofhazardous materials in FCL Further research has to be doneat least in two directions distribution of hazardous materialsin FCL with multiple distribution centers and transportationwith different volumes vehicle will be focused on

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (nos 61064012 61364026

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 9

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

2500 3000 3500 40002000Transportation cost$

TSAGNA

(a) Iteration number = 5

2500 3000 3500 40002000Transportation cost$

TSAGNA

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k(b) Iteration number = 10

040

050

060

070

080

090

100

Tran

spor

tatio

n ris

k

2500 3000 3500 40002000Transportation cost$

TSAGNA

(c) Iteration number = 20

2500 3000 3500 40002000Transportation cost$

040

050

060

070

080

090

100Tr

ansp

orta

tion

risk

TSAGNA

(d) Iteration number = 50

Figure 6 Pareto solutions obtained in path optimization

5 Case Study

The proposed methodology was implemented for the urbantransportation network in Lanzhou China Daily trafficvolumes of all road sections in realistic road networks ofmain districts are used to estimate vehicle travel time for allroad sections In addition other factors such as distributionof population and facilities as well as probabilities andconsequences of traffic accident along the road sectionsinvolved are also considered to establish a traditional model[15] to estimate the risk value of each road section for vehicletransportation Thus a transport network for hazardous

materials (see Figure 7) in main districts of Lanzhou isformed containing 170 nodes and 272 edges Distribution ofgasoline 92 (permil) for 16 gas stations (e) under PetroChinain this network is taken as an example to calculate vehiclescheduling plans for oil product transportation

Table 7 shows the positions of 16 gas stations in thenetwork indicated by their nearest nodes (destination site119889) as well as their needs of gasoline and the time windowof distribution The position of the distribution center isindicated by node 1 The volume of a transportation vehicle(tank car) is 135m3 Traveling fee of a tank car in FCL inurban highways is 200 RMBh while that with no load is

10 Discrete Dynamics in Nature and Society

1

2

3

4

56

78

910

11

12

13

14

15

1617

181920

21

22

23

24

25

26

27

28

29

30

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49 50

51

5255

53

5456

57

58

5960

6162

63

64

65

66

67

69

70

7172

7374

75

7677 78

79

80

81

8283

84

85

86

8788

899091 92

93

94

9596

979899100101102

103

104105

106

107108

109110

111112 113

114115116

117

118

119

120121

122

123124125

126127128

129

130

131 132133

134

135

136

137139138

140141

142 143

144145

146147

148

149

150 151

152153

154

155

156157

158

159

160

161

162163

164

165

166 167

68

168

169

170

Figure 7 A transport network for hazardous materials in main districts of Lanzhou

Table 6 Vehicle scheduling plan (for solution 14)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

11 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

34 0715 0800 0840 0925 10061 1006 1051 1144 1229 13187 1318 1403 1646 1731 1820

52 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

32 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

41 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

6 8 0729 0814 1000 1045 11307 1130 1215 1457 1542 1631

150 RMBh eliminating cost of loading and unloading Timefor either loading or unloading is 075 h

C is the programming language to implement the algo-rithmThe running environment is PC i5-3470 320GHz 4GParameters setting in the program and CPU times of vehiclesare shown in Table 8 A total of 164 path selection plans areobtained When a certain plan is selected the correspondingtimetable for vehicle operation can be obtained by the vehiclescheduling program

A path plan is a choice based on game between trans-portation cost and risk by a decision-maker with a specific

risk preference For example in plan 1 which has the lowesttransportation cost but the highest transportation risk thetransportation time is 439658 RMB while the transporta-tion risk is 32262 8 vehicles are needed to complete jobsAs another example in plan 164 which has the lowesttransportation risk but the highest transportation cost thetransportation time is 460983 RMB while the transporta-tion risk is 25586 also 8 vehicles are needed to complete jobsThe traveling path and transport timetable for all vehicles inplan 164 are shown in Tables 9 and 10

Discrete Dynamics in Nature and Society 11

Table 7 Nodes representing 16 gas stations as well as their needs of gasoline and the time window of distribution

Number Destination Required volumem3 Time window1 2 15 [1200 1600]2 12 26 [830 1330]3 16 12 [1030 1600]4 25 16 [800 1200]5 43 16 [800 1600]6 44 29 [1000 1600]7 47 32 [900 1500]8 52 25 [900 1200]9 62 24 [800 1000]10 63 10 [800 1500]11 94 18 [1400 1800]12 138 25 [800 1500]13 146 40 [1200 1800]14 150 23 [1000 1600]15 167 9 [1200 1800]16 170 25 [800 1600]

Table 8 Transport parameters setting in the program and CPU times

Stage Parameters setting CPU timessPopulation size Iteration

Path selectionStage 1 the pulse algorithm is used to obtain the Pareto solutions of all pathsfrom the distribution center to each destination mdash mdash 17

Stage 2 get all paths selection solution by NSGA-II algorithm 500 100 32Vehicle scheduling

The vehicle scheduling plan under the current path selection plan is acquired byUMDA 200 100 9

6 Conclusion

Distribution of hazardous materials in FCL is essentially amultistage combinatorial optimization problem In this studya decision approach for distribution of hazardous materialsin FCL with a single distribution center is formed which canprovide a decision support for transportation enterprises andadministrative authorities

In this study mathematical models for path planningand vehicle scheduling problems are established respectivelywith corresponding problem-solving algorithms designedIn order to solve the path planning model a two-stagealgorithm is designed Based on screening of paths in thefirst stage the computational complexity of the second stageinvolved in the global path optimization is significantlyreduced which improves calculation efficiency In obtainingthe vehicle scheduling plan the operating process of atransporting vehicle is converted to single objective VRPTWThen the problem-solvingmethod forVRPTWon the basis ofestimation of distribution is adopted to obtain the timetablefor all vehicles

The proposed approach was applied to a hypotheticalnumerical example as well as an application to a real casein distribution of gasoline of Lanzhou ChinaThe numerical

example showed that for small-scale problems the two-stagealgorithm can obtain Pareto-optimal front faster in the sameiterationHowever for large-scale problems due to the lengthof coding it is difficult to obtain Pareto-optimal path set in alimited iterative step in the second stage the problem size ofVRPTW is only related to the number of destination nodesnot to the scale of the transportation network Therefore thevehicle scheduling method has high efficiency whether it issmall-scale or large-scale transportation network

Our purpose is to design a decision support systemfor path planning and vehicle scheduling of distribution ofhazardous materials in FCL Further research has to be doneat least in two directions distribution of hazardous materialsin FCL with multiple distribution centers and transportationwith different volumes vehicle will be focused on

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (nos 61064012 61364026

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Discrete Dynamics in Nature and Society

1

2

3

4

56

78

910

11

12

13

14

15

1617

181920

21

22

23

24

25

26

27

28

29

30

31

32

3334

35

36

37

38

39

40

41

42

43

44

45

46

47

48

49 50

51

5255

53

5456

57

58

5960

6162

63

64

65

66

67

69

70

7172

7374

75

7677 78

79

80

81

8283

84

85

86

8788

899091 92

93

94

9596

979899100101102

103

104105

106

107108

109110

111112 113

114115116

117

118

119

120121

122

123124125

126127128

129

130

131 132133

134

135

136

137139138

140141

142 143

144145

146147

148

149

150 151

152153

154

155

156157

158

159

160

161

162163

164

165

166 167

68

168

169

170

Figure 7 A transport network for hazardous materials in main districts of Lanzhou

Table 6 Vehicle scheduling plan (for solution 14)

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

11 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

34 0715 0800 0840 0925 10061 1006 1051 1144 1229 13187 1318 1403 1646 1731 1820

52 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

32 0730 0815 0900 0945 10308 1030 1115 1301 1346 14318 1431 1516 1701 1746 1831

41 0715 0800 0854 0939 10274 1027 1112 1153 1238 13187 1318 1403 1646 1731 1820

6 8 0729 0814 1000 1045 11307 1130 1215 1457 1542 1631

150 RMBh eliminating cost of loading and unloading Timefor either loading or unloading is 075 h

C is the programming language to implement the algo-rithmThe running environment is PC i5-3470 320GHz 4GParameters setting in the program and CPU times of vehiclesare shown in Table 8 A total of 164 path selection plans areobtained When a certain plan is selected the correspondingtimetable for vehicle operation can be obtained by the vehiclescheduling program

A path plan is a choice based on game between trans-portation cost and risk by a decision-maker with a specific

risk preference For example in plan 1 which has the lowesttransportation cost but the highest transportation risk thetransportation time is 439658 RMB while the transporta-tion risk is 32262 8 vehicles are needed to complete jobsAs another example in plan 164 which has the lowesttransportation risk but the highest transportation cost thetransportation time is 460983 RMB while the transporta-tion risk is 25586 also 8 vehicles are needed to complete jobsThe traveling path and transport timetable for all vehicles inplan 164 are shown in Tables 9 and 10

Discrete Dynamics in Nature and Society 11

Table 7 Nodes representing 16 gas stations as well as their needs of gasoline and the time window of distribution

Number Destination Required volumem3 Time window1 2 15 [1200 1600]2 12 26 [830 1330]3 16 12 [1030 1600]4 25 16 [800 1200]5 43 16 [800 1600]6 44 29 [1000 1600]7 47 32 [900 1500]8 52 25 [900 1200]9 62 24 [800 1000]10 63 10 [800 1500]11 94 18 [1400 1800]12 138 25 [800 1500]13 146 40 [1200 1800]14 150 23 [1000 1600]15 167 9 [1200 1800]16 170 25 [800 1600]

Table 8 Transport parameters setting in the program and CPU times

Stage Parameters setting CPU timessPopulation size Iteration

Path selectionStage 1 the pulse algorithm is used to obtain the Pareto solutions of all pathsfrom the distribution center to each destination mdash mdash 17

Stage 2 get all paths selection solution by NSGA-II algorithm 500 100 32Vehicle scheduling

The vehicle scheduling plan under the current path selection plan is acquired byUMDA 200 100 9

6 Conclusion

Distribution of hazardous materials in FCL is essentially amultistage combinatorial optimization problem In this studya decision approach for distribution of hazardous materialsin FCL with a single distribution center is formed which canprovide a decision support for transportation enterprises andadministrative authorities

In this study mathematical models for path planningand vehicle scheduling problems are established respectivelywith corresponding problem-solving algorithms designedIn order to solve the path planning model a two-stagealgorithm is designed Based on screening of paths in thefirst stage the computational complexity of the second stageinvolved in the global path optimization is significantlyreduced which improves calculation efficiency In obtainingthe vehicle scheduling plan the operating process of atransporting vehicle is converted to single objective VRPTWThen the problem-solvingmethod forVRPTWon the basis ofestimation of distribution is adopted to obtain the timetablefor all vehicles

The proposed approach was applied to a hypotheticalnumerical example as well as an application to a real casein distribution of gasoline of Lanzhou ChinaThe numerical

example showed that for small-scale problems the two-stagealgorithm can obtain Pareto-optimal front faster in the sameiterationHowever for large-scale problems due to the lengthof coding it is difficult to obtain Pareto-optimal path set in alimited iterative step in the second stage the problem size ofVRPTW is only related to the number of destination nodesnot to the scale of the transportation network Therefore thevehicle scheduling method has high efficiency whether it issmall-scale or large-scale transportation network

Our purpose is to design a decision support systemfor path planning and vehicle scheduling of distribution ofhazardous materials in FCL Further research has to be doneat least in two directions distribution of hazardous materialsin FCL with multiple distribution centers and transportationwith different volumes vehicle will be focused on

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (nos 61064012 61364026

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 11

Table 7 Nodes representing 16 gas stations as well as their needs of gasoline and the time window of distribution

Number Destination Required volumem3 Time window1 2 15 [1200 1600]2 12 26 [830 1330]3 16 12 [1030 1600]4 25 16 [800 1200]5 43 16 [800 1600]6 44 29 [1000 1600]7 47 32 [900 1500]8 52 25 [900 1200]9 62 24 [800 1000]10 63 10 [800 1500]11 94 18 [1400 1800]12 138 25 [800 1500]13 146 40 [1200 1800]14 150 23 [1000 1600]15 167 9 [1200 1800]16 170 25 [800 1600]

Table 8 Transport parameters setting in the program and CPU times

Stage Parameters setting CPU timessPopulation size Iteration

Path selectionStage 1 the pulse algorithm is used to obtain the Pareto solutions of all pathsfrom the distribution center to each destination mdash mdash 17

Stage 2 get all paths selection solution by NSGA-II algorithm 500 100 32Vehicle scheduling

The vehicle scheduling plan under the current path selection plan is acquired byUMDA 200 100 9

6 Conclusion

Distribution of hazardous materials in FCL is essentially amultistage combinatorial optimization problem In this studya decision approach for distribution of hazardous materialsin FCL with a single distribution center is formed which canprovide a decision support for transportation enterprises andadministrative authorities

In this study mathematical models for path planningand vehicle scheduling problems are established respectivelywith corresponding problem-solving algorithms designedIn order to solve the path planning model a two-stagealgorithm is designed Based on screening of paths in thefirst stage the computational complexity of the second stageinvolved in the global path optimization is significantlyreduced which improves calculation efficiency In obtainingthe vehicle scheduling plan the operating process of atransporting vehicle is converted to single objective VRPTWThen the problem-solvingmethod forVRPTWon the basis ofestimation of distribution is adopted to obtain the timetablefor all vehicles

The proposed approach was applied to a hypotheticalnumerical example as well as an application to a real casein distribution of gasoline of Lanzhou ChinaThe numerical

example showed that for small-scale problems the two-stagealgorithm can obtain Pareto-optimal front faster in the sameiterationHowever for large-scale problems due to the lengthof coding it is difficult to obtain Pareto-optimal path set in alimited iterative step in the second stage the problem size ofVRPTW is only related to the number of destination nodesnot to the scale of the transportation network Therefore thevehicle scheduling method has high efficiency whether it issmall-scale or large-scale transportation network

Our purpose is to design a decision support systemfor path planning and vehicle scheduling of distribution ofhazardous materials in FCL Further research has to be doneat least in two directions distribution of hazardous materialsin FCL with multiple distribution centers and transportationwith different volumes vehicle will be focused on

Conflicts of Interest

The authors declare no conflicts of interest

Acknowledgments

This research was jointly supported by the National NaturalScience Foundation of China (nos 61064012 61364026

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Discrete Dynamics in Nature and Society

Table9Path

selectionplanso

fplan164

Nod

ePath

Total

cost

RMB

Totalrisk

21rarr

2rarr1

1983

00056

121rarr

3rarr4rarr

5rarr6rarr

12rarr6rarr

5rarr4rarr

3rarr1

14817

01098

161rarr

3rarr4rarr

5rarr6rarr

12rarr17rarr16rarr11rarr5rarr

4rarr3rarr

17758

00592

251rarr

170rarr

168rarr

28rarr25rarr28rarr168rarr

170rarr

11516

70119

443

1rarr2rarr

39rarr41rarr43rarr41rarr39rarr2rarr

12210

001284

441rarr

2rarr39rarr41rarr43rarr44rarr46rarr47rarr45rarr13rarr7rarr

3rarr1

33825

02013

471rarr

2rarr45rarr47rarr45rarr13rarr7rarr

3rarr1

27525

01806

521rarr

2rarr45rarr47rarr46rarr49rarr50rarr51rarr52rarr51rarr50rarr49rarr48rarr29rarr13rarr7rarr

3rarr1

2913

302028

621rarr

3rarr7rarr

13rarr29rarr30rarr33rarr36rarr38rarr55rarr62rarr55rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

29283

01972

631rarr

2rarr39rarr53rarr63rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

18275

00787

941rarr

170rarr

168rarr

169rarr

94rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

39250

01870

138

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

142rarr

141rarr

140rarr

139rarr

138rarr

137rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

50850

02660

146

1rarr170rarr

168rarr

169rarr

145rarr

144rarr

143rarr

148rarr

147rarr

146rarr

147rarr

141rarr

133rarr

132rarr

131rarr

128rarr

113rarr

112rarr

102rarr

92rarr93rarr80rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

7875

003807

150

1rarr170rarr

168rarr

169rarr

145rarr

136rarr

121rarr

130rarr

129rarr

128rarr

127rarr

126rarr

150rarr

149rarr

123rarr

116rarr105rarr

86rarr81rarr76rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

5790

002914

167

1rarr2rarr

39rarr53rarr63rarr67rarr71rarr103rarr

167rarr

103rarr

71rarr72rarr68rarr64rarr65rarr66rarr38rarr36rarr33rarr30rarr29rarr13rarr7rarr

3rarr1

31800

01451

170

1rarr170rarr

12567

00054

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Discrete Dynamics in Nature and Society 13

Table 10 Vehicle scheduling plan of plan 164

Vehicle Node Vehicle scheduling schemeLS DD US RU AC

1

47 0758 0843 0900 0945 100012 1000 1045 1058 1143 1155138 1155 1240 1329 1414 145494 1454 1539 1614 1659 1732

2

47 0758 0843 0900 0945 1000150 1000 1045 1142 1227 13112 1311 1356 1358 1443 144594 1445 1530 1605 1650 1722

3

25 0715 0800 0821 0906 092147 0921 1006 1023 1108 1123167 1123 1208 1307 1352 144344 1443 1528 1549 1634 1655

4

43 0715 0800 0817 0902 091952 0919 1004 1031 1116 1140138 1140 1225 1313 1358 143844 1438 1523 1544 1630 1651

5

62 0715 0800 0825 0910 093563 0935 1020 1056 1141 12092 1209 1254 1256 1341 1342146 1342 1427 1514 1559 1643

6

62 0715 0800 0825 0910 093552 0935 1020 1047 1132 115644 1156 1241 1302 1347 1408146 1408 1453 1540 1625 1709

7

170 0715 0800 0802 0847 084925 0849 0934 0956 1041 105612 1056 1141 1153 1238 125143 1251 1336 1353 1438 145516 1455 1540 1554 1639 1652

8170 0715 0800 0802 0847 0849146 0849 0934 1200 1245 1329150 1329 1414 1510 1555 1640

and 51408288) Natural Science Foundation of Gansu(1610RJZA048 1610RJZA037) Youth Foundation of LanzhouJiaotong University (2015026) and LZJTU (201604) EP

References

[1] MO Ball B L Golden AAAssad and LD Bodin ldquoPlanningfor Truck Fleet Size in the Presence of a Common-CarrierOptionrdquo Decision Sciences vol 14 no 1 pp 103ndash120 1983

[2] A Imai E Nishimura and J Current ldquoA Lagrangian relaxation-based heuristic for the vehicle routing with full container loadrdquoEuropean Journal of Operational Research vol 176 no 1 pp 87ndash105 2007

[3] J J Bartholdi III and K R Gue ldquoReducing labor costs in an LTLcrossdocking terminalrdquo Operations Research vol 48 no 6 pp823ndash832 2000

[4] N Jozefowiez F Semet and E-G Talbi ldquoMulti-objective vehi-cle routing problemsrdquo European Journal of Operational Researchvol 189 no 2 pp 293ndash309 2008

[5] K G Zografos and K N Androutsopoulos ldquoA decision sup-port system for integrated hazardous materials routing andemergency response decisionsrdquo Transportation Research Part CEmerging Technologies vol 16 no 6 pp 684ndash703 2008

[6] R Pradhananga S Hanaoka and W Sattayaprasert ldquoOpti-misation model for hazardous material transport routing inThailandrdquo in Proceedings of the International Journal of LogisticsSystems amp Management vol 9 pp 22ndash42 2011

[7] R Pradhananga E Taniguchi T Yamada and A G QureshildquoBi-objective decision support system for routing and schedul-ing of hazardous materialsrdquo Socio-Economic Planning Sciencesvol 48 no 2 pp 135ndash148 2014

[8] A Raith and M Ehrgott ldquoA comparison of solution strategiesfor biobjective shortest path problemsrdquo Computers amp Opera-tions Research vol 36 no 4 pp 1299ndash1331 2009

[9] S Demeyer J Goedgebeur P Audenaert M Pickavet andP Demeester ldquoSpeeding up Martinsrsquo algorithm for multipleobjective shortest path problemsrdquo 4OR vol 11 no 4 pp 323ndash348 2013

[10] J C Clımaco and M M Pascoal ldquoMulticriteria path and treeproblems discussion on exact algorithms and applicationsrdquoInternational Transactions in Operational Research vol 19 no1-2 pp 63ndash98 2012

[11] D Duque L Lozano and A L Medaglia ldquoAn exact methodfor the biobjective shortest path problem for large-scale roadnetworksrdquo European Journal of Operational Research vol 242no 3 pp 788ndash797 2015

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andelitist multiobjective genetic algorithm NSGA-IIrdquo IEEE Trans-actions on Evolutionary Computation vol 6 no 2 pp 182ndash1972002

[13] H Chai R CHe C XMa andC J Dai ldquoA univariatemarginaldistribution algorithm hybridized with insertion heuristicsfor the vehicle routing problem with hard time windowsrdquoJournal of Transportation Systems Engineering and InformationTechnology vol 16 no 2 pp 176ndash182 2016

[14] C Chitra and P Subbaraj ldquoA nondominated sorting geneticalgorithm solution for shortest path routing problem in com-puter networksrdquo Expert Systems with Applications vol 39 no 1pp 1518ndash1525 2012

[15] R Batta and S S Chiu ldquoOptimal obnoxious paths on a networkTransportation of hazardous materialsrdquo Operations Researchvol 36 no 1 pp 84ndash92 1988

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 201

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of