a multiobjective route robust optimization model and algorithm...

Research ArticleA Multiobjective Route Robust Optimization Model andAlgorithm for Hazmat Transportation

Changxi Ma 1 Wei Hao 23 Ruichun He 1 and BahmanMoghimi4

1 School of Traffic and Transportation Lanzhou Jiaotong University Lanzhou Gansu 730070 China2Key Laboratory of Road Traffic Engineering of the Ministry of Education Changsha University of Science and TechnologyChangsha Hunan 410114 China

3Hunan Provincial Key Laboratory of Intelligent Processing of Big Data on TransportationChangsha University of Science and Technology Changsha Hunan 410114 China

4Department of Civil Engineering City College of New York NY 10031 USA

Correspondence should be addressed to Changxi Ma machangximaillzjtucn

Received 12 June 2018 Accepted 13 September 2018 Published 9 October 2018

Guest Editor Xiaobo Qu

Copyright copy 2018 Changxi Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Aiming at route optimization problem of hazardous materials transportation in uncertain environment this paper presents amultiobjective robust optimization model by taking robust control parameters into consideration The objective of the model isto minimize not only transportation risk but also transportation time and a robust counterpart of the model is introduced throughapplying the Bertsimas-Sim robust optimization theory Moreover a fuzzy C-means clustering-particle swarm optimization(FCMC-PSO) algorithm is designed and the FCMC algorithm is used to cluster the demand points In addition the PSOalgorithm with the adaptive archives grid is used to calculate the robust optimization route of hazmat transportation Finallythe computational results show the multiobjective route robust optimization model with 3 centers and 20 demand pointsrsquo samplestudied and FCMC-PSO algorithm for hazmat transportation can obtain different robustness Pareto solution sets As a result thisstudy will provide basic theory support for hazmat transportation safeguarding

1 Introduction

Hazardous materials (hazmat) refer to products withflammable poisonous and corrosive properties that cancause casualties damage to properties and environmentalpollution and require special protection in the processof transportation loading unloading and storage Inrecent years the demand for hazmat has increased itsfreight volume has increased year by year and the potentialtransportation risk is also expanding Practice has provedthat the optimization of the transportation route of hazmatcan effectively reduce the transportation risk and it hassignificant influence to ensure the safety of people alongroute and protect the surrounding ecological environment

Many scholars have studied the transportation routeoptimization problem for hazmat Rhyne (1994) conducteda statistical analysis of the hazmat transport accident usingthe diffusion formula [1] Gordon et al (2000) used the

British transport accident to prove the different risk levelin different places which provided a realistic basis for theevacuation of the population and the balance of risks duringthe accident [2] Power et al (2000) established a risk-costanalysis model and conducted a systematic analysis to findout the transport plan [3]Wu et al (2002) studied the vehiclerouting optimization problem of hazmat transportation withmultidistribution center and the clustering algorithm wasused to solve the complex model [4] Kara et al (2003)presented a riskmodel based on the primitive roads to reducethe error of risk estimation and then they proposed the min-imum risk route selection algorithm [5] Fabiano et al (2005)used the experimental data to calculate the probability andconsequence of the transport accident on the specific routeof the experimental area and got the optimal transportationscheme [6] Erkut and Ingolfsson (2005) studied the hazmattransport route by considering the shortest distance thesmallest population exposure and the smallest accident risk

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 2916391 12 pageshttpsdoiorg10115520182916391

2 Discrete Dynamics in Nature and Society

[7] Bubbico et al (2006) analyzed the transportation risk ofhazmat and obtained the safety route algorithm by using theexperimental data from Italy [8] Wei et al (2006) analyzedthe route selectionmodel under the influence of time-varyingconditions without considering the uncertain risk [9] Verma(2009) presented a biobjective optimization model consider-ing the cost and the risk and used the boundary algorithmto solve the model [10] Wang et al (2009) established ahazmat transportation path optimization model based ona geographic information system [11] Jassbi et al (2010)developed a multiobjective optimization framework for thehazmat transportation by minimizing the transport mileagethe number of affected residents social risks the probabilityof accidents and so on [12] Pradhananga et al (2014) createda dual-objective optimization model with time windows forthe hazmat transportation and designed a heuristic algorithmto solve this model [13] Suh-Wen Chiou (2016) proposeda dual-objective and dual-level signal control policy for thehazmat transportation [14] Assadipour et al (2016) proposeda toll-based dual-level programming method for the haz-mat transportation and designed hybrid speed constraintsfor multiobjective particle swarm optimization algorithm[15] Pamucar et al (2016) proposed a multiobjective routeplanning method for hazmat transportation and designeda solution algorithm combining neurofuzzy and artificialbee colony approach [16] Mohammadi et al (2017) studiedthe hazmat transportation under uncertain conditions usinga mixed integer nonlinear programming model and themetaheuristic algorithm was utilized to solve this model [17]Kheirkhah et al (2017) established a bilevel optimizationmodel for the hazmat transportation two heuristic algo-rithms were designed to solve the dual-level optimizationmodel and some randomly generated problems were usedto verify the applicability and validity of the model [18]Bula et al (2017) focused on the heterogeneous fleet vehiclerouting problem and designed a variant of the variableneighborhood search algorithm to solve the problem [19]Ma et al (2013) studied the route optimization models andalgorithms for hazardous materials transportation underdifferent environments [20] Ma et al (2018) proposed aroad screening algorithm and created a distribution routemultiobjective robust optimization for hazardous materials[21] Obviously although the route optimization of hazmattransportation has made many achievements the robustnessof solution is rarely considered in this field

Ben-Tal and Neimirovski (1998) proposed robust opti-mization theory based on ellipsoid uncertainty set [22] Bert-simas and Sim (2003) further put forward adjustable robust-ness robust optimal theory [23] Based on the adjustablerobust optimization theory this paper will propose a routerobust optimization model for hazmat transportation withmultiple distribution centers and the author also designed akind of improved PSO algorithm

The rest of this paper is structured as follows Section 2introduces the hazmat transportation route problem andestablishes a multiobjective route robust optimization modelof hazmat transportation Section 3 presents the FCMC-PSOalgorithm Section 4 is the case study At the end of the paperthe conclusion is proposed

2 Multiobjective Route Robust OptimizationModel of Hazmat Transportation

21 Problem Definition Hazmat transportation route robustoptimization for multidistribution center is defined as fol-lows there are several hazmat distribution centers and eachdistribution center owns enough hazmat transport vehiclesmeanwhile multiple need points exist which should beassigned to the relevant hazmat distribution center Vehi-cles from distribution center will service the correspondingdemand points Each vehicle can service several customerdemand points while each customer demand point only canbe serviced by one vehicle After completing the transportmission the vehicles must return to the distribution center[24]

Uncertainty of hazmat transport refers to the uncertaintyof the transportation time and transportation risk whichmay be caused by the traffic accident weather and trafficdensity of the road Compared to ordinary goods transporthazmat transport is more complicated and it needs moresecurity demands Therefore it is needed to set the goalof minimizing the total hazmat transportation risk In theprocess of hazmat transportation transport time reductionis also necessary As a consequence this paper will targetminimizing the total transportation time In conclusion thescientific transportation routes should be found to guaranteethe hazmat transported safely and quickly

22 Model221 Assumption There are a few assumptions in this study

(1) Multiple hazmat distribution centers are existent(2) The supply of hazmat distribution centers is adequate(3) Vehicle loading capacity is provided and the demand

of each customer is specified(4)Multiple vehicles of the distribution center can service

the customer(5) The transportation risk and transportation time are

identified among the customer demand points but they areuncertain number as interval number

222 Symbol Definition 1198780 set of the Hazmat distributioncenters where 1198780 = 119894 | 119894 = 1 2 119898 shows that thenumber distribution center is 119898 and the sequence numberof nodes set is 1 1198981198781 set of customer demand points where 1198781 = 119894 | 119894 =1 119899 shows that the number of customer demand pointsis 119899 and the sequence number of nodes set is 1 119899119878 all nodes set in the transportation network where 119878 =1198780 cup 1198781119881119889 available transportation vehicle set in the hazmatdistribution center where 119881 = 119896 | 119896 = 1 2 119870 119889 isin 1198780119864 road section set among nodes119902119894 demand of customer demand point 119894119871119889119896 maximum load of transport vehicle k from hazmatdistribution center d119903119894119895 variable transport risk from customer demand pointsi and j where 119903119894119895 isin [119903119894119895 119903119894119895 + 1006704119903119894119895](1006704119903119894119895 ge 0)119903119894119895 transportation risk nominal value from customerdemand points i and j

Discrete Dynamics in Nature and Society 3

1006704119903119894119895 deviation of the variable transport risk to its nominalvalue from customer demand points i and j where 1006704119903119894119895 ge 0119905119894119895 travel time nominal value from customer demandpoints i and j1006704119905119894119895 deviation of variable travel time to its nominal valuefrom customer demand points 119894 and 119895 where 1006704119905119894119895 ge 0119894119895 variable transport risk from customer demand pointsi and j where 119894119895 isin [119905119894119895 119905119894119895 + 1006704119905119894119895](1006704119905119894119895 ge 0)119869119903119894 the set of columns which all uncertain data 119903119894119895belonging to the ith row of the variable riskmatrix are in here|119869119903119894 | le 119899Γ119903119894 parameter Γ119903119894 isin [0 |119869119903119894 |] to adjust robust risk of robustdiscrete optimization method and control the risk degree ofconservatism where decimal is permitted

lfloorΓ119903119894 rfloor maximum integer less than Γ119903119894 120595ri the set of column subscripts j of uncertain data 119903119894119895 of

line 119894 in the variable risk matrix 119903119894119895119869119894119905 the set of columns with all uncertain data 119894119895 belongingto the ith row of the variable time matrix where |119869119905119894 | lenΓti parameter Γ119905119894 isin [0 |119869119905119894 |] to adjust robust time of robustdiscrete optimization method and control the time degree ofconservatism where decimal is permittedlfloorΓ119905119894 rfloor the maximum integer less than Γ119905119894 120595t

i the set of column subscript j of uncertain data 119903119894119895 ofline i in variable time matrix 119894119895223 Multiobjective Route Robust Optimization Model

min 1198851= sum119889isin1198780

sum119896isin119881119889

sum119894isin119878

sum119895isin1198781

119903119894119895119909119889119894119895119896+ maxΨ119903119894cup119898119903|Ψ119903

119894sube119869119903119894|Ψ119903119894|=lfloorΓ119903119894rfloor119898119903isin119869119903

119894Ψ119903119894

sum119889isin1198780sum119896isin119881119889sum119894isin119878sum119895isinΨ119903119894 1006704119903119894119895119909119889119894119895119896 + sum119889isin1198780

sum119896isin119881119889

sum119894isin119878

sum119898119903isin119869119903119894Ψ119903119894

(Γ119894 minus lfloorΓ119894rfloor) 1006704119903119894119898119903119909119889119894119898119903119896minus sum119889isin1198780

sum119896isin119881119889

sum119894isin119878

11990311989401199091198891198940119896

(1)

min 1198852= sum119889isin1198780

sum119896isin119881119889

sum119894isin119878

sum119895isin119878

119905119894119895119909119889119894119895119896+ maxΨ119905119894cup119898119905|Ψ119905

119894sube119869119894|Ψ

119905

119894|=lfloorΓ119905119894rfloor119898119905isin119869119905

119894Ψ119905119894

sum119889isin1198780 sum119896isin119881119889sum119894isin119878 sum119895isinΨ1199051198941006704119905119894119895119909119889119894119895119896 + sum

119889isin1198780

sum119896isin119881119889

sum119894isin119878

sum119898119905isin119869119905119894Ψ119905119894

(Γ119894 minus lfloorΓ119894rfloor)1006704119905119894119898119905119909119889119894119898119905119896(2)

st sum119894isin119878

sum119895isin1198781

sum119889isin1198780

119909119889119894119895119896119902119895 le 119871119889119896 forall119896 isin 119881119889 (3)

sum119895isin1198781

sum119896isin119881119889

119909119889119894119895119896 le 10038161003816100381610038161198811198891003816100381610038161003816 forall119894 isin 1198780 forall119889 isin 1198780 (4)

max(119894119895)isin119864

119903119894119895119909119889119894119895119896 le 119903max forall119894 isin 119878 forall119895 isin 119878 forall119889 isin 1198780 forall119896 isin 119881119889 (5)

max(119894119895)isin119864

sum119894isin1198780cup1198781

sum119895isin1198780cup1198781

119903119894119895119909119889119894119895119896 minus sum119894isin1198781

sum119895isin1198780


sum119895isin1198781

119909119889119894119895119896 minus sum119895isin1198781

119909119889119895119894119896 = 0 forall119894 isin 1198780 forall119889 isin 1198780 forall119896 isin 119881119889 (7)

sum119894isin119878

sum119889isin1198780

sum119896isin119881119889

119909119889119894119895119896 = 1 forall119895 isin 1198781 (8)

sum119895isin119878

sum119889isin1198780

sum119896isin119881119889

119909119889119894119895119896 = 1 forall119894 isin 1198781 (9)

sum119895isin1198780

119909119889119894119895119896 = sum119895isin1198780


119909119889119894119895119896 = 1 0 forall119894 isin 119878 forall119895 isin 119878 forall119889 isin 1198780 forall119896 isin 119881119889 (11)


where the objective function (1) minimizes the total trans-portation risk of hazmat [25] and the objective function(2) minimizes the total transportation time of hazmat Con-straint (3) as the load constraint that means any vehicleof any distribution center should satisfy the correspondingload constraint namely cannot overload Constraint (4)means that vehicle number of the distribution center islimited and vehicles arranged to transport the hazmat shouldnot exceed the distribution center owning Constraint (5)expresses that the transportation risk of each section must beless than or equal to threshold rmax set by decision makersConstraint (6) expresses that the transportation risk of eachroute must be less than or equal to threshold Rmax setby decision makers Constraint (7) indicates every hazmatvehicle departing fromdistribution center should return backto the original distribution center after finishing the trans-portation task Constraint (8) and Constraint (9) guaranteethat each demand point is served once and by one vehicleform a distribution center Constraint (10) indicates hazmattransport vehicles cannot depart fromone distribution centerbut back to another one Constraint (11) defines the 0-1 integervariable 119909119889119894119895119896 if the route of Hazmat transport vehicle k fromdistribution center 119889 containing the segment from node i tonode j 119909119889119894119895119896 =1 else 119909119889119894119895119896 =0224 Robust Counterpart Model Each objective functionof the above multiobjective robust model corresponds toparameter Γ The purpose is to control the degree of con-servatism of the solution Objective functions (1) and (2)of the robust optimization model contain ldquomaxrdquo extremevalue problem Set feasible solution set Xvrp to satisfy allconstraints and robust discrete optimization criterion is usedto transform the multiobjective route robust optimizationmodel and a new robust counterpart of the model is asfollows [26 27]

Objective function is

R (119903) = Γ119903119897 +min( 1198992sum119898=1

119903119898119909119898 + 119897sum119898=1

(119903119898 minus 119903119897) 119909119898) (12)

T (119905) = Γ119897 +min( 1198992sum119898=1

119905119898119909119898 + 119897sum119898=1

(119898 minus 119897) 119909119898) (13)

Constraint condition is

119909 isin 119883V119903119901 (14)

Then the optimal objective function value can beobtained as Rlowast = min119897=1sdotsdotsdot1198992+1R(119897) and Tlowast = min119897=1sdotsdotsdot1198992+1T(119897)3 Algorithm

In this section we propose FCMC-PSO algorithm to solvethe multiobjective route optimization problem of hazmattransportation in uncertain environment The demand pointsis clustered by the fuzzy C means algorithm and the trans-portation route for each demand points is determined basedon the adaptive archives grid multiobjective particle swarmoptimization [28ndash30]

31 Fuzzy C-Means Clustering Suppose that n data sam-ples are 119883 = 1198831 1198832 sdot sdot sdot 119883119899 C(2leClen) is the numberof types into which the data samples are to be divided1198601 1198602 sdot sdot sdot 119860119888 indicating the corresponding C categoriesU is its similar classification matrix the clustering centers ofall categories are 1198811 1198812 sdot sdot sdot 119881119888 and 120583119896(119883119894) is the member-ship degree of sample 119883119894 to category Ak (abbreviated as 120583119894119896)Then the objective function can be expressed as follows

119869119887 (119880119881) = 119899sum119894=1

119888sum119896=1

(120583119894119896)119887 (119889119894119896)2 (15)

where 119889119894119896 = 119889(119883119894 minus 119881119896) = radicsum119898119895=1(119883119894119895 minus 119881119896119895)2 it is thesynthetic weighted value of the transport risk and time afternondimensionalization between the i-th sample Xi and thek-th category center point m is the characteristics numberof the sample b is the weighting parameter and the valuerange is 1 le b le infin The fuzzy C-means clustering method isto find an optimal classification so that the classification canproduce the smallest function value Jb It requires a sample forthe sum of the membership degree of each cluster is 1 whichis satisfied

119888sum119895=1

120583119895 (119883119894) = 1 119894 = 1 2 119899 (16)

Formulas (17) and (18) are used to calculate separately themembership degree 120583119894119896 of the sample Xi for the category Akand C clustering centers 119881119894

120583119894119896 = 1sum119888119895=1 (119889119894119896119889119895119896)2(119887minus1) (17)

Let 119868119896 = 119894 | 2 le 119862 lt 119899 119889119894119896 = 0 for all the i categories119894 isin 119868119896 120583119894119896 = 0119881119894119895 = sum

119899119896=1 (120583119894119896)119887119883119896119895sum119899119896=1 (120583119894119896)119887 (18)

Using formulas (17) and (18) to repeatedly modify thecluster center data membership and classification when thealgorithm is convergent in theory we can get themembershipdegree of the cluster center and each sample for each patternclass thus the division of the fuzzy clustering is completed

32 Multiobject PSO Algorithm Particle swarm algorithm isderived from the study of the predatory behavior of birdsIt is used to solve the problem of path optimization Eachparticle in the algorithm represents a potential solution andthe fitness value for each particle is determined by the fitnessfunction and the value of fitness determines the pros andcons of the particle The particle moves in the N-dimensionalsolution space and updates the individual position by theindividual extremum and the group extremum In the algo-rithm the velocity position and fitness value are used torepresent the characteristics of the particleThe velocity of theparticle determines the direction and distance of the particle


Individuals [9 2 1 4 3 6 7 5 8] rarr New 1 [9 2 2 4 1 5 9 5 8]Extremum [8 3 2 4 1 5 9 7 6] rarr New 2 [8 3 1 4 3 6 7 7 6]

Figure 1 Crossover operation

movement and the velocity is dynamically adjusted with themoving experience of its own and other particles Once theposition of the particle is updated the fitness value will becalculated and the individual extremum and the populationextremum are updated by the fitness values of the new parti-cles the individual extremum and the population extremumMultiobjective particle swarm optimization algorithm is amethod based on particle swarm optimization algorithm tosolve multiobjective problem At the same time the bestlocation of multiple populations exists in the population andthe optimal positions of multiple particles themselves are alsofound in the iterative process Therefore gbest and pbest alsoneed to adopt certain strategies to choose Aiming at therobust optimization model of hazmat transportation the keyelements ofmultiobjective particle swarm optimization are asfollows

(1) Individual Coding In this paper the method of particleencoding adopts integer encoding and each particle rep-resents the experienced demand point For example whenthe number of required points is 9 the individual codingis [9 2 1 4 3 6 7 5 8] indicating that the requirementpoint traversal starting from the distribution center followedby 9 2 1 4 3 6 7 5 8 and ultimately return to the distri-bution center

(2) Fitness Value In the hybrid particle swarm algorithmthe fitness value is the criterion of judging the quality of theparticle And the fitness function is to facilitate the searchand improve the performance of the algorithm In the paperthe fitness value of the particle is expressed by the objectivefunction of the built model

(3) Crossover Operation Crossover operation is the processof replacing the partial structure of the parent individual andreorganizing the new individual The design of the crossoveroperation is related to the representation of the codingthe cross-operation design based on the coarranged codingmethod of the demand point and the distribution center[31 32] The method of integer crossing is adopted Set thetwo individuals of the parent as [9 2 1 4 3 6 7 5 8] and[8 3 2 4 1 5 9 7 6] Firstly two crossover positions areselected and then the individual is crossed the operationprocess can be seen in Figure 1

The new individuals need to be adjusted if there is aduplicate position and the adjustment method is to replacethe repeated demand points by using the absence of demandpoints in individuals For the new individual 1 there aremappings about 2 to 3 9 to 6 and 5 to 7 The specificadjustments process can be seen in Figure 2

The strategy of retaining outstanding individuals is usedfor the owned new individuals and the particles are updatedonly when the new particle fitness is better than the oldparticles

New individuals [9 2 2 4 1 5 9 5 8] rarr [9 2 3 4 1 5 6 7 8]New individuals [8 3 1 4 3 6 7 7 6] rarr [8 3 1 4 2 6 7 5 9]

Figure 2 Adjustment operation

Individuals [9 2 3 4 1 5 6 7 8] rarr [9 6 3 4 1 5 2 7 8]Individuals [8 3 1 4 2 6 7 5 9] rarr [8 3 1 4 6 2 7 5 9]

Figure 3 Mutation operation

(4) Mutation Operation The mutation of the particles isto make some changes in some genes of the particle themutation can increase the ability of searching particles andincrease the diversity of the populations to avoid falling intothe local optimal situation

The variation is also related to the way the particle isencoded Based on the lease point and the dispatch centerthere are many methods about the coarranged coding andthe variation In this paper the variation method adoptsthe individual internal exchange method For example foran individual [9 2 3 4 1 5 6 7 8] at first the mutatedpositions pos1 and pos2 are selected randomly and thenthe positions of two variants are swapped Assuming thatthe selected mutation positions are 2 and 6 the mutationoperation process can be seen in Figure 3

The strategy of retaining outstanding individuals is usedfor the owned new individuals and the particles are updatedonly when the new particle fitness is better than the oldparticles [33]

(5) Multiobject PSO Algorithm Based on Adaptive ArchivesGrid Multiobjective PSO based on adaptive archives grid isa particle swarm optimization algorithm proposed by Coelloand Lechuga to solve multiobjective problem [34] Its basicidea is to divide the target space into several hypercubesand to judge the number of noninferiority contained in eachhypercube to maintain the external files In each iterationif the file does not exceed the given size then a newnondominated solution will be added to the file If thefile has been filled the file is maintained according to thedensity of noninferior solution contained in the hypercubethe noninferior solution is removed from the high-densityhypercube and the noninferior solution with low density isadded to ensure the diversity of the populationThe algorithmsteps are as follows

Step 1 Create and initialize a group so that the ex archivesexternal file is empty

Step 2 Evaluate all particles and add the noninferior solutionto the external file

Step 3 Maintain external files according to the adaptive gridmethod

Step 4 Select gbest and pbest for each particle

Step 5 Updating the velocity and position of the particlesaccording to the speed formula and the position formula ofthe particle swarm


Table 1 Customer demands

Demand point 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20Demand (ton) 2 15 45 3 15 4 25 3 3 45 2 15 25 35 2 2 25 35 25 3

Table 2 The nominal transportation risk value of hazmat

119903119894119895 a b c 1 2 3 18 19 20a 0 39 71 59 38 80 52b 0 30 30 49 74 67 78c 0 77 57 37 49 34 641 39 30 77 0 32 30 35 60 492 71 30 57 32 0 67 33 65 553 59 49 37 30 67 0 39 39 71 18 38 74 49 35 33 39 0 55 3419 80 67 34 60 65 39 55 0 6720 52 78 64 49 55 71 34 67 0

Table 3 The nominal transportation time value of hazamt

119903119894119895 a b c 1 2 3 18 19 20a 0 92 103 43 107 97 44b 0 37 36 88 95 57 115c 0 41 35 101 40 74 641 92 37 41 0 75 102 64 101 802 103 36 35 75 0 62 108 84 953 43 88 101 102 62 0 103 77 85 18 107 95 40 64 108 103 0 52 4319 97 57 74 101 84 77 52 0 8620 44 115 64 80 95 85 43 86 0

Step 6 Make sure the particles exist in the search space

Step 7 If the termination condition is satisfied the outputresult algorithm is terminated if it is not satisfied Step 2 isexecuted and the execution is continued

4 Case Study

There are 3 distribution centers and 20 demand points themaximum load for each transport vehicle is 8 ton and eachdistribution center has adequate hazmat The distributioncenters are marked as a b and c and the hazmat demandpoints are marked as 1 2 20 The demand amount of eachdemand point is shown in Table 1 and transportation riskand time from each distribution center to demand points andbetween the demand points are respectively shown in Tables2 and 3The risk and time nominal values are given in Tables 2and 3 The transportation risk deviation 119903119898(0 le 119903119898 lt 05119903119898)and transportation time deviation 119898(0 le 119898 lt 05119903119898) areprovided

We use FCMC algorithm to calculate the demand pointsclustering results and the results can be seen in Table 4

Table 4 Demand points clustering result

Distribution center Demand pointsa 3914151620b 1268101112c 45713171819

Based on the cluster results we use the multiobjectivePSO to solve the robust optimization problem for eachdistribution center The parameters of the algorithm areset as follows population size is 100 maximum evolutiongeneration is 1000 inertia weight is 06 accelerated factor is17 crossover rate is 095 andmutation rate is 009TheParetosolution set with different robust control parameters can beobtained by calculation which are showed in Tables 5ndash13 andFigures 4ndash6

In Table 5 the Pareto optimal solutions sets are obtainedby the program running on the condition that uncertaintransportation risk and uncertain time are taken the nominalvalues It is known from Table 5 that when Γ = 0 the programfinds 6 Pareto solutions in the encoding sequence the first


Table 5 Pareto solution set of robust control parameters Γ=0 for distribution center a

Encoding Decoding Total risk Total time9-20-14-3-15-16 a-9-20-a-14-3-a-15-16 237 5369-16-14-3-15-20 a-9-16-a-14-3-a-15-20 261 44815-20-9-14-3-16 a-15-20-9-a-14-3-a-16 247 52014-3-16-9-20-15 a-14-3-a-16-9-20-a-15 253 46915-14-16-9-3-20 a-15-14-16-a-9-3-a-20 291 40414-3-15-9-16-20 a-14-3-a-15-9-16-a-20 285 425


Encoding Decoding Total risk Total time3-14-20-9-16-15 a-3-14-a-20-9-16-a-15 317817 637819-20-16-14-3-15 a-9-20-16-a-14-3-a-15 319893 6217499-20-14-16-3-15 a-9-20-a-14-16-a-3-15 377505 52001820-9-16-14-3-15 a-20-9-16-a-14-3-a-15 337903 5321419-3-15-14-16-20 a-9-3-a-15-14-16-a-20 385542 4638039-3-20-15-14-16 a-9-3-a-20-15-a-14-16 354151 5273069-20-14-3-15-16 a-9-20-a-14-3-a-15-16 30549 65757720-15-14-3-9-16 a-20-15-a-14-3-a-9-16 359107 525349

200

250

300

350

400

450

350 450 550 650 750 850Time

Risk

Γ=0Γ=10Γ=20

Figure 4 Pareto optimal solution distribution of robust controlparameters Γ=0 Γ=10 and Γ=20 for distribution center a

number a represents the distribution center While in thedecoding sequence every alphabet represents one vehiclethe figures behind alphabet show the customer demandpoints and the corresponding service order Such as inTable 5 the first decoding sequence indicates that 3 vehiclesare needed and each vehicle corresponds to a subroutethey are respectively a997888rarr9997888rarr20997888rarra a997888rarr14997888rarr3997888rarra anda997888rarr15997888rarr16997888rarra and it is clearly known that all transportvehicles from distribution center a after serving the allocatedcustomer demand points eventually return to distributioncenter a When the robustness control parameters Γ = 10and Γ = 20 Pareto optimal solutions sets obtained by theprogram running are given in Tables 6 and 7 respectivelyTables 8 9 and 10 present the optimal Pareto solutions sets fordistribution center b when the robustness control parametersare taken 0 10 and 20 Tables 11 12 and 13 present the

200

250

300

350

400

450

500

550

500 600 700 800 900 1000 1100Time

Γ=0Γ=10Γ=20

Risk

Figure 5 Pareto optimal solution distribution of robust controlparameters Γ=0 Γ=10 and Γ=20 for distribution center b

200

250

300

350

400

450

500

550

600

500 600 700 800 900 1000 1100 1200 1300 1400Time

Risk

Γ=0Γ=10Γ=20

Figure 6 Pareto optimal solution distribution of robust controlparameters Γ=0 Γ=10 and Γ=20 for distribution center c



Encoding Decoding Total risk Total time3-14-20-9-16-15 a-3-14-a-20-9-16-a-15 323453 7614239-3-15-20-14-16 a-9-3-a-15-20-a-14-16 381844 5600879-20-16-14-3-15 a-9-20-16-a-14-3-a-15 35798 6449-16-20-14-3-15 a-9-16-20-a-14-3-a-15 391378 5580439-20-14-16-3-15 a-9-20-a-14-16-a-3-15 415592 5422699-3-15-14-16-20 a-9-3-a-15-14-16-a-20 423629 4860549-20-15-14-3-16 a-9-20-15-a-14-3-a-16 37869 606483-14-9-20-16-15 a-3-14-a-9-20-16-a-15 316995 8667859-3-20-15-14-16 a-9-3-a-20-15-a-14-16 392239 5495573-14-16-9-20-15 a-3-14-a-16-9-20-a-15 336135 7509439-20-14-3-15-16 a-9-20-a-14-3-a-15-16 343577 679828

Table 8 Pareto solution set of robust control parameters Γ=0 for distribution center b

Encoding Decoding Total risk Total time12-10-11-1-2-6-8 b-12-10-11-b-1-2-6-b-8 289 58211-10-1-12-6-2-8 b-11-10-b-1-12-6-b-2-8 303 47112-6-10-11-1-2-8 b-12-6-b-10-11-b-1-2-8 302 53012-10-11-1-6-8-2 b-12-10-11-b-1-6-b-8-2 293 5731-12-10-2-6-8-11 b-1-12-10-b-2-6-b-8-11 296 5728-6-12-10-11-1-2 b-8-6-b-12-10-11-b-1-2 299 5492-1-6-12-10-11-8 b-2-1-6-b-12-10-11-b-8 284 6278-6-1-12-2-10-11 b-8-6-b-1-12-2-b-10-11 363 458


Encoding Decoding Total risk Total time2-6-8-12-1-11-10 b-2-6-b-8-12-1-b-11-10 415472 5521671-6-2-11-10-12-8 b-1-6-2-b-11-10-12-b-8 348014 7996952-8-6-1-12-11-10 b-2-8-b-6-1-12-b-11-10 399364 6312191-12-6-11-10-8-2 b-1-12-6-b-11-10-b-8-2 375264 6600622-12-1-6-8-11-10 b-2-12-1-b-6-8-b-11-10 48504 5335892-8-6-12-1-11-10 b-2-8-b-6-12-1-b-11-10 402203 5578811-6-2-12-10-8-11 b-1-6-2-b-12-10-b-8-11 346521 8724171-6-12-11-10-8-2 b-1-6-12-b-11-10-b-8-2 356381 7242742-6-1-11-10-12-8 b-2-6-1-b-11-10-12-b-8 381268 645953


Encoding Decoding Total risk Total time2-6-1-12-10-8-11 b-2-6-1-b-12-10-b-8-11 417863 7782971-6-2-12-10-8-11 b-1-6-2-b-12-10-b-8-11 352157 10282372-6-1-11-10-12-8 b-2-6-1-b-11-10-12-b-8 419356 6777321-6-2-11-10-12-8 b-1-6-2-b-11-10-12-b-8 35365 9555162-8-6-1-12-11-10 b-2-8-b-6-1-12-b-11-10 437452 6629972-8-6-12-1-11-10 b-2-8-b-6-12-1-b-11-10 440291 5896592-6-8-12-1-11-10 b-2-6-b-8-12-1-b-11-10 453559 5839461-6-12-11-10-8-2 b-1-6-12-b-11-10-b-8-2 362017 8800941-12-6-11-10-8-2 b-1-12-6-b-11-10-b-8-2 3809 8158832-12-1-6-8-10-11 b-2-12-1-b-6-8-b-10-11 532501 576115


Table 11 Pareto solution set of robust control parameters Γ=0 for distribution center c

Encoding Decoding Total risk Total time17-4-19-7-13-18-5 c-17-4-19-c-7-13-c-18-5 278 6787-5-18-17-4-19-13 c-7-5-18-c-17-4-19-c-13 294 56919-4-13-7-5-18-17 c-19-4-13-c-7-5-18-c-17 298 5667-4-13-17-19-5-18 c-7-4-13-c-17-19-5-c-18 382 5247-5-18-17-19-4-13 c-7-5-18-c-17-19-4-c-13 305 54517-4-13-19-5-7-18 c-17-4-13-c-19-5-7-c-18 361 53917-18-19-4-13-7-5 c-17-18-c-19-4-13-c-7-5 320 54418-5-19-17-4-13-7 c-18-5-19-c-17-4-13-c-7 359 54319-5-7-17-18-13-4 c-19-5-7-c-17-18-c-13-4 379 530


Encoding Decoding Total risk Total time18-17-13-4-19-7-5 c-18-17-c-13-4-19-c-7-5 433965 64085518-5-7-17-4-19-13 c-18-5-7-c-17-4-19-c-13 374262 7144954-19-13-7-5-18-17 c-4-19-13-c-7-5-18-c-17 328774 106864418-17-19-4-13-7-5 c-18-17-c-19-4-13-c-7-5 432616 64964518-17-13-4-19-5-7 c-18-17-c-13-4-19-c-5-7 442263 632415-18-7-17-4-19-13 c-5-18-7-c-17-4-19-c-13 361711 81337618-17-7-5-19-13-4 c-18-17-c-7-5-19-c-13-4 565935 62952218-5-13-17-4-19-7 c-18-5-13-c-17-4-19-c-7 369796 74180313-4-18-17-19-5-7 c-13-4-c-18-17-c-19-5-7 538139 63200618-17-13-19-4-7-5 c-18-17-c-13-19-4-c-7-5 415204 6611224-19-13-17-18-5-7 c-4-19-13-c-17-18-5-c-7 345113 93732318-17-13-19-4-5-7 c-18-17-c-13-19-4-c-5-7 421737 6526774-19-13-17-5-18-7 c-4-19-13-c-17-5-18-c-7 333451 9757384-19-13-17-18-7-5 c-4-19-13-c-17-18-c-7-5 351435 928318-5-7-19-4-17-13 c-18-5-7-c-19-4-17-c-13 388454 682357


Encoding Decoding Total risk Total time18-5-13-17-4-19-7 c-18-5-13-c-17-4-19-c-7 418386 7974913-4-19-17-18-7-5 c-13-4-19-c-17-18-c-7-5 565673 67746213-4-19-17-18-5-7 c-13-4-19-c-17-18-5-c-7 558373 68648513-4-19-18-17-7-5 c-13-4-19-c-18-17-c-7-5 588331 67418118-17-13-19-4-7-5 c-18-17-c-13-19-4-c-7-5 463794 7168094-19-13-17-5-18-7 c-4-19-13-c-17-5-18-c-7 339087 130155918-17-19-4-13-7-5 c-18-17-c-19-4-13-c-7-5 481205 7053324-19-13-17-18-5-7 c-4-19-13-c-17-18-5-c-7 350749 126314318-17-13-4-19-7-5 c-18-17-c-13-4-19-c-7-5 482554 69654318-17-13-19-4-5-7 c-18-17-c-13-19-4-c-5-7 470326 7083644-19-13-18-17-7-5 c-4-19-13-c-18-17-c-7-5 379729 1250844-19-13-17-18-7-5 c-4-19-13-c-17-18-c-7-5 357071 12541214-19-13-7-5-18-17 c-4-19-13-c-7-5-18-c-17 33441 13944644-17-19-18-5-7-13 c-4-17-19-c-18-5-7-c-13 405101 124559118-5-7-17-19-4-13 c-18-5-7-c-17-19-4-c-13 449149 73819218-17-13-4-19-5-7 c-18-17-c-13-4-19-c-5-7 490853 6880975-18-7-17-4-19-13 c-5-18-7-c-17-4-19-c-13 409798 89515518-5-7-17-4-19-13 c-18-5-7-c-17-4-19-c-13 422852 770182


Table 14 Pareto optimal solutions for distribution center a when Γ= 0 Γ = 10 and Γ = 20

Γ Distribution center aRisk optimal route Time optimal route

0a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

10a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

20a-9-20-16-a a-15-14-16-aa-14-3-a a-9-3-aa-15-a a-20-a

Table 15 Pareto optimal solutions for distribution center b when Γ= 0 Γ = 10 and Γ = 20

Γ Distribution center bRisk optimal route Time optimal route

0b-2-1-6-b b-8-6-bb-12-10-11-b b-1-12-2-b

b-8-b b-10-11-b

10b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b

20b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b

Table 16 Pareto optimal solutions for distribution center c when Γ= 0 Γ = 10 and Γ = 20

Γ Distribution center cRisk optimal route Time optimal route

0c-17-4-19-c c-7-4-13-cc-7-13-c c-17-19-5-cc-18-5-c c-18-c

10c-4-19-13-c c-13-4-cc-7-5-18-c c-7-5-19-cc-17-c c-18-17-c

20c-4-19-13-c c-13-4-19-cc-7-5-18-c c-7-5-cc-17-c c-18-17-c

optimal Pareto solutions sets for distribution center c whenthe robustness control parameters are taken 0 10 and 20The above tables are analyzed similar to Table 5 Figures 45 and 6 respectively show the Pareto optimal solutions fordistribution centers a b and c when the robustness controlparameters Γ = 0 Γ = 10 and Γ = 20

When the robustness control parameters Γ = 0 Γ = 10and Γ = 20 each objective optimal distribution route for allvehicles from distribution centers a b c is shown in Tables14 15 and 16 Aiming at distribution center a when therobustness control parameters Γ = 0 and Γ = 10 the total risk

Table 17 Performance comparison between FCMC-PSO algorithmand SPEA

Index FCMC-PSO SPEAΓ value 0 20 40 0 20 40Convergence Iterations 33 39 51 58 72 97Run time(s) 21 33 49 40 52 88

optimal routes have not changed when Γ = 20 the total riskoptimal route has changed relatively small however the totaltime optimal routes have not changed when the robustnesscontrol parameters Γ = 0 Γ = 10 and Γ = 20 which meansthat uncertainty data are taken nominal value and the totalrisk and time optimal routes always have some robustnessAiming at distribution center b when the robustness controlparameters Γ = 10 and Γ =20 the total risk optimal routeshave not changed when the robustness control parametersΓ = 0 Γ=10 and Γ =20 the total time optimal vehicle routesdo not have any change which shows the total risk optimaldistribution route is relatively stable Aiming at distributioncenter c when the robustness control parameters Γ = 10 andΓ = 20 the total risk optimal vehicle routes do not have anychange but the total time optimal vehicle routes are changedwhen the robustness control parameters Γ = 0 Γ = 10 andΓ = 20 which shows when Γ = 10 the total risk optimaldistribution route is relatively stable to a certain extentwhich can be selected as distribution route and the total timeoptimal vehicle route has relatively weak robustness and ifmore stable distribution route is needed it will be expected toincrease the robustness control parameter to find the strongrobustness distribution route

For solution robustness as the robustness control param-eters get bigger the corresponding Pareto solution robustnessshould be enhanced in theory After a large number ofanalyses for the actual situation and basic data we candetermine the robustness control parameter value and obtainthe corresponding candidate route set

The strength Pareto genetic algorithm (SPEA) is usedto test the efficiency of the FCMC-PSO algorithm Theparameters of the SPEA are set as follows population size is50 maximum evolution generation is 300 crossover rate is08 andmutation rate is 005The solution set can be obtainedby calculation The results are shown in Table 17 Comparedwith SPEA the convergence iterations and operation timeof FCMC-PSO algorithm are reduced The results show thatthe FCMC-PSO algorithm designed in this paper not onlycan obtain a more satisfactory solution but also has fasterconvergence speed compared with the SPEA

5 Conclusion

Hazmat transportation route optimization is an importantlink to ensure transportation safety of hazmat In thispaper we take the hazmat transportation route problemwith multidistribution center as the research object con-sidering the transportation risk and transportation timeIn addition an adjustable robustness transportation routemultiobjective robust optimization model is established in


the end Speaking of the solution the FCMC-PSO algorithmis designed in this researchThedemand points were assignedthrough FCMC algorithm in which transportation time andtransportation risk are considered The multiobjective routerobust optimization model is solved by multiobject PSOalgorithm based on adaptive archives grid In the end theexample shows that the robust optimization model andFCMC-PSO algorithm can obtain different robustness Paretosolution sets The robust optimization transportation routesof hazmat will provide basic theory support for safeguardingthe transportation safety of hazmat

In this paper we only consider the two uncertain-ties including transportation risk and transportation timeHowever there may be some other uncertainties such ascustomer demand and service time window in the realworld According to this situation we need to establishthe corresponding robust model for future study Althoughmost of the hazmats are transported by road transportationthe hazmat transportation risks induced by other modescannot be ignored The optimization research of multimodaltransport modes for the hazmat needs to be further studied

Data Availability

The data used to support the findings of this study areincluded within the article

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

Acknowledgments

This research was funded by National Natural Science Foun-dation of China (no 71861023 andno 51808057) the Programof Humanities and Social Science of Education Ministryof China (no 18YJC630118) Lanzhou Jiaotong University(no 201804) and Hunan Key Laboratory of Smart Road-way and Cooperative Vehicle-Infrastructure Systems (no2017TP1016)

References

[1] W R RhyneHazmat Transportation RiskAnalysis QuantitativeApproaches for Truck and Train Wiley New York NY USA1994

[2] GWalker JMooney andD Pratts ldquoThe people and the hazardthe spatial context of major accident hazard management inBritainrdquo Applied Geography vol 20 no 2 pp 119ndash135 2000

[3] M Power and L S McCarty ldquoRisk-cost trade-offs in envi-ronmental risk management decision-makingrdquo EnvironmentalScience amp Policy vol 3 no 1 pp 31ndash38 2000

[4] T Wu C Low and J Bai ldquoHeuristic solutions to multi-depotlocation-routing problemsrdquo Computers amp Operations Researchvol 29 no 10 pp 1393ndash1415 2002

[5] B Y Kara E Erkut and V Verter ldquoAccurate calculationof hazardous materials transport risksrdquo Operations ResearchLetters vol 31 no 4 pp 285ndash292 2003

[6] B Fabiano F Curro A P Reverberi and R Pastorino ldquoDan-gerous good transportation by road From risk analysis toemergency planningrdquo Journal of Loss Prevention in the ProcessIndustries vol 18 no 4ndash6 pp 403ndash413 2005

[7] E Erkut and A Ingolfsson ldquoTransport risk models for haz-ardous materials revisitedrdquoOperations Research Letters vol 33no 1 pp 81ndash89 2005

[8] R Bubbico G Maschio B Mazzarotta M F Milazzo andE Parisi ldquoRisk management of road and rail transport ofhazardous materials in Sicilyrdquo Journal of Loss Prevention in theProcess Industries vol 19 no 1 pp 32ndash38 2006

[9] S R Lin L P Cai and D Y Lin ldquoEffects of electroacupunctureof ldquoZusanlirdquo (ST 36) on gastric mucosal blood flow NO and ETcontents in gastric mucosal injury ratsrdquo Acupuncture Researchvol 31 no 2 pp 110ndash112 2006

[10] M Verma ldquoA cost and expected consequence approach toplanning and managing railroad transportation of hazardousmaterialsrdquo Transportation Research Part D Transport and Envi-ronment vol 14 no 5 pp 300ndash308 2009

[11] Y P Wang W C Sun and S W Li ldquoRoute optimization modelfor urban hazardousmaterial transportation based onArc GISrdquoJournal of Jilin University vol 39 no 1 pp 45ndash49 2009

[12] J Jassbi andPMakvandi ldquoRoute selection based on softMODMFramework in transportation of hazmatrdquoApplied MathematicalSciences vol 63 no 4 pp 3121ndash3132 2010

[13] 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

[14] S-W Chiou ldquoA bi-objective bi-level signal control policy fortransport of hazardous materials in urban road networksrdquoTransportation Research Part D Transport and Environmentvol 42 pp 16ndash44 2016

[15] G Assadipour G Y Ke and M Verma ldquoA toll-based bi-level programming approach to managing hazardous materialsshipments over an intermodal transportation networkrdquo Trans-portation Research Part D Transport and Environment vol 47pp 208ndash221 2016

[16] D Pamucar S Ljubojevic D Kostadinovic and B ETHorovicldquoCost and risk aggregation inmulti-objective route planning forhazardous materials transportationmdashA neuro-fuzzy and artifi-cial bee colony approachrdquo Expert Systems with Applications vol65 pp 1ndash15 2016

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

[18] A KheirkhahHNavidi andMM Bidgoli ldquoA bi-level networkinterdiction model for solving the hazmat routing problemrdquoInternational Journal of Production Research vol 54 no 2 pp459ndash471 2016

[19] G A Bula C Prodhon F A Gonzalez H M Afsar and NVelasco ldquoVariable neighborhood search to solve the vehiclerouting problem for hazardous materials transportationrdquo Jour-nal of Hazardous Materials vol 324 pp 472ndash480 2017

[20] CMa Y Li R He FWu B Qi and Q Ye ldquoRoute optimisationmodels and algorithms for hazardous materials transportationunder different environmentsrdquo International Journal of Bio-Inspired Computation vol 5 no 4 pp 252ndash265 2013

[21] C Ma W Hao F Pan W Xiang and X Hu ldquoRoad screeningand distribution route multi-objective robust optimization for


hazardous materials based on neural network and geneticalgorithmrdquo PLoS ONE vol 13 no 6 pp 1ndash22 2018

[22] A Ben-Tal and A Nemirovski ldquoRobust convex optimizationrdquoMathematics of Operations Research vol 23 no 4 pp 769ndash8051998

[23] D Bertsimas and M Sim ldquoRobust discrete optimization andnetwork flowsrdquoMathematical Programming vol 98 no 1ndash3 pp49ndash71 2003

[24] C MaW Hao R He et al ldquoDistribution path robust optimiza-tion of electric vehicle with multiple distribution centersrdquo PLoSONE vol 13 no 3 pp 1ndash16 2018

[25] R Yu XWang andMAbdel-Aty ldquoA hybrid latent class analysismodeling approach to analyze urban expressway crash riskrdquoAccident Analysis and Prevention vol 101 pp 37ndash43 2017

[26] C Ma C Ma Q Ye et al ldquoAn improved genetic algorithmfor the large-scale rural highway network layoutrdquoMathematicalProblems in Engineering vol 2014 Article ID 267851 6 pages2014

[27] Y Wang X Ma Z Li Y Liu M Xu and Y Wang ldquoProfitdistribution in collaborative multiple centers vehicle routingproblemrdquo Journal of Cleaner Production vol 144 pp 203ndash2192017

[28] C Ma and R He ldquoGreen wave traffic control system optimiza-tion based on adaptive genetic-artificial fish swarm algorithmrdquoNeural Computing and Applications vol 26 no 5 pp 1ndash11 2015

[29] Y Wang K Assogba Y Liu X Ma M Xu and Y WangldquoTwo-echelon location-routing optimization with time win-dows based on customer clusteringrdquo Expert Systems withApplications vol 104 pp 244ndash260 2018

[30] Z He W Guan and S Ma ldquoA traffic-condition-based routeguidance strategy for a single destination road networkrdquo Trans-portation Research Part C Emerging Technologies vol 32 pp89ndash102 2013

[31] C Ma R He W Zhang and X Ma ldquoPath optimization of taxicarpoolingrdquo PLoS ONE vol 13 no 8 pp 1ndash15 2018

[32] X Yang A Chen B Ning and T Tang ldquoA stochastic modelfor the integrated optimization on metro timetable and speedprofile with uncertain train massrdquo Transportation Research PartB Methodological vol 91 pp 424ndash445 2016

[33] C X Ma Y Z Li R C He et al ldquoBus-priority intersectionsignal control system based on wireless sensor network andimproved particle swarm optimization algorithmrdquo Sensor Let-ters vol 10 no 8 pp 1823ndash1829 2012

[34] C A Coello andM S Lechuga ldquoMPSO a proposal for multipleobjective particle swarm optimizationrdquo in Proceedings of IEEECongress on Evolutionary Computation (CEC) pp 1051ndash1056IEEE Computer Society New Jersey NJ USA May 2002

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[7] Bubbico et al (2006) analyzed the transportation risk ofhazmat and obtained the safety route algorithm by using theexperimental data from Italy [8] Wei et al (2006) analyzedthe route selectionmodel under the influence of time-varyingconditions without considering the uncertain risk [9] Verma(2009) presented a biobjective optimization model consider-ing the cost and the risk and used the boundary algorithmto solve the model [10] Wang et al (2009) established ahazmat transportation path optimization model based ona geographic information system [11] Jassbi et al (2010)developed a multiobjective optimization framework for thehazmat transportation by minimizing the transport mileagethe number of affected residents social risks the probabilityof accidents and so on [12] Pradhananga et al (2014) createda dual-objective optimization model with time windows forthe hazmat transportation and designed a heuristic algorithmto solve this model [13] Suh-Wen Chiou (2016) proposeda dual-objective and dual-level signal control policy for thehazmat transportation [14] Assadipour et al (2016) proposeda toll-based dual-level programming method for the haz-mat transportation and designed hybrid speed constraintsfor multiobjective particle swarm optimization algorithm[15] Pamucar et al (2016) proposed a multiobjective routeplanning method for hazmat transportation and designeda solution algorithm combining neurofuzzy and artificialbee colony approach [16] Mohammadi et al (2017) studiedthe hazmat transportation under uncertain conditions usinga mixed integer nonlinear programming model and themetaheuristic algorithm was utilized to solve this model [17]Kheirkhah et al (2017) established a bilevel optimizationmodel for the hazmat transportation two heuristic algo-rithms were designed to solve the dual-level optimizationmodel and some randomly generated problems were usedto verify the applicability and validity of the model [18]Bula et al (2017) focused on the heterogeneous fleet vehiclerouting problem and designed a variant of the variableneighborhood search algorithm to solve the problem [19]Ma et al (2013) studied the route optimization models andalgorithms for hazardous materials transportation underdifferent environments [20] Ma et al (2018) proposed aroad screening algorithm and created a distribution routemultiobjective robust optimization for hazardous materials[21] Obviously although the route optimization of hazmattransportation has made many achievements the robustnessof solution is rarely considered in this field

Ben-Tal and Neimirovski (1998) proposed robust opti-mization theory based on ellipsoid uncertainty set [22] Bert-simas and Sim (2003) further put forward adjustable robust-ness robust optimal theory [23] Based on the adjustablerobust optimization theory this paper will propose a routerobust optimization model for hazmat transportation withmultiple distribution centers and the author also designed akind of improved PSO algorithm

The rest of this paper is structured as follows Section 2introduces the hazmat transportation route problem andestablishes a multiobjective route robust optimization modelof hazmat transportation Section 3 presents the FCMC-PSOalgorithm Section 4 is the case study At the end of the paperthe conclusion is proposed

2 Multiobjective Route Robust OptimizationModel of Hazmat Transportation

21 Problem Definition Hazmat transportation route robustoptimization for multidistribution center is defined as fol-lows there are several hazmat distribution centers and eachdistribution center owns enough hazmat transport vehiclesmeanwhile multiple need points exist which should beassigned to the relevant hazmat distribution center Vehi-cles from distribution center will service the correspondingdemand points Each vehicle can service several customerdemand points while each customer demand point only canbe serviced by one vehicle After completing the transportmission the vehicles must return to the distribution center[24]

Uncertainty of hazmat transport refers to the uncertaintyof the transportation time and transportation risk whichmay be caused by the traffic accident weather and trafficdensity of the road Compared to ordinary goods transporthazmat transport is more complicated and it needs moresecurity demands Therefore it is needed to set the goalof minimizing the total hazmat transportation risk In theprocess of hazmat transportation transport time reductionis also necessary As a consequence this paper will targetminimizing the total transportation time In conclusion thescientific transportation routes should be found to guaranteethe hazmat transported safely and quickly

22 Model221 Assumption There are a few assumptions in this study

(1) Multiple hazmat distribution centers are existent(2) The supply of hazmat distribution centers is adequate(3) Vehicle loading capacity is provided and the demand

of each customer is specified(4)Multiple vehicles of the distribution center can service

the customer(5) The transportation risk and transportation time are

identified among the customer demand points but they areuncertain number as interval number

222 Symbol Definition 1198780 set of the Hazmat distributioncenters where 1198780 = 119894 | 119894 = 1 2 119898 shows that thenumber distribution center is 119898 and the sequence numberof nodes set is 1 1198981198781 set of customer demand points where 1198781 = 119894 | 119894 =1 119899 shows that the number of customer demand pointsis 119899 and the sequence number of nodes set is 1 119899119878 all nodes set in the transportation network where 119878 =1198780 cup 1198781119881119889 available transportation vehicle set in the hazmatdistribution center where 119881 = 119896 | 119896 = 1 2 119870 119889 isin 1198780119864 road section set among nodes119902119894 demand of customer demand point 119894119871119889119896 maximum load of transport vehicle k from hazmatdistribution center d119903119894119895 variable transport risk from customer demand pointsi and j where 119903119894119895 isin [119903119894119895 119903119894119895 + 1006704119903119894119895](1006704119903119894119895 ge 0)119903119894119895 transportation risk nominal value from customerdemand points i and j






min 1198851= sum119889isin1198780

sum119896isin119881119889

sum119894isin119878

sum119895isin1198781

119903119894119895119909119889119894119895119896+ maxΨ119903119894cup119898119903|Ψ119903


119894Ψ119903119894


sum119896isin119881119889

sum119894isin119878

sum119898119903isin119869119903119894Ψ119903119894


sum119896isin119881119889

sum119894isin119878

11990311989401199091198891198940119896

(1)

min 1198852= sum119889isin1198780

sum119896isin119881119889

sum119894isin119878

sum119895isin119878

119905119894119895119909119889119894119895119896+ maxΨ119905119894cup119898119905|Ψ119905

119894sube119869119894|Ψ

119905


119894Ψ119905119894


119889isin1198780

sum119896isin119881119889

sum119894isin119878

sum119898119905isin119869119905119894Ψ119905119894



sum119895isin1198781

sum119889isin1198780

119909119889119894119895119896119902119895 le 119871119889119896 forall119896 isin 119881119889 (3)

sum119895isin1198781

sum119896isin119881119889


max(119894119895)isin119864


max(119894119895)isin119864

sum119894isin1198780cup1198781

sum119895isin1198780cup1198781

119903119894119895119909119889119894119895119896 minus sum119894isin1198781

sum119895isin1198780


sum119895isin1198781

119909119889119894119895119896 minus sum119895isin1198781


sum119894isin119878

sum119889isin1198780

sum119896isin119881119889

119909119889119894119895119896 = 1 forall119895 isin 1198781 (8)

sum119895isin119878

sum119889isin1198780

sum119896isin119881119889

119909119889119894119895119896 = 1 forall119894 isin 1198781 (9)

sum119895isin1198780

119909119889119894119895119896 = sum119895isin1198780






R (119903) = Γ119903119897 +min( 1198992sum119898=1

119903119898119909119898 + 119897sum119898=1

(119903119898 minus 119903119897) 119909119898) (12)

T (119905) = Γ119897 +min( 1198992sum119898=1

119905119898119909119898 + 119897sum119898=1

(119898 minus 119897) 119909119898) (13)


119909 isin 119883V119903119901 (14)




119869119887 (119880119881) = 119899sum119894=1

119888sum119896=1

(120583119894119896)119887 (119889119894119896)2 (15)


119888sum119895=1

120583119895 (119883119894) = 1 119894 = 1 2 119899 (16)


120583119894119896 = 1sum119888119895=1 (119889119894119896119889119895119896)2(119887minus1) (17)


119899119896=1 (120583119894119896)119887119883119896119895sum119899119896=1 (120583119894119896)119887 (18)





























119903119894119895 a b c 1 2 3 18 19 20a 0 39 71 59 38 80 52b 0 30 30 49 74 67 78c 0 77 57 37 49 34 641 39 30 77 0 32 30 35 60 492 71 30 57 32 0 67 33 65 553 59 49 37 30 67 0 39 39 71 18 38 74 49 35 33 39 0 55 3419 80 67 34 60 65 39 55 0 6720 52 78 64 49 55 71 34 67 0


119903119894119895 a b c 1 2 3 18 19 20a 0 92 103 43 107 97 44b 0 37 36 88 95 57 115c 0 41 35 101 40 74 641 92 37 41 0 75 102 64 101 802 103 36 35 75 0 62 108 84 953 43 88 101 102 62 0 103 77 85 18 107 95 40 64 108 103 0 52 4319 97 57 74 101 84 77 52 0 8620 44 115 64 80 95 85 43 86 0



4 Case Study












200

250

300

350

400

450

350 450 550 650 750 850Time

Risk

Γ=0Γ=10Γ=20



200

250

300

350

400

450

500

550

500 600 700 800 900 1000 1100Time

Γ=0Γ=10Γ=20

Risk


200

250

300

350

400

450

500

550

600

500 600 700 800 900 1000 1100 1200 1300 1400Time

Risk

Γ=0Γ=10Γ=20





















0a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

10a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

20a-9-20-16-a a-15-14-16-aa-14-3-a a-9-3-aa-15-a a-20-a



0b-2-1-6-b b-8-6-bb-12-10-11-b b-1-12-2-b

b-8-b b-10-11-b

10b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b

20b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b



0c-17-4-19-c c-7-4-13-cc-7-13-c c-17-19-5-cc-18-5-c c-18-c

10c-4-19-13-c c-13-4-cc-7-5-18-c c-7-5-19-cc-17-c c-18-17-c

20c-4-19-13-c c-13-4-19-cc-7-5-18-c c-7-5-cc-17-c c-18-17-c








5 Conclusion





Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018













min 1198851= sum119889isin1198780

sum119896isin119881119889

sum119894isin119878

sum119895isin1198781

119903119894119895119909119889119894119895119896+ maxΨ119903119894cup119898119903|Ψ119903


119894Ψ119903119894


sum119896isin119881119889

sum119894isin119878

sum119898119903isin119869119903119894Ψ119903119894


sum119896isin119881119889

sum119894isin119878

11990311989401199091198891198940119896

(1)

min 1198852= sum119889isin1198780

sum119896isin119881119889

sum119894isin119878

sum119895isin119878

119905119894119895119909119889119894119895119896+ maxΨ119905119894cup119898119905|Ψ119905

119894sube119869119894|Ψ

119905


119894Ψ119905119894


119889isin1198780

sum119896isin119881119889

sum119894isin119878

sum119898119905isin119869119905119894Ψ119905119894



sum119895isin1198781

sum119889isin1198780

119909119889119894119895119896119902119895 le 119871119889119896 forall119896 isin 119881119889 (3)

sum119895isin1198781

sum119896isin119881119889


max(119894119895)isin119864


max(119894119895)isin119864

sum119894isin1198780cup1198781

sum119895isin1198780cup1198781

119903119894119895119909119889119894119895119896 minus sum119894isin1198781

sum119895isin1198780


sum119895isin1198781

119909119889119894119895119896 minus sum119895isin1198781


sum119894isin119878

sum119889isin1198780

sum119896isin119881119889

119909119889119894119895119896 = 1 forall119895 isin 1198781 (8)

sum119895isin119878

sum119889isin1198780

sum119896isin119881119889

119909119889119894119895119896 = 1 forall119894 isin 1198781 (9)

sum119895isin1198780

119909119889119894119895119896 = sum119895isin1198780






R (119903) = Γ119903119897 +min( 1198992sum119898=1

119903119898119909119898 + 119897sum119898=1

(119903119898 minus 119903119897) 119909119898) (12)

T (119905) = Γ119897 +min( 1198992sum119898=1

119905119898119909119898 + 119897sum119898=1

(119898 minus 119897) 119909119898) (13)


119909 isin 119883V119903119901 (14)




119869119887 (119880119881) = 119899sum119894=1

119888sum119896=1

(120583119894119896)119887 (119889119894119896)2 (15)


119888sum119895=1

120583119895 (119883119894) = 1 119894 = 1 2 119899 (16)


120583119894119896 = 1sum119888119895=1 (119889119894119896119889119895119896)2(119887minus1) (17)


119899119896=1 (120583119894119896)119887119883119896119895sum119899119896=1 (120583119894119896)119887 (18)





























119903119894119895 a b c 1 2 3 18 19 20a 0 39 71 59 38 80 52b 0 30 30 49 74 67 78c 0 77 57 37 49 34 641 39 30 77 0 32 30 35 60 492 71 30 57 32 0 67 33 65 553 59 49 37 30 67 0 39 39 71 18 38 74 49 35 33 39 0 55 3419 80 67 34 60 65 39 55 0 6720 52 78 64 49 55 71 34 67 0


119903119894119895 a b c 1 2 3 18 19 20a 0 92 103 43 107 97 44b 0 37 36 88 95 57 115c 0 41 35 101 40 74 641 92 37 41 0 75 102 64 101 802 103 36 35 75 0 62 108 84 953 43 88 101 102 62 0 103 77 85 18 107 95 40 64 108 103 0 52 4319 97 57 74 101 84 77 52 0 8620 44 115 64 80 95 85 43 86 0



4 Case Study












200

250

300

350

400

450

350 450 550 650 750 850Time

Risk

Γ=0Γ=10Γ=20



200

250

300

350

400

450

500

550

500 600 700 800 900 1000 1100Time

Γ=0Γ=10Γ=20

Risk


200

250

300

350

400

450

500

550

600

500 600 700 800 900 1000 1100 1200 1300 1400Time

Risk

Γ=0Γ=10Γ=20





















0a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

10a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

20a-9-20-16-a a-15-14-16-aa-14-3-a a-9-3-aa-15-a a-20-a



0b-2-1-6-b b-8-6-bb-12-10-11-b b-1-12-2-b

b-8-b b-10-11-b

10b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b

20b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b



0c-17-4-19-c c-7-4-13-cc-7-13-c c-17-19-5-cc-18-5-c c-18-c

10c-4-19-13-c c-13-4-cc-7-5-18-c c-7-5-19-cc-17-c c-18-17-c

20c-4-19-13-c c-13-4-19-cc-7-5-18-c c-7-5-cc-17-c c-18-17-c








5 Conclusion





Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018











R (119903) = Γ119903119897 +min( 1198992sum119898=1

119903119898119909119898 + 119897sum119898=1

(119903119898 minus 119903119897) 119909119898) (12)

T (119905) = Γ119897 +min( 1198992sum119898=1

119905119898119909119898 + 119897sum119898=1

(119898 minus 119897) 119909119898) (13)


119909 isin 119883V119903119901 (14)




119869119887 (119880119881) = 119899sum119894=1

119888sum119896=1

(120583119894119896)119887 (119889119894119896)2 (15)


119888sum119895=1

120583119895 (119883119894) = 1 119894 = 1 2 119899 (16)


120583119894119896 = 1sum119888119895=1 (119889119894119896119889119895119896)2(119887minus1) (17)


119899119896=1 (120583119894119896)119887119883119896119895sum119899119896=1 (120583119894119896)119887 (18)





























119903119894119895 a b c 1 2 3 18 19 20a 0 39 71 59 38 80 52b 0 30 30 49 74 67 78c 0 77 57 37 49 34 641 39 30 77 0 32 30 35 60 492 71 30 57 32 0 67 33 65 553 59 49 37 30 67 0 39 39 71 18 38 74 49 35 33 39 0 55 3419 80 67 34 60 65 39 55 0 6720 52 78 64 49 55 71 34 67 0


119903119894119895 a b c 1 2 3 18 19 20a 0 92 103 43 107 97 44b 0 37 36 88 95 57 115c 0 41 35 101 40 74 641 92 37 41 0 75 102 64 101 802 103 36 35 75 0 62 108 84 953 43 88 101 102 62 0 103 77 85 18 107 95 40 64 108 103 0 52 4319 97 57 74 101 84 77 52 0 8620 44 115 64 80 95 85 43 86 0



4 Case Study












200

250

300

350

400

450

350 450 550 650 750 850Time

Risk

Γ=0Γ=10Γ=20



200

250

300

350

400

450

500

550

500 600 700 800 900 1000 1100Time

Γ=0Γ=10Γ=20

Risk


200

250

300

350

400

450

500

550

600

500 600 700 800 900 1000 1100 1200 1300 1400Time

Risk

Γ=0Γ=10Γ=20





















0a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

10a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

20a-9-20-16-a a-15-14-16-aa-14-3-a a-9-3-aa-15-a a-20-a



0b-2-1-6-b b-8-6-bb-12-10-11-b b-1-12-2-b

b-8-b b-10-11-b

10b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b

20b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b



0c-17-4-19-c c-7-4-13-cc-7-13-c c-17-19-5-cc-18-5-c c-18-c

10c-4-19-13-c c-13-4-cc-7-5-18-c c-7-5-19-cc-17-c c-18-17-c

20c-4-19-13-c c-13-4-19-cc-7-5-18-c c-7-5-cc-17-c c-18-17-c








5 Conclusion





Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018


































119903119894119895 a b c 1 2 3 18 19 20a 0 39 71 59 38 80 52b 0 30 30 49 74 67 78c 0 77 57 37 49 34 641 39 30 77 0 32 30 35 60 492 71 30 57 32 0 67 33 65 553 59 49 37 30 67 0 39 39 71 18 38 74 49 35 33 39 0 55 3419 80 67 34 60 65 39 55 0 6720 52 78 64 49 55 71 34 67 0


119903119894119895 a b c 1 2 3 18 19 20a 0 92 103 43 107 97 44b 0 37 36 88 95 57 115c 0 41 35 101 40 74 641 92 37 41 0 75 102 64 101 802 103 36 35 75 0 62 108 84 953 43 88 101 102 62 0 103 77 85 18 107 95 40 64 108 103 0 52 4319 97 57 74 101 84 77 52 0 8620 44 115 64 80 95 85 43 86 0



4 Case Study












200

250

300

350

400

450

350 450 550 650 750 850Time

Risk

Γ=0Γ=10Γ=20



200

250

300

350

400

450

500

550

500 600 700 800 900 1000 1100Time

Γ=0Γ=10Γ=20

Risk


200

250

300

350

400

450

500

550

600

500 600 700 800 900 1000 1100 1200 1300 1400Time

Risk

Γ=0Γ=10Γ=20





















0a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

10a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

20a-9-20-16-a a-15-14-16-aa-14-3-a a-9-3-aa-15-a a-20-a



0b-2-1-6-b b-8-6-bb-12-10-11-b b-1-12-2-b

b-8-b b-10-11-b

10b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b

20b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b



0c-17-4-19-c c-7-4-13-cc-7-13-c c-17-19-5-cc-18-5-c c-18-c

10c-4-19-13-c c-13-4-cc-7-5-18-c c-7-5-19-cc-17-c c-18-17-c

20c-4-19-13-c c-13-4-19-cc-7-5-18-c c-7-5-cc-17-c c-18-17-c








5 Conclusion





Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018













200

250

300

350

400

450

350 450 550 650 750 850Time

Risk

Γ=0Γ=10Γ=20



200

250

300

350

400

450

500

550

500 600 700 800 900 1000 1100Time

Γ=0Γ=10Γ=20

Risk


200

250

300

350

400

450

500

550

600

500 600 700 800 900 1000 1100 1200 1300 1400Time

Risk

Γ=0Γ=10Γ=20





















0a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

10a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

20a-9-20-16-a a-15-14-16-aa-14-3-a a-9-3-aa-15-a a-20-a



0b-2-1-6-b b-8-6-bb-12-10-11-b b-1-12-2-b

b-8-b b-10-11-b

10b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b

20b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b



0c-17-4-19-c c-7-4-13-cc-7-13-c c-17-19-5-cc-18-5-c c-18-c

10c-4-19-13-c c-13-4-cc-7-5-18-c c-7-5-19-cc-17-c c-18-17-c

20c-4-19-13-c c-13-4-19-cc-7-5-18-c c-7-5-cc-17-c c-18-17-c








5 Conclusion





Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018



























0a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

10a-9-20-a a-15-14-16-aa-14-3-a a-9-3-aa-15-16-a a-20-a

20a-9-20-16-a a-15-14-16-aa-14-3-a a-9-3-aa-15-a a-20-a



0b-2-1-6-b b-8-6-bb-12-10-11-b b-1-12-2-b

b-8-b b-10-11-b

10b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b

20b-1-6-2-b b-8-6-bb-12-10-b b-2-12-1-bb-8-11-b b-10-11-b



0c-17-4-19-c c-7-4-13-cc-7-13-c c-17-19-5-cc-18-5-c c-18-c

10c-4-19-13-c c-13-4-cc-7-5-18-c c-7-5-19-cc-17-c c-18-17-c

20c-4-19-13-c c-13-4-19-cc-7-5-18-c c-7-5-cc-17-c c-18-17-c








5 Conclusion





Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018











Data Availability




Acknowledgments


References












































Journal of












Journal of







Volume 2018






Volume 2018






























Journal of












Journal of







Volume 2018






Volume 2018








a multiobjective route robust optimization model and algorithm...

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