research article decision of multimodal transportation ...multimodal transportation is a kind of...

Research ArticleDecision of Multimodal Transportation Scheme Based onSwarm Intelligence

Kai Lei1 Xiaoning Zhu1 Jianfei Hou2 and Wencheng Huang1

1 School of Traffic amp Transportation Beijing Jiaotong University Beijing 100044 China2 China Academy of Transportation Science Beijing 100029 China

Correspondence should be addressed to Xiaoning Zhu leikaibjtueducn

Received 10 January 2014 Revised 20 March 2014 Accepted 20 March 2014 Published 24 April 2014

Academic Editor Rui Mu

Copyright copy 2014 Kai Lei et al This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper some basic concepts of multimodal transportation and swarm intelligence were described and reviewed and analyzedrelated literatures of multimodal transportation scheme decision and swarm intelligence methods application areas Then thispaper established a multimodal transportation scheme decision optimization mathematical model based on transportation coststransportation time and transportation risks explained relevant parameters and the constraints of themodel in detail and used theweight coefficient to transform themultiobjective optimization problems into a single objective optimization transportation schemedecision problem Then this paper is proposed by combining particle swarm optimization algorithm and ant colony algorithm(PSACO) to solve the combinatorial optimization problem of multimodal transportation scheme decision for the first time thisalgorithm effectively combines the advantages of particle swarm optimization algorithm and ant colony algorithm The solutionshows that the PSACO algorithm has two algorithmsrsquo advantages and makes up their own problems PSACO algorithm is betterthan ant colony algorithm in time efficiency and its accuracy is better than that of the particle swarm optimization algorithm whichis proved to be an effective heuristic algorithm to solve the problem about multimodal transportation scheme decision and it canprovide economical reasonable and safe transportation plan reference for the transportation decision makers

1 Introduction

11 Basic Concepts

(1) Swarm Intelligence Swarm intelligence refers to the overallintelligent behavior emerging from interaction of manysimple acts of individual the swarm intelligence can producea number of different individual collective behaviors Swarmintelligence is the creative application and development ofthe collective behaviors and forms a more advanced wholewisdom Swarm intelligence is a hot research topic in artificialintelligence researches in recent years with no centralizedcontrol does not provide a global model and provides acomplex distributed solution

(2) Multimodal Transportation Multimodal transportation isdefined as completing the transportation process by usingat least two transportation tools to connect and transport

together it is a kind of transportation organization formwhich uses optimum efficiency as the goal There are no newtransportation channels and tools in the multimodal trans-portation process it is a combination ofmodern organizationmeans and single transportation mode It has important andrealistic meaning for saving transportation costs and trans-portation time and improving transportation service qualityby researching the decision of multimodal transportationscheme in the process of transportation

12 The Current Analysis Situation of the Domestic andForeign Research

(1) Swarm Intelligence In recent years the scholars at homeand abroad carried out a lot of research works about swarmintelligence the swarm intelligence theory in transportationrelated fields has been applied extensively Chen and Ma

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 932832 10 pageshttpdxdoiorg1011552014932832

2 Mathematical Problems in Engineering

[1] gave a new algorithm of glowworm swarm optimizationalgorithm which was based on swarm intelligence optimiza-tion principle and successfully applied it to the 0-1 knapsackproblem Li et al [2] proposed the Harvard structure andswarm intelligence optimization system based on the CTLSmodel which used swarm intelligence based on calculationand source of agent in agent technology and the simulationbased optimization theory and the feasibility and reliability ofthemodeling optimizationmethod are verified by simulationanalysis of CTLS half bridge scheduling problem Mi [3]proposed an optimization algorithm for execution efficiencyand convergence problems of integrated design process usinggenetic algorithm for task scheduling the genetic algorithmwas improved and combined with the particle swarm algo-rithm by considering the effect of coupling relationshipbetween tasks to the task execution results It is verifiedwith an example the results showed that the algorithm hasa fast convergence speed and stable result Li et al [4]established automated warehouse single pock horizontallyrotating storage shelf system mathematical model and usedthe ant colony optimization algorithm to solve the pathplanning problem based on the goods selectionThis methodis capable of rotating storage rack system for rapid selectionof goods and finds the optimal goods picking path in theglobal high quality solution and short calculation time Wuand Shi [5] established ant colony algorithm based on theimproved meeting algorithm and proposed ant colony algo-rithm ant touring quality meeting algorithm and a parallelalgorithm which is combined with the proposed strategythe experimental results show that this algorithm has bettereffectiveness Xiao et al [6] proposed the mathematicalmodel of concurrent tolerance optimization design and thenmapped into a special traveling salesman problem ordermultiple traveling salesman problem so as to reduce thedifficulty of problem solving they proposed a hybrid swarmintelligence algorithm for concurrent tolerance optimizationdesign problemThrough calculation examples the efficiencywould be hybrid swarm intelligence algorithm with geneticalgorithm and simulated annealing algorithm has bettersearch ability and higher Ma et al [7] minimized the energyconsumption and delay time improved the evolutionaryalgorithms for solving the traveling salesman problem basedon artificial neural network algorithm and established theswarm intelligence analysis model and used it to analyzethe lines planning for public travel The experimental resultsshowed that the improved hybrid intelligent computingmethod was simple and effective and helped to overcomethe blindness of algorithm selection to further expand theresearch direction of computational intelligence From theabove research literatures it is not difficult to see that theswarm intelligence has not only attracted much attention inrecent years but also has a lot of space in the field of trans-portation applications This paper was proposed by combin-ing particle swarm optimization algorithm and ant colonyalgorithm (PSACO) to solve the combinatorial optimizationproblem of multimodal transportation scheme decision forthe first time which can make up shortcomings of their ownat the same time

(2) Decision of Multimodal Transportation Scheme In recentyears scholars at home and abroad did a lot of researchworks on how to choose the proper transportation mode inthe multimodal transportation access Among them Lozanoand Storchirsquos [8] research was about the problem of shortestfeasible path in the multimodal transportation and used thesequential algorithm to solve the problem Calvete et al [9]studied theVRPwith soft timewindowvehicles path problemand used the goal programming method to solve the prob-lem Li and Zhao [10] used the minimum total cost as theoptimization objective which included transportation costtransfer cost and penalty cost the goods delivery time andtransportation capacity were used as constraint conditionsthey built mixed integer linear programming model anddesigned the integer coding genetic algorithm Wang et al[11] studied and established themultimodal transportation 0-1 integer programming model under the time and capacityconstraints designed the natural number coding geneticalgorithm based on the characteristics of the model andfinally tested validity of the model and the algorithm withexamples Feng andZhang [12] considered the environmentalpollution energy consumption and safety factors into mul-timodal transportation sustainable development and thenintroduced the concept of total society cost of multimodaltransportation used this minimum cost value as a wholeobject and built a multimodal transportation collaborativeoptimization model Tong and Nie [13] transformed inter-modal path optimization problem into a generalized shortestpath problem used time and cost as optimization objectiveestablished multimodal transportation mathematical modelfor the optimization of path changes in volume and solvedand verified actual problem by using ant colony algorithmZhang [14] deeply studied the whole container multimodaltransportation network expounded the key elements and theoperation mechanism of multimodal container transporta-tion network in the whole system and in accordance withthe existing problems put forward the basic countermeasuresrationalization Li and Zeng [15] improved path shortest timemodel under the multimodal transportation in the fourthparty logistics gave the transportation cost model based onsatisfactory time path under the condition of multimodaltransportation and improved genetic algorithm to solve theproblemWen [16] usedmultimodal container transportationoptimization as the research object and discussed the opti-mization method of container multimodal transportationroutes transportation and time constraints modes Li [17]discussed the development of transportation mode choice aswell as the main transportation mode selection method andthen put forward the concept of generalized transportationcost Wang [18] in order to achieve time delivery reducedthe cost according to the actual situation of the key problemof logistics distribution in the process of transportationand traffic network of multimodal transportation with timewindow ofminimum cost problem Zhang [19] analyzed Chi-nese railway container transportation features and operationprocess used the transportation path selection as the goal toimprove the efficiency of container transport and shorten thecontainer transport cycle and reduced transportation costsfor the purpose then he used the optimization theory and

Mathematical Problems in Engineering 3

operations research theory to establish the railway containertransportation route choice model

There are some shortcomings existing in the domesticand foreign documents all above about the multimodaltransportation modes selection the main problem is thatthe impact of transportation cost and time selection onmultimodal transportationmodeswas considered in the pastonly a single target situation for multimodal transportationmode optimal selection failed to consider other factors inthe problems But in the actual decision process selectionin multimodal transportation modes in order to ensure thecontinuity and safety we need to consider transportationsafety and transportation risk factors comprehensively onlyin this way we can meet the actual needs of transportation

Multimodal transportation is a kind of holistic and inte-grated operation process about the mutual coordination andcooperation between multimodal transportation networknodes node enterprises lines social economic conditionsand natural environment factors Risk factors existed inseveral aspects of the multimodal transportation networkincludingmultimodal transportation enterprises transporta-tion channels external economic social environment andnatural factors these risk factors are not only the necessaryconditions for the formation of multimodal transportationnetwork risk but also an important prerequisite for theexistence of multimodal transportation network risk

The transportation risk is one of the important factorsof the multimodal transportation selection in this paper butit can be found from the domestic and foreign documentsrsquoresearches that few scholars considered transportation riskas the important factor when they conducted researches inthe choice of transportation mode and multimodal trans-portation problems so the present researches cannot be welladapted to the actual situation of multimodal transportationmodes selection it is also difficult to give correct guidance topractice and reference

2 Establishment of the Optimization Model

21 Problem Description We suppose that a company hasa batch of goods to transport from origin point O todestination point D through a multimodal transportationnetwork a plurality of hub nodes composed the multimodaltransportation network several transportation modes can bechosen between two arbitrary adjacent nodes each node isa transit hub for different transportation modes when weswitched the goods from onemode to anothermode we needto spend some time and cost but total transportation timecannot exceed the scope of delivery timeWe hoped to deliverthe goods safely economically and conveniently through theproper choice of the transportation modes during the agreedperiod

22 Making the Assumption

(1) We have known and fixed all the relevant constants(2) there is one transportation mode existent at least on

any section of known delivery path

(3) the same batch of goods cannot be separated duringthe transport process that is to say we can onlychoose one transportation mode in the known arbi-trary section pathwhendelivering one batch of goods

(4) the goods are transited at hub nodes and the transitfor each batch of goods in each hub node occurs onetime at most

(5) when choosing the transportation model just con-sider the influence of the transportation operationrisk not including the others

23 Description of the Model Parameters

119862119896

119894119894+1is the per unit cost of the selected 119896 transporta-

tion mode from the node 119894 to the node 119894 + 1

119889119896119897

119894is the unit transfer cost from the transportation

mode 119896 to transportation mode 119897 in the node 119894

119905119896

119894119894+1is transportation time of the selected 119896 trans-

portation mode from the node 119894 to the node 119894 + 1

119891119896119897

119894is transfer time from the transportation mode 119896

to transportation mode 119897 in the node 119894

119877119879 is the risk factors of the delivery goods

119877119862119896

119894119894+1is the transportation risk factor of the selected

119896 transportation mode from the node 119894 to the node119894 + 1

119877119865119896119897

119894is the transfer risk factor from the transportation


119860119896

119894119894+1is the transportation ability of the selected 119896

transportationmode from the node 119894 to the node 119894+1Consider

119909119896

119894119894+1=

1 select 119896 transportation mode fromthe node 119894 to the node 119894 + 1

0 otherwise

119910119896119897

119894=

1 from the transportation mode119896 to transportation mode 119897 in the node 119894

0 otherwise(1)

119879 is the actual goods delivery time 119908 is the earliestgoods delivery time and V is the latest goods deliverytime V ge 119879 ge 119908 gt 0

119868119886 is each unit goods warehousing cost when deliveryadvanced and119881119889 is each unit goods penalty cost whendelivery is late


119869119886 is all goods warehousing cost when deliveryadvanced 119869119886 = max[0 (119908 minus 119879)119868119886]

119878119889 is all goods penalty cost when delivery is late 119878119889 =max[0 (119879 minus V)119881119889]

119872 is a sufficiently large penalty factor

24 Establishment of the Model

(1) Objective 1 the least transportation cost for the deci-sion of multimodal transportation scheme is as fol-lows

Min119885 =

119899

sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894

+max [0 (119908 minus 119879) 119868119886] +max [0 (119879 minus V) 119881119889]

(2)

(2) Objective 2 the least transportation time for the deci-sion of multimodal transportation scheme is as fol-lows

Min119879 =

119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894 (3)

(3) Objective 3 the lowest transportation risk for thedecision of multimodal transportation scheme is asfollows

Min119877 = 119877119879 +

119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894

(4)

25 Constraints of the Model

(1) We can only choose one transportation modebetween two nodes Consider

119898

sum

119896=1

119909119896

119894119894+1= 1 forall119894 isin 119899 (5)

(2) Only transportationmodes can be changed in a nodeConsider

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894= 1 forall119894 isin 119899 (6)

(3) Ensure continuity during the transportation processConsider

119909119896

119894minus1119894+ 119909119897

119894119894+1ge 2119910119896119897

119894 forall119894 isin 119899 forall119896 isin 119898 forall119897 isin 119898 (7)

(4) Goods transportation time should meet the actualtime range allowed Consider

119908 le

119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894le V V gt 119908 gt 0

(8)

(5) Cargo weight should not exceed the transportationcapacity Consider

119876119909119896

119894119894+1le 119860119896

119894119894+1 forall119894 isin 119899 forall119896 isin 119898 forall119897 isin 119898 (9)

(6) The decision variable is 0 or 1 Consider

119909119896

119894119894+1isin 0 1 forall119894 isin 119899 forall119896 isin 119898 forall119897 isin 119898

119910119896119897

119894isin 0 1 forall119894 isin 119899 forall119896 isin 119898 forall119897 isin 119898

(10)

(7) Nonnegative constraints of variables Consider

119905119896

119894119894+1ge 0 forall119894 isin 119899 forall119896 isin 119898 forall119897 isin 119898

119891119896119897

119894ge 0 forall119894 isin 119899 forall119896 isin 119898 forall119897 isin 119898

(11)

26 Model Processing Multiobjective programming modelis difficult to solve because of the incoordination of thetarget vectors and the existence of constraints in this paperthe objective function was simplified into a single objectivefunction usually we use the linear weighted method directlyand sum multiple objective functions into a single objectivefunction but this method is only applicable to the transfor-mation of the dimensionless objective function or uniformdimension objective function it must be dimensionless ora dimensionally unified treatment for the multiobjectiveproblem whose dimension is not uniformWe should changethe original objective functions (2) (3) and (4) into thefollowing

Max1198851015840 = 119885max minus119899

sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894minus 119869119886 minus 119878119889

Max1198791015840 = 119879max minus119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894

Max1198771015840 = 119877max minus 119877119879 minus

119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894

(12)


According to the evaluation function method of multipleobjective function optimization method referring to theweight degree of each target weight we gave respectively thetransportation cost transportation time and transportationrisk weight as 120572 120573 and 120575 and 120572 + 120573 + 120575 = 1 120572 120573 120575 isin [0 1]

At the same time in order to solve the dimensionalconsistency problem of the objective function we need totransform the target function (12)

Max119885lowast = (119885max minus119899

sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894minus 119869119886 minus 119878119889)

times (119885max)minus1


sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894)


Max119877lowast = (119877max minus 119877119879 minus

119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894)


(13)

In the objective function (13) 119885lowast 119879lowast and 119877lowastare the goal

function after dimensionless treatment the range for thethree is [0 1) Among them 119885max 119879max and 119877max are therespective maxima of target functions (2) (3) and (4) ineach generation of the genetic algorithm so in this paper weestablished the comprehensive objective function

119891(119885 119879 119877) = 120572 times 119885lowast+ 120573 times 119879

lowast+ 120575 times 119877

lowast (14)

The states and effects of the three indexes weights in thecomprehensive evaluation results are not the same in thecomprehensive objective function (14) it is so important forthe model solution when choosing the weight values Gen-erally speaking in the decision of intermodal transportationscheme the selected weight valuesmostly depend on the typeof goods and the decision preference

3 Particle Swarm Optimization (PSO)

31 Introduction of the Algorithm Particle swarm optimiza-tion is an evolutionary computation technique proposed in1995 by Dr Eberhart and Dr Kennedy from the behavior of

birdsrsquo predationThe algorithm is initially rules by birds clus-ter activities of the enlightenment using swarm intelligenceto establish a simplified model Particle swarm optimizationalgorithm is based on the observation of animal collectivebehavior because of information sharing the whole popula-tion movement from disorderly to orderly evolution processin the problem space by the individual in the group thenobtained the optimal solution PSO is similar to geneticalgorithm it is an optimization algorithm based on iterationThe system is initialized with a set of random solutionssearching for the optimal value through the iterative but ithas no genetic algorithm with the crossover and mutationthe particles search in the solution space to follow the optimalparticle PSO is simple and easy to realize and there is no needto adjust many parameters

32 The Advantages of PSO Algorithm

(1) The algorithm is simple with less adjustable parame-ters and easy to implement

(2) The random initialization population has strongglobal searching ability which is similar to geneticalgorithm

(3) It uses the evaluation function to measure the indi-vidual searching speed

(4) It has strong scalability

33 The Disadvantages of PSO Algorithm

(1) The algorithm cannot make full use of the feedbackinformation in the system

(2) Its ability to solve combinatorial optimization prob-lems is not strong

(3) Its ability to solve the optimization problem is notvery well

(4) It is easy for this algorithm to obtain local optimalsolution

34 The Basic Principle of the Algorithm The algorithm isdescribed as follows hypothesis in a 119863 dimension targetsearching space119873 particle to form a community the 119894 parti-cle vector to represent a119863 dimension and the position vectorConsider

119883119894 = (1199091198941 1199091198942 119909119894119863) 119894 = 1 2 119873 (15)

The velocity of vector 119894 particle is also a 119863 dimension thevelocity vector

119881119894 = (V1198941 V1198942 V119894119863) 119894 = 1 2 119873 (16)


The 119894 particle is the searched best position called individualextreme value by far and is denoted as

119875best = (1199011198941 1199011198942 119901119894119863) 119894 = 1 2 119873 (17)

The optimal position of the whole particle swarm searched sofar is called global extreme and is denoted as

119892best = (1199011198921 1199011198922 119901119892119863) (18)

After finding out these two optimal values particles updatedtheir velocity and position according to the following formu-las

V119894119889 = 119908 lowast V119894119889 + 11988811199031 (119901119894119889 minus 119909119894119889) + 11988821199032 (119901119892119889 minus 119909119894119889) (19)

119909119894119889 = 119909119894119889 + V119894119889 (20)

Among them119908 is the inertia factor 1198881 and 1198882 are the learningfactors and 1199031 and 1199032 are the ranges of [0 1] for the uniformrandom numbers The formula (19) is composed of threeparts on the right the first part reflects the movement ofparticles and represents particles which have maintainedtheir previous velocity trend the second part reflectsmemoryparticles on the historical experience and represents particleswhich are close to the optimal position of its historical trendthe third part reflects the cooperation and knowledge sharinggroup history experience synergy between particles

4 Ant Colony Algorithm (ACO)

41 Introduction of the Algorithm Ant colony algorithm(ACO) is a new intelligent optimization algorithm proposedby Italian scholars M Dorigo V Maniezzo and A Colorniin the twentieth century and early 90s it has been appliedto the TSP secondary distribution and quadratic assignmentproblems graph coloring problem and many classical com-bination optimization problems and gained very good results

42 The Advantages of ACO Algorithm

(1) It uses a kind of positive feedback mechanism andgets converges to the optimal path on the objectivethrough the pheromone update ceaselessly

(2) It is a kind of optimization method of distribution itis easy to imply in parallel

(3) It is a kind of method for global optimization usednot only for the single objective optimization prob-lems but also for solving themultiobjective optimiza-tion problems

(4) It is suitable to solve the discretization problems

(5) It has strong robustness

43 The Disadvantages of ACO Algorithm

(1) The solving speed is slow because of the shortage inthe initial stage of information algorithm

(2) The algorithm based on the discrete problem cannotdirectly solve continuous optimization problems

(3) This algorithm lacks strong searching ability

44 The Basic Principle of the Algorithm The basic principleof the algorithm is using119870 ants jointly to construct a feasiblesolution each ant is an independent one for the constructionprocess of solution the ants between each other exchangeinformation through information value and constantly opti-mize the cooperative solution and at the same time give theants distribution list (recorded the visited point of this ant)introduce the other one ant to follow the first antrsquos behaviorcontinue to search in the remaining points The antrsquos task isunder many constraints look for communicating nodes oneby one and leave the pheromone in the connected node arcUse arc pheromone and heuristic information to determineassignment order and use local optimization of pheromoneto update after all the 119896 ants are constructed completing theapplication then apply to the global pheromone update

120591119894119895(119905) stated at the 119905 moment in the node 119894 119895 connec-tion information 120572 is the importance degree of residualinformation in ant path 119894 119895 120578119894119895 is the heuristic informationthe expected degree of ants from node 119894 to node 119895 in thispaper 120578119894119895 = 1119889119894119895 (119889119894119895 is the distance between node 119894

and node 119895) 120573 is the importance degree of the heuristicinformation The initial time of nodes each ant was placedon the random selection path construction according to thetransition probabilities the 119870 antrsquos transfer probability ofnode 119894 to node 119895 at 119905 time is 119875119896

119894119895 which is feasible in the ant

119870 sentinel node 119894 set Consider

119901119896

119894119895(119905) =

0 otherwise

[120591119894119895(119905)]120572

[120578119894119895]120573

sum119894isin119873119896119894

[120591119894119895(119905)]120572

[120578119894119895]120573 119895 isin 119873

119896

119894

(21)

According to the rules and update time determined bydifferent update quantity of information all the ants of eachgroup after the completion of travel in the set the pheromoneshould be adjusted the ant density model was used in thispaper that is to say ants moved from node 119894 to node 119895

and updated pheromone of the path 119894 119895 Among them Δ120591119896119894119895

shows the number of unit length of messages of ant 119870 in thepath from 119894 to 119895 120574 is evaporation coefficient of pheromoneconstant 119876 is ants leaving tracks number Consider

Δ120591119896

119894119895(119905 119905 + 1) =

119876 ant 119896 transfers from node 119894 tonode 119895 at the 119905 time to 119905 + 1 time

0 otherwise(22)


Update rules corresponding information is as follows

120591119894119895 (119905 + 1) = (1 minus 120574) 120591119894119895 (119905) + Δ120591119894119895 (119905 119905 + 1) (23)

Δ120591119894119895 (119905 119905 + 1) =

119898

sum

119896=1

Δ120591119896

119894119895(119905 119905 + 1) (24)

5 Particle Swarm and Ant ColonyAlgorithm (PSACO)

51 Introduction of the Algorithm In this paper PSACOalgorithm which combines the particle swarm algorithm andant colony algorithm is proposed combining the advantagesof two kinds of algorithms andmake up for the disadvantagesof the two algorithms Particle swarm algorithm is moresuitable for solving continuous optimization problems andis slightly inferior in solving combinatorial optimizationproblems but due to the random distribution of the initialparticle we use it to solve the combinatorial optimizationproblems the algorithm has better global search ability andfast solution speed the ant colony algorithm is better in solv-ing combinatorial optimization problems than the particleswarm optimization algorithm because the initial distri-bution is uniform which makes the ant colony algorithmwith blindness at the beginning of the algorithm so it doesnot converge very quickly PSACO algorithm combinesparticle swarm algorithm and ant colony algorithm absorbsthe advantages of the two algorithms and overcomes theshortcomings the advantage is complementary and betterthan that of ant colony algorithm in time efficiency and theparticle swarm algorithm in accuracy and efficiency

52 The Basic Ideas and Steps of PSACO Algorithm ThePSACO algorithm combines advantages of particle swarmalgorithm and ant colony algorithm and its basic idea isas follows in the first stage the PSACO algorithm makesfull use of random rapid global features of the particleswarm optimization algorithm and after a certain numberof iterations finds out the suboptimal solution of the problemand adjusts the initial distribution in the second stage of thealgorithm PSACO algorithm uses information distributionin the first stage makes full use of parallelism positivefeedback and high accuracy advantages of ant colony algo-rithm and completes the entire problem solving progressSpecifically according to the characteristics of the solutionproblem combining two groups of algorithms is effective tosolve the combinatorial optimization problem of multimodaltransportation choice mode

The PSACO algorithm is divided into two parts firstlyparticle swarm algorithm generates transportation connec-tivity nodes from node start to node finish and then uses theant colony algorithm to find the optimal combination of thesequence nodes of the transportation mode the basic stepsare as follows

D

8

7

6

5

4

3

2

1

O

Figure 1 Schematic diagram of a multimodal transportation net-work

Step 1 Particle swarmalgorithmparameters and variables areinitialized

Step 2 Set the ant colony algorithm parameters initializationand each antrsquos position as follows

(1) according to the selection probability formula (21)and the roulette wheel method choose the path foreach ant

(2) when structural path each ant updates pheromonelocally at the same time according to the formula (23)

(3) execute (1) (2) and (3) circularly until every antgenerates a complete path

(4) record and calculate the objective function value ofeach ant according to the tabu list and give the optimaland worst ant

(5) update global pheromone for the best ant and worstant according to formula (23)

(6) execute (2)ndash(6) until they meet the end condition

Step 3 Call the ant colony algorithm to get each particlersquosfitness and update the historical optimum particle positionvector and global optimum particle position vector

Step 4 Update the particle velocity vector and the positionvector according to the formula (19) and (20)

Step 5 Execute Steps 2ndash4 to satisfy the end conditions anduse the ant colony algorithm to get the optimal solution

Step 6 End

6 Example

For example a transportation enterprise needs to transport100 tons of goods from city 119874 to city 119863 each passing sectionhas four kinds of transportation modes to select Figure 1shows the schematic diagram of a multimodal transportationnetwork Table 1 shows the transportation modes that we canchoose between two adjacent city nodes the transportationcosts transportation time and risk values about differenttransportation modes Table 2 shows the unit transfer costunit transit time and transfer risk about different transporta-tionmodesThe cargo delivery time interval was [130 h 165 h]


Table 1 Transportation coststransportation timetransportation risktransportation capacity of different transportation modes

Start node Terminal node Transportationmode

Transportationcostsyuan

Transportationtimehour Transportation risk Transportation

capacityO 1 Railway 35 1 1 120O 1 Road 20 2 3 115O 1 Waterway 18 15 1 90O 1 Aviation 40 05 2 120O 2 Railway 25 08 1 100O 2 Road 15 15 2 115O 2 Waterway 13 1 1 120O 2 Aviation 35 03 2 125O 3 Railway 30 12 1 120O 3 Road 18 22 25 95O 3 Waterway 13 18 12 110O 3 Aviation 36 05 18 1001 4 Railway 45 15 1 901 4 Road 25 25 3 1101 4 Waterway 20 2 1 1201 4 Aviation 42 07 2 1102 4 Railway 50 2 2 1002 4 Road 30 3 4 1052 4 Waterway 30 25 2 1052 4 Aviation 44 09 2 902 5 Railway 55 25 2 1002 5 Road 35 35 4 1002 5 Waterway 32 28 2 1302 5 Aviation 46 12 2 952 6 Railway 65 35 3 952 6 Road 45 45 45 1002 6 Waterway 33 35 3 1102 6 Aviation 50 18 25 1102 8 Railway 70 4 4 1102 8 Road 50 5 5 1002 8 Waterway 58 4 3 1502 8 Aviation 55 2 25 1003 5 Railway 43 14 1 1003 5 Road 22 23 25 1103 5 Waterway 20 2 1 1203 5 Aviation 40 08 18 1104 6 Railway 24 1 1 1004 6 Road 16 15 2 1154 6 Waterway 24 1 1 1204 6 Aviation 30 03 2 1254 7 Railway 35 2 3 1054 7 Road 25 3 4 954 7 Waterway 20 2 2 1204 7 Aviation 38 15 25 1005 7 Railway 35 2 35 1005 7 Road 24 25 35 1005 7 Waterway 30 18 2 115


Table 1 Continued




capacity5 7 Aviation 36 15 25 1005 8 Railway 35 1 1 1205 8 Road 20 2 3 1155 8 Waterway 32 15 1 905 8 Aviation 40 05 2 1206 D Railway 30 15 2 1006 D Road 20 2 3 956 D Waterway 18 15 2 1206 D Aviation 35 1 2 1257 6 Railway 40 15 2 1007 6 Road 20 25 25 1107 6 Waterway 24 2 1 1207 6 Aviation 43 1 2 1107 8 Railway 42 2 3 1007 8 Road 20 25 3 1107 8 Waterway 25 25 2 1207 8 Aviation 45 15 4 1108 D Railway 45 15 1 908 D Road 25 25 3 1108 D Waterway 20 2 1 1208 D Aviation 42 07 2 110

Table 2 Unit transfer costs (yuan)unit transit time (second)trans-fer risk of different transportation modes

Railway Road Waterway AviationRailway 000 222 343 443Road 222 000 344 434Waterway 343 344 000 433Aviation 443 434 433 000

in the contract the goodsrsquo penalty cost coefficient was 6yuanhour when delivery advanced and 10 yuanhour whendelivery is late the risk factor to send the goods was 3 Howto choose the multimodal transportation path and let thetransportation enterprise find the optimal scheme to savetransportation costs reduce the total transportation timeand reduce the total risk

According to the example by combining decisionmakersrsquomultimodal transportation needs the weights of transporta-tion costs transportation time and transportation risk valuewere set to 06 01 and 03 (different decisions will lead to dif-ferent weights setting) this paper adoptedMATLAB to verifythe effectiveness of the proposed algorithm first the 20 itera-tions of particle swarm algorithm were the initial solution ofthe problem then we used the results of the initial solutionby ant colony algorithm for distribution 120574 = 002 in antcolony algorithm the number of ants was equal to the multi-modal transportation city node number 120572 = 2 and 120573 = 5

Table 3 shows the comparison of PSACO algorithm and basicant colony algorithm in solving ability and efficiency

It can be seen from the simulation results that the PSACOalgorithm is compared with the basic ACO algorithm ithas not only better results than the ant colony algorithmbut also higher computation efficiency that is because afterthe particle swarm algorithm initial pheromone distributionwas improved avoiding the basic ACO algorithm for blindinitial pheromone distribution searching thus it is helpful forant colony algorithm to get more accurate searches Table 4shows the optimal transport scheme

7 Conclusion

This paper is about the decision problem of multimodaltransportation scheme it searched and analyzed relevantresearch literatures at home and abroad combined themwiththe diverse needs of transportation scheme decision andproposed and established combinatorial optimization math-ematical model of intermodal transportation plan decisionbased on transportation time and transportation cost andrisk at the same time it used heuristic algorithm whichcombined particle swarm optimization algorithm and antcolony algorithm to solve the model and got the trans-portation scheme optimization The experiments showedthat the algorithm has achieved very good results in timeperformance and optimization performance it was a feasiblealgorithm The content of this paper enriched and perfected


Table 3 The results and comparison of PSACO algorithm

Algorithm The comprehensiveevaluation value

Transportationtimehour


Transportationrisks

Operatetimesecond

PSACO algorithm 0438 1506 4258 67 28ACO algorithm 0486 1568 4456 73 43

Table 4 The optimal transport scheme

The optimal transport schemeTransportation route O-2-4-7-8-DTransportation mode Railway-road-waterway-railway-aviation

the theories and methods of the hot issues in the study ofintermodal transportation scheme decision and the resultsprovided the references and reasonable arrangements for thetransportation process and transportation project decisionmaking

Conflict of Interests

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

References

[1] K Chen and LMa ldquoGlowworm swarm optimization algorithmfor 0-1 knapsack problemrdquo Application Research of Computersvol 4 no 30 pp 993ndash995 2013

[2] B Li X Q Yan and J X Hu ldquoModel of container terminallogistics optimization system based on the Harvard structureand swarm intelligencerdquo Journal of Jiangsu University of Scienceand Technology vol 3 no 25 pp 282ndash286 2011

[3] J Mi ldquoImproved swarm intelligence optimization algorithmused in the task sortingrdquo Journal of Beijing Information Scienceand Technology University vol 3 no 25 pp 13ndash18 2010

[4] H Li X Chen and M Zhang ldquoThe path planning problembased on swarm intelligence algorithmrdquo Journal of TsinghuaUniversity vol 3 pp 1770ndash1773 2007

[5] BWu andZ Z Shi ldquoA kind of TSP problembased on ant colonyalgorithmpartition algorithmrdquo Journal of Computers vol 12 pp1328ndash1333 2001

[6] R-B Xiao Z-W Tao and H-F Zou ldquoConcurrent toleranceoptimization design based on hybrid swarm intelligence algo-rithmrdquo Computer Integrated Manufacturing Systems vol 13 no4 pp 668ndash674 2007

[7] Q L Ma W N Liu D H Sun and Y F Dan ldquoTravel routeplanning model based on fusion of multi intelligent algorithmrdquoComputer Science vol 10 pp 211ndash221 2010

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

[9] H I Calvete C Gale M-J Oliveros and B Sanchez-ValverdeldquoA goal programming approach to vehicle routing problemswith soft time windowsrdquo European Journal of OperationalResearch vol 177 no 3 pp 1720ndash1733 2007

[10] Y Li and J Zhao ldquoChoice of multimodal transportation modalwith fixed freightrdquo Journal of Southwest Jiaotong University vol47 no 5 pp 881ndash887 2012

[11] X Wang Z B Chi and X L Ge ldquoWhole vehicle multimodaltransportationmodel research and analysiswith timewindowsrdquoComputer Research and Application vol 28 no 2 pp 563ndash5652011

[12] F L Feng and Q Y Zhang ldquoOrganization collaborative opti-mization model of multimodal transportation based on envi-ronmentrdquo Journal of Railway Science and Engineering vol 9 no6 pp 95ndash100 2012

[13] L Tong and L Nie ldquoMultimodal transportation path optimiza-tion model and method researchrdquo Logistics Technology vol 29no 5 pp 57ndash60 2010

[14] J W Zhang ldquoMultimodal container transportation networkpath rationalization processrdquo Journal of Beijing Jiaotong Univer-sity 2011

[15] L Li and Y C Zeng ldquoMultimodal transportation model andalgorithm in logistics transportationrdquo Statistics and DecisionMaking vol 20 pp 27ndash29 2009

[16] J Wen ldquoOptimization model to control the cost of containermultimodal transportationrdquo Changsha Science and TechnologyUniversity 2010

[17] L Li ldquoMultimodal transportation modes generalized cost opti-mal choice under conditionsrdquo Journal of Beijing Jiaotong Uni-versity 2006

[18] X X Wang ldquoThe multimodal transportation minimum costproblem with time windowrdquo Beijing Posts and Telecommunica-tions University 2009

[19] J Zhang ldquoRailway container transportation networkmodel andalgorithmrdquo Central South University 2007

[20] Z J Jia ldquoThe risks analysis and control of the goodsmultimodaltransportationrdquo Chongqing Jiaotong University 2009

[21] F Liu ldquoThe containermultimodal transportation corridor plan-ning and transportation modes selectionrdquo Journal of SouthwestJiaotong University 2011

[22] X K Luo ldquoOptimization algorithm research of logistics andmultimodal transportationrdquo Shanghai JiaotongUniversity 2004

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


[1] gave a new algorithm of glowworm swarm optimizationalgorithm which was based on swarm intelligence optimiza-tion principle and successfully applied it to the 0-1 knapsackproblem Li et al [2] proposed the Harvard structure andswarm intelligence optimization system based on the CTLSmodel which used swarm intelligence based on calculationand source of agent in agent technology and the simulationbased optimization theory and the feasibility and reliability ofthemodeling optimizationmethod are verified by simulationanalysis of CTLS half bridge scheduling problem Mi [3]proposed an optimization algorithm for execution efficiencyand convergence problems of integrated design process usinggenetic algorithm for task scheduling the genetic algorithmwas improved and combined with the particle swarm algo-rithm by considering the effect of coupling relationshipbetween tasks to the task execution results It is verifiedwith an example the results showed that the algorithm hasa fast convergence speed and stable result Li et al [4]established automated warehouse single pock horizontallyrotating storage shelf system mathematical model and usedthe ant colony optimization algorithm to solve the pathplanning problem based on the goods selectionThis methodis capable of rotating storage rack system for rapid selectionof goods and finds the optimal goods picking path in theglobal high quality solution and short calculation time Wuand Shi [5] established ant colony algorithm based on theimproved meeting algorithm and proposed ant colony algo-rithm ant touring quality meeting algorithm and a parallelalgorithm which is combined with the proposed strategythe experimental results show that this algorithm has bettereffectiveness Xiao et al [6] proposed the mathematicalmodel of concurrent tolerance optimization design and thenmapped into a special traveling salesman problem ordermultiple traveling salesman problem so as to reduce thedifficulty of problem solving they proposed a hybrid swarmintelligence algorithm for concurrent tolerance optimizationdesign problemThrough calculation examples the efficiencywould be hybrid swarm intelligence algorithm with geneticalgorithm and simulated annealing algorithm has bettersearch ability and higher Ma et al [7] minimized the energyconsumption and delay time improved the evolutionaryalgorithms for solving the traveling salesman problem basedon artificial neural network algorithm and established theswarm intelligence analysis model and used it to analyzethe lines planning for public travel The experimental resultsshowed that the improved hybrid intelligent computingmethod was simple and effective and helped to overcomethe blindness of algorithm selection to further expand theresearch direction of computational intelligence From theabove research literatures it is not difficult to see that theswarm intelligence has not only attracted much attention inrecent years but also has a lot of space in the field of trans-portation applications This paper was proposed by combin-ing particle swarm optimization algorithm and ant colonyalgorithm (PSACO) to solve the combinatorial optimizationproblem of multimodal transportation scheme decision forthe first time which can make up shortcomings of their ownat the same time

(2) Decision of Multimodal Transportation Scheme In recentyears scholars at home and abroad did a lot of researchworks on how to choose the proper transportation mode inthe multimodal transportation access Among them Lozanoand Storchirsquos [8] research was about the problem of shortestfeasible path in the multimodal transportation and used thesequential algorithm to solve the problem Calvete et al [9]studied theVRPwith soft timewindowvehicles path problemand used the goal programming method to solve the prob-lem Li and Zhao [10] used the minimum total cost as theoptimization objective which included transportation costtransfer cost and penalty cost the goods delivery time andtransportation capacity were used as constraint conditionsthey built mixed integer linear programming model anddesigned the integer coding genetic algorithm Wang et al[11] studied and established themultimodal transportation 0-1 integer programming model under the time and capacityconstraints designed the natural number coding geneticalgorithm based on the characteristics of the model andfinally tested validity of the model and the algorithm withexamples Feng andZhang [12] considered the environmentalpollution energy consumption and safety factors into mul-timodal transportation sustainable development and thenintroduced the concept of total society cost of multimodaltransportation used this minimum cost value as a wholeobject and built a multimodal transportation collaborativeoptimization model Tong and Nie [13] transformed inter-modal path optimization problem into a generalized shortestpath problem used time and cost as optimization objectiveestablished multimodal transportation mathematical modelfor the optimization of path changes in volume and solvedand verified actual problem by using ant colony algorithmZhang [14] deeply studied the whole container multimodaltransportation network expounded the key elements and theoperation mechanism of multimodal container transporta-tion network in the whole system and in accordance withthe existing problems put forward the basic countermeasuresrationalization Li and Zeng [15] improved path shortest timemodel under the multimodal transportation in the fourthparty logistics gave the transportation cost model based onsatisfactory time path under the condition of multimodaltransportation and improved genetic algorithm to solve theproblemWen [16] usedmultimodal container transportationoptimization as the research object and discussed the opti-mization method of container multimodal transportationroutes transportation and time constraints modes Li [17]discussed the development of transportation mode choice aswell as the main transportation mode selection method andthen put forward the concept of generalized transportationcost Wang [18] in order to achieve time delivery reducedthe cost according to the actual situation of the key problemof logistics distribution in the process of transportationand traffic network of multimodal transportation with timewindow ofminimum cost problem Zhang [19] analyzed Chi-nese railway container transportation features and operationprocess used the transportation path selection as the goal toimprove the efficiency of container transport and shorten thecontainer transport cycle and reduced transportation costsfor the purpose then he used the optimization theory and















119862119896



119889119896119897



119905119896



119891119896119897




119877119862119896



119877119865119896119897



119860119896



119909119896

119894119894+1=


0 otherwise

119910119896119897

119894=


0 otherwise(1)









Min119885 =

119899

sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894


(2)


Min119879 =

119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894 (3)


Min119877 = 119877119879 +

119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894

(4)



119898

sum

119896=1

119909119896

119894119894+1= 1 forall119894 isin 119899 (5)


119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894= 1 forall119894 isin 119899 (6)


119909119896

119894minus1119894+ 119909119897

119894119894+1ge 2119910119896119897



119908 le

119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894le V V gt 119908 gt 0

(8)


119876119909119896

119894119894+1le 119860119896



119909119896


119910119896119897


(10)


119905119896


119891119896119897


(11)


Max1198851015840 = 119885max minus119899

sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894minus 119869119886 minus 119878119889

Max1198791015840 = 119879max minus119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894


119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894

(12)





sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894minus 119869119886 minus 119878119889)



sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894)



119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894)


(13)





lowast (14)
















119883119894 = (1199091198941 1199091198942 119909119894119863) 119894 = 1 2 119873 (15)


119881119894 = (V1198941 V1198942 V119894119863) 119894 = 1 2 119873 (16)



119875best = (1199011198941 1199011198942 119901119894119863) 119894 = 1 2 119873 (17)


119892best = (1199011198921 1199011198922 119901119892119863) (18)



119909119894119889 = 119909119894119889 + V119894119889 (20)



















119901119896

119894119895(119905) =

0 otherwise

[120591119894119895(119905)]120572

[120578119894119895]120573

sum119894isin119873119896119894

[120591119894119895(119905)]120572

[120578119894119895]120573 119895 isin 119873

119896

119894

(21)




Δ120591119896

119894119895(119905 119905 + 1) =


0 otherwise(22)



120591119894119895 (119905 + 1) = (1 minus 120574) 120591119894119895 (119905) + Δ120591119894119895 (119905 119905 + 1) (23)

Δ120591119894119895 (119905 119905 + 1) =

119898

sum

119896=1

Δ120591119896

119894119895(119905 119905 + 1) (24)





D

8

7

6

5

4

3

2

1

O













Step 6 End

6 Example









Table 1 Continued











7 Conclusion







Transportationrisks

Operatetimesecond







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra























119862119896



119889119896119897



119905119896



119891119896119897




119877119862119896



119877119865119896119897



119860119896



119909119896

119894119894+1=


0 otherwise

119910119896119897

119894=


0 otherwise(1)









Min119885 =

119899

sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894


(2)


Min119879 =

119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894 (3)


Min119877 = 119877119879 +

119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894

(4)



119898

sum

119896=1

119909119896

119894119894+1= 1 forall119894 isin 119899 (5)


119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894= 1 forall119894 isin 119899 (6)


119909119896

119894minus1119894+ 119909119897

119894119894+1ge 2119910119896119897



119908 le

119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894le V V gt 119908 gt 0

(8)


119876119909119896

119894119894+1le 119860119896



119909119896


119910119896119897


(10)


119905119896


119891119896119897


(11)


Max1198851015840 = 119885max minus119899

sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894minus 119869119886 minus 119878119889

Max1198791015840 = 119879max minus119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894


119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894

(12)





sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894minus 119869119886 minus 119878119889)



sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894)



119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894)


(13)





lowast (14)
















119883119894 = (1199091198941 1199091198942 119909119894119863) 119894 = 1 2 119873 (15)


119881119894 = (V1198941 V1198942 V119894119863) 119894 = 1 2 119873 (16)



119875best = (1199011198941 1199011198942 119901119894119863) 119894 = 1 2 119873 (17)


119892best = (1199011198921 1199011198922 119901119892119863) (18)



119909119894119889 = 119909119894119889 + V119894119889 (20)



















119901119896

119894119895(119905) =

0 otherwise

[120591119894119895(119905)]120572

[120578119894119895]120573

sum119894isin119873119896119894

[120591119894119895(119905)]120572

[120578119894119895]120573 119895 isin 119873

119896

119894

(21)




Δ120591119896

119894119895(119905 119905 + 1) =


0 otherwise(22)



120591119894119895 (119905 + 1) = (1 minus 120574) 120591119894119895 (119905) + Δ120591119894119895 (119905 119905 + 1) (23)

Δ120591119894119895 (119905 119905 + 1) =

119898

sum

119896=1

Δ120591119896

119894119895(119905 119905 + 1) (24)





D

8

7

6

5

4

3

2

1

O













Step 6 End

6 Example









Table 1 Continued











7 Conclusion







Transportationrisks

Operatetimesecond







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra















Min119885 =

119899

sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894


(2)


Min119879 =

119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894 (3)


Min119877 = 119877119879 +

119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894

(4)



119898

sum

119896=1

119909119896

119894119894+1= 1 forall119894 isin 119899 (5)


119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894= 1 forall119894 isin 119899 (6)


119909119896

119894minus1119894+ 119909119897

119894119894+1ge 2119910119896119897



119908 le

119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

+

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894le V V gt 119908 gt 0

(8)


119876119909119896

119894119894+1le 119860119896



119909119896


119910119896119897


(10)


119905119896


119891119896119897


(11)


Max1198851015840 = 119885max minus119899

sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894minus 119869119886 minus 119878119889

Max1198791015840 = 119879max minus119899

sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894


119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894

(12)





sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894minus 119869119886 minus 119878119889)



sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894)



119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894)


(13)





lowast (14)
















119883119894 = (1199091198941 1199091198942 119909119894119863) 119894 = 1 2 119873 (15)


119881119894 = (V1198941 V1198942 V119894119863) 119894 = 1 2 119873 (16)



119875best = (1199011198941 1199011198942 119901119894119863) 119894 = 1 2 119873 (17)


119892best = (1199011198921 1199011198922 119901119892119863) (18)



119909119894119889 = 119909119894119889 + V119894119889 (20)



















119901119896

119894119895(119905) =

0 otherwise

[120591119894119895(119905)]120572

[120578119894119895]120573

sum119894isin119873119896119894

[120591119894119895(119905)]120572

[120578119894119895]120573 119895 isin 119873

119896

119894

(21)




Δ120591119896

119894119895(119905 119905 + 1) =


0 otherwise(22)



120591119894119895 (119905 + 1) = (1 minus 120574) 120591119894119895 (119905) + Δ120591119894119895 (119905 119905 + 1) (23)

Δ120591119894119895 (119905 119905 + 1) =

119898

sum

119896=1

Δ120591119896

119894119895(119905 119905 + 1) (24)





D

8

7

6

5

4

3

2

1

O













Step 6 End

6 Example









Table 1 Continued











7 Conclusion







Transportationrisks

Operatetimesecond







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













sum

119894=1

119898

sum

119896=1

119862119896

119894119894+1119883119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119889119896119897

119894119910119896119897

119894minus 119869119886 minus 119878119889)



sum

119894=1

119898

sum

119896=1

119905119896

119894119894+1119909119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119891119896119897

119894119910119896119897

119894)



119899

sum

119894=1

119898

sum

119896=1

119909119896

119894119894+1119877119862119896

119894119894+1

minus

119899

sum

119894=1

119898

sum

119896=1

119898

sum

119897=1

119910119896119897

119894119877119865119896119897

119894)


(13)





lowast (14)
















119883119894 = (1199091198941 1199091198942 119909119894119863) 119894 = 1 2 119873 (15)


119881119894 = (V1198941 V1198942 V119894119863) 119894 = 1 2 119873 (16)



119875best = (1199011198941 1199011198942 119901119894119863) 119894 = 1 2 119873 (17)


119892best = (1199011198921 1199011198922 119901119892119863) (18)



119909119894119889 = 119909119894119889 + V119894119889 (20)



















119901119896

119894119895(119905) =

0 otherwise

[120591119894119895(119905)]120572

[120578119894119895]120573

sum119894isin119873119896119894

[120591119894119895(119905)]120572

[120578119894119895]120573 119895 isin 119873

119896

119894

(21)




Δ120591119896

119894119895(119905 119905 + 1) =


0 otherwise(22)



120591119894119895 (119905 + 1) = (1 minus 120574) 120591119894119895 (119905) + Δ120591119894119895 (119905 119905 + 1) (23)

Δ120591119894119895 (119905 119905 + 1) =

119898

sum

119896=1

Δ120591119896

119894119895(119905 119905 + 1) (24)





D

8

7

6

5

4

3

2

1

O













Step 6 End

6 Example









Table 1 Continued











7 Conclusion







Transportationrisks

Operatetimesecond







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119875best = (1199011198941 1199011198942 119901119894119863) 119894 = 1 2 119873 (17)


119892best = (1199011198921 1199011198922 119901119892119863) (18)



119909119894119889 = 119909119894119889 + V119894119889 (20)



















119901119896

119894119895(119905) =

0 otherwise

[120591119894119895(119905)]120572

[120578119894119895]120573

sum119894isin119873119896119894

[120591119894119895(119905)]120572

[120578119894119895]120573 119895 isin 119873

119896

119894

(21)




Δ120591119896

119894119895(119905 119905 + 1) =


0 otherwise(22)



120591119894119895 (119905 + 1) = (1 minus 120574) 120591119894119895 (119905) + Δ120591119894119895 (119905 119905 + 1) (23)

Δ120591119894119895 (119905 119905 + 1) =

119898

sum

119896=1

Δ120591119896

119894119895(119905 119905 + 1) (24)





D

8

7

6

5

4

3

2

1

O













Step 6 End

6 Example









Table 1 Continued











7 Conclusion







Transportationrisks

Operatetimesecond







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











120591119894119895 (119905 + 1) = (1 minus 120574) 120591119894119895 (119905) + Δ120591119894119895 (119905 119905 + 1) (23)

Δ120591119894119895 (119905 119905 + 1) =

119898

sum

119896=1

Δ120591119896

119894119895(119905 119905 + 1) (24)





D

8

7

6

5

4

3

2

1

O













Step 6 End

6 Example









Table 1 Continued











7 Conclusion







Transportationrisks

Operatetimesecond







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Table 1 Continued











7 Conclusion







Transportationrisks

Operatetimesecond







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














Transportationrisks

Operatetimesecond







References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article decision of multimodal transportation ...multimodal transportation is a kind of...

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