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Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2013 Article ID 672078 12 pageshttpdxdoiorg1011552013672078

Research ArticleDynamic Vehicle Routing Using an Improved VariableNeighborhood Search Algorithm

Yingcheng Xu12 Li Wang3 and Yuexiang Yang1

1Branch of Quality Management China National Institute of Standardization Beijing 100088 China2Department of Industrial Engineering Tsinghua University Beijing 100084 China3School of Economics and Management Beihang University Beijing 100191 China

Correspondence should be addressed to Yuexiang Yang yxyangcnis163com

Received 14 September 2012 Revised 5 December 2012 Accepted 25 December 2012

Academic Editor Qiuhong Zhao

Copyright copy 2013 Yingcheng Xu 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

In order to effectively solve the dynamic vehicle routing problem with time windows the mathematical model is established and animproved variable neighborhood search algorithm is proposed In the algorithm allocation customers and planning routes for theinitial solution are completed by the clustering method Hybrid operators of insert and exchange are used to achieve the shakingprocess the later optimization process is presented to improve the solution space and the best-improvement strategy is adoptedwhich make the algorithm can achieve a better balance in the solution quality and running time The idea of simulated annealingis introduced to take control of the acceptance of new solutions and the influences of arrival time distribution of geographicallocation and time window range on route selection are analyzed In the experiment the proposed algorithm is applied to solve thedifferent sizesrsquo problems of DVRP Comparing to other algorithms on the results shows that the algorithm is effective and feasible

1 Introduction

Dynamic Vehicle Routing Problem (DVRP) is a variant ofthe Vehicle Routing Problems (VRPs) that has arisen due torecent advances in real-time communication and informa-tion technologiesTheVRP is a nondeterministic polynomialhard (NP-hard) problem that calls for the determinationof the optimal set of routes to be performed by a fleet ofvehicles to serve a given set of customers However amajorityof the information is unpredictable before path optimiza-tion such as the customerrsquos geographic position customerdemand and vehicle travel and service timeThis informationis dynamic and new information may appear or existinginformation changes and so forth Many different factorsmust be considered when a decision about the allocation andscheduling of a new request is taken the current locationof each vehicle their current planned route and schedulecharacteristics of the new request travel times betweenthe service points characteristics of the underlying roadnetwork service policy of the company and other relatedconstraints The DVRP is a complex problem compared to

the classic VRP and variable neighborhood search (VNS) isproposed as a means to effectively and efficiently tackle thedynamic problem and optimize the planned routes betweenthe occurrences of new events VNS was initially proposedby Hansen and Mladenovic [1 2] for solving combinatorialand global optimization problemsThemain reasoning of thismetaheuristic is based on the idea of a systematic change ofneighborhoods within a local search method

This paper has twomain contributions First according tothe characteristic of DVRP we gave the graph expression andformulated the mathematical model Second the proposedalgorithm is improved in initial solution shaking and localsearch based on classic VNS The later optimization processis presented and the results show that the algorithm is feasibleand competitive with other existing algorithms The remain-der of the paper is organized as follows literature reviewsare illustrated in Section 2 and the mathematical formulationproblem is discussed in Section 3 Section 4 introduces themain ideas of the improved variable neighborhood searchwhile computational results are presented and discussed inSection 5 Section 6 concludes the paper

2 Journal of Applied Mathematics

2 Literature Reviews

TheDVRP problem is closely related to the actual productionand life In recent years DVRP gradually becomes a hot topicBoth domestic and foreign scholars mainly focus on the con-struction of different scheduling models and the designing ofsimple and efficient heuristic algorithms A survey of resultsachieved on the different types of DVRPs can be found inGendreau and Potvin [3]The dynamic full-truckload pickupand delivery problem has been studied by Fleischmann et al[4] and Yang et al [5] Montemanni et al [6] proposed thevehicle schedulingmodel with randomdynamic demand andsolved it using an ant colony algorithm In Gendreau et al[7] neighborhood search heuristics for dial-a-ride problemsare finally presented Goel and Gruhn [8] studied the math-ematical model of dynamic vehicle routing problems underthe conditions of the real-life information Schonbergerand Kopfer [9] studied the real-time decision-making andautonomous decision-making of DVRP Novoa and Storer[10] introduced the approximate solution algorithm withstochastic demands Branchini et al [11] presented localsearch algorithm for a dynamic vehicle routing problemMuller [12] studied DVRP with time window and analyzedthe algorithm for the model suboptimal solution

In the past ten decades a tremendous amount of workin the field of vehicle routing problems has been publishedespecially literature based on VNS The Braysy [13] gavethe internal design of the VND and RVNS algorithm indetail analyzed the VRPTW problem and indicated theVND algorithm as one of the most effective ways to solveVRPTW problems Polacek et al [14] designed VNS to solvethe multidepot vehicle routing problem with time windowsMDVRPTWHis algorithm used the neighborhood structureof swap and cross to do shaking operation for the currentsolution to do local search with a constrained 3-opt operatorto accept the part of the poor solution to avoid getting intoa local optimum for the algorithm by Threshold AcceptingKytojoki et al [15] designed the guided VNS algorithmto handle the 32 existing large-scale VRP problems andcompared it to the TS algorithm The result showed thatthe VNS algorithm was more effective than TS algorithmin solving time Goel and Gruhn [16] introduced the RVNSto solve the general VRP problem including time windowsvehicle constraints path constraints travel departure timeconstraints capacity constraints the order models compati-bility constraintsmultisupplier point of the orders and trans-port and service position constraints Hemmelmayr et al [17]proposed the VNS algorithm for periodical VRP problemadopted the saving algorithm for the construction of theinitial solution designed the move and cross neighborhoodused 3-opt operator as local search strategies and contrastedit with other research results Fleszar et al [18] adopted VNSalgorithm to solve the open-loop VRP problem and tested 16benchmark problems

Due to the complexity of the problem the current solvingquality and efficiency for the DVRP are far from the practicalrequirements So there are many problems need to make in-depth research such as how to seek feasible solutions howto prevent falling into local optimum and how to control

Advance request customer (static)Immediate request customer (dynamic)Depot

Depot

New route segmentPlanned routeCurrent positionof vehicle

Figure 1 An example for dynamic vehicle routing problem

the solution within the acceptable range This paper presentsan improved variable neighborhood search algorithm tosolve DVRP it integrates local search operator optimizationprocess and the simulated annealing algorithm into the VNSalgorithm framework Through the comparison with otheralgorithms it shows that the proposed algorithm can get thebetter solution

3 Problem Descriptions

31 Problem Definition Larsen [19] defined DVRP with twoaspects not all information relevant to the planning of theroutes is known by the planner when the routing processbegins information can change after the initial routes havebeen constructed And he illustrated a dynamic vehiclerouting situation The simple example is shown in Figure 1As seen from it two uncapacitated vehicles must serviceboth advance and immediate request customers without timewindows The advance request customers are represented byyellow nodes while those that are immediate requests aredepicted by black nodesThe solid blue lines represent the tworoutes the dispatcher has planned prior to the vehicles leavingthe depot The two thick arcs indicate the vehicle positions atthe time the dynamic requests are received Ideally the newcustomers should be inserted into the already planned routeswithout the order of the nonvisited customers being changedandwithminimal delayThis is the case depicted on the right-hand side route However in practice the insertion of newcustomers will usually be a much more complicated task andwill imply a replanning of the nonvisited part of the routesystem This is illustrated by the left-hand side route whereservicing the new customer creates a large detour

We can also describe the DVRP with a mathematicapproach The problem is defined on a complete graph 119866 =

(119881 119864) where 119881 = 1199070 119907

119871 is the vertex set and 119864 =

(119907119894 119907119895) 119907119894 119907119895isin 119881 119894 = 119895 is the arc set The verticesrsquo set 119863 =

1199070 corresponds to the depots Each vertex 119907

119894isin 119881 has several

nonnegative weights associated with it namely a demand 119889119894

a arrival time 119886119894 the waiting time 119908

119894 service time 119904

119894 and an

earliest 119890119894and latest 119897

119894possible start time for the service which

define a time window [119890119894 119897119894] 119862119905119894119895is the transportation cost

Journal of Applied Mathematics 3

from customer 119894 to customer 119895 in the period 119905 Each vehicle119896 has associated with a nonnegative capacity 119902

119896 119879 refers to

the number of time periods 119888119891is the fixed cost for a vehicle

119888119905is the traveling cost per unit time and 119872 is a very large

numberThe variables are defined as follows

119910119894119896=

1 if customer 119894 is delivered by vehicle 119896

0 other

119909119905

119894119895

=

1 if the vehicle visits the customer 119895 fromcustomer 119894 in the 119905 period

0 other(1)

32 Mathematical Formulation

321 Objective Function The dynamic vehicle routing prob-lem with time windows is formulated in this section as amixed integer linear programming problemThe objective ofthe formulation is to minimize the total cost that consists ofthe fixed costs of used vehicles and the routing costs

min119865 = 119888119891times sum

119894isin119881

sum

119895isin1198811198810

119879

sum

119905=1

119909119905

119894119895+ 119888119905

times sum

119894isin119881

sum

119895isin1198811198810

119879

sum

119905=1

(119888119905

119894119895times 119909119905

119894119895)

(2)

322 Problem Constraints The constraints of the problemconsist of vehicle constraints demand constraints routingconstraints and other constraintsAssignment of Nodes to Vehicles Equation (3) ensure that eachcustomer has a vehicle service for only one time and thevehicle does not return to the yard

119871

sum

119894=0119894 = 119895

119879

sum

119905=1

119909119905

119894119895= 1 119895 = 1 2 119871

119871

sum

119895=1119895 = 119894

119879

sum

119905=1

119909119905

119894119895= 1 119894 = 1 2 119871

(3)

Relationship between the Vehicle and Depot Constraint (4)ensures that the number of vehicles departed from theyard does not exceed the maximum number 119870 of vehiclesbelonging to the distribution center

119871

sum

119895=1

119879

sum

119905=1

119909119905

0119895le 119870 (4)

Relationship between the Vehicle and Route Constraint (5)ensures that customers on the same route are delivered by thesame vehicle

119870

sum

119896=1

119896 (119910119894119896minus 119910119895119896) ge 119872(

119879

sum

119905=1

119909119905

119894119895minus 1) forall119894 119895 119894 = 119895

119870

sum

119896=1

119896 (119910119894119896minus 119910119895119896) le 119872(1 minus

119879

sum

119905=1

119909119905

119894119895) forall119894 119895 119894 = 119895

(5)

Assignment of Nodes to Vehicles Equation (6) states that everycustomer node must be serviced by a single vehicle

119870

sum

119896=1

119910119894119896= 1 forall119894 = 1 2 119871 (6)

Capacity Constraints Constraint (7) states that the overallload to deliver to customer sites serviced by a used vehicle119907 should never exceed its capacity

119871

sum

119894=1

119889119894119910119894119896le 119902119894 forall119896 = 1 2 119870 (7)

Subcircuit Constraints Equation (8) ensures to eliminate sub-circuit

sum

119894119895isin119878times119878119894 = 119895

119879

sum

119905=1

119909119905

119894119895le |119878| minus 1 119878 isin 1 2 119871 forall119894 119895 119894 = 119895

(8)

Time Constraint Violations due to EarlyLate Services atCustomer Sites One has

119886119894+ 119908119894+ 119904119894+ 119888119905

119894119895minus 119872(1 minus 119909

119905

119894119895) le 119886119895

119886119895le 119897119895

119908119894= max 119890

119894minus 119886119894 0

(9)

Other Constraints One has

119909119905

119894119895= 0 1 forall119894 119895 119896

119910119894119896= 0 1 forall119894 119896

(10)

4 An Improved Variable NeighborhoodSearch Algorithm

VNS is a metaheuristic for solving combinatorial and globaloptimization problems proposed by Hansen andMladenovic[1 2] Starting from any initial solution a so-called shakingstep is performed by randomly selecting a solution from thefirst neighborhood This is followed by applying an iterativeimprovement algorithmThis procedure is repeated as long asa new incumbent solution is found If not one switches to thenext neighborhood (which is typically larger) and performsa shaking step followed by the iterative improvement Ifa new incumbent solution is found one starts with thefirst neighborhood otherwise one proceeds with the next


Main Clarke And Wright savings algorithmInput the number of customers and vehicles the capacity of vehiclesOutput set of initial solutionBeginfor each day do

while number of routes gt number of vehicles doshortest route= find route with fewest number of customersfor each customer isin shortest route do

delete current routeinsert in cheapest position of the remaining routes

end forend while

end forend Begin

Algorithm 1 The initial solution based on CW

neighborhood and so forth The description consists of thebuilding of an initial solution the shaking phase the localsearchmethod and the acceptance decisionThe flow of VNSis shown in Figure 2

41 Initial Solution Using variable neighborhood searchalgorithm first it needs to build one or more initial feasiblesolution a clustering algorithm for an initial feasible solutionmainly completes two tasks customer allocation and pathplanning For obtaining an initial solution each customeris assigned a visit day combination randomly Routes areconstructed by solving a vehicle routing problem for eachday using the Clarke andWright savingsrsquo algorithm [20] TheClarke and Wright savingsrsquo algorithm terminates when notwo routes can feasibly be merged that is no two routes canbe merged without violating the route duration or capacityconstraints shown in Algorithm 1 As a result the numberof routes may exceed the number of available vehicles Inthat case a route with the fewest customers is identifiedand the customers in this route are moved to other routes(minimizing the increase in costs) Note that this may resultin routes that no longer satisfy the duration or capacityconstraintsThis step is repeated until the number of routes isequal to the number of vehicles Since the initial solutionmaynot be feasible the VNS needs to incorporate techniques thatdrive the search to a feasible solution

The initial solution obtained by the above method canbasically meet the needs of the follow-up work buildingthe foundation to achieve optimal feasible solution in thefollowing algorithm

42 Shaking Shaking is a key process in the variable neigh-borhood search algorithm design The main purpose of theshaking process is to extend the current solution search spaceto reduce the possibility that the algorithm falls into the localoptimal solution in the follow-solving process and to get thebetter solution The set of neighborhood structures used forshaking is the core of the VNS The primary difficulty is tofind a balance between effectiveness and the chance to getout of local optimal In the shaking execution it first selects

a neighborhood structure 119873119896from the set of neighborhood

structures of current solution 119909 then according to thedefinition of119873

119896 119909 corresponds to change and generate a new

solution 119909lowast

There are two neighborhood structures to achieve theshaking insert and exchange Insert operator denotes acertain period of consecutive nodes moving from the currentpath to another path exchange operator refers to interchangethe two-stage continuous nodes belonging to different pathsThe insert and exchange operators are shown in Figure 3The cross-exchange operator was developed by Taillard et al[21] The main idea of this exchange is to take two segmentsof different routes and exchange them Compared with theVNS by Polacek et al [14] the selection criterion is slightlychangedNow it is possible to select the same route twiceThisallows exploring more customer visit combinations withinone route An extension to the CROSS exchange operator isintroduced by Braysy [13] this operator is called improvedcross-exchangemdash119894Cross exchange for short Both operatorsare used to define a set of neighborhood structures for theimproved VNS

In each neighborhood the insert operator is appliedwith a probability 119901insert to both routes to further increasethe extent of the perturbation then the probability of theexchange operator is 1 minus 119901insert IVNS selects randomly anexchange operator to change path for each shaking executionThe shaking process is somewhat similar to the crossoveroperation of the genetic algorithm When the process isfinished the only twopaths have the exchange of informationmost of the features of the current solution will be preservedto speed the convergence of the algorithm

43 Local Search In a VNS algorithm local search pro-cedures will search the neighborhood of a new solutionspace obtained through shaking in order to achieve a locallyoptimal solution Local search is the most time-consumingpart in the entire VNS algorithm framework and decides thefinal solution quality so computational efficiency must beconsidered in the design process of local search algorithmTwomain aspects are considered in the design of local search


Start

Define the set of the neighborhoodstructure for local research

To construct the initial solution 119909 assume

Shaking process in the 119896th neighborhoodstructure of current solution 119909 randomly

generates a 119909s solution and the evaluationfunction of 119909 is 119891(119909)

Local search process for 119909119904 to adopt somelocal search algorithm to solve local optimalsolution 119909119897 evaluation function of 119909119897 is 119891(119909119897)

Later optimization process get the

To accept the new solution

If meet the acceptancedecision of new solution

Yes

Yes

Yes

No

No

Yes

No

EndNo

the optimal solution 119909119887larr119909 the evaluationfunction of 119909119887 is 119891(119909119887) The number of

iterations is 119899119905 and 119894 = 1 119896 = 1

119891(119909119897) lt 119891(119909119887)

optimal solution 119909op119909119897larr119909op 119909119887larr119909op 119891(119909119887)larr119891(119909119897)

119896 larr (119896 mod 119873max ) + 1

119891(119909119897) lt 119891(119909)

119909 larr 119909119897 119896 larr 1 119894 larr 119894 + 1

119894 larr 119894 + 1

119894 le 119899119905

119873119896 (119896 = 1 2 119873max )

Figure 2 The flow of IVNS


119894119894 minus 1

119894 + 1 119895 + 1

119894 + 1 119895 + 1

119895 + 2

119895 + 2 119895 + 3

119896

119894

119896

119896 minus 1 119897

119897

119897 + 1

119894 minus 1 119895 + 3

119896 minus 1 119897 + 1

Depot

Depot

Depot

Depot

(a) Insert

119894 minus 1

119894 minus 1

119894 + 2

119894 + 2

119895

119895

119895 minus 1

119895 minus 1

119894 + 1

119894 + 1

119895 + 1

119895 + 1 119895 + 2

119895 + 2

119894

119894

119894 minus 2

119894 minus 2

119895 minus 2

119895 minus 2

Depot

Depot

Depot

Depot

(b) Cross

119894 minus 1

119894 minus 1

119895

119895

119895 + 1

119895 + 1

119894

119894

119896 minus 1

119896 minus 1

119896 119897

119896119897

119897 + 1

119897 + 1

DepotDepot

Depot Depot

Customer nodeRoute

119894

(c) 119894Cross

Figure 3 Insert and exchange operator


119895 119895119895 + 1119895 + 1

119894 119894 119894 + 1119894 + 1

DepotDepot

(a) 2 minus 119900119901119905

Customer nodeRoute

119894 + 1 119894 + 1

119895119895

119895 + 1119895 + 1

119896 119896119896 + 1119896 + 1

119894

119894

119894

Depot Depot

(b) 3 minus 119900119901119905

Figure 4 2 minus 119900119901119905 and 3 minus 119900119901119905 strategy

algorithms local search operator and the search strategyBased on the previous studies this paper selects 2 minus 119900119901119905 and3 minus 119900119901119905 as a local search operator in order to obtain thegood quality local optimal solution in a short period theyare shown in Figure 4 According to the probability one ofthe two operators is selected in each local search processTheparameter 119901

2minus119900119901119905represents the probability of selection for

2 minus 119900119901119905 similarly the probability of selection for 3 minus 119900119901119905 canbe expressed as 1 minus 119901

2minus119900119901119905 This mixed operator can develop

optimization ability for 2 minus 119900119901119905 and 3 minus 119900119901119905 and expand thesolution space of the algorithm

There aremainly two search strategies first-improvementand best-improvement in local search algorithmThe formerrefers to access the neighborhood solution of the current 119909solution successively in the solution process if thev currentaccess neighborhood solution 119909

119899is better than 119909 tomake 119909 =

119909119899and update neighborhood solution We repeat these steps

until all the neighborhood solutions of 119909 are accessed Finally119909 will be obtained as a local optimal solution The latterrefers to traverse all of the neighborhood solution of current119909 solution in the solution process to select the optimumneighborhood solution 119909

119899as a local optimal solution In this

paper we adopt the best-improvement strategy it enables thealgorithm to achieve a better balance in the solution qualityand run time

44 Later Optimization Process In order to accelerate theconvergence speed and improve the solution quality the lateroptimization process is proposed in the IVNS algorithmAfter the local search is completed if the local optimalsolution 119909119897 is better than the global optimal solution 119909119887 thatis 119891(119909119897) lt 119891(119909119887) the later optimization process will be

continued to be implemented in order to seek a better globaloptimal solution [22] The algorithm of later optimizationprocess which was proposed by Gendreau et al is suitablefor solving the traveling salesman problem and the vehiclerouting problemwith timewindowsThe algorithmprocessescan be simply described as follows

Step 1 There are some assumptions The path that will beoptimized is 119903 its length is 119899 and its value of the evaluationfunction is 119888 The final optimized path is 119903lowast the value of theevaluation function is 119888lowast and 119903

lowast= 119903 119888lowast = 119888 and 119896 = 1

Step 2 The Unstring and a String processes [23] are respec-tively performed for the 119896th customer in the 119903 the optimizedpath 119903

1015840 can be obtained and its value of the evaluationfunction value is 1198881015840

Step 3 If 1198881015840 lt 119888lowast some processes are carried out they are

119903lowast

= 1199031015840 119903 = 119903

1015840 119888lowast = 1198881015840 119888 = 119888

1015840 and 119896 = 1 jump to Step 2otherwise 119896 = 119896 + 1

Step 4 If 119896 = 119899 + 1 the algorithm will be terminatedotherwise jump to Step 2

45 Acceptance Decision The last part of the heuristic con-cerns the acceptance criterion Here we have to decidewhether the solution produced by VNS will be acceptedas a starting solution for the next iteration To avoid thatthe VNS becomes too easily trapped in local optima dueto the cost function guiding towards feasible solutions andmost likely complicating the escape of basins surrounded byinfeasible solutions we also allow to accept worse solutionsunder certain conditions This is accomplished by utilizing a


Table 1 The position and demand of static customers

No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30X 29 47 45 32 6 34 44 36 38 5 27 32 17 40 37 26 34 18 43 19 25 26 29 29 14 50 47 8 42 46Y 44 9 13 16 1 6 6 46 39 28 7 3 14 47 47 12 19 40 26 3 49 49 3 37 37 10 26 10 27 34Demand 05 02 04 2 17 03 17 07 02 12 17 05 03 15 04 08 14 14 1 17 02 09 03 02 05 03 09 05 14 2

Metropolis criterion like in simulated annealing Kirkpatrick[24] for inferior solutions 119909

lowast and accepting them with aprobability of (11) depending on the cost difference to theactual solution 119909 of the VNS process and the temperature119879 We update 119879 every 119899

119879iterations by an amount of (119879 times

119899119879)120575max where 1199020 denotes a random number on the interval

[0 1] where 120575max denotes the maximal VNS iterations andan initial temperature value is 119879

0= 10

119878119860 (119909lowast 119909) =

119909lowast if 119902

0gt exp(

119891 (119909lowast) minus 119891 (119909)

119879)

119909 if 1199020le exp(

119891 (119909lowast) minus 119891 (119909)

119879)

(11)

5 Numerical Experiments

In order to assess the performance of the improved variableneighborhood search algorithm to solve DVRP three testproblems with respect to different sizes (small medium andlarge) have been done We analyze the solution quality andefficiency of our proposed algorithm IVNS algorithm isimplemented by theC language and themain configurationof the computer is an Intel Core i3 18 GHz 2GB RAMrunning Windows XP

51 Case 1

511 Experimental Data and Setting In order to assess theperformance of the improved variable neighborhood searchalgorithm to solveDVRP the data sets from the literature [25]are used Firstly the VNS algorithm is applied to solve theDVRP and then the results are analyzed and compared withother existing algorithms

The dynamic distribution network is randomly gener-ated by the computer The distribution area is a square of50 km times 50 km 30 static demand customers and 10 thedynamic demand customers are randomly generated Eachcustomerrsquos demand is a randomnumber of [0 2] the vehiclersquoscapacity of distribution centers is 8 t and the maximumdriving distance of vehicle once is 100 kmThese informationincluding the coordinates and demand of 30 static customersand 10 dynamic customers are randomly generated by thecomputer the location of the distribution center is 119874 (25 km25 km) and the number of vehicles is 3 The objective isto arrange the delivery route of the vehicle reasonably sothat the distribution mileage is the shortest For simplicitythe distance between customer and distribution center usesthe straight-line distance The coordinates and demandsof static customers and dynamic customers are shown in

0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

Figure 5 The position relationship of customer and depot

Table 2 The position and demand of dynamic customers

No A B C D E F G H I JX 8 47 2 9 21 41 39 4 44 13Y 19 10 15 9 12 43 25 8 41 49Period 1 2 2 1 1 1 1 2 1 2Demand 1 15 16 3 14 03 02 22 28 1

Tables 1 and 2 respectively the specific position relationshipfor the customer and the customer and the customer anddistribution center are shown in Figure 5

The initial values of the various parameters for IVNSalgorithm are set as follows

(1) The parameter settings for simulated annealingaccepted criteria are initial temperature 119879

0= 10

every 119899119879

= 11989910 generation to update temperature119879119899+1

= 09 times 119879119899 119899119905= 1000 to end the algorithm

(2) The parameters value of the Shaking operation are asfollows 119901insert = 02 119901cross = 015 and 119901

119894cross = 01(3) The 119901

2minus119900119901119905value is 05 in local search

512 Numerical Results First distribution of fixed-demandcustomer is optimized solved and generated initial distribu-tion route as shown in Table 3

The customersrsquo demand information is updated at period1 at this moment the service requests are put forward by


Table 3 Initial distribution route of fixed-demand customer

Vehicle Route Utilization Mileagekm1 0-17-7-3-2-26-27-19-29-0 90 69922 0-4-6-12-23-11-16-0 70 50823 0-24-1-22-21-18-25-10-0 6125 75834 0-13-28-5-20-0 5250 68635 0-30-9-14-15-8-0 60 6865

Table 4 Scheduling plan at the period 1

Vehicle Route Utilization Mileagekm1 0-17-7-3-2-26-27-19-29-G-0 9379 67312 0-4-6-12-23-11-16-0 9025 48353 0-24-1-22-21-18-25-10-0 7443 75474 0-A-D-28-5-20-E-12-0 9132 84115 0-30-I-F-14-15-8-9-0 9917 7143


Vehicle Route Utilization Mileagekm1 0-17-7-3-B-2-26-27-19-29-G-0 9454 70612 0-4-6-12-23-11-16-0 9025 48353 0-24-1-22-21-18-J-25-10-0 8382 87934 0-13-E-20-5-28-D-H-C-A-0 9319 98795 0-30-9-I-F-14-15-8-0 9917 7143

Table 6 The comparison results of different algorithms

Algorithm IVNS GA TS TPAThe optimal value 47085 47085 47085 47085The worst value 749 1089 1367 879The average value 558 823 919 548Success rate 40 147 112 285The number of iterations 2547 4765 2817 3169

six dynamic customers of A D E F G and I Accordingto the known information on real-time optimization stagethe improved variable neighborhood search algorithm is usedto solve the distribution network at period 1 and outputscheduling plan as shown in Table 4 and the specific routeis shown in Figure 6

At period 2 the four dynamic demand customers of BC H and J have the service request and now accordingto customer requests we update the relevant informationBased on known real-time information on the dynamicdistribution network the improved variable neighborhoodsearch algorithm is applied to solve the distribution networkat period 2 and then the scheduling plan is output as shownin Table 5 while specific vehicle route is shown in Figure 7

These results show that the improved variable neigh-borhood search algorithm can solve the real-time require-ments of the dynamic vehicle routing problem the best-improvement search strategy and the insert and exchangeoperators speed the convergence of the algorithm and obtainbetter solution

0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

19 27

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B

2

26

Figure 6 The route at period 1

0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

1927

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B2

26


In order to verify that the improved variable neighbor-hood search for solving the dynamic vehicle routing problemwe compare it with the genetic algorithms (GAs) tabu search(TS) and a two-stage algorithm (TPA) based on the aboveexample The parameters setting of GA TS and TPA canbe seen in the relevant literature [22 25] A comprehensivecomparison is made from the optimal value the worst valuethe average value search success rate and the number ofiterationsThe results of the experiment are shown in Table 6

As can be seen from Table 6 the four algorithms consis-tently find an optimal solution but the worst and averagevalues have obvious differences there are differences insearch success rate and the number of iterationsThe order of


Table 7 The comparison results of Lackner and IVNS

No Dod () Average vehicle number Average driving distancekm Average calculating timesIVNS Lackner ARE () IVNS Lackner ARE () IVNS Lackner ARE()

R1

90 1425 1515 minus594 134382 127833 512 1687 1785 minus54970 1423 1502 minus526 133135 13361 minus036 1906 1987 minus40850 1523 1419 733 129381 132998 minus272 2569 2746 minus64530 1347 1422 minus527 128663 133786 minus383 4834 4697 29210 148 139 647 125918 127806 minus148 7001 6801 294

C1

90 1125 1178 minus450 123547 14796 minus1650 631 647 minus24770 1126 1187 minus514 103178 12613 minus1820 1108 1081 25050 1221 1197 201 107232 123606 minus1325 1777 1832 minus30030 1057 1054 028 9708 106689 minus901 2806 2708 36210 1023 1078 minus510 89568 99635 minus1010 4246 4425 minus405

RC1

90 15 15 000 150643 147521 212 1766 1704 36470 1358 1465 minus730 151323 148844 167 2412 285 minus153750 1319 1257 493 151912 144807 491 4984 4798 38830 1298 1343 minus335 148996 143971 349 4412 4621 minus45210 1288 1315 minus205 143179 142689 034 8678 8821 minus162

R2

90 323 356 minus927 104589 119333 minus1236 141 139 14470 379 361 499 103422 111693 minus741 1932 2175 minus111750 333 365 minus877 101211 113878 minus1112 3211 3208 00930 422 487 minus1335 98742 108542 minus903 5522 5829 minus52710 622 61 197 951 105285 minus967 7101 7011 128

C2

90 314 302 397 6955 79246 minus1224 621 552 125070 367 39 minus590 67186 74378 minus967 914 991 minus77750 388 37 486 61098 68925 minus1136 18 18 00030 319 35 minus886 65523 63233 362 2718 2835 minus41310 4 4 000 58232 62908 minus743 6282 6367 minus134

RC2

90 5 5 000 135121 147676 minus850 996 1089 minus85470 398 41 minus293 125632 134676 minus672 1903 1898 02650 432 445 minus292 118913 126929 minus632 2578 2796 minus78030 521 565 minus779 115135 124485 minus751 3867 3911 minus11310 68 71 minus423 118311 12209 minus310 571 5729 minus033

Average 8637 881 minus201 111863 118372 minus550 3246 3303 minus172

the four algorithms of search success rate from the smallest tolargest is tabu search algorithm two-stage algorithm geneticalgorithm and IVNS the order for the worst value is asfollows the average value from small to big is IVNS two-stagealgorithm genetic algorithm and tabu search algorithm itreflects the IVNS has the better global search capabilityThe number of iterations for four algorithms from thesmallest to the largest is IVNS tabu search algorithm two-stage algorithm and genetic algorithm this also shows thatconvergence rate of IVNS is faster than the other algorithmsand it can be more appropriate to solve dynamic vehiclerouting problem to some extent

52 Case 2 The experimental test uses the benchmarkdata which was 100-node VRPTW calculation example andcompiled by Solomon in 1987 Every sample contains 100nodes and distributes into 100 times 100 Euclidean plane Thesample is divided into six categories R1 R2 C1 C2 RC1and RC2 DVRPTW adopts the Lackner dynamic test data

set which is designed in 2004 based on the Solomon exampleFor each question in the Solomon example there are five datasets corresponding to it They are 90 70 50 30 and10 five dynamic degree

The dynamic degree is described as follows

Dod =119873119889

119873119889+ 119873119904

times 100 (12)

119873119889is the number of dynamic customer demand and 119873

119904is

the number of static customer demandTable 7 gives the comparison results of the Solomon

problem to IVNS and Lackner under different dynamicdegrees From the number of vehicles the average drivingdistance and the average calculation time we compared thecalculation results and the relative error For five dynamicdegrees the average values of the number of vehiclesrespectively are 8637 and 881 for IVNS and Lackner the


Table 8 The comparison results of Gehring and IVNS

Instance set Objective The number of vehicles The calculation times Total demand New demand Reject demandGehring IVNS Gehring IVNS Gehring IVNS

C1 10 1 434942 432659 100 95 207 201 720 122 0C2 10 1 217784 21763 31 30 216 212 640 114 2R1 10 1 563898 563831 103 100 1896 1903 701 90 0R2 10 1 370147 369822 19 20 237 228 550 49 2RC1 10 1 480561 480057 97 98 3168 3135 853 80 3RC2 10 1 335343 331561 23 21 5076 4998 690 189 0Average 4004458 3992600 6217 6067 27900 27410 69233 10733 117

relative error is minus201 the average values of driving distancerespectively are 111863 and 118372 for IVNS and Lackner andthe relative error is minus550 the average values of calculationtime respectively are 3246 and 3303 for IVNS and Lacknerand the relative error is minus172 According to Table 7 thefollowing conclusions can be drawn

(1) In terms of the total driving distance relative tothe best results which are obtained by Lacknerrsquosvarious algorithms except for the RC1 R1 and C2the proposed algorithm can obtain better calculationresults for other typersquos dynamic degrees

(2) The present algorithm in which the average com-putation time is less than 90 seconds fully meetsthe requirements of real-time schedulingThe averagecalculation time of IVNS is lower than Lackner itdemonstrates that the algorithm has better searchcapabilities

53 Case 3 In order to verify the proposed variableneighborhood search algorithm to solve the large-scaledynamic vehicle routing problem effectively this paperextends to large-scale test instances with 1000 customerwhich are presented by Gehring and Homberger Table 8gives the comparison results of the first path adjustmentfor the part instance based on Gehring and our proposedalgorithm

As shown in Table 8 the results of the objective value thenumber of vehicles and the calculation time are comparedbased on Gehring and IVNS 10733 average new demands areoccurred and 69233 total demands are handled in the pathadjustment instance only 117 customer demands failed to besatisfied while the service level is close to 100 In dealingwith a large-scale problem the algorithm running time is animportant evaluation index the average running time of ourproposed algorithm is 27410 s it is smaller than GehringrsquosalgorithmThe number of vehicle is closely related to the costand it is decreased from 6217 to 6067 The objective valueis reduced from 4004458 to 3992600 it demonstrates thatthe results have been improved According to the analysis tosome extent the proposed algorithm can solve the large-scaledynamic VRP and improve the results

6 Conclusions

In this research we proposed a formulation for a dynamicvehicle routing problemWe also presented an improved vari-able neighborhood search (IVNS) metaheuristic for DVRPIn the initial solution the routes are constructed by solving avehicle routing problem for each day using the Clarke andWright savings algorithm Clarke and Wright [20] In theshaking execution there are two neighborhood structures toachieve the shaking insert and exchange This paper selects2minus119900119901119905 and 3minus119900119901119905 as a local search operator in order to obtainthe good quality local optimal solution in a short period Inorder to accelerate the convergence speed and improve thesolution quality the later optimization process is proposed inthe IVNS algorithm To test the performance of the proposedimproved variable neighborhood search algorithm we testthree problemswith respect to different sizes (small mediumand large) and compare the results with other algorithmsTheresults show that the proposed model and the algorithm areeffective and can solve theDVRPwithin a very short time andimprove the quality of solution to some extent

Acknowledgments

Funding for this research is supported by theNational ScienceFoundation of China under Grants nos 90924020 70971005and 71271013 the Societal Science Foundation of China underGrant no 11AZD096China Postdoctoral Science Foundationfunded project no 2012M520008 and Quality InspectionProjects no 200910088 and no 201010268

References

[1] P Hansen and N Mladenovic ldquoVariable neighborhood searchfor the p-medianrdquo Location Science vol 5 no 4 pp 207ndash2261997

[2] P Hansen N Mladenovic and D Perez-Britos ldquoVariableneighborhood decomposition searchrdquo Journal of Heuristics vol7 no 4 pp 335ndash350 2001

[3] M Gendreau and J-Y Potvin ldquoDynamic vehicle routing anddispatchingrdquo in Fleet Management and Logistic T G Crainicand G Laporte Eds pp 115ndash226 Kluwer Academic 1998

[4] B Fleischmann S Gnutzmann and E Sandvoszlig ldquoDynamicvehicle routing based on online traffic informationrdquoTransporta-tion Science vol 38 no 4 pp 420ndash433 2004


[5] J Yang P Jaillet and H Mahmassani ldquoReal-time multivehicletruckload pickup and delivery problemsrdquo Transportation Sci-ence vol 38 no 2 pp 135ndash148 2004

[6] R Montemanni L M Gambardella A E Rizzoli and AV Donati ldquoAnt colony system for a dynamic vehicle routingproblemrdquo Journal of Combinatorial Optimization vol 10 no 4pp 327ndash343 2005

[7] M Gendreau F Guertin J Y Potvin and R Seguin ldquoNeigh-borhood search heuristics for a dynamic vehicle dispatchingproblem with pick-ups and deliveriesrdquo Transportation ResearchC vol 14 no 3 pp 157ndash174 2006

[8] A Goel and V Gruhn ldquoSolving a dynamic real-life vehiclerouting problemrdquo Operations Research Proceedings vol 2005no 8 pp 367ndash372 2006

[9] J Schonberger and H Kopfer ldquoOnline decision making andautomatic decision model adaptationrdquo Computers and Opera-tions Research vol 36 no 6 pp 1740ndash1750 2009

[10] C Novoa and R Storer ldquoAn approximate dynamic program-ming approach for the vehicle routing problem with stochasticdemandsrdquo European Journal of Operational Research vol 196no 2 pp 509ndash515 2009

[11] R M Branchini V A Armentano and A LoslashkketangenldquoAdaptive granular local search heuristic for a dynamic vehiclerouting problemrdquo Computers and Operations Research vol 36no 11 pp 2955ndash2968 2009

[12] J Muller ldquoApproximative solutions to the bicriterion vehiclerouting problem with time windowsrdquo European Journal ofOperational Research vol 202 no 1 pp 223ndash231 2010

[13] O Braysy ldquoA reactive variable neighborhood search for thevehicle-routing problem with time windowsrdquo INFORMS Jour-nal on Computing vol 15 no 4 pp 347ndash368 2003

[14] M Polacek R F Hartl K Doerner and M Reimann ldquoAvariable neighborhood search for the multi depot vehiclerouting problem with time windowsrdquo Journal of Heuristics vol10 no 6 pp 613ndash627 2004

[15] J Kytojoki T Nuortio O Braysy and M Gendreau ldquoAn effi-cient variable neighborhood search heuristic for very large scalevehicle routing problemsrdquo Computers and Operations Researchvol 34 no 9 pp 2743ndash2757 2007

[16] A Goel and V Gruhn ldquoA general vehicle routing problemrdquoEuropean Journal of Operational Research vol 191 no 3 pp650ndash660 2008

[17] V C Hemmelmayr K F Doerner and R F Hartl ldquoA variableneighborhood search heuristic for periodic routing problemsrdquoEuropean Journal of Operational Research vol 195 no 3 pp791ndash802 2009

[18] K Fleszar I H Osman and K S Hindi ldquoA variable neighbour-hood search algorithm for the open vehicle routing problemrdquoEuropean Journal of Operational Research vol 195 no 3 pp803ndash809 2009

[19] A Larsen ldquoThe dynamic vehicle routing problemrdquo IMM DTU2001

[20] G Clarke and J W Wright ldquoScheduling of vehicles froma central depot to a number of delivery pointsrdquo OperationsResearch vol 12 no 4 pp 568ndash581 1964

[21] E Taillard P Badeau M Gendreau F Guertin and J Y PotvinldquoA tabu search heuristic for the vehicle routing problem withsoft time windowsrdquo Transportation Science vol 31 no 2 pp170ndash186 1997

[22] ZWang J Zhang andXWang ldquoAmodified variable neighbor-hood search algorithm for multi depot vehicle routing problem

with time windowsrdquo Chinese Journal of Management Sciencevol 19 no 2 pp 99ndash108 2011

[23] M Gendreau A Hertz G Laporte andM Stan ldquoA generalizedinsertion heuristic for the traveling salesman problemwith timewindowsrdquoOperations Research vol 46 no 3 pp 330ndash335 1998

[24] S Kirkpatrick C D Gelatt and M P Vecchi ldquoOptimization bysimulated annealingrdquo Science vol 220 no 4598 pp 671ndash6801983

[25] W Xu G Xianlong and D Ying ldquoResearch on dynamic vehiclerouting problem based on two-phase algorithmrdquo Control andDecision vol 27 no 2 pp 175ndash181 2012

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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


2 Literature Reviews

TheDVRP problem is closely related to the actual productionand life In recent years DVRP gradually becomes a hot topicBoth domestic and foreign scholars mainly focus on the con-struction of different scheduling models and the designing ofsimple and efficient heuristic algorithms A survey of resultsachieved on the different types of DVRPs can be found inGendreau and Potvin [3]The dynamic full-truckload pickupand delivery problem has been studied by Fleischmann et al[4] and Yang et al [5] Montemanni et al [6] proposed thevehicle schedulingmodel with randomdynamic demand andsolved it using an ant colony algorithm In Gendreau et al[7] neighborhood search heuristics for dial-a-ride problemsare finally presented Goel and Gruhn [8] studied the math-ematical model of dynamic vehicle routing problems underthe conditions of the real-life information Schonbergerand Kopfer [9] studied the real-time decision-making andautonomous decision-making of DVRP Novoa and Storer[10] introduced the approximate solution algorithm withstochastic demands Branchini et al [11] presented localsearch algorithm for a dynamic vehicle routing problemMuller [12] studied DVRP with time window and analyzedthe algorithm for the model suboptimal solution

In the past ten decades a tremendous amount of workin the field of vehicle routing problems has been publishedespecially literature based on VNS The Braysy [13] gavethe internal design of the VND and RVNS algorithm indetail analyzed the VRPTW problem and indicated theVND algorithm as one of the most effective ways to solveVRPTW problems Polacek et al [14] designed VNS to solvethe multidepot vehicle routing problem with time windowsMDVRPTWHis algorithm used the neighborhood structureof swap and cross to do shaking operation for the currentsolution to do local search with a constrained 3-opt operatorto accept the part of the poor solution to avoid getting intoa local optimum for the algorithm by Threshold AcceptingKytojoki et al [15] designed the guided VNS algorithmto handle the 32 existing large-scale VRP problems andcompared it to the TS algorithm The result showed thatthe VNS algorithm was more effective than TS algorithmin solving time Goel and Gruhn [16] introduced the RVNSto solve the general VRP problem including time windowsvehicle constraints path constraints travel departure timeconstraints capacity constraints the order models compati-bility constraintsmultisupplier point of the orders and trans-port and service position constraints Hemmelmayr et al [17]proposed the VNS algorithm for periodical VRP problemadopted the saving algorithm for the construction of theinitial solution designed the move and cross neighborhoodused 3-opt operator as local search strategies and contrastedit with other research results Fleszar et al [18] adopted VNSalgorithm to solve the open-loop VRP problem and tested 16benchmark problems

Due to the complexity of the problem the current solvingquality and efficiency for the DVRP are far from the practicalrequirements So there are many problems need to make in-depth research such as how to seek feasible solutions howto prevent falling into local optimum and how to control

Advance request customer (static)Immediate request customer (dynamic)Depot

Depot

New route segmentPlanned routeCurrent positionof vehicle

Figure 1 An example for dynamic vehicle routing problem

the solution within the acceptable range This paper presentsan improved variable neighborhood search algorithm tosolve DVRP it integrates local search operator optimizationprocess and the simulated annealing algorithm into the VNSalgorithm framework Through the comparison with otheralgorithms it shows that the proposed algorithm can get thebetter solution

3 Problem Descriptions

31 Problem Definition Larsen [19] defined DVRP with twoaspects not all information relevant to the planning of theroutes is known by the planner when the routing processbegins information can change after the initial routes havebeen constructed And he illustrated a dynamic vehiclerouting situation The simple example is shown in Figure 1As seen from it two uncapacitated vehicles must serviceboth advance and immediate request customers without timewindows The advance request customers are represented byyellow nodes while those that are immediate requests aredepicted by black nodesThe solid blue lines represent the tworoutes the dispatcher has planned prior to the vehicles leavingthe depot The two thick arcs indicate the vehicle positions atthe time the dynamic requests are received Ideally the newcustomers should be inserted into the already planned routeswithout the order of the nonvisited customers being changedandwithminimal delayThis is the case depicted on the right-hand side route However in practice the insertion of newcustomers will usually be a much more complicated task andwill imply a replanning of the nonvisited part of the routesystem This is illustrated by the left-hand side route whereservicing the new customer creates a large detour

We can also describe the DVRP with a mathematicapproach The problem is defined on a complete graph 119866 =

(119881 119864) where 119881 = 1199070 119907

119871 is the vertex set and 119864 =

(119907119894 119907119895) 119907119894 119907119895isin 119881 119894 = 119895 is the arc set The verticesrsquo set 119863 =

1199070 corresponds to the depots Each vertex 119907

119894isin 119881 has several

nonnegative weights associated with it namely a demand 119889119894

a arrival time 119886119894 the waiting time 119908

119894 service time 119904

119894 and an

earliest 119890119894and latest 119897

119894possible start time for the service which

define a time window [119890119894 119897119894] 119862119905119894119895is the transportation cost



119896 119879 refers to




119910119894119896=


0 other

119909119905

119894119895

=


0 other(1)



min119865 = 119888119891times sum

119894isin119881

sum

119895isin1198811198810

119879

sum

119905=1

119909119905

119894119895+ 119888119905

times sum

119894isin119881

sum

119895isin1198811198810

119879

sum

119905=1

(119888119905

119894119895times 119909119905

119894119895)

(2)


119871

sum

119894=0119894 = 119895

119879

sum

119905=1

119909119905

119894119895= 1 119895 = 1 2 119871

119871

sum

119895=1119895 = 119894

119879

sum

119905=1

119909119905

119894119895= 1 119894 = 1 2 119871

(3)


119871

sum

119895=1

119879

sum

119905=1

119909119905

0119895le 119870 (4)


119870

sum

119896=1

119896 (119910119894119896minus 119910119895119896) ge 119872(

119879

sum

119905=1

119909119905

119894119895minus 1) forall119894 119895 119894 = 119895

119870

sum

119896=1

119896 (119910119894119896minus 119910119895119896) le 119872(1 minus

119879

sum

119905=1

119909119905

119894119895) forall119894 119895 119894 = 119895

(5)


119870

sum

119896=1

119910119894119896= 1 forall119894 = 1 2 119871 (6)


119871

sum

119894=1

119889119894119910119894119896le 119902119894 forall119896 = 1 2 119870 (7)


sum

119894119895isin119878times119878119894 = 119895

119879

sum

119905=1

119909119905


(8)


119886119894+ 119908119894+ 119904119894+ 119888119905

119894119895minus 119872(1 minus 119909

119905

119894119895) le 119886119895

119886119895le 119897119895

119908119894= max 119890

119894minus 119886119894 0

(9)


119909119905

119894119895= 0 1 forall119894 119895 119896

119910119894119896= 0 1 forall119894 119896

(10)







end forend while

end forend Begin














Start









Yes

Yes

Yes

No

No

Yes

No

EndNo



119891(119909119897) lt 119891(119909119887)


119896 larr (119896 mod 119873max ) + 1

119891(119909119897) lt 119891(119909)

119909 larr 119909119897 119896 larr 1 119894 larr 119894 + 1

119894 larr 119894 + 1

119894 le 119899119905

119873119896 (119896 = 1 2 119873max )



119894119894 minus 1

119894 + 1 119895 + 1

119894 + 1 119895 + 1

119895 + 2

119895 + 2 119895 + 3

119896

119894

119896

119896 minus 1 119897

119897

119897 + 1

119894 minus 1 119895 + 3

119896 minus 1 119897 + 1

Depot

Depot

Depot

Depot

(a) Insert

119894 minus 1

119894 minus 1

119894 + 2

119894 + 2

119895

119895

119895 minus 1

119895 minus 1

119894 + 1

119894 + 1

119895 + 1

119895 + 1 119895 + 2

119895 + 2

119894

119894

119894 minus 2

119894 minus 2

119895 minus 2

119895 minus 2

Depot

Depot

Depot

Depot

(b) Cross

119894 minus 1

119894 minus 1

119895

119895

119895 + 1

119895 + 1

119894

119894

119896 minus 1

119896 minus 1

119896 119897

119896119897

119897 + 1

119897 + 1

DepotDepot

Depot Depot

Customer nodeRoute

119894

(c) 119894Cross



119895 119895119895 + 1119895 + 1

119894 119894 119894 + 1119894 + 1

DepotDepot

(a) 2 minus 119900119901119905

Customer nodeRoute

119894 + 1 119894 + 1

119895119895

119895 + 1119895 + 1

119896 119896119896 + 1119896 + 1

119894

119894

119894

Depot Depot

(b) 3 minus 119900119901119905




















119903lowast

= 1199031015840 119903 = 119903

1015840 119888lowast = 1198881015840 119888 = 119888






No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30X 29 47 45 32 6 34 44 36 38 5 27 32 17 40 37 26 34 18 43 19 25 26 29 29 14 50 47 8 42 46Y 44 9 13 16 1 6 6 46 39 28 7 3 14 47 47 12 19 40 26 3 49 49 3 37 37 10 26 10 27 34Demand 05 02 04 2 17 03 17 07 02 12 17 05 03 15 04 08 14 14 1 17 02 09 03 02 05 03 09 05 14 2






0= 10

119878119860 (119909lowast 119909) =

119909lowast if 119902

0gt exp(

119891 (119909lowast) minus 119891 (119909)

119879)

119909 if 1199020le exp(

119891 (119909lowast) minus 119891 (119909)

119879)

(11)



51 Case 1



0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot







0= 10

every 119899119879




119894cross = 01(3) The 119901
















0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

19 27

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B

2

26


0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

1927

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B2

26







R1


C1


RC1


R2


C2


RC2







Dod =119873119889

119873119889+ 119873119904

times 100 (12)


119904is












6 Conclusions


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119896 119879 refers to




119910119894119896=


0 other

119909119905

119894119895

=


0 other(1)



min119865 = 119888119891times sum

119894isin119881

sum

119895isin1198811198810

119879

sum

119905=1

119909119905

119894119895+ 119888119905

times sum

119894isin119881

sum

119895isin1198811198810

119879

sum

119905=1

(119888119905

119894119895times 119909119905

119894119895)

(2)


119871

sum

119894=0119894 = 119895

119879

sum

119905=1

119909119905

119894119895= 1 119895 = 1 2 119871

119871

sum

119895=1119895 = 119894

119879

sum

119905=1

119909119905

119894119895= 1 119894 = 1 2 119871

(3)


119871

sum

119895=1

119879

sum

119905=1

119909119905

0119895le 119870 (4)


119870

sum

119896=1

119896 (119910119894119896minus 119910119895119896) ge 119872(

119879

sum

119905=1

119909119905

119894119895minus 1) forall119894 119895 119894 = 119895

119870

sum

119896=1

119896 (119910119894119896minus 119910119895119896) le 119872(1 minus

119879

sum

119905=1

119909119905

119894119895) forall119894 119895 119894 = 119895

(5)


119870

sum

119896=1

119910119894119896= 1 forall119894 = 1 2 119871 (6)


119871

sum

119894=1

119889119894119910119894119896le 119902119894 forall119896 = 1 2 119870 (7)


sum

119894119895isin119878times119878119894 = 119895

119879

sum

119905=1

119909119905


(8)


119886119894+ 119908119894+ 119904119894+ 119888119905

119894119895minus 119872(1 minus 119909

119905

119894119895) le 119886119895

119886119895le 119897119895

119908119894= max 119890

119894minus 119886119894 0

(9)


119909119905

119894119895= 0 1 forall119894 119895 119896

119910119894119896= 0 1 forall119894 119896

(10)







end forend while

end forend Begin














Start









Yes

Yes

Yes

No

No

Yes

No

EndNo



119891(119909119897) lt 119891(119909119887)


119896 larr (119896 mod 119873max ) + 1

119891(119909119897) lt 119891(119909)

119909 larr 119909119897 119896 larr 1 119894 larr 119894 + 1

119894 larr 119894 + 1

119894 le 119899119905

119873119896 (119896 = 1 2 119873max )



119894119894 minus 1

119894 + 1 119895 + 1

119894 + 1 119895 + 1

119895 + 2

119895 + 2 119895 + 3

119896

119894

119896

119896 minus 1 119897

119897

119897 + 1

119894 minus 1 119895 + 3

119896 minus 1 119897 + 1

Depot

Depot

Depot

Depot

(a) Insert

119894 minus 1

119894 minus 1

119894 + 2

119894 + 2

119895

119895

119895 minus 1

119895 minus 1

119894 + 1

119894 + 1

119895 + 1

119895 + 1 119895 + 2

119895 + 2

119894

119894

119894 minus 2

119894 minus 2

119895 minus 2

119895 minus 2

Depot

Depot

Depot

Depot

(b) Cross

119894 minus 1

119894 minus 1

119895

119895

119895 + 1

119895 + 1

119894

119894

119896 minus 1

119896 minus 1

119896 119897

119896119897

119897 + 1

119897 + 1

DepotDepot

Depot Depot

Customer nodeRoute

119894

(c) 119894Cross



119895 119895119895 + 1119895 + 1

119894 119894 119894 + 1119894 + 1

DepotDepot

(a) 2 minus 119900119901119905

Customer nodeRoute

119894 + 1 119894 + 1

119895119895

119895 + 1119895 + 1

119896 119896119896 + 1119896 + 1

119894

119894

119894

Depot Depot

(b) 3 minus 119900119901119905




















119903lowast

= 1199031015840 119903 = 119903

1015840 119888lowast = 1198881015840 119888 = 119888






No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30X 29 47 45 32 6 34 44 36 38 5 27 32 17 40 37 26 34 18 43 19 25 26 29 29 14 50 47 8 42 46Y 44 9 13 16 1 6 6 46 39 28 7 3 14 47 47 12 19 40 26 3 49 49 3 37 37 10 26 10 27 34Demand 05 02 04 2 17 03 17 07 02 12 17 05 03 15 04 08 14 14 1 17 02 09 03 02 05 03 09 05 14 2






0= 10

119878119860 (119909lowast 119909) =

119909lowast if 119902

0gt exp(

119891 (119909lowast) minus 119891 (119909)

119879)

119909 if 1199020le exp(

119891 (119909lowast) minus 119891 (119909)

119879)

(11)



51 Case 1



0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot







0= 10

every 119899119879




119894cross = 01(3) The 119901
















0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

19 27

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B

2

26


0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

1927

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B2

26







R1


C1


RC1


R2


C2


RC2







Dod =119873119889

119873119889+ 119873119904

times 100 (12)


119904is












6 Conclusions


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra













end forend while

end forend Begin














Start









Yes

Yes

Yes

No

No

Yes

No

EndNo



119891(119909119897) lt 119891(119909119887)


119896 larr (119896 mod 119873max ) + 1

119891(119909119897) lt 119891(119909)

119909 larr 119909119897 119896 larr 1 119894 larr 119894 + 1

119894 larr 119894 + 1

119894 le 119899119905

119873119896 (119896 = 1 2 119873max )



119894119894 minus 1

119894 + 1 119895 + 1

119894 + 1 119895 + 1

119895 + 2

119895 + 2 119895 + 3

119896

119894

119896

119896 minus 1 119897

119897

119897 + 1

119894 minus 1 119895 + 3

119896 minus 1 119897 + 1

Depot

Depot

Depot

Depot

(a) Insert

119894 minus 1

119894 minus 1

119894 + 2

119894 + 2

119895

119895

119895 minus 1

119895 minus 1

119894 + 1

119894 + 1

119895 + 1

119895 + 1 119895 + 2

119895 + 2

119894

119894

119894 minus 2

119894 minus 2

119895 minus 2

119895 minus 2

Depot

Depot

Depot

Depot

(b) Cross

119894 minus 1

119894 minus 1

119895

119895

119895 + 1

119895 + 1

119894

119894

119896 minus 1

119896 minus 1

119896 119897

119896119897

119897 + 1

119897 + 1

DepotDepot

Depot Depot

Customer nodeRoute

119894

(c) 119894Cross



119895 119895119895 + 1119895 + 1

119894 119894 119894 + 1119894 + 1

DepotDepot

(a) 2 minus 119900119901119905

Customer nodeRoute

119894 + 1 119894 + 1

119895119895

119895 + 1119895 + 1

119896 119896119896 + 1119896 + 1

119894

119894

119894

Depot Depot

(b) 3 minus 119900119901119905




















119903lowast

= 1199031015840 119903 = 119903

1015840 119888lowast = 1198881015840 119888 = 119888






No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30X 29 47 45 32 6 34 44 36 38 5 27 32 17 40 37 26 34 18 43 19 25 26 29 29 14 50 47 8 42 46Y 44 9 13 16 1 6 6 46 39 28 7 3 14 47 47 12 19 40 26 3 49 49 3 37 37 10 26 10 27 34Demand 05 02 04 2 17 03 17 07 02 12 17 05 03 15 04 08 14 14 1 17 02 09 03 02 05 03 09 05 14 2






0= 10

119878119860 (119909lowast 119909) =

119909lowast if 119902

0gt exp(

119891 (119909lowast) minus 119891 (119909)

119879)

119909 if 1199020le exp(

119891 (119909lowast) minus 119891 (119909)

119879)

(11)



51 Case 1



0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot







0= 10

every 119899119879




119894cross = 01(3) The 119901
















0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

19 27

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B

2

26


0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

1927

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B2

26







R1


C1


RC1


R2


C2


RC2







Dod =119873119889

119873119889+ 119873119904

times 100 (12)


119904is












6 Conclusions


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start









Yes

Yes

Yes

No

No

Yes

No

EndNo



119891(119909119897) lt 119891(119909119887)


119896 larr (119896 mod 119873max ) + 1

119891(119909119897) lt 119891(119909)

119909 larr 119909119897 119896 larr 1 119894 larr 119894 + 1

119894 larr 119894 + 1

119894 le 119899119905

119873119896 (119896 = 1 2 119873max )



119894119894 minus 1

119894 + 1 119895 + 1

119894 + 1 119895 + 1

119895 + 2

119895 + 2 119895 + 3

119896

119894

119896

119896 minus 1 119897

119897

119897 + 1

119894 minus 1 119895 + 3

119896 minus 1 119897 + 1

Depot

Depot

Depot

Depot

(a) Insert

119894 minus 1

119894 minus 1

119894 + 2

119894 + 2

119895

119895

119895 minus 1

119895 minus 1

119894 + 1

119894 + 1

119895 + 1

119895 + 1 119895 + 2

119895 + 2

119894

119894

119894 minus 2

119894 minus 2

119895 minus 2

119895 minus 2

Depot

Depot

Depot

Depot

(b) Cross

119894 minus 1

119894 minus 1

119895

119895

119895 + 1

119895 + 1

119894

119894

119896 minus 1

119896 minus 1

119896 119897

119896119897

119897 + 1

119897 + 1

DepotDepot

Depot Depot

Customer nodeRoute

119894

(c) 119894Cross



119895 119895119895 + 1119895 + 1

119894 119894 119894 + 1119894 + 1

DepotDepot

(a) 2 minus 119900119901119905

Customer nodeRoute

119894 + 1 119894 + 1

119895119895

119895 + 1119895 + 1

119896 119896119896 + 1119896 + 1

119894

119894

119894

Depot Depot

(b) 3 minus 119900119901119905




















119903lowast

= 1199031015840 119903 = 119903

1015840 119888lowast = 1198881015840 119888 = 119888






No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30X 29 47 45 32 6 34 44 36 38 5 27 32 17 40 37 26 34 18 43 19 25 26 29 29 14 50 47 8 42 46Y 44 9 13 16 1 6 6 46 39 28 7 3 14 47 47 12 19 40 26 3 49 49 3 37 37 10 26 10 27 34Demand 05 02 04 2 17 03 17 07 02 12 17 05 03 15 04 08 14 14 1 17 02 09 03 02 05 03 09 05 14 2






0= 10

119878119860 (119909lowast 119909) =

119909lowast if 119902

0gt exp(

119891 (119909lowast) minus 119891 (119909)

119879)

119909 if 1199020le exp(

119891 (119909lowast) minus 119891 (119909)

119879)

(11)



51 Case 1



0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot







0= 10

every 119899119879




119894cross = 01(3) The 119901
















0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

19 27

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B

2

26


0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

1927

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B2

26







R1


C1


RC1


R2


C2


RC2







Dod =119873119889

119873119889+ 119873119904

times 100 (12)


119904is












6 Conclusions


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119894119894 minus 1

119894 + 1 119895 + 1

119894 + 1 119895 + 1

119895 + 2

119895 + 2 119895 + 3

119896

119894

119896

119896 minus 1 119897

119897

119897 + 1

119894 minus 1 119895 + 3

119896 minus 1 119897 + 1

Depot

Depot

Depot

Depot

(a) Insert

119894 minus 1

119894 minus 1

119894 + 2

119894 + 2

119895

119895

119895 minus 1

119895 minus 1

119894 + 1

119894 + 1

119895 + 1

119895 + 1 119895 + 2

119895 + 2

119894

119894

119894 minus 2

119894 minus 2

119895 minus 2

119895 minus 2

Depot

Depot

Depot

Depot

(b) Cross

119894 minus 1

119894 minus 1

119895

119895

119895 + 1

119895 + 1

119894

119894

119896 minus 1

119896 minus 1

119896 119897

119896119897

119897 + 1

119897 + 1

DepotDepot

Depot Depot

Customer nodeRoute

119894

(c) 119894Cross



119895 119895119895 + 1119895 + 1

119894 119894 119894 + 1119894 + 1

DepotDepot

(a) 2 minus 119900119901119905

Customer nodeRoute

119894 + 1 119894 + 1

119895119895

119895 + 1119895 + 1

119896 119896119896 + 1119896 + 1

119894

119894

119894

Depot Depot

(b) 3 minus 119900119901119905




















119903lowast

= 1199031015840 119903 = 119903

1015840 119888lowast = 1198881015840 119888 = 119888






No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30X 29 47 45 32 6 34 44 36 38 5 27 32 17 40 37 26 34 18 43 19 25 26 29 29 14 50 47 8 42 46Y 44 9 13 16 1 6 6 46 39 28 7 3 14 47 47 12 19 40 26 3 49 49 3 37 37 10 26 10 27 34Demand 05 02 04 2 17 03 17 07 02 12 17 05 03 15 04 08 14 14 1 17 02 09 03 02 05 03 09 05 14 2






0= 10

119878119860 (119909lowast 119909) =

119909lowast if 119902

0gt exp(

119891 (119909lowast) minus 119891 (119909)

119879)

119909 if 1199020le exp(

119891 (119909lowast) minus 119891 (119909)

119879)

(11)



51 Case 1



0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot







0= 10

every 119899119879




119894cross = 01(3) The 119901
















0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

19 27

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B

2

26


0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

1927

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B2

26







R1


C1


RC1


R2


C2


RC2







Dod =119873119889

119873119889+ 119873119904

times 100 (12)


119904is












6 Conclusions


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119895 119895119895 + 1119895 + 1

119894 119894 119894 + 1119894 + 1

DepotDepot

(a) 2 minus 119900119901119905

Customer nodeRoute

119894 + 1 119894 + 1

119895119895

119895 + 1119895 + 1

119896 119896119896 + 1119896 + 1

119894

119894

119894

Depot Depot

(b) 3 minus 119900119901119905




















119903lowast

= 1199031015840 119903 = 119903

1015840 119888lowast = 1198881015840 119888 = 119888






No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30X 29 47 45 32 6 34 44 36 38 5 27 32 17 40 37 26 34 18 43 19 25 26 29 29 14 50 47 8 42 46Y 44 9 13 16 1 6 6 46 39 28 7 3 14 47 47 12 19 40 26 3 49 49 3 37 37 10 26 10 27 34Demand 05 02 04 2 17 03 17 07 02 12 17 05 03 15 04 08 14 14 1 17 02 09 03 02 05 03 09 05 14 2






0= 10

119878119860 (119909lowast 119909) =

119909lowast if 119902

0gt exp(

119891 (119909lowast) minus 119891 (119909)

119879)

119909 if 1199020le exp(

119891 (119909lowast) minus 119891 (119909)

119879)

(11)



51 Case 1



0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot







0= 10

every 119899119879




119894cross = 01(3) The 119901
















0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

19 27

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B

2

26


0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

1927

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B2

26







R1


C1


RC1


R2


C2


RC2







Dod =119873119889

119873119889+ 119873119904

times 100 (12)


119904is












6 Conclusions


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30X 29 47 45 32 6 34 44 36 38 5 27 32 17 40 37 26 34 18 43 19 25 26 29 29 14 50 47 8 42 46Y 44 9 13 16 1 6 6 46 39 28 7 3 14 47 47 12 19 40 26 3 49 49 3 37 37 10 26 10 27 34Demand 05 02 04 2 17 03 17 07 02 12 17 05 03 15 04 08 14 14 1 17 02 09 03 02 05 03 09 05 14 2






0= 10

119878119860 (119909lowast 119909) =

119909lowast if 119902

0gt exp(

119891 (119909lowast) minus 119891 (119909)

119879)

119909 if 1199020le exp(

119891 (119909lowast) minus 119891 (119909)

119879)

(11)



51 Case 1



0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot







0= 10

every 119899119879




119894cross = 01(3) The 119901
















0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

19 27

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B

2

26


0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

1927

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B2

26







R1


C1


RC1


R2


C2


RC2







Dod =119873119889

119873119889+ 119873119904

times 100 (12)


119904is












6 Conclusions


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

19 27

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B

2

26


0

5

10

15

20

25

30

35

40

45

50

55

0 5 10 15 20 25 30 35 40 45 50 55(km)

(km

)

Depot

24

1

2221J

1825

10

A

C 13E 16 4

17G

3

1927

30

29

9I

F8

15 14

D

H 28

520 11 23

12

67

B2

26







R1


C1


RC1


R2


C2


RC2







Dod =119873119889

119873119889+ 119873119904

times 100 (12)


119904is












6 Conclusions


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












R1


C1


RC1


R2


C2


RC2







Dod =119873119889

119873119889+ 119873119904

times 100 (12)


119904is












6 Conclusions


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















6 Conclusions


Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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