Research ArticleOptimization of Vehicle Routing with Pickup Based onMultibatch Production

Hongtao Hu1 JiaoMo 1 and Chengle Ma 2

1Logistics Engineering College Shanghai Maritime University 201306 Shanghai China2School of Management Shanghai University 200444 Shanghai China

Correspondence should be addressed to Chengle Ma machenglefoxmailcom

Received 1 August 2018 Accepted 11 October 2018 Published 1 November 2018

Guest Editor XinchangWang

Copyright copy 2018 Hongtao Hu 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

To reduce the inventory cost and ensure product quality while meeting the diverse demands of customers manufacturers yieldproducts in batchesHowever the rawmaterials required for manufacturing need to be obtained from suppliers in advance makingit necessary to understand beforehand how to best structure the pickup routes so as to reduce the cost of picking up and stockingwhile also ensuring the supply of raw materials required for each batch of production To reduce the transportation and inventorycosts therefore this paper establishes a mixed integer programming model for the joint optimization of multibatch productionand vehicle routing problems involving a pickup Following this a two-stage hybrid heuristic algorithm is proposed to solve thismodel In the first stage an integrated algorithm combining the Clarke-Wright (CW) algorithm and the Record to Record (RTR)travel algorithm was used to solve vehicle routing problem In the second stage the Particle Swarm Optimization (PSO) algorithmwas used to allocate vehicles to each production batchMultiple sets of numerical experiments were then performed to validate theeffectiveness of the proposed model and the performance efficiency of the two-stage hybrid heuristic algorithm

1 Introduction

The ldquotruck dispatching problemrdquo was first introduced byDantzig and Ramser [1] Then Clarke and Wright [2] gen-eralized the ldquovehicle routing problemrdquo (VRP) to the domainof logistics and transport that is to how to serve a setof customers who are geographically dispersed around thecentral depot using a fleet of trucks with varying capacitiesTaking into account real-life complexities this paper expandsthe traditional VRP by combining VRP with productionin order to consider the vehicle routing problem with rawmaterial pickup under multibatch production (VRPPMP)The problem at hand is defined as follows The manufacturerdivides the production in each cycle intomultiple batches anduses vehicles to undertake the pickup from their suppliersin order to ensure sufficient raw materials prior to the pro-duction of each batch Since the inseparable pickup demandsof dispersed suppliers must be met the manufacturer thusneeds to design both the vehicle pickup routes and the batchallocation of the routes in such a way that they satisfy thedemand for the raw materials needed for production whilealso minimizing the total inventory and transportation costs

By designing the vehicle pickup routes in the context ofmultibatch production the manufacturer can reduce trans-portation and inventory costs in the production process andensure product quality In the automobile assembly industryfor example automobile producers need to retrieve the partsneeded from the auto parts manufacturer separately If theproducers retrieve all the parts needed for production duringeach period this not only will greatly increase the inventorycost but also cannot guarantee the quality of the partsTherefore multibatch production is needed for automobileassembly production with the VRPPMP of automobileassembly production being investigated in order to plan thevehicle pickup path for each batch so as to ensure the qualityof the raw material and the products and reduce the costs ofinventory and transportation

Prior to initiating periodic production the manufacturermust pick up raw materials from geographically dispersedlocations (suppliers) with the raw materials being picked upperiodically at production batches In single-cycle produc-tion planning there are often multiple batches of productionoperations Before each batch of production and processing

HindawiDiscrete Dynamics in Nature and SocietyVolume 2018 Article ID 2804589 9 pageshttpsdoiorg10115520182804589

2 Discrete Dynamics in Nature and Society

begins the manufacturer needs sufficient raw materials inorder to complete the set production volume If the purchasequantity of raw materials is too large it will lead to a certaindegree of inventory accumulation This will in turn generatethe problems of a high inventory management cost andthe low quality of raw materials Thus according to theproduction demands of each batch the problem lies in therebeing a reasonable arrangement of vehicle pickup routesand also in matching these routes with production batchesin order to minimize the total costs of transportation andinventory

In line with the above description this paper proposesan optimization of vehicle routing involving pickup based onmultibatch productionThe contributions of this paper are asfollows(1) Combining production with the vehicle pickup rout-ing problem this paper studies the optimization of vehi-cle routing involving pickup in the context of multibatchproduction We design vehicle pickup routes based on theraw material demands of each batch in such a way as tominimize transportation costs and inventory costs whilemeeting production demands(2)Weput forward a two-stage hybrid heuristic algorithmto solve the proposed model In the first stage of thealgorithm the Clarke-Wright (CW) algorithm is proposed togenerate the initial vehicle routes followingwhich the Recordto Record (RTR) travel algorithm is used to further optimizethe generated routes In the second stage a Particle SwarmOptimization (PSO) algorithm is developed to allocate thevehicles to each production batch

The remainder of this paper is organized as followsSection 2 reviews the related literature Section 3 details thevehicle routing involving pickup based on multibatch pro-duction and develops themixed integer programmingmodelSection 4 introduces a two-stage hybrid heuristic algorithmto solve this model In Section 5 multiple sets of numericalexperiments are performed to validate the effectiveness ofthe proposed model and the performance efficiency of thetwo-stage hybrid heuristic algorithm Section 6 presents theconclusions of this study and recommendations for futurework

2 Literature Review

The vehicle path problem (VRP) is a key aspect of logisticsand transportation research Selecting the appropriate vehiclepathmethod can speed up the response to customer demandsand reduce operation costs Since first being proposed byDantzig and Ramser [1] research on VRP has extended intonumerous avenues after decades of exploration and researchsuch as the capacitated vehicle routing problem (CVRP) [3]the vehicle routing problem with time windows (VRPTW)[4] and vehicle routing problems with pickup and delivery(VRPPD) [5 6] among others This paper focuses on theangle of VRP with pickup and vehicle capacity limitation

Many scholars have conducted in-depth research on theissues of VRPPD Savelsbergh and Sol [7] discussed severalcharacteristics distinguishing the latter from standard VRPand presented a survey of the problem types and solution

methods A VRPPDwith multiple time windows and vehicletypes was considered by Xu et al [8] where computationalresults showed that the computational times required by theproposed column-generation-based solution were accept-able Gabor Nagya [9] proposed a heuristic algorithm thattakes the pickup and delivery stages as awhole andwhichwasused to solve the VRPPD in the context of single andmultiplecar parks Kalina andVokrinek [10] presented a parallel solverfor VRPTW and the pickup and delivery problem with timewindows which was based on the parallel competition ofparticular solvers solving the given problem

The problem of combining VRPPD with productionwhich is a type of VRPPD has been studied extensively Suchproblems involve coordinating production inventory anddelivery operations to meet customer needs and minimizecosts Zheng et al [11] investigated the vendor-managedcyclic inventory routing problem under constant customerdemand rates To minimize inventory costs without causingany lack of stock at the customers a heuristic solutionapproach was proposed The latter proved well capable offinding the appropriate cost trade-off under varying circum-stances Shiguemoto andArmentano [12] addressed the prob-lem of optimally coordinating a production-distribution sys-tem with a fleet of homogeneous vehicles over a multiperiodfinite horizon proposing a Tabu search procedure for solvingthe problem Adulyasak et al [13] introduced multivehicleproduction routing problem and inventory routing problemformulations with and without a vehicle index to solve theseproblems under both the maximum level and the order-up-to level inventory replenishment policies To minimize thetotal distribution cost Sainathuni et al [14] introduced thewarehouse inventory transportation problem of determiningan optimal distribution plan from vendors to customers viaone or more warehouses

Although the research on combining VRPPD with pro-duction has made some progress there is still some room fordevelopment To date existing studies that have consideredcombining multibatch production with vehicle pickup rout-ing issues have been rather limited To enrich the research inthis area based on this combination the current paper con-siders two aspects from the perspective of the manufacturer(i) VRP ndash the vehicle routing problem from an upstreamsupplier to the manufacturers procurement link and (ii)production and processing problems considering the single-cycle and multibatch production and processing problems ofa single manufacturer how the allocation of vehicles can bestbe undertaken so as to correspond to the production batchesUnder the premise of completing each batch of productionand processing operations the inventory cost should be aslow as possible

3 Problem Description and Modeling

31 Problem Description The VRPPMP can be defined asfollows Let119866 = (119878 119864) be a complete oriented graphwith a setof vertices 1198781015840 = 0 1 |1198781015840| where the vertex 0 representsthe manufacturer and the remaining ones represent thesuppliers Each edge 119894 119895 isin 119864 for 119894 119895 isin 1198781015840 has a nonnegativecost 119862119894119895 and each supplier 119894 isin 119878 = 1198781015840 minus 0 has known

Discrete Dynamics in Nature and Society 3

Production batch

Amount of material

Batch 1

D

1

Manufacturer

52

3 4

Amount of material picked up

2D

inventory

material of route 1 for

batch 1

0

material of route 2 for

batch 2

Batch 2

Supplier

Manufacturer

Route 1

Route 2

Amount of consumed materialFigure 1 A schematic diagram of the VRPPMP

nonnegative demands 119901119894 with regard to the pickup Let 119879 =(1 119905 |119879|) be a set of production batches 119863 representsthe manufacturerrsquos fixed output for each production batchLet 119881 = 1 119898 |119881| be a set of homogeneous vehicleswith capacity 119876 The VRPPMP is depicted in Figure 1 Thevehicles visit the corresponding suppliers for each productionbatch and then return to the manufacturer The retrieved rawmaterial is put into production If there is any unfinished rawmaterial this will be used as raw material inventory for thenext batch of production For each batch of access routes theVRPPMP consists of constructing a structure for the routesin such a way that (i) every route starts and ends at themanufacturer (ii) all pickup demands are met (iii) a supplieris visited only by a single vehicle and (iv) the sumof inventorycosts and vehicle transportation costs is minimized

In addition this paper makes the following assumptionsunderlying the VRPPMP(1) The manufacturer conducts multiple batches of pro-duction and processing operations with the total amount ofmaterials supplied by the suppliers equaling the total amountacquired for production(2) The remaining raw material inventory of each batchcan be used for the next batch of production tasks theinventory capacity of the manufacturer is not considered(3) The raw materials for each batch can be delivered tothe manufacturer in time for the batch to be produced(4)The problem considers vehicle capacity limitations

In the model we also assumed that each vehicle can onlyperform one trip for each production batch It should be

noted that a vehiclersquos multiple trips for a production batch(the multitrip vehicle routing problem VRPMT) and theconsistency of a vehicle in terms of pickup from materialsuppliers for different batches (the consistent vehicle routingproblem CVRP) are not considered In real productionactivities the production plan is often based on a day a weekor a longer time interval while the vehicles can completea trip within a relatively shorter time span Therefore if avehicle performs more than one trip for different batches itcan be considered several dummy vehicles

32 Mathematical Model

ParametersS Set of suppliers 119878 = 1 2 119894 |119878|119878+ 0 represents trucks departing from the manufac-turer 119878+ = 119878 cup 0119878minus |119878|+1 represents vehicles arriving to the manufac-turer 119878minus = 119878 cup |119878| + 1T Set of production batches 119879 = 1 2 119905 |119879|V Set of vehicles 119881 = 1 2 119898 |119881|119901119894 The amount of raw material for pickup fromsupplier i119888119894119895 The cost of transportation from supplier i tosupplier j119888119906119899119901119903119900The inventory cost for each unit of unprocessedraw material during unit production batch

4 Discrete Dynamics in Nature and Society

Q The capacity of homogeneous vehiclesDThe amount of planned product in each productionbatchM A sufficient large positive number

Decision Variables

119909119894119898 1 if vehicle m provides supplier i with pickupservice 0 otherwise119911119898119905 1 if vehicle m is assigned to the production batcht 0 otherwise119910119894119895119898 1 if suppliers i and j are successively serviced byvehicle m 0 otherwise119892119898119905 The amount of raw material transported byvehicle m for production batch t119902119894119898 The total amount of raw material transported byvehicle m after servicing supplier i120593119905The remaining inventory of the rawmaterials afterthe tth production batch

Based on the above definition the VRPPMP is established asfollows

min (sum119894isin119878+

sum119895isin119878minus

sum119898isin119881

119888119894119895 sdot 119910119894119895119898 + 119888119906119899119901119903119900 sdot sum119905isin119879

120593119905) (1)

119904119905 1199090119898 = 1 forall119898 isin 119881 (2)

sum119898isin119881

119909119894119898 = 1 forall119894 isin 119878 (3)

sum119894isin119878+119894 =ℎ

119910119894ℎ119898 = sum119895isin119878minus119895 =ℎ

119910ℎ119895119898 = 119909ℎ119898forallℎ isin 119878 forall119898 isin 119881

(4)

sum119895isin119878

1199100119895119898 minus sum119905isin119879

119911119898119905 = 0 forall119898 isin 119881 (5)

sum119905isin119879

119911119898119905 le 1 forall119898 isin 119881 (6)

119892119898119905 le 119876 sdot 119911119898119905 forall119905 isin 119879 forall119898 isin 119881 (7)

119892119898119905 le sum119895isin119878

119901119895 sdot 119909119895119898 + 119876 sdot (1 minus 119911119898119905)forall119905 isin 119879 forall119898 isin 119881

(8)

119892119898119905 ge sum119895isin119878

119901119895 sdot 119909119895119898 minus 119876 sdot (1 minus 119911119898119905)forall119905 isin 119879 forall119898 isin 119881

(9)

sum119898isin119881

119892119898119905 ge 119863 119905 = 1 (10)

sum119898isin119881

119892119898119905 + 120593119905minus1 ge 119863 forall119905 isin 119879 1 (11)

119902119895119898 ge 119901119895 minus 119876 sdot (1 minus 1199100119895119898) forall119895 isin 119878 forall119898 isin 119881 (12)

119902119895119898 le 119901119895 + 119876 sdot (1 minus 1199100119895119898) forall119895 isin 119878 forall119898 isin 119881 (13)

119902119895119898 ge 119902119894119898 + 119901119895 minus 119876 sdot (1 minus 119910119894119895119898)forall119894 isin 119878 forall119895 isin 119878minus 119894 = 119895 forall119898 isin 119881 (14)

119902119895119898 le 119902119894119898 + 119901119895 + 119876 sdot (1 minus 119910119894119895119898)forall119894 isin 119878 forall119895 isin 119878minus 119894 = 119895 forall119898 isin 119881 (15)

120593119905 = sum119898isin119881

119892119898119905 minus 119863 119905 = 1 (16)

120593119905 = 120593119905minus1 + sum119898isin119881

119892119898119905 minus 119863 forall119905 isin 119879 1 (17)

Objective function (1) maximizes the total costs of themanufacturer which are equal to the sum of the inventorycosts and shipping costs incurred Constraints (2) state thateach vehicle starts at the manufacturer and ends at the man-ufacturer Constraints (3) ensure that each supplier will beaccessed once Flow conservation is ensured by Constraints(4) Constraints (5) indicate that each vehicle that providessuppliers with a pickup service must serve in exactly oneproduction batch Constraints (6) state that each vehicle canat most serve at most one production batch Constraints(7) ensure that the raw material transported by each vehicledoes not exceed the capacity of vehicle Constraints (8) and(9) represent the total amount of raw material delivery byeach vehicle in each production batch Constraints (10) and(11) indicate the constraint of amount of raw materials thatcan be put into production in each batch Constraints (12)to (15) state the relationship between the vehicle load andthe quantity supplied Constraints (16) and (17) represent theraw material inventory after each batch of production andprocessing

4 Solution Algorithm

The above model can be directly solved by using commercialoptimization packages such as CPLEX when the scale issmall However when the scale increases by a certain extentusing CPLEX is time consuming or unable to complete thesolution Therefore the algorithm needs to be added to solvelarge-scale problems

Taking the two aspects into account when making adecision that is vehicle routing as well as the matchingbetween the vehicle and the production batch the VRPMPmentioned above cannot be solved directly In light of thisthe VRPPMP was divided into two stages In the first stagethe problem of vehicle routing in the process of pickingup cargoes was solved using the Clarke-Wright and theRecord-to-Record travel algorithms In the second stage weused the PSO algorithm to allocate each vehicle that needsto be picking up to a certain production batch so as tosatisfy the demands of the manufacturer in each productionbatch

Discrete Dynamics in Nature and Society 5

Manufacturer

Supplier i

Supplier j

Manufacturer

Supplier i

Supplier j

Figure 2 Changes in vehicle pickup routes

41 Routes Initialization (CW) For the purposes of thispaper the CW algorithm was used to generate the initialvehicle path for the first stage problem The algorithm wasfirst proposed by Clarke and Wright [2] and is applicable toCVRP It forms a route in the following ways First it yieldsthe shortest route between every two suppliers in the systemThe desired aim here is to allocate loads to vehicles in such amanner that all the raw materials are assigned and the totalmileage covered is at a minimum

Consider a general savings value S given by linking twosuppliers i and j as shown in Figure 2

119904119894119895 = 1198890119894 + 1198890119895 minus 120582119889119894119895 (18)

Where119904= savings value1198890119894= distance from supplier i to the manufacturer119889119894119895= distance between points i and j120582= route shape parameter Special cases 120582 = 1 thesavings approach and 120582 = 2 Gaskellrsquos120587 method [15] Withan increase in 120582 greater emphasis is placed on the distancebetween the suppliers rather than on the position relative tothe manufacturer in selecting an addition to a route

In line with the savings value S between every twosuppliers the algorithm yields the CW table In the case ofsatisfying the on-vehicle capacity the manufacturer prefer-entially links a pair of suppliers with a larger S-value and usesone vehicle to provide pickup services for these suppliersFollowing the linking method until all the suppliers areserviced the manufacturer then obtains the initial solutionfor the pickup vehicle routes The procedure given is simplebut effective in producing a near-optimal solution and hasbeen programmed for several digital computers

42 Routes Improvement (RTR) After the initial vehicleroute was generated we further optimized it using the RTRtravel method which was first proposed by Dueck [16] Weimproved the solutions using the operations specified by Liet al ([17] namely a one-point move and two-point move Inthe one-point move we attempted to move each supplier inthe existing solution to a newposition on the same route or ona different route In the two-point move we tried to exchangethe positions of two suppliers These improvement moves areshown in Figure 3

The routes were improved by repeatedly applying thesetwo local search operators in two phases diversification andimprovement In the diversification phase we attempted toexplore new areas in the solution space by accepting bothimproving and deteriorating moves In the improvement

phase we attempted to improve the current solution as muchas possible by accepting only improving moves until wereached a local minimum

By combining the CW algorithm and RTR travel methodwe were able to obtain a better pickup vehicle routingsolution Based on the routes we then solved the decisionmaking problems pertaining to vehicle allocation in eachproduction batch by using the PSO algorithm in order tosatisfy the manufacturerrsquos demands in each production batchand lower the inventory cost as far as possible

43 Batch Allocation (PSO) The first stage enabled us toobtain all the routes that the vehicles needed to cover thenmaking it necessary to assign the routes to each productionbatch in the second stage Consequently the PSO algorithmdeveloped by Eberhart and Kennedy [18] was employed tosolve the route allocation problem and can be seen as anefficient stochastic global optimization technique

In the PSO algorithm the position of each particlerepresents a feasible solution in the search domain and eachparticle changes its position and velocity depending on itsflying experience The decision variables 119909119894119898 119910119894119895119898 and 119902119894119898which are related to the pickup service were determined inthe first stage 119892119898119905 and 120593119905 could be calculated according toconstraints (16) and (17) as long as 119911119898119905 was determined inthe second stage The core variable 119911119898119905 which determineswhether the raw material transported by vehicle m wasassigned to production batch t 119898 isin 119881 119905 isin 119879 was coded asfollows All the utilized vehicles were grouped in set 1198811015840 1198811015840 isin119881 Each particle with |1198811015840| dimensions was represented by119865 =1198911 1198912 119891119898 119891|1198811015840| Each dimension 119891119898 was coded as apositive number After the vehicle sequence was reorderedaccording to the nondescending order of the 119891119898 vehicleswere assigned to the production batches The main ideabehind the distribution rule is as follows vehicles at the frontof the sorted sequence have priority in terms of assignationto the previous production batch

For instance if the amount of materials acquired by aproduction batch is 10 10 10 the utilized vehicles wouldbe 1 2 3 4 5 and the quantity of materials correspondingto the vehicles would be 4 6 8 7 5 If the sorted vehiclesequence is 2 3 1 5 4 the amount of materials would be6 8 5 4 7 accordingly Vehicle 2 was first assigned toproduction batch 1 given that the amount of materialacquired in this batch was 10 which is greater than theamount that vehicle 2 carriedThe next vehicle in the sortedsequence that is vehicle 3was assigned to batch 1 thusgiving the storage material of production batch 1 a value of4 Accordingly vehicles 4 and 5 were assigned to batch 2with the storage value of 3 and vehicle 7 was assigned tobatch 3 with no storage material in the last

In applying the PSO algorithm we assumed that a swarmhas k particles 119881119899119896 = V119899119896119898119905 was the velocity of particle k atiteration n and119875119899119896 = 119901119899119896119898119905 the position of particle k at itera-tion n Each particle had its personal best position119875119871119861119890119904119905119899119896119898119905 at iteration n on dimensions m and t 119875119866119861119890119904119905119899119896119898119905 is the globalbest positon of the entire swarm on dimensions m and tup until iteration n The swarm flies through hyperspaceaccording to the experience of its neighbors The ordinary

6 Discrete Dynamics in Nature and Society

(a) One-point move (b) Two-point move

Figure 3 Improvement operators used in VRPPMP

updating formulates used to calculate speed and positionwere as follows

V119899+1119896119898119905 = 119908 sdot V119899119896119898119905 + 11988811199031 (119875119871119861119890119904119905119899119896119898119905 minus 119901119899119896119898119905)+ 11988821199032 (119875119866119861119890119904119905119899119896119898119905 minus 119901119899119896119898119905) (19)

119901119899+1119896119898119905 = 119901119899119896119898119905 + V119899+1119896119898119905 (20)

Where w is an inertia weight parameter which affects theconvergence procedure of the optimal solution 1198881and 1198882 arethe cognitive and social learning parameters respectively 1199031and 1199032 are random numbers generated between [0 1]

Based on the above description we now present the PSOalgorithm process for solving the route allocation problem

Step 1 Set the iteration number n = 1 Initialize k particleswhose positions and velocities are randomly generated Theparticlesrsquo positions represent the result of the route allocationand determine their qualities

Step 2 Evaluate the fitness value of each particle using thecommercial software CPLEX More specifically we used theroute allocation result obtained by the particles to fix thevariables 119911119898119905 and then used CPLEX to solve the originalmodel

Step 3 For each particle compare its fitness value andthe global best optimal solution119875119866119861119890119904119905119899119896119898119905 if 119875119871119861119890119904119905119899119896119898119905 gt119875119866119861119890119904119905119899119896119898119905 replace the value of 119875119866119861119890119904119905119899119896119898119905 using the value of119875119871119861119890119904119905119899119896119898119905Step 4 According to formulas (19) and (20) update theparticle velocity and positon

Step 5 If n reaches the preset maximum iteration or119875119866119861119890119904119905119899119896119898119905 has not been improved during a preset numberof iterations end the procedure otherwise set 119899 = 119899 + 1andthen go on to Step 2

5 Numerical Experiments

To validate the performance of the proposed two-stageheuristic numerical experiments were carried out on theVRPPMP For the PSO algorithm the maximum number ofiterations was set to 100 while the population size was 30According to the previous tests we set 1198881 = 1198882 = 1 Frompreliminary testing of 119908 = 0 05 1 15 and 2 it emergedthat 119908 = 1 performed the best thus we adopted 119908 = 1 for

the next experiments Based on the same decision obtainedin the first stage another nature inspired algorithm the Tabusearch (TS) algorithm was applied for comparison purposesWe listed the optimal solutions obtained by CPLEX in small-scale instances

All the experiments were performed on a computer with360 gigahertz Intel (R) Core (TM) i7-4790 CPU and 360gigabyte RAM The model and solution methods includingthe CW algorithm RTR travel algorithm and the PSOalgorithm were implemented using CPLEX 126 with C(VS2015) concert technology

51 Generation of Test Instance In this section we demon-strate the implementation of numerical experiments to val-idate the effectiveness of the proposed model and the effi-ciency of the proposed algorithm For small-scale instancesthe results of the proposed two-stage heuristic algorithmwerecompared with the optimal solutions obtained byCPLEX andthe solutions obtained using the TS algorithm Based on thesame decision obtained in the first stage the results obtainedusing the PSO heuristic were compared with those of the TSalgorithm in medium- and large-scale instances given thatCPLEX would have been time consuming or unable to solvethe problem

For the numerical experiments on the VRPPMP in-stances were randomly generated in the three scales Wesimulated the cases of 10 suppliers and two productionbatches and 15 suppliers and three production batches forthe small-scale 30 suppliers and six production batches and50 suppliers and 10 production batches for the medium scale70 suppliers and 14 production batches 90 suppliers and 18production batches 100 suppliers and 18 production batchesfor the large scale Information pertaining to the coordinatepositions and pickup demands of the 10 suppliers of thefirst small-scale instance (ie 10-2-1) are shown in Table 1The coordinate position of the manufacturer was (5 5) andthe cost of unit distance traveled by vehicle was $2 Themanufacturer had sufficient pickup vehicles with the capacityof 10 tons to provide a service to suppliersThe volume of rawmaterials required for each production batch was 10 tons andthe unit inventory cost of unprocessed raw materials in eachbatch was $20

52 Analysis of Calculation Results We tested examples atdifferent scales and used CPLEX the PSO algorithm andthe TS algorithm respectively to solve the VRPPMP Table 2yields the following conclusions In the small-scale examples(10-2) the exact solutions (7345 17317 and 18465) were

Discrete Dynamics in Nature and Society 7

Table 1 Pickup demands (tons) and coordinate positions (km) of instances 10-2-1

Supplier Pickup demands Coordinate axis X Coordinate axis Y1 15577 62804 834392 14597 56732 598063 22747 20603 558884 19595 90603 442185 15754 97755 273706 25402 29191 467317 20115 63266 469518 15807 98215 030379 23763 86237 9953510 27253 67718 31459

Table 2 Computational results of small-scale instances

Instance IDCPLEX PSO algorithm TS algorithm

Result($)

Time(sec)

Result($)

Time(sec)

Gap()

Result($)

Time(sec)

Gap()

10-2-1 73 45 545 7345 447 0 7345 755 010-2-2 17317 366 17657 307 196 17657 708 19610-2-3 18465 510 18919 355 246 18919 760 24615-3-1 - gt12hours 24854 1424 - 24854 1730 -15-3-2 - gt12hours 43527 1369 - 43527 1747 -15-3-3 - gt12hours 23667 1373 - 23667 1739 -

Table 3 Computational results of medium-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

30-6-1 57288 8346 57288 51069 0 5119030-6-2 56730 13409 55166 51958 -276 2874930-6-3 67289 3792 67289 59352 0 14651950-10-1 124389 23773 119875 224667 -363 8450550-10-2 164245 61163 157761 229187 -395 2747250-10-3 250116 25234 242029 229583 -323 80982

obtained byCPLEX and the accuracy of themodel is verifiedCompared with the TS algorithm the PSO algorithm canobtain the approximate solutions (7345 17657 and 18919)in a shorter time In the small-scale examples (15-3) CPLEXhad difficultly in getting the results but the PSO algorithmand TS algorithm can still obtain the results in a shorttime

However after the scale of the study increases as shownin Tables 3 and 4 the PSO algorithm can take less timeto get a relatively appropriate solution than TS algorithmIn the medium-scale examples the relative value deviationsbetween the PSO algorithm and the TS algorithm are within4 However the relative time deviations between the PSOalgorithmand theTS algorithm reach 699on average In thelarge-scale examples although the relative value deviationsbetween the PSO algorithm and the TS algorithm reached

nearly 7 the PSO calculation results are still acceptableand it holds great advantages in terms of computing timeThis proves that the two-stage heuristic algorithm can helpus solve such problems well

6 Conclusion

This paper discusses a novel and practical research issuearising in the manufacturing industry namely the prob-lem of vehicle routing involving pickup in the context ofmultibatch production In order to reduce transportationcosts and inventory costs in the production process a mixedinteger programming model was developed Following thisa two-stage hybrid heuristic algorithm was put forward tosolve this model In the first stage the CW algorithm wasproposed in order to generate the initial vehicle routes The

8 Discrete Dynamics in Nature and Society

Table 4 Computational results of large-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

70-14-1 226615 112267 210941 706980 -692 5297370-14-2 372110 56183 356346 601707 -424 9709870-14-3 453408 94929 431432 627870 -485 5614190-18-1 422180 456409 401415 2122673 -492 3650890-18-2 427458 164882 401614 1371370 -605 7317390-18-3 777256 369100 740527 1960738 -473 43122100-18-1 516445 698274 493579 2861304 -443 30977100-18-2 804758 723314 776298 2856240 -354 29488100-18-3 1132139 692266 1086673 2868246 -402 31433

RTR travel algorithm was then used further to optimize thegenerated routes In the second stage of the algorithm thePSO algorithm was developed so as to allocate the vehiclesto each production batch This process ensured that themanufacturer first assigned each supplier to the appropri-ate route and then assigned each route to the appropriatebatch

By testing examples at different scales we were able tovalidate the effectiveness of the proposed model and theperformance efficiency of the two-stage hybrid heuristicalgorithm First in the small-scale examples the exact solu-tion was obtained using CPLEX and the accuracy of themodel was verified Second after the scale was expandedit was found that CPLEX had difficulty in solving thisover a short period of time and that the PSO algorithmtook less time to obtain a relatively appropriate resultthan the TS algorithm This proves that the two-stageheuristic algorithm is well placed to help us solve suchproblems

There are certain limitations to the current study First weassumed that the amount of the single raw material requiredfor each production batch would be the same in practicehowever the amount of multiple rawmaterials varies for eachproduction batch Second this paper only considered thecombination of the pickup stage and the production stagewithout considering product delivery The joint optimizationof multibatch production and the vehicle routing problemwith pickup and delivery considering inventory will thus beaddressed in future studies

Data Availability

The data used to support this study have been deposited inthe Baidu Netdisk readers can access the data at httpspanbaiducoms1ZSkGF-syiNTk FN9JNShbA

Conflicts of Interest

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

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China [no 71771143 and no 71501125] andShanghai Young Eastern Scholar Programme [QD2015041]

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] 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

[3] R Fukasawa H Longo J Lysgaard et al ldquoRobust branch-and-cut-and-price for the capacitated vehicle routing problemrdquoMathematical Programming vol 106 no 3 pp 491ndash511 2006

[4] M LGambardella D E Taillard andG Agazzi ldquoAmultiple antcolony system for vehicle routing problemswith timewindowsrdquo1999

[5] P Sombuntham and V Kachitvichyanukul ldquoMulti-depot vehi-cle routing problem with pickup and delivery requestsrdquo inProceedings of the International MultiConference of Engineersand Computer Scientists IMECS 2010 pp 71ndash85 China March2010

[6] C K Y Lin ldquoA vehicle routing problem with pickup anddelivery time windows and coordination of transportableresourcesrdquoComputers amp Operations Research vol 38 no 11 pp1596ndash1609 2011

[7] MW Savelsbergh andM Sol ldquoThe general pickup and deliveryproblemrdquo Transportation Science vol 29 no 1 pp 17ndash29 1995

[8] H Xu Z-L Chen S Rajagopal and S Arunapuram ldquoSolving apractical pickup and delivery problemrdquo Transportation Sciencevol 37 no 3 pp 347ndash364 2003

[9] G Nagy and S Salhi ldquoHeuristic algorithms for single andmultiple depot vehicle routing problems with pickups anddeliveriesrdquo European Journal of Operational Research vol 162no 1 pp 126ndash141 2005

[10] P Kalina and J Vokrınek ldquoParallel solver for vehicle routingand pickup and delivery problems with time windows based onagent negotiationrdquo in Proceedings of the 2012 IEEE InternationalConference on Systems Man and Cybernetics SMC 2012 pp1558ndash1563 Republic of Korea October 2012

Discrete Dynamics in Nature and Society 3

Production batch

Amount of material

Batch 1

D

1

Manufacturer

52

3 4

Amount of material picked up

2D

inventory

material of route 1 for

batch 1

0

material of route 2 for

batch 2

Batch 2

Supplier

Manufacturer

Route 1

Route 2

Amount of consumed materialFigure 1 A schematic diagram of the VRPPMP

nonnegative demands 119901119894 with regard to the pickup Let 119879 =(1 119905 |119879|) be a set of production batches 119863 representsthe manufacturerrsquos fixed output for each production batchLet 119881 = 1 119898 |119881| be a set of homogeneous vehicleswith capacity 119876 The VRPPMP is depicted in Figure 1 Thevehicles visit the corresponding suppliers for each productionbatch and then return to the manufacturer The retrieved rawmaterial is put into production If there is any unfinished rawmaterial this will be used as raw material inventory for thenext batch of production For each batch of access routes theVRPPMP consists of constructing a structure for the routesin such a way that (i) every route starts and ends at themanufacturer (ii) all pickup demands are met (iii) a supplieris visited only by a single vehicle and (iv) the sumof inventorycosts and vehicle transportation costs is minimized

In addition this paper makes the following assumptionsunderlying the VRPPMP(1) The manufacturer conducts multiple batches of pro-duction and processing operations with the total amount ofmaterials supplied by the suppliers equaling the total amountacquired for production(2) The remaining raw material inventory of each batchcan be used for the next batch of production tasks theinventory capacity of the manufacturer is not considered(3) The raw materials for each batch can be delivered tothe manufacturer in time for the batch to be produced(4)The problem considers vehicle capacity limitations

In the model we also assumed that each vehicle can onlyperform one trip for each production batch It should be

noted that a vehiclersquos multiple trips for a production batch(the multitrip vehicle routing problem VRPMT) and theconsistency of a vehicle in terms of pickup from materialsuppliers for different batches (the consistent vehicle routingproblem CVRP) are not considered In real productionactivities the production plan is often based on a day a weekor a longer time interval while the vehicles can completea trip within a relatively shorter time span Therefore if avehicle performs more than one trip for different batches itcan be considered several dummy vehicles

32 Mathematical Model

ParametersS Set of suppliers 119878 = 1 2 119894 |119878|119878+ 0 represents trucks departing from the manufac-turer 119878+ = 119878 cup 0119878minus |119878|+1 represents vehicles arriving to the manufac-turer 119878minus = 119878 cup |119878| + 1T Set of production batches 119879 = 1 2 119905 |119879|V Set of vehicles 119881 = 1 2 119898 |119881|119901119894 The amount of raw material for pickup fromsupplier i119888119894119895 The cost of transportation from supplier i tosupplier j119888119906119899119901119903119900The inventory cost for each unit of unprocessedraw material during unit production batch

4 Discrete Dynamics in Nature and Society

Q The capacity of homogeneous vehiclesDThe amount of planned product in each productionbatchM A sufficient large positive number

Decision Variables

119909119894119898 1 if vehicle m provides supplier i with pickupservice 0 otherwise119911119898119905 1 if vehicle m is assigned to the production batcht 0 otherwise119910119894119895119898 1 if suppliers i and j are successively serviced byvehicle m 0 otherwise119892119898119905 The amount of raw material transported byvehicle m for production batch t119902119894119898 The total amount of raw material transported byvehicle m after servicing supplier i120593119905The remaining inventory of the rawmaterials afterthe tth production batch

Based on the above definition the VRPPMP is established asfollows

min (sum119894isin119878+

sum119895isin119878minus

sum119898isin119881

119888119894119895 sdot 119910119894119895119898 + 119888119906119899119901119903119900 sdot sum119905isin119879

120593119905) (1)

119904119905 1199090119898 = 1 forall119898 isin 119881 (2)

sum119898isin119881

119909119894119898 = 1 forall119894 isin 119878 (3)

sum119894isin119878+119894 =ℎ

119910119894ℎ119898 = sum119895isin119878minus119895 =ℎ

119910ℎ119895119898 = 119909ℎ119898forallℎ isin 119878 forall119898 isin 119881

(4)

sum119895isin119878

1199100119895119898 minus sum119905isin119879

119911119898119905 = 0 forall119898 isin 119881 (5)

sum119905isin119879

119911119898119905 le 1 forall119898 isin 119881 (6)

119892119898119905 le 119876 sdot 119911119898119905 forall119905 isin 119879 forall119898 isin 119881 (7)

119892119898119905 le sum119895isin119878

119901119895 sdot 119909119895119898 + 119876 sdot (1 minus 119911119898119905)forall119905 isin 119879 forall119898 isin 119881

(8)

119892119898119905 ge sum119895isin119878

119901119895 sdot 119909119895119898 minus 119876 sdot (1 minus 119911119898119905)forall119905 isin 119879 forall119898 isin 119881

(9)

sum119898isin119881

119892119898119905 ge 119863 119905 = 1 (10)

sum119898isin119881

119892119898119905 + 120593119905minus1 ge 119863 forall119905 isin 119879 1 (11)

119902119895119898 ge 119901119895 minus 119876 sdot (1 minus 1199100119895119898) forall119895 isin 119878 forall119898 isin 119881 (12)

119902119895119898 le 119901119895 + 119876 sdot (1 minus 1199100119895119898) forall119895 isin 119878 forall119898 isin 119881 (13)

119902119895119898 ge 119902119894119898 + 119901119895 minus 119876 sdot (1 minus 119910119894119895119898)forall119894 isin 119878 forall119895 isin 119878minus 119894 = 119895 forall119898 isin 119881 (14)

119902119895119898 le 119902119894119898 + 119901119895 + 119876 sdot (1 minus 119910119894119895119898)forall119894 isin 119878 forall119895 isin 119878minus 119894 = 119895 forall119898 isin 119881 (15)

120593119905 = sum119898isin119881

119892119898119905 minus 119863 119905 = 1 (16)

120593119905 = 120593119905minus1 + sum119898isin119881

119892119898119905 minus 119863 forall119905 isin 119879 1 (17)

Objective function (1) maximizes the total costs of themanufacturer which are equal to the sum of the inventorycosts and shipping costs incurred Constraints (2) state thateach vehicle starts at the manufacturer and ends at the man-ufacturer Constraints (3) ensure that each supplier will beaccessed once Flow conservation is ensured by Constraints(4) Constraints (5) indicate that each vehicle that providessuppliers with a pickup service must serve in exactly oneproduction batch Constraints (6) state that each vehicle canat most serve at most one production batch Constraints(7) ensure that the raw material transported by each vehicledoes not exceed the capacity of vehicle Constraints (8) and(9) represent the total amount of raw material delivery byeach vehicle in each production batch Constraints (10) and(11) indicate the constraint of amount of raw materials thatcan be put into production in each batch Constraints (12)to (15) state the relationship between the vehicle load andthe quantity supplied Constraints (16) and (17) represent theraw material inventory after each batch of production andprocessing

4 Solution Algorithm

The above model can be directly solved by using commercialoptimization packages such as CPLEX when the scale issmall However when the scale increases by a certain extentusing CPLEX is time consuming or unable to complete thesolution Therefore the algorithm needs to be added to solvelarge-scale problems

Taking the two aspects into account when making adecision that is vehicle routing as well as the matchingbetween the vehicle and the production batch the VRPMPmentioned above cannot be solved directly In light of thisthe VRPPMP was divided into two stages In the first stagethe problem of vehicle routing in the process of pickingup cargoes was solved using the Clarke-Wright and theRecord-to-Record travel algorithms In the second stage weused the PSO algorithm to allocate each vehicle that needsto be picking up to a certain production batch so as tosatisfy the demands of the manufacturer in each productionbatch

Discrete Dynamics in Nature and Society 5

Manufacturer

Supplier i

Supplier j

Manufacturer

Supplier i

Supplier j

Figure 2 Changes in vehicle pickup routes

41 Routes Initialization (CW) For the purposes of thispaper the CW algorithm was used to generate the initialvehicle path for the first stage problem The algorithm wasfirst proposed by Clarke and Wright [2] and is applicable toCVRP It forms a route in the following ways First it yieldsthe shortest route between every two suppliers in the systemThe desired aim here is to allocate loads to vehicles in such amanner that all the raw materials are assigned and the totalmileage covered is at a minimum

Consider a general savings value S given by linking twosuppliers i and j as shown in Figure 2

119904119894119895 = 1198890119894 + 1198890119895 minus 120582119889119894119895 (18)

Where119904= savings value1198890119894= distance from supplier i to the manufacturer119889119894119895= distance between points i and j120582= route shape parameter Special cases 120582 = 1 thesavings approach and 120582 = 2 Gaskellrsquos120587 method [15] Withan increase in 120582 greater emphasis is placed on the distancebetween the suppliers rather than on the position relative tothe manufacturer in selecting an addition to a route

In line with the savings value S between every twosuppliers the algorithm yields the CW table In the case ofsatisfying the on-vehicle capacity the manufacturer prefer-entially links a pair of suppliers with a larger S-value and usesone vehicle to provide pickup services for these suppliersFollowing the linking method until all the suppliers areserviced the manufacturer then obtains the initial solutionfor the pickup vehicle routes The procedure given is simplebut effective in producing a near-optimal solution and hasbeen programmed for several digital computers

42 Routes Improvement (RTR) After the initial vehicleroute was generated we further optimized it using the RTRtravel method which was first proposed by Dueck [16] Weimproved the solutions using the operations specified by Liet al ([17] namely a one-point move and two-point move Inthe one-point move we attempted to move each supplier inthe existing solution to a newposition on the same route or ona different route In the two-point move we tried to exchangethe positions of two suppliers These improvement moves areshown in Figure 3

The routes were improved by repeatedly applying thesetwo local search operators in two phases diversification andimprovement In the diversification phase we attempted toexplore new areas in the solution space by accepting bothimproving and deteriorating moves In the improvement

phase we attempted to improve the current solution as muchas possible by accepting only improving moves until wereached a local minimum

By combining the CW algorithm and RTR travel methodwe were able to obtain a better pickup vehicle routingsolution Based on the routes we then solved the decisionmaking problems pertaining to vehicle allocation in eachproduction batch by using the PSO algorithm in order tosatisfy the manufacturerrsquos demands in each production batchand lower the inventory cost as far as possible

43 Batch Allocation (PSO) The first stage enabled us toobtain all the routes that the vehicles needed to cover thenmaking it necessary to assign the routes to each productionbatch in the second stage Consequently the PSO algorithmdeveloped by Eberhart and Kennedy [18] was employed tosolve the route allocation problem and can be seen as anefficient stochastic global optimization technique

In the PSO algorithm the position of each particlerepresents a feasible solution in the search domain and eachparticle changes its position and velocity depending on itsflying experience The decision variables 119909119894119898 119910119894119895119898 and 119902119894119898which are related to the pickup service were determined inthe first stage 119892119898119905 and 120593119905 could be calculated according toconstraints (16) and (17) as long as 119911119898119905 was determined inthe second stage The core variable 119911119898119905 which determineswhether the raw material transported by vehicle m wasassigned to production batch t 119898 isin 119881 119905 isin 119879 was coded asfollows All the utilized vehicles were grouped in set 1198811015840 1198811015840 isin119881 Each particle with |1198811015840| dimensions was represented by119865 =1198911 1198912 119891119898 119891|1198811015840| Each dimension 119891119898 was coded as apositive number After the vehicle sequence was reorderedaccording to the nondescending order of the 119891119898 vehicleswere assigned to the production batches The main ideabehind the distribution rule is as follows vehicles at the frontof the sorted sequence have priority in terms of assignationto the previous production batch

For instance if the amount of materials acquired by aproduction batch is 10 10 10 the utilized vehicles wouldbe 1 2 3 4 5 and the quantity of materials correspondingto the vehicles would be 4 6 8 7 5 If the sorted vehiclesequence is 2 3 1 5 4 the amount of materials would be6 8 5 4 7 accordingly Vehicle 2 was first assigned toproduction batch 1 given that the amount of materialacquired in this batch was 10 which is greater than theamount that vehicle 2 carriedThe next vehicle in the sortedsequence that is vehicle 3was assigned to batch 1 thusgiving the storage material of production batch 1 a value of4 Accordingly vehicles 4 and 5 were assigned to batch 2with the storage value of 3 and vehicle 7 was assigned tobatch 3 with no storage material in the last

In applying the PSO algorithm we assumed that a swarmhas k particles 119881119899119896 = V119899119896119898119905 was the velocity of particle k atiteration n and119875119899119896 = 119901119899119896119898119905 the position of particle k at itera-tion n Each particle had its personal best position119875119871119861119890119904119905119899119896119898119905 at iteration n on dimensions m and t 119875119866119861119890119904119905119899119896119898119905 is the globalbest positon of the entire swarm on dimensions m and tup until iteration n The swarm flies through hyperspaceaccording to the experience of its neighbors The ordinary

6 Discrete Dynamics in Nature and Society

(a) One-point move (b) Two-point move

Figure 3 Improvement operators used in VRPPMP

updating formulates used to calculate speed and positionwere as follows

V119899+1119896119898119905 = 119908 sdot V119899119896119898119905 + 11988811199031 (119875119871119861119890119904119905119899119896119898119905 minus 119901119899119896119898119905)+ 11988821199032 (119875119866119861119890119904119905119899119896119898119905 minus 119901119899119896119898119905) (19)

119901119899+1119896119898119905 = 119901119899119896119898119905 + V119899+1119896119898119905 (20)

Where w is an inertia weight parameter which affects theconvergence procedure of the optimal solution 1198881and 1198882 arethe cognitive and social learning parameters respectively 1199031and 1199032 are random numbers generated between [0 1]

Based on the above description we now present the PSOalgorithm process for solving the route allocation problem

Step 1 Set the iteration number n = 1 Initialize k particleswhose positions and velocities are randomly generated Theparticlesrsquo positions represent the result of the route allocationand determine their qualities

Step 2 Evaluate the fitness value of each particle using thecommercial software CPLEX More specifically we used theroute allocation result obtained by the particles to fix thevariables 119911119898119905 and then used CPLEX to solve the originalmodel

Step 3 For each particle compare its fitness value andthe global best optimal solution119875119866119861119890119904119905119899119896119898119905 if 119875119871119861119890119904119905119899119896119898119905 gt119875119866119861119890119904119905119899119896119898119905 replace the value of 119875119866119861119890119904119905119899119896119898119905 using the value of119875119871119861119890119904119905119899119896119898119905Step 4 According to formulas (19) and (20) update theparticle velocity and positon

Step 5 If n reaches the preset maximum iteration or119875119866119861119890119904119905119899119896119898119905 has not been improved during a preset numberof iterations end the procedure otherwise set 119899 = 119899 + 1andthen go on to Step 2

5 Numerical Experiments

To validate the performance of the proposed two-stageheuristic numerical experiments were carried out on theVRPPMP For the PSO algorithm the maximum number ofiterations was set to 100 while the population size was 30According to the previous tests we set 1198881 = 1198882 = 1 Frompreliminary testing of 119908 = 0 05 1 15 and 2 it emergedthat 119908 = 1 performed the best thus we adopted 119908 = 1 for

the next experiments Based on the same decision obtainedin the first stage another nature inspired algorithm the Tabusearch (TS) algorithm was applied for comparison purposesWe listed the optimal solutions obtained by CPLEX in small-scale instances

All the experiments were performed on a computer with360 gigahertz Intel (R) Core (TM) i7-4790 CPU and 360gigabyte RAM The model and solution methods includingthe CW algorithm RTR travel algorithm and the PSOalgorithm were implemented using CPLEX 126 with C(VS2015) concert technology

51 Generation of Test Instance In this section we demon-strate the implementation of numerical experiments to val-idate the effectiveness of the proposed model and the effi-ciency of the proposed algorithm For small-scale instancesthe results of the proposed two-stage heuristic algorithmwerecompared with the optimal solutions obtained byCPLEX andthe solutions obtained using the TS algorithm Based on thesame decision obtained in the first stage the results obtainedusing the PSO heuristic were compared with those of the TSalgorithm in medium- and large-scale instances given thatCPLEX would have been time consuming or unable to solvethe problem

For the numerical experiments on the VRPPMP in-stances were randomly generated in the three scales Wesimulated the cases of 10 suppliers and two productionbatches and 15 suppliers and three production batches forthe small-scale 30 suppliers and six production batches and50 suppliers and 10 production batches for the medium scale70 suppliers and 14 production batches 90 suppliers and 18production batches 100 suppliers and 18 production batchesfor the large scale Information pertaining to the coordinatepositions and pickup demands of the 10 suppliers of thefirst small-scale instance (ie 10-2-1) are shown in Table 1The coordinate position of the manufacturer was (5 5) andthe cost of unit distance traveled by vehicle was $2 Themanufacturer had sufficient pickup vehicles with the capacityof 10 tons to provide a service to suppliersThe volume of rawmaterials required for each production batch was 10 tons andthe unit inventory cost of unprocessed raw materials in eachbatch was $20

52 Analysis of Calculation Results We tested examples atdifferent scales and used CPLEX the PSO algorithm andthe TS algorithm respectively to solve the VRPPMP Table 2yields the following conclusions In the small-scale examples(10-2) the exact solutions (7345 17317 and 18465) were

Discrete Dynamics in Nature and Society 7

Table 1 Pickup demands (tons) and coordinate positions (km) of instances 10-2-1

Supplier Pickup demands Coordinate axis X Coordinate axis Y1 15577 62804 834392 14597 56732 598063 22747 20603 558884 19595 90603 442185 15754 97755 273706 25402 29191 467317 20115 63266 469518 15807 98215 030379 23763 86237 9953510 27253 67718 31459

Table 2 Computational results of small-scale instances

Instance IDCPLEX PSO algorithm TS algorithm

Result($)

Time(sec)

Result($)

Time(sec)

Gap()

Result($)

Time(sec)

Gap()

10-2-1 73 45 545 7345 447 0 7345 755 010-2-2 17317 366 17657 307 196 17657 708 19610-2-3 18465 510 18919 355 246 18919 760 24615-3-1 - gt12hours 24854 1424 - 24854 1730 -15-3-2 - gt12hours 43527 1369 - 43527 1747 -15-3-3 - gt12hours 23667 1373 - 23667 1739 -

Table 3 Computational results of medium-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

30-6-1 57288 8346 57288 51069 0 5119030-6-2 56730 13409 55166 51958 -276 2874930-6-3 67289 3792 67289 59352 0 14651950-10-1 124389 23773 119875 224667 -363 8450550-10-2 164245 61163 157761 229187 -395 2747250-10-3 250116 25234 242029 229583 -323 80982

obtained byCPLEX and the accuracy of themodel is verifiedCompared with the TS algorithm the PSO algorithm canobtain the approximate solutions (7345 17657 and 18919)in a shorter time In the small-scale examples (15-3) CPLEXhad difficultly in getting the results but the PSO algorithmand TS algorithm can still obtain the results in a shorttime

However after the scale of the study increases as shownin Tables 3 and 4 the PSO algorithm can take less timeto get a relatively appropriate solution than TS algorithmIn the medium-scale examples the relative value deviationsbetween the PSO algorithm and the TS algorithm are within4 However the relative time deviations between the PSOalgorithmand theTS algorithm reach 699on average In thelarge-scale examples although the relative value deviationsbetween the PSO algorithm and the TS algorithm reached

nearly 7 the PSO calculation results are still acceptableand it holds great advantages in terms of computing timeThis proves that the two-stage heuristic algorithm can helpus solve such problems well

6 Conclusion

This paper discusses a novel and practical research issuearising in the manufacturing industry namely the prob-lem of vehicle routing involving pickup in the context ofmultibatch production In order to reduce transportationcosts and inventory costs in the production process a mixedinteger programming model was developed Following thisa two-stage hybrid heuristic algorithm was put forward tosolve this model In the first stage the CW algorithm wasproposed in order to generate the initial vehicle routes The

8 Discrete Dynamics in Nature and Society

Table 4 Computational results of large-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

70-14-1 226615 112267 210941 706980 -692 5297370-14-2 372110 56183 356346 601707 -424 9709870-14-3 453408 94929 431432 627870 -485 5614190-18-1 422180 456409 401415 2122673 -492 3650890-18-2 427458 164882 401614 1371370 -605 7317390-18-3 777256 369100 740527 1960738 -473 43122100-18-1 516445 698274 493579 2861304 -443 30977100-18-2 804758 723314 776298 2856240 -354 29488100-18-3 1132139 692266 1086673 2868246 -402 31433

RTR travel algorithm was then used further to optimize thegenerated routes In the second stage of the algorithm thePSO algorithm was developed so as to allocate the vehiclesto each production batch This process ensured that themanufacturer first assigned each supplier to the appropri-ate route and then assigned each route to the appropriatebatch

By testing examples at different scales we were able tovalidate the effectiveness of the proposed model and theperformance efficiency of the two-stage hybrid heuristicalgorithm First in the small-scale examples the exact solu-tion was obtained using CPLEX and the accuracy of themodel was verified Second after the scale was expandedit was found that CPLEX had difficulty in solving thisover a short period of time and that the PSO algorithmtook less time to obtain a relatively appropriate resultthan the TS algorithm This proves that the two-stageheuristic algorithm is well placed to help us solve suchproblems

There are certain limitations to the current study First weassumed that the amount of the single raw material requiredfor each production batch would be the same in practicehowever the amount of multiple rawmaterials varies for eachproduction batch Second this paper only considered thecombination of the pickup stage and the production stagewithout considering product delivery The joint optimizationof multibatch production and the vehicle routing problemwith pickup and delivery considering inventory will thus beaddressed in future studies

Data Availability

The data used to support this study have been deposited inthe Baidu Netdisk readers can access the data at httpspanbaiducoms1ZSkGF-syiNTk FN9JNShbA

Conflicts of Interest

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

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China [no 71771143 and no 71501125] andShanghai Young Eastern Scholar Programme [QD2015041]

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] 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

[3] R Fukasawa H Longo J Lysgaard et al ldquoRobust branch-and-cut-and-price for the capacitated vehicle routing problemrdquoMathematical Programming vol 106 no 3 pp 491ndash511 2006

[4] M LGambardella D E Taillard andG Agazzi ldquoAmultiple antcolony system for vehicle routing problemswith timewindowsrdquo1999

[5] P Sombuntham and V Kachitvichyanukul ldquoMulti-depot vehi-cle routing problem with pickup and delivery requestsrdquo inProceedings of the International MultiConference of Engineersand Computer Scientists IMECS 2010 pp 71ndash85 China March2010

[6] C K Y Lin ldquoA vehicle routing problem with pickup anddelivery time windows and coordination of transportableresourcesrdquoComputers amp Operations Research vol 38 no 11 pp1596ndash1609 2011

[7] MW Savelsbergh andM Sol ldquoThe general pickup and deliveryproblemrdquo Transportation Science vol 29 no 1 pp 17ndash29 1995

[8] H Xu Z-L Chen S Rajagopal and S Arunapuram ldquoSolving apractical pickup and delivery problemrdquo Transportation Sciencevol 37 no 3 pp 347ndash364 2003

[9] G Nagy and S Salhi ldquoHeuristic algorithms for single andmultiple depot vehicle routing problems with pickups anddeliveriesrdquo European Journal of Operational Research vol 162no 1 pp 126ndash141 2005

[10] P Kalina and J Vokrınek ldquoParallel solver for vehicle routingand pickup and delivery problems with time windows based onagent negotiationrdquo in Proceedings of the 2012 IEEE InternationalConference on Systems Man and Cybernetics SMC 2012 pp1558ndash1563 Republic of Korea October 2012

4 Discrete Dynamics in Nature and Society

Q The capacity of homogeneous vehiclesDThe amount of planned product in each productionbatchM A sufficient large positive number

Decision Variables

119909119894119898 1 if vehicle m provides supplier i with pickupservice 0 otherwise119911119898119905 1 if vehicle m is assigned to the production batcht 0 otherwise119910119894119895119898 1 if suppliers i and j are successively serviced byvehicle m 0 otherwise119892119898119905 The amount of raw material transported byvehicle m for production batch t119902119894119898 The total amount of raw material transported byvehicle m after servicing supplier i120593119905The remaining inventory of the rawmaterials afterthe tth production batch

Based on the above definition the VRPPMP is established asfollows

min (sum119894isin119878+

sum119895isin119878minus

sum119898isin119881

119888119894119895 sdot 119910119894119895119898 + 119888119906119899119901119903119900 sdot sum119905isin119879

120593119905) (1)

119904119905 1199090119898 = 1 forall119898 isin 119881 (2)

sum119898isin119881

119909119894119898 = 1 forall119894 isin 119878 (3)

sum119894isin119878+119894 =ℎ

119910119894ℎ119898 = sum119895isin119878minus119895 =ℎ

119910ℎ119895119898 = 119909ℎ119898forallℎ isin 119878 forall119898 isin 119881

(4)

sum119895isin119878

1199100119895119898 minus sum119905isin119879

119911119898119905 = 0 forall119898 isin 119881 (5)

sum119905isin119879

119911119898119905 le 1 forall119898 isin 119881 (6)

119892119898119905 le 119876 sdot 119911119898119905 forall119905 isin 119879 forall119898 isin 119881 (7)

119892119898119905 le sum119895isin119878

119901119895 sdot 119909119895119898 + 119876 sdot (1 minus 119911119898119905)forall119905 isin 119879 forall119898 isin 119881

(8)

119892119898119905 ge sum119895isin119878

119901119895 sdot 119909119895119898 minus 119876 sdot (1 minus 119911119898119905)forall119905 isin 119879 forall119898 isin 119881

(9)

sum119898isin119881

119892119898119905 ge 119863 119905 = 1 (10)

sum119898isin119881

119892119898119905 + 120593119905minus1 ge 119863 forall119905 isin 119879 1 (11)

119902119895119898 ge 119901119895 minus 119876 sdot (1 minus 1199100119895119898) forall119895 isin 119878 forall119898 isin 119881 (12)

119902119895119898 le 119901119895 + 119876 sdot (1 minus 1199100119895119898) forall119895 isin 119878 forall119898 isin 119881 (13)

119902119895119898 ge 119902119894119898 + 119901119895 minus 119876 sdot (1 minus 119910119894119895119898)forall119894 isin 119878 forall119895 isin 119878minus 119894 = 119895 forall119898 isin 119881 (14)

119902119895119898 le 119902119894119898 + 119901119895 + 119876 sdot (1 minus 119910119894119895119898)forall119894 isin 119878 forall119895 isin 119878minus 119894 = 119895 forall119898 isin 119881 (15)

120593119905 = sum119898isin119881

119892119898119905 minus 119863 119905 = 1 (16)

120593119905 = 120593119905minus1 + sum119898isin119881

119892119898119905 minus 119863 forall119905 isin 119879 1 (17)

Objective function (1) maximizes the total costs of themanufacturer which are equal to the sum of the inventorycosts and shipping costs incurred Constraints (2) state thateach vehicle starts at the manufacturer and ends at the man-ufacturer Constraints (3) ensure that each supplier will beaccessed once Flow conservation is ensured by Constraints(4) Constraints (5) indicate that each vehicle that providessuppliers with a pickup service must serve in exactly oneproduction batch Constraints (6) state that each vehicle canat most serve at most one production batch Constraints(7) ensure that the raw material transported by each vehicledoes not exceed the capacity of vehicle Constraints (8) and(9) represent the total amount of raw material delivery byeach vehicle in each production batch Constraints (10) and(11) indicate the constraint of amount of raw materials thatcan be put into production in each batch Constraints (12)to (15) state the relationship between the vehicle load andthe quantity supplied Constraints (16) and (17) represent theraw material inventory after each batch of production andprocessing

4 Solution Algorithm

The above model can be directly solved by using commercialoptimization packages such as CPLEX when the scale issmall However when the scale increases by a certain extentusing CPLEX is time consuming or unable to complete thesolution Therefore the algorithm needs to be added to solvelarge-scale problems

Taking the two aspects into account when making adecision that is vehicle routing as well as the matchingbetween the vehicle and the production batch the VRPMPmentioned above cannot be solved directly In light of thisthe VRPPMP was divided into two stages In the first stagethe problem of vehicle routing in the process of pickingup cargoes was solved using the Clarke-Wright and theRecord-to-Record travel algorithms In the second stage weused the PSO algorithm to allocate each vehicle that needsto be picking up to a certain production batch so as tosatisfy the demands of the manufacturer in each productionbatch

Discrete Dynamics in Nature and Society 5

Manufacturer

Supplier i

Supplier j

Manufacturer

Supplier i

Supplier j

Figure 2 Changes in vehicle pickup routes

41 Routes Initialization (CW) For the purposes of thispaper the CW algorithm was used to generate the initialvehicle path for the first stage problem The algorithm wasfirst proposed by Clarke and Wright [2] and is applicable toCVRP It forms a route in the following ways First it yieldsthe shortest route between every two suppliers in the systemThe desired aim here is to allocate loads to vehicles in such amanner that all the raw materials are assigned and the totalmileage covered is at a minimum

Consider a general savings value S given by linking twosuppliers i and j as shown in Figure 2

119904119894119895 = 1198890119894 + 1198890119895 minus 120582119889119894119895 (18)

Where119904= savings value1198890119894= distance from supplier i to the manufacturer119889119894119895= distance between points i and j120582= route shape parameter Special cases 120582 = 1 thesavings approach and 120582 = 2 Gaskellrsquos120587 method [15] Withan increase in 120582 greater emphasis is placed on the distancebetween the suppliers rather than on the position relative tothe manufacturer in selecting an addition to a route

In line with the savings value S between every twosuppliers the algorithm yields the CW table In the case ofsatisfying the on-vehicle capacity the manufacturer prefer-entially links a pair of suppliers with a larger S-value and usesone vehicle to provide pickup services for these suppliersFollowing the linking method until all the suppliers areserviced the manufacturer then obtains the initial solutionfor the pickup vehicle routes The procedure given is simplebut effective in producing a near-optimal solution and hasbeen programmed for several digital computers

42 Routes Improvement (RTR) After the initial vehicleroute was generated we further optimized it using the RTRtravel method which was first proposed by Dueck [16] Weimproved the solutions using the operations specified by Liet al ([17] namely a one-point move and two-point move Inthe one-point move we attempted to move each supplier inthe existing solution to a newposition on the same route or ona different route In the two-point move we tried to exchangethe positions of two suppliers These improvement moves areshown in Figure 3

The routes were improved by repeatedly applying thesetwo local search operators in two phases diversification andimprovement In the diversification phase we attempted toexplore new areas in the solution space by accepting bothimproving and deteriorating moves In the improvement

phase we attempted to improve the current solution as muchas possible by accepting only improving moves until wereached a local minimum

By combining the CW algorithm and RTR travel methodwe were able to obtain a better pickup vehicle routingsolution Based on the routes we then solved the decisionmaking problems pertaining to vehicle allocation in eachproduction batch by using the PSO algorithm in order tosatisfy the manufacturerrsquos demands in each production batchand lower the inventory cost as far as possible

43 Batch Allocation (PSO) The first stage enabled us toobtain all the routes that the vehicles needed to cover thenmaking it necessary to assign the routes to each productionbatch in the second stage Consequently the PSO algorithmdeveloped by Eberhart and Kennedy [18] was employed tosolve the route allocation problem and can be seen as anefficient stochastic global optimization technique

In the PSO algorithm the position of each particlerepresents a feasible solution in the search domain and eachparticle changes its position and velocity depending on itsflying experience The decision variables 119909119894119898 119910119894119895119898 and 119902119894119898which are related to the pickup service were determined inthe first stage 119892119898119905 and 120593119905 could be calculated according toconstraints (16) and (17) as long as 119911119898119905 was determined inthe second stage The core variable 119911119898119905 which determineswhether the raw material transported by vehicle m wasassigned to production batch t 119898 isin 119881 119905 isin 119879 was coded asfollows All the utilized vehicles were grouped in set 1198811015840 1198811015840 isin119881 Each particle with |1198811015840| dimensions was represented by119865 =1198911 1198912 119891119898 119891|1198811015840| Each dimension 119891119898 was coded as apositive number After the vehicle sequence was reorderedaccording to the nondescending order of the 119891119898 vehicleswere assigned to the production batches The main ideabehind the distribution rule is as follows vehicles at the frontof the sorted sequence have priority in terms of assignationto the previous production batch

For instance if the amount of materials acquired by aproduction batch is 10 10 10 the utilized vehicles wouldbe 1 2 3 4 5 and the quantity of materials correspondingto the vehicles would be 4 6 8 7 5 If the sorted vehiclesequence is 2 3 1 5 4 the amount of materials would be6 8 5 4 7 accordingly Vehicle 2 was first assigned toproduction batch 1 given that the amount of materialacquired in this batch was 10 which is greater than theamount that vehicle 2 carriedThe next vehicle in the sortedsequence that is vehicle 3was assigned to batch 1 thusgiving the storage material of production batch 1 a value of4 Accordingly vehicles 4 and 5 were assigned to batch 2with the storage value of 3 and vehicle 7 was assigned tobatch 3 with no storage material in the last

In applying the PSO algorithm we assumed that a swarmhas k particles 119881119899119896 = V119899119896119898119905 was the velocity of particle k atiteration n and119875119899119896 = 119901119899119896119898119905 the position of particle k at itera-tion n Each particle had its personal best position119875119871119861119890119904119905119899119896119898119905 at iteration n on dimensions m and t 119875119866119861119890119904119905119899119896119898119905 is the globalbest positon of the entire swarm on dimensions m and tup until iteration n The swarm flies through hyperspaceaccording to the experience of its neighbors The ordinary

6 Discrete Dynamics in Nature and Society

(a) One-point move (b) Two-point move

Figure 3 Improvement operators used in VRPPMP

updating formulates used to calculate speed and positionwere as follows

V119899+1119896119898119905 = 119908 sdot V119899119896119898119905 + 11988811199031 (119875119871119861119890119904119905119899119896119898119905 minus 119901119899119896119898119905)+ 11988821199032 (119875119866119861119890119904119905119899119896119898119905 minus 119901119899119896119898119905) (19)

119901119899+1119896119898119905 = 119901119899119896119898119905 + V119899+1119896119898119905 (20)

Where w is an inertia weight parameter which affects theconvergence procedure of the optimal solution 1198881and 1198882 arethe cognitive and social learning parameters respectively 1199031and 1199032 are random numbers generated between [0 1]

Based on the above description we now present the PSOalgorithm process for solving the route allocation problem

Step 1 Set the iteration number n = 1 Initialize k particleswhose positions and velocities are randomly generated Theparticlesrsquo positions represent the result of the route allocationand determine their qualities

Step 2 Evaluate the fitness value of each particle using thecommercial software CPLEX More specifically we used theroute allocation result obtained by the particles to fix thevariables 119911119898119905 and then used CPLEX to solve the originalmodel

Step 3 For each particle compare its fitness value andthe global best optimal solution119875119866119861119890119904119905119899119896119898119905 if 119875119871119861119890119904119905119899119896119898119905 gt119875119866119861119890119904119905119899119896119898119905 replace the value of 119875119866119861119890119904119905119899119896119898119905 using the value of119875119871119861119890119904119905119899119896119898119905Step 4 According to formulas (19) and (20) update theparticle velocity and positon

Step 5 If n reaches the preset maximum iteration or119875119866119861119890119904119905119899119896119898119905 has not been improved during a preset numberof iterations end the procedure otherwise set 119899 = 119899 + 1andthen go on to Step 2

5 Numerical Experiments

To validate the performance of the proposed two-stageheuristic numerical experiments were carried out on theVRPPMP For the PSO algorithm the maximum number ofiterations was set to 100 while the population size was 30According to the previous tests we set 1198881 = 1198882 = 1 Frompreliminary testing of 119908 = 0 05 1 15 and 2 it emergedthat 119908 = 1 performed the best thus we adopted 119908 = 1 for

the next experiments Based on the same decision obtainedin the first stage another nature inspired algorithm the Tabusearch (TS) algorithm was applied for comparison purposesWe listed the optimal solutions obtained by CPLEX in small-scale instances

All the experiments were performed on a computer with360 gigahertz Intel (R) Core (TM) i7-4790 CPU and 360gigabyte RAM The model and solution methods includingthe CW algorithm RTR travel algorithm and the PSOalgorithm were implemented using CPLEX 126 with C(VS2015) concert technology

51 Generation of Test Instance In this section we demon-strate the implementation of numerical experiments to val-idate the effectiveness of the proposed model and the effi-ciency of the proposed algorithm For small-scale instancesthe results of the proposed two-stage heuristic algorithmwerecompared with the optimal solutions obtained byCPLEX andthe solutions obtained using the TS algorithm Based on thesame decision obtained in the first stage the results obtainedusing the PSO heuristic were compared with those of the TSalgorithm in medium- and large-scale instances given thatCPLEX would have been time consuming or unable to solvethe problem

For the numerical experiments on the VRPPMP in-stances were randomly generated in the three scales Wesimulated the cases of 10 suppliers and two productionbatches and 15 suppliers and three production batches forthe small-scale 30 suppliers and six production batches and50 suppliers and 10 production batches for the medium scale70 suppliers and 14 production batches 90 suppliers and 18production batches 100 suppliers and 18 production batchesfor the large scale Information pertaining to the coordinatepositions and pickup demands of the 10 suppliers of thefirst small-scale instance (ie 10-2-1) are shown in Table 1The coordinate position of the manufacturer was (5 5) andthe cost of unit distance traveled by vehicle was $2 Themanufacturer had sufficient pickup vehicles with the capacityof 10 tons to provide a service to suppliersThe volume of rawmaterials required for each production batch was 10 tons andthe unit inventory cost of unprocessed raw materials in eachbatch was $20

52 Analysis of Calculation Results We tested examples atdifferent scales and used CPLEX the PSO algorithm andthe TS algorithm respectively to solve the VRPPMP Table 2yields the following conclusions In the small-scale examples(10-2) the exact solutions (7345 17317 and 18465) were

Discrete Dynamics in Nature and Society 7

Table 1 Pickup demands (tons) and coordinate positions (km) of instances 10-2-1

Supplier Pickup demands Coordinate axis X Coordinate axis Y1 15577 62804 834392 14597 56732 598063 22747 20603 558884 19595 90603 442185 15754 97755 273706 25402 29191 467317 20115 63266 469518 15807 98215 030379 23763 86237 9953510 27253 67718 31459

Table 2 Computational results of small-scale instances

Instance IDCPLEX PSO algorithm TS algorithm

Result($)

Time(sec)

Result($)

Time(sec)

Gap()

Result($)

Time(sec)

Gap()

10-2-1 73 45 545 7345 447 0 7345 755 010-2-2 17317 366 17657 307 196 17657 708 19610-2-3 18465 510 18919 355 246 18919 760 24615-3-1 - gt12hours 24854 1424 - 24854 1730 -15-3-2 - gt12hours 43527 1369 - 43527 1747 -15-3-3 - gt12hours 23667 1373 - 23667 1739 -

Table 3 Computational results of medium-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

30-6-1 57288 8346 57288 51069 0 5119030-6-2 56730 13409 55166 51958 -276 2874930-6-3 67289 3792 67289 59352 0 14651950-10-1 124389 23773 119875 224667 -363 8450550-10-2 164245 61163 157761 229187 -395 2747250-10-3 250116 25234 242029 229583 -323 80982

obtained byCPLEX and the accuracy of themodel is verifiedCompared with the TS algorithm the PSO algorithm canobtain the approximate solutions (7345 17657 and 18919)in a shorter time In the small-scale examples (15-3) CPLEXhad difficultly in getting the results but the PSO algorithmand TS algorithm can still obtain the results in a shorttime

However after the scale of the study increases as shownin Tables 3 and 4 the PSO algorithm can take less timeto get a relatively appropriate solution than TS algorithmIn the medium-scale examples the relative value deviationsbetween the PSO algorithm and the TS algorithm are within4 However the relative time deviations between the PSOalgorithmand theTS algorithm reach 699on average In thelarge-scale examples although the relative value deviationsbetween the PSO algorithm and the TS algorithm reached

nearly 7 the PSO calculation results are still acceptableand it holds great advantages in terms of computing timeThis proves that the two-stage heuristic algorithm can helpus solve such problems well

6 Conclusion

This paper discusses a novel and practical research issuearising in the manufacturing industry namely the prob-lem of vehicle routing involving pickup in the context ofmultibatch production In order to reduce transportationcosts and inventory costs in the production process a mixedinteger programming model was developed Following thisa two-stage hybrid heuristic algorithm was put forward tosolve this model In the first stage the CW algorithm wasproposed in order to generate the initial vehicle routes The

8 Discrete Dynamics in Nature and Society

Table 4 Computational results of large-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

70-14-1 226615 112267 210941 706980 -692 5297370-14-2 372110 56183 356346 601707 -424 9709870-14-3 453408 94929 431432 627870 -485 5614190-18-1 422180 456409 401415 2122673 -492 3650890-18-2 427458 164882 401614 1371370 -605 7317390-18-3 777256 369100 740527 1960738 -473 43122100-18-1 516445 698274 493579 2861304 -443 30977100-18-2 804758 723314 776298 2856240 -354 29488100-18-3 1132139 692266 1086673 2868246 -402 31433

RTR travel algorithm was then used further to optimize thegenerated routes In the second stage of the algorithm thePSO algorithm was developed so as to allocate the vehiclesto each production batch This process ensured that themanufacturer first assigned each supplier to the appropri-ate route and then assigned each route to the appropriatebatch

By testing examples at different scales we were able tovalidate the effectiveness of the proposed model and theperformance efficiency of the two-stage hybrid heuristicalgorithm First in the small-scale examples the exact solu-tion was obtained using CPLEX and the accuracy of themodel was verified Second after the scale was expandedit was found that CPLEX had difficulty in solving thisover a short period of time and that the PSO algorithmtook less time to obtain a relatively appropriate resultthan the TS algorithm This proves that the two-stageheuristic algorithm is well placed to help us solve suchproblems

There are certain limitations to the current study First weassumed that the amount of the single raw material requiredfor each production batch would be the same in practicehowever the amount of multiple rawmaterials varies for eachproduction batch Second this paper only considered thecombination of the pickup stage and the production stagewithout considering product delivery The joint optimizationof multibatch production and the vehicle routing problemwith pickup and delivery considering inventory will thus beaddressed in future studies

Data Availability

The data used to support this study have been deposited inthe Baidu Netdisk readers can access the data at httpspanbaiducoms1ZSkGF-syiNTk FN9JNShbA

Conflicts of Interest

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

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China [no 71771143 and no 71501125] andShanghai Young Eastern Scholar Programme [QD2015041]

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] 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

[3] R Fukasawa H Longo J Lysgaard et al ldquoRobust branch-and-cut-and-price for the capacitated vehicle routing problemrdquoMathematical Programming vol 106 no 3 pp 491ndash511 2006

[4] M LGambardella D E Taillard andG Agazzi ldquoAmultiple antcolony system for vehicle routing problemswith timewindowsrdquo1999

[5] P Sombuntham and V Kachitvichyanukul ldquoMulti-depot vehi-cle routing problem with pickup and delivery requestsrdquo inProceedings of the International MultiConference of Engineersand Computer Scientists IMECS 2010 pp 71ndash85 China March2010

[6] C K Y Lin ldquoA vehicle routing problem with pickup anddelivery time windows and coordination of transportableresourcesrdquoComputers amp Operations Research vol 38 no 11 pp1596ndash1609 2011

[7] MW Savelsbergh andM Sol ldquoThe general pickup and deliveryproblemrdquo Transportation Science vol 29 no 1 pp 17ndash29 1995

[8] H Xu Z-L Chen S Rajagopal and S Arunapuram ldquoSolving apractical pickup and delivery problemrdquo Transportation Sciencevol 37 no 3 pp 347ndash364 2003

[9] G Nagy and S Salhi ldquoHeuristic algorithms for single andmultiple depot vehicle routing problems with pickups anddeliveriesrdquo European Journal of Operational Research vol 162no 1 pp 126ndash141 2005

[10] P Kalina and J Vokrınek ldquoParallel solver for vehicle routingand pickup and delivery problems with time windows based onagent negotiationrdquo in Proceedings of the 2012 IEEE InternationalConference on Systems Man and Cybernetics SMC 2012 pp1558ndash1563 Republic of Korea October 2012

Discrete Dynamics in Nature and Society 5

Manufacturer

Supplier i

Supplier j

Manufacturer

Supplier i

Supplier j

Figure 2 Changes in vehicle pickup routes

41 Routes Initialization (CW) For the purposes of thispaper the CW algorithm was used to generate the initialvehicle path for the first stage problem The algorithm wasfirst proposed by Clarke and Wright [2] and is applicable toCVRP It forms a route in the following ways First it yieldsthe shortest route between every two suppliers in the systemThe desired aim here is to allocate loads to vehicles in such amanner that all the raw materials are assigned and the totalmileage covered is at a minimum

Consider a general savings value S given by linking twosuppliers i and j as shown in Figure 2

119904119894119895 = 1198890119894 + 1198890119895 minus 120582119889119894119895 (18)

Where119904= savings value1198890119894= distance from supplier i to the manufacturer119889119894119895= distance between points i and j120582= route shape parameter Special cases 120582 = 1 thesavings approach and 120582 = 2 Gaskellrsquos120587 method [15] Withan increase in 120582 greater emphasis is placed on the distancebetween the suppliers rather than on the position relative tothe manufacturer in selecting an addition to a route

In line with the savings value S between every twosuppliers the algorithm yields the CW table In the case ofsatisfying the on-vehicle capacity the manufacturer prefer-entially links a pair of suppliers with a larger S-value and usesone vehicle to provide pickup services for these suppliersFollowing the linking method until all the suppliers areserviced the manufacturer then obtains the initial solutionfor the pickup vehicle routes The procedure given is simplebut effective in producing a near-optimal solution and hasbeen programmed for several digital computers

42 Routes Improvement (RTR) After the initial vehicleroute was generated we further optimized it using the RTRtravel method which was first proposed by Dueck [16] Weimproved the solutions using the operations specified by Liet al ([17] namely a one-point move and two-point move Inthe one-point move we attempted to move each supplier inthe existing solution to a newposition on the same route or ona different route In the two-point move we tried to exchangethe positions of two suppliers These improvement moves areshown in Figure 3

The routes were improved by repeatedly applying thesetwo local search operators in two phases diversification andimprovement In the diversification phase we attempted toexplore new areas in the solution space by accepting bothimproving and deteriorating moves In the improvement

phase we attempted to improve the current solution as muchas possible by accepting only improving moves until wereached a local minimum

By combining the CW algorithm and RTR travel methodwe were able to obtain a better pickup vehicle routingsolution Based on the routes we then solved the decisionmaking problems pertaining to vehicle allocation in eachproduction batch by using the PSO algorithm in order tosatisfy the manufacturerrsquos demands in each production batchand lower the inventory cost as far as possible

43 Batch Allocation (PSO) The first stage enabled us toobtain all the routes that the vehicles needed to cover thenmaking it necessary to assign the routes to each productionbatch in the second stage Consequently the PSO algorithmdeveloped by Eberhart and Kennedy [18] was employed tosolve the route allocation problem and can be seen as anefficient stochastic global optimization technique

In the PSO algorithm the position of each particlerepresents a feasible solution in the search domain and eachparticle changes its position and velocity depending on itsflying experience The decision variables 119909119894119898 119910119894119895119898 and 119902119894119898which are related to the pickup service were determined inthe first stage 119892119898119905 and 120593119905 could be calculated according toconstraints (16) and (17) as long as 119911119898119905 was determined inthe second stage The core variable 119911119898119905 which determineswhether the raw material transported by vehicle m wasassigned to production batch t 119898 isin 119881 119905 isin 119879 was coded asfollows All the utilized vehicles were grouped in set 1198811015840 1198811015840 isin119881 Each particle with |1198811015840| dimensions was represented by119865 =1198911 1198912 119891119898 119891|1198811015840| Each dimension 119891119898 was coded as apositive number After the vehicle sequence was reorderedaccording to the nondescending order of the 119891119898 vehicleswere assigned to the production batches The main ideabehind the distribution rule is as follows vehicles at the frontof the sorted sequence have priority in terms of assignationto the previous production batch

For instance if the amount of materials acquired by aproduction batch is 10 10 10 the utilized vehicles wouldbe 1 2 3 4 5 and the quantity of materials correspondingto the vehicles would be 4 6 8 7 5 If the sorted vehiclesequence is 2 3 1 5 4 the amount of materials would be6 8 5 4 7 accordingly Vehicle 2 was first assigned toproduction batch 1 given that the amount of materialacquired in this batch was 10 which is greater than theamount that vehicle 2 carriedThe next vehicle in the sortedsequence that is vehicle 3was assigned to batch 1 thusgiving the storage material of production batch 1 a value of4 Accordingly vehicles 4 and 5 were assigned to batch 2with the storage value of 3 and vehicle 7 was assigned tobatch 3 with no storage material in the last

In applying the PSO algorithm we assumed that a swarmhas k particles 119881119899119896 = V119899119896119898119905 was the velocity of particle k atiteration n and119875119899119896 = 119901119899119896119898119905 the position of particle k at itera-tion n Each particle had its personal best position119875119871119861119890119904119905119899119896119898119905 at iteration n on dimensions m and t 119875119866119861119890119904119905119899119896119898119905 is the globalbest positon of the entire swarm on dimensions m and tup until iteration n The swarm flies through hyperspaceaccording to the experience of its neighbors The ordinary

6 Discrete Dynamics in Nature and Society

(a) One-point move (b) Two-point move

Figure 3 Improvement operators used in VRPPMP

updating formulates used to calculate speed and positionwere as follows

V119899+1119896119898119905 = 119908 sdot V119899119896119898119905 + 11988811199031 (119875119871119861119890119904119905119899119896119898119905 minus 119901119899119896119898119905)+ 11988821199032 (119875119866119861119890119904119905119899119896119898119905 minus 119901119899119896119898119905) (19)

119901119899+1119896119898119905 = 119901119899119896119898119905 + V119899+1119896119898119905 (20)

Where w is an inertia weight parameter which affects theconvergence procedure of the optimal solution 1198881and 1198882 arethe cognitive and social learning parameters respectively 1199031and 1199032 are random numbers generated between [0 1]

Based on the above description we now present the PSOalgorithm process for solving the route allocation problem

Step 1 Set the iteration number n = 1 Initialize k particleswhose positions and velocities are randomly generated Theparticlesrsquo positions represent the result of the route allocationand determine their qualities

Step 2 Evaluate the fitness value of each particle using thecommercial software CPLEX More specifically we used theroute allocation result obtained by the particles to fix thevariables 119911119898119905 and then used CPLEX to solve the originalmodel

Step 3 For each particle compare its fitness value andthe global best optimal solution119875119866119861119890119904119905119899119896119898119905 if 119875119871119861119890119904119905119899119896119898119905 gt119875119866119861119890119904119905119899119896119898119905 replace the value of 119875119866119861119890119904119905119899119896119898119905 using the value of119875119871119861119890119904119905119899119896119898119905Step 4 According to formulas (19) and (20) update theparticle velocity and positon

Step 5 If n reaches the preset maximum iteration or119875119866119861119890119904119905119899119896119898119905 has not been improved during a preset numberof iterations end the procedure otherwise set 119899 = 119899 + 1andthen go on to Step 2

5 Numerical Experiments

To validate the performance of the proposed two-stageheuristic numerical experiments were carried out on theVRPPMP For the PSO algorithm the maximum number ofiterations was set to 100 while the population size was 30According to the previous tests we set 1198881 = 1198882 = 1 Frompreliminary testing of 119908 = 0 05 1 15 and 2 it emergedthat 119908 = 1 performed the best thus we adopted 119908 = 1 for

the next experiments Based on the same decision obtainedin the first stage another nature inspired algorithm the Tabusearch (TS) algorithm was applied for comparison purposesWe listed the optimal solutions obtained by CPLEX in small-scale instances

All the experiments were performed on a computer with360 gigahertz Intel (R) Core (TM) i7-4790 CPU and 360gigabyte RAM The model and solution methods includingthe CW algorithm RTR travel algorithm and the PSOalgorithm were implemented using CPLEX 126 with C(VS2015) concert technology

51 Generation of Test Instance In this section we demon-strate the implementation of numerical experiments to val-idate the effectiveness of the proposed model and the effi-ciency of the proposed algorithm For small-scale instancesthe results of the proposed two-stage heuristic algorithmwerecompared with the optimal solutions obtained byCPLEX andthe solutions obtained using the TS algorithm Based on thesame decision obtained in the first stage the results obtainedusing the PSO heuristic were compared with those of the TSalgorithm in medium- and large-scale instances given thatCPLEX would have been time consuming or unable to solvethe problem

For the numerical experiments on the VRPPMP in-stances were randomly generated in the three scales Wesimulated the cases of 10 suppliers and two productionbatches and 15 suppliers and three production batches forthe small-scale 30 suppliers and six production batches and50 suppliers and 10 production batches for the medium scale70 suppliers and 14 production batches 90 suppliers and 18production batches 100 suppliers and 18 production batchesfor the large scale Information pertaining to the coordinatepositions and pickup demands of the 10 suppliers of thefirst small-scale instance (ie 10-2-1) are shown in Table 1The coordinate position of the manufacturer was (5 5) andthe cost of unit distance traveled by vehicle was $2 Themanufacturer had sufficient pickup vehicles with the capacityof 10 tons to provide a service to suppliersThe volume of rawmaterials required for each production batch was 10 tons andthe unit inventory cost of unprocessed raw materials in eachbatch was $20

52 Analysis of Calculation Results We tested examples atdifferent scales and used CPLEX the PSO algorithm andthe TS algorithm respectively to solve the VRPPMP Table 2yields the following conclusions In the small-scale examples(10-2) the exact solutions (7345 17317 and 18465) were

Discrete Dynamics in Nature and Society 7

Table 1 Pickup demands (tons) and coordinate positions (km) of instances 10-2-1

Supplier Pickup demands Coordinate axis X Coordinate axis Y1 15577 62804 834392 14597 56732 598063 22747 20603 558884 19595 90603 442185 15754 97755 273706 25402 29191 467317 20115 63266 469518 15807 98215 030379 23763 86237 9953510 27253 67718 31459

Table 2 Computational results of small-scale instances

Instance IDCPLEX PSO algorithm TS algorithm

Result($)

Time(sec)

Result($)

Time(sec)

Gap()

Result($)

Time(sec)

Gap()

10-2-1 73 45 545 7345 447 0 7345 755 010-2-2 17317 366 17657 307 196 17657 708 19610-2-3 18465 510 18919 355 246 18919 760 24615-3-1 - gt12hours 24854 1424 - 24854 1730 -15-3-2 - gt12hours 43527 1369 - 43527 1747 -15-3-3 - gt12hours 23667 1373 - 23667 1739 -

Table 3 Computational results of medium-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

30-6-1 57288 8346 57288 51069 0 5119030-6-2 56730 13409 55166 51958 -276 2874930-6-3 67289 3792 67289 59352 0 14651950-10-1 124389 23773 119875 224667 -363 8450550-10-2 164245 61163 157761 229187 -395 2747250-10-3 250116 25234 242029 229583 -323 80982

obtained byCPLEX and the accuracy of themodel is verifiedCompared with the TS algorithm the PSO algorithm canobtain the approximate solutions (7345 17657 and 18919)in a shorter time In the small-scale examples (15-3) CPLEXhad difficultly in getting the results but the PSO algorithmand TS algorithm can still obtain the results in a shorttime

However after the scale of the study increases as shownin Tables 3 and 4 the PSO algorithm can take less timeto get a relatively appropriate solution than TS algorithmIn the medium-scale examples the relative value deviationsbetween the PSO algorithm and the TS algorithm are within4 However the relative time deviations between the PSOalgorithmand theTS algorithm reach 699on average In thelarge-scale examples although the relative value deviationsbetween the PSO algorithm and the TS algorithm reached

nearly 7 the PSO calculation results are still acceptableand it holds great advantages in terms of computing timeThis proves that the two-stage heuristic algorithm can helpus solve such problems well

6 Conclusion

This paper discusses a novel and practical research issuearising in the manufacturing industry namely the prob-lem of vehicle routing involving pickup in the context ofmultibatch production In order to reduce transportationcosts and inventory costs in the production process a mixedinteger programming model was developed Following thisa two-stage hybrid heuristic algorithm was put forward tosolve this model In the first stage the CW algorithm wasproposed in order to generate the initial vehicle routes The

8 Discrete Dynamics in Nature and Society

Table 4 Computational results of large-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

70-14-1 226615 112267 210941 706980 -692 5297370-14-2 372110 56183 356346 601707 -424 9709870-14-3 453408 94929 431432 627870 -485 5614190-18-1 422180 456409 401415 2122673 -492 3650890-18-2 427458 164882 401614 1371370 -605 7317390-18-3 777256 369100 740527 1960738 -473 43122100-18-1 516445 698274 493579 2861304 -443 30977100-18-2 804758 723314 776298 2856240 -354 29488100-18-3 1132139 692266 1086673 2868246 -402 31433

RTR travel algorithm was then used further to optimize thegenerated routes In the second stage of the algorithm thePSO algorithm was developed so as to allocate the vehiclesto each production batch This process ensured that themanufacturer first assigned each supplier to the appropri-ate route and then assigned each route to the appropriatebatch

By testing examples at different scales we were able tovalidate the effectiveness of the proposed model and theperformance efficiency of the two-stage hybrid heuristicalgorithm First in the small-scale examples the exact solu-tion was obtained using CPLEX and the accuracy of themodel was verified Second after the scale was expandedit was found that CPLEX had difficulty in solving thisover a short period of time and that the PSO algorithmtook less time to obtain a relatively appropriate resultthan the TS algorithm This proves that the two-stageheuristic algorithm is well placed to help us solve suchproblems

There are certain limitations to the current study First weassumed that the amount of the single raw material requiredfor each production batch would be the same in practicehowever the amount of multiple rawmaterials varies for eachproduction batch Second this paper only considered thecombination of the pickup stage and the production stagewithout considering product delivery The joint optimizationof multibatch production and the vehicle routing problemwith pickup and delivery considering inventory will thus beaddressed in future studies

Data Availability

The data used to support this study have been deposited inthe Baidu Netdisk readers can access the data at httpspanbaiducoms1ZSkGF-syiNTk FN9JNShbA

Conflicts of Interest

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

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China [no 71771143 and no 71501125] andShanghai Young Eastern Scholar Programme [QD2015041]

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] 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

[3] R Fukasawa H Longo J Lysgaard et al ldquoRobust branch-and-cut-and-price for the capacitated vehicle routing problemrdquoMathematical Programming vol 106 no 3 pp 491ndash511 2006

[4] M LGambardella D E Taillard andG Agazzi ldquoAmultiple antcolony system for vehicle routing problemswith timewindowsrdquo1999

[5] P Sombuntham and V Kachitvichyanukul ldquoMulti-depot vehi-cle routing problem with pickup and delivery requestsrdquo inProceedings of the International MultiConference of Engineersand Computer Scientists IMECS 2010 pp 71ndash85 China March2010

[6] C K Y Lin ldquoA vehicle routing problem with pickup anddelivery time windows and coordination of transportableresourcesrdquoComputers amp Operations Research vol 38 no 11 pp1596ndash1609 2011

[7] MW Savelsbergh andM Sol ldquoThe general pickup and deliveryproblemrdquo Transportation Science vol 29 no 1 pp 17ndash29 1995

[8] H Xu Z-L Chen S Rajagopal and S Arunapuram ldquoSolving apractical pickup and delivery problemrdquo Transportation Sciencevol 37 no 3 pp 347ndash364 2003

[9] G Nagy and S Salhi ldquoHeuristic algorithms for single andmultiple depot vehicle routing problems with pickups anddeliveriesrdquo European Journal of Operational Research vol 162no 1 pp 126ndash141 2005

[10] P Kalina and J Vokrınek ldquoParallel solver for vehicle routingand pickup and delivery problems with time windows based onagent negotiationrdquo in Proceedings of the 2012 IEEE InternationalConference on Systems Man and Cybernetics SMC 2012 pp1558ndash1563 Republic of Korea October 2012

6 Discrete Dynamics in Nature and Society

(a) One-point move (b) Two-point move

Figure 3 Improvement operators used in VRPPMP

updating formulates used to calculate speed and positionwere as follows

V119899+1119896119898119905 = 119908 sdot V119899119896119898119905 + 11988811199031 (119875119871119861119890119904119905119899119896119898119905 minus 119901119899119896119898119905)+ 11988821199032 (119875119866119861119890119904119905119899119896119898119905 minus 119901119899119896119898119905) (19)

119901119899+1119896119898119905 = 119901119899119896119898119905 + V119899+1119896119898119905 (20)

Where w is an inertia weight parameter which affects theconvergence procedure of the optimal solution 1198881and 1198882 arethe cognitive and social learning parameters respectively 1199031and 1199032 are random numbers generated between [0 1]

Based on the above description we now present the PSOalgorithm process for solving the route allocation problem

Step 1 Set the iteration number n = 1 Initialize k particleswhose positions and velocities are randomly generated Theparticlesrsquo positions represent the result of the route allocationand determine their qualities

Step 2 Evaluate the fitness value of each particle using thecommercial software CPLEX More specifically we used theroute allocation result obtained by the particles to fix thevariables 119911119898119905 and then used CPLEX to solve the originalmodel

Step 3 For each particle compare its fitness value andthe global best optimal solution119875119866119861119890119904119905119899119896119898119905 if 119875119871119861119890119904119905119899119896119898119905 gt119875119866119861119890119904119905119899119896119898119905 replace the value of 119875119866119861119890119904119905119899119896119898119905 using the value of119875119871119861119890119904119905119899119896119898119905Step 4 According to formulas (19) and (20) update theparticle velocity and positon

Step 5 If n reaches the preset maximum iteration or119875119866119861119890119904119905119899119896119898119905 has not been improved during a preset numberof iterations end the procedure otherwise set 119899 = 119899 + 1andthen go on to Step 2

5 Numerical Experiments

To validate the performance of the proposed two-stageheuristic numerical experiments were carried out on theVRPPMP For the PSO algorithm the maximum number ofiterations was set to 100 while the population size was 30According to the previous tests we set 1198881 = 1198882 = 1 Frompreliminary testing of 119908 = 0 05 1 15 and 2 it emergedthat 119908 = 1 performed the best thus we adopted 119908 = 1 for

the next experiments Based on the same decision obtainedin the first stage another nature inspired algorithm the Tabusearch (TS) algorithm was applied for comparison purposesWe listed the optimal solutions obtained by CPLEX in small-scale instances

All the experiments were performed on a computer with360 gigahertz Intel (R) Core (TM) i7-4790 CPU and 360gigabyte RAM The model and solution methods includingthe CW algorithm RTR travel algorithm and the PSOalgorithm were implemented using CPLEX 126 with C(VS2015) concert technology

51 Generation of Test Instance In this section we demon-strate the implementation of numerical experiments to val-idate the effectiveness of the proposed model and the effi-ciency of the proposed algorithm For small-scale instancesthe results of the proposed two-stage heuristic algorithmwerecompared with the optimal solutions obtained byCPLEX andthe solutions obtained using the TS algorithm Based on thesame decision obtained in the first stage the results obtainedusing the PSO heuristic were compared with those of the TSalgorithm in medium- and large-scale instances given thatCPLEX would have been time consuming or unable to solvethe problem

For the numerical experiments on the VRPPMP in-stances were randomly generated in the three scales Wesimulated the cases of 10 suppliers and two productionbatches and 15 suppliers and three production batches forthe small-scale 30 suppliers and six production batches and50 suppliers and 10 production batches for the medium scale70 suppliers and 14 production batches 90 suppliers and 18production batches 100 suppliers and 18 production batchesfor the large scale Information pertaining to the coordinatepositions and pickup demands of the 10 suppliers of thefirst small-scale instance (ie 10-2-1) are shown in Table 1The coordinate position of the manufacturer was (5 5) andthe cost of unit distance traveled by vehicle was $2 Themanufacturer had sufficient pickup vehicles with the capacityof 10 tons to provide a service to suppliersThe volume of rawmaterials required for each production batch was 10 tons andthe unit inventory cost of unprocessed raw materials in eachbatch was $20

52 Analysis of Calculation Results We tested examples atdifferent scales and used CPLEX the PSO algorithm andthe TS algorithm respectively to solve the VRPPMP Table 2yields the following conclusions In the small-scale examples(10-2) the exact solutions (7345 17317 and 18465) were

Discrete Dynamics in Nature and Society 7

Table 1 Pickup demands (tons) and coordinate positions (km) of instances 10-2-1

Supplier Pickup demands Coordinate axis X Coordinate axis Y1 15577 62804 834392 14597 56732 598063 22747 20603 558884 19595 90603 442185 15754 97755 273706 25402 29191 467317 20115 63266 469518 15807 98215 030379 23763 86237 9953510 27253 67718 31459

Table 2 Computational results of small-scale instances

Instance IDCPLEX PSO algorithm TS algorithm

Result($)

Time(sec)

Result($)

Time(sec)

Gap()

Result($)

Time(sec)

Gap()

10-2-1 73 45 545 7345 447 0 7345 755 010-2-2 17317 366 17657 307 196 17657 708 19610-2-3 18465 510 18919 355 246 18919 760 24615-3-1 - gt12hours 24854 1424 - 24854 1730 -15-3-2 - gt12hours 43527 1369 - 43527 1747 -15-3-3 - gt12hours 23667 1373 - 23667 1739 -

Table 3 Computational results of medium-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

30-6-1 57288 8346 57288 51069 0 5119030-6-2 56730 13409 55166 51958 -276 2874930-6-3 67289 3792 67289 59352 0 14651950-10-1 124389 23773 119875 224667 -363 8450550-10-2 164245 61163 157761 229187 -395 2747250-10-3 250116 25234 242029 229583 -323 80982

obtained byCPLEX and the accuracy of themodel is verifiedCompared with the TS algorithm the PSO algorithm canobtain the approximate solutions (7345 17657 and 18919)in a shorter time In the small-scale examples (15-3) CPLEXhad difficultly in getting the results but the PSO algorithmand TS algorithm can still obtain the results in a shorttime

However after the scale of the study increases as shownin Tables 3 and 4 the PSO algorithm can take less timeto get a relatively appropriate solution than TS algorithmIn the medium-scale examples the relative value deviationsbetween the PSO algorithm and the TS algorithm are within4 However the relative time deviations between the PSOalgorithmand theTS algorithm reach 699on average In thelarge-scale examples although the relative value deviationsbetween the PSO algorithm and the TS algorithm reached

nearly 7 the PSO calculation results are still acceptableand it holds great advantages in terms of computing timeThis proves that the two-stage heuristic algorithm can helpus solve such problems well

6 Conclusion

This paper discusses a novel and practical research issuearising in the manufacturing industry namely the prob-lem of vehicle routing involving pickup in the context ofmultibatch production In order to reduce transportationcosts and inventory costs in the production process a mixedinteger programming model was developed Following thisa two-stage hybrid heuristic algorithm was put forward tosolve this model In the first stage the CW algorithm wasproposed in order to generate the initial vehicle routes The

8 Discrete Dynamics in Nature and Society

Table 4 Computational results of large-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

70-14-1 226615 112267 210941 706980 -692 5297370-14-2 372110 56183 356346 601707 -424 9709870-14-3 453408 94929 431432 627870 -485 5614190-18-1 422180 456409 401415 2122673 -492 3650890-18-2 427458 164882 401614 1371370 -605 7317390-18-3 777256 369100 740527 1960738 -473 43122100-18-1 516445 698274 493579 2861304 -443 30977100-18-2 804758 723314 776298 2856240 -354 29488100-18-3 1132139 692266 1086673 2868246 -402 31433

RTR travel algorithm was then used further to optimize thegenerated routes In the second stage of the algorithm thePSO algorithm was developed so as to allocate the vehiclesto each production batch This process ensured that themanufacturer first assigned each supplier to the appropri-ate route and then assigned each route to the appropriatebatch

By testing examples at different scales we were able tovalidate the effectiveness of the proposed model and theperformance efficiency of the two-stage hybrid heuristicalgorithm First in the small-scale examples the exact solu-tion was obtained using CPLEX and the accuracy of themodel was verified Second after the scale was expandedit was found that CPLEX had difficulty in solving thisover a short period of time and that the PSO algorithmtook less time to obtain a relatively appropriate resultthan the TS algorithm This proves that the two-stageheuristic algorithm is well placed to help us solve suchproblems

There are certain limitations to the current study First weassumed that the amount of the single raw material requiredfor each production batch would be the same in practicehowever the amount of multiple rawmaterials varies for eachproduction batch Second this paper only considered thecombination of the pickup stage and the production stagewithout considering product delivery The joint optimizationof multibatch production and the vehicle routing problemwith pickup and delivery considering inventory will thus beaddressed in future studies

Data Availability

The data used to support this study have been deposited inthe Baidu Netdisk readers can access the data at httpspanbaiducoms1ZSkGF-syiNTk FN9JNShbA

Conflicts of Interest

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

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China [no 71771143 and no 71501125] andShanghai Young Eastern Scholar Programme [QD2015041]

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] 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

[3] R Fukasawa H Longo J Lysgaard et al ldquoRobust branch-and-cut-and-price for the capacitated vehicle routing problemrdquoMathematical Programming vol 106 no 3 pp 491ndash511 2006

[4] M LGambardella D E Taillard andG Agazzi ldquoAmultiple antcolony system for vehicle routing problemswith timewindowsrdquo1999

[5] P Sombuntham and V Kachitvichyanukul ldquoMulti-depot vehi-cle routing problem with pickup and delivery requestsrdquo inProceedings of the International MultiConference of Engineersand Computer Scientists IMECS 2010 pp 71ndash85 China March2010

[6] C K Y Lin ldquoA vehicle routing problem with pickup anddelivery time windows and coordination of transportableresourcesrdquoComputers amp Operations Research vol 38 no 11 pp1596ndash1609 2011

[7] MW Savelsbergh andM Sol ldquoThe general pickup and deliveryproblemrdquo Transportation Science vol 29 no 1 pp 17ndash29 1995

[8] H Xu Z-L Chen S Rajagopal and S Arunapuram ldquoSolving apractical pickup and delivery problemrdquo Transportation Sciencevol 37 no 3 pp 347ndash364 2003

[9] G Nagy and S Salhi ldquoHeuristic algorithms for single andmultiple depot vehicle routing problems with pickups anddeliveriesrdquo European Journal of Operational Research vol 162no 1 pp 126ndash141 2005

[10] P Kalina and J Vokrınek ldquoParallel solver for vehicle routingand pickup and delivery problems with time windows based onagent negotiationrdquo in Proceedings of the 2012 IEEE InternationalConference on Systems Man and Cybernetics SMC 2012 pp1558ndash1563 Republic of Korea October 2012

Discrete Dynamics in Nature and Society 7

Table 1 Pickup demands (tons) and coordinate positions (km) of instances 10-2-1

Supplier Pickup demands Coordinate axis X Coordinate axis Y1 15577 62804 834392 14597 56732 598063 22747 20603 558884 19595 90603 442185 15754 97755 273706 25402 29191 467317 20115 63266 469518 15807 98215 030379 23763 86237 9953510 27253 67718 31459

Table 2 Computational results of small-scale instances

Instance IDCPLEX PSO algorithm TS algorithm

Result($)

Time(sec)

Result($)

Time(sec)

Gap()

Result($)

Time(sec)

Gap()

10-2-1 73 45 545 7345 447 0 7345 755 010-2-2 17317 366 17657 307 196 17657 708 19610-2-3 18465 510 18919 355 246 18919 760 24615-3-1 - gt12hours 24854 1424 - 24854 1730 -15-3-2 - gt12hours 43527 1369 - 43527 1747 -15-3-3 - gt12hours 23667 1373 - 23667 1739 -

Table 3 Computational results of medium-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

30-6-1 57288 8346 57288 51069 0 5119030-6-2 56730 13409 55166 51958 -276 2874930-6-3 67289 3792 67289 59352 0 14651950-10-1 124389 23773 119875 224667 -363 8450550-10-2 164245 61163 157761 229187 -395 2747250-10-3 250116 25234 242029 229583 -323 80982

obtained byCPLEX and the accuracy of themodel is verifiedCompared with the TS algorithm the PSO algorithm canobtain the approximate solutions (7345 17657 and 18919)in a shorter time In the small-scale examples (15-3) CPLEXhad difficultly in getting the results but the PSO algorithmand TS algorithm can still obtain the results in a shorttime

However after the scale of the study increases as shownin Tables 3 and 4 the PSO algorithm can take less timeto get a relatively appropriate solution than TS algorithmIn the medium-scale examples the relative value deviationsbetween the PSO algorithm and the TS algorithm are within4 However the relative time deviations between the PSOalgorithmand theTS algorithm reach 699on average In thelarge-scale examples although the relative value deviationsbetween the PSO algorithm and the TS algorithm reached

nearly 7 the PSO calculation results are still acceptableand it holds great advantages in terms of computing timeThis proves that the two-stage heuristic algorithm can helpus solve such problems well

6 Conclusion

This paper discusses a novel and practical research issuearising in the manufacturing industry namely the prob-lem of vehicle routing involving pickup in the context ofmultibatch production In order to reduce transportationcosts and inventory costs in the production process a mixedinteger programming model was developed Following thisa two-stage hybrid heuristic algorithm was put forward tosolve this model In the first stage the CW algorithm wasproposed in order to generate the initial vehicle routes The

8 Discrete Dynamics in Nature and Society

Table 4 Computational results of large-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

70-14-1 226615 112267 210941 706980 -692 5297370-14-2 372110 56183 356346 601707 -424 9709870-14-3 453408 94929 431432 627870 -485 5614190-18-1 422180 456409 401415 2122673 -492 3650890-18-2 427458 164882 401614 1371370 -605 7317390-18-3 777256 369100 740527 1960738 -473 43122100-18-1 516445 698274 493579 2861304 -443 30977100-18-2 804758 723314 776298 2856240 -354 29488100-18-3 1132139 692266 1086673 2868246 -402 31433

RTR travel algorithm was then used further to optimize thegenerated routes In the second stage of the algorithm thePSO algorithm was developed so as to allocate the vehiclesto each production batch This process ensured that themanufacturer first assigned each supplier to the appropri-ate route and then assigned each route to the appropriatebatch

By testing examples at different scales we were able tovalidate the effectiveness of the proposed model and theperformance efficiency of the two-stage hybrid heuristicalgorithm First in the small-scale examples the exact solu-tion was obtained using CPLEX and the accuracy of themodel was verified Second after the scale was expandedit was found that CPLEX had difficulty in solving thisover a short period of time and that the PSO algorithmtook less time to obtain a relatively appropriate resultthan the TS algorithm This proves that the two-stageheuristic algorithm is well placed to help us solve suchproblems

There are certain limitations to the current study First weassumed that the amount of the single raw material requiredfor each production batch would be the same in practicehowever the amount of multiple rawmaterials varies for eachproduction batch Second this paper only considered thecombination of the pickup stage and the production stagewithout considering product delivery The joint optimizationof multibatch production and the vehicle routing problemwith pickup and delivery considering inventory will thus beaddressed in future studies

Data Availability

The data used to support this study have been deposited inthe Baidu Netdisk readers can access the data at httpspanbaiducoms1ZSkGF-syiNTk FN9JNShbA

Conflicts of Interest

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

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China [no 71771143 and no 71501125] andShanghai Young Eastern Scholar Programme [QD2015041]

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] 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

[3] R Fukasawa H Longo J Lysgaard et al ldquoRobust branch-and-cut-and-price for the capacitated vehicle routing problemrdquoMathematical Programming vol 106 no 3 pp 491ndash511 2006

[4] M LGambardella D E Taillard andG Agazzi ldquoAmultiple antcolony system for vehicle routing problemswith timewindowsrdquo1999

[5] P Sombuntham and V Kachitvichyanukul ldquoMulti-depot vehi-cle routing problem with pickup and delivery requestsrdquo inProceedings of the International MultiConference of Engineersand Computer Scientists IMECS 2010 pp 71ndash85 China March2010

[6] C K Y Lin ldquoA vehicle routing problem with pickup anddelivery time windows and coordination of transportableresourcesrdquoComputers amp Operations Research vol 38 no 11 pp1596ndash1609 2011

[7] MW Savelsbergh andM Sol ldquoThe general pickup and deliveryproblemrdquo Transportation Science vol 29 no 1 pp 17ndash29 1995

[8] H Xu Z-L Chen S Rajagopal and S Arunapuram ldquoSolving apractical pickup and delivery problemrdquo Transportation Sciencevol 37 no 3 pp 347ndash364 2003

[9] G Nagy and S Salhi ldquoHeuristic algorithms for single andmultiple depot vehicle routing problems with pickups anddeliveriesrdquo European Journal of Operational Research vol 162no 1 pp 126ndash141 2005

[10] P Kalina and J Vokrınek ldquoParallel solver for vehicle routingand pickup and delivery problems with time windows based onagent negotiationrdquo in Proceedings of the 2012 IEEE InternationalConference on Systems Man and Cybernetics SMC 2012 pp1558ndash1563 Republic of Korea October 2012

8 Discrete Dynamics in Nature and Society

Table 4 Computational results of large-scale instances

Instance IDPSO algorithm TS algorithm Relative deviation

(TS - PSO)PSOResult Time Result Time Result Time($) (sec) ($) (sec) () ()

70-14-1 226615 112267 210941 706980 -692 5297370-14-2 372110 56183 356346 601707 -424 9709870-14-3 453408 94929 431432 627870 -485 5614190-18-1 422180 456409 401415 2122673 -492 3650890-18-2 427458 164882 401614 1371370 -605 7317390-18-3 777256 369100 740527 1960738 -473 43122100-18-1 516445 698274 493579 2861304 -443 30977100-18-2 804758 723314 776298 2856240 -354 29488100-18-3 1132139 692266 1086673 2868246 -402 31433

RTR travel algorithm was then used further to optimize thegenerated routes In the second stage of the algorithm thePSO algorithm was developed so as to allocate the vehiclesto each production batch This process ensured that themanufacturer first assigned each supplier to the appropri-ate route and then assigned each route to the appropriatebatch

By testing examples at different scales we were able tovalidate the effectiveness of the proposed model and theperformance efficiency of the two-stage hybrid heuristicalgorithm First in the small-scale examples the exact solu-tion was obtained using CPLEX and the accuracy of themodel was verified Second after the scale was expandedit was found that CPLEX had difficulty in solving thisover a short period of time and that the PSO algorithmtook less time to obtain a relatively appropriate resultthan the TS algorithm This proves that the two-stageheuristic algorithm is well placed to help us solve suchproblems

There are certain limitations to the current study First weassumed that the amount of the single raw material requiredfor each production batch would be the same in practicehowever the amount of multiple rawmaterials varies for eachproduction batch Second this paper only considered thecombination of the pickup stage and the production stagewithout considering product delivery The joint optimizationof multibatch production and the vehicle routing problemwith pickup and delivery considering inventory will thus beaddressed in future studies

Data Availability

The data used to support this study have been deposited inthe Baidu Netdisk readers can access the data at httpspanbaiducoms1ZSkGF-syiNTk FN9JNShbA

Conflicts of Interest

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

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China [no 71771143 and no 71501125] andShanghai Young Eastern Scholar Programme [QD2015041]

References

[1] G B Dantzig and J H Ramser ldquoThe truck dispatchingproblemrdquoManagement Science vol 6 no 1 pp 80ndash91 1959

[2] 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

[3] R Fukasawa H Longo J Lysgaard et al ldquoRobust branch-and-cut-and-price for the capacitated vehicle routing problemrdquoMathematical Programming vol 106 no 3 pp 491ndash511 2006

[4] M LGambardella D E Taillard andG Agazzi ldquoAmultiple antcolony system for vehicle routing problemswith timewindowsrdquo1999

[5] P Sombuntham and V Kachitvichyanukul ldquoMulti-depot vehi-cle routing problem with pickup and delivery requestsrdquo inProceedings of the International MultiConference of Engineersand Computer Scientists IMECS 2010 pp 71ndash85 China March2010

[6] C K Y Lin ldquoA vehicle routing problem with pickup anddelivery time windows and coordination of transportableresourcesrdquoComputers amp Operations Research vol 38 no 11 pp1596ndash1609 2011

[7] MW Savelsbergh andM Sol ldquoThe general pickup and deliveryproblemrdquo Transportation Science vol 29 no 1 pp 17ndash29 1995

[8] H Xu Z-L Chen S Rajagopal and S Arunapuram ldquoSolving apractical pickup and delivery problemrdquo Transportation Sciencevol 37 no 3 pp 347ndash364 2003

[9] G Nagy and S Salhi ldquoHeuristic algorithms for single andmultiple depot vehicle routing problems with pickups anddeliveriesrdquo European Journal of Operational Research vol 162no 1 pp 126ndash141 2005

[10] P Kalina and J Vokrınek ldquoParallel solver for vehicle routingand pickup and delivery problems with time windows based onagent negotiationrdquo in Proceedings of the 2012 IEEE InternationalConference on Systems Man and Cybernetics SMC 2012 pp1558ndash1563 Republic of Korea October 2012

Discrete Dynamics in Nature and Society 9

[11] H-Z Zheng H-Y Guo and X-D Zhang ldquoModeling andapproach for vmi cyclic inventory routing problemrdquo in Proceed-ings of the 2009 International Conference on Machine Learningand Cybernetics pp 1393ndash1398 China July 2009

[12] A L Shiguemoto and V A Armentano ldquoA tabu search proce-dure for coordinating production inventory and distributionrouting problemsrdquo International Transactions in OperationalResearch vol 17 no 2 pp 179ndash195 2010

[13] Y Adulyasak J-F Cordeau and R Jans ldquoFormulations andbranch-and-cut algorithms for multivehicle production andinventory routing problemsrdquo INFORMS Journal on Computingvol 26 no 1 pp 103ndash120 2014

[14] B Sainathuni P J Parikh X Zhang and N Kong ldquoThe ware-house-inventory-transportation problem for supply chainsrdquoEuropean Journal of Operational Research vol 237 no 2 pp690ndash700 2014

[15] P C Yellow ldquoA Computational Modification to the SavingsMethod of Vehicle Schedulingrdquo Journal of the OperationalResearch Society vol 21 no 2 pp 281ndash283 2017

[16] G Dueck ldquoNew optimization heuristics the great delugealgorithm and the record-to-record travelrdquo Journal of Compu-tational Physics vol 104 no 1 pp 86ndash92 1993

[17] F Li B Golden and E Wasil ldquoVery large-scale vehicle routingnew test problems algorithms and resultsrdquo Computers ampOperations Research vol 32 no 5 pp 1165ndash1179 2005

[18] R C Eberhart and J Kennedy ldquoA new optimizer using particleswarm theoryrdquo in Proceedings of the 6th International Sympo-sium on Micromachine and Human Science pp 39ndash43 NagoyaJapan October 1995

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Mathematical PhysicsAdvances in

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Hindawiwwwhindawicom Volume 2018

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International Journal of

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Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

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Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom