Research ArticleOmni-Channel Product Distribution Network Design by Usingthe Improved Particle Swarm Optimization Algorithm

Si Zhang Heying Zhu Xia Li and YanWang

School of Management Shanghai University Shanghai 200444 China

Correspondence should be addressed to Si Zhang zhangsi817sinacom

Received 30 November 2018 Accepted 13 February 2019 Published 4 April 2019

Guest Editor Xiaobo Qu

Copyright copy 2019 Si Zhang et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Distribution network under Omni-Channel integration contains many levels There are one or more dealers at each level whichforms a many-to-many distribution network Consumers purchase a wide variety of products and their demands are uncertainwhich constitutes a complex demand network and increases the complexity of the supply chain network This paper focuses onthe integrated optimization of supply chain distribution network and demand network and constructs the joint randomizationplanning model of location and routing The goal is to minimize the total costs of the supply chain network under uncertaincustomer demands Based on the traditional particle swarm optimization (PSO) this study introduces the collaborative idea toreduce the coding dimension improves the boundary processing strategy and adopts the mutation operator to expand the searchspace A case study of distribution under Omni-Channel integration in a large enterprise was done The validity of the model andthe effectiveness of the proposed method were verified by numerical experiments

1 Introduction

Over the last two decades the product transportation of theretail industry was largely driven by the Internet Technologydevelopment More and more retailers not only have thetraditional offline stores but also start to have the onlinestores to satisfy the demands of customers from the InternetAs the emergence of new online channels retailers beginto transform from traditionally storefront-based retailers tomulti-channel retailers ([1ndash3]) and consider the operationmanagement on two channels This gave birth to the conceptof Omni-Channel integration

Omni-Channel integration is the deep integration ofonline services offline experience and modern logistics innew retail era It makes the supply chain more complex Theoperation goal of Omni-Channel integration supply chainis to meet customer demands with the lowest costs Saghiriet al [4] proposed that it is a challenging task to analyzethe distribution strategy of Omni-Channel integration Thesupply chain usually contains many levels and each level hasmany sellers which leads to the complexity of the supplychain network Besides the geographical locations of thesellers are spread all over the country These factors have

a significant impact on transportation costs In additioncustomer demands are often uncertain which will increasethe costs of supply chain Therefore the optimization forthe supply chain with Omni-Channel is a challenge taskand it is necessary to have some efficient approaches tohelp retailers reduce their costs For example Sursquoning oneof the largest retailers with Omni-Channel supply chainin China adopted a distribution method which is city-centered distributed to the county and rural areas stepby step and constructed a three-level logistics distributionsystem This system is ldquoNational Logistics CentermdashRegionalLogistics CentermdashCity Logistics Centerrdquo As of June 2017 ithas covered 297 prefecture-level cities and above with about2079 authorized chain stores and 833 authorized serviceoutlets It launched ldquo100-City Half-Dayrdquo service to requireproducts that are delivered to the customer within half dayafter the order is placed which poses greater challenges tothe distribution of logistics

Therefore facing such a large supply chain networkbuilding an efficient high-quality and low-cost distributionnetwork is the key to improve competitiveness and sus-tainable development of enterprises The main aim of this

HindawiDiscrete Dynamics in Nature and SocietyVolume 2019 Article ID 1520213 15 pageshttpsdoiorg10115520191520213

2 Discrete Dynamics in Nature and Society

paper is to construct a highly efficient supply chain systemto minimize costs and improve service quality

This paper is organized as follows Section 1 introducesthe background of Omni-Channel distribution The relatedliterature is reviewed in Section 2 from the perspective ofLRP uncertain demand and solvingmethods respectively InSection 3 a distribution network for Omni-Channel supplychain is proposed and a model for this problem is built Tosolve this model one algorithm based on the traditional PSOis developed in Section 4 by introducing the collaborative ideato reduce the coding dimension improving the boundaryprocessing strategy and introducing themutation operator toexpand the search space Sections 5 and 6 are the numericalresults and sensitivity analysis based on one case studySection 7 concludes the whole paper

2 Literature Review

This paper mainly focuses on the location and routingproblem (LRP) for Omni-Channel supply chain Early LRPissues were limited to the location of a single facility Maran-zana [5] studied the problem of supplier location with thegoal of minimizing transportation costs Subsequently manyresearchers conducted further research on the systematic andnetworking nature of transportation and location decisionsGovindan et al [6] introduced a two-tier supply chain witha time window based on the supply chain network distri-bution of perishable food which determined the number ofdistribution centers and facility locations Abdulrazik et al[7] optimized the tertiary supply chain under the backgroundof producing oil palm inMalaysia In addition from the typesof product distribution in the LRP products transportationhas expanded from single-product distribution to multi-product distribution Geoffrion and Graves [8] developeda multi-product distribution model using Mixed-IntegerProgramming (MIP) Tiwari et al [9] proposed an algorithmfor solving multi-stage multi-product supply chain networkdesign Viswanathan and Mathur [10] studied a distributionsystem consisting of a warehouse multiple products andmultiple retailers Arntzen [11] et al used a mixed inte-ger programming to establish a multi-product and multi-stage supply chain model for large multinational companiesAlthough research on the LRP changes from the initial singlelocation planning to more consideration of the systemic andnetworking feature of the supply chain the current researcheson supply chain network are mainly focused on two-levelor three-level system and the distribution route planning ismostly one-to-many or many-to-one delivery For the retailindustry it usually has many supply chain levels and manytypes of products in the distributionTherefore it is necessaryto study the supply chainwithmore levels andmore productsThis paper is to design a five-level supply chain network forthe LRP with multi-product distribution in retail industry

One another factor that cannot be ignored in Omni-Channel retail supply chain study is the uncertainties ofcustomer demands Different scholars used different meth-ods to research uncertainty Wang and He [12] introducedthe uncertainty of demand and studied the problem of

distribution route and warehouse location under the condi-tion of uncertain demand Zhen et al [13] used the scenariomethod to solve the problem of uncertainty in the productionprocess of automobiles Tapiero and Soliman [14] appliedoptimal control theory to solve the problem of multi-producttransportation multi-regional production and cross-timeinventory planning under demand uncertainty Baghalianet al [15] used multi-objective optimization to model thelocation and routing problem in low-carbon supply chainand explored the impact of demand uncertainty In generalcustomers have great differences in the demand for differentproducts It is necessary to form a personalized demandnetwork To describe customer demands more accurately weset each customerrsquos uncertain demand for different kinds ofproducts to follow different normal distributions instead ofletting all customers uniformly meet a normal distributionfor all products uniformly

The LRP with multi-level multi-product and personal-ized demand is a very complex optimization problem Thereare two main streams solving the large-scale optimizationproblem The first stream is to use new evolutionary algo-rithms or add local search strategies (eg tabu search) tooriginal algorithm Zhang et al [16] adopted tabu searchalgorithm to improve the accuracy and efficiency of coldchain product distribution Javid and Azad [17] proposed amethod of synchronous optimization of location and routingTwo-stage heuristic algorithm based on tabu search andsimulated annealing was used to improve the solution spaceXia et al [18] established the dual-objective programmingmodel and designed an adaptive tabu search algorithm Thesecond stream of literature studies is to decompose thehigh-dimensional of large-scale complex problems into low-dimensional simple problems and optimize them separatelyHu et al [19] studied the large-scale refrigerated truckscheduling problem and combined a variable neighborhoodsearch with a particle swarm optimization to develop atwo-stage decomposition algorithm In 2009 Li and Yao[20] proposed a new co-evolutionary particle swarm opti-mization algorithm (CPSO) which added random groupingand adaptive weighting strategy to prove that it is effectivein high-dimensional indivisible problems The optimized30-dimensional CPSO algorithm which has been proposedearlier can be extended to 1000-dimensional problems nowCompared with manufacture industry distribution in retailindustry usually have more supply chain levels which resultsin the high dimensions of decision variables We develop animproved PSO algorithm based on CPSO to efficiently solvethe optimization problem

Therefore in our paper we aim to contribute to thecurrent studies on this area from the following three aspectsFirstly under the Omni-Channel distribution the supplychain network level has been increased and a many-to-many distribution network has been formed which makesthe supply chain network distribution more in line with theactual situation Secondly we allow each customerrsquos demandsto follow different normal distributions for each product andform a demand network accordingly The demand networkcan better predict customer demands Thirdly based on thetraditional PSO the algorithm introduces the collaborative

Discrete Dynamics in Nature and Society 3

Nationallogistics center

Regionallogistics center

City logisticscenter

Dealer Client

Figure 1 Supply chain distribution network

idea to reduce the coding dimension improves the boundaryprocessing strategy and introduces the mutation operator toexpand the search spaceThismethod improves the efficiencyof the algorithm and provides a solution for solving highdimensional problems

3 Model

31 Model Description This model includes some commoncharacteristics of distribution under Omni-Channel inte-gration so the conclusions of the model can be useful formany enterprises of supply-chain distribution under Omni-Channel integration There are five levels in the entire supplychain These levels include National Logistics Center ARegional Logistics Center B City Logistics Center C DealerD and Customer E Distribution network shall distributem-type products to two types of customers The first typeof customers E1 is large customers such as companiesthe demand of such customers is delivered directly by thenational logistics center AThe second type of customers E2 issmall supermarkets like storesThe orders of such customerswill be delivered through the regional logistics center B thecity logistics center C and the dealer DWe can intuitively seethe model in Figure 1

According to the supply chain network we built a mixedinteger programming model to minimize the total supplychain costsThe total costs include three items transportationcosts of the products fixed operating costs of logistics centerand carbon treatment costs which is to deal with carbonemission generated from the product transportation process

32 e Index of the Model

a The number of national logistics centersb The number of regional logistics centersc The number of city logistics centersd The number of dealers1198901 The number of first type of customers1198902 The number of second type of customersm The number of the category of products

33 Parameter Symbols

1198661198981198861198901 The unit transportation cost of product mdelivered to customer 1198901 through national logisticscenter a119866119898119886119887 The unit transportation cost of product mdelivered to regional logistics center b throughnational logistics center a119866119898119887119888 The unit transportation cost of product mdelivered to city logistics center c through regionallogistics center b119866119898119888119889 The unit transportation cost of product mdelivered to dealer d through city logistics center c1198661198981198891198902 The unit transportation cost of product mdelivered to customer 1198902 through dealer d1198891198941199041198861198901 The distance between national logistics cen-ter a and customer 1198901

4 Discrete Dynamics in Nature and Society

119889119894119904119886119887 Thedistance betweennational logistics centera and regional logistics center b

119889119894119904119887119888 The distance between regional logistics centerb and city logistics center c

119889119894119904119888119889 The distance between city logistics center cand dealer d

1198891198941199041198891198902 The distance between dealer d and customer1198902F119886 The fixed operating costs of national logisticscenter located at point a

F119887 The fixed operating costs of regional logisticscenter located at point b

F119888 The fixed operating costs of city logistics centerlocated at point c

F119889 The fixed operating costs of dealer located atpoint d

119874119898119886 The average inventory of productm in nationallogistics center located at point a

1198771198981198901 The demand of customer 1198901 for product m1198771198981198902 The demand of customer 1198902 for product mT Unit carbon treatment cost

N A large enough positive number

34 Decision Variables

1198761198981198861198901 The quantity of product m from nationallogistics center a to customer 1198901119876119898119886119887 The quantity of productm from national logis-tics center a to regional logistics center b

119876119898119887119888 The quantity of product m from regional logis-tics center b to city logistics center c

119876119898119888119889 The quantity of product m from city logisticscenter c to dealer d

1198761198981198891198902 The quantity of product m from dealer d tocustomer 1198902119910119886 Whether the national logistics center a operatesif it operates then 119910119886 equals 1 otherwise the value is0

119910119887 Whether the regional logistics center b operatesif it operates then 119910119887 equals 1 otherwise the value is 0119910119888 Whether the city logistics center c operates if itoperates then 119910119888 equals 1 otherwise the value is 0119910119889 Whether the dealer d operates if it operatesthen 119910119889 equals 1 otherwise the value is 0

35 Data Model

min f (x) (1)

f (x) = sumAsumE1sumMGm

ae1Qmae1disae1 +sum

AsumBsumMGm

abQmabdisab

+sumBsumCsumMGm

bcQmbcdisbc +sum

CsumDsumMGm

cdQmcddiscd

+sumDsumE2sumMGm

de2Qmde2disde2 +sum

AFaya +sum

BFbyb

+sumCFcyc +sum

DFdyd +sum

AsumE1sumMTQm

ae1disae1

+sumAsumBsumMTQm

abdisab +sumBsumCsumMTQm

bcdisbc

+sumCsumDsumMTQm

cddiscd +sumDsumE2sumMTQm

de2disde2

(2)

sum119860

1198761198981198861198901 = 1198771198981198901 forall1198901 isin 1198641 forallm isin M (3)

sum119863

1198761198981198891198902 = 1198771198981198902 forall1198902 isin 1198642 forallm isin M (4)

sum119860

119876119898119886119887 ge sum119862

119876119898119887119888 forallb isin B forallm isin M (5)

sum119861

119876119898119887119888 ge sum119863

119876119898119888119889 forallc isin C forallm isin M (6)

sum119862

119876119898119888119889 ge sum1198642

1198761198981198891198902 foralld isin D forallm isin M (7)

sum1198641

1198761198981198861198901 +sum119861

119876119898119886119887 le 119874119898119886 lowast 119910119886 foralla isin A forallm isin M (8)

sum119860

119876119898119886119887 minus 119873 times 119910119887 le 0 forallb isin B forallm isin M (9)

sum119861

119876119898119887119888 minus 119873 times 119910119888 le 0 forallc isin C forallm isin M (10)

sum119862

119876119898119888119889 minus 119873 times 119910119889 le 0 foralld isin D forallm isin M (11)

119910119886119910119887 119910119888 119910119889 isin 0 1foralla isin A forallb isin B forallc isin C foralld isin D

(12)

1198761198981198861198901 119876119898119886119887 119876119898119887119888 119876119898119888119889 1198761198981198891198902 ge 0foralla isin A forallb isin B forallc isin C foralld isin D forall1198901 isin 1198641 forall1198902 isin 1198642

(13)

The objective function (2) aims to minimize the totalcosts of the supply chain network The constraint (3) requiresthat the demands of the customer 1198901 can only be directlydelivered by the national logistics center a and the constraint

Discrete Dynamics in Nature and Society 5

(4) states that the product can only be provided from thedealer d to the customer 1198902 Equations (5)-(7) specify thatthe volume of each logistics center is greater than or equalto the shipment at that point Constraint (8) ensures that theinventory from the national logistics center a is greater thanthe total volume of shipped products and the operation of thenational logistics center a is represented by the 0-1 variableConstraint (9) donates that if the regional logistics center bis not in operation the quantity of product shipped to theregional logistics center b is zero While constraint (10) andconstraint (11) indicate that if city logistics center c and thedealer d do not operate the volume of product shipped to thecity logistics center c and the dealer d is zero Constraint (12)and constraint (13) are the value constraints of the decisionvariable

4 Proposed Heuristic Method

The model of LRP presented in Section 3 aims to minimizethe total costs in product transportation process In a large-scale transportation network the code dimension is highmaking it difficult to find an optimal solution in a limitedtime To solve the problem we resort to heuristic methodssuch as particle swarm optimization (PSO) PSO is a stochas-tic parallel optimization algorithm that does not requirethe properties of the optimized function to be divisiblesteerable and continuous In addition to further improve theaccuracy and efficiency of the algorithm the improved PSOis proposed

41 Particle Swarm Optimization PSO was proposed byKennedy and Eberhart [21] and Lian [22] pointed out thatthis method was a stochastic optimization method based onswarm intelligence and PSOwas inspired by the social behav-ior of bird flocking and their means of information exchangeDue to its easy implementation and fast convergence PSOhas been successfully applied to solve nonlinear optimizationproblems Ai and Kachitvichyanukul [23] Goksal et al [24]and Alinaghian et al [25] designed an improved randomtopology particle swarm optimization algorithm (RT-PSO)for time-dependent vehicle routing problems Hu et al [19]developed a two-stage decomposition algorithm by combin-ing variable neighborhood search algorithm and PSO to solvethe large-scale refrigerated truck scheduling problem

In PSO each particle represents a solution to the opti-mization problem Particle initialization is generated in arandom way and each particle iteratively updates its positionthrough personal best and global best Each particle searchesfor the optimal fitness value in the D-dimensional feasiblesolution spaceThere are N particles in a group In iteration tthe position of particle i is represented as a D-dimensionalvector 119883119905119894119889 = (1199091199051198941 1199091199051198942 119909119905119894119863) 119894 = 1 2 119873 At thesame time the velocity of the particle i is also a D minusdimensional vector denoted as 119881119905119894119889 = (V1199051198941 V1199051198942 V119905119894119863) 119894 =1 2 119873 The optimal position searched by the particle iin iteration t is called the personal best which is denotedas 119875119887119890119904119905 = (1199011198941 1199011198942 119901119894119863) 119894 = 1 2 119873 The optimalposition searched by the group in iteration t is the global best

denoted as 119866119887119890119904119905 = (1199011198921 1199011198922 119901119892119863) After finding the twooptimal values each particle updates its velocity and positionaccording to the following formula

V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)

(14)

119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889 (15)

In the standard PSO there are some shortcomings forexample (1)when solving the high dimensional optimizationproblem the increase of the dimension makes the searchspace expand into an exponential form which is prone topremature convergence That is the particle group aggregatesprematurely to the local optimal solution (2) Particles mayaggregate to local optimum too early in the process of gather-ing If there is no escaping mechanism the optimal solutionmay not be obtained Therefore we proposed improvedstrategy accordingly

42 Strategies for Improving PSO The improved PSO focuseson three aspects (1) facing the ldquodimensional disasterrdquo inlarge-scale problems cooperative particle swarm optimiza-tion (CPSO) is proposed to transform large-scale complexproblems into low-dimensional simple problems (2) Whenparticle flies out of the boundary the boundary processingstrategy is adopted to change the flight trajectory of theparticles (3) When the particles are aggregated in order toprevent premature convergence to the local optimum thevariation factor is introduced to further explore the searchspace

421 CPSO for Dimensionality Reduction As the dimensionof the optimization problem increases the performance ofall algorithm drops dramatically An effective solution isthe cooperative evolution strategy Bergh and Engelbrecht[26] combined cooperative evolution strategy with PSO andproposed cooperative particle swarm optimization (CPSO)The basic idea of CPSO is using K independent particlegroups to search different dimension directions in the D-dimensional search space Each particle swarm is updatedindependently and no information is shared between theparticle swarms When calculating the fitness value theposition vectors of the optimal particles in the particle swarmare combined together to form a D-dimensional vector tocalculate the fitness value

In this paper to calculate the traffic volume betweensupply chain levels the particle coding in standard PSOis 11-dimensional X(ABCD 1198641 1198642m 1198601015840 1198611015840 1198621015840 1198631015840)Thefirst 6 dimensions are the supply chain routing betweenneighboring level the 7th dimension demonstrates differentproducts and the last 4 dimensions represent the nodeoperation decision According to CPSO decision variableis classified by supply chain levels Decision variable canbe split into five types of 3-dimensional routing decisionQ1(A 1198641m)Q2(ABm)Q3(BCm)Q4(CDm)Q5(D 1198642 119898) and four types of 1-dimensional node operationdecision Y1(A)Y2(B)Y3(C)Y4(D) The process is shown

6 Discrete Dynamics in Nature and Society

A

B

C

D

E2 E1

Q2(ABm)

Q3(BCm)

Q4(CDm)

Q5(DE2m)

Q1(AE1m)

Y1(A)

Y2(B)

Y3(C)

Y4(D)

X(ABCDE1E2mA BI I C DI I

A

B

C

D

E2 E1

Drop dimension Splice

)

Figure 2 Collaborative dimension reduction

in Figure 2 Coding dimension is reduced and the algorithmspeed is improved through CPSO

422 Improved Boundary Processing Strategy In the searchprocess in order to improve the convergence efficiencyand reduce the invalid search the boundary conditions areusually set for the position and velocity to narrow the searchrange In the traditional PSO when a particle goes beyond therange the general processing method is to set the particle onthe boundary That is

If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 (16)

Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 (17)

Through this method after several iterations a pluralityof particles will aggregate toward the boundary in a pluralityof dimensions and the flight path of the particles will tendto be the same So the efficiency of the particle group isreduced Thus the following improvement strategies havebeen proposed

If 119883119894119889 lt 119883119898119894119899119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899

(18)

Else if 119883119894119889 gt 119883119898119886119909119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909

(19)

1199033 is a random number distributed between [0 1) and1198883 is a constant on [001 05) The value of 1198883 will changeaccording to the algorithm the objective function etc Thispaper sets it to 005 to enhance the ability of the algorithm tojump out of the boundary region so that the particles returnto the feasible search space After using the boundary strategyon the one hand it can ensure that the particles are located inthe feasible search space On the other hand it can preventthe particles from accumulating too much to the boundaryresulting in local optimization on the boundary of the searchspace It can maximize the search space and improve thequality of the solution A comparison of the two strategies isshown in Figure 3

423 Mutation Operators to Expand Search Space WhenPSO appears as local convergence or global convergencethe particles will have an aggregation phenomenon thatis the particles have the same fitness The fitness varianceis the judgment of the convergence degree of the wholegroup The larger the fitness variance is the more likelythe particle swarm is in the search state When the fitnessvariance becomes smaller the particle swarm is tendingto converge when the fitness variance is 0 PSO achieveslocal optimization or global optimization The formula forcalculating the fitness variance is as follows

1205902 =119898

sum119894=1

(119891119894 minus 119891119886V119891 )2

(20)

And f = max1max1le119894le119873|119891119894minus119891119886V| the function off is to limit 1205902N is the total number of particles in the particle group119891119894 is the fitness value of particle i119891119886V = (1119873)sum119873119894=1 119891 is average fitness function value ofeach particle

If it is local optimum the algorithm is premature Whenthe algorithm is premature it will affect the accuracy of thealgorithm So 119866119887119890119904119905119905119889 will be the personal best The variationfactor Pr is introduced thereby changing the forward direc-tion of the particle So 119866119887119890119904119905119905119889 and 119875119887119890119904119905119905119894119889 will be updated

If Pr gt rand (d) then update particle (21)

The probability of mutation is relatively small Pr gener-ally takes a number between [0 1) in this algorithm Pr =01 rand(d) is a random number distributed between [0 1)The specific approach is to mutate the particles when theyare within 10 of the probability of mutation Firstly all theparticles are sorted according to the fitness value secondlym particles with the best fitness value (m is usually half ofthe total number of particles) are obtained then the formerPr lowastm particles are mutated as follows

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578) (22)

Discrete Dynamics in Nature and Society 7

Feasible solution space

Feasible solution space

Before

After

Figure 3 Improved boundary processing strategy

Table 1 Parameters setting for the experiments

Parameter Distribution type Distributions UnitGm

ae1 Gmab Gm

bc Gmcd Gm

de2 Uniform distribution U(190 250) RMBtonkmFa Fb Fc Fd Uniform distribution U(150000 350000) RMByearOm

a Uniform distribution U(150000 200000) tonR Normal distribution N(120583119901119895 1205902) ton year

where 120578 is a randomvariable obeying the standard normaldistribution N(0 1)The purpose of using themutation factoris to avoid the particle group falling into local optimum andbetter approaching the global optimum When the algorithmappears premature the mutation can change the direction ofthe entire particle swarm and the particle swarm reaches anew field for search By introducing the mutation operatorthe algorithm can expand the search space when the search isstagnant The improved PSO flow is shown in Figure 4 Thepseudocode of improved PSO is presented in Box 1

5 Numerical Experiments

In this section some numerical experiments are performedto verify the effectiveness of the proposed model of multi-product distribution and location problem based on uncer-tain demandnetworksTheprogramming language is C andthe experiments are completed on a PC (Intel Core i7 26GHz Memory 8G)

51 Parameter Setting According to the background of thedistribution network under Omni-Channel integration theparameters of this experiment are set The location androuting process is the same as the model description inSection 3 To demonstrate the broad applicability of themodel some of the parameters of the experiment weregenerated based on a random probability distribution shownin Table 1

The unit transportation cost G is uniformly generatedaccording to different road conditions The operating costsF of dealers for different levels are uniformly generatedaccording to the land price and scale The capacity O ofthe distribution center is uniformly distributed according tothe product category and scale This article deals with theuncertain customer demand as the following two steps

Step 1 Each customer has different demands for differentproducts A set of scenarios with different demands are setaccording to the characteristics of the normal distributionfunction

8 Discrete Dynamics in Nature and Society

Update ０＜ＭＮ and ＜ＭＮ

Output ＜ＭＮ

Start

Initialization

Adjust based on constraints and boundary

Update position and velocity

tgtMaxItr

Variation

End

Yes

Satisfies the constraint

No

Evaluate the particles and calculate the fitness values

t=t+1

Yes

No

Figure 4 Particle swarm optimization flowchart

Step 2 Let different customer demands for different productsmeet different normal distributions This creates a networkof demand of each customer for each product rather thanallowing all customers to meet only one normal distributiondemand for each product which could predict customerdemands more accurately

52 Experimental Results This paper studies multi-productdistribution under the uncertainty of customer demand sothe experiment is divided into four groups according to thechanges of customer demand Each group of experiments isdivided into five different situations according to the numberof suppliers and customers The results in Table 2 show 5experimental scales with different changes of demand Theobjective function value and running time are calculatedby the improved PSO and CPLEX solver ldquo1-1-2-4-10 (2+8)rdquorepresents one national logistics center one regional logisticscenter two city logistics centers four dealers and ten cus-tomers (two of them are first-type customers and eight ofthem are second-type customers)

In small-scale cases the running time of CPLEX is lessthan the improved PSO As the scale increases the running

time of CPLEX increases significantly For example in thefirst three experiments improved PSO and CPLEX havesimilar effects When the scale increased to 85 customersthe running time of CPLEX increased quickly while therunning time of improved PSO is less increased In generalthe results of the CPLEX solver are considered to be theoptimal solution Therefore the CPLEX solver solves themodel faster than the improved PSO method on a smallscale However when the network exceeds a certain scale theimproved PSO gets results faster than the CPLEX solver Inaddition the average deviation between the improved PSOand the optimal result is about 053 which indicates thatthe result is accurate and the proposed algorithm is suitablefor solving the model

6 Case Analysis

Based on the background of the distribution of three typesof products in Chengdu China the scale is ldquo2-4-12-22-85(7+78)rdquo M Company is a typical retail enterprise andproducts transportation is under Omni-Channel integration

Discrete Dynamics in Nature and Society 9

Table 2 Comparison of different kinds of algorithms

CPLEX Improved PSO GapScale of the study Obj CPU(s) Obj CPU(s) [(4)-(2)](2)

(1) (2) (3) (4) (5) (6)

Change in demand =0

1-1-2-4-10 (2+8) 1551183119 995 1551183119 1331 0002-2-4-10-35 (4+31) 4850885922 268 4861980932 2903 0232-2-6-14-55 (5+50) 5181467988 45 5213955899 42 0632-4-12-22-85 (7+78) 5318725918 3041 5363403216 7706 084

3-5-18-70-205 (10+195) 5619609011 gt3600 5672995297 503 095

Change in demand =001

1-1-2-4-10 (2+8) 1554786933 995 1554786933 1171 0002-2-4-10-35 (4+31) 4905668616 2345 4913027119 2857 0152-2-6-14-55 (5+50) 5193458934 49 5224634879 4454 0602-4-12-22-85 (7+78) 5304694312 325 5350314683 8386 086

3-5-18-70-205 (10+195) 5608465011 gt3600 5666793047 524 104

Change in demand =005

1-1-2-4-10 (2+8) 1554468433 99 1554468433 863 0002-2-4-10-35 (4+31) 4902814227 2375 4913110137 2477 0212-2-6-14-55 (5+50) 5220507224 4573 5231902083 3829 0222-4-12-22-85 (7+78) 5342902440 3416 5392591433 768 093

3-5-18-70-205 (10+195) 5621433450 gt3600 5685517791 537 114

Change in demand =01

1-1-2-4-10 (2+8) 1558676503 995 1558676503 1271 0002-2-4-10-35 (4+31) 4928341567 299 4941648089 2794 0272-2-6-14-55 (5+50) 5210996332 5791 5233955899 4836 0442-4-12-22-85 (7+78) 5339691338 3647 5389350467 7474 093

3-5-18-70-205 (10+195) 5620433450 gt3600 5691250911 556 126Average value 053

Table 3 Annual average demand of 3 products for 85 customers

No D1 D2 D3 No D1 D2 D3 No D1 D2 D3 No D1 D2 D31 99 9 12 23 41 27 3 45 24 4 5 67 29 5 12 21 7 1 24 44 1 1 46 14 4 4 68 10 2 23 68 18 3 25 6 5 2 47 4 2 2 69 33 8 14 4 2 2 26 2 0 1 48 15 12 2 70 56 9 25 55 2 1 27 104 23 1 49 19 7 4 71 4 1 36 16 6 3 28 12 4 2 50 2 1 0 72 43 23 27 14 5 1 29 1 1 1 51 9 7 0 73 16 5 28 3 1 0 30 1 1 1 52 4 1 1 74 9 1 19 17 3 1 31 5 2 1 53 16 7 7 75 24 2 310 1 0 0 32 8 1 1 54 14 8 1 76 2 1 111 4 1 7 33 48 9 7 55 22 7 2 77 7 3 212 18 1 6 34 23 15 1 56 4 3 1 78 23 5 213 67 14 7 35 65 8 4 57 6 2 2 79 163 19 114 3 1 2 36 31 15 1 58 1 1 2 80 429 4 115 9 5 1 37 31 3 4 59 4 3 0 81 417 54 116 7 3 1 38 9 1 1 60 13 7 3 82 501 25 117 43 3 15 39 16 2 1 61 13 3 1 83 286 22 418 1 0 7 40 45 11 2 62 12 2 2 84 455 25 219 29 2 3 41 86 8 1 63 45 8 3 85 393 45 220 28 8 4 42 39 5 4 64 39 14 321 38 6 1 43 16 3 1 65 23 6 122 5 2 3 44 6 2 1 66 67 7 1No represents the number of customers D1-D3 represent demand of products 1-3

10 Discrete Dynamics in Nature and Society

InitializationSet the parameters MaxItrNw 1198881 1198882 1199031 r2 c3 1199033 PrInitialize 9 independent particle swarms1198761198981198861198901(a 1198901m) 119876119898119886119887(a bm) 119876119898119887119888(b cm) 119876119898119888119889(c dm) 1198761198981198891198902(d 1198902m)119910119886(a) 119910119887(b) 119910119888(c) 119910119889(d)which are in the 11-dimensional solution space X(a b c d 1198901 1198902m 1198861015840 1198871015840 1198881015840 1198891015840)Define position (x) and velocity (v) for each particleAssume individual best (119875119887119890119904119905119905119894119889) equal to the initialized particlesSelect the best particle as global best (119866119887119890119904119905119905119889)

UpdateLet 9 particle swarms be updated independently as follows

For t = 1 to MaxItrUpdate position (x) and velocity (v)V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032 lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909RespectivelyEndEvaluate the particles and calculate the fitness valuesSort the fitness valuesIf Pr gt rand(d)

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578)EndUpdate personal best (119875119887119890119904119905119905119894119889) of 9 independent particle swarmsUpdate global best (119866119887119890119904119905119905119889) of 9 independent particle swarms

EndReport

Global best (119866119887119890119904119905119905119889) of 9 independent particle swarms119866119887119890119904119905119905119889 found by splicing 9 independent particle swarmsFitness values calculated by 119866119887119890119904119905119905119889

Box 1 The improved PSO

Products include electrical appliances household productsand daily chemical products The average annual demand ofthree products for 85 customers is shown in Table 3 Theirlocation is shown in Figure 5

61 Location Decision Results The locations of the logis-tics center and customers (blue icon) are shown in Fig-ure 5 After the location decision three regional logis-tics centers eight city logistics centers nine dealers maynot operate The non-operating logistics centers are iden-tified by red rectangle Reasons for not operating areanalyzed

Two national logistics centers (numbered 119860 119894 yellowicon) Since the national distribution center needs to dis-tribute to regional logistics centers and large customers onenational distribution centerrsquos capacity is not enough both ofthem need to operate

Four regional logistics centers (numbered 119861119894 purpleicon) Regional logistics center B2 needs to operate B2 is clos-est to the two national logistics centers and is closer to the citylogistics center compared with B1 B3 and B4 In the vicinityof B1 B3 and B4 the city logistics centers (numbered 119862119894)are less distributed and dispersed Considering the operatingcosts B1 B3 and B4 may not operate

Twelve city logistics centers (numbered 119862119894 green icon)City logistics centers C1 C4 C8 and C11 need to oper-ate C2 C9 and C10 are far away from B2 and havehigh transportation costs compared with C4 and C8 sothey are not operated C3 C5 C6 C7 and C12 areremote and scattered and the demand of dealer is smallIt does not operate due to transportation and operatingcosts

Twenty-two dealers (numbered 119863119894 red icon) Throughthe calculation of location and routing 9 dealers do not needto operate The remaining 13 dealers can fulfill the productdistribution demands of 78 second-class customers

The above analyses show that M companyrsquos ldquo2-4-12-22rdquosupply chain network is too large The current demand canbe satisfied by the ldquo2-1-4-13rdquo scale network The location-routing decision can reduce the operations of three regionaldistribution centers eight city distribution centers and ninedealers

62 Transportation Route Decision Results Distributionroute of the three products for 85 customers is shown inTable 4

If distribution routes of the three products are the samethe route decision follows the customer number and the first

Discrete Dynamics in Nature and Society 11

Table4Deliveryrouteo

f3prod

uctsfor8

5custo

mers(To

n)

No

Deliverypath

No

Deliverypath

No

Deliverypath

No

Deliverypath

1A1-B

2-C1-D

8-E1

23A1-B

2-C1-D

7-E2

343

A1-B

2-C1

1-D19-E43

67A1-B

2-C1-D

1-E67

2A1-B

2-C1-D

1-E2

24A1-B

2-C1

1-D2-E2

444

A1-B

2-C1-D

1-E44

68A1-B

2-C8

-D21-E68

3A1-B

2-C1-D

1-E3

25A1-B

2-C1

1-D2-E2

545

A1-B

2-C1

1-D2-E4

569

A1-B

2-C1-D

1-E69

4A1-B

2-C1

1-D19-E4

26A1-B

2-C1-D

1-E26

46A1-B

2-C1-D

8-E4

670

A1-B

2-C1

1-D19-E70

5A1-B

2-C1-D

1-E5

27A1-B

2-C1

1-D2-E2

747

A1-B

2-C1

1-D19-E47

71A1-B

2-C1-D

1-E71

6A1-B

2-C1

1-D19-E6

28A1-B

2-C1

1-D19-E28

48A1-B

2-C1

1-D2-E4

872

A1-B

2-C1-D

1-E72

7A1-B

2-C1-D

10-E7

29A1-B

2-C1-D

8-E2

949

A1-B

2-C1-D

1-E49

73A1-B

2-C1-D

1-E73

9A1-B

2-C1-D

1-E9

30A1-B

2-C1-D

1-E30

51A1-B

2-C8

-D21-E51

74A1-B

2-C1-D

1-E74

10A1-B

2-C1-D

1-E10

31A1-B

2-C1-D

1-E31

52A1-B

2-C1-D

7-E5

275

A1-B

2-C1-D

1-E75

11A1-B

2-C1-D

22-E11

32A1-B

2-C1-D

1-E32

53A1-B

2-C1-D

1-E53

76A1-B

2-C1-D

1-E76

12A1-B

2-C1-D

1-E12

33A1-B

2-C1

1-D2-E3

354

A1-B

2-C1-D

1-E54

77A1-B

2-C1-D

1-E77

13A1-B

2-C8

-D21-E13

34A1-B

2-C1

1-D19-E34

55A1-B

2-C1-D

1-E55

78A1-B

2-C1-D

1-E78

14A1-B

2-C1-D

1-E14

35A1-B

2-C1-D

22-E35

56A1-B

2-C1-D

1-E56

79A1-E

7915

A1-B

2-C1-D

1-E15

36A1-B

2-C1-D

16-E36

57A1-B

2-C1

1-D17-E57

80A2-E8

016

A1-B

2-C1

1-D2-E16

37A1-B

2-C1

1-D2-E3

758

A1-B

2-C1-D

1-E58

81A2-E8

117

A1-B

2-C1-D

22-E17

38A1-B

2-C1

1-D2-E3

860

A1-B

2-C1-D

8-E6

082

A2-E8

219

A1-B

2-C1-D

10-E19

39A1-B

2-C1

1-D19-E39

61A1-B

2-C1-D

1-E61

83A1-E

8320

A1-B

2-C8

-D21-E20

40A1-B

2-C4

-D18-E40

62A1-B

2-C8

-D21-E62

84A2-E8

421

A1-B

2-C8

-D21-E21

41A1-B

2-C4

-D18-E41

63A1-B

2-C1-D

1-E63

85A2-E8

522

A1-B

2-C1-D

8-E2

242

A1-B

2-C1-D

1-E42

64A1-B

2-C1-D

4-E6

48

A1-B

2-C1-D

8-E8

8A1-B

2-C1-D

8-E8

8

18

18A1-B

2-C1-D

8-E18

18A1-B

2-C1-D

8-E18

50A1-B

2-C8

-D21-E50

50A1-B

2-C8

-D21-E50

50

59A1-B

2-C4

-D12-E59

59A1-B

2-C4

-D12-E59

59

65A1-B

2-C1-D

4-E6

565

A1-B

2-C1-D

4-E6

565

66

A1-B

2-C4

-D18-E66

66A1-B

2-C4

-D18-E66

66

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

Discrete Dynamics in Nature and Society 3

Nationallogistics center

Regionallogistics center

City logisticscenter

Dealer Client

Figure 1 Supply chain distribution network

idea to reduce the coding dimension improves the boundaryprocessing strategy and introduces the mutation operator toexpand the search spaceThismethod improves the efficiencyof the algorithm and provides a solution for solving highdimensional problems

3 Model

31 Model Description This model includes some commoncharacteristics of distribution under Omni-Channel inte-gration so the conclusions of the model can be useful formany enterprises of supply-chain distribution under Omni-Channel integration There are five levels in the entire supplychain These levels include National Logistics Center ARegional Logistics Center B City Logistics Center C DealerD and Customer E Distribution network shall distributem-type products to two types of customers The first typeof customers E1 is large customers such as companiesthe demand of such customers is delivered directly by thenational logistics center AThe second type of customers E2 issmall supermarkets like storesThe orders of such customerswill be delivered through the regional logistics center B thecity logistics center C and the dealer DWe can intuitively seethe model in Figure 1

According to the supply chain network we built a mixedinteger programming model to minimize the total supplychain costsThe total costs include three items transportationcosts of the products fixed operating costs of logistics centerand carbon treatment costs which is to deal with carbonemission generated from the product transportation process

32 e Index of the Model

a The number of national logistics centersb The number of regional logistics centersc The number of city logistics centersd The number of dealers1198901 The number of first type of customers1198902 The number of second type of customersm The number of the category of products

33 Parameter Symbols

1198661198981198861198901 The unit transportation cost of product mdelivered to customer 1198901 through national logisticscenter a119866119898119886119887 The unit transportation cost of product mdelivered to regional logistics center b throughnational logistics center a119866119898119887119888 The unit transportation cost of product mdelivered to city logistics center c through regionallogistics center b119866119898119888119889 The unit transportation cost of product mdelivered to dealer d through city logistics center c1198661198981198891198902 The unit transportation cost of product mdelivered to customer 1198902 through dealer d1198891198941199041198861198901 The distance between national logistics cen-ter a and customer 1198901

4 Discrete Dynamics in Nature and Society

119889119894119904119886119887 Thedistance betweennational logistics centera and regional logistics center b

119889119894119904119887119888 The distance between regional logistics centerb and city logistics center c

119889119894119904119888119889 The distance between city logistics center cand dealer d

1198891198941199041198891198902 The distance between dealer d and customer1198902F119886 The fixed operating costs of national logisticscenter located at point a

F119887 The fixed operating costs of regional logisticscenter located at point b

F119888 The fixed operating costs of city logistics centerlocated at point c

F119889 The fixed operating costs of dealer located atpoint d

119874119898119886 The average inventory of productm in nationallogistics center located at point a

1198771198981198901 The demand of customer 1198901 for product m1198771198981198902 The demand of customer 1198902 for product mT Unit carbon treatment cost

N A large enough positive number

34 Decision Variables

1198761198981198861198901 The quantity of product m from nationallogistics center a to customer 1198901119876119898119886119887 The quantity of productm from national logis-tics center a to regional logistics center b

119876119898119887119888 The quantity of product m from regional logis-tics center b to city logistics center c

119876119898119888119889 The quantity of product m from city logisticscenter c to dealer d

1198761198981198891198902 The quantity of product m from dealer d tocustomer 1198902119910119886 Whether the national logistics center a operatesif it operates then 119910119886 equals 1 otherwise the value is0

119910119887 Whether the regional logistics center b operatesif it operates then 119910119887 equals 1 otherwise the value is 0119910119888 Whether the city logistics center c operates if itoperates then 119910119888 equals 1 otherwise the value is 0119910119889 Whether the dealer d operates if it operatesthen 119910119889 equals 1 otherwise the value is 0

35 Data Model

min f (x) (1)

f (x) = sumAsumE1sumMGm

ae1Qmae1disae1 +sum

AsumBsumMGm

abQmabdisab

+sumBsumCsumMGm

bcQmbcdisbc +sum

CsumDsumMGm

cdQmcddiscd

+sumDsumE2sumMGm

de2Qmde2disde2 +sum

AFaya +sum

BFbyb

+sumCFcyc +sum

DFdyd +sum

AsumE1sumMTQm

ae1disae1

+sumAsumBsumMTQm

abdisab +sumBsumCsumMTQm

bcdisbc

+sumCsumDsumMTQm

cddiscd +sumDsumE2sumMTQm

de2disde2

(2)

sum119860

1198761198981198861198901 = 1198771198981198901 forall1198901 isin 1198641 forallm isin M (3)

sum119863

1198761198981198891198902 = 1198771198981198902 forall1198902 isin 1198642 forallm isin M (4)

sum119860

119876119898119886119887 ge sum119862

119876119898119887119888 forallb isin B forallm isin M (5)

sum119861

119876119898119887119888 ge sum119863

119876119898119888119889 forallc isin C forallm isin M (6)

sum119862

119876119898119888119889 ge sum1198642

1198761198981198891198902 foralld isin D forallm isin M (7)

sum1198641

1198761198981198861198901 +sum119861

119876119898119886119887 le 119874119898119886 lowast 119910119886 foralla isin A forallm isin M (8)

sum119860

119876119898119886119887 minus 119873 times 119910119887 le 0 forallb isin B forallm isin M (9)

sum119861

119876119898119887119888 minus 119873 times 119910119888 le 0 forallc isin C forallm isin M (10)

sum119862

119876119898119888119889 minus 119873 times 119910119889 le 0 foralld isin D forallm isin M (11)

119910119886119910119887 119910119888 119910119889 isin 0 1foralla isin A forallb isin B forallc isin C foralld isin D

(12)

1198761198981198861198901 119876119898119886119887 119876119898119887119888 119876119898119888119889 1198761198981198891198902 ge 0foralla isin A forallb isin B forallc isin C foralld isin D forall1198901 isin 1198641 forall1198902 isin 1198642

(13)

The objective function (2) aims to minimize the totalcosts of the supply chain network The constraint (3) requiresthat the demands of the customer 1198901 can only be directlydelivered by the national logistics center a and the constraint

Discrete Dynamics in Nature and Society 5

(4) states that the product can only be provided from thedealer d to the customer 1198902 Equations (5)-(7) specify thatthe volume of each logistics center is greater than or equalto the shipment at that point Constraint (8) ensures that theinventory from the national logistics center a is greater thanthe total volume of shipped products and the operation of thenational logistics center a is represented by the 0-1 variableConstraint (9) donates that if the regional logistics center bis not in operation the quantity of product shipped to theregional logistics center b is zero While constraint (10) andconstraint (11) indicate that if city logistics center c and thedealer d do not operate the volume of product shipped to thecity logistics center c and the dealer d is zero Constraint (12)and constraint (13) are the value constraints of the decisionvariable

4 Proposed Heuristic Method

The model of LRP presented in Section 3 aims to minimizethe total costs in product transportation process In a large-scale transportation network the code dimension is highmaking it difficult to find an optimal solution in a limitedtime To solve the problem we resort to heuristic methodssuch as particle swarm optimization (PSO) PSO is a stochas-tic parallel optimization algorithm that does not requirethe properties of the optimized function to be divisiblesteerable and continuous In addition to further improve theaccuracy and efficiency of the algorithm the improved PSOis proposed

41 Particle Swarm Optimization PSO was proposed byKennedy and Eberhart [21] and Lian [22] pointed out thatthis method was a stochastic optimization method based onswarm intelligence and PSOwas inspired by the social behav-ior of bird flocking and their means of information exchangeDue to its easy implementation and fast convergence PSOhas been successfully applied to solve nonlinear optimizationproblems Ai and Kachitvichyanukul [23] Goksal et al [24]and Alinaghian et al [25] designed an improved randomtopology particle swarm optimization algorithm (RT-PSO)for time-dependent vehicle routing problems Hu et al [19]developed a two-stage decomposition algorithm by combin-ing variable neighborhood search algorithm and PSO to solvethe large-scale refrigerated truck scheduling problem

In PSO each particle represents a solution to the opti-mization problem Particle initialization is generated in arandom way and each particle iteratively updates its positionthrough personal best and global best Each particle searchesfor the optimal fitness value in the D-dimensional feasiblesolution spaceThere are N particles in a group In iteration tthe position of particle i is represented as a D-dimensionalvector 119883119905119894119889 = (1199091199051198941 1199091199051198942 119909119905119894119863) 119894 = 1 2 119873 At thesame time the velocity of the particle i is also a D minusdimensional vector denoted as 119881119905119894119889 = (V1199051198941 V1199051198942 V119905119894119863) 119894 =1 2 119873 The optimal position searched by the particle iin iteration t is called the personal best which is denotedas 119875119887119890119904119905 = (1199011198941 1199011198942 119901119894119863) 119894 = 1 2 119873 The optimalposition searched by the group in iteration t is the global best

denoted as 119866119887119890119904119905 = (1199011198921 1199011198922 119901119892119863) After finding the twooptimal values each particle updates its velocity and positionaccording to the following formula

V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)

(14)

119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889 (15)

In the standard PSO there are some shortcomings forexample (1)when solving the high dimensional optimizationproblem the increase of the dimension makes the searchspace expand into an exponential form which is prone topremature convergence That is the particle group aggregatesprematurely to the local optimal solution (2) Particles mayaggregate to local optimum too early in the process of gather-ing If there is no escaping mechanism the optimal solutionmay not be obtained Therefore we proposed improvedstrategy accordingly

42 Strategies for Improving PSO The improved PSO focuseson three aspects (1) facing the ldquodimensional disasterrdquo inlarge-scale problems cooperative particle swarm optimiza-tion (CPSO) is proposed to transform large-scale complexproblems into low-dimensional simple problems (2) Whenparticle flies out of the boundary the boundary processingstrategy is adopted to change the flight trajectory of theparticles (3) When the particles are aggregated in order toprevent premature convergence to the local optimum thevariation factor is introduced to further explore the searchspace

421 CPSO for Dimensionality Reduction As the dimensionof the optimization problem increases the performance ofall algorithm drops dramatically An effective solution isthe cooperative evolution strategy Bergh and Engelbrecht[26] combined cooperative evolution strategy with PSO andproposed cooperative particle swarm optimization (CPSO)The basic idea of CPSO is using K independent particlegroups to search different dimension directions in the D-dimensional search space Each particle swarm is updatedindependently and no information is shared between theparticle swarms When calculating the fitness value theposition vectors of the optimal particles in the particle swarmare combined together to form a D-dimensional vector tocalculate the fitness value

In this paper to calculate the traffic volume betweensupply chain levels the particle coding in standard PSOis 11-dimensional X(ABCD 1198641 1198642m 1198601015840 1198611015840 1198621015840 1198631015840)Thefirst 6 dimensions are the supply chain routing betweenneighboring level the 7th dimension demonstrates differentproducts and the last 4 dimensions represent the nodeoperation decision According to CPSO decision variableis classified by supply chain levels Decision variable canbe split into five types of 3-dimensional routing decisionQ1(A 1198641m)Q2(ABm)Q3(BCm)Q4(CDm)Q5(D 1198642 119898) and four types of 1-dimensional node operationdecision Y1(A)Y2(B)Y3(C)Y4(D) The process is shown

6 Discrete Dynamics in Nature and Society

A

B

C

D

E2 E1

Q2(ABm)

Q3(BCm)

Q4(CDm)

Q5(DE2m)

Q1(AE1m)

Y1(A)

Y2(B)

Y3(C)

Y4(D)

X(ABCDE1E2mA BI I C DI I

A

B

C

D

E2 E1

Drop dimension Splice

)

Figure 2 Collaborative dimension reduction

in Figure 2 Coding dimension is reduced and the algorithmspeed is improved through CPSO

422 Improved Boundary Processing Strategy In the searchprocess in order to improve the convergence efficiencyand reduce the invalid search the boundary conditions areusually set for the position and velocity to narrow the searchrange In the traditional PSO when a particle goes beyond therange the general processing method is to set the particle onthe boundary That is

If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 (16)

Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 (17)

Through this method after several iterations a pluralityof particles will aggregate toward the boundary in a pluralityof dimensions and the flight path of the particles will tendto be the same So the efficiency of the particle group isreduced Thus the following improvement strategies havebeen proposed

If 119883119894119889 lt 119883119898119894119899119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899

(18)

Else if 119883119894119889 gt 119883119898119886119909119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909

(19)

1199033 is a random number distributed between [0 1) and1198883 is a constant on [001 05) The value of 1198883 will changeaccording to the algorithm the objective function etc Thispaper sets it to 005 to enhance the ability of the algorithm tojump out of the boundary region so that the particles returnto the feasible search space After using the boundary strategyon the one hand it can ensure that the particles are located inthe feasible search space On the other hand it can preventthe particles from accumulating too much to the boundaryresulting in local optimization on the boundary of the searchspace It can maximize the search space and improve thequality of the solution A comparison of the two strategies isshown in Figure 3

423 Mutation Operators to Expand Search Space WhenPSO appears as local convergence or global convergencethe particles will have an aggregation phenomenon thatis the particles have the same fitness The fitness varianceis the judgment of the convergence degree of the wholegroup The larger the fitness variance is the more likelythe particle swarm is in the search state When the fitnessvariance becomes smaller the particle swarm is tendingto converge when the fitness variance is 0 PSO achieveslocal optimization or global optimization The formula forcalculating the fitness variance is as follows

1205902 =119898

sum119894=1

(119891119894 minus 119891119886V119891 )2

(20)

And f = max1max1le119894le119873|119891119894minus119891119886V| the function off is to limit 1205902N is the total number of particles in the particle group119891119894 is the fitness value of particle i119891119886V = (1119873)sum119873119894=1 119891 is average fitness function value ofeach particle

If it is local optimum the algorithm is premature Whenthe algorithm is premature it will affect the accuracy of thealgorithm So 119866119887119890119904119905119905119889 will be the personal best The variationfactor Pr is introduced thereby changing the forward direc-tion of the particle So 119866119887119890119904119905119905119889 and 119875119887119890119904119905119905119894119889 will be updated

If Pr gt rand (d) then update particle (21)

The probability of mutation is relatively small Pr gener-ally takes a number between [0 1) in this algorithm Pr =01 rand(d) is a random number distributed between [0 1)The specific approach is to mutate the particles when theyare within 10 of the probability of mutation Firstly all theparticles are sorted according to the fitness value secondlym particles with the best fitness value (m is usually half ofthe total number of particles) are obtained then the formerPr lowastm particles are mutated as follows

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578) (22)

Discrete Dynamics in Nature and Society 7

Feasible solution space

Feasible solution space

Before

After

Figure 3 Improved boundary processing strategy

Table 1 Parameters setting for the experiments

Parameter Distribution type Distributions UnitGm

ae1 Gmab Gm

bc Gmcd Gm

de2 Uniform distribution U(190 250) RMBtonkmFa Fb Fc Fd Uniform distribution U(150000 350000) RMByearOm

a Uniform distribution U(150000 200000) tonR Normal distribution N(120583119901119895 1205902) ton year

where 120578 is a randomvariable obeying the standard normaldistribution N(0 1)The purpose of using themutation factoris to avoid the particle group falling into local optimum andbetter approaching the global optimum When the algorithmappears premature the mutation can change the direction ofthe entire particle swarm and the particle swarm reaches anew field for search By introducing the mutation operatorthe algorithm can expand the search space when the search isstagnant The improved PSO flow is shown in Figure 4 Thepseudocode of improved PSO is presented in Box 1

5 Numerical Experiments

In this section some numerical experiments are performedto verify the effectiveness of the proposed model of multi-product distribution and location problem based on uncer-tain demandnetworksTheprogramming language is C andthe experiments are completed on a PC (Intel Core i7 26GHz Memory 8G)

51 Parameter Setting According to the background of thedistribution network under Omni-Channel integration theparameters of this experiment are set The location androuting process is the same as the model description inSection 3 To demonstrate the broad applicability of themodel some of the parameters of the experiment weregenerated based on a random probability distribution shownin Table 1

The unit transportation cost G is uniformly generatedaccording to different road conditions The operating costsF of dealers for different levels are uniformly generatedaccording to the land price and scale The capacity O ofthe distribution center is uniformly distributed according tothe product category and scale This article deals with theuncertain customer demand as the following two steps

Step 1 Each customer has different demands for differentproducts A set of scenarios with different demands are setaccording to the characteristics of the normal distributionfunction

8 Discrete Dynamics in Nature and Society

Update ０＜ＭＮ and ＜ＭＮ

Output ＜ＭＮ

Start

Initialization

Adjust based on constraints and boundary

Update position and velocity

tgtMaxItr

Variation

End

Yes

Satisfies the constraint

No

Evaluate the particles and calculate the fitness values

t=t+1

Yes

No

Figure 4 Particle swarm optimization flowchart

Step 2 Let different customer demands for different productsmeet different normal distributions This creates a networkof demand of each customer for each product rather thanallowing all customers to meet only one normal distributiondemand for each product which could predict customerdemands more accurately

52 Experimental Results This paper studies multi-productdistribution under the uncertainty of customer demand sothe experiment is divided into four groups according to thechanges of customer demand Each group of experiments isdivided into five different situations according to the numberof suppliers and customers The results in Table 2 show 5experimental scales with different changes of demand Theobjective function value and running time are calculatedby the improved PSO and CPLEX solver ldquo1-1-2-4-10 (2+8)rdquorepresents one national logistics center one regional logisticscenter two city logistics centers four dealers and ten cus-tomers (two of them are first-type customers and eight ofthem are second-type customers)

In small-scale cases the running time of CPLEX is lessthan the improved PSO As the scale increases the running

time of CPLEX increases significantly For example in thefirst three experiments improved PSO and CPLEX havesimilar effects When the scale increased to 85 customersthe running time of CPLEX increased quickly while therunning time of improved PSO is less increased In generalthe results of the CPLEX solver are considered to be theoptimal solution Therefore the CPLEX solver solves themodel faster than the improved PSO method on a smallscale However when the network exceeds a certain scale theimproved PSO gets results faster than the CPLEX solver Inaddition the average deviation between the improved PSOand the optimal result is about 053 which indicates thatthe result is accurate and the proposed algorithm is suitablefor solving the model

6 Case Analysis

Based on the background of the distribution of three typesof products in Chengdu China the scale is ldquo2-4-12-22-85(7+78)rdquo M Company is a typical retail enterprise andproducts transportation is under Omni-Channel integration

Discrete Dynamics in Nature and Society 9

Table 2 Comparison of different kinds of algorithms

CPLEX Improved PSO GapScale of the study Obj CPU(s) Obj CPU(s) [(4)-(2)](2)

(1) (2) (3) (4) (5) (6)

Change in demand =0

1-1-2-4-10 (2+8) 1551183119 995 1551183119 1331 0002-2-4-10-35 (4+31) 4850885922 268 4861980932 2903 0232-2-6-14-55 (5+50) 5181467988 45 5213955899 42 0632-4-12-22-85 (7+78) 5318725918 3041 5363403216 7706 084

3-5-18-70-205 (10+195) 5619609011 gt3600 5672995297 503 095

Change in demand =001

1-1-2-4-10 (2+8) 1554786933 995 1554786933 1171 0002-2-4-10-35 (4+31) 4905668616 2345 4913027119 2857 0152-2-6-14-55 (5+50) 5193458934 49 5224634879 4454 0602-4-12-22-85 (7+78) 5304694312 325 5350314683 8386 086

3-5-18-70-205 (10+195) 5608465011 gt3600 5666793047 524 104

Change in demand =005

1-1-2-4-10 (2+8) 1554468433 99 1554468433 863 0002-2-4-10-35 (4+31) 4902814227 2375 4913110137 2477 0212-2-6-14-55 (5+50) 5220507224 4573 5231902083 3829 0222-4-12-22-85 (7+78) 5342902440 3416 5392591433 768 093

3-5-18-70-205 (10+195) 5621433450 gt3600 5685517791 537 114

Change in demand =01

1-1-2-4-10 (2+8) 1558676503 995 1558676503 1271 0002-2-4-10-35 (4+31) 4928341567 299 4941648089 2794 0272-2-6-14-55 (5+50) 5210996332 5791 5233955899 4836 0442-4-12-22-85 (7+78) 5339691338 3647 5389350467 7474 093

3-5-18-70-205 (10+195) 5620433450 gt3600 5691250911 556 126Average value 053

Table 3 Annual average demand of 3 products for 85 customers

No D1 D2 D3 No D1 D2 D3 No D1 D2 D3 No D1 D2 D31 99 9 12 23 41 27 3 45 24 4 5 67 29 5 12 21 7 1 24 44 1 1 46 14 4 4 68 10 2 23 68 18 3 25 6 5 2 47 4 2 2 69 33 8 14 4 2 2 26 2 0 1 48 15 12 2 70 56 9 25 55 2 1 27 104 23 1 49 19 7 4 71 4 1 36 16 6 3 28 12 4 2 50 2 1 0 72 43 23 27 14 5 1 29 1 1 1 51 9 7 0 73 16 5 28 3 1 0 30 1 1 1 52 4 1 1 74 9 1 19 17 3 1 31 5 2 1 53 16 7 7 75 24 2 310 1 0 0 32 8 1 1 54 14 8 1 76 2 1 111 4 1 7 33 48 9 7 55 22 7 2 77 7 3 212 18 1 6 34 23 15 1 56 4 3 1 78 23 5 213 67 14 7 35 65 8 4 57 6 2 2 79 163 19 114 3 1 2 36 31 15 1 58 1 1 2 80 429 4 115 9 5 1 37 31 3 4 59 4 3 0 81 417 54 116 7 3 1 38 9 1 1 60 13 7 3 82 501 25 117 43 3 15 39 16 2 1 61 13 3 1 83 286 22 418 1 0 7 40 45 11 2 62 12 2 2 84 455 25 219 29 2 3 41 86 8 1 63 45 8 3 85 393 45 220 28 8 4 42 39 5 4 64 39 14 321 38 6 1 43 16 3 1 65 23 6 122 5 2 3 44 6 2 1 66 67 7 1No represents the number of customers D1-D3 represent demand of products 1-3

10 Discrete Dynamics in Nature and Society

InitializationSet the parameters MaxItrNw 1198881 1198882 1199031 r2 c3 1199033 PrInitialize 9 independent particle swarms1198761198981198861198901(a 1198901m) 119876119898119886119887(a bm) 119876119898119887119888(b cm) 119876119898119888119889(c dm) 1198761198981198891198902(d 1198902m)119910119886(a) 119910119887(b) 119910119888(c) 119910119889(d)which are in the 11-dimensional solution space X(a b c d 1198901 1198902m 1198861015840 1198871015840 1198881015840 1198891015840)Define position (x) and velocity (v) for each particleAssume individual best (119875119887119890119904119905119905119894119889) equal to the initialized particlesSelect the best particle as global best (119866119887119890119904119905119905119889)

UpdateLet 9 particle swarms be updated independently as follows

For t = 1 to MaxItrUpdate position (x) and velocity (v)V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032 lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909RespectivelyEndEvaluate the particles and calculate the fitness valuesSort the fitness valuesIf Pr gt rand(d)

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578)EndUpdate personal best (119875119887119890119904119905119905119894119889) of 9 independent particle swarmsUpdate global best (119866119887119890119904119905119905119889) of 9 independent particle swarms

EndReport

Global best (119866119887119890119904119905119905119889) of 9 independent particle swarms119866119887119890119904119905119905119889 found by splicing 9 independent particle swarmsFitness values calculated by 119866119887119890119904119905119905119889

Box 1 The improved PSO

Products include electrical appliances household productsand daily chemical products The average annual demand ofthree products for 85 customers is shown in Table 3 Theirlocation is shown in Figure 5

61 Location Decision Results The locations of the logis-tics center and customers (blue icon) are shown in Fig-ure 5 After the location decision three regional logis-tics centers eight city logistics centers nine dealers maynot operate The non-operating logistics centers are iden-tified by red rectangle Reasons for not operating areanalyzed

Two national logistics centers (numbered 119860 119894 yellowicon) Since the national distribution center needs to dis-tribute to regional logistics centers and large customers onenational distribution centerrsquos capacity is not enough both ofthem need to operate

Four regional logistics centers (numbered 119861119894 purpleicon) Regional logistics center B2 needs to operate B2 is clos-est to the two national logistics centers and is closer to the citylogistics center compared with B1 B3 and B4 In the vicinityof B1 B3 and B4 the city logistics centers (numbered 119862119894)are less distributed and dispersed Considering the operatingcosts B1 B3 and B4 may not operate

Twelve city logistics centers (numbered 119862119894 green icon)City logistics centers C1 C4 C8 and C11 need to oper-ate C2 C9 and C10 are far away from B2 and havehigh transportation costs compared with C4 and C8 sothey are not operated C3 C5 C6 C7 and C12 areremote and scattered and the demand of dealer is smallIt does not operate due to transportation and operatingcosts

Twenty-two dealers (numbered 119863119894 red icon) Throughthe calculation of location and routing 9 dealers do not needto operate The remaining 13 dealers can fulfill the productdistribution demands of 78 second-class customers

The above analyses show that M companyrsquos ldquo2-4-12-22rdquosupply chain network is too large The current demand canbe satisfied by the ldquo2-1-4-13rdquo scale network The location-routing decision can reduce the operations of three regionaldistribution centers eight city distribution centers and ninedealers

62 Transportation Route Decision Results Distributionroute of the three products for 85 customers is shown inTable 4

If distribution routes of the three products are the samethe route decision follows the customer number and the first

Discrete Dynamics in Nature and Society 11

Table4Deliveryrouteo

f3prod

uctsfor8

5custo

mers(To

n)

No

Deliverypath

No

Deliverypath

No

Deliverypath

No

Deliverypath

1A1-B

2-C1-D

8-E1

23A1-B

2-C1-D

7-E2

343

A1-B

2-C1

1-D19-E43

67A1-B

2-C1-D

1-E67

2A1-B

2-C1-D

1-E2

24A1-B

2-C1

1-D2-E2

444

A1-B

2-C1-D

1-E44

68A1-B

2-C8

-D21-E68

3A1-B

2-C1-D

1-E3

25A1-B

2-C1

1-D2-E2

545

A1-B

2-C1

1-D2-E4

569

A1-B

2-C1-D

1-E69

4A1-B

2-C1

1-D19-E4

26A1-B

2-C1-D

1-E26

46A1-B

2-C1-D

8-E4

670

A1-B

2-C1

1-D19-E70

5A1-B

2-C1-D

1-E5

27A1-B

2-C1

1-D2-E2

747

A1-B

2-C1

1-D19-E47

71A1-B

2-C1-D

1-E71

6A1-B

2-C1

1-D19-E6

28A1-B

2-C1

1-D19-E28

48A1-B

2-C1

1-D2-E4

872

A1-B

2-C1-D

1-E72

7A1-B

2-C1-D

10-E7

29A1-B

2-C1-D

8-E2

949

A1-B

2-C1-D

1-E49

73A1-B

2-C1-D

1-E73

9A1-B

2-C1-D

1-E9

30A1-B

2-C1-D

1-E30

51A1-B

2-C8

-D21-E51

74A1-B

2-C1-D

1-E74

10A1-B

2-C1-D

1-E10

31A1-B

2-C1-D

1-E31

52A1-B

2-C1-D

7-E5

275

A1-B

2-C1-D

1-E75

11A1-B

2-C1-D

22-E11

32A1-B

2-C1-D

1-E32

53A1-B

2-C1-D

1-E53

76A1-B

2-C1-D

1-E76

12A1-B

2-C1-D

1-E12

33A1-B

2-C1

1-D2-E3

354

A1-B

2-C1-D

1-E54

77A1-B

2-C1-D

1-E77

13A1-B

2-C8

-D21-E13

34A1-B

2-C1

1-D19-E34

55A1-B

2-C1-D

1-E55

78A1-B

2-C1-D

1-E78

14A1-B

2-C1-D

1-E14

35A1-B

2-C1-D

22-E35

56A1-B

2-C1-D

1-E56

79A1-E

7915

A1-B

2-C1-D

1-E15

36A1-B

2-C1-D

16-E36

57A1-B

2-C1

1-D17-E57

80A2-E8

016

A1-B

2-C1

1-D2-E16

37A1-B

2-C1

1-D2-E3

758

A1-B

2-C1-D

1-E58

81A2-E8

117

A1-B

2-C1-D

22-E17

38A1-B

2-C1

1-D2-E3

860

A1-B

2-C1-D

8-E6

082

A2-E8

219

A1-B

2-C1-D

10-E19

39A1-B

2-C1

1-D19-E39

61A1-B

2-C1-D

1-E61

83A1-E

8320

A1-B

2-C8

-D21-E20

40A1-B

2-C4

-D18-E40

62A1-B

2-C8

-D21-E62

84A2-E8

421

A1-B

2-C8

-D21-E21

41A1-B

2-C4

-D18-E41

63A1-B

2-C1-D

1-E63

85A2-E8

522

A1-B

2-C1-D

8-E2

242

A1-B

2-C1-D

1-E42

64A1-B

2-C1-D

4-E6

48

A1-B

2-C1-D

8-E8

8A1-B

2-C1-D

8-E8

8

18

18A1-B

2-C1-D

8-E18

18A1-B

2-C1-D

8-E18

50A1-B

2-C8

-D21-E50

50A1-B

2-C8

-D21-E50

50

59A1-B

2-C4

-D12-E59

59A1-B

2-C4

-D12-E59

59

65A1-B

2-C1-D

4-E6

565

A1-B

2-C1-D

4-E6

565

66

A1-B

2-C4

-D18-E66

66A1-B

2-C4

-D18-E66

66

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

4 Discrete Dynamics in Nature and Society

119889119894119904119886119887 Thedistance betweennational logistics centera and regional logistics center b

119889119894119904119887119888 The distance between regional logistics centerb and city logistics center c

119889119894119904119888119889 The distance between city logistics center cand dealer d

1198891198941199041198891198902 The distance between dealer d and customer1198902F119886 The fixed operating costs of national logisticscenter located at point a

F119887 The fixed operating costs of regional logisticscenter located at point b

F119888 The fixed operating costs of city logistics centerlocated at point c

F119889 The fixed operating costs of dealer located atpoint d

119874119898119886 The average inventory of productm in nationallogistics center located at point a

1198771198981198901 The demand of customer 1198901 for product m1198771198981198902 The demand of customer 1198902 for product mT Unit carbon treatment cost

N A large enough positive number

34 Decision Variables

1198761198981198861198901 The quantity of product m from nationallogistics center a to customer 1198901119876119898119886119887 The quantity of productm from national logis-tics center a to regional logistics center b

119876119898119887119888 The quantity of product m from regional logis-tics center b to city logistics center c

119876119898119888119889 The quantity of product m from city logisticscenter c to dealer d

1198761198981198891198902 The quantity of product m from dealer d tocustomer 1198902119910119886 Whether the national logistics center a operatesif it operates then 119910119886 equals 1 otherwise the value is0

119910119887 Whether the regional logistics center b operatesif it operates then 119910119887 equals 1 otherwise the value is 0119910119888 Whether the city logistics center c operates if itoperates then 119910119888 equals 1 otherwise the value is 0119910119889 Whether the dealer d operates if it operatesthen 119910119889 equals 1 otherwise the value is 0

35 Data Model

min f (x) (1)

f (x) = sumAsumE1sumMGm

ae1Qmae1disae1 +sum

AsumBsumMGm

abQmabdisab

+sumBsumCsumMGm

bcQmbcdisbc +sum

CsumDsumMGm

cdQmcddiscd

+sumDsumE2sumMGm

de2Qmde2disde2 +sum

AFaya +sum

BFbyb

+sumCFcyc +sum

DFdyd +sum

AsumE1sumMTQm

ae1disae1

+sumAsumBsumMTQm

abdisab +sumBsumCsumMTQm

bcdisbc

+sumCsumDsumMTQm

cddiscd +sumDsumE2sumMTQm

de2disde2

(2)

sum119860

1198761198981198861198901 = 1198771198981198901 forall1198901 isin 1198641 forallm isin M (3)

sum119863

1198761198981198891198902 = 1198771198981198902 forall1198902 isin 1198642 forallm isin M (4)

sum119860

119876119898119886119887 ge sum119862

119876119898119887119888 forallb isin B forallm isin M (5)

sum119861

119876119898119887119888 ge sum119863

119876119898119888119889 forallc isin C forallm isin M (6)

sum119862

119876119898119888119889 ge sum1198642

1198761198981198891198902 foralld isin D forallm isin M (7)

sum1198641

1198761198981198861198901 +sum119861

119876119898119886119887 le 119874119898119886 lowast 119910119886 foralla isin A forallm isin M (8)

sum119860

119876119898119886119887 minus 119873 times 119910119887 le 0 forallb isin B forallm isin M (9)

sum119861

119876119898119887119888 minus 119873 times 119910119888 le 0 forallc isin C forallm isin M (10)

sum119862

119876119898119888119889 minus 119873 times 119910119889 le 0 foralld isin D forallm isin M (11)

119910119886119910119887 119910119888 119910119889 isin 0 1foralla isin A forallb isin B forallc isin C foralld isin D

(12)

1198761198981198861198901 119876119898119886119887 119876119898119887119888 119876119898119888119889 1198761198981198891198902 ge 0foralla isin A forallb isin B forallc isin C foralld isin D forall1198901 isin 1198641 forall1198902 isin 1198642

(13)

The objective function (2) aims to minimize the totalcosts of the supply chain network The constraint (3) requiresthat the demands of the customer 1198901 can only be directlydelivered by the national logistics center a and the constraint

Discrete Dynamics in Nature and Society 5

(4) states that the product can only be provided from thedealer d to the customer 1198902 Equations (5)-(7) specify thatthe volume of each logistics center is greater than or equalto the shipment at that point Constraint (8) ensures that theinventory from the national logistics center a is greater thanthe total volume of shipped products and the operation of thenational logistics center a is represented by the 0-1 variableConstraint (9) donates that if the regional logistics center bis not in operation the quantity of product shipped to theregional logistics center b is zero While constraint (10) andconstraint (11) indicate that if city logistics center c and thedealer d do not operate the volume of product shipped to thecity logistics center c and the dealer d is zero Constraint (12)and constraint (13) are the value constraints of the decisionvariable

4 Proposed Heuristic Method

The model of LRP presented in Section 3 aims to minimizethe total costs in product transportation process In a large-scale transportation network the code dimension is highmaking it difficult to find an optimal solution in a limitedtime To solve the problem we resort to heuristic methodssuch as particle swarm optimization (PSO) PSO is a stochas-tic parallel optimization algorithm that does not requirethe properties of the optimized function to be divisiblesteerable and continuous In addition to further improve theaccuracy and efficiency of the algorithm the improved PSOis proposed

41 Particle Swarm Optimization PSO was proposed byKennedy and Eberhart [21] and Lian [22] pointed out thatthis method was a stochastic optimization method based onswarm intelligence and PSOwas inspired by the social behav-ior of bird flocking and their means of information exchangeDue to its easy implementation and fast convergence PSOhas been successfully applied to solve nonlinear optimizationproblems Ai and Kachitvichyanukul [23] Goksal et al [24]and Alinaghian et al [25] designed an improved randomtopology particle swarm optimization algorithm (RT-PSO)for time-dependent vehicle routing problems Hu et al [19]developed a two-stage decomposition algorithm by combin-ing variable neighborhood search algorithm and PSO to solvethe large-scale refrigerated truck scheduling problem

In PSO each particle represents a solution to the opti-mization problem Particle initialization is generated in arandom way and each particle iteratively updates its positionthrough personal best and global best Each particle searchesfor the optimal fitness value in the D-dimensional feasiblesolution spaceThere are N particles in a group In iteration tthe position of particle i is represented as a D-dimensionalvector 119883119905119894119889 = (1199091199051198941 1199091199051198942 119909119905119894119863) 119894 = 1 2 119873 At thesame time the velocity of the particle i is also a D minusdimensional vector denoted as 119881119905119894119889 = (V1199051198941 V1199051198942 V119905119894119863) 119894 =1 2 119873 The optimal position searched by the particle iin iteration t is called the personal best which is denotedas 119875119887119890119904119905 = (1199011198941 1199011198942 119901119894119863) 119894 = 1 2 119873 The optimalposition searched by the group in iteration t is the global best

denoted as 119866119887119890119904119905 = (1199011198921 1199011198922 119901119892119863) After finding the twooptimal values each particle updates its velocity and positionaccording to the following formula

V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)

(14)

119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889 (15)

In the standard PSO there are some shortcomings forexample (1)when solving the high dimensional optimizationproblem the increase of the dimension makes the searchspace expand into an exponential form which is prone topremature convergence That is the particle group aggregatesprematurely to the local optimal solution (2) Particles mayaggregate to local optimum too early in the process of gather-ing If there is no escaping mechanism the optimal solutionmay not be obtained Therefore we proposed improvedstrategy accordingly

42 Strategies for Improving PSO The improved PSO focuseson three aspects (1) facing the ldquodimensional disasterrdquo inlarge-scale problems cooperative particle swarm optimiza-tion (CPSO) is proposed to transform large-scale complexproblems into low-dimensional simple problems (2) Whenparticle flies out of the boundary the boundary processingstrategy is adopted to change the flight trajectory of theparticles (3) When the particles are aggregated in order toprevent premature convergence to the local optimum thevariation factor is introduced to further explore the searchspace

421 CPSO for Dimensionality Reduction As the dimensionof the optimization problem increases the performance ofall algorithm drops dramatically An effective solution isthe cooperative evolution strategy Bergh and Engelbrecht[26] combined cooperative evolution strategy with PSO andproposed cooperative particle swarm optimization (CPSO)The basic idea of CPSO is using K independent particlegroups to search different dimension directions in the D-dimensional search space Each particle swarm is updatedindependently and no information is shared between theparticle swarms When calculating the fitness value theposition vectors of the optimal particles in the particle swarmare combined together to form a D-dimensional vector tocalculate the fitness value

In this paper to calculate the traffic volume betweensupply chain levels the particle coding in standard PSOis 11-dimensional X(ABCD 1198641 1198642m 1198601015840 1198611015840 1198621015840 1198631015840)Thefirst 6 dimensions are the supply chain routing betweenneighboring level the 7th dimension demonstrates differentproducts and the last 4 dimensions represent the nodeoperation decision According to CPSO decision variableis classified by supply chain levels Decision variable canbe split into five types of 3-dimensional routing decisionQ1(A 1198641m)Q2(ABm)Q3(BCm)Q4(CDm)Q5(D 1198642 119898) and four types of 1-dimensional node operationdecision Y1(A)Y2(B)Y3(C)Y4(D) The process is shown

6 Discrete Dynamics in Nature and Society

A

B

C

D

E2 E1

Q2(ABm)

Q3(BCm)

Q4(CDm)

Q5(DE2m)

Q1(AE1m)

Y1(A)

Y2(B)

Y3(C)

Y4(D)

X(ABCDE1E2mA BI I C DI I

A

B

C

D

E2 E1

Drop dimension Splice

)

Figure 2 Collaborative dimension reduction

in Figure 2 Coding dimension is reduced and the algorithmspeed is improved through CPSO

422 Improved Boundary Processing Strategy In the searchprocess in order to improve the convergence efficiencyand reduce the invalid search the boundary conditions areusually set for the position and velocity to narrow the searchrange In the traditional PSO when a particle goes beyond therange the general processing method is to set the particle onthe boundary That is

If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 (16)

Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 (17)

Through this method after several iterations a pluralityof particles will aggregate toward the boundary in a pluralityof dimensions and the flight path of the particles will tendto be the same So the efficiency of the particle group isreduced Thus the following improvement strategies havebeen proposed

If 119883119894119889 lt 119883119898119894119899119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899

(18)

Else if 119883119894119889 gt 119883119898119886119909119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909

(19)

1199033 is a random number distributed between [0 1) and1198883 is a constant on [001 05) The value of 1198883 will changeaccording to the algorithm the objective function etc Thispaper sets it to 005 to enhance the ability of the algorithm tojump out of the boundary region so that the particles returnto the feasible search space After using the boundary strategyon the one hand it can ensure that the particles are located inthe feasible search space On the other hand it can preventthe particles from accumulating too much to the boundaryresulting in local optimization on the boundary of the searchspace It can maximize the search space and improve thequality of the solution A comparison of the two strategies isshown in Figure 3

423 Mutation Operators to Expand Search Space WhenPSO appears as local convergence or global convergencethe particles will have an aggregation phenomenon thatis the particles have the same fitness The fitness varianceis the judgment of the convergence degree of the wholegroup The larger the fitness variance is the more likelythe particle swarm is in the search state When the fitnessvariance becomes smaller the particle swarm is tendingto converge when the fitness variance is 0 PSO achieveslocal optimization or global optimization The formula forcalculating the fitness variance is as follows

1205902 =119898

sum119894=1

(119891119894 minus 119891119886V119891 )2

(20)

And f = max1max1le119894le119873|119891119894minus119891119886V| the function off is to limit 1205902N is the total number of particles in the particle group119891119894 is the fitness value of particle i119891119886V = (1119873)sum119873119894=1 119891 is average fitness function value ofeach particle

If it is local optimum the algorithm is premature Whenthe algorithm is premature it will affect the accuracy of thealgorithm So 119866119887119890119904119905119905119889 will be the personal best The variationfactor Pr is introduced thereby changing the forward direc-tion of the particle So 119866119887119890119904119905119905119889 and 119875119887119890119904119905119905119894119889 will be updated

If Pr gt rand (d) then update particle (21)

The probability of mutation is relatively small Pr gener-ally takes a number between [0 1) in this algorithm Pr =01 rand(d) is a random number distributed between [0 1)The specific approach is to mutate the particles when theyare within 10 of the probability of mutation Firstly all theparticles are sorted according to the fitness value secondlym particles with the best fitness value (m is usually half ofthe total number of particles) are obtained then the formerPr lowastm particles are mutated as follows

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578) (22)

Discrete Dynamics in Nature and Society 7

Feasible solution space

Feasible solution space

Before

After

Figure 3 Improved boundary processing strategy

Table 1 Parameters setting for the experiments

Parameter Distribution type Distributions UnitGm

ae1 Gmab Gm

bc Gmcd Gm

de2 Uniform distribution U(190 250) RMBtonkmFa Fb Fc Fd Uniform distribution U(150000 350000) RMByearOm

a Uniform distribution U(150000 200000) tonR Normal distribution N(120583119901119895 1205902) ton year

where 120578 is a randomvariable obeying the standard normaldistribution N(0 1)The purpose of using themutation factoris to avoid the particle group falling into local optimum andbetter approaching the global optimum When the algorithmappears premature the mutation can change the direction ofthe entire particle swarm and the particle swarm reaches anew field for search By introducing the mutation operatorthe algorithm can expand the search space when the search isstagnant The improved PSO flow is shown in Figure 4 Thepseudocode of improved PSO is presented in Box 1

5 Numerical Experiments

In this section some numerical experiments are performedto verify the effectiveness of the proposed model of multi-product distribution and location problem based on uncer-tain demandnetworksTheprogramming language is C andthe experiments are completed on a PC (Intel Core i7 26GHz Memory 8G)

51 Parameter Setting According to the background of thedistribution network under Omni-Channel integration theparameters of this experiment are set The location androuting process is the same as the model description inSection 3 To demonstrate the broad applicability of themodel some of the parameters of the experiment weregenerated based on a random probability distribution shownin Table 1

The unit transportation cost G is uniformly generatedaccording to different road conditions The operating costsF of dealers for different levels are uniformly generatedaccording to the land price and scale The capacity O ofthe distribution center is uniformly distributed according tothe product category and scale This article deals with theuncertain customer demand as the following two steps

Step 1 Each customer has different demands for differentproducts A set of scenarios with different demands are setaccording to the characteristics of the normal distributionfunction

8 Discrete Dynamics in Nature and Society

Update ０＜ＭＮ and ＜ＭＮ

Output ＜ＭＮ

Start

Initialization

Adjust based on constraints and boundary

Update position and velocity

tgtMaxItr

Variation

End

Yes

Satisfies the constraint

No

Evaluate the particles and calculate the fitness values

t=t+1

Yes

No

Figure 4 Particle swarm optimization flowchart

Step 2 Let different customer demands for different productsmeet different normal distributions This creates a networkof demand of each customer for each product rather thanallowing all customers to meet only one normal distributiondemand for each product which could predict customerdemands more accurately

52 Experimental Results This paper studies multi-productdistribution under the uncertainty of customer demand sothe experiment is divided into four groups according to thechanges of customer demand Each group of experiments isdivided into five different situations according to the numberof suppliers and customers The results in Table 2 show 5experimental scales with different changes of demand Theobjective function value and running time are calculatedby the improved PSO and CPLEX solver ldquo1-1-2-4-10 (2+8)rdquorepresents one national logistics center one regional logisticscenter two city logistics centers four dealers and ten cus-tomers (two of them are first-type customers and eight ofthem are second-type customers)

In small-scale cases the running time of CPLEX is lessthan the improved PSO As the scale increases the running

time of CPLEX increases significantly For example in thefirst three experiments improved PSO and CPLEX havesimilar effects When the scale increased to 85 customersthe running time of CPLEX increased quickly while therunning time of improved PSO is less increased In generalthe results of the CPLEX solver are considered to be theoptimal solution Therefore the CPLEX solver solves themodel faster than the improved PSO method on a smallscale However when the network exceeds a certain scale theimproved PSO gets results faster than the CPLEX solver Inaddition the average deviation between the improved PSOand the optimal result is about 053 which indicates thatthe result is accurate and the proposed algorithm is suitablefor solving the model

6 Case Analysis

Based on the background of the distribution of three typesof products in Chengdu China the scale is ldquo2-4-12-22-85(7+78)rdquo M Company is a typical retail enterprise andproducts transportation is under Omni-Channel integration

Discrete Dynamics in Nature and Society 9

Table 2 Comparison of different kinds of algorithms

CPLEX Improved PSO GapScale of the study Obj CPU(s) Obj CPU(s) [(4)-(2)](2)

(1) (2) (3) (4) (5) (6)

Change in demand =0

1-1-2-4-10 (2+8) 1551183119 995 1551183119 1331 0002-2-4-10-35 (4+31) 4850885922 268 4861980932 2903 0232-2-6-14-55 (5+50) 5181467988 45 5213955899 42 0632-4-12-22-85 (7+78) 5318725918 3041 5363403216 7706 084

3-5-18-70-205 (10+195) 5619609011 gt3600 5672995297 503 095

Change in demand =001

1-1-2-4-10 (2+8) 1554786933 995 1554786933 1171 0002-2-4-10-35 (4+31) 4905668616 2345 4913027119 2857 0152-2-6-14-55 (5+50) 5193458934 49 5224634879 4454 0602-4-12-22-85 (7+78) 5304694312 325 5350314683 8386 086

3-5-18-70-205 (10+195) 5608465011 gt3600 5666793047 524 104

Change in demand =005

1-1-2-4-10 (2+8) 1554468433 99 1554468433 863 0002-2-4-10-35 (4+31) 4902814227 2375 4913110137 2477 0212-2-6-14-55 (5+50) 5220507224 4573 5231902083 3829 0222-4-12-22-85 (7+78) 5342902440 3416 5392591433 768 093

3-5-18-70-205 (10+195) 5621433450 gt3600 5685517791 537 114

Change in demand =01

1-1-2-4-10 (2+8) 1558676503 995 1558676503 1271 0002-2-4-10-35 (4+31) 4928341567 299 4941648089 2794 0272-2-6-14-55 (5+50) 5210996332 5791 5233955899 4836 0442-4-12-22-85 (7+78) 5339691338 3647 5389350467 7474 093

3-5-18-70-205 (10+195) 5620433450 gt3600 5691250911 556 126Average value 053

Table 3 Annual average demand of 3 products for 85 customers

No D1 D2 D3 No D1 D2 D3 No D1 D2 D3 No D1 D2 D31 99 9 12 23 41 27 3 45 24 4 5 67 29 5 12 21 7 1 24 44 1 1 46 14 4 4 68 10 2 23 68 18 3 25 6 5 2 47 4 2 2 69 33 8 14 4 2 2 26 2 0 1 48 15 12 2 70 56 9 25 55 2 1 27 104 23 1 49 19 7 4 71 4 1 36 16 6 3 28 12 4 2 50 2 1 0 72 43 23 27 14 5 1 29 1 1 1 51 9 7 0 73 16 5 28 3 1 0 30 1 1 1 52 4 1 1 74 9 1 19 17 3 1 31 5 2 1 53 16 7 7 75 24 2 310 1 0 0 32 8 1 1 54 14 8 1 76 2 1 111 4 1 7 33 48 9 7 55 22 7 2 77 7 3 212 18 1 6 34 23 15 1 56 4 3 1 78 23 5 213 67 14 7 35 65 8 4 57 6 2 2 79 163 19 114 3 1 2 36 31 15 1 58 1 1 2 80 429 4 115 9 5 1 37 31 3 4 59 4 3 0 81 417 54 116 7 3 1 38 9 1 1 60 13 7 3 82 501 25 117 43 3 15 39 16 2 1 61 13 3 1 83 286 22 418 1 0 7 40 45 11 2 62 12 2 2 84 455 25 219 29 2 3 41 86 8 1 63 45 8 3 85 393 45 220 28 8 4 42 39 5 4 64 39 14 321 38 6 1 43 16 3 1 65 23 6 122 5 2 3 44 6 2 1 66 67 7 1No represents the number of customers D1-D3 represent demand of products 1-3

10 Discrete Dynamics in Nature and Society

InitializationSet the parameters MaxItrNw 1198881 1198882 1199031 r2 c3 1199033 PrInitialize 9 independent particle swarms1198761198981198861198901(a 1198901m) 119876119898119886119887(a bm) 119876119898119887119888(b cm) 119876119898119888119889(c dm) 1198761198981198891198902(d 1198902m)119910119886(a) 119910119887(b) 119910119888(c) 119910119889(d)which are in the 11-dimensional solution space X(a b c d 1198901 1198902m 1198861015840 1198871015840 1198881015840 1198891015840)Define position (x) and velocity (v) for each particleAssume individual best (119875119887119890119904119905119905119894119889) equal to the initialized particlesSelect the best particle as global best (119866119887119890119904119905119905119889)

UpdateLet 9 particle swarms be updated independently as follows

For t = 1 to MaxItrUpdate position (x) and velocity (v)V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032 lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909RespectivelyEndEvaluate the particles and calculate the fitness valuesSort the fitness valuesIf Pr gt rand(d)

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578)EndUpdate personal best (119875119887119890119904119905119905119894119889) of 9 independent particle swarmsUpdate global best (119866119887119890119904119905119905119889) of 9 independent particle swarms

EndReport

Global best (119866119887119890119904119905119905119889) of 9 independent particle swarms119866119887119890119904119905119905119889 found by splicing 9 independent particle swarmsFitness values calculated by 119866119887119890119904119905119905119889

Box 1 The improved PSO

Products include electrical appliances household productsand daily chemical products The average annual demand ofthree products for 85 customers is shown in Table 3 Theirlocation is shown in Figure 5

61 Location Decision Results The locations of the logis-tics center and customers (blue icon) are shown in Fig-ure 5 After the location decision three regional logis-tics centers eight city logistics centers nine dealers maynot operate The non-operating logistics centers are iden-tified by red rectangle Reasons for not operating areanalyzed

Two national logistics centers (numbered 119860 119894 yellowicon) Since the national distribution center needs to dis-tribute to regional logistics centers and large customers onenational distribution centerrsquos capacity is not enough both ofthem need to operate

Four regional logistics centers (numbered 119861119894 purpleicon) Regional logistics center B2 needs to operate B2 is clos-est to the two national logistics centers and is closer to the citylogistics center compared with B1 B3 and B4 In the vicinityof B1 B3 and B4 the city logistics centers (numbered 119862119894)are less distributed and dispersed Considering the operatingcosts B1 B3 and B4 may not operate

Twelve city logistics centers (numbered 119862119894 green icon)City logistics centers C1 C4 C8 and C11 need to oper-ate C2 C9 and C10 are far away from B2 and havehigh transportation costs compared with C4 and C8 sothey are not operated C3 C5 C6 C7 and C12 areremote and scattered and the demand of dealer is smallIt does not operate due to transportation and operatingcosts

Twenty-two dealers (numbered 119863119894 red icon) Throughthe calculation of location and routing 9 dealers do not needto operate The remaining 13 dealers can fulfill the productdistribution demands of 78 second-class customers

The above analyses show that M companyrsquos ldquo2-4-12-22rdquosupply chain network is too large The current demand canbe satisfied by the ldquo2-1-4-13rdquo scale network The location-routing decision can reduce the operations of three regionaldistribution centers eight city distribution centers and ninedealers

62 Transportation Route Decision Results Distributionroute of the three products for 85 customers is shown inTable 4

If distribution routes of the three products are the samethe route decision follows the customer number and the first

Discrete Dynamics in Nature and Society 11

Table4Deliveryrouteo

f3prod

uctsfor8

5custo

mers(To

n)

No

Deliverypath

No

Deliverypath

No

Deliverypath

No

Deliverypath

1A1-B

2-C1-D

8-E1

23A1-B

2-C1-D

7-E2

343

A1-B

2-C1

1-D19-E43

67A1-B

2-C1-D

1-E67

2A1-B

2-C1-D

1-E2

24A1-B

2-C1

1-D2-E2

444

A1-B

2-C1-D

1-E44

68A1-B

2-C8

-D21-E68

3A1-B

2-C1-D

1-E3

25A1-B

2-C1

1-D2-E2

545

A1-B

2-C1

1-D2-E4

569

A1-B

2-C1-D

1-E69

4A1-B

2-C1

1-D19-E4

26A1-B

2-C1-D

1-E26

46A1-B

2-C1-D

8-E4

670

A1-B

2-C1

1-D19-E70

5A1-B

2-C1-D

1-E5

27A1-B

2-C1

1-D2-E2

747

A1-B

2-C1

1-D19-E47

71A1-B

2-C1-D

1-E71

6A1-B

2-C1

1-D19-E6

28A1-B

2-C1

1-D19-E28

48A1-B

2-C1

1-D2-E4

872

A1-B

2-C1-D

1-E72

7A1-B

2-C1-D

10-E7

29A1-B

2-C1-D

8-E2

949

A1-B

2-C1-D

1-E49

73A1-B

2-C1-D

1-E73

9A1-B

2-C1-D

1-E9

30A1-B

2-C1-D

1-E30

51A1-B

2-C8

-D21-E51

74A1-B

2-C1-D

1-E74

10A1-B

2-C1-D

1-E10

31A1-B

2-C1-D

1-E31

52A1-B

2-C1-D

7-E5

275

A1-B

2-C1-D

1-E75

11A1-B

2-C1-D

22-E11

32A1-B

2-C1-D

1-E32

53A1-B

2-C1-D

1-E53

76A1-B

2-C1-D

1-E76

12A1-B

2-C1-D

1-E12

33A1-B

2-C1

1-D2-E3

354

A1-B

2-C1-D

1-E54

77A1-B

2-C1-D

1-E77

13A1-B

2-C8

-D21-E13

34A1-B

2-C1

1-D19-E34

55A1-B

2-C1-D

1-E55

78A1-B

2-C1-D

1-E78

14A1-B

2-C1-D

1-E14

35A1-B

2-C1-D

22-E35

56A1-B

2-C1-D

1-E56

79A1-E

7915

A1-B

2-C1-D

1-E15

36A1-B

2-C1-D

16-E36

57A1-B

2-C1

1-D17-E57

80A2-E8

016

A1-B

2-C1

1-D2-E16

37A1-B

2-C1

1-D2-E3

758

A1-B

2-C1-D

1-E58

81A2-E8

117

A1-B

2-C1-D

22-E17

38A1-B

2-C1

1-D2-E3

860

A1-B

2-C1-D

8-E6

082

A2-E8

219

A1-B

2-C1-D

10-E19

39A1-B

2-C1

1-D19-E39

61A1-B

2-C1-D

1-E61

83A1-E

8320

A1-B

2-C8

-D21-E20

40A1-B

2-C4

-D18-E40

62A1-B

2-C8

-D21-E62

84A2-E8

421

A1-B

2-C8

-D21-E21

41A1-B

2-C4

-D18-E41

63A1-B

2-C1-D

1-E63

85A2-E8

522

A1-B

2-C1-D

8-E2

242

A1-B

2-C1-D

1-E42

64A1-B

2-C1-D

4-E6

48

A1-B

2-C1-D

8-E8

8A1-B

2-C1-D

8-E8

8

18

18A1-B

2-C1-D

8-E18

18A1-B

2-C1-D

8-E18

50A1-B

2-C8

-D21-E50

50A1-B

2-C8

-D21-E50

50

59A1-B

2-C4

-D12-E59

59A1-B

2-C4

-D12-E59

59

65A1-B

2-C1-D

4-E6

565

A1-B

2-C1-D

4-E6

565

66

A1-B

2-C4

-D18-E66

66A1-B

2-C4

-D18-E66

66

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

Discrete Dynamics in Nature and Society 5

(4) states that the product can only be provided from thedealer d to the customer 1198902 Equations (5)-(7) specify thatthe volume of each logistics center is greater than or equalto the shipment at that point Constraint (8) ensures that theinventory from the national logistics center a is greater thanthe total volume of shipped products and the operation of thenational logistics center a is represented by the 0-1 variableConstraint (9) donates that if the regional logistics center bis not in operation the quantity of product shipped to theregional logistics center b is zero While constraint (10) andconstraint (11) indicate that if city logistics center c and thedealer d do not operate the volume of product shipped to thecity logistics center c and the dealer d is zero Constraint (12)and constraint (13) are the value constraints of the decisionvariable

4 Proposed Heuristic Method

The model of LRP presented in Section 3 aims to minimizethe total costs in product transportation process In a large-scale transportation network the code dimension is highmaking it difficult to find an optimal solution in a limitedtime To solve the problem we resort to heuristic methodssuch as particle swarm optimization (PSO) PSO is a stochas-tic parallel optimization algorithm that does not requirethe properties of the optimized function to be divisiblesteerable and continuous In addition to further improve theaccuracy and efficiency of the algorithm the improved PSOis proposed

41 Particle Swarm Optimization PSO was proposed byKennedy and Eberhart [21] and Lian [22] pointed out thatthis method was a stochastic optimization method based onswarm intelligence and PSOwas inspired by the social behav-ior of bird flocking and their means of information exchangeDue to its easy implementation and fast convergence PSOhas been successfully applied to solve nonlinear optimizationproblems Ai and Kachitvichyanukul [23] Goksal et al [24]and Alinaghian et al [25] designed an improved randomtopology particle swarm optimization algorithm (RT-PSO)for time-dependent vehicle routing problems Hu et al [19]developed a two-stage decomposition algorithm by combin-ing variable neighborhood search algorithm and PSO to solvethe large-scale refrigerated truck scheduling problem

In PSO each particle represents a solution to the opti-mization problem Particle initialization is generated in arandom way and each particle iteratively updates its positionthrough personal best and global best Each particle searchesfor the optimal fitness value in the D-dimensional feasiblesolution spaceThere are N particles in a group In iteration tthe position of particle i is represented as a D-dimensionalvector 119883119905119894119889 = (1199091199051198941 1199091199051198942 119909119905119894119863) 119894 = 1 2 119873 At thesame time the velocity of the particle i is also a D minusdimensional vector denoted as 119881119905119894119889 = (V1199051198941 V1199051198942 V119905119894119863) 119894 =1 2 119873 The optimal position searched by the particle iin iteration t is called the personal best which is denotedas 119875119887119890119904119905 = (1199011198941 1199011198942 119901119894119863) 119894 = 1 2 119873 The optimalposition searched by the group in iteration t is the global best

denoted as 119866119887119890119904119905 = (1199011198921 1199011198922 119901119892119863) After finding the twooptimal values each particle updates its velocity and positionaccording to the following formula

V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)

(14)

119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889 (15)

In the standard PSO there are some shortcomings forexample (1)when solving the high dimensional optimizationproblem the increase of the dimension makes the searchspace expand into an exponential form which is prone topremature convergence That is the particle group aggregatesprematurely to the local optimal solution (2) Particles mayaggregate to local optimum too early in the process of gather-ing If there is no escaping mechanism the optimal solutionmay not be obtained Therefore we proposed improvedstrategy accordingly

42 Strategies for Improving PSO The improved PSO focuseson three aspects (1) facing the ldquodimensional disasterrdquo inlarge-scale problems cooperative particle swarm optimiza-tion (CPSO) is proposed to transform large-scale complexproblems into low-dimensional simple problems (2) Whenparticle flies out of the boundary the boundary processingstrategy is adopted to change the flight trajectory of theparticles (3) When the particles are aggregated in order toprevent premature convergence to the local optimum thevariation factor is introduced to further explore the searchspace

421 CPSO for Dimensionality Reduction As the dimensionof the optimization problem increases the performance ofall algorithm drops dramatically An effective solution isthe cooperative evolution strategy Bergh and Engelbrecht[26] combined cooperative evolution strategy with PSO andproposed cooperative particle swarm optimization (CPSO)The basic idea of CPSO is using K independent particlegroups to search different dimension directions in the D-dimensional search space Each particle swarm is updatedindependently and no information is shared between theparticle swarms When calculating the fitness value theposition vectors of the optimal particles in the particle swarmare combined together to form a D-dimensional vector tocalculate the fitness value

In this paper to calculate the traffic volume betweensupply chain levels the particle coding in standard PSOis 11-dimensional X(ABCD 1198641 1198642m 1198601015840 1198611015840 1198621015840 1198631015840)Thefirst 6 dimensions are the supply chain routing betweenneighboring level the 7th dimension demonstrates differentproducts and the last 4 dimensions represent the nodeoperation decision According to CPSO decision variableis classified by supply chain levels Decision variable canbe split into five types of 3-dimensional routing decisionQ1(A 1198641m)Q2(ABm)Q3(BCm)Q4(CDm)Q5(D 1198642 119898) and four types of 1-dimensional node operationdecision Y1(A)Y2(B)Y3(C)Y4(D) The process is shown

6 Discrete Dynamics in Nature and Society

A

B

C

D

E2 E1

Q2(ABm)

Q3(BCm)

Q4(CDm)

Q5(DE2m)

Q1(AE1m)

Y1(A)

Y2(B)

Y3(C)

Y4(D)

X(ABCDE1E2mA BI I C DI I

A

B

C

D

E2 E1

Drop dimension Splice

)

Figure 2 Collaborative dimension reduction

in Figure 2 Coding dimension is reduced and the algorithmspeed is improved through CPSO

422 Improved Boundary Processing Strategy In the searchprocess in order to improve the convergence efficiencyand reduce the invalid search the boundary conditions areusually set for the position and velocity to narrow the searchrange In the traditional PSO when a particle goes beyond therange the general processing method is to set the particle onthe boundary That is

If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 (16)

Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 (17)

Through this method after several iterations a pluralityof particles will aggregate toward the boundary in a pluralityof dimensions and the flight path of the particles will tendto be the same So the efficiency of the particle group isreduced Thus the following improvement strategies havebeen proposed

If 119883119894119889 lt 119883119898119894119899119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899

(18)

Else if 119883119894119889 gt 119883119898119886119909119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909

(19)

1199033 is a random number distributed between [0 1) and1198883 is a constant on [001 05) The value of 1198883 will changeaccording to the algorithm the objective function etc Thispaper sets it to 005 to enhance the ability of the algorithm tojump out of the boundary region so that the particles returnto the feasible search space After using the boundary strategyon the one hand it can ensure that the particles are located inthe feasible search space On the other hand it can preventthe particles from accumulating too much to the boundaryresulting in local optimization on the boundary of the searchspace It can maximize the search space and improve thequality of the solution A comparison of the two strategies isshown in Figure 3

423 Mutation Operators to Expand Search Space WhenPSO appears as local convergence or global convergencethe particles will have an aggregation phenomenon thatis the particles have the same fitness The fitness varianceis the judgment of the convergence degree of the wholegroup The larger the fitness variance is the more likelythe particle swarm is in the search state When the fitnessvariance becomes smaller the particle swarm is tendingto converge when the fitness variance is 0 PSO achieveslocal optimization or global optimization The formula forcalculating the fitness variance is as follows

1205902 =119898

sum119894=1

(119891119894 minus 119891119886V119891 )2

(20)

And f = max1max1le119894le119873|119891119894minus119891119886V| the function off is to limit 1205902N is the total number of particles in the particle group119891119894 is the fitness value of particle i119891119886V = (1119873)sum119873119894=1 119891 is average fitness function value ofeach particle

If it is local optimum the algorithm is premature Whenthe algorithm is premature it will affect the accuracy of thealgorithm So 119866119887119890119904119905119905119889 will be the personal best The variationfactor Pr is introduced thereby changing the forward direc-tion of the particle So 119866119887119890119904119905119905119889 and 119875119887119890119904119905119905119894119889 will be updated

If Pr gt rand (d) then update particle (21)

The probability of mutation is relatively small Pr gener-ally takes a number between [0 1) in this algorithm Pr =01 rand(d) is a random number distributed between [0 1)The specific approach is to mutate the particles when theyare within 10 of the probability of mutation Firstly all theparticles are sorted according to the fitness value secondlym particles with the best fitness value (m is usually half ofthe total number of particles) are obtained then the formerPr lowastm particles are mutated as follows

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578) (22)

Discrete Dynamics in Nature and Society 7

Feasible solution space

Feasible solution space

Before

After

Figure 3 Improved boundary processing strategy

Table 1 Parameters setting for the experiments

Parameter Distribution type Distributions UnitGm

ae1 Gmab Gm

bc Gmcd Gm

de2 Uniform distribution U(190 250) RMBtonkmFa Fb Fc Fd Uniform distribution U(150000 350000) RMByearOm

a Uniform distribution U(150000 200000) tonR Normal distribution N(120583119901119895 1205902) ton year

where 120578 is a randomvariable obeying the standard normaldistribution N(0 1)The purpose of using themutation factoris to avoid the particle group falling into local optimum andbetter approaching the global optimum When the algorithmappears premature the mutation can change the direction ofthe entire particle swarm and the particle swarm reaches anew field for search By introducing the mutation operatorthe algorithm can expand the search space when the search isstagnant The improved PSO flow is shown in Figure 4 Thepseudocode of improved PSO is presented in Box 1

5 Numerical Experiments

In this section some numerical experiments are performedto verify the effectiveness of the proposed model of multi-product distribution and location problem based on uncer-tain demandnetworksTheprogramming language is C andthe experiments are completed on a PC (Intel Core i7 26GHz Memory 8G)

51 Parameter Setting According to the background of thedistribution network under Omni-Channel integration theparameters of this experiment are set The location androuting process is the same as the model description inSection 3 To demonstrate the broad applicability of themodel some of the parameters of the experiment weregenerated based on a random probability distribution shownin Table 1

The unit transportation cost G is uniformly generatedaccording to different road conditions The operating costsF of dealers for different levels are uniformly generatedaccording to the land price and scale The capacity O ofthe distribution center is uniformly distributed according tothe product category and scale This article deals with theuncertain customer demand as the following two steps

Step 1 Each customer has different demands for differentproducts A set of scenarios with different demands are setaccording to the characteristics of the normal distributionfunction

8 Discrete Dynamics in Nature and Society

Update ０＜ＭＮ and ＜ＭＮ

Output ＜ＭＮ

Start

Initialization

Adjust based on constraints and boundary

Update position and velocity

tgtMaxItr

Variation

End

Yes

Satisfies the constraint

No

Evaluate the particles and calculate the fitness values

t=t+1

Yes

No

Figure 4 Particle swarm optimization flowchart

Step 2 Let different customer demands for different productsmeet different normal distributions This creates a networkof demand of each customer for each product rather thanallowing all customers to meet only one normal distributiondemand for each product which could predict customerdemands more accurately

52 Experimental Results This paper studies multi-productdistribution under the uncertainty of customer demand sothe experiment is divided into four groups according to thechanges of customer demand Each group of experiments isdivided into five different situations according to the numberof suppliers and customers The results in Table 2 show 5experimental scales with different changes of demand Theobjective function value and running time are calculatedby the improved PSO and CPLEX solver ldquo1-1-2-4-10 (2+8)rdquorepresents one national logistics center one regional logisticscenter two city logistics centers four dealers and ten cus-tomers (two of them are first-type customers and eight ofthem are second-type customers)

In small-scale cases the running time of CPLEX is lessthan the improved PSO As the scale increases the running

time of CPLEX increases significantly For example in thefirst three experiments improved PSO and CPLEX havesimilar effects When the scale increased to 85 customersthe running time of CPLEX increased quickly while therunning time of improved PSO is less increased In generalthe results of the CPLEX solver are considered to be theoptimal solution Therefore the CPLEX solver solves themodel faster than the improved PSO method on a smallscale However when the network exceeds a certain scale theimproved PSO gets results faster than the CPLEX solver Inaddition the average deviation between the improved PSOand the optimal result is about 053 which indicates thatthe result is accurate and the proposed algorithm is suitablefor solving the model

6 Case Analysis

Based on the background of the distribution of three typesof products in Chengdu China the scale is ldquo2-4-12-22-85(7+78)rdquo M Company is a typical retail enterprise andproducts transportation is under Omni-Channel integration

Discrete Dynamics in Nature and Society 9

Table 2 Comparison of different kinds of algorithms

CPLEX Improved PSO GapScale of the study Obj CPU(s) Obj CPU(s) [(4)-(2)](2)

(1) (2) (3) (4) (5) (6)

Change in demand =0

1-1-2-4-10 (2+8) 1551183119 995 1551183119 1331 0002-2-4-10-35 (4+31) 4850885922 268 4861980932 2903 0232-2-6-14-55 (5+50) 5181467988 45 5213955899 42 0632-4-12-22-85 (7+78) 5318725918 3041 5363403216 7706 084

3-5-18-70-205 (10+195) 5619609011 gt3600 5672995297 503 095

Change in demand =001

1-1-2-4-10 (2+8) 1554786933 995 1554786933 1171 0002-2-4-10-35 (4+31) 4905668616 2345 4913027119 2857 0152-2-6-14-55 (5+50) 5193458934 49 5224634879 4454 0602-4-12-22-85 (7+78) 5304694312 325 5350314683 8386 086

3-5-18-70-205 (10+195) 5608465011 gt3600 5666793047 524 104

Change in demand =005

1-1-2-4-10 (2+8) 1554468433 99 1554468433 863 0002-2-4-10-35 (4+31) 4902814227 2375 4913110137 2477 0212-2-6-14-55 (5+50) 5220507224 4573 5231902083 3829 0222-4-12-22-85 (7+78) 5342902440 3416 5392591433 768 093

3-5-18-70-205 (10+195) 5621433450 gt3600 5685517791 537 114

Change in demand =01

1-1-2-4-10 (2+8) 1558676503 995 1558676503 1271 0002-2-4-10-35 (4+31) 4928341567 299 4941648089 2794 0272-2-6-14-55 (5+50) 5210996332 5791 5233955899 4836 0442-4-12-22-85 (7+78) 5339691338 3647 5389350467 7474 093

3-5-18-70-205 (10+195) 5620433450 gt3600 5691250911 556 126Average value 053

Table 3 Annual average demand of 3 products for 85 customers

No D1 D2 D3 No D1 D2 D3 No D1 D2 D3 No D1 D2 D31 99 9 12 23 41 27 3 45 24 4 5 67 29 5 12 21 7 1 24 44 1 1 46 14 4 4 68 10 2 23 68 18 3 25 6 5 2 47 4 2 2 69 33 8 14 4 2 2 26 2 0 1 48 15 12 2 70 56 9 25 55 2 1 27 104 23 1 49 19 7 4 71 4 1 36 16 6 3 28 12 4 2 50 2 1 0 72 43 23 27 14 5 1 29 1 1 1 51 9 7 0 73 16 5 28 3 1 0 30 1 1 1 52 4 1 1 74 9 1 19 17 3 1 31 5 2 1 53 16 7 7 75 24 2 310 1 0 0 32 8 1 1 54 14 8 1 76 2 1 111 4 1 7 33 48 9 7 55 22 7 2 77 7 3 212 18 1 6 34 23 15 1 56 4 3 1 78 23 5 213 67 14 7 35 65 8 4 57 6 2 2 79 163 19 114 3 1 2 36 31 15 1 58 1 1 2 80 429 4 115 9 5 1 37 31 3 4 59 4 3 0 81 417 54 116 7 3 1 38 9 1 1 60 13 7 3 82 501 25 117 43 3 15 39 16 2 1 61 13 3 1 83 286 22 418 1 0 7 40 45 11 2 62 12 2 2 84 455 25 219 29 2 3 41 86 8 1 63 45 8 3 85 393 45 220 28 8 4 42 39 5 4 64 39 14 321 38 6 1 43 16 3 1 65 23 6 122 5 2 3 44 6 2 1 66 67 7 1No represents the number of customers D1-D3 represent demand of products 1-3

10 Discrete Dynamics in Nature and Society

InitializationSet the parameters MaxItrNw 1198881 1198882 1199031 r2 c3 1199033 PrInitialize 9 independent particle swarms1198761198981198861198901(a 1198901m) 119876119898119886119887(a bm) 119876119898119887119888(b cm) 119876119898119888119889(c dm) 1198761198981198891198902(d 1198902m)119910119886(a) 119910119887(b) 119910119888(c) 119910119889(d)which are in the 11-dimensional solution space X(a b c d 1198901 1198902m 1198861015840 1198871015840 1198881015840 1198891015840)Define position (x) and velocity (v) for each particleAssume individual best (119875119887119890119904119905119905119894119889) equal to the initialized particlesSelect the best particle as global best (119866119887119890119904119905119905119889)

UpdateLet 9 particle swarms be updated independently as follows

For t = 1 to MaxItrUpdate position (x) and velocity (v)V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032 lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909RespectivelyEndEvaluate the particles and calculate the fitness valuesSort the fitness valuesIf Pr gt rand(d)

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578)EndUpdate personal best (119875119887119890119904119905119905119894119889) of 9 independent particle swarmsUpdate global best (119866119887119890119904119905119905119889) of 9 independent particle swarms

EndReport

Global best (119866119887119890119904119905119905119889) of 9 independent particle swarms119866119887119890119904119905119905119889 found by splicing 9 independent particle swarmsFitness values calculated by 119866119887119890119904119905119905119889

Box 1 The improved PSO

Products include electrical appliances household productsand daily chemical products The average annual demand ofthree products for 85 customers is shown in Table 3 Theirlocation is shown in Figure 5

61 Location Decision Results The locations of the logis-tics center and customers (blue icon) are shown in Fig-ure 5 After the location decision three regional logis-tics centers eight city logistics centers nine dealers maynot operate The non-operating logistics centers are iden-tified by red rectangle Reasons for not operating areanalyzed

Two national logistics centers (numbered 119860 119894 yellowicon) Since the national distribution center needs to dis-tribute to regional logistics centers and large customers onenational distribution centerrsquos capacity is not enough both ofthem need to operate

Four regional logistics centers (numbered 119861119894 purpleicon) Regional logistics center B2 needs to operate B2 is clos-est to the two national logistics centers and is closer to the citylogistics center compared with B1 B3 and B4 In the vicinityof B1 B3 and B4 the city logistics centers (numbered 119862119894)are less distributed and dispersed Considering the operatingcosts B1 B3 and B4 may not operate

Twelve city logistics centers (numbered 119862119894 green icon)City logistics centers C1 C4 C8 and C11 need to oper-ate C2 C9 and C10 are far away from B2 and havehigh transportation costs compared with C4 and C8 sothey are not operated C3 C5 C6 C7 and C12 areremote and scattered and the demand of dealer is smallIt does not operate due to transportation and operatingcosts

Twenty-two dealers (numbered 119863119894 red icon) Throughthe calculation of location and routing 9 dealers do not needto operate The remaining 13 dealers can fulfill the productdistribution demands of 78 second-class customers

The above analyses show that M companyrsquos ldquo2-4-12-22rdquosupply chain network is too large The current demand canbe satisfied by the ldquo2-1-4-13rdquo scale network The location-routing decision can reduce the operations of three regionaldistribution centers eight city distribution centers and ninedealers

62 Transportation Route Decision Results Distributionroute of the three products for 85 customers is shown inTable 4

If distribution routes of the three products are the samethe route decision follows the customer number and the first

Discrete Dynamics in Nature and Society 11

Table4Deliveryrouteo

f3prod

uctsfor8

5custo

mers(To

n)

No

Deliverypath

No

Deliverypath

No

Deliverypath

No

Deliverypath

1A1-B

2-C1-D

8-E1

23A1-B

2-C1-D

7-E2

343

A1-B

2-C1

1-D19-E43

67A1-B

2-C1-D

1-E67

2A1-B

2-C1-D

1-E2

24A1-B

2-C1

1-D2-E2

444

A1-B

2-C1-D

1-E44

68A1-B

2-C8

-D21-E68

3A1-B

2-C1-D

1-E3

25A1-B

2-C1

1-D2-E2

545

A1-B

2-C1

1-D2-E4

569

A1-B

2-C1-D

1-E69

4A1-B

2-C1

1-D19-E4

26A1-B

2-C1-D

1-E26

46A1-B

2-C1-D

8-E4

670

A1-B

2-C1

1-D19-E70

5A1-B

2-C1-D

1-E5

27A1-B

2-C1

1-D2-E2

747

A1-B

2-C1

1-D19-E47

71A1-B

2-C1-D

1-E71

6A1-B

2-C1

1-D19-E6

28A1-B

2-C1

1-D19-E28

48A1-B

2-C1

1-D2-E4

872

A1-B

2-C1-D

1-E72

7A1-B

2-C1-D

10-E7

29A1-B

2-C1-D

8-E2

949

A1-B

2-C1-D

1-E49

73A1-B

2-C1-D

1-E73

9A1-B

2-C1-D

1-E9

30A1-B

2-C1-D

1-E30

51A1-B

2-C8

-D21-E51

74A1-B

2-C1-D

1-E74

10A1-B

2-C1-D

1-E10

31A1-B

2-C1-D

1-E31

52A1-B

2-C1-D

7-E5

275

A1-B

2-C1-D

1-E75

11A1-B

2-C1-D

22-E11

32A1-B

2-C1-D

1-E32

53A1-B

2-C1-D

1-E53

76A1-B

2-C1-D

1-E76

12A1-B

2-C1-D

1-E12

33A1-B

2-C1

1-D2-E3

354

A1-B

2-C1-D

1-E54

77A1-B

2-C1-D

1-E77

13A1-B

2-C8

-D21-E13

34A1-B

2-C1

1-D19-E34

55A1-B

2-C1-D

1-E55

78A1-B

2-C1-D

1-E78

14A1-B

2-C1-D

1-E14

35A1-B

2-C1-D

22-E35

56A1-B

2-C1-D

1-E56

79A1-E

7915

A1-B

2-C1-D

1-E15

36A1-B

2-C1-D

16-E36

57A1-B

2-C1

1-D17-E57

80A2-E8

016

A1-B

2-C1

1-D2-E16

37A1-B

2-C1

1-D2-E3

758

A1-B

2-C1-D

1-E58

81A2-E8

117

A1-B

2-C1-D

22-E17

38A1-B

2-C1

1-D2-E3

860

A1-B

2-C1-D

8-E6

082

A2-E8

219

A1-B

2-C1-D

10-E19

39A1-B

2-C1

1-D19-E39

61A1-B

2-C1-D

1-E61

83A1-E

8320

A1-B

2-C8

-D21-E20

40A1-B

2-C4

-D18-E40

62A1-B

2-C8

-D21-E62

84A2-E8

421

A1-B

2-C8

-D21-E21

41A1-B

2-C4

-D18-E41

63A1-B

2-C1-D

1-E63

85A2-E8

522

A1-B

2-C1-D

8-E2

242

A1-B

2-C1-D

1-E42

64A1-B

2-C1-D

4-E6

48

A1-B

2-C1-D

8-E8

8A1-B

2-C1-D

8-E8

8

18

18A1-B

2-C1-D

8-E18

18A1-B

2-C1-D

8-E18

50A1-B

2-C8

-D21-E50

50A1-B

2-C8

-D21-E50

50

59A1-B

2-C4

-D12-E59

59A1-B

2-C4

-D12-E59

59

65A1-B

2-C1-D

4-E6

565

A1-B

2-C1-D

4-E6

565

66

A1-B

2-C4

-D18-E66

66A1-B

2-C4

-D18-E66

66

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

6 Discrete Dynamics in Nature and Society

A

B

C

D

E2 E1

Q2(ABm)

Q3(BCm)

Q4(CDm)

Q5(DE2m)

Q1(AE1m)

Y1(A)

Y2(B)

Y3(C)

Y4(D)

X(ABCDE1E2mA BI I C DI I

A

B

C

D

E2 E1

Drop dimension Splice

)

Figure 2 Collaborative dimension reduction

in Figure 2 Coding dimension is reduced and the algorithmspeed is improved through CPSO

422 Improved Boundary Processing Strategy In the searchprocess in order to improve the convergence efficiencyand reduce the invalid search the boundary conditions areusually set for the position and velocity to narrow the searchrange In the traditional PSO when a particle goes beyond therange the general processing method is to set the particle onthe boundary That is

If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 (16)

Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 (17)

Through this method after several iterations a pluralityof particles will aggregate toward the boundary in a pluralityof dimensions and the flight path of the particles will tendto be the same So the efficiency of the particle group isreduced Thus the following improvement strategies havebeen proposed

If 119883119894119889 lt 119883119898119894119899119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899

(18)

Else if 119883119894119889 gt 119883119898119886119909119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909

(19)

1199033 is a random number distributed between [0 1) and1198883 is a constant on [001 05) The value of 1198883 will changeaccording to the algorithm the objective function etc Thispaper sets it to 005 to enhance the ability of the algorithm tojump out of the boundary region so that the particles returnto the feasible search space After using the boundary strategyon the one hand it can ensure that the particles are located inthe feasible search space On the other hand it can preventthe particles from accumulating too much to the boundaryresulting in local optimization on the boundary of the searchspace It can maximize the search space and improve thequality of the solution A comparison of the two strategies isshown in Figure 3

423 Mutation Operators to Expand Search Space WhenPSO appears as local convergence or global convergencethe particles will have an aggregation phenomenon thatis the particles have the same fitness The fitness varianceis the judgment of the convergence degree of the wholegroup The larger the fitness variance is the more likelythe particle swarm is in the search state When the fitnessvariance becomes smaller the particle swarm is tendingto converge when the fitness variance is 0 PSO achieveslocal optimization or global optimization The formula forcalculating the fitness variance is as follows

1205902 =119898

sum119894=1

(119891119894 minus 119891119886V119891 )2

(20)

And f = max1max1le119894le119873|119891119894minus119891119886V| the function off is to limit 1205902N is the total number of particles in the particle group119891119894 is the fitness value of particle i119891119886V = (1119873)sum119873119894=1 119891 is average fitness function value ofeach particle

If it is local optimum the algorithm is premature Whenthe algorithm is premature it will affect the accuracy of thealgorithm So 119866119887119890119904119905119905119889 will be the personal best The variationfactor Pr is introduced thereby changing the forward direc-tion of the particle So 119866119887119890119904119905119905119889 and 119875119887119890119904119905119905119894119889 will be updated

If Pr gt rand (d) then update particle (21)

The probability of mutation is relatively small Pr gener-ally takes a number between [0 1) in this algorithm Pr =01 rand(d) is a random number distributed between [0 1)The specific approach is to mutate the particles when theyare within 10 of the probability of mutation Firstly all theparticles are sorted according to the fitness value secondlym particles with the best fitness value (m is usually half ofthe total number of particles) are obtained then the formerPr lowastm particles are mutated as follows

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578) (22)

Discrete Dynamics in Nature and Society 7

Feasible solution space

Feasible solution space

Before

After

Figure 3 Improved boundary processing strategy

Table 1 Parameters setting for the experiments

Parameter Distribution type Distributions UnitGm

ae1 Gmab Gm

bc Gmcd Gm

de2 Uniform distribution U(190 250) RMBtonkmFa Fb Fc Fd Uniform distribution U(150000 350000) RMByearOm

a Uniform distribution U(150000 200000) tonR Normal distribution N(120583119901119895 1205902) ton year

where 120578 is a randomvariable obeying the standard normaldistribution N(0 1)The purpose of using themutation factoris to avoid the particle group falling into local optimum andbetter approaching the global optimum When the algorithmappears premature the mutation can change the direction ofthe entire particle swarm and the particle swarm reaches anew field for search By introducing the mutation operatorthe algorithm can expand the search space when the search isstagnant The improved PSO flow is shown in Figure 4 Thepseudocode of improved PSO is presented in Box 1

5 Numerical Experiments

In this section some numerical experiments are performedto verify the effectiveness of the proposed model of multi-product distribution and location problem based on uncer-tain demandnetworksTheprogramming language is C andthe experiments are completed on a PC (Intel Core i7 26GHz Memory 8G)

51 Parameter Setting According to the background of thedistribution network under Omni-Channel integration theparameters of this experiment are set The location androuting process is the same as the model description inSection 3 To demonstrate the broad applicability of themodel some of the parameters of the experiment weregenerated based on a random probability distribution shownin Table 1

The unit transportation cost G is uniformly generatedaccording to different road conditions The operating costsF of dealers for different levels are uniformly generatedaccording to the land price and scale The capacity O ofthe distribution center is uniformly distributed according tothe product category and scale This article deals with theuncertain customer demand as the following two steps

Step 1 Each customer has different demands for differentproducts A set of scenarios with different demands are setaccording to the characteristics of the normal distributionfunction

8 Discrete Dynamics in Nature and Society

Update ０＜ＭＮ and ＜ＭＮ

Output ＜ＭＮ

Start

Initialization

Adjust based on constraints and boundary

Update position and velocity

tgtMaxItr

Variation

End

Yes

Satisfies the constraint

No

Evaluate the particles and calculate the fitness values

t=t+1

Yes

No

Figure 4 Particle swarm optimization flowchart

Step 2 Let different customer demands for different productsmeet different normal distributions This creates a networkof demand of each customer for each product rather thanallowing all customers to meet only one normal distributiondemand for each product which could predict customerdemands more accurately

52 Experimental Results This paper studies multi-productdistribution under the uncertainty of customer demand sothe experiment is divided into four groups according to thechanges of customer demand Each group of experiments isdivided into five different situations according to the numberof suppliers and customers The results in Table 2 show 5experimental scales with different changes of demand Theobjective function value and running time are calculatedby the improved PSO and CPLEX solver ldquo1-1-2-4-10 (2+8)rdquorepresents one national logistics center one regional logisticscenter two city logistics centers four dealers and ten cus-tomers (two of them are first-type customers and eight ofthem are second-type customers)

In small-scale cases the running time of CPLEX is lessthan the improved PSO As the scale increases the running

time of CPLEX increases significantly For example in thefirst three experiments improved PSO and CPLEX havesimilar effects When the scale increased to 85 customersthe running time of CPLEX increased quickly while therunning time of improved PSO is less increased In generalthe results of the CPLEX solver are considered to be theoptimal solution Therefore the CPLEX solver solves themodel faster than the improved PSO method on a smallscale However when the network exceeds a certain scale theimproved PSO gets results faster than the CPLEX solver Inaddition the average deviation between the improved PSOand the optimal result is about 053 which indicates thatthe result is accurate and the proposed algorithm is suitablefor solving the model

6 Case Analysis

Based on the background of the distribution of three typesof products in Chengdu China the scale is ldquo2-4-12-22-85(7+78)rdquo M Company is a typical retail enterprise andproducts transportation is under Omni-Channel integration

Discrete Dynamics in Nature and Society 9

Table 2 Comparison of different kinds of algorithms

CPLEX Improved PSO GapScale of the study Obj CPU(s) Obj CPU(s) [(4)-(2)](2)

(1) (2) (3) (4) (5) (6)

Change in demand =0

1-1-2-4-10 (2+8) 1551183119 995 1551183119 1331 0002-2-4-10-35 (4+31) 4850885922 268 4861980932 2903 0232-2-6-14-55 (5+50) 5181467988 45 5213955899 42 0632-4-12-22-85 (7+78) 5318725918 3041 5363403216 7706 084

3-5-18-70-205 (10+195) 5619609011 gt3600 5672995297 503 095

Change in demand =001

1-1-2-4-10 (2+8) 1554786933 995 1554786933 1171 0002-2-4-10-35 (4+31) 4905668616 2345 4913027119 2857 0152-2-6-14-55 (5+50) 5193458934 49 5224634879 4454 0602-4-12-22-85 (7+78) 5304694312 325 5350314683 8386 086

3-5-18-70-205 (10+195) 5608465011 gt3600 5666793047 524 104

Change in demand =005

1-1-2-4-10 (2+8) 1554468433 99 1554468433 863 0002-2-4-10-35 (4+31) 4902814227 2375 4913110137 2477 0212-2-6-14-55 (5+50) 5220507224 4573 5231902083 3829 0222-4-12-22-85 (7+78) 5342902440 3416 5392591433 768 093

3-5-18-70-205 (10+195) 5621433450 gt3600 5685517791 537 114

Change in demand =01

1-1-2-4-10 (2+8) 1558676503 995 1558676503 1271 0002-2-4-10-35 (4+31) 4928341567 299 4941648089 2794 0272-2-6-14-55 (5+50) 5210996332 5791 5233955899 4836 0442-4-12-22-85 (7+78) 5339691338 3647 5389350467 7474 093

3-5-18-70-205 (10+195) 5620433450 gt3600 5691250911 556 126Average value 053

Table 3 Annual average demand of 3 products for 85 customers

No D1 D2 D3 No D1 D2 D3 No D1 D2 D3 No D1 D2 D31 99 9 12 23 41 27 3 45 24 4 5 67 29 5 12 21 7 1 24 44 1 1 46 14 4 4 68 10 2 23 68 18 3 25 6 5 2 47 4 2 2 69 33 8 14 4 2 2 26 2 0 1 48 15 12 2 70 56 9 25 55 2 1 27 104 23 1 49 19 7 4 71 4 1 36 16 6 3 28 12 4 2 50 2 1 0 72 43 23 27 14 5 1 29 1 1 1 51 9 7 0 73 16 5 28 3 1 0 30 1 1 1 52 4 1 1 74 9 1 19 17 3 1 31 5 2 1 53 16 7 7 75 24 2 310 1 0 0 32 8 1 1 54 14 8 1 76 2 1 111 4 1 7 33 48 9 7 55 22 7 2 77 7 3 212 18 1 6 34 23 15 1 56 4 3 1 78 23 5 213 67 14 7 35 65 8 4 57 6 2 2 79 163 19 114 3 1 2 36 31 15 1 58 1 1 2 80 429 4 115 9 5 1 37 31 3 4 59 4 3 0 81 417 54 116 7 3 1 38 9 1 1 60 13 7 3 82 501 25 117 43 3 15 39 16 2 1 61 13 3 1 83 286 22 418 1 0 7 40 45 11 2 62 12 2 2 84 455 25 219 29 2 3 41 86 8 1 63 45 8 3 85 393 45 220 28 8 4 42 39 5 4 64 39 14 321 38 6 1 43 16 3 1 65 23 6 122 5 2 3 44 6 2 1 66 67 7 1No represents the number of customers D1-D3 represent demand of products 1-3

10 Discrete Dynamics in Nature and Society

InitializationSet the parameters MaxItrNw 1198881 1198882 1199031 r2 c3 1199033 PrInitialize 9 independent particle swarms1198761198981198861198901(a 1198901m) 119876119898119886119887(a bm) 119876119898119887119888(b cm) 119876119898119888119889(c dm) 1198761198981198891198902(d 1198902m)119910119886(a) 119910119887(b) 119910119888(c) 119910119889(d)which are in the 11-dimensional solution space X(a b c d 1198901 1198902m 1198861015840 1198871015840 1198881015840 1198891015840)Define position (x) and velocity (v) for each particleAssume individual best (119875119887119890119904119905119905119894119889) equal to the initialized particlesSelect the best particle as global best (119866119887119890119904119905119905119889)

UpdateLet 9 particle swarms be updated independently as follows

For t = 1 to MaxItrUpdate position (x) and velocity (v)V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032 lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909RespectivelyEndEvaluate the particles and calculate the fitness valuesSort the fitness valuesIf Pr gt rand(d)

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578)EndUpdate personal best (119875119887119890119904119905119905119894119889) of 9 independent particle swarmsUpdate global best (119866119887119890119904119905119905119889) of 9 independent particle swarms

EndReport

Global best (119866119887119890119904119905119905119889) of 9 independent particle swarms119866119887119890119904119905119905119889 found by splicing 9 independent particle swarmsFitness values calculated by 119866119887119890119904119905119905119889

Box 1 The improved PSO

Products include electrical appliances household productsand daily chemical products The average annual demand ofthree products for 85 customers is shown in Table 3 Theirlocation is shown in Figure 5

61 Location Decision Results The locations of the logis-tics center and customers (blue icon) are shown in Fig-ure 5 After the location decision three regional logis-tics centers eight city logistics centers nine dealers maynot operate The non-operating logistics centers are iden-tified by red rectangle Reasons for not operating areanalyzed

Two national logistics centers (numbered 119860 119894 yellowicon) Since the national distribution center needs to dis-tribute to regional logistics centers and large customers onenational distribution centerrsquos capacity is not enough both ofthem need to operate

Four regional logistics centers (numbered 119861119894 purpleicon) Regional logistics center B2 needs to operate B2 is clos-est to the two national logistics centers and is closer to the citylogistics center compared with B1 B3 and B4 In the vicinityof B1 B3 and B4 the city logistics centers (numbered 119862119894)are less distributed and dispersed Considering the operatingcosts B1 B3 and B4 may not operate

Twelve city logistics centers (numbered 119862119894 green icon)City logistics centers C1 C4 C8 and C11 need to oper-ate C2 C9 and C10 are far away from B2 and havehigh transportation costs compared with C4 and C8 sothey are not operated C3 C5 C6 C7 and C12 areremote and scattered and the demand of dealer is smallIt does not operate due to transportation and operatingcosts

Twenty-two dealers (numbered 119863119894 red icon) Throughthe calculation of location and routing 9 dealers do not needto operate The remaining 13 dealers can fulfill the productdistribution demands of 78 second-class customers

The above analyses show that M companyrsquos ldquo2-4-12-22rdquosupply chain network is too large The current demand canbe satisfied by the ldquo2-1-4-13rdquo scale network The location-routing decision can reduce the operations of three regionaldistribution centers eight city distribution centers and ninedealers

62 Transportation Route Decision Results Distributionroute of the three products for 85 customers is shown inTable 4

If distribution routes of the three products are the samethe route decision follows the customer number and the first

Discrete Dynamics in Nature and Society 11

Table4Deliveryrouteo

f3prod

uctsfor8

5custo

mers(To

n)

No

Deliverypath

No

Deliverypath

No

Deliverypath

No

Deliverypath

1A1-B

2-C1-D

8-E1

23A1-B

2-C1-D

7-E2

343

A1-B

2-C1

1-D19-E43

67A1-B

2-C1-D

1-E67

2A1-B

2-C1-D

1-E2

24A1-B

2-C1

1-D2-E2

444

A1-B

2-C1-D

1-E44

68A1-B

2-C8

-D21-E68

3A1-B

2-C1-D

1-E3

25A1-B

2-C1

1-D2-E2

545

A1-B

2-C1

1-D2-E4

569

A1-B

2-C1-D

1-E69

4A1-B

2-C1

1-D19-E4

26A1-B

2-C1-D

1-E26

46A1-B

2-C1-D

8-E4

670

A1-B

2-C1

1-D19-E70

5A1-B

2-C1-D

1-E5

27A1-B

2-C1

1-D2-E2

747

A1-B

2-C1

1-D19-E47

71A1-B

2-C1-D

1-E71

6A1-B

2-C1

1-D19-E6

28A1-B

2-C1

1-D19-E28

48A1-B

2-C1

1-D2-E4

872

A1-B

2-C1-D

1-E72

7A1-B

2-C1-D

10-E7

29A1-B

2-C1-D

8-E2

949

A1-B

2-C1-D

1-E49

73A1-B

2-C1-D

1-E73

9A1-B

2-C1-D

1-E9

30A1-B

2-C1-D

1-E30

51A1-B

2-C8

-D21-E51

74A1-B

2-C1-D

1-E74

10A1-B

2-C1-D

1-E10

31A1-B

2-C1-D

1-E31

52A1-B

2-C1-D

7-E5

275

A1-B

2-C1-D

1-E75

11A1-B

2-C1-D

22-E11

32A1-B

2-C1-D

1-E32

53A1-B

2-C1-D

1-E53

76A1-B

2-C1-D

1-E76

12A1-B

2-C1-D

1-E12

33A1-B

2-C1

1-D2-E3

354

A1-B

2-C1-D

1-E54

77A1-B

2-C1-D

1-E77

13A1-B

2-C8

-D21-E13

34A1-B

2-C1

1-D19-E34

55A1-B

2-C1-D

1-E55

78A1-B

2-C1-D

1-E78

14A1-B

2-C1-D

1-E14

35A1-B

2-C1-D

22-E35

56A1-B

2-C1-D

1-E56

79A1-E

7915

A1-B

2-C1-D

1-E15

36A1-B

2-C1-D

16-E36

57A1-B

2-C1

1-D17-E57

80A2-E8

016

A1-B

2-C1

1-D2-E16

37A1-B

2-C1

1-D2-E3

758

A1-B

2-C1-D

1-E58

81A2-E8

117

A1-B

2-C1-D

22-E17

38A1-B

2-C1

1-D2-E3

860

A1-B

2-C1-D

8-E6

082

A2-E8

219

A1-B

2-C1-D

10-E19

39A1-B

2-C1

1-D19-E39

61A1-B

2-C1-D

1-E61

83A1-E

8320

A1-B

2-C8

-D21-E20

40A1-B

2-C4

-D18-E40

62A1-B

2-C8

-D21-E62

84A2-E8

421

A1-B

2-C8

-D21-E21

41A1-B

2-C4

-D18-E41

63A1-B

2-C1-D

1-E63

85A2-E8

522

A1-B

2-C1-D

8-E2

242

A1-B

2-C1-D

1-E42

64A1-B

2-C1-D

4-E6

48

A1-B

2-C1-D

8-E8

8A1-B

2-C1-D

8-E8

8

18

18A1-B

2-C1-D

8-E18

18A1-B

2-C1-D

8-E18

50A1-B

2-C8

-D21-E50

50A1-B

2-C8

-D21-E50

50

59A1-B

2-C4

-D12-E59

59A1-B

2-C4

-D12-E59

59

65A1-B

2-C1-D

4-E6

565

A1-B

2-C1-D

4-E6

565

66

A1-B

2-C4

-D18-E66

66A1-B

2-C4

-D18-E66

66

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

Discrete Dynamics in Nature and Society 7

Feasible solution space

Feasible solution space

Before

After

Figure 3 Improved boundary processing strategy

Table 1 Parameters setting for the experiments

Parameter Distribution type Distributions UnitGm

ae1 Gmab Gm

bc Gmcd Gm

de2 Uniform distribution U(190 250) RMBtonkmFa Fb Fc Fd Uniform distribution U(150000 350000) RMByearOm

a Uniform distribution U(150000 200000) tonR Normal distribution N(120583119901119895 1205902) ton year

where 120578 is a randomvariable obeying the standard normaldistribution N(0 1)The purpose of using themutation factoris to avoid the particle group falling into local optimum andbetter approaching the global optimum When the algorithmappears premature the mutation can change the direction ofthe entire particle swarm and the particle swarm reaches anew field for search By introducing the mutation operatorthe algorithm can expand the search space when the search isstagnant The improved PSO flow is shown in Figure 4 Thepseudocode of improved PSO is presented in Box 1

5 Numerical Experiments

In this section some numerical experiments are performedto verify the effectiveness of the proposed model of multi-product distribution and location problem based on uncer-tain demandnetworksTheprogramming language is C andthe experiments are completed on a PC (Intel Core i7 26GHz Memory 8G)

51 Parameter Setting According to the background of thedistribution network under Omni-Channel integration theparameters of this experiment are set The location androuting process is the same as the model description inSection 3 To demonstrate the broad applicability of themodel some of the parameters of the experiment weregenerated based on a random probability distribution shownin Table 1

The unit transportation cost G is uniformly generatedaccording to different road conditions The operating costsF of dealers for different levels are uniformly generatedaccording to the land price and scale The capacity O ofthe distribution center is uniformly distributed according tothe product category and scale This article deals with theuncertain customer demand as the following two steps

Step 1 Each customer has different demands for differentproducts A set of scenarios with different demands are setaccording to the characteristics of the normal distributionfunction

8 Discrete Dynamics in Nature and Society

Update ０＜ＭＮ and ＜ＭＮ

Output ＜ＭＮ

Start

Initialization

Adjust based on constraints and boundary

Update position and velocity

tgtMaxItr

Variation

End

Yes

Satisfies the constraint

No

Evaluate the particles and calculate the fitness values

t=t+1

Yes

No

Figure 4 Particle swarm optimization flowchart

Step 2 Let different customer demands for different productsmeet different normal distributions This creates a networkof demand of each customer for each product rather thanallowing all customers to meet only one normal distributiondemand for each product which could predict customerdemands more accurately

52 Experimental Results This paper studies multi-productdistribution under the uncertainty of customer demand sothe experiment is divided into four groups according to thechanges of customer demand Each group of experiments isdivided into five different situations according to the numberof suppliers and customers The results in Table 2 show 5experimental scales with different changes of demand Theobjective function value and running time are calculatedby the improved PSO and CPLEX solver ldquo1-1-2-4-10 (2+8)rdquorepresents one national logistics center one regional logisticscenter two city logistics centers four dealers and ten cus-tomers (two of them are first-type customers and eight ofthem are second-type customers)

In small-scale cases the running time of CPLEX is lessthan the improved PSO As the scale increases the running

time of CPLEX increases significantly For example in thefirst three experiments improved PSO and CPLEX havesimilar effects When the scale increased to 85 customersthe running time of CPLEX increased quickly while therunning time of improved PSO is less increased In generalthe results of the CPLEX solver are considered to be theoptimal solution Therefore the CPLEX solver solves themodel faster than the improved PSO method on a smallscale However when the network exceeds a certain scale theimproved PSO gets results faster than the CPLEX solver Inaddition the average deviation between the improved PSOand the optimal result is about 053 which indicates thatthe result is accurate and the proposed algorithm is suitablefor solving the model

6 Case Analysis

Based on the background of the distribution of three typesof products in Chengdu China the scale is ldquo2-4-12-22-85(7+78)rdquo M Company is a typical retail enterprise andproducts transportation is under Omni-Channel integration

Discrete Dynamics in Nature and Society 9

Table 2 Comparison of different kinds of algorithms

CPLEX Improved PSO GapScale of the study Obj CPU(s) Obj CPU(s) [(4)-(2)](2)

(1) (2) (3) (4) (5) (6)

Change in demand =0

1-1-2-4-10 (2+8) 1551183119 995 1551183119 1331 0002-2-4-10-35 (4+31) 4850885922 268 4861980932 2903 0232-2-6-14-55 (5+50) 5181467988 45 5213955899 42 0632-4-12-22-85 (7+78) 5318725918 3041 5363403216 7706 084

3-5-18-70-205 (10+195) 5619609011 gt3600 5672995297 503 095

Change in demand =001

1-1-2-4-10 (2+8) 1554786933 995 1554786933 1171 0002-2-4-10-35 (4+31) 4905668616 2345 4913027119 2857 0152-2-6-14-55 (5+50) 5193458934 49 5224634879 4454 0602-4-12-22-85 (7+78) 5304694312 325 5350314683 8386 086

3-5-18-70-205 (10+195) 5608465011 gt3600 5666793047 524 104

Change in demand =005

1-1-2-4-10 (2+8) 1554468433 99 1554468433 863 0002-2-4-10-35 (4+31) 4902814227 2375 4913110137 2477 0212-2-6-14-55 (5+50) 5220507224 4573 5231902083 3829 0222-4-12-22-85 (7+78) 5342902440 3416 5392591433 768 093

3-5-18-70-205 (10+195) 5621433450 gt3600 5685517791 537 114

Change in demand =01

1-1-2-4-10 (2+8) 1558676503 995 1558676503 1271 0002-2-4-10-35 (4+31) 4928341567 299 4941648089 2794 0272-2-6-14-55 (5+50) 5210996332 5791 5233955899 4836 0442-4-12-22-85 (7+78) 5339691338 3647 5389350467 7474 093

3-5-18-70-205 (10+195) 5620433450 gt3600 5691250911 556 126Average value 053

Table 3 Annual average demand of 3 products for 85 customers

No D1 D2 D3 No D1 D2 D3 No D1 D2 D3 No D1 D2 D31 99 9 12 23 41 27 3 45 24 4 5 67 29 5 12 21 7 1 24 44 1 1 46 14 4 4 68 10 2 23 68 18 3 25 6 5 2 47 4 2 2 69 33 8 14 4 2 2 26 2 0 1 48 15 12 2 70 56 9 25 55 2 1 27 104 23 1 49 19 7 4 71 4 1 36 16 6 3 28 12 4 2 50 2 1 0 72 43 23 27 14 5 1 29 1 1 1 51 9 7 0 73 16 5 28 3 1 0 30 1 1 1 52 4 1 1 74 9 1 19 17 3 1 31 5 2 1 53 16 7 7 75 24 2 310 1 0 0 32 8 1 1 54 14 8 1 76 2 1 111 4 1 7 33 48 9 7 55 22 7 2 77 7 3 212 18 1 6 34 23 15 1 56 4 3 1 78 23 5 213 67 14 7 35 65 8 4 57 6 2 2 79 163 19 114 3 1 2 36 31 15 1 58 1 1 2 80 429 4 115 9 5 1 37 31 3 4 59 4 3 0 81 417 54 116 7 3 1 38 9 1 1 60 13 7 3 82 501 25 117 43 3 15 39 16 2 1 61 13 3 1 83 286 22 418 1 0 7 40 45 11 2 62 12 2 2 84 455 25 219 29 2 3 41 86 8 1 63 45 8 3 85 393 45 220 28 8 4 42 39 5 4 64 39 14 321 38 6 1 43 16 3 1 65 23 6 122 5 2 3 44 6 2 1 66 67 7 1No represents the number of customers D1-D3 represent demand of products 1-3

10 Discrete Dynamics in Nature and Society

InitializationSet the parameters MaxItrNw 1198881 1198882 1199031 r2 c3 1199033 PrInitialize 9 independent particle swarms1198761198981198861198901(a 1198901m) 119876119898119886119887(a bm) 119876119898119887119888(b cm) 119876119898119888119889(c dm) 1198761198981198891198902(d 1198902m)119910119886(a) 119910119887(b) 119910119888(c) 119910119889(d)which are in the 11-dimensional solution space X(a b c d 1198901 1198902m 1198861015840 1198871015840 1198881015840 1198891015840)Define position (x) and velocity (v) for each particleAssume individual best (119875119887119890119904119905119905119894119889) equal to the initialized particlesSelect the best particle as global best (119866119887119890119904119905119905119889)

UpdateLet 9 particle swarms be updated independently as follows

For t = 1 to MaxItrUpdate position (x) and velocity (v)V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032 lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909RespectivelyEndEvaluate the particles and calculate the fitness valuesSort the fitness valuesIf Pr gt rand(d)

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578)EndUpdate personal best (119875119887119890119904119905119905119894119889) of 9 independent particle swarmsUpdate global best (119866119887119890119904119905119905119889) of 9 independent particle swarms

EndReport

Global best (119866119887119890119904119905119905119889) of 9 independent particle swarms119866119887119890119904119905119905119889 found by splicing 9 independent particle swarmsFitness values calculated by 119866119887119890119904119905119905119889

Box 1 The improved PSO

Products include electrical appliances household productsand daily chemical products The average annual demand ofthree products for 85 customers is shown in Table 3 Theirlocation is shown in Figure 5

61 Location Decision Results The locations of the logis-tics center and customers (blue icon) are shown in Fig-ure 5 After the location decision three regional logis-tics centers eight city logistics centers nine dealers maynot operate The non-operating logistics centers are iden-tified by red rectangle Reasons for not operating areanalyzed

Two national logistics centers (numbered 119860 119894 yellowicon) Since the national distribution center needs to dis-tribute to regional logistics centers and large customers onenational distribution centerrsquos capacity is not enough both ofthem need to operate

Four regional logistics centers (numbered 119861119894 purpleicon) Regional logistics center B2 needs to operate B2 is clos-est to the two national logistics centers and is closer to the citylogistics center compared with B1 B3 and B4 In the vicinityof B1 B3 and B4 the city logistics centers (numbered 119862119894)are less distributed and dispersed Considering the operatingcosts B1 B3 and B4 may not operate

Twelve city logistics centers (numbered 119862119894 green icon)City logistics centers C1 C4 C8 and C11 need to oper-ate C2 C9 and C10 are far away from B2 and havehigh transportation costs compared with C4 and C8 sothey are not operated C3 C5 C6 C7 and C12 areremote and scattered and the demand of dealer is smallIt does not operate due to transportation and operatingcosts

Twenty-two dealers (numbered 119863119894 red icon) Throughthe calculation of location and routing 9 dealers do not needto operate The remaining 13 dealers can fulfill the productdistribution demands of 78 second-class customers

The above analyses show that M companyrsquos ldquo2-4-12-22rdquosupply chain network is too large The current demand canbe satisfied by the ldquo2-1-4-13rdquo scale network The location-routing decision can reduce the operations of three regionaldistribution centers eight city distribution centers and ninedealers

62 Transportation Route Decision Results Distributionroute of the three products for 85 customers is shown inTable 4

If distribution routes of the three products are the samethe route decision follows the customer number and the first

Discrete Dynamics in Nature and Society 11

Table4Deliveryrouteo

f3prod

uctsfor8

5custo

mers(To

n)

No

Deliverypath

No

Deliverypath

No

Deliverypath

No

Deliverypath

1A1-B

2-C1-D

8-E1

23A1-B

2-C1-D

7-E2

343

A1-B

2-C1

1-D19-E43

67A1-B

2-C1-D

1-E67

2A1-B

2-C1-D

1-E2

24A1-B

2-C1

1-D2-E2

444

A1-B

2-C1-D

1-E44

68A1-B

2-C8

-D21-E68

3A1-B

2-C1-D

1-E3

25A1-B

2-C1

1-D2-E2

545

A1-B

2-C1

1-D2-E4

569

A1-B

2-C1-D

1-E69

4A1-B

2-C1

1-D19-E4

26A1-B

2-C1-D

1-E26

46A1-B

2-C1-D

8-E4

670

A1-B

2-C1

1-D19-E70

5A1-B

2-C1-D

1-E5

27A1-B

2-C1

1-D2-E2

747

A1-B

2-C1

1-D19-E47

71A1-B

2-C1-D

1-E71

6A1-B

2-C1

1-D19-E6

28A1-B

2-C1

1-D19-E28

48A1-B

2-C1

1-D2-E4

872

A1-B

2-C1-D

1-E72

7A1-B

2-C1-D

10-E7

29A1-B

2-C1-D

8-E2

949

A1-B

2-C1-D

1-E49

73A1-B

2-C1-D

1-E73

9A1-B

2-C1-D

1-E9

30A1-B

2-C1-D

1-E30

51A1-B

2-C8

-D21-E51

74A1-B

2-C1-D

1-E74

10A1-B

2-C1-D

1-E10

31A1-B

2-C1-D

1-E31

52A1-B

2-C1-D

7-E5

275

A1-B

2-C1-D

1-E75

11A1-B

2-C1-D

22-E11

32A1-B

2-C1-D

1-E32

53A1-B

2-C1-D

1-E53

76A1-B

2-C1-D

1-E76

12A1-B

2-C1-D

1-E12

33A1-B

2-C1

1-D2-E3

354

A1-B

2-C1-D

1-E54

77A1-B

2-C1-D

1-E77

13A1-B

2-C8

-D21-E13

34A1-B

2-C1

1-D19-E34

55A1-B

2-C1-D

1-E55

78A1-B

2-C1-D

1-E78

14A1-B

2-C1-D

1-E14

35A1-B

2-C1-D

22-E35

56A1-B

2-C1-D

1-E56

79A1-E

7915

A1-B

2-C1-D

1-E15

36A1-B

2-C1-D

16-E36

57A1-B

2-C1

1-D17-E57

80A2-E8

016

A1-B

2-C1

1-D2-E16

37A1-B

2-C1

1-D2-E3

758

A1-B

2-C1-D

1-E58

81A2-E8

117

A1-B

2-C1-D

22-E17

38A1-B

2-C1

1-D2-E3

860

A1-B

2-C1-D

8-E6

082

A2-E8

219

A1-B

2-C1-D

10-E19

39A1-B

2-C1

1-D19-E39

61A1-B

2-C1-D

1-E61

83A1-E

8320

A1-B

2-C8

-D21-E20

40A1-B

2-C4

-D18-E40

62A1-B

2-C8

-D21-E62

84A2-E8

421

A1-B

2-C8

-D21-E21

41A1-B

2-C4

-D18-E41

63A1-B

2-C1-D

1-E63

85A2-E8

522

A1-B

2-C1-D

8-E2

242

A1-B

2-C1-D

1-E42

64A1-B

2-C1-D

4-E6

48

A1-B

2-C1-D

8-E8

8A1-B

2-C1-D

8-E8

8

18

18A1-B

2-C1-D

8-E18

18A1-B

2-C1-D

8-E18

50A1-B

2-C8

-D21-E50

50A1-B

2-C8

-D21-E50

50

59A1-B

2-C4

-D12-E59

59A1-B

2-C4

-D12-E59

59

65A1-B

2-C1-D

4-E6

565

A1-B

2-C1-D

4-E6

565

66

A1-B

2-C4

-D18-E66

66A1-B

2-C4

-D18-E66

66

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

8 Discrete Dynamics in Nature and Society

Update ０＜ＭＮ and ＜ＭＮ

Output ＜ＭＮ

Start

Initialization

Adjust based on constraints and boundary

Update position and velocity

tgtMaxItr

Variation

End

Yes

Satisfies the constraint

No

Evaluate the particles and calculate the fitness values

t=t+1

Yes

No

Figure 4 Particle swarm optimization flowchart

Step 2 Let different customer demands for different productsmeet different normal distributions This creates a networkof demand of each customer for each product rather thanallowing all customers to meet only one normal distributiondemand for each product which could predict customerdemands more accurately

52 Experimental Results This paper studies multi-productdistribution under the uncertainty of customer demand sothe experiment is divided into four groups according to thechanges of customer demand Each group of experiments isdivided into five different situations according to the numberof suppliers and customers The results in Table 2 show 5experimental scales with different changes of demand Theobjective function value and running time are calculatedby the improved PSO and CPLEX solver ldquo1-1-2-4-10 (2+8)rdquorepresents one national logistics center one regional logisticscenter two city logistics centers four dealers and ten cus-tomers (two of them are first-type customers and eight ofthem are second-type customers)

In small-scale cases the running time of CPLEX is lessthan the improved PSO As the scale increases the running

time of CPLEX increases significantly For example in thefirst three experiments improved PSO and CPLEX havesimilar effects When the scale increased to 85 customersthe running time of CPLEX increased quickly while therunning time of improved PSO is less increased In generalthe results of the CPLEX solver are considered to be theoptimal solution Therefore the CPLEX solver solves themodel faster than the improved PSO method on a smallscale However when the network exceeds a certain scale theimproved PSO gets results faster than the CPLEX solver Inaddition the average deviation between the improved PSOand the optimal result is about 053 which indicates thatthe result is accurate and the proposed algorithm is suitablefor solving the model

6 Case Analysis

Based on the background of the distribution of three typesof products in Chengdu China the scale is ldquo2-4-12-22-85(7+78)rdquo M Company is a typical retail enterprise andproducts transportation is under Omni-Channel integration

Discrete Dynamics in Nature and Society 9

Table 2 Comparison of different kinds of algorithms

CPLEX Improved PSO GapScale of the study Obj CPU(s) Obj CPU(s) [(4)-(2)](2)

(1) (2) (3) (4) (5) (6)

Change in demand =0

1-1-2-4-10 (2+8) 1551183119 995 1551183119 1331 0002-2-4-10-35 (4+31) 4850885922 268 4861980932 2903 0232-2-6-14-55 (5+50) 5181467988 45 5213955899 42 0632-4-12-22-85 (7+78) 5318725918 3041 5363403216 7706 084

3-5-18-70-205 (10+195) 5619609011 gt3600 5672995297 503 095

Change in demand =001

1-1-2-4-10 (2+8) 1554786933 995 1554786933 1171 0002-2-4-10-35 (4+31) 4905668616 2345 4913027119 2857 0152-2-6-14-55 (5+50) 5193458934 49 5224634879 4454 0602-4-12-22-85 (7+78) 5304694312 325 5350314683 8386 086

3-5-18-70-205 (10+195) 5608465011 gt3600 5666793047 524 104

Change in demand =005

1-1-2-4-10 (2+8) 1554468433 99 1554468433 863 0002-2-4-10-35 (4+31) 4902814227 2375 4913110137 2477 0212-2-6-14-55 (5+50) 5220507224 4573 5231902083 3829 0222-4-12-22-85 (7+78) 5342902440 3416 5392591433 768 093

3-5-18-70-205 (10+195) 5621433450 gt3600 5685517791 537 114

Change in demand =01

1-1-2-4-10 (2+8) 1558676503 995 1558676503 1271 0002-2-4-10-35 (4+31) 4928341567 299 4941648089 2794 0272-2-6-14-55 (5+50) 5210996332 5791 5233955899 4836 0442-4-12-22-85 (7+78) 5339691338 3647 5389350467 7474 093

3-5-18-70-205 (10+195) 5620433450 gt3600 5691250911 556 126Average value 053

Table 3 Annual average demand of 3 products for 85 customers

No D1 D2 D3 No D1 D2 D3 No D1 D2 D3 No D1 D2 D31 99 9 12 23 41 27 3 45 24 4 5 67 29 5 12 21 7 1 24 44 1 1 46 14 4 4 68 10 2 23 68 18 3 25 6 5 2 47 4 2 2 69 33 8 14 4 2 2 26 2 0 1 48 15 12 2 70 56 9 25 55 2 1 27 104 23 1 49 19 7 4 71 4 1 36 16 6 3 28 12 4 2 50 2 1 0 72 43 23 27 14 5 1 29 1 1 1 51 9 7 0 73 16 5 28 3 1 0 30 1 1 1 52 4 1 1 74 9 1 19 17 3 1 31 5 2 1 53 16 7 7 75 24 2 310 1 0 0 32 8 1 1 54 14 8 1 76 2 1 111 4 1 7 33 48 9 7 55 22 7 2 77 7 3 212 18 1 6 34 23 15 1 56 4 3 1 78 23 5 213 67 14 7 35 65 8 4 57 6 2 2 79 163 19 114 3 1 2 36 31 15 1 58 1 1 2 80 429 4 115 9 5 1 37 31 3 4 59 4 3 0 81 417 54 116 7 3 1 38 9 1 1 60 13 7 3 82 501 25 117 43 3 15 39 16 2 1 61 13 3 1 83 286 22 418 1 0 7 40 45 11 2 62 12 2 2 84 455 25 219 29 2 3 41 86 8 1 63 45 8 3 85 393 45 220 28 8 4 42 39 5 4 64 39 14 321 38 6 1 43 16 3 1 65 23 6 122 5 2 3 44 6 2 1 66 67 7 1No represents the number of customers D1-D3 represent demand of products 1-3

10 Discrete Dynamics in Nature and Society

InitializationSet the parameters MaxItrNw 1198881 1198882 1199031 r2 c3 1199033 PrInitialize 9 independent particle swarms1198761198981198861198901(a 1198901m) 119876119898119886119887(a bm) 119876119898119887119888(b cm) 119876119898119888119889(c dm) 1198761198981198891198902(d 1198902m)119910119886(a) 119910119887(b) 119910119888(c) 119910119889(d)which are in the 11-dimensional solution space X(a b c d 1198901 1198902m 1198861015840 1198871015840 1198881015840 1198891015840)Define position (x) and velocity (v) for each particleAssume individual best (119875119887119890119904119905119905119894119889) equal to the initialized particlesSelect the best particle as global best (119866119887119890119904119905119905119889)

UpdateLet 9 particle swarms be updated independently as follows

For t = 1 to MaxItrUpdate position (x) and velocity (v)V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032 lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909RespectivelyEndEvaluate the particles and calculate the fitness valuesSort the fitness valuesIf Pr gt rand(d)

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578)EndUpdate personal best (119875119887119890119904119905119905119894119889) of 9 independent particle swarmsUpdate global best (119866119887119890119904119905119905119889) of 9 independent particle swarms

EndReport

Global best (119866119887119890119904119905119905119889) of 9 independent particle swarms119866119887119890119904119905119905119889 found by splicing 9 independent particle swarmsFitness values calculated by 119866119887119890119904119905119905119889

Box 1 The improved PSO

Products include electrical appliances household productsand daily chemical products The average annual demand ofthree products for 85 customers is shown in Table 3 Theirlocation is shown in Figure 5

61 Location Decision Results The locations of the logis-tics center and customers (blue icon) are shown in Fig-ure 5 After the location decision three regional logis-tics centers eight city logistics centers nine dealers maynot operate The non-operating logistics centers are iden-tified by red rectangle Reasons for not operating areanalyzed

Two national logistics centers (numbered 119860 119894 yellowicon) Since the national distribution center needs to dis-tribute to regional logistics centers and large customers onenational distribution centerrsquos capacity is not enough both ofthem need to operate

Four regional logistics centers (numbered 119861119894 purpleicon) Regional logistics center B2 needs to operate B2 is clos-est to the two national logistics centers and is closer to the citylogistics center compared with B1 B3 and B4 In the vicinityof B1 B3 and B4 the city logistics centers (numbered 119862119894)are less distributed and dispersed Considering the operatingcosts B1 B3 and B4 may not operate

Twelve city logistics centers (numbered 119862119894 green icon)City logistics centers C1 C4 C8 and C11 need to oper-ate C2 C9 and C10 are far away from B2 and havehigh transportation costs compared with C4 and C8 sothey are not operated C3 C5 C6 C7 and C12 areremote and scattered and the demand of dealer is smallIt does not operate due to transportation and operatingcosts

Twenty-two dealers (numbered 119863119894 red icon) Throughthe calculation of location and routing 9 dealers do not needto operate The remaining 13 dealers can fulfill the productdistribution demands of 78 second-class customers

The above analyses show that M companyrsquos ldquo2-4-12-22rdquosupply chain network is too large The current demand canbe satisfied by the ldquo2-1-4-13rdquo scale network The location-routing decision can reduce the operations of three regionaldistribution centers eight city distribution centers and ninedealers

62 Transportation Route Decision Results Distributionroute of the three products for 85 customers is shown inTable 4

If distribution routes of the three products are the samethe route decision follows the customer number and the first

Discrete Dynamics in Nature and Society 11

Table4Deliveryrouteo

f3prod

uctsfor8

5custo

mers(To

n)

No

Deliverypath

No

Deliverypath

No

Deliverypath

No

Deliverypath

1A1-B

2-C1-D

8-E1

23A1-B

2-C1-D

7-E2

343

A1-B

2-C1

1-D19-E43

67A1-B

2-C1-D

1-E67

2A1-B

2-C1-D

1-E2

24A1-B

2-C1

1-D2-E2

444

A1-B

2-C1-D

1-E44

68A1-B

2-C8

-D21-E68

3A1-B

2-C1-D

1-E3

25A1-B

2-C1

1-D2-E2

545

A1-B

2-C1

1-D2-E4

569

A1-B

2-C1-D

1-E69

4A1-B

2-C1

1-D19-E4

26A1-B

2-C1-D

1-E26

46A1-B

2-C1-D

8-E4

670

A1-B

2-C1

1-D19-E70

5A1-B

2-C1-D

1-E5

27A1-B

2-C1

1-D2-E2

747

A1-B

2-C1

1-D19-E47

71A1-B

2-C1-D

1-E71

6A1-B

2-C1

1-D19-E6

28A1-B

2-C1

1-D19-E28

48A1-B

2-C1

1-D2-E4

872

A1-B

2-C1-D

1-E72

7A1-B

2-C1-D

10-E7

29A1-B

2-C1-D

8-E2

949

A1-B

2-C1-D

1-E49

73A1-B

2-C1-D

1-E73

9A1-B

2-C1-D

1-E9

30A1-B

2-C1-D

1-E30

51A1-B

2-C8

-D21-E51

74A1-B

2-C1-D

1-E74

10A1-B

2-C1-D

1-E10

31A1-B

2-C1-D

1-E31

52A1-B

2-C1-D

7-E5

275

A1-B

2-C1-D

1-E75

11A1-B

2-C1-D

22-E11

32A1-B

2-C1-D

1-E32

53A1-B

2-C1-D

1-E53

76A1-B

2-C1-D

1-E76

12A1-B

2-C1-D

1-E12

33A1-B

2-C1

1-D2-E3

354

A1-B

2-C1-D

1-E54

77A1-B

2-C1-D

1-E77

13A1-B

2-C8

-D21-E13

34A1-B

2-C1

1-D19-E34

55A1-B

2-C1-D

1-E55

78A1-B

2-C1-D

1-E78

14A1-B

2-C1-D

1-E14

35A1-B

2-C1-D

22-E35

56A1-B

2-C1-D

1-E56

79A1-E

7915

A1-B

2-C1-D

1-E15

36A1-B

2-C1-D

16-E36

57A1-B

2-C1

1-D17-E57

80A2-E8

016

A1-B

2-C1

1-D2-E16

37A1-B

2-C1

1-D2-E3

758

A1-B

2-C1-D

1-E58

81A2-E8

117

A1-B

2-C1-D

22-E17

38A1-B

2-C1

1-D2-E3

860

A1-B

2-C1-D

8-E6

082

A2-E8

219

A1-B

2-C1-D

10-E19

39A1-B

2-C1

1-D19-E39

61A1-B

2-C1-D

1-E61

83A1-E

8320

A1-B

2-C8

-D21-E20

40A1-B

2-C4

-D18-E40

62A1-B

2-C8

-D21-E62

84A2-E8

421

A1-B

2-C8

-D21-E21

41A1-B

2-C4

-D18-E41

63A1-B

2-C1-D

1-E63

85A2-E8

522

A1-B

2-C1-D

8-E2

242

A1-B

2-C1-D

1-E42

64A1-B

2-C1-D

4-E6

48

A1-B

2-C1-D

8-E8

8A1-B

2-C1-D

8-E8

8

18

18A1-B

2-C1-D

8-E18

18A1-B

2-C1-D

8-E18

50A1-B

2-C8

-D21-E50

50A1-B

2-C8

-D21-E50

50

59A1-B

2-C4

-D12-E59

59A1-B

2-C4

-D12-E59

59

65A1-B

2-C1-D

4-E6

565

A1-B

2-C1-D

4-E6

565

66

A1-B

2-C4

-D18-E66

66A1-B

2-C4

-D18-E66

66

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

Discrete Dynamics in Nature and Society 9

Table 2 Comparison of different kinds of algorithms

CPLEX Improved PSO GapScale of the study Obj CPU(s) Obj CPU(s) [(4)-(2)](2)

(1) (2) (3) (4) (5) (6)

Change in demand =0

1-1-2-4-10 (2+8) 1551183119 995 1551183119 1331 0002-2-4-10-35 (4+31) 4850885922 268 4861980932 2903 0232-2-6-14-55 (5+50) 5181467988 45 5213955899 42 0632-4-12-22-85 (7+78) 5318725918 3041 5363403216 7706 084

3-5-18-70-205 (10+195) 5619609011 gt3600 5672995297 503 095

Change in demand =001

1-1-2-4-10 (2+8) 1554786933 995 1554786933 1171 0002-2-4-10-35 (4+31) 4905668616 2345 4913027119 2857 0152-2-6-14-55 (5+50) 5193458934 49 5224634879 4454 0602-4-12-22-85 (7+78) 5304694312 325 5350314683 8386 086

3-5-18-70-205 (10+195) 5608465011 gt3600 5666793047 524 104

Change in demand =005

1-1-2-4-10 (2+8) 1554468433 99 1554468433 863 0002-2-4-10-35 (4+31) 4902814227 2375 4913110137 2477 0212-2-6-14-55 (5+50) 5220507224 4573 5231902083 3829 0222-4-12-22-85 (7+78) 5342902440 3416 5392591433 768 093

3-5-18-70-205 (10+195) 5621433450 gt3600 5685517791 537 114

Change in demand =01

1-1-2-4-10 (2+8) 1558676503 995 1558676503 1271 0002-2-4-10-35 (4+31) 4928341567 299 4941648089 2794 0272-2-6-14-55 (5+50) 5210996332 5791 5233955899 4836 0442-4-12-22-85 (7+78) 5339691338 3647 5389350467 7474 093

3-5-18-70-205 (10+195) 5620433450 gt3600 5691250911 556 126Average value 053

Table 3 Annual average demand of 3 products for 85 customers

No D1 D2 D3 No D1 D2 D3 No D1 D2 D3 No D1 D2 D31 99 9 12 23 41 27 3 45 24 4 5 67 29 5 12 21 7 1 24 44 1 1 46 14 4 4 68 10 2 23 68 18 3 25 6 5 2 47 4 2 2 69 33 8 14 4 2 2 26 2 0 1 48 15 12 2 70 56 9 25 55 2 1 27 104 23 1 49 19 7 4 71 4 1 36 16 6 3 28 12 4 2 50 2 1 0 72 43 23 27 14 5 1 29 1 1 1 51 9 7 0 73 16 5 28 3 1 0 30 1 1 1 52 4 1 1 74 9 1 19 17 3 1 31 5 2 1 53 16 7 7 75 24 2 310 1 0 0 32 8 1 1 54 14 8 1 76 2 1 111 4 1 7 33 48 9 7 55 22 7 2 77 7 3 212 18 1 6 34 23 15 1 56 4 3 1 78 23 5 213 67 14 7 35 65 8 4 57 6 2 2 79 163 19 114 3 1 2 36 31 15 1 58 1 1 2 80 429 4 115 9 5 1 37 31 3 4 59 4 3 0 81 417 54 116 7 3 1 38 9 1 1 60 13 7 3 82 501 25 117 43 3 15 39 16 2 1 61 13 3 1 83 286 22 418 1 0 7 40 45 11 2 62 12 2 2 84 455 25 219 29 2 3 41 86 8 1 63 45 8 3 85 393 45 220 28 8 4 42 39 5 4 64 39 14 321 38 6 1 43 16 3 1 65 23 6 122 5 2 3 44 6 2 1 66 67 7 1No represents the number of customers D1-D3 represent demand of products 1-3

10 Discrete Dynamics in Nature and Society

InitializationSet the parameters MaxItrNw 1198881 1198882 1199031 r2 c3 1199033 PrInitialize 9 independent particle swarms1198761198981198861198901(a 1198901m) 119876119898119886119887(a bm) 119876119898119887119888(b cm) 119876119898119888119889(c dm) 1198761198981198891198902(d 1198902m)119910119886(a) 119910119887(b) 119910119888(c) 119910119889(d)which are in the 11-dimensional solution space X(a b c d 1198901 1198902m 1198861015840 1198871015840 1198881015840 1198891015840)Define position (x) and velocity (v) for each particleAssume individual best (119875119887119890119904119905119905119894119889) equal to the initialized particlesSelect the best particle as global best (119866119887119890119904119905119905119889)

UpdateLet 9 particle swarms be updated independently as follows

For t = 1 to MaxItrUpdate position (x) and velocity (v)V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032 lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909RespectivelyEndEvaluate the particles and calculate the fitness valuesSort the fitness valuesIf Pr gt rand(d)

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578)EndUpdate personal best (119875119887119890119904119905119905119894119889) of 9 independent particle swarmsUpdate global best (119866119887119890119904119905119905119889) of 9 independent particle swarms

EndReport

Global best (119866119887119890119904119905119905119889) of 9 independent particle swarms119866119887119890119904119905119905119889 found by splicing 9 independent particle swarmsFitness values calculated by 119866119887119890119904119905119905119889

Box 1 The improved PSO

Products include electrical appliances household productsand daily chemical products The average annual demand ofthree products for 85 customers is shown in Table 3 Theirlocation is shown in Figure 5

61 Location Decision Results The locations of the logis-tics center and customers (blue icon) are shown in Fig-ure 5 After the location decision three regional logis-tics centers eight city logistics centers nine dealers maynot operate The non-operating logistics centers are iden-tified by red rectangle Reasons for not operating areanalyzed

Two national logistics centers (numbered 119860 119894 yellowicon) Since the national distribution center needs to dis-tribute to regional logistics centers and large customers onenational distribution centerrsquos capacity is not enough both ofthem need to operate

Four regional logistics centers (numbered 119861119894 purpleicon) Regional logistics center B2 needs to operate B2 is clos-est to the two national logistics centers and is closer to the citylogistics center compared with B1 B3 and B4 In the vicinityof B1 B3 and B4 the city logistics centers (numbered 119862119894)are less distributed and dispersed Considering the operatingcosts B1 B3 and B4 may not operate

Twelve city logistics centers (numbered 119862119894 green icon)City logistics centers C1 C4 C8 and C11 need to oper-ate C2 C9 and C10 are far away from B2 and havehigh transportation costs compared with C4 and C8 sothey are not operated C3 C5 C6 C7 and C12 areremote and scattered and the demand of dealer is smallIt does not operate due to transportation and operatingcosts

Twenty-two dealers (numbered 119863119894 red icon) Throughthe calculation of location and routing 9 dealers do not needto operate The remaining 13 dealers can fulfill the productdistribution demands of 78 second-class customers

The above analyses show that M companyrsquos ldquo2-4-12-22rdquosupply chain network is too large The current demand canbe satisfied by the ldquo2-1-4-13rdquo scale network The location-routing decision can reduce the operations of three regionaldistribution centers eight city distribution centers and ninedealers

62 Transportation Route Decision Results Distributionroute of the three products for 85 customers is shown inTable 4

If distribution routes of the three products are the samethe route decision follows the customer number and the first

Discrete Dynamics in Nature and Society 11

Table4Deliveryrouteo

f3prod

uctsfor8

5custo

mers(To

n)

No

Deliverypath

No

Deliverypath

No

Deliverypath

No

Deliverypath

1A1-B

2-C1-D

8-E1

23A1-B

2-C1-D

7-E2

343

A1-B

2-C1

1-D19-E43

67A1-B

2-C1-D

1-E67

2A1-B

2-C1-D

1-E2

24A1-B

2-C1

1-D2-E2

444

A1-B

2-C1-D

1-E44

68A1-B

2-C8

-D21-E68

3A1-B

2-C1-D

1-E3

25A1-B

2-C1

1-D2-E2

545

A1-B

2-C1

1-D2-E4

569

A1-B

2-C1-D

1-E69

4A1-B

2-C1

1-D19-E4

26A1-B

2-C1-D

1-E26

46A1-B

2-C1-D

8-E4

670

A1-B

2-C1

1-D19-E70

5A1-B

2-C1-D

1-E5

27A1-B

2-C1

1-D2-E2

747

A1-B

2-C1

1-D19-E47

71A1-B

2-C1-D

1-E71

6A1-B

2-C1

1-D19-E6

28A1-B

2-C1

1-D19-E28

48A1-B

2-C1

1-D2-E4

872

A1-B

2-C1-D

1-E72

7A1-B

2-C1-D

10-E7

29A1-B

2-C1-D

8-E2

949

A1-B

2-C1-D

1-E49

73A1-B

2-C1-D

1-E73

9A1-B

2-C1-D

1-E9

30A1-B

2-C1-D

1-E30

51A1-B

2-C8

-D21-E51

74A1-B

2-C1-D

1-E74

10A1-B

2-C1-D

1-E10

31A1-B

2-C1-D

1-E31

52A1-B

2-C1-D

7-E5

275

A1-B

2-C1-D

1-E75

11A1-B

2-C1-D

22-E11

32A1-B

2-C1-D

1-E32

53A1-B

2-C1-D

1-E53

76A1-B

2-C1-D

1-E76

12A1-B

2-C1-D

1-E12

33A1-B

2-C1

1-D2-E3

354

A1-B

2-C1-D

1-E54

77A1-B

2-C1-D

1-E77

13A1-B

2-C8

-D21-E13

34A1-B

2-C1

1-D19-E34

55A1-B

2-C1-D

1-E55

78A1-B

2-C1-D

1-E78

14A1-B

2-C1-D

1-E14

35A1-B

2-C1-D

22-E35

56A1-B

2-C1-D

1-E56

79A1-E

7915

A1-B

2-C1-D

1-E15

36A1-B

2-C1-D

16-E36

57A1-B

2-C1

1-D17-E57

80A2-E8

016

A1-B

2-C1

1-D2-E16

37A1-B

2-C1

1-D2-E3

758

A1-B

2-C1-D

1-E58

81A2-E8

117

A1-B

2-C1-D

22-E17

38A1-B

2-C1

1-D2-E3

860

A1-B

2-C1-D

8-E6

082

A2-E8

219

A1-B

2-C1-D

10-E19

39A1-B

2-C1

1-D19-E39

61A1-B

2-C1-D

1-E61

83A1-E

8320

A1-B

2-C8

-D21-E20

40A1-B

2-C4

-D18-E40

62A1-B

2-C8

-D21-E62

84A2-E8

421

A1-B

2-C8

-D21-E21

41A1-B

2-C4

-D18-E41

63A1-B

2-C1-D

1-E63

85A2-E8

522

A1-B

2-C1-D

8-E2

242

A1-B

2-C1-D

1-E42

64A1-B

2-C1-D

4-E6

48

A1-B

2-C1-D

8-E8

8A1-B

2-C1-D

8-E8

8

18

18A1-B

2-C1-D

8-E18

18A1-B

2-C1-D

8-E18

50A1-B

2-C8

-D21-E50

50A1-B

2-C8

-D21-E50

50

59A1-B

2-C4

-D12-E59

59A1-B

2-C4

-D12-E59

59

65A1-B

2-C1-D

4-E6

565

A1-B

2-C1-D

4-E6

565

66

A1-B

2-C4

-D18-E66

66A1-B

2-C4

-D18-E66

66

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

10 Discrete Dynamics in Nature and Society

InitializationSet the parameters MaxItrNw 1198881 1198882 1199031 r2 c3 1199033 PrInitialize 9 independent particle swarms1198761198981198861198901(a 1198901m) 119876119898119886119887(a bm) 119876119898119887119888(b cm) 119876119898119888119889(c dm) 1198761198981198891198902(d 1198902m)119910119886(a) 119910119887(b) 119910119888(c) 119910119889(d)which are in the 11-dimensional solution space X(a b c d 1198901 1198902m 1198861015840 1198871015840 1198881015840 1198891015840)Define position (x) and velocity (v) for each particleAssume individual best (119875119887119890119904119905119905119894119889) equal to the initialized particlesSelect the best particle as global best (119866119887119890119904119905119905119889)

UpdateLet 9 particle swarms be updated independently as follows

For t = 1 to MaxItrUpdate position (x) and velocity (v)V119905+1119894119889 = 119908 lowast V119905119894119889 + 1198881 lowast 1199031 lowast (119875119887119890119904119905119905119894119889 minus 119909119905119894119889) + 1198882 lowast 1199032 lowast (119866119887119890119904119905119905119889 minus 119909119905119894119889)119909119905+1119894119889 = 119909119905119894119889 + V119905+1119894119889If 119883119894119889 lt 119883119898119894119899 119905ℎ119890119899 119883119894119889 = 119883119898119894119899 + 1198883 lowast 1199033 lowast 119883119898119894119899Else if 119883119894119889 gt 119883119898119886119909 119905ℎ119890119899 119883119894119889 = 119883119898119886119909 minus 1198883 lowast 1199033 lowast 119883119898119886119909RespectivelyEndEvaluate the particles and calculate the fitness valuesSort the fitness valuesIf Pr gt rand(d)

119909119905+1119894119889 = 119909119905119894119889 lowast (1 + 05120578)EndUpdate personal best (119875119887119890119904119905119905119894119889) of 9 independent particle swarmsUpdate global best (119866119887119890119904119905119905119889) of 9 independent particle swarms

EndReport

Global best (119866119887119890119904119905119905119889) of 9 independent particle swarms119866119887119890119904119905119905119889 found by splicing 9 independent particle swarmsFitness values calculated by 119866119887119890119904119905119905119889

Box 1 The improved PSO

Products include electrical appliances household productsand daily chemical products The average annual demand ofthree products for 85 customers is shown in Table 3 Theirlocation is shown in Figure 5

61 Location Decision Results The locations of the logis-tics center and customers (blue icon) are shown in Fig-ure 5 After the location decision three regional logis-tics centers eight city logistics centers nine dealers maynot operate The non-operating logistics centers are iden-tified by red rectangle Reasons for not operating areanalyzed

Two national logistics centers (numbered 119860 119894 yellowicon) Since the national distribution center needs to dis-tribute to regional logistics centers and large customers onenational distribution centerrsquos capacity is not enough both ofthem need to operate

Four regional logistics centers (numbered 119861119894 purpleicon) Regional logistics center B2 needs to operate B2 is clos-est to the two national logistics centers and is closer to the citylogistics center compared with B1 B3 and B4 In the vicinityof B1 B3 and B4 the city logistics centers (numbered 119862119894)are less distributed and dispersed Considering the operatingcosts B1 B3 and B4 may not operate

Twelve city logistics centers (numbered 119862119894 green icon)City logistics centers C1 C4 C8 and C11 need to oper-ate C2 C9 and C10 are far away from B2 and havehigh transportation costs compared with C4 and C8 sothey are not operated C3 C5 C6 C7 and C12 areremote and scattered and the demand of dealer is smallIt does not operate due to transportation and operatingcosts

Twenty-two dealers (numbered 119863119894 red icon) Throughthe calculation of location and routing 9 dealers do not needto operate The remaining 13 dealers can fulfill the productdistribution demands of 78 second-class customers

The above analyses show that M companyrsquos ldquo2-4-12-22rdquosupply chain network is too large The current demand canbe satisfied by the ldquo2-1-4-13rdquo scale network The location-routing decision can reduce the operations of three regionaldistribution centers eight city distribution centers and ninedealers

62 Transportation Route Decision Results Distributionroute of the three products for 85 customers is shown inTable 4

If distribution routes of the three products are the samethe route decision follows the customer number and the first

Discrete Dynamics in Nature and Society 11

Table4Deliveryrouteo

f3prod

uctsfor8

5custo

mers(To

n)

No

Deliverypath

No

Deliverypath

No

Deliverypath

No

Deliverypath

1A1-B

2-C1-D

8-E1

23A1-B

2-C1-D

7-E2

343

A1-B

2-C1

1-D19-E43

67A1-B

2-C1-D

1-E67

2A1-B

2-C1-D

1-E2

24A1-B

2-C1

1-D2-E2

444

A1-B

2-C1-D

1-E44

68A1-B

2-C8

-D21-E68

3A1-B

2-C1-D

1-E3

25A1-B

2-C1

1-D2-E2

545

A1-B

2-C1

1-D2-E4

569

A1-B

2-C1-D

1-E69

4A1-B

2-C1

1-D19-E4

26A1-B

2-C1-D

1-E26

46A1-B

2-C1-D

8-E4

670

A1-B

2-C1

1-D19-E70

5A1-B

2-C1-D

1-E5

27A1-B

2-C1

1-D2-E2

747

A1-B

2-C1

1-D19-E47

71A1-B

2-C1-D

1-E71

6A1-B

2-C1

1-D19-E6

28A1-B

2-C1

1-D19-E28

48A1-B

2-C1

1-D2-E4

872

A1-B

2-C1-D

1-E72

7A1-B

2-C1-D

10-E7

29A1-B

2-C1-D

8-E2

949

A1-B

2-C1-D

1-E49

73A1-B

2-C1-D

1-E73

9A1-B

2-C1-D

1-E9

30A1-B

2-C1-D

1-E30

51A1-B

2-C8

-D21-E51

74A1-B

2-C1-D

1-E74

10A1-B

2-C1-D

1-E10

31A1-B

2-C1-D

1-E31

52A1-B

2-C1-D

7-E5

275

A1-B

2-C1-D

1-E75

11A1-B

2-C1-D

22-E11

32A1-B

2-C1-D

1-E32

53A1-B

2-C1-D

1-E53

76A1-B

2-C1-D

1-E76

12A1-B

2-C1-D

1-E12

33A1-B

2-C1

1-D2-E3

354

A1-B

2-C1-D

1-E54

77A1-B

2-C1-D

1-E77

13A1-B

2-C8

-D21-E13

34A1-B

2-C1

1-D19-E34

55A1-B

2-C1-D

1-E55

78A1-B

2-C1-D

1-E78

14A1-B

2-C1-D

1-E14

35A1-B

2-C1-D

22-E35

56A1-B

2-C1-D

1-E56

79A1-E

7915

A1-B

2-C1-D

1-E15

36A1-B

2-C1-D

16-E36

57A1-B

2-C1

1-D17-E57

80A2-E8

016

A1-B

2-C1

1-D2-E16

37A1-B

2-C1

1-D2-E3

758

A1-B

2-C1-D

1-E58

81A2-E8

117

A1-B

2-C1-D

22-E17

38A1-B

2-C1

1-D2-E3

860

A1-B

2-C1-D

8-E6

082

A2-E8

219

A1-B

2-C1-D

10-E19

39A1-B

2-C1

1-D19-E39

61A1-B

2-C1-D

1-E61

83A1-E

8320

A1-B

2-C8

-D21-E20

40A1-B

2-C4

-D18-E40

62A1-B

2-C8

-D21-E62

84A2-E8

421

A1-B

2-C8

-D21-E21

41A1-B

2-C4

-D18-E41

63A1-B

2-C1-D

1-E63

85A2-E8

522

A1-B

2-C1-D

8-E2

242

A1-B

2-C1-D

1-E42

64A1-B

2-C1-D

4-E6

48

A1-B

2-C1-D

8-E8

8A1-B

2-C1-D

8-E8

8

18

18A1-B

2-C1-D

8-E18

18A1-B

2-C1-D

8-E18

50A1-B

2-C8

-D21-E50

50A1-B

2-C8

-D21-E50

50

59A1-B

2-C4

-D12-E59

59A1-B

2-C4

-D12-E59

59

65A1-B

2-C1-D

4-E6

565

A1-B

2-C1-D

4-E6

565

66

A1-B

2-C4

-D18-E66

66A1-B

2-C4

-D18-E66

66

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

Discrete Dynamics in Nature and Society 11

Table4Deliveryrouteo

f3prod

uctsfor8

5custo

mers(To

n)

No

Deliverypath

No

Deliverypath

No

Deliverypath

No

Deliverypath

1A1-B

2-C1-D

8-E1

23A1-B

2-C1-D

7-E2

343

A1-B

2-C1

1-D19-E43

67A1-B

2-C1-D

1-E67

2A1-B

2-C1-D

1-E2

24A1-B

2-C1

1-D2-E2

444

A1-B

2-C1-D

1-E44

68A1-B

2-C8

-D21-E68

3A1-B

2-C1-D

1-E3

25A1-B

2-C1

1-D2-E2

545

A1-B

2-C1

1-D2-E4

569

A1-B

2-C1-D

1-E69

4A1-B

2-C1

1-D19-E4

26A1-B

2-C1-D

1-E26

46A1-B

2-C1-D

8-E4

670

A1-B

2-C1

1-D19-E70

5A1-B

2-C1-D

1-E5

27A1-B

2-C1

1-D2-E2

747

A1-B

2-C1

1-D19-E47

71A1-B

2-C1-D

1-E71

6A1-B

2-C1

1-D19-E6

28A1-B

2-C1

1-D19-E28

48A1-B

2-C1

1-D2-E4

872

A1-B

2-C1-D

1-E72

7A1-B

2-C1-D

10-E7

29A1-B

2-C1-D

8-E2

949

A1-B

2-C1-D

1-E49

73A1-B

2-C1-D

1-E73

9A1-B

2-C1-D

1-E9

30A1-B

2-C1-D

1-E30

51A1-B

2-C8

-D21-E51

74A1-B

2-C1-D

1-E74

10A1-B

2-C1-D

1-E10

31A1-B

2-C1-D

1-E31

52A1-B

2-C1-D

7-E5

275

A1-B

2-C1-D

1-E75

11A1-B

2-C1-D

22-E11

32A1-B

2-C1-D

1-E32

53A1-B

2-C1-D

1-E53

76A1-B

2-C1-D

1-E76

12A1-B

2-C1-D

1-E12

33A1-B

2-C1

1-D2-E3

354

A1-B

2-C1-D

1-E54

77A1-B

2-C1-D

1-E77

13A1-B

2-C8

-D21-E13

34A1-B

2-C1

1-D19-E34

55A1-B

2-C1-D

1-E55

78A1-B

2-C1-D

1-E78

14A1-B

2-C1-D

1-E14

35A1-B

2-C1-D

22-E35

56A1-B

2-C1-D

1-E56

79A1-E

7915

A1-B

2-C1-D

1-E15

36A1-B

2-C1-D

16-E36

57A1-B

2-C1

1-D17-E57

80A2-E8

016

A1-B

2-C1

1-D2-E16

37A1-B

2-C1

1-D2-E3

758

A1-B

2-C1-D

1-E58

81A2-E8

117

A1-B

2-C1-D

22-E17

38A1-B

2-C1

1-D2-E3

860

A1-B

2-C1-D

8-E6

082

A2-E8

219

A1-B

2-C1-D

10-E19

39A1-B

2-C1

1-D19-E39

61A1-B

2-C1-D

1-E61

83A1-E

8320

A1-B

2-C8

-D21-E20

40A1-B

2-C4

-D18-E40

62A1-B

2-C8

-D21-E62

84A2-E8

421

A1-B

2-C8

-D21-E21

41A1-B

2-C4

-D18-E41

63A1-B

2-C1-D

1-E63

85A2-E8

522

A1-B

2-C1-D

8-E2

242

A1-B

2-C1-D

1-E42

64A1-B

2-C1-D

4-E6

48

A1-B

2-C1-D

8-E8

8A1-B

2-C1-D

8-E8

8

18

18A1-B

2-C1-D

8-E18

18A1-B

2-C1-D

8-E18

50A1-B

2-C8

-D21-E50

50A1-B

2-C8

-D21-E50

50

59A1-B

2-C4

-D12-E59

59A1-B

2-C4

-D12-E59

59

65A1-B

2-C1-D

4-E6

565

A1-B

2-C1-D

4-E6

565

66

A1-B

2-C4

-D18-E66

66A1-B

2-C4

-D18-E66

66

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

12 Discrete Dynamics in Nature and Society

Table 5 Comparison of costs and product volume when demand changes

Scale of the study1-1-2-4-10 2-2-4-10-35 2-2-6-14-55 2-4-12-22-85 3-5-18-70-205(2+8) (4+31) (5+50) (7+78) (10+195)

120590 = 0 1551183119 4861980932 5213955899 5363403216 5672995297120590 = 001 1554786933 4913027119 5224634879 5350314683 5666793047120590 = 005 1554468433 4913110137 5231902083 5392591433 5685517791120590 = 01 1558676503 4941648089 5233955899 5389350467 5691250911Mean 1554778747 4907441569 5226112190 5373914950 5679139262max-min 7493384 79667157 20000000 42276750 24457864(max-min)mean 048 162 038 079 043Average value 074

Figure 5 Logistics center and customer location in Chengdu

20 lines show the delivery of three products for 79 customersThe delivery routes of three products for 6 customers in thelast 6 lines are not completely consistent and are listed fromleft to right in the order of delivery paths of products 1-3This example shows that the mathematical model and thesolution algorithm can complete both operational decisionand routing decision of three products through five supplychain levels

63 Sensitivity Analysis In the case of 1 5 and 10change in demand the impact of uncertain demand ontotal costs was analyzed The total operating costs of thefour demand changes under each experimental scale werecounted Demand change is represented by 120590 Each row ofTable 5 shows the total costs of different situations

As the demand changes the total costs of the fourscenarios change As a result the ratio of the maximum andminimum differences to the average cost is 074Thereforeconsidering each customerrsquos demands for each product sep-arately can reduce the impact of demand uncertainty on thetotal costs of supply chain network and reasonably predict thedemand distribution

Based on the location decision results in Section 61another sensitivity analysis on location decision is conducted

Assuming that the model does not contain location decisionall dealers need to operate

The total costs and capacity without location decision areshown in Table 6

The comparison of Table 6 shows that the locationdecision can effectively save the total costs which is3700000 RMB compared with the non-site selection deci-sion 3700000 RMB is exactly the sum of the operation costsof three regional logistics centers eight city logistics and ninedealers Specific operational decisions are in Section 61

7 Conclusion

This paper studies the joint optimization of location androuting problem with the background of distribution underOmni-Channel integration In our research amany-to-manydistribution of supply chain network is designedWe considereach customerrsquos uncertain demands for different productsthen build a demand network and complete the integrationoptimization of distribution network and demand networkThe progress and contributions of this research include thefollowing

(1) The customers are classified in this designed supplychain network This study increases product categories andincreases supply chain levels Many-to-many distribution is

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

Discrete Dynamics in Nature and Society 13

Table6Com

paris

onof

non-sitelocationandsitelocation

Totalcosts

Totalcostsof

custo

mer119890 1

Totalcostsof

custo

mer119890 2

Prod

uct1

Prod

uct2

Prod

uct3

Prod

uct1

Prod

uct2

Prod

uct3

Time

Volumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fVo

lumeo

fCu

stomer119890 1

Custo

mer119890 1

Custo

mer119890 1

Custo

mer119890 2

Custo

mer119890 2

Custo

mer119890 2

Non

-site

locatio

n5367103216

3422230239

1944

872977

2644

194

121815

427

192

68Sitelocatio

n5363403216

3422230239

1941172977

2644

194

121815

427

192

75Difference

3700

000

03700

000

00

00

00

7

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

14 Discrete Dynamics in Nature and Society

achieved between the distribution network levels and thedemand network considers the different demands of cus-tomers for different products Less attention is concentratedon integration optimization issues of distribution networkand demand network in existing research

(2) Through three improvements of particle swarmoptimization such as collaborative dimension reductionboundary processing and introduction of mutation factorsthe performance of the algorithm is improved The paperprovides an algorithmic solution to solve high-dimensionalproblems

In future study the research results of this paper can beextended to the scale economy of the distribution networkwhile increasing the uncertainty factors and solving largerscale examples

Data Availability

The data used to support this study have been uploaded toBaiduCloudDisk Readers can access the data by the followinglink httpspanbaiducoms1FFECqz8yP7y13j6cj-W4mA

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China [Grant no 71701123]References

[1] P C Verhoef P Kannan and J J Inman ldquoFrom multi-channelretailing to omni-channel retailingrdquo Journal of Retailing vol 91no 2 pp 174ndash181 2015

[2] S Min and M Wolfinbarger ldquoMarket share profit margin andmarketing efficiency of early movers bricks and clicks andspecialists in e-commercerdquo Journal of Business Research vol 58no 8 pp 1030ndash1039 2005

[3] J Zhang P W Farris J W Irvin T Kushwaha T J Steenburghand B A Weitz ldquoCrafting integrated multichannel retailingstrategiesrdquo Journal of Interactive Marketing vol 24 no 2 pp168ndash180 2010

[4] S Saghiri R Wilding C Mena and M Bourlakis ldquoTowarda three-dimensional framework for omni-channelrdquo Journal ofBusiness Research vol 77 pp 53ndash67 2017

[5] F E Maranzana ldquoOn the location of supply points to minimizetransportation costsrdquo Journal of the Operational Research Soci-ety vol 15 no 3 pp 261ndash270 1964

[6] K Govindan A Jafarian R Khodaverdi and K DevikaldquoTwo-echelon multiple-vehicle location-routing problem withtime windows for optimization of sustainable supply chainnetwork of perishable foodrdquo International Journal of ProductionEconomics vol 152 no 2 pp 9ndash28 2014

[7] A Abdulrazik M Elsholkami A Elkamel and L SimonldquoMulti-products productions from Malaysian oil palm emptyfruit bunch (EFB) analyzing economic potentials from theoptimal biomass supply chainrdquo Journal of Cleaner Productionvol 168 no 2 pp 131ndash148 2017

[8] A M Geoffrion and G W Graves ldquoMulticommodity distri-bution system design by benders decompositionrdquoManagementScience vol 26 no 8 pp 855-856 1980

[9] A Tiwari P C Chang and M K Tiwari ldquoA highly optimisedtolerance-based approach formulti-stagemulti-product supplychain network designrdquo International Journal of ProductionResearch vol 50 no 19 pp 5430ndash5444 2012

[10] S Viswanathan and K Mathur ldquoIntegrating routing and inven-tory decisions in one-warehouse multiretailer multiproductdistribution systemsrdquo Management Science vol 43 no 3 pp294ndash312 1997

[11] B C Arntzen G G Brown T P Harrison and L L TraftonldquoGlobal supply chain management at digital equipment corpo-rationrdquo Interfaces vol 25 no 1 pp 69ndash93 1995

[12] B H Wang and S W He ldquoRobust optimization model andalgorithm for logistics center location and allocation underuncertain environmentrdquo Journal of Transportation SystemsEngineering and Information Technology vol 9 no 2 pp 69ndash74 2009

[13] L Zhen D Zhuge and J Lei ldquoSupply chain optimization incontext of production flow networkrdquo Journal of Systems Scienceand Systems Engineering vol 25 no 3 pp 351ndash369 2016

[14] C S Tapiero and M A Soliman ldquoMulti-commodities trans-portation schedules over timerdquo Networks vol 2 pp 311ndash3271972

[15] A Baghalian S Rezapour and R Z Farahani ldquoRobust supplychain network design with service level against disruptionsand demand uncertainties a real-life caserdquo European Journal ofOperational Research vol 227 no 1 pp 199ndash215 2013

[16] G Zhang W Habenicht and W E L Spieszlig ldquoImproving thestructure of deep frozen and chilled food chainwith tabu searchprocedurerdquo Journal of Food Engineering vol 60 no 1 pp 67ndash792003

[17] A Ahmadi Javid and N Azad ldquoIncorporating location routingand inventory decisions in supply chain network designrdquoTransportation Research Part E Logistics and TransportationReview vol 46 no 5 pp 582ndash597 2010

[18] Y Xia Z Fu L Pan and F Duan ldquoTabu search algorithmfor the distance-constrained vehicle routing problem with splitdeliveries by orderrdquo PLoS ONE vol 13 no 5 Article IDe0195457 2018

[19] H Hu Y Zhang and L Zhen ldquoA two-stage decompositionmethod on fresh product distribution problemrdquo InternationalJournal of Production Research vol 55 no 16 pp 4729ndash47522017

[20] X Li and X Yao ldquoCooperatively coevolving particle swarmsfor large scale optimizationrdquo IEEE Transactions on EvolutionaryComputation vol 16 no 2 pp 210ndash224 2012

[21] J Kennedy and R Eberhart ldquoParticle swarm optimizationrdquoin Proceedings of the IEEE International Conference on NeuralNetworks pp 39ndash43 Nagoya Japan 1995

[22] Z Lian ldquoA united search particle swarmoptimization algorithmfor multi-objective scheduling problemrdquo Applied MathematicalModelling vol 34 no 11 pp 3518ndash3526 2010

[23] T J Ai and V Kachitvichyanukul ldquoA particle swarm optimiza-tion for the vehicle routing problem with simultaneous pickupand deliveryrdquo Computers amp Operations Research vol 36 no 5pp 1693ndash1702 2009

[24] F P Goksal I Karaoglan and F Altiparmak ldquoA hybrid discreteparticle swarm optimization for vehicle routing problem withsimultaneous pickup and deliveryrdquo Computers amp IndustrialEngineering vol 65 no 1 pp 39ndash53 2013

Discrete Dynamics in Nature and Society 15

[25] M Alinaghian M Ghazanfari N Norouzi and H Noural-izadeh ldquoA novel model for the time dependent competitivevehicle routing problem modified random topology particleswarm optimizationrdquo Networks and Spatial Economics vol 17no 4 pp 1ndash27 2017

[26] F van denBergh andA P Engelbrecht ldquoA cooperative approachto participle swam optimizationrdquo IEEE Transactions on Evolu-tionary Computation vol 8 no 3 pp 225ndash239 2004

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Applied MathematicsJournal of

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Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

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OptimizationJournal of

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Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

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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

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