research article a two-stage algorithm for the closed-loop...

Research ArticleA Two-Stage Algorithm for the Closed-Loop Location-InventoryProblem Model Considering Returns in E-Commerce

Yanhui Li Mengmeng Lu and Bailing Liu

School of Information Management Central China Normal University Wuhan 430079 China

Correspondence should be addressed to Bailing Liu bailingcsgmailcom

Received 27 March 2014 Revised 12 August 2014 Accepted 22 August 2014 Published 28 September 2014

Academic Editor Albert Victoire

Copyright copy 2014 Yanhui Li et alThis is an open access article distributed under theCreative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Facility location and inventory control are critical and highly related problems in the design of logistics system for e-commerceMeanwhile the return ratio in Internet sales was significantly higher than in the traditional business Focusing on the existingproblem in e-commerce logistics system we formulate a closed-loop location-inventory problem model considering returnedmerchandise to minimize the total cost which is produced in both forward and reverse logistics networks To solve this nonlinearmixed programming model an effective two-stage heuristic algorithm named LRCAC is designed by combining Lagrangianrelaxation with ant colony algorithm (AC) Results of numerical examples show that LRCAC outperforms ant colony algorithm(AC) on optimal solution and computing stability The proposed model is able to help managers make the right decisions undere-commerce environment

1 Introduction

The increasing progress of information and prevalence ofinternet in the 21st century has forced the e-commerce todevelop in a world-wide range In 2012 B2C e-commercesales grew 211 to top $1 trillion for the first time inhistory in the whole world [1] Comparing with traditionalcommerce customers are more liable to return goods undere-commerce environment Note that many customer returnsonline account for 35 of original orders [2 3] Thereforelogistics systems as an important support system in e-commerce need to be adjusted and improved To adapt tothe reality of e-commerce market environment it is criticalto conduct the research on the reverse logistics network andhighly integrated logistics process

Facility location and inventory control are critical prob-lems in the design of logistics systemThere is much previouswork in these two areas In fact there is amutually dependentrelationship among these problems in logistics system Com-prehensive optimizing and logistics activities managementshould be based on this relationship [4] According tothis idea besides location allocation problem and inventoryoptimization the location-inventory problem (LIP) starts tobe researched

Many papers about the LIP are studied deeply and havemade some abundant achievements In recent years intelli-gent algorithms and heuristic algorithm have been used tosolve LIP model [5ndash7] In the reverse logistics research fieldLIP attracts researchersrsquo attention Lieckens and Vandaele[8] applied a queuing mode in reverse logistics networkto solve the facility location problem while consideringthe impact of inventory costs Srivastava [9] established areverse logistics network optimization model to optimizethe location-allocation problem and capacity decisions andhe used heuristic algorithm to solve the model Wang etal [10] proposed a location-inventory policy in ChineseB2C electronic market as a bilevel programming modelTancrez et al [11] studied the LIP in three-level supplychain networks including reverse logistics they developed aniterative heuristics approach to solve the model Diabat et al[12] built a mixed integer nonlinear programming (MINLP)model to minimize the total reverse logistics cost by findingout the number and location of initial collection point andcentralized return center considering the inventory cost Twosolution approaches namely genetic algorithm (GA) andartificial immune system are implemented and comparedHowever research on the LIP of closed-loop logistics system

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 260869 9 pageshttpdxdoiorg1011552014260869

2 Mathematical Problems in Engineering

is limited Sahyouni et al [13] designed three generic facilitylocation models that account for the integrated distributionand collection of products in the closed-loop supply chainnetworks the authors described a Lagrangian relaxation-based solution algorithm to solve the models Easwaran andUster [14] offered amixed integer linear programmingmodelto optimize the total cost that consists of location processingand transportation costs of the multimerchandise in closed-loop supply chains they introduced a heuristics solutionapproach that combines Benders decomposition and tabusearch to solve the model Abdallah et al [15] presentedthe uncapacitated closed-loop location-inventory model asensitivity analysis for different parameters of the modelreveals that the value of recovered products is a major factorin the economic feasibility of the closed-loop network Fordealing with returnedmerchandise without quality problemsin e-commerce Li et al [16] developed a practical LIP modelwith considering the vehicle routing under e-supply chainenvironment and provided a new hybrid heuristic algorithmto solve this model

Previous researches on the closed-loop logistics sys-tem optimization mainly focus on the minimization ofthe total cost of the network To our best knowledgefew researches on manufacturingremanufacturing systemconsider returns and concept of green logistics recyclingin logistics network Since customers may be dissatis-fied with merchandise and return it the cost of process-ing returns the cost of inventory and shipping ordertime and size are changed Furthermore research on theLIP with return of closed-loop logistics system is lim-ited

The aim of this study is to develop a practical LIPmodel with the consideration of returns in e-commerce andprovide a new two-stage heuristics algorithm To our bestknowledge this work is the first step to introduce returnsinto the LIP in e-commerce which makes it become morepractical We also provide an effective algorithm namedLagrangian relaxation combined with ant colony algorithm(LRCAC) to solve this model Lagrangian relaxation algo-rithm (LR) can obtain a near-optimal solution by analyzingthe upper bound and lower bound of objective functionBut its effectiveness mainly relies on the performance ofsubgradient optimization algorithm On the other handAC has great ability of local searching If there is anappropriate initial solution the performance of AC willbe good To adopt their strong points while overcomingtheir weak points we combine the two algorithms Resultsof numerical examples show that LRCAC outperforms antcolony algorithm (AC) on optimal solution and computingstability

The remainder of the paper is structured as follows InSection 2 a nonlinear integrated programming model aboutLIP considering returns in e-commerce is designed Section 3proposes the heuristic algorithm named LRCAC based onLagrangian relaxation and ant colony algorithm Section 4shows and analyzes the results of different experimentsSection 5 concludes this paper and discusses the futureresearch directions

2 Problem and Mathematic Model

21 Problem Description In e-commerce some returnedmerchandise has a high integrity which makes it usuallynot in need of being repaired and can reenter the saleschannels after simply repackaging [17] Some returned goodshave quality problems they have to be sent back to factoryfor repair Therefore we merge the recycling center withdistribution center as merchandise centers (MCs) with anadditional inspection function MC is responsible for dis-tributing normal goods to the sale regions meanwhile thereturned goods are collected toMCsAfter inspecting atMCsthe returned goods with quality problems are sent back tofactory and the other returned goods are resalable as normalgoods after simply repackaging Customers can choose toreturn goods in e-commerce and quantity of the returnsis uncertain [18] However for a certain sale region (SR)the quantity of the returns can be usually seen as stochasticvariable

The objective of this paper is to determine the quantitylocations order times and order size of MCs in the closed-loop logistics network in e-commerce The final target isto minimize the total cost and improve the efficiency oflogistics operations The involved decisions are as follows (1)location decisions the optimal number of MCs and theirlocations (2) allocation decisions the corresponding servicerelationship between MCs and sale regions (3) inventorydecisions the optimal order times and order size

22 Assumptions (1) There is a single type of merchandise(2) the capability of factory is unlimited (3) the capabilityof MCs is unlimited (4) the demand and return of eachsale region comply with the normal distribution whoseparameters are fixed (5) the demands of regions are mutuallyindependent (6) returned merchandise is inspected andrepackaged at MCs

23 Notations

Sets

I Set of SRJ Set of candidate MC

Constants

119891119895 Fixed cost (annual) administrative and operationalcost of MC119895

119905119895 Shipping cost per unit of merchandise between fac-tory and MC119895

119888119894119895 The delivering cost per unit of merchandise betweenSR119894 and MC119895

ℎ119895 The inventory holding cost per unit of merchandiseper year at MC119895

119901119895 Ordering cost per time at MC119895119897119895 Lead time at MC119895120583119894 Mean of annual demand at SR119894

Mathematical Problems in Engineering 3

1205902119894 Variance of annual demand at SR119894

119911120572119895 Standard normal deviate such that 119901 (119911 le 119911120572119895) = 120572119895

120572119895 Service level of MC119895

119903119894 The quantity of return at SR119894

119908 The probability of quality problem product in returngoods

119889 Repacking cost per unit returned merchandise

Decision Variables

119883119895 1 if the candidate MC119895 is selected as a MC and 0otherwise

119884119894119895 1 if SR119894 is served by MC119895 and 0 otherwise

119876119895 Optimal order size at MC119895

119873119895 Optimal order times at MC119895

24 Model Formula

(1) Location Cost The construction cost of MC119895 is given bysum119899119895=1 119891119895119883119895

(2) Inventory Cost The total annual inventory cost consistsof ordering cost and inventory holding cost according toliteratures [19 20] it is given by 119901119895119873119895 + ℎ119895(1198631198952119873119895)

(3) Safety Stock CostThe demand in the lead time 119897119895 atMC119895 isradic119897119895sum119894isinI 120590

2119894 119884119894119895 so the safety stock is 119911120572119895radic119897119895sum119894isinI 120590

2119894 119884119894119895 and the

safety stock cost is given by ℎ119895119911120572119895radic119897119895sum119894isinI 1205902119894 119884119894119895

(4) Transportation Cost The transportation cost consists ofcost from factory to MC cost from MC to customer regionfor forward logistics cost from customer region to MC andcost from MC to factory for reverse logistics So the trans-portation cost is given by sum119895isinJ 119905119895sum119894isinI[120583119894 minus 119903119894(1 minus 119908)]119883119895119884119894119895 +

sum119895isinJ sum119894isinI 119888119894119895119908119903119894119883119895 + sum119895isinJ sum119894isinI 119905119895119908119903119894119883119895 + sum119895isinJ sum119894isinI 119888119894119895120583119894119884119894119895

(5) Repacking Cost The returned goods without qualityproblems need to be repacked before reentering to salechannel so the repacking cost is given by sum119895isinJ sum119894isinI 119889(1 minus

119908)119903119894119884119894119895To sum up the location-inventory model with returned

merchandise (RLIP) is

min119885 = sum119895isinJ

(119891119895119883119895 + 119901119895119873119895119883119895 + ℎ119895119863119895

2119873119895119883119895

+ℎ119895119911120572119895radic119897119895sum119894isinI

1205902119894 119884119894119895)

+ sum119895isinJ

119905119895sum119894isinI

[120583119894 minus 119903119894 (1 minus 119908)]119883119895119884119894119895

+ sum119895isinJ

sum119894isinI

119888119894119895119908119903119894119883119895 + sum119895isinJ

sum119894isinI

119905119895119908119903119894119883119895

+ sum119895isinJ

sum119894isinI

119888119894119895120583119894119884119894119895 + sum119895isinJ

sum119894isinI

119889 (1 minus 119908) 119903119894119884119894119895

(1)st

sum119895isinJ

119884119894119895 = 1 119894 isin I (2)

119884119894119895 minus 119883119895 le 0 119894 isin I 119895 isin J (3)

119883119895 = 0 1 119895 isin J (4)

119884119894119895 = 0 1 119894 isin I 119895 isin J (5)

119873119895 ge 0 119895 isin J (6)

119876119895 ge 0 119895 isin J (7)The objective function (1) is to minimize the systemrsquos

total cost Constraint (2) ensures that each sale region mustbe assigned to a MC Constraint (3) stipulates that theassignment can only bemade to the selectedMC Constraints(4) and (5) are standard integrality constraints Constraints(6) and (7) are nonnegative constraints

3 Solution Approach

On the one hand Lagrangian relaxation algorithm (LR)is used to solve the complex optimization problem veryoften It can obtain a near-optimal solution by analyzing theupper bound and lower bound of objective function But itseffectiveness mainly relies on the performance of subgradientoptimization algorithm The speed of convergence becomesmore and more slow with the increasing of the numberof iterations On the other hand AC has great ability oflocal searching If there is an appropriate initial solution theperformance of ACwill be good To adopt their strong pointswhile overcoming their weak points we design a two-stagealgorithm In the first stage we use LR algorithm to get anear-optimum solution In the second stage let the solutionobtained from the first stage be the initial solution we use ACto further improve it

The abstract idea of solution approach is described asfollows Firstly we give the formula for solving optimal orderquantity 119876119895 and optimal order times 119873119895 which also relyon the decision variables 119883119895 and 119884119894119895 Secondly we use LRalgorithm to get a near-optimal solution by computing thelower bound and upper bound of objective function Thenlet the near-optimum solution obtained fromLR be the initialsolution we use AC to further improve it

31 Finding the Optimal Order Quantity and Optimal OrderTimes In the model (1)ndash(7) the decision variable 119873119895 onlyhas appeared in the objective function Also the objectivefunction is convex for 119873119895 gt 0 Consequently we can obtainthe optimal value of 119873119895 by taking the derivative of theobjective function with respect to119873119895 as119873119895 = radicℎ1198951198631198952119901119895119883119895where119863119895 = sum119894isinI[120583119894 minus 119903119894(1 minus 119908)]119884119894119895


As we know the optimal order quantity 119876119895 = 119863119895119873119895 sothere is

119876119895 =119863119895

119873119895=

119863119895

radicℎ1198951198631198952119901119895

= radic2119901119895119863119895

ℎ119895

= radic2119901119895sum119894isinI [120583119894 minus 119903119894 (1 minus 119908)] 119884119894119895

ℎ119895

(8)

32 Transforming the Objective Function In order to applythe LR algorithm we transform the objective function aslinear teams and nonlinear teams separately The objectivefunction can be rearranged as follows


[119891119895 + (119905119895 + 119888119894119895)sum119894isinI

119908119903119894]119883119895

+ sum119895isinJ

(119901119895119873119895 + ℎ119895119863119895

2119873119895)119883119895

+ sum119895isinJ

sum119894isinI

[119888119894119895120583119894 + 119889 (1 minus 119908) 119903119894

+ 119905119895120583119894 minus 119905119895119903119894 (1 minus 119908)] 119884119894119895

+ sum119895isinJ

ℎ119895119911120572119895radic119897119895radicsum119894isinI

1205902119894 119884119894119895

= sum119895isinJ

(1198911015840119895119883119895 + sum119894isinI

1198881015840119894119895119884119894119895 + ℎ1015840119895119883119895 + 120587119895radicsum119894isinI

1205902119894 119884119894119895)

(9)

where 1198911015840119895 = 119891119895 + (119905119895 + 119888119894119895) sum119894isinI 119908119903119894 1198881015840119894119895 = 119888119894119895120583119894 + 119889(1 minus 119908)119903119894 +

119905119895120583119894minus119905119895119903119894(1minus119908) ℎ1015840119895 = 119901119895119873119895+ℎ119895(1198631198952119873119895) and 120587119895 = ℎ119895119911120572119895radic119897119895

33 Lagrangian Relaxation

331 Finding a Lower Bound To solve this problem weintend to use Lagrangian relaxation embedded in branch andbound In particular we relax constraint (2) to obtain thefollowing Lagrangian dual problem

max120582

min119885

997888rarr sum119895isinJ

(1198911015840119895119883119895 + sum119894isinI

1198881015840119894119895119884119894119895 + ℎ1015840119895119883119895

+ 120587119895radicsum119894isinI

1205902119894 119884119894119895) + 120582119894sum119894isinI

(1 minus sum119895isinJ

119884119894119895)

=119899

sum119895=1

(1198911015840119895119883119895 + sum119894isinI

(1198881015840119894119895 minus 120582119894) 119884119894119895

+ ℎ1015840119895119883119895 + 120587119895radicsum119894isinI

1205902119894 119884119894119895) + sum119894isinI

120582119894

st (3) ndash (7) (10)

For fixed values of the Lagrange multipliers 120582119894 we wantto minimize (10) over the location variables 119883119895 and theassignment variables 119884119894119895 We separate the linear teams andnonlinear teams

(1) For each MC119895 let119881119895 = 1198911015840119895 +sum119894isin119868min(0 1198881015840119894119895 minus120582119894) + ℎ1015840119895and let

119883119895 = 1 119881119895 le 0

0 119881119895 gt 0(11)

If all 119881119895 values are positive we identify the smallestpositive 119881119895 and set the corresponding 119883119895 = 1 Theassignment variables are then easy to determinesetting as follows

119884119894119895 = 1 119883119895 = 1 1198881015840119894119895 minus 120582119894 le 0

0 otherwise(12)

(2) However the presence of the nonlinear terms makesfinding an appropriate value of 119881119895 difficult So weneed to solve a subproblem as the following form foreach candidate MC119895

SP (119895) 1198811015840119895 = min sum119894isinI

119887119894119885119894 + radicsum119894isinI

120593119894119885119894

st 119885119894 isin 0 1 119894 isin I

(13)

where 119887119894 = 1198881015840119894119895 minus 120582119894 120593119894 = 12058721198951205902119894 ge 0

In (13) we use 119885119894 to substitute 119884119894119895The solution of subproblem SP(119895) refers to literature

[19] the solution of (10) is the summary of SP(119895) and1198911015840119895 To get the lower bound we need to find the optimalLagrange multipliers We do so using a standard subgradientoptimization procedure as illustrated in literatures [21 22]The optimal value of (10) is a lower bound of the objectivefunction (1)

332 Finding an Upper Bound We find an upper bound asfollows

We initially fix the MC locations at those sites for which119883119895 = 1 in the current Lagrangian solutionThenwe assign SRto MCs in a two-phased process

119878119905119890119901 1 For each SR119894 for whichsum119895isinJ 119884119894119895 ge 1 we assign the SR119894to the MC119895 for which 119884119894119895 = 1 and that increases the least costbased on the assignments made so far

119878119905119890119901 2 We process SR119894 for whichsum119895isinJ 119884119894119895 = 0 we assign eachSR to the open MC which increases the least total cost basedon the assignments made so far

Hence for these SRs we consider all possible assignmentsto open MCs and the cost of this stage is the upper bound

34 Ant Colony Clustering According to the clusteringbehavior of ant colony we set the clustering probability 119901119896119894119895(119905)


Start

Transform the objective function

Finding the lower bound ofthe objective function

Finding the upper bound of

Let the near-optimum solution as theinitial solution of AC

Lagr

angi

an re

laxa

tion

Firs

t sta

ge

Initialize the tabu search matrix

Set the MCj as the ant nest Ant k selects a i to its antnest with and taboos the SR

(t) is fullAll the SRs areclustered to MC

Remove theMC of the least

SRs

Conditions ofconvergence are

meeting

End and outputthe final solution

Y

Y

Y

N

N

N

Seco

nd st

age

Get the formula for solving Qj and Nj

Select the near-optimum solution

Impr

oved

ant c

olon

y al

gorit

hm

the objective function

SR

Tub(t)

Tubki

Tubki(t) updates to null matrix update the amount of pheromone

120591ij(t + h) = 120588120591ij(t) + Δ120591ij Δ120591ij = summ

k=1Δ120591kij and record the optimal solution

Figure 1 The working process of the integral two-stage algorithm

to represent the probability of the SR119894 and clustering center 119895at time 119905 The formula of 119901119896119894119895(119905) is shown as follows

119901119896119894119895 (119905) =

120591120572119894119895120578120573

119894119895

sumTub119896119904(119905)= 0 120591120572119894119904120578120573

119904119895

Tub119896119895 (119905) = 0

0 otherwise

(14)

where Tub119896119895(119905) = 0 represents that ant 119896 can cluster SR119894in next step 120591120572119894119895 is the amount of pheromone deposited for

transition from state 119894 to 119895 120572 is the parameter used to controlthe influence of 120591120572119894119895 120578

120573

119894119895 is the desirability of state transition 119894

and 119895 120573 is the parameter of controlling the influence of 120578120573119894119895 119889119894119895is the distance from 119894 to 119895

And the following relationship exists

120578119894119895 =

1

119889119894119895 if 119889119894119895 = 0

1 if 119889119894119895 = 0

(15)


Table 1 Parameters of MCs

MC Coordinate (km) Fixed construction cost (Yuan)Wuhan (j1) (3342 38529) 50Xiangyang (j2) (3322 37609) 45Xiaogan (j3) (3533 38491) 40Yichang (j4) (3397 37528) 45Jingzhou (j5) (3356 37619) 40Huanggang (j6) (3369 38583) 35

Table 2 Parameters of SRs

SR Coordinate (km) Demand (unit)Wuhan (i1) (3342 38529) 673Xiangyang (i2) (3322 37609) 514Xiaogan (i3) (3533 38491) 500Yichang (i4) (3397 37528) 465Jingzhou (i5) (3356 37619) 520Huanggang (i6) (3369 38583) 440Huangshi (i7) (3342 38604) 360Shiyan (i8) (3614 37480) 400Suizhou (i9) (3468 38361) 350Xianning (i10) (3305 38527) 400Enshi (i11) (3271 37357) 410Jingmen (i12) (3433 37613) 510Ezhou (i13) (3362 38583) 400

Table 3 Optimal results

Number of MCs 119873 (unit) 119876 (unit) The SRs assigned to MCj1 2 1410 i1 i3 i6 i7 i10 i13j2 2 930 i2 i8 i9j4 2 642 i4 i11j5 4 776 i5 i12

35 Algorithm Step The integral two-stagealgorithm stepsare shown below

119878119905119890119901 0 We give the formula for solving119876119895 and119873119895 which alsorely on the decision variables119883119895 and 119884119894119895

First Stage

119878119905119890119901 1 Transform the objective function as linear teams andnonlinear teams separately

119878119905119890119901 2 Find the lower bound of objective function by usingthe LR

119878119905119890119901 3 Find the upper bound of objective function by usingthe LR

119878119905119890119901 4 Select the solution whose value is equal or approx-imately equal to the average value of the lower bound andupper bound as near-optimum solution

Xiangyang

EnshiYichang Jingzhou

Jingmen

Wuhan

Xiaogan

Xianning

Huangshi

HuanggangEzhou

ShiyanSuizhou

Hubei

Figure 2 The logistics network obtained by LRCAC

0 50 100 150 200 250 300 350 400 4502

25

3

35

4

45

5

55

times105 Total cost curve

Iteration

Tota

l cos

t val

ue

Figure 3 Trends of optimal objective function value by LRCAC

2

25

3

35

4

45

5


Iteration

Tota

l cos

t val

ue

100 200 300 400 500 600 700 800 900 1000 1100

Figure 4 Trends of optimal objective function value by AC


1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

The objective valueThe mean value

231500

231000

230500

230000

229500

229000

228500

228000

Figure 5 Fluctuation curve of total cost by LRCAC algorithm

Second Stage

119878119905119890119901 5 Let the near-optimum solution be the initial solutionof AC

119878119905119890119901 6 Initialize the tabu search matrix Tub(119905) which is usedto record the served SRs in the time [119905 119905 + 1] Additionallythe Tub(119905) is a 0-1 matrix Tub119896119894(119905) = 1 and SR119894 is tabooedTub119896119894(119905) = 0 SR119894 is free

119878119905119890119901 7 Set the MC119895 as the ant nest 119911119895 Ant 119896 selects a SR119894 toits ant nest 119911119895 with 119901119896119894119895(119905) and taboos the SR If Tub119896119894(119905) is fullgo to Step 8 else repeat Step 7

119878119905119890119901 8 If all the SRs are clustered toMC the Tub119896119894(119905) updatesto null matrix and goes to Step 9 else go to Step 6

119878119905119890119901 9 Update the amount of pheromone 120591119894119895(119905+ℎ) = 120588120591119894119895(119905)+

Δ120591119894119895 Δ120591119894119895 = sum119898119896=1 Δ120591119896119894119895 and record the optimal solution

119878119905119890119901 10 If the conditions of convergence are meeting ter-minate the procedure and output the optimal solution elseremove the MC of the least SRs and go to Step 6

The flowchart for our algorithm is shown in Figure 1

4 Computational Experiments andAlgorithm Analysis

41 Computational Experiment We refer to the logisticsnetwork of company 119870 in Hubei province of China as anexample We convert the latitude and longitude coordinates

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


220000

225000

230000

235000

240000

245000

250000

255000

260000

265000

270000

Figure 6 Fluctuation curve of total cost by AC

Table 4 Statistical results of optimal objective function value of twoalgorithms

Max Min Mean Standarddeviation

Coefficient ofvariation

AC 263678 235791 244118 135673 056LRCAC 231104 224965 230092 93876 041

of some cities in Hubei province and the central meridian toXirsquoan 80 geographic coordinate They are shown in Tables 1and 2 in which the values represent the actual kilometersAnd other parametersrsquo values are as follows randomly gen-erate the values between 100 and 160 as the 120583119894 and assumethat the 1205902119894 is equal to 120583119894 119905119895 = 1 119901119895 = 2 ℎ119895 = 1 119897119895 = 7 (day)120572119895 = 975 119908 = 02 and 119889 = 2

Based on MATLAB 70 platform we programmed theLRCACalgorithm and run it 30 times on the computer (CPUIntel Core2 P7570 226GHz 227GHz RAM 20GB OSWindows 7) the optimal result is in Table 3

The optimal cost is 224965 yuan and logistics network isshown in Figure 2

For comparison we programmed AC algorithm in thesame platform and run 30 times on the same computer Theoptimal objective function values of these two algorithms areshown in Table 4

The two optimization trends of the two algorithms areshown in Figures 3 and 4

The fluctuation curves of optimal objective function in 30times are shown in Figures 5 and 6 respectively

We can see that the LRCAC algorithm can convergemorequickly than AC from Figures 3 and 4 Moreover LRCAC


Table 5 Optimal objective function values of two algorithms

Instance Algorithm Max Min Mean Standard deviation Coefficient of variation

Srivastava 86-8times2 AC 402316 336543 365784 196736 0537847LRCAC 355215 306842 335765 158962 0473432

Perl 183-12times2 AC 598173 528538 563184 257649 0457486LRCAC 553785 498037 528764 149717 0283145

Christofides 69-50times5 AC 696271 620975 667864 456287 0683203LRCAC 633762 582867 619458 287392 04639411


Daskin 95-88times8 AC 1102873 898483 926586 537862 05804771LRCAC 921587 873672 896926 407816 04546819


has better stability than AC which can be easily found fromTable 4 and Figures 5 and 6

42 Algorithms Analysis In this section all the data in ourexperiments come from LRP database of the University ofAveiro [23] A series of experiments show that LRCAC ismore efficient and stable than AC Results of numericalexample in Section 41 show that the related parameters ofLRCAC are reasonable Thus we employ these parameters inthe remainder of this section Each instance was calculated 30times by LRCAC and AC respectively the results are shownin Table 5 In this table Srivastava 86 is the name of thisinstance 8 times 2 means there are 8 SRs and 2 candidate MCsso do others The coordinate of all nodes and the demands ofSRs are given by the database Table 5 shows that LRCAC canobtain better objective function value and stability than AC

5 Conclusion

Customers have a higher return rate in the e-commerceenvironment Some returned goods have quality problemsand need to be sent back to the factory for repair Theothers without quality problems can be reentered in thesales channels just after a simple repackaging process Thisphenomenon puts forward high requirements to the logisticssystem that supports the operation of e-commerce Thisstudy handles the above interesting problem and provides aneffective heuristic The main contributions are as follows

(1) In reality the cost of processing returned merchan-dise is produced by considering the condition thatcustomers are not satisfied with products and returnthem We firstly design a closed-loop LIP model tominimize the total cost which is produced in both for-ward and reverse logistics networks It is able to helpmanagers make the right decision in e-commercedecreasing the cost of logistics and improving theoperational efficiency of e-commerce

(2) The above closed-loop LIP model with returns isdifficult to be solved by analytical method Thusa two-stage heuristic algorithm named LRCAC is

designed by integrating Lagrangian relaxation withAC to solve the model

(3) Results of our experiments show that LRCAC outper-forms AC on both optimal solution and computingstability LRCAC is a good candidate to effectivelysolve the proposed LIP model with returns

However some extensions should be considered in fur-ther work Considering the dynamic of the demand adynamic model should be established Considering the fuzzydemand of customs or related fuzzy costs more practical LIPmodel should be developed Moreover differential evolutionalgorithms (DEs) have turned out to be one of the bestevolutionary algorithms in a variety of fields [24 25] Inthe future we may use an improved DE to find bettersolutions for the LIPs The integration research and practiceof the management of e-commerce logistics system can beconstantly improved

Conflict of Interests

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

Acknowledgments

This work was supported by the National Natural ScienceFoundation of China (nos 71171093 and 71101061) and theFundamental Research Funds for the Central Universities ofChina (nos CCNU13A05049 and CCNU13F024)

References

[1] ldquoEcommerce sales topped $1 trillion for the first time in 2012rdquohttpwwwemarketercomArticleEcommerce-Sales-Topped-1-Trillion-First-Time-20121009649

[2] H Meyer ldquoMany happy returnsrdquo Journal of Business Strategyvol 20 no 4 pp 27ndash31 1999

[3] C R Gentry ldquoReducing the cost of returnsrdquo Chain Store Agevol 75 no 10 pp 124ndash126 1999


[4] C D T Watson-Gandy and P J Dohrn ldquoDepot location withvan salesmenmdasha practical approachrdquo Omega vol 1 no 3 pp321ndash329 1973

[5] A R Singh R Jain and P K Mishra ldquoCapacities-based supplychain network design considering demand uncertainty usingtwo-stage stochastic programmingrdquo International Journal ofAdvanced Manufacturing Technology vol 69 no 1ndash4 pp 555ndash562 2013

[6] O Berman D Krass and M M Tajbakhsh ldquoA coordinatedlocation-inventory modelrdquo European Journal of OperationalResearch vol 217 no 3 pp 500ndash508 2012

[7] M Shahabi S Akbarinasaji A Unnikrishnan and R JamesldquoIntegrated inventory control and facility location decisions inamulti-echelon supply chain network with hubsrdquoNetworks andSpatial Economics vol 13 pp 497ndash514 2013

[8] K Lieckens and N Vandaele ldquoReverse logistics network designwith stochastic lead timesrdquoComputers andOperations Researchvol 34 no 2 pp 395ndash416 2007

[9] S K Srivastava ldquoNetwork design for reverse logisticsrdquo Omegavol 36 no 4 pp 535ndash548 2008

[10] Z Wang D-Q Yao and P Huang ldquoA new location-inventorypolicy with reverse logistics applied to B2C e-markets of ChinardquoInternational Journal of Production Economics vol 107 no 2 pp350ndash363 2007

[11] J-S Tancrez J-C Lange and P Semal ldquoA location-inventorymodel for large three-level supply chainsrdquo TransportationResearch E Logistics and Transportation Review vol 48 no 2pp 485ndash502 2012

[12] A Diabat D Kannan M Kaliyan and D Svetinovic ldquoA opti-mization model for product returns using genetic algorithmsand artificial immune systemrdquo Resources Conservation andRecycling vol 74 pp 156ndash169 2013

[13] K Sahyouni R C Savaskan and M S Daskin ldquoA facilitylocation model for bidirectional flowsrdquo Transportation Sciencevol 41 no 4 pp 484ndash499 2007

[14] G Easwaran and H Uster ldquoTabu search and Benders decom-position approaches for a capacitated closed-loop supply chainnetwork design problemrdquo Transportation Science vol 43 no 3pp 301ndash320 2009

[15] T Abdallah A Diabat and D Simchi-Levi ldquoSustainable supplychain design a closed-loop formulation and sensitivity analy-sisrdquo Production Planning and Control vol 23 no 2-3 pp 120ndash133 2012

[16] Y Li H Guo L Wang and J Fu ldquoA hybrid genetic-simulatedannealing algorithm for the location-inventory-routing prob-lem considering returns under e-supply chain environmentrdquoThe Scientific World Journal vol 2013 Article ID 125893 10pages 2013

[17] D Vlachos and R Dekker ldquoReturn handling options and orderquantities for single period productsrdquo European Journal ofOperational Research vol 151 no 1 pp 38ndash52 2003

[18] Z Lu and N Bostel ldquoA facility location model for logisticssystems including reverse flows the case of remanufacturingactivitiesrdquo Computers and Operations Research vol 34 no 2pp 299ndash323 2007

[19] M S Daskin C R Coullard and Z-J M Shen ldquoAn inventory-location model formulation solution algorithm and computa-tional resultsrdquo Annals of Operations Research vol 110 pp 83ndash106 2002

[20] Z-J M Shen C R Coullard and M S Daskin ldquoA jointlocation-inventory modelrdquo Transportation Science vol 37 no1 pp 40ndash55 2003

[21] M L Fisher ldquoThe Lagrangian relaxation method for solvinginteger programming problemsrdquo Management Science vol 27no 1 pp 1ndash18 1981

[22] M L Fisher ldquoAn applications oriented guide to Lagrangianrelaxationrdquo Interfaces vol 15 no 2 pp 10ndash21 1985

[23] ldquoLocation-Routing Problems (LRP)rdquo httpsweetuaptsimiscf143 privateSergioBarreto

[24] LWang H Qu Y Li and J He ldquoModeling and optimization ofstochastic joint replenishment and delivery scheduling problemwith uncertain costsrdquo Discrete Dynamics in Nature and Societyvol 2013 Article ID 657465 12 pages 2013

[25] H Qu L Wang and Y-R Zeng ldquoModeling and optimizationfor the joint replenishment and delivery problem with hetero-geneous itemsrdquo Knowledge-Based Systems vol 54 pp 207ndash2152013

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


is limited Sahyouni et al [13] designed three generic facilitylocation models that account for the integrated distributionand collection of products in the closed-loop supply chainnetworks the authors described a Lagrangian relaxation-based solution algorithm to solve the models Easwaran andUster [14] offered amixed integer linear programmingmodelto optimize the total cost that consists of location processingand transportation costs of the multimerchandise in closed-loop supply chains they introduced a heuristics solutionapproach that combines Benders decomposition and tabusearch to solve the model Abdallah et al [15] presentedthe uncapacitated closed-loop location-inventory model asensitivity analysis for different parameters of the modelreveals that the value of recovered products is a major factorin the economic feasibility of the closed-loop network Fordealing with returnedmerchandise without quality problemsin e-commerce Li et al [16] developed a practical LIP modelwith considering the vehicle routing under e-supply chainenvironment and provided a new hybrid heuristic algorithmto solve this model

Previous researches on the closed-loop logistics sys-tem optimization mainly focus on the minimization ofthe total cost of the network To our best knowledgefew researches on manufacturingremanufacturing systemconsider returns and concept of green logistics recyclingin logistics network Since customers may be dissatis-fied with merchandise and return it the cost of process-ing returns the cost of inventory and shipping ordertime and size are changed Furthermore research on theLIP with return of closed-loop logistics system is lim-ited

The aim of this study is to develop a practical LIPmodel with the consideration of returns in e-commerce andprovide a new two-stage heuristics algorithm To our bestknowledge this work is the first step to introduce returnsinto the LIP in e-commerce which makes it become morepractical We also provide an effective algorithm namedLagrangian relaxation combined with ant colony algorithm(LRCAC) to solve this model Lagrangian relaxation algo-rithm (LR) can obtain a near-optimal solution by analyzingthe upper bound and lower bound of objective functionBut its effectiveness mainly relies on the performance ofsubgradient optimization algorithm On the other handAC has great ability of local searching If there is anappropriate initial solution the performance of AC willbe good To adopt their strong points while overcomingtheir weak points we combine the two algorithms Resultsof numerical examples show that LRCAC outperforms antcolony algorithm (AC) on optimal solution and computingstability

The remainder of the paper is structured as follows InSection 2 a nonlinear integrated programming model aboutLIP considering returns in e-commerce is designed Section 3proposes the heuristic algorithm named LRCAC based onLagrangian relaxation and ant colony algorithm Section 4shows and analyzes the results of different experimentsSection 5 concludes this paper and discusses the futureresearch directions

2 Problem and Mathematic Model

21 Problem Description In e-commerce some returnedmerchandise has a high integrity which makes it usuallynot in need of being repaired and can reenter the saleschannels after simply repackaging [17] Some returned goodshave quality problems they have to be sent back to factoryfor repair Therefore we merge the recycling center withdistribution center as merchandise centers (MCs) with anadditional inspection function MC is responsible for dis-tributing normal goods to the sale regions meanwhile thereturned goods are collected toMCsAfter inspecting atMCsthe returned goods with quality problems are sent back tofactory and the other returned goods are resalable as normalgoods after simply repackaging Customers can choose toreturn goods in e-commerce and quantity of the returnsis uncertain [18] However for a certain sale region (SR)the quantity of the returns can be usually seen as stochasticvariable

The objective of this paper is to determine the quantitylocations order times and order size of MCs in the closed-loop logistics network in e-commerce The final target isto minimize the total cost and improve the efficiency oflogistics operations The involved decisions are as follows (1)location decisions the optimal number of MCs and theirlocations (2) allocation decisions the corresponding servicerelationship between MCs and sale regions (3) inventorydecisions the optimal order times and order size

22 Assumptions (1) There is a single type of merchandise(2) the capability of factory is unlimited (3) the capabilityof MCs is unlimited (4) the demand and return of eachsale region comply with the normal distribution whoseparameters are fixed (5) the demands of regions are mutuallyindependent (6) returned merchandise is inspected andrepackaged at MCs

23 Notations

Sets

I Set of SRJ Set of candidate MC

Constants

119891119895 Fixed cost (annual) administrative and operationalcost of MC119895

119905119895 Shipping cost per unit of merchandise between fac-tory and MC119895

119888119894119895 The delivering cost per unit of merchandise betweenSR119894 and MC119895

ℎ119895 The inventory holding cost per unit of merchandiseper year at MC119895

119901119895 Ordering cost per time at MC119895119897119895 Lead time at MC119895120583119894 Mean of annual demand at SR119894








Decision Variables





24 Model Formula





2119894 119884119894119895 and the








(119891119895119883119895 + 119901119895119873119895119883119895 + ℎ119895119863119895

2119873119895119883119895

+ℎ119895119911120572119895radic119897119895sum119894isinI

1205902119894 119884119894119895)

+ sum119895isinJ

119905119895sum119894isinI

[120583119894 minus 119903119894 (1 minus 119908)]119883119895119884119894119895

+ sum119895isinJ

sum119894isinI

119888119894119895119908119903119894119883119895 + sum119895isinJ

sum119894isinI

119905119895119908119903119894119883119895

+ sum119895isinJ

sum119894isinI

119888119894119895120583119894119884119894119895 + sum119895isinJ

sum119894isinI

119889 (1 minus 119908) 119903119894119884119894119895

(1)st

sum119895isinJ

119884119894119895 = 1 119894 isin I (2)


119883119895 = 0 1 119895 isin J (4)

119884119894119895 = 0 1 119894 isin I 119895 isin J (5)

119873119895 ge 0 119895 isin J (6)



3 Solution Approach






119876119895 =119863119895

119873119895=

119863119895

radicℎ1198951198631198952119901119895

= radic2119901119895119863119895

ℎ119895


ℎ119895

(8)



[119891119895 + (119905119895 + 119888119894119895)sum119894isinI

119908119903119894]119883119895

+ sum119895isinJ

(119901119895119873119895 + ℎ119895119863119895

2119873119895)119883119895

+ sum119895isinJ

sum119894isinI

[119888119894119895120583119894 + 119889 (1 minus 119908) 119903119894

+ 119905119895120583119894 minus 119905119895119903119894 (1 minus 119908)] 119884119894119895

+ sum119895isinJ


1205902119894 119884119894119895

= sum119895isinJ

(1198911015840119895119883119895 + sum119894isinI

1198881015840119894119895119884119894119895 + ℎ1015840119895119883119895 + 120587119895radicsum119894isinI

1205902119894 119884119894119895)

(9)





max120582

min119885


(1198911015840119895119883119895 + sum119894isinI

1198881015840119894119895119884119894119895 + ℎ1015840119895119883119895

+ 120587119895radicsum119894isinI

1205902119894 119884119894119895) + 120582119894sum119894isinI


119884119894119895)

=119899

sum119895=1

(1198911015840119895119883119895 + sum119894isinI

(1198881015840119894119895 minus 120582119894) 119884119894119895

+ ℎ1015840119895119883119895 + 120587119895radicsum119894isinI

1205902119894 119884119894119895) + sum119894isinI

120582119894

st (3) ndash (7) (10)



119883119895 = 1 119881119895 le 0

0 119881119895 gt 0(11)


119884119894119895 = 1 119883119895 = 1 1198881015840119894119895 minus 120582119894 le 0

0 otherwise(12)


SP (119895) 1198811015840119895 = min sum119894isinI

119887119894119885119894 + radicsum119894isinI

120593119894119885119894

st 119885119894 isin 0 1 119894 isin I

(13)

where 119887119894 = 1198881015840119894119895 minus 120582119894 120593119894 = 12058721198951205902119894 ge 0










Start





Lagr

angi

an re

laxa

tion

Firs

t sta

ge





SRs


meeting


Y

Y

Y

N

N

N

Seco

nd st

age



Impr

oved

ant c

olon

y al

gorit

hm


SR

Tub(t)

Tubki






119901119896119894119895 (119905) =

120591120572119894119895120578120573

119894119895

sumTub119896119904(119905)= 0 120591120572119894119904120578120573

119904119895

Tub119896119895 (119905) = 0

0 otherwise

(14)



120573




120578119894119895 =

1

119889119894119895 if 119889119894119895 = 0

1 if 119889119894119895 = 0

(15)










First Stage





Xiangyang


Jingmen

Wuhan

Xiaogan

Xianning

Huangshi

HuanggangEzhou

ShiyanSuizhou

Hubei


0 50 100 150 200 250 300 350 400 4502

25

3

35

4

45

5

55


Iteration

Tota

l cos

t val

ue


2

25

3

35

4

45

5


Iteration

Tota

l cos

t val

ue

100 200 300 400 500 600 700 800 900 1000 1100



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


231500

231000

230500

230000

229500

229000

228500

228000


Second Stage











1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


220000

225000

230000

235000

240000

245000

250000

255000

260000

265000

270000





AC 263678 235791 244118 135673 056LRCAC 231104 224965 230092 93876 041



















5 Conclusion









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Decision Variables





24 Model Formula





2119894 119884119894119895 and the








(119891119895119883119895 + 119901119895119873119895119883119895 + ℎ119895119863119895

2119873119895119883119895

+ℎ119895119911120572119895radic119897119895sum119894isinI

1205902119894 119884119894119895)

+ sum119895isinJ

119905119895sum119894isinI

[120583119894 minus 119903119894 (1 minus 119908)]119883119895119884119894119895

+ sum119895isinJ

sum119894isinI

119888119894119895119908119903119894119883119895 + sum119895isinJ

sum119894isinI

119905119895119908119903119894119883119895

+ sum119895isinJ

sum119894isinI

119888119894119895120583119894119884119894119895 + sum119895isinJ

sum119894isinI

119889 (1 minus 119908) 119903119894119884119894119895

(1)st

sum119895isinJ

119884119894119895 = 1 119894 isin I (2)


119883119895 = 0 1 119895 isin J (4)

119884119894119895 = 0 1 119894 isin I 119895 isin J (5)

119873119895 ge 0 119895 isin J (6)



3 Solution Approach






119876119895 =119863119895

119873119895=

119863119895

radicℎ1198951198631198952119901119895

= radic2119901119895119863119895

ℎ119895


ℎ119895

(8)



[119891119895 + (119905119895 + 119888119894119895)sum119894isinI

119908119903119894]119883119895

+ sum119895isinJ

(119901119895119873119895 + ℎ119895119863119895

2119873119895)119883119895

+ sum119895isinJ

sum119894isinI

[119888119894119895120583119894 + 119889 (1 minus 119908) 119903119894

+ 119905119895120583119894 minus 119905119895119903119894 (1 minus 119908)] 119884119894119895

+ sum119895isinJ


1205902119894 119884119894119895

= sum119895isinJ

(1198911015840119895119883119895 + sum119894isinI

1198881015840119894119895119884119894119895 + ℎ1015840119895119883119895 + 120587119895radicsum119894isinI

1205902119894 119884119894119895)

(9)





max120582

min119885


(1198911015840119895119883119895 + sum119894isinI

1198881015840119894119895119884119894119895 + ℎ1015840119895119883119895

+ 120587119895radicsum119894isinI

1205902119894 119884119894119895) + 120582119894sum119894isinI


119884119894119895)

=119899

sum119895=1

(1198911015840119895119883119895 + sum119894isinI

(1198881015840119894119895 minus 120582119894) 119884119894119895

+ ℎ1015840119895119883119895 + 120587119895radicsum119894isinI

1205902119894 119884119894119895) + sum119894isinI

120582119894

st (3) ndash (7) (10)



119883119895 = 1 119881119895 le 0

0 119881119895 gt 0(11)


119884119894119895 = 1 119883119895 = 1 1198881015840119894119895 minus 120582119894 le 0

0 otherwise(12)


SP (119895) 1198811015840119895 = min sum119894isinI

119887119894119885119894 + radicsum119894isinI

120593119894119885119894

st 119885119894 isin 0 1 119894 isin I

(13)

where 119887119894 = 1198881015840119894119895 minus 120582119894 120593119894 = 12058721198951205902119894 ge 0










Start





Lagr

angi

an re

laxa

tion

Firs

t sta

ge





SRs


meeting


Y

Y

Y

N

N

N

Seco

nd st

age



Impr

oved

ant c

olon

y al

gorit

hm


SR

Tub(t)

Tubki






119901119896119894119895 (119905) =

120591120572119894119895120578120573

119894119895

sumTub119896119904(119905)= 0 120591120572119894119904120578120573

119904119895

Tub119896119895 (119905) = 0

0 otherwise

(14)



120573




120578119894119895 =

1

119889119894119895 if 119889119894119895 = 0

1 if 119889119894119895 = 0

(15)










First Stage





Xiangyang


Jingmen

Wuhan

Xiaogan

Xianning

Huangshi

HuanggangEzhou

ShiyanSuizhou

Hubei


0 50 100 150 200 250 300 350 400 4502

25

3

35

4

45

5

55


Iteration

Tota

l cos

t val

ue


2

25

3

35

4

45

5


Iteration

Tota

l cos

t val

ue

100 200 300 400 500 600 700 800 900 1000 1100



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


231500

231000

230500

230000

229500

229000

228500

228000


Second Stage











1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


220000

225000

230000

235000

240000

245000

250000

255000

260000

265000

270000





AC 263678 235791 244118 135673 056LRCAC 231104 224965 230092 93876 041



















5 Conclusion









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119876119895 =119863119895

119873119895=

119863119895

radicℎ1198951198631198952119901119895

= radic2119901119895119863119895

ℎ119895


ℎ119895

(8)



[119891119895 + (119905119895 + 119888119894119895)sum119894isinI

119908119903119894]119883119895

+ sum119895isinJ

(119901119895119873119895 + ℎ119895119863119895

2119873119895)119883119895

+ sum119895isinJ

sum119894isinI

[119888119894119895120583119894 + 119889 (1 minus 119908) 119903119894

+ 119905119895120583119894 minus 119905119895119903119894 (1 minus 119908)] 119884119894119895

+ sum119895isinJ


1205902119894 119884119894119895

= sum119895isinJ

(1198911015840119895119883119895 + sum119894isinI

1198881015840119894119895119884119894119895 + ℎ1015840119895119883119895 + 120587119895radicsum119894isinI

1205902119894 119884119894119895)

(9)





max120582

min119885


(1198911015840119895119883119895 + sum119894isinI

1198881015840119894119895119884119894119895 + ℎ1015840119895119883119895

+ 120587119895radicsum119894isinI

1205902119894 119884119894119895) + 120582119894sum119894isinI


119884119894119895)

=119899

sum119895=1

(1198911015840119895119883119895 + sum119894isinI

(1198881015840119894119895 minus 120582119894) 119884119894119895

+ ℎ1015840119895119883119895 + 120587119895radicsum119894isinI

1205902119894 119884119894119895) + sum119894isinI

120582119894

st (3) ndash (7) (10)



119883119895 = 1 119881119895 le 0

0 119881119895 gt 0(11)


119884119894119895 = 1 119883119895 = 1 1198881015840119894119895 minus 120582119894 le 0

0 otherwise(12)


SP (119895) 1198811015840119895 = min sum119894isinI

119887119894119885119894 + radicsum119894isinI

120593119894119885119894

st 119885119894 isin 0 1 119894 isin I

(13)

where 119887119894 = 1198881015840119894119895 minus 120582119894 120593119894 = 12058721198951205902119894 ge 0










Start





Lagr

angi

an re

laxa

tion

Firs

t sta

ge





SRs


meeting


Y

Y

Y

N

N

N

Seco

nd st

age



Impr

oved

ant c

olon

y al

gorit

hm


SR

Tub(t)

Tubki






119901119896119894119895 (119905) =

120591120572119894119895120578120573

119894119895

sumTub119896119904(119905)= 0 120591120572119894119904120578120573

119904119895

Tub119896119895 (119905) = 0

0 otherwise

(14)



120573




120578119894119895 =

1

119889119894119895 if 119889119894119895 = 0

1 if 119889119894119895 = 0

(15)










First Stage





Xiangyang


Jingmen

Wuhan

Xiaogan

Xianning

Huangshi

HuanggangEzhou

ShiyanSuizhou

Hubei


0 50 100 150 200 250 300 350 400 4502

25

3

35

4

45

5

55


Iteration

Tota

l cos

t val

ue


2

25

3

35

4

45

5


Iteration

Tota

l cos

t val

ue

100 200 300 400 500 600 700 800 900 1000 1100



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


231500

231000

230500

230000

229500

229000

228500

228000


Second Stage











1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


220000

225000

230000

235000

240000

245000

250000

255000

260000

265000

270000





AC 263678 235791 244118 135673 056LRCAC 231104 224965 230092 93876 041



















5 Conclusion









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start





Lagr

angi

an re

laxa

tion

Firs

t sta

ge





SRs


meeting


Y

Y

Y

N

N

N

Seco

nd st

age



Impr

oved

ant c

olon

y al

gorit

hm


SR

Tub(t)

Tubki






119901119896119894119895 (119905) =

120591120572119894119895120578120573

119894119895

sumTub119896119904(119905)= 0 120591120572119894119904120578120573

119904119895

Tub119896119895 (119905) = 0

0 otherwise

(14)



120573




120578119894119895 =

1

119889119894119895 if 119889119894119895 = 0

1 if 119889119894119895 = 0

(15)










First Stage





Xiangyang


Jingmen

Wuhan

Xiaogan

Xianning

Huangshi

HuanggangEzhou

ShiyanSuizhou

Hubei


0 50 100 150 200 250 300 350 400 4502

25

3

35

4

45

5

55


Iteration

Tota

l cos

t val

ue


2

25

3

35

4

45

5


Iteration

Tota

l cos

t val

ue

100 200 300 400 500 600 700 800 900 1000 1100



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


231500

231000

230500

230000

229500

229000

228500

228000


Second Stage











1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


220000

225000

230000

235000

240000

245000

250000

255000

260000

265000

270000





AC 263678 235791 244118 135673 056LRCAC 231104 224965 230092 93876 041



















5 Conclusion









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















First Stage





Xiangyang


Jingmen

Wuhan

Xiaogan

Xianning

Huangshi

HuanggangEzhou

ShiyanSuizhou

Hubei


0 50 100 150 200 250 300 350 400 4502

25

3

35

4

45

5

55


Iteration

Tota

l cos

t val

ue


2

25

3

35

4

45

5


Iteration

Tota

l cos

t val

ue

100 200 300 400 500 600 700 800 900 1000 1100



1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


231500

231000

230500

230000

229500

229000

228500

228000


Second Stage











1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


220000

225000

230000

235000

240000

245000

250000

255000

260000

265000

270000





AC 263678 235791 244118 135673 056LRCAC 231104 224965 230092 93876 041



















5 Conclusion









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


231500

231000

230500

230000

229500

229000

228500

228000


Second Stage











1 3 5 7 9 11 13 15 17 19 21 23 25 27 29


220000

225000

230000

235000

240000

245000

250000

255000

260000

265000

270000





AC 263678 235791 244118 135673 056LRCAC 231104 224965 230092 93876 041



















5 Conclusion









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















5 Conclusion









Acknowledgments


References


































Volume 2014




Journal of











Journal of


Function Spaces






Algebra







































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a two-stage algorithm for the closed-loop...

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