Facility location and inventory control are critical and highly related problems in the design of logistics system for ecommerce. Meanwhile, the return ratio in Internet sales was significantly higher than in the traditional business. Focusing on the existing problem in ecommerce logistics system, we formulate a closedloop locationinventory problem model considering returned merchandise to minimize the total cost which is produced in both forward and reverse logistics networks. To solve this nonlinear mixed programming model, an effective twostage heuristic algorithm named LRCAC is designed by combining Lagrangian relaxation 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 under ecommerce environment.
The increasing progress of information and prevalence of internet in the 21st century has forced the ecommerce to develop in a worldwide range. In 2012, B2C ecommerce sales grew 21.1% to top $1 trillion for the first time in history in the whole world [
Facility location and inventory control are critical problems in the design of logistics system. There is much previous work in these two areas. In fact, there is a mutually dependent relationship among these problems in logistics system. Comprehensive optimizing and logistics activities management should be based on this relationship [
Many papers about the LIP are studied deeply and have made some abundant achievements. In recent years, intelligent algorithms and heuristic algorithm have been used to solve LIP model [
Previous researches on the closedloop logistics system optimization mainly focus on the minimization of the total cost of the network. To our best knowledge, few researches on manufacturing/remanufacturing system consider returns and concept of green logistics recycling in logistics network. Since customers may be dissatisfied with merchandise and return it, the cost of processing returns, the cost of inventory and shipping, order time, and size are changed. Furthermore, research on the LIP with return of closedloop logistics system is limited.
The aim of this study is to develop a practical LIP model with the consideration of returns in ecommerce and provide a new twostage heuristics algorithm. To our best knowledge, this work is the first step to introduce returns into the LIP in ecommerce, which makes it become more practical. We also provide an effective algorithm named Lagrangian relaxation combined with ant colony algorithm (LRCAC) to solve this model. Lagrangian relaxation algorithm (LR) can obtain a nearoptimal solution by analyzing the upper bound and lower bound of objective function. But its effectiveness mainly relies on the performance of subgradient optimization algorithm. On the other hand, AC has great ability of local searching. If there is an appropriate initial solution, the performance of AC will be good. To adopt their strong points while overcoming their weak points, we combine the two algorithms. Results of numerical examples show that LRCAC outperforms ant colony algorithm (AC) on optimal solution and computing stability.
The remainder of the paper is structured as follows. In Section
In ecommerce, some returned merchandise has a high integrity, which makes it usually not in need of being repaired and can reenter the sales channels after simply repackaging [
The objective of this paper is to determine the quantity, locations, order times, and order size of MCs in the closedloop logistics network in ecommerce. The final target is to minimize the total cost and improve the efficiency of logistics operations. The involved decisions are as follows: (1) location decisions, the optimal number of MCs, and their locations; (2) allocation decisions, the corresponding service relationship between MCs, and sale regions; (3) inventory decisions, the optimal order times, and order size.
(1) There is a single type of merchandise; (2) the capability of factory is unlimited; (3) the capability of MCs is unlimited; (4) the demand and return of each sale region comply with the normal distribution, whose parameters are fixed; (5) the demands of regions are mutually independent; (6) returned merchandise is inspected and repackaged at MCs.
Set of SR;
Set of candidate MC.
Fixed cost (annual) administrative and operational cost of
Shipping cost per unit of merchandise between factory and
The delivering cost per unit of merchandise between
The inventory holding cost per unit of merchandise per year at
Ordering cost per time at
Lead time at
Mean of annual demand at
Variance of annual demand at
Standard normal deviate such that
Service level of
The quantity of return at
The probability of quality problem product in return goods;
Repacking cost per unit returned merchandise.
1, if the candidate
1, if
Optimal order size at
Optimal order times at
To sum up, the locationinventory model with returned merchandise (RLIP) is
The objective function (
On the one hand, Lagrangian relaxation algorithm (LR) is used to solve the complex optimization problem very often. It can obtain a nearoptimal solution by analyzing the upper bound and lower bound of objective function. But its effectiveness mainly relies on the performance of subgradient optimization algorithm. The speed of convergence becomes more and more slow with the increasing of the number of iterations. On the other hand, AC has great ability of local searching. If there is an appropriate initial solution, the performance of AC will be good. To adopt their strong points while overcoming their weak points, we design a twostage algorithm. In the first stage, we use LR algorithm to get a nearoptimum solution. In the second stage, let the solution obtained from the first stage be the initial solution; we use AC to further improve it.
The abstract idea of solution approach is described as follows. Firstly, we give the formula for solving optimal order quantity
In the model (
As we know the optimal order quantity
In order to apply the LR algorithm, we transform the objective function as linear teams and nonlinear teams separately. The objective function can be rearranged as follows:
To solve this problem, we intend to use Lagrangian relaxation embedded in branch and bound. In particular, we relax constraint (
For fixed values of the Lagrange multipliers
For each
However, the presence of the nonlinear terms makes finding an appropriate value of
In (
The solution of subproblem SP(
We find an upper bound as follows.
We initially fix the MC locations at those sites for which
Hence, for these SRs, we consider all possible assignments to open MCs, and the cost of this stage is the upper bound.
According to the clustering behavior of ant colony, we set the clustering probability
And the following relationship exists:
The integral twostagealgorithm steps are shown below.
The flowchart for our algorithm is shown in Figure
The working process of the integral twostage algorithm.
We refer to the logistics network of company
Parameters of MCs.
MC  Coordinate (km)  Fixed construction cost (Yuan) 

Wuhan ( 
(3342, 38529)  50 
Xiangyang ( 
(3322, 37609)  45 
Xiaogan ( 
(3533, 38491)  40 
Yichang ( 
(3397, 37528)  45 
Jingzhou ( 
(3356, 37619)  40 
Huanggang ( 
(3369, 38583)  35 
Parameters of SRs.
SR  Coordinate (km)  Demand (unit) 

Wuhan ( 
(3342, 38529)  673 
Xiangyang ( 
(3322, 37609)  514 
Xiaogan ( 
(3533, 38491)  500 
Yichang ( 
(3397, 37528)  465 
Jingzhou ( 
(3356, 37619)  520 
Huanggang ( 
(3369, 38583)  440 
Huangshi ( 
(3342, 38604)  360 
Shiyan ( 
(3614, 37480)  400 
Suizhou ( 
(3468, 38361)  350 
Xianning ( 
(3305, 38527)  400 
Enshi ( 
(3271, 37357)  410 
Jingmen ( 
(3433, 37613)  510 
Ezhou ( 
(3362, 38583)  400 
Based on MATLAB 7.0 platform, we programmed the LRCAC algorithm and run it 30 times on the computer (CPU: Intel Core2 P7570 @2.26 GHz 2.27 GHz; RAM: 2.0 GB; OS: Windows 7); the optimal result is in Table
Optimal results.
Number of MCs 


The SRs assigned to MC 


2  1410 


2  930 


2  642 


4  776 

The optimal cost is 224965 yuan, and logistics network is shown in Figure
The logistics network obtained by LRCAC.
For comparison, we programmed AC algorithm in the same platform and run 30 times on the same computer. The optimal objective function values of these two algorithms are shown in Table
Statistical results of optimal objective function value of two algorithms.
Max  Min  Mean  Standard deviation  Coefficient of variation  

AC  263678  235791  244118  135673  0.56 
LRCAC  231104  224965  230092  93876  0.41 
The two optimization trends of the two algorithms are shown in Figures
Trends of optimal objective function value by LRCAC.
Trends of optimal objective function value by AC.
The fluctuation curves of optimal objective function in 30 times are shown in Figures
Fluctuation curve of total cost by LRCAC algorithm.
Fluctuation curve of total cost by AC.
We can see that the LRCAC algorithm can converge more quickly than AC from Figures
In this section, all the data in our experiments come from LRP database of the University of Aveiro [
Optimal objective function values of two algorithms.
Instance  Algorithm  Max  Min  Mean  Standard deviation  Coefficient of variation 


AC  402316  336543  365784  196736  0.537847 
LRCAC  355215  306842  335765  158962  0.473432  



AC  598173  528538  563184  257649  0.457486 
LRCAC  553785  498037  528764  149717  0.283145  



AC  696271  620975  667864  456287  0.683203 
LRCAC  633762  582867  619458  287392  0.4639411  



AC  890756  821789  867246  564782  0.651236 
LRCAC  843685  783804  810473  395647  0.488168  



AC  1102873  898483  926586  537862  0.5804771 
LRCAC  921587  873672  896926  407816  0.4546819  



AC  1516428  1256731  1478356  624638  0.422522 
LRCAC  1258904  1078396  1157832  428373  0.3699785 
Customers have a higher return rate in the ecommerce environment. Some returned goods have quality problems and need to be sent back to the factory for repair. The others without quality problems can be reentered in the sales channels just after a simple repackaging process. This phenomenon puts forward high requirements to the logistics system that supports the operation of ecommerce. This study handles the above interesting problem and provides an effective heuristic. The main contributions are as follows.
In reality, the cost of processing returned merchandise is produced by considering the condition that customers are not satisfied with products and return them. We firstly design a closedloop LIP model to minimize the total cost which is produced in both forward and reverse logistics networks. It is able to help managers make the right decision in ecommerce, decreasing the cost of logistics and improving the operational efficiency of ecommerce.
The above closedloop LIP model with returns is difficult to be solved by analytical method. Thus, a twostage heuristic algorithm named LRCAC is designed by integrating Lagrangian relaxation with AC to solve the model.
Results of our experiments show that LRCAC outperforms AC on both optimal solution and computing stability. LRCAC is a good candidate to effectively solve the proposed LIP model with returns.
However, some extensions should be considered in further work. Considering the dynamic of the demand, a dynamic model should be established. Considering the fuzzy demand of customs or related fuzzy costs, more practical LIP model should be developed. Moreover, differential evolution algorithms (DEs) have turned out to be one of the best evolutionary algorithms in a variety of fields [
The authors declare that there is no conflict of interests regarding the publishing of this paper.
This work was supported by the National Natural Science Foundation of China (nos. 71171093 and 71101061) and the Fundamental Research Funds for the Central Universities of China (nos. CCNU13A05049 and CCNU13F024).