Omni-Channel Product Distribution Network Design by Using the Improved Particle Swarm Optimization Algorithm

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


Introduction
Over the last two decades, the product transportation of the retail industry was largely driven by the Internet Technology development.More and more retailers not only have the traditional offline stores but also start to have the online stores to satisfy the demands of customers from the Internet.As the emergence of new online channels, retailers begin to transform from traditionally storefront-based retailers to multi-channel retailers ( [1][2][3]) and consider the operation management on two channels.This gave birth to the concept of Omni-Channel integration.
Omni-Channel integration is the deep integration of online services, offline experience, and modern logistics in new retail era.It makes the supply chain more complex.The operation goal of Omni-Channel integration supply chain is to meet customer demands with the lowest costs.Saghiri et al. [4] proposed that it is a challenging task to analyze the distribution strategy of Omni-Channel integration.The supply chain usually contains many levels and each level has many sellers, which leads to the complexity of the supply chain network.Besides, the geographical locations of the sellers are spread all over the country.These factors have a significant impact on transportation costs.In addition, customer demands are often uncertain which will increase the costs of supply chain.Therefore the optimization for the supply chain with Omni-Channel is a challenge task and it is necessary to have some efficient approaches to help retailers reduce their costs.For example, Su'ning, one of the largest retailers with Omni-Channel supply chain in China, adopted a distribution method which is citycentered distributed to the county and rural areas step by step and constructed a three-level logistics distribution system.This system is "National Logistics Center-Regional Logistics Center-City Logistics Center".As of June 2017, it has covered 297 prefecture-level cities and above, with about 2079 authorized chain stores and 833 authorized service outlets.It launched "100-City Half-Day" service to require products that are delivered to the customer within half day after the order is placed, which poses greater challenges to the distribution of logistics.
Therefore, facing such a large supply chain network, building an efficient, high-quality, and low-cost distribution network is the key to improve competitiveness and sustainable development of enterprises.The main aim of this paper is to construct a highly efficient supply chain system to minimize costs and improve service quality.
This paper is organized as follows.Section 1 introduces the background of Omni-Channel distribution.The related literature is reviewed in Section 2 from the perspective of LRP, uncertain demand, and solving methods, respectively.In Section 3, a distribution network for Omni-Channel supply chain is proposed and a model for this problem is built.To solve this model, one algorithm based on the traditional PSO is developed in Section 4 by introducing the collaborative idea to reduce the coding dimension, improving the boundary processing strategy, and introducing the mutation operator to expand the search space.Sections 5 and 6 are the numerical results and sensitivity analysis based on one case study.Section 7 concludes the whole paper.

Literature Review
This paper mainly focuses on the location and routing problem (LRP) for Omni-Channel supply chain.Early LRP issues were limited to the location of a single facility.Maranzana [5] studied the problem of supplier location with the goal of minimizing transportation costs.Subsequently, many researchers conducted further research on the systematic and networking nature of transportation and location decisions.Govindan et al. [6] introduced a two-tier supply chain with a time window based on the supply chain network distribution of perishable food, which determined the number of distribution centers and facility locations.Abdulrazik et al. [7] optimized the tertiary supply chain under the background of producing oil palm in Malaysia.In addition, from the types of product distribution in the LRP, products transportation has expanded from single-product distribution to multiproduct distribution.Geoffrion and Graves [8] developed a multi-product distribution model using Mixed-Integer Programming (MIP).Tiwari et al. [9] proposed an algorithm for solving multi-stage, multi-product supply chain network design.Viswanathan and Mathur [10] studied a distribution system consisting of a warehouse multiple products and multiple retailers.Arntzen [11] et al. used a mixed integer programming to establish a multi-product and multistage supply chain model for large multinational companies.Although research on the LRP changes from the initial single location planning to more consideration of the systemic and networking feature of the supply chain, the current researches on supply chain network are mainly focused on two-level or three-level system, and the distribution route planning is mostly one-to-many or many-to-one delivery.For the retail industry, it usually has many supply chain levels and many types of products in the distribution.Therefore, it is necessary to study the supply chain with more levels and more products.This paper is to design a five-level supply chain network for the 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 of customer demands.Different scholars used different methods to research uncertainty.Wang and He [12] introduced the uncertainty of demand and studied the problem of distribution route and warehouse location under the condition of uncertain demand.Zhen et al. [13] used the scenario method to solve the problem of uncertainty in the production process of automobiles.Tapiero and Soliman [14] applied optimal control theory to solve the problem of multi-product transportation, multi-regional production, and cross-time inventory planning under demand uncertainty.Baghalian et al. [15] used multi-objective optimization to model the location and routing problem in low-carbon supply chain and explored the impact of demand uncertainty.In general, customers have great differences in the demand for different products.It is necessary to form a personalized demand network.To describe customer demands more accurately, we set each customer's uncertain demand for different kinds of products to follow different normal distributions instead of letting all customers uniformly meet a normal distribution for all products uniformly.
The LRP with multi-level, multi-product, and personalized demand is a very complex optimization problem.There are two main streams solving the large-scale optimization problem.The first stream is to use new evolutionary algorithms or add local search strategies (e.g.tabu search) to original algorithm.Zhang et al. [16] adopted tabu search algorithm to improve the accuracy and efficiency of cold chain product distribution.Javid and Azad [17] proposed a method of synchronous optimization of location and routing.Two-stage heuristic algorithm based on tabu search and simulated annealing was used to improve the solution space.Xia et al. [18] established the dual-objective programming model and designed an adaptive tabu search algorithm.The second stream of literature studies is to decompose the high-dimensional of large-scale complex problems into lowdimensional simple problems and optimize them separately.Hu et al. [19] studied the large-scale refrigerated truck scheduling problem and combined a variable neighborhood search with a particle swarm optimization to develop a two-stage decomposition algorithm.In 2009, Li and Yao [20] proposed a new co-evolutionary particle swarm optimization algorithm (CPSO), which added random grouping and adaptive weighting strategy to prove that it is effective in high-dimensional indivisible problems.The optimized 30-dimensional CPSO algorithm which has been proposed earlier can be extended to 1000-dimensional problems now.Compared with manufacture industry, distribution in retail industry usually have more supply chain levels, which results in the high dimensions of decision variables.We develop an improved PSO algorithm based on CPSO to efficiently solve the optimization problem.
Therefore, in our paper, we aim to contribute to the current studies on this area from the following three aspects.Firstly, under the Omni-Channel distribution, the supply chain network level has been increased, and a many-tomany distribution network has been formed, which makes the supply chain network distribution more in line with the actual situation.Secondly, we allow each customer's demands to follow different normal distributions for each product and form a demand network accordingly.The demand network can better predict customer demands.Thirdly, based on the traditional PSO, the algorithm introduces the collaborative idea to reduce the coding dimension, improves the boundary processing strategy, and introduces the mutation operator to expand the search space.This method improves the efficiency of the algorithm and provides a solution for solving high dimensional problems.

Model
. .Model Description.This model includes some common characteristics of distribution under Omni-Channel integration, so the conclusions of the model can be useful for many enterprises of supply-chain distribution under Omni-Channel integration.There are five levels in the entire supply chain.These levels include National Logistics Center A, Regional Logistics Center B, City Logistics Center C, Dealer D, and Customer E. Distribution network shall distribute m-type products to two types of customers.The first type of customers E1 is large customers such as companies; the demand of such customers is delivered directly by the national logistics center A. The second type of customers E2 is small supermarkets, like stores.The orders of such customers will be delivered through the regional logistics center B, the city logistics center C, and the dealer D. We can intuitively see the model in Figure 1.
According to the supply chain network, we built a mixed integer programming model to minimize the total supply chain costs.The total costs include three items: transportation costs of the products, fixed operating costs of logistics center, and carbon treatment costs which is to deal with carbon emission generated from the product transportation process.  : Whether the national logistics center a operates, if it operates, then   equals 1; otherwise the value is 0   : Whether the regional logistics center b operates, if it operates, then   equals 1; otherwise the value is 0   : Whether the city logistics center c operates, if it operates, then   equals 1; otherwise the value is 0   : Whether the dealer d operates, if it operates, then   equals 1; otherwise the value is 0 The objective function (2) aims to minimize the total costs of the supply chain network.The constraint (3) requires that the demands of the customer  1 can only be directly delivered by the national logistics center a, and the constraint (4) states that the product can only be provided from the dealer d to the customer  2 .Equations ( 5)- (7) specify that the volume of each logistics center is greater than or equal to the shipment at that point.Constraint (8) ensures that the inventory from the national logistics center a is greater than the total volume of shipped products, and the operation of the national logistics center a is represented by the 0-1 variable.Constraint (9) donates that if the regional logistics center b is not in operation, the quantity of product shipped to the regional logistics center b is zero.While constraint (10) and constraint (11) indicate that if city logistics center c and the dealer d do not operate, the volume of product shipped to the city logistics center c and the dealer d is zero.Constraint (12) and constraint (13) are the value constraints of the decision variable.

Proposed Heuristic Method
The model of LRP presented in Section 3 aims to minimize the total costs in product transportation process.In a largescale transportation network, the code dimension is high, making it difficult to find an optimal solution in a limited time.To solve the problem, we resort to heuristic methods, such as particle swarm optimization (PSO).PSO is a stochastic, parallel optimization algorithm that does not require the properties of the optimized function to be divisible, steerable, and continuous.In addition, to further improve the accuracy and efficiency of the algorithm, the improved PSO is proposed.
. .Particle Swarm Optimization.PSO was proposed by Kennedy and Eberhart [21] and Lian [22] pointed out that this method was a stochastic optimization method based on swarm intelligence and PSO was inspired by the social behavior of bird flocking and their means of information exchange.Due to its easy implementation and fast convergence, PSO has been successfully applied to solve nonlinear optimization problems.Ai and Kachitvichyanukul [23], Goksal et al. [24], and Alinaghian et al. [25] designed an improved random topology particle swarm optimization algorithm (RT-PSO) for time-dependent vehicle routing problems.Hu et al. [19] developed a two-stage decomposition algorithm by combining variable neighborhood search algorithm and PSO to solve the large-scale refrigerated truck scheduling problem.
In PSO, each particle represents a solution to the optimization problem.Particle initialization is generated in a random way, and each particle iteratively updates its position through personal best and global best.Each particle searches for the optimal fitness value in the D-dimensional feasible solution space.There are N particles in a group.In iteration t, the position of particle i is represented as a D-dimensional vector    = (  1 ,   2 , . . .,    ),  = 1, 2, . . ., .At the same time, the velocity of the particle i is also a D − dimensional vector, denoted as    = (V  1 , V  2 , . . ., V   ),  = 1, 2, . . ., .The optimal position searched by the particle i in iteration t is called the personal best, which is denoted as   = ( 1 ,  2 , . . .,   ),  = 1, 2, . . ., .The optimal position searched by the group in iteration t is the global best, denoted as   = ( 1 ,  2 , . . .,   ).After finding the two optimal values, each particle updates its velocity and position according to the following formula: In the standard PSO, there are some shortcomings; for example, (1) when solving the high dimensional optimization problem, the increase of the dimension makes the search space expand into an exponential form, which is prone to premature convergence.That is, the particle group aggregates prematurely to the local optimal solution.(2) Particles may aggregate to local optimum too early in the process of gathering.If there is no escaping mechanism, the optimal solution may not be obtained.Therefore, we proposed improved strategy accordingly.
. .Strategies for Improving PSO.The improved PSO focuses on three aspects: (1) facing the "dimensional disaster" in large-scale problems, cooperative particle swarm optimization (CPSO) is proposed to transform large-scale complex problems into low-dimensional simple problems.(2) When particle flies out of the boundary, the boundary processing strategy is adopted to change the flight trajectory of the particles.(3) When the particles are aggregated, in order to prevent premature convergence to the local optimum, the variation factor is introduced to further explore the search space.
. . .CPSO for Dimensionality Reduction.As the dimension of the optimization problem increases, the performance of all algorithm drops dramatically.An effective solution is the cooperative evolution strategy.Bergh and Engelbrecht [26] combined cooperative evolution strategy with PSO and proposed cooperative particle swarm optimization (CPSO).The basic idea of CPSO is using K independent particle groups to search different dimension directions in the Ddimensional search space.Each particle swarm is updated independently, and no information is shared between the particle swarms.When calculating the fitness value, the position vectors of the optimal particles in the particle swarm are combined together to form a D-dimensional vector to calculate the fitness value.
In this paper, to calculate the traffic volume between supply chain levels, the particle coding in standard PSO is 11-dimensional X(A, B, C, D,  1 ,  2 , m,   ,   ,   ,   ).The first 6 dimensions are the supply chain routing between neighboring level, the 7th dimension demonstrates different products, and the last 4 dimensions represent the node operation decision.According to CPSO, decision variable is classified by supply chain levels.Decision variable can be split into five types of 3-dimensional routing decision Q1(A,  1 , m), Q2(A, B, m), Q3(B, C, m), Q4(C, D, m), Q5(D,  2 , ) and four types of 1-dimensional node operation decision Y1(A), Y2(B), Y3(C), Y4(D).The process is shown in Figure 2. Coding dimension is reduced and the algorithm speed is improved through CPSO.
. . .Improved Boundary Processing Strategy.In the search process, in order to improve the convergence efficiency and reduce the invalid search, the boundary conditions are usually set for the position and velocity to narrow the search range.In the traditional PSO, when a particle goes beyond the range, the general processing method is to set the particle on the boundary.That is, Else if   >   ℎ   =   (17) Through this method, after several iterations, a plurality of particles will aggregate toward the boundary in a plurality of dimensions and the flight path of the particles will tend to be the same.So the efficiency of the particle group is reduced.Thus, the following improvement strategies have been proposed: 3 is a random number distributed between [0, 1), and  3 is a constant on [0.01, 0.5).The value of  3 will change according to the algorithm, the objective function, etc.This paper sets it to 0.05 to enhance the ability of the algorithm to jump out of the boundary region, so that the particles return to the feasible search space.After using the boundary strategy, on the one hand, it can ensure that the particles are located in the feasible search space.On the other hand, it can prevent the particles from accumulating too much to the boundary, resulting in local optimization on the boundary of the search space.It can maximize the search space and improve the quality of the solution.A comparison of the two strategies is shown in Figure 3.
. . .Mutation Operators to Expand Search Space.When PSO appears as local convergence or global convergence, the particles will have an aggregation phenomenon; that is, the particles have the same fitness.The fitness variance is the judgment of the convergence degree of the whole group.The larger the fitness variance is, the more likely the particle swarm is in the search state.When the fitness variance becomes smaller, the particle swarm is tending to converge; when the fitness variance is 0, PSO achieves local optimization or global optimization.The formula for calculating the fitness variance is as follows: And, f = max{1, max 1≤≤ {|  −  V |}}, the function of f is to limit  2 .
N is the total number of particles in the particle group.  is the fitness value of particle i.
If it is local optimum, the algorithm is premature.When the algorithm is premature, it will affect the accuracy of the algorithm.So,    will be the personal best.The variation factor Pr is introduced, thereby changing the forward direction of the particle.So    and    will be updated: The probability of mutation is relatively small; Pr generally takes a number between [0, 1); in this algorithm, Pr = 0.1; rand(d) is a random number distributed between [0, 1).The specific approach is to mutate the particles when they are within 10% of the probability of mutation.Firstly, all the particles are sorted according to the fitness value; secondly, m particles with the best fitness value (m is usually half of the total number of particles) are obtained; then, the former Pr * m particles are mutated as follows:  where  is a random variable obeying the standard normal distribution N(0, 1).The purpose of using the mutation factor is to avoid the particle group falling into local optimum and better approaching the global optimum.When the algorithm appears premature, the mutation can change the direction of the entire particle swarm, and the particle swarm reaches a new field for search.By introducing the mutation operator, the algorithm can expand the search space when the search is stagnant.The improved PSO flow is shown in Figure 4.The pseudocode of improved PSO is presented in Box 1.

Numerical Experiments
In this section, some numerical experiments are performed to verify the effectiveness of the proposed model of multiproduct distribution and location problem based on uncertain demand networks.The programming language is C#, and the experiments are completed on a PC (Intel Core i7, 2.6 GHz; Memory, 8G).
. .Parameter Setting.According to the background of the distribution network under Omni-Channel integration, the parameters of this experiment are set.The location and routing process is the same as the model description in Section 3. To demonstrate the broad applicability of the model, some of the parameters of the experiment were generated based on a random probability distribution shown in Table 1.
The unit transportation cost G is uniformly generated according to different road conditions.The operating costs F of dealers for different levels are uniformly generated according to the land price and scale.The capacity O of the distribution center is uniformly distributed according to the product category and scale.This article deals with the uncertain customer demand as the following two steps.
Step .Each customer has different demands for different products.A set of scenarios with different demands are set according to the characteristics of the normal distribution function.Step .Let different customer demands for different products meet different normal distributions.This creates a network of demand of each customer for each product, rather than allowing all customers to meet only one normal distribution demand for each product, which could predict customer demands more accurately.
. .Experimental Results.This paper studies multi-product distribution under the uncertainty of customer demand, so the experiment is divided into four groups according to the changes of customer demand.Each group of experiments is divided into five different situations according to the number of suppliers and customers.The results in Table 2 show 5 experimental scales with different changes of demand.The objective function value and running time are calculated by the improved PSO and CPLEX solver."1-1-2-4-10 (2+8)" represents one national logistics center, one regional logistics center, two city logistics centers, four dealers, and ten customers (two of them are first-type customers and eight of them are second-type customers).
In small-scale cases, the running time of CPLEX is less than the improved PSO.As the scale increases, the running time of CPLEX increases significantly.For example, in the first three experiments, improved PSO and CPLEX have similar effects.When the scale increased to 85 customers, the running time of CPLEX increased quickly, while the running time of improved PSO is less increased.In general, the results of the CPLEX solver are considered to be the optimal solution.Therefore, the CPLEX solver solves the model faster than the improved PSO method on a small scale.However, when the network exceeds a certain scale, the improved PSO gets results faster than the CPLEX solver.In addition, the average deviation between the improved PSO and the optimal result is about 0.53%, which indicates that the result is accurate and the proposed algorithm is suitable for solving the model.

Case Analysis
Based on the background of the distribution of three types of products in Chengdu, China, the scale is "2-4-12-22-85(7+78)". M Company is a typical retail enterprise and products transportation is under Omni-Channel integration.No. represents the number of customers; D1-D3 represent demand of products 1-3.
Table 4: Delivery route of 3 products for 85 customers (Ton).
No. (2) Through three improvements of particle swarm optimization, such as collaborative dimension reduction, boundary processing, and introduction of mutation factors, the performance of the algorithm is improved.The paper provides an algorithmic solution to solve high-dimensional problems.
In future study, the research results of this paper can be extended to the scale economy of the distribution network, while increasing the uncertainty factors and solving larger scale examples.

. . e 2 F
Index of the Model a: The number of national logistics centers b: The number of regional logistics centers c: The number of city logistics centers d: The number of dealers  1 : The number of first type of customers  2 : The number of second type of customers m: The number of the category of products . .Parameter Symbols   1 : The unit transportation cost of product m delivered to customer  1 through national logistics center a    : The unit transportation cost of product m delivered to regional logistics center b through national logistics center a    : The unit transportation cost of product m delivered to city logistics center c through regional logistics center b    : The unit transportation cost of product m delivered to dealer d through city logistics center c   2 : The unit transportation cost of product m delivered to customer  2 through dealer d  1 : The distance between national logistics center a and customer  1   : The distance between national logistics center a and regional logistics center b   : The distance between regional logistics center b and city logistics center c   : The distance between city logistics center c and dealer d  2 : The distance between dealer d and customer   : The fixed operating costs of national logistics center located at point a F  : The fixed operating costs of regional logistics center located at point b F  : The fixed operating costs of city logistics center located at point c F  : The fixed operating costs of dealer located at point d    : The average inventory of product m in national logistics center located at point a   1 : The demand of customer  1 for product m   2 : The demand of customer  2 for product m T: Unit carbon treatment cost N: A large enough positive number . .Decision Variables   1 : The quantity of product m from national logistics center a to customer  1    : The quantity of product m from national logistics center a to regional logistics center b    : The quantity of product m from regional logistics center b to city logistics center c    : The quantity of product m from city logistics center c to dealer d   2 : The quantity of product m from dealer d to customer  2

14 Discrete
Dynamics in Nature and Society achieved between the distribution network levels, and the demand network considers the different demands of customers for different products.Less attention is concentrated on integration optimization issues of distribution network and demand network in existing research.

Table 1 :
Parameters setting for the experiments.

Table 2 :
Comparison of different kinds of algorithms.

Table 3 :
Annual average demand of 3 products for 85 customers.