Research on an Optimization Model for Logistics Nodes System Layout and Its Solution Algorithm

Based on the logistics nodes system consisting of first-degree logistics node (logistics park) secondary logistics nodes (including logistics center and distribution center), a dynamic logistics nodes location model of multi-period , multi-type cargo flow and multiple logistics nodes is given. The optimization model considers the factors including fixed cost for logistics opening, handling cost and economic of scale of different type logistics nodes. An effective algorithm based on the improved minimum cost-maximal flow algorithm and genetic algorithm is presented according to the characteristic of optimization problem. Finally, a numerical example is provided to validate the proposed model and solution algorithm. The findings indicate that the model proposed in this paper is a useful tool for the investigation of the Multi-period and Multiple Logistics Node Dynamic Location Problem.


Introduction
The logistics network design of a city is very important to reduce the social distribution costs and improve the efficiency of distribution.The literature on the city logistics network design problem is extensive with a wide variety of solution methods.Ballou (1968) was the first to address the dynamic location problem and suggested a heuristic algorithm for its solution.Geoffrion and Graves (1974) provided a solution procedure based on Bender's decomposition and applied it to a real situation for a major food company.Denis (1976) investigated the multi-commodity and multi-period e facility location problem, and gave the heuristics algorithm based on dynamic programming.Alexander (1991) studied the multi-period and multi-stages location problem, and solves the model by the Lagrangian heuristics combined with dynamic programming.John (1997) proposed a dynamic location model with uncertain quantity facilities by applying the heuristics based on decision criteria: the minimization of expected opportunity loss and the minimization of maximum regret.Nozick (2001) presents a combined location model considering facility costs, inventory costs, transportation costs, and customer's responsiveness.A comprehensive review on the facility location model can be found in Andreas Klose et al. (2005).
In this paper, we present an optimization model to solve the capacitated, multi-commodity, multi-period, multi-class logistics node location problem.And an effective algorithm is given which is based on hybrid algorithm and heuristic rules.
The rest of this paper is organized as follows.In Section 2 we present an optimal mathematical model of the capacitated, multi-stage, multi-commodity dynamic logistics nodes location problem.Section 3 provides the development of the algorithm for solving this problem.The fourth section presents an illustrated application, while the last section presents conclusions and suggestions for further research.

Model formulation
In this paper, we consider a city long-term logistics nodes system planning according to its logistics need changes in the future 2 years.The logistics nodes systems planning consists of three stages, taking from 1 to 5 as stage 1, from 5 to 10 as stage 2, and from 10 to 20 as stage 3.The logistics nodes are made up of the first class logistics nodes and the second class logistics nodes.The first class logistics nodes have the logistics hub function, such as Logistics Park.While the second class logistics nodes have the distribution function, such as Logistics Center and Distribution Centers.We denote the first class logistics nodes as LN (I), the second logistics nodes as LN (II).The planning aim is to enhance distribution efficiency and cut down the society logistics cost by optimizing the layout logistics nodes step by step.
The assumptions to the model are as follows.
(1) There are n1 candidate locations of LN(I), n2 candidate locations of LN(II) and m customer demand spots, where LN(I), LN(II) represents the first class logistics nodes.
(2) The candidate logistics nodes are satisfied on traffic conditions and others necessary location conditions.
(3) The design parameters and relative construction cost of each potential logistics nodes are known in advance.
(4) The transportation costs among LN (I) nodes, LN (II) nodes and customers are given.
(5) The demand of customer zones is obtained in advance., and the less the value of agglomeration factors, the more the benefit of scale economy )

Inputs and Sets
) ( 2 t R j : the total shipment of LN(I) i and LN(II) j in time t respectively, I i , J j t i a , t j b : the expanding cost of LN(I) i and LN(II) j in time t respectively, I i , J j The capacitated, multi-stage, multi-commodity, dynamic logistics nodes location problem (CMDLNLP) can be formulated as follows: In the above formulation, the objective function involves three types of costs: transport costs, handle and fixed costs, and the transitional costs related to the dynamic nature of the problem.It aims to minimize the sum of the costs including: the costs of transport commodity from LN (I) to LN (II) and from LN (II) to customer; the fixed cost associated with locating and operating logistics nodes; the transitional costs of logistics nodes due to expanding and closing.
Constraint set (2) stipulates that all shipments from LN (I) to LN (II) must not exceed its capacity.The capacity restriction of LN (II) is ensured by constraint set (3). Constraint set (4) indicates a conservation of flow at each logistics node, while (5) requires that all customer demands must be met.Constraint sets ( 6), ( 7) indicate that the total shipment processed by LN (I) nodes and LN (II) nodes respectively.Constraint set (8), ( 9) imply the LN (I) nodes and LN (II) nodes to be expanded or not respectively.The non-negativity on each shipment is imposed by constraint sets (10)-( 12).Constraint set (13) on ) (t Y k restricts every logistics node to be either open or closed.

Solution algorithm
The above model is a nonlinear optimization model extended from the static location problem.Thus, to this optimization model, it is hard to obtain the optimization solution by classical optimization methods.An effective heuristic algorithm is given which is based on genetic algorithm and heuristic rules in the paper.
The proposed algorithm is segmented into two phases.In phase I, the optimal static solution for the first planning period is obtained by hybrid genetic algorithm.In phase II, based on the above optimal solution, the optimal solutions for the second and third planning period can be acquired by heuristic rules.
(1) the description of heuristic rule ) (t DE j be the change value of the sum of transport costs and operator costs in the objective function due to logistics node j opening ,closing and expanding in period t, respectively.

heuristic rules
(1) opening rule: if the savings in opening logistics node j at time t is more than additional investment with opening, it is rational to open logistics node j.It can be expressed as: (2) closing rule: if the savings in closing logistics node j at time t is more than additional costs of transporting and operating with closing, it is rational to close logistics node j.It can be expressed as: (3) expanding rule: if the savings in expanding logistics node j at time t is more than extra costs with closing, it is rational to expand logistics node j.It can be expressed as: (2) solution algorithm Step 1: sorting by the capacity of logistics nodes and calculating the upper bound of the number of logistics nodes needed.
Step 2: Constructing the virtual network based on the above logistics nodes needed.
Step 3: Applying the hybrid genetic algorithm based on revised ford-fakon algorithm to obtain the static optima solution under T=1(i.e.in the first period planning), and denoting its optimal objective value ) , ( ,where * * , j i n n is the number of LN(I) nodes and LN(II) nodes needed opening respectively.
Step 4: loading the flow of the second planning period into the optimal network of the first planning period, if the capacity of network can be meet the flow demand completely, goto Setp8 ;otherwise goto Step 5.
Step 5: Estimating the logistics nodes closed to the customers without fully satisfied according the above expanding rule, and calculating the minimum costs of expanding for all unsatisfied customers, denoting the extra expanding cost Step 6: Estimating the logistics nodes closed to the customers without fully satisfied according the above opening rule, and calculating the minimum costs of opening for all unsatisfied customers, denoting the extra opening cost , it is reasonable to expand; otherwise, it should take the plan of opening.By this method, the optimal solution under T=2 (i.e.in the first period planning) can be obtained.Go to Step9.
Step 8: Calculating and sorting ascend the ratio of the using of all logistics nodes opening of the optimal solution under T=1, and closing the logistics nodes with less ratio of the using according to the above opening rule.By this method, the optimal solution in T=2 (i.e.in the first period planning) can be obtained.Go to Step9.
Step 9: Loading the flow of the second planning period into the optimal network of the first planning period, and by imitating the operations of Step4-Setp8, the optimal network of the third planning period can be obtained.
Step 10: Getting the finally optimal dynamic solution.
(3) Transformation of virtual network Insert Figure 1 here As Fig. 1 shown, if there are m candidate nodes of LN(I), n nodes of LN(II) and p customer zones, a virtual network can be constructed by adding a original node s and destination node t.
: the distance of node I and node j, where M J I j i , ij cap : the throughput limit of the arc ij (from node i to j) In the above virtual network, there exist the following rules: (1) where C ij denotes the transport costs between node i and j, U ij denotes the maximum of the design capacity of node i and j. (3) where C jk denotes the transport costs between node j and k, U jk denotes the maximum between the design capacity of node j and the demand of customer zone k.
It indicates that the candidate LN(I) i will be opened if ui is equal to 1 and the candidate LN(I) i will be opened if vj is equal to 1.
Based on the revised fork-fankson algorithm, the whole network flow assignment can be obtained and the objective function value is used to calculate the fitness of the associated chromosome.
Crossover operator is applied to the method of partially matched crossover (PMC).In order to ensure the feasibility of the offspring chromosome, it is necessary to fulfill remedy strategy.Zhao (2001) has given a detail description on the remedy strategy.The inversion operator is adopted for mutation operator in this paper.

A Numerical example
Suppose a city logistics nodes system planning is made according to its intending 20 years logistics demand changes, consisting of 3 stages: stage1 is from the first year to fifth year, stage 2 is from the sixth year to tenth year, and stage 3 is from the eleventh year to the twentieth year.
There are 6 potential LN (I) nodes (I), 22 LN (II) nodes (J) and 40 customer demand zones.Other input data is shown in the table1-5.
Insert Table 1 here Insert Table 2 here Insert Table 3 here Insert Figure 2 here The figure2 shows the change curve of objective function in the evolvement of GA algorithm.
It is very important on how to represent the solution of the investigating problem as chromosome and to design genetic operator.In general, the binary code is a good alternative.The representation of chromosome is denoted as

j
DC denotes the logistics node j belonging to the type of LN(II); k c denotes the customer zone k(2)the unit of flow is ten thousand ton/ per year.Table7.The optimization configure of logistics nodes in the second period (total cost every year=75934 ten thousand; where distribution cost =46129 ten thousand, fixed cost=13344 ten thousand, operator cost=13401 ten thousand, the cost of logistics nodes changing=3060 ten thousand(2)Total cost=316634 ten thousand in T=2 notes The logistics node configure changes of the second period is to add the DC4 and DC21, comparing with that of the first period.Table8.The optimization configure of logistics nodes in the third period (total cost every year=86002 ten thousand; where distribution cost =58214 ten thousand, fixed cost=6910 ten thousand, operator cost=16218 ten thousand, the cost of logistics nodes changing=4660 ten thousand,(2)Total cost=3,586,160,000.8yuan ten thousand in T=3 notes The logistics node configure changes of the second period is to expand the capacity of the DC2, DC4, DC14; DC21, LP4; to add new nodes DC6, DC7, DC16, DC22.

Figure 1 .
Figure 1.Virtual logistics network I : set of the candidate locations of first class logistics nodes, indexed by i J: set of the candidate locations of second class logistics nodes, indexed by j jn: the unit shipment cost of commodity m from LN(II) j to customer n in period t i : the initialization design capacity of LN(I) i for commodity m in time t, I i be the sum of transport costs and operator costs in the objective function value (i.e. the sum of TC and OC in the objective function) before the logistics node i expanding and after expanding in period t, respectively.

Table 1 .
The design parameter of potentialLN(I)

Table 3 .
The demand of customer zones (10,000t/year)

Table 4 .
The unit transport cost from LN (I) to LN(II) (yuan/t)

Table 5 .
The unit transport cost from LN(II) to customer zones(yuan/t) LN(II) No.the transport cost from LN(II) to 1-40 customer zones ,respectively.

Table 6 .
The optimization configure of logistics nodes in the first period (T=1) i LP denotes the logistics node i belonging to the type of LN(I);