is the set of m feasible sites of candidate facility,

is the set of N customers to be served,

is the set of all feasible sites and customers (it is also referred to nodes),

is the set of K vehicles available for routing from the facilities, average annual cost per distance traveling from

annual cost of establishing and operating a facility at site, average number of units demands by customer capacity of vehicle distance from node i to node j, traveling by vehicle k from node r to customer i or to customer j, respectively.

The objective function of LRP is

Subject to

In this model, the objective function minimizes the total cost of routing, and establishing and operating the facilities. Constraint (2) ensures that each customer is served by one and only one vehicle. Constraint (3) ensures that the vehicle capacity constraints are not exceeded for any of the vehicles used in routing, while (4) is the route continuity constraint, which implies that the same vehicle should leave every point that is entered by a vehicle. Constraint (5) guarantees that each vehicle is routed from one depot. Constraint (6) guarantees that there are no links between any two depots. Constraints (7) and (8) require that a vehicle can only from a depot if that depot is opened. The last two of constraints are the integer constraints.

Clustering analysis is one of the major techniques in pattern recognition. It is a technique for grouping data and finding where similar data are assigned to the same cluster whereas dissimilar data should belong to different clusters. The conventional clustering methods generate partitions, in a partition; each pattern belongs to one and only one cluster.

The customer aggregate C is divided to cluster by the euclidean measure. The steps of optimal facility location are performed in the following way, which have extensively analyzed and used in location theory to approximate distances between two spatial coordinates. For the sake of simplicity, it is assumed that each facility has its own set m of their coordinates of candidate sites for facility location, and no two facilities share the same location, which are denoted by

Every two coordinates of candidates sites for facility locations are denoted by and, and which satisfied with

where R_ij is one half of the distance between and.

Rules for classification are given as follows.

Rule1             (12)

Ｃ_p can only be divided into PF_i if formula (12) is true.

Rule 2 The coordinates of customers on the circumferences are satisfied that the distance to the nearest two customers is less than that to the sub nearest two points. sub-nearest meaning that they are not included a cluster by reason of distance).

Rule 3 If                      (13)

where r_jq is the distance from any customer C_q to any facility PF_j, maxR_ij is half of the most distance between any two facilities. Either these customers will not be served, or else the candidate’s facility location will be abolished. Last two rules only relate to abnormal customers points.

where Ｎ is the numbers of customers,

are expressed the points and their coordinates, respectively.

r_ip is the distance between any candidate facility PF_i and any customer p. The tangent cycles O₁, O₂, …, O_i, O_j, …, O_m are drawn with the radius R_ij , where R₁, R₂, …, R_i, R_j, …, R_m are their radiuses, respectively.

The fundamental idea is that chaotic sequences belonging to the ergodic points in [0,1] are constructed by the chaos system [9]. They transform to chaotic ergodic variables in solution space of optimization and then search for a good solution. Chaotic sequences are produced by the chaotic equation, of which the Logistic difference equation is one of the most typical models [10]. The Logistic difference equation is defined as.

where is the so called driving parameter(a parameter capturing such factors as fertility rate) initial living area, etc. The initial value, or seed x is in., There if From about x is restricted to and to.

The characteristic of Logistic map is given as follow:

1) When λ to, there are stable points in Logistic map.

2) When λ to, two sequences with seeds only slightly apart diverge very rapidly, that is to say, there exists in the period-doubling bifurcation of the solutions.

3) When λ to, chaos start to come out, and Logistic map isn’t in stable cycling orbit. It is difficult to decide which points with Lebesgueis more than zero in chaotic sequences.

4) When λ to 4, there is a stable chaotic sequence.

In the stable chaotic sequence, any orbit f is dense, that is to say, the points of one orbit f is included in open ball, when and. On the contrary, all points that belong to are pressed on towards any orbit f as precision as possible. In this paper when, the population sequence is chaotic and ergodic and gives search optimal solutions (or near optimal solutions) solutions. A chaotic system is defined as one where two different, but close, seeds result in widely different population sequences. An ergodic system is one where the sequences will approach every possible point arbitrarily, little by little (more formally, for every pair of element of, there is an xk between them). Thus, an ergodic system is one that is “trying to converge to everything”. The Logistic model can be solved not only for the continuous variable optimization, but also discrete variable optimization.

The hybrid algorithm includes two steps. First, preference-fitting rules to solve location-allocation cluster the customers. Second, the optimal routes is obtained by chaos search to be minimized the function. The Logistic chaotic model is transformed for discrete optimization of LRP [11]. A legal route will be determined with the numbers customer of r in one cluster, that is to say, the order of the customer clusters will be arranged [12]. The variable d indicates the route distance in any cluster to be minimized.

where

denoted the distance between customer and customer.

Steps with the hybrid algorithm based on clustering analysis and chaos search are described as follow.

Step 1 Customers are clustered by preference-fitting rules.

1) Initial calculations for R_ij and r_ip are performed.

2) If r_ip < R_ij, the customer p is divided into cluster O_i.

3) Otherwise, return step (1.1).

Step 2 is given by initial random value. The best optimal value of the function, given with the control parameter k gifted with a larger value.

Step 3 Put the results of order arranged into. It is obtainable to a current searching route, and to a current legal route by hybrid algorithm, compared with.

Step 4 Calculate the function values f of the current legal route.

Step 5 Remember the minimum function values and and also remembering the present legal route.

Step 6 Decide whether the cycle is stopped or not. If step = k, stop.

Step 7 A set of is generated by Logistic map, step = step + 1.

Step 8 Return steps 3, and cycling.

Test type (C, T, K), The letters of C, T, K means intercept, Trend, lagging numbers. The first four are contains intercept and trend; The after four are not contains trend.

From the Table 3, it is known that the original data in time trend, after an order difference or second order difference, the data is smooth.

The Johansen co-integration test is accepted for GDP, WLCZ, HZL, WLC. This test results is presented in Table 4.

From Table 4, refused to no co-integration variables of the null hypothesis in 5% level of trace statistics.

Table 3. ADF unit root test results of each variable.

Table 4 Johansen test results.

Note: The letters of r means numbers of co-integration equation, ^*means reject null hypothesis.

This demonstrates that the variable is at least one cointegration equation between the relationships, that there is a long-term equilibrium relationship between variables.

               (2.22)      (0.88)      (0.24)    (17)

The above co-integration equation reflects the longrun equilibrium relationship between variables, but cannot fully reflect the short-term changes relationship between variables. Therefore, it is necessary to establish the error correction model (ECM), to estimate short-term changes between variables. Estimated ECM is give as following:

                (3.01)         (6.84)                       (1.90)         (–2.58)  (18)

From the equation, T statistics test of all variables are significant in level 10%, the error correction of ECMt-1 coefficient is negative, accord with reverse revision mechanism, thus further shows that the error correction model is good, with residual sequence is the white noise.

By comparing the co-integration equation and ECM, We can conclusion that t compared on short-term, freight turnover (HZL) have negative influence on the GDP in long term, but its elasticity coefficient is greater than the short-term elastic coefficient; Output value of logistics (WLCZ) have short-term elastic significantly greater than its long-term flexibility to GDP, and both have different symbols, changes of direction that does not accordance, it indicates that in recent years the logistics infrastructure investment lead to the logistics industry close to a saturation stage in Fujian province, although output value of logistics (WLCZ) promote the growth of GDP in short-term, but due to the expansion of the regular, the GDP growth plays a role of obstacles in the long term.