As an important constituent of wireless network planning, location area planning (LAP) directly affects the stability, security, and performance of wireless network. This work proposes a novel evolutionary algorithm (EA) to solve the LAP problem. The difference between the proposed algorithm and the previous EA is mainly how to encode. The new coding method is inspired by the famous fourcolor theorem in graph theory. Only four numbers are needed to encode all chromosomes by this method. The encoding and decoding process is fast and easy to implement. What is more, illegal solutions can be processed easily in the process of decoding. The design of effective and efficient genetic operators can also benefit from this coding method. The modified evolutionary algorithm with this coding method is especially effective for LAP problem. The use of the principle of fuzzy clustering in initialization can effectively compress the search space in this new algorithm. The computer simulation has been conducted, and the quality of proposed algorithm is confirmed by comparing the results of proposed algorithm with EA and simulated annealing (SA).
The requirement for wireless communication, both realtime and nonrealtime, has grown tremendously recently. As the foundation of mobile communication system, the management of the users’ mobility plays an important role in the design of wireless mobile networks. When the user (which is also called mobile station or mobile terminal) roams in the entire mobile network, the network must transmit data or voice information to the user quickly and accurately. So it is necessary for the network to locate the users by tracing their location. These requirements result in the management of mobile communication. In the management of the users’ mobility in a mobile communication system, LAs mainly have two functions: location inquiry and location update. Location inquiry is performed by the network itself. When the network launches a phone call to a certain user, all the base stations in the LA will page it. In the process, the network tries to locate the user based on its last known location information. In the process of location update, every user updates its location in the network and notifies the network its current location. Whenever the users’ movement of entering a new LA service is checked, the records of home location register (HLR) and visitor location register (VLR) are updated timely. In the GSM network, cells are bounded together to form a series of LAs. It is assumed that the user reports to the system as soon as it crosses the LA boundary. The user’s report is called location update (LU). So when the user enters a new LA, its location is updated, or the location will never be updated. Obviously, the higher the frequency of LA boundary crossing is, the larger the number of LU is. Consequently, frequent LU leads to a high updating cost in location register. It seems to be a good idea to simply increase the size of LA if we want to reduce LU cost. However, it does not mean that only lowering the number of LU can result in a better network performance. The size of the LA is limited by many other factors such as the paging capacity of an mobile switching center (MSC) and the number of available channels. Moreover, there is another constraint in terms of paging cost. Paging cost is caused by the network during a location inquiry when the network tries to locate a user. As the size of LA increases, the cost of paging will also increase because more cells need to be checked to find the called user in a bigger LA. It implies that the size of LA tends to be small. The problem is that LAs with small size will lead to high frequency of boundary crossing. The results are frequent LU and the waste of signal resource. Both paging and LU consume scarce resources such as wireless network bandwidth. Therefore, the size of the LA should not tend to be too small or too big. So a reasonable planning of LA, which aims to minimize users’ LA boundary crossing and guarantee the quality of a signal received on currently allocated channel at the same time, can effectively reduce the total cost of a mobile management system. To build a rational mathematical model, it is not only the number of LAs, carrier frequency, telephone traffic load, paging, and LU frequency but also the assurance of a certain margin for system expansion that we should take comprehensive consideration of. Furthermore, there are many other variables and complicated constraints involved in LAP. It is obvious that the LAP is kind of combination optimization problem with constraints [
LAP is highly valued by network operators. A lot of heuristic algorithms such as artificial neural network (ANN), SA, and EA have been proposed to solve the problem. Lots of research literatures have emerged [
In this paper, an EA with a new coding method is proposed to solve LAP problem. As we discussed previously, a proper coding method can significantly increase the performance of EA. Although a variety of different coding methods for LAP have been presented in previous literatures, the coding method to solve LAP problem still needs to be explored deeply. The existing coding methods in previous works of literature could not be regarded to really satisfy LAP. The processes of encoding and decoding based on them may be very hard to deal with. In this paper, we creatively design a coding method based on the famous fourcolor theorem in graph theory. According to this coding method, only four numbers are needed to encode all the solutions of LAP. The encoding and decoding process is fast and easy to implement when corresponding evolutionary algorithm is used to solve LAP problem. This design of the chromosome decoding method can effectively avoid the generation of illegal solutions. Unfortunately, illegal solutions cannot be avoided completely. Based on this new coding method, we design a repair strategy which can easily handle illegal solutions. The corresponding evolutionary algorithm of this new coding method is used to search the good solutions of LAP problem. To avoid the shortcomings of some algorithms described previouly and make the model more practical, we use a LAP model integrated with the road distribution information and traffic flow. In the process of solving this model, the multiple constraints are integrated into a single constrain. A new constrain that cells adjacent but without road connection should belong to different LAs is added. This strategy can effectively compress the solution space and increase search efficiency. A reasonable coding method can benefit not only the process of encoding and decoding but also the design of genetic operators. In this algorithm, fixed length chromosomes with a size equal to the number of cells in the network are used. Owing to the novel coding method, only four numbers are needed to encode all chromosomes. Therefore, the crossover and mutation of the chromosomes can be very convenient. What is more, in the process of initialization, the use of fuzzy cluster method based on the known information can produce solutions with lower LA boundary crossing. It can effectively compress the search space and enhance the practical value of the algorithm.
The remainder of this paper is organized as follows. Section
In the GSM network, the geographical coverage area is partitioned into cells, while each cell is served by a base station (BS). The essential task of LAP is to group those cells to form a series of LAs. The LU cost and paging cost of the network are all related to the size of the LA. Therefore, the goals of LAP are to lower the user LA boundary crossing and improve the quality of service at the same time. So the division of cells in the network planning should be scientific and reasonable. The study of intelligence optimization algorithms to solve LAP focuses on mathematical model and corresponding algorithm. It is of great significance both in theory study and practical application.
To satisfy the goals of LAP, LA should be divided neither too big nor too small. We hope that the paging cost and the LU cost can be minimized simultaneously when we plan a network. But the fact is that those two objectives are conflicting, and the situation in which these two goals both achieve their smallest value at the same time does not exist. We can only get a set of tradeoff solutions. In order to achieve a satisfactory result, some proper transformation of the two goals should be made. Regarding that the task is to find a balance between the paging cost and LU cost, we take the paging load as a constraint in this paper. Therefore, the problem of finding tradeoff solutions between the two goals is converted to finding a minimum LU cost satisfying corresponding constraints.
When solving optimization problem, it is very important to properly model the problem. However, almost all the mathematic models about LAP commonly used previously only consider the theoretical distribution of cells. In other words, the information provided by the real geographic environment such as the location of mountains, rivers, and streets is ignored. In fact, this environmental factors can directly affect the mobility of users in the real situation. Those effects could be decisive in some situations. It is a common sense that the regions with connective roads have a higher frequency of user mobility, while the regions without connection roads have a relatively smaller users’ movements. For example, if there are mountains or rivers in some area, the users’ movements in this area tend to be small. The LU among LAs is generated by the users’ LA boundary crossing. So the geographic information is crucial in LAP. If we know that there is a river or something else between two cells which hinders people’s movements, we should avoid to assign the two cells to the same LA in the process of dividing. Considering the impact of streets and roads, the boundaries of LA should avoid contacting with the road or paralleling the road to reduce the socalled pingpong effect.
This model is based on an important hypothesis that the user movement between two cells is realized by roads. It means that the LA boundary has no LU if there are no roads crossing the boundary because of rivers or something else hindering the user movement. Figure
The LU between cells in LA.
For reasons of simplicity, roads are roughly divided into three different types: main road, street, and alley. The main notations used in the following equations are listed as follows:
For the entire network system, the total number of LA boundary crossing in busy hour is
Here,
Those constraints are relevant to paging capacity and call traffic capacity. The meanings of (
In mathematics, the fourcolor theorem states that, given any separation of a plane into contiguous regions, no more than four colors are needed to color all the regions of the plane so that no two adjacent regions have the same color. Inspired by the fourcolor theorem, we use four different numbers
cell
cells with different codes belong to different LAs. That is, if
cells with the same code belong to the same LAs. That is, if
if cell
According to the previously described coding method and the method to divide LA, the cells can be divided into different LAs using the following steps.
Set
Set
Check all the cells adjacent to cell
If all the elements in
Divide all the cells in
If those constraints are handled directly, the implementation of EA can be very difficult. So constraints violation adjustment strategy is adopted in this paper. As we know, every cell must belong to an LA, and LA is divided in the interior of MSC. Therefore, the constraints presented in (
For (
For (
For (
The initialization of GA plays an important role in finding the solution effectively. We must make sure that all possible solutions can be generated from the initial population. This paper integrates fuzzy clustering into the initialization. The fuzzy clustering algorithm procedure is very simple and easy to implement. It will be described as follows in detail. At first, fuzzy similarity matrix is determined. Secondly, fuzzy equivalence matrix is calculated. At last, a threshold to the equivalent matrix is set to get equivalence class. The traffic flow between cell
Considering the different distances between cells, we define an operator
Every individual in the population represents a configuration of LA, the population is initialized using the following steps.
Generate a random sequence
Find cells which should not be divided into the same LA according to (
Calculate the equivalence matrix
If
According to the steps above, it can be ensured that the cells with large number of boundary crossing are divided into the same LA. What is more, the situation that two cells which should not be assigned to the same LA are divided into the same LA can be avoided.
Evolutionary algorithm is implemented to solve the LAP problem in this paper. Each individual represents a configuration of LA. The population size
the best individual update: in every generation of the evolution, every newly generated individual by crossover and mutation is compared with the best individual in current population. First of all, their constraints violation values are compared. The individual with a lower constraints violation value is set as the new best individual. If the constraints violation values of the two are equal, the individuals with a lower LU is set as the new best individual,
the population update: the constraints violation values of every individual both in parent population and offspring population are calculated using the following steps.
The violation values
Set
The best individual in current population can maintain the optimal state by means of comparing with every newly generated individual and updating according to the above best individual update method in time. The crossover and mutation with the best individual in the current population can make best use of information kept by the best individual. In the process of dealing with constraints, the normalization method used in this algorithm can evaluate different constrains by giving a qualitative standard. It is a feasible way to avoid the incomparability of different constraints. As a consequence, the new population keeps the highquality individuals from both parent population and the new population.
The crossover and mutation operation with the best individual play an important role in exploring the good individuals in solution space. The method of mutation and crossover is described in detail as follows.
Generate a random number
The algorithm can be implemented by the steps described as follows.
Set the population size
Select the best individual
Make every individual
If the newly generated individual violates (
cell
only one of the two cells supposed as cell
both the two cells have adjacent cells with different codes. Identify the LAs where the adjacent cells are located, and calculate the sum of the traffic flow between the two cells within these LAs. The cell with larger traffic flow will be retained, while the cell with smaller traffic flow is divided into a new LA using the method described in (b).
New individuals are processed according to the coding method in Section
If
In this part, two different test networks are generated to examine the validity of the proposed algorithm. Figures
The roads distribution in
Road type  Number  Traffic flow  Cells crossed by roads 

( 
4 

{11, 12, 17, 22, 23, 24, 25} 
 
( 
7 

{2, 3, 4, 5} 
 
( 
4 

{6, 11, 16, 21, 22} 
The roads distribution in
Road type  Number  Traffic flow  Cells crossed by roads 

( 
4 

{6, 7, 12, 17, 22, 23, 24, 25} 
 
( 
7 

{1, 2, 3, 4} 
 
( 
4 

{6, 11, 16, 21, 22} 
EA results of LAP for given
Data set number  The cost of proposed EA  The cost of EA  The cost of SA 

1 

6302  9934 
2 

5932  10878 
3 

6564  9770 
4 

5928  10326 
5 

5310  9406 
6 

5842  10372 
7 

6278  10686 
8 

6298  8880 
9 

7152  10850 
10 

6242  8226 
EA results of LAP for given
Data set number  The cost of proposed EA  The cost of EA  The cost of SA 

1 

12176  12606 
2 

6632  13426 
3 

7210  12316 
4 

7469  13564 
5 

7224  14040 
6 

7186  12774 
7 

7698  14972 
8 

7728  10484 
9 

7648  12256 
10 

7934  14892 
The result of LAP for given
The result of LAP for given
The result of LAP for given
We use the EA without fourcolor coding method and a SA to solve this model as a contrast to identify the validity of the algorithm in this work. To be fair, the initialization, mutation, and crossover of the EA and SA are the same with the proposed EA. The only difference between the proposed EA and two algorithms compared is the coding method.
The other basic parameters are supposed as follows:
the paging capacity of every cell
the traffic load capacity of single LA is
the total paging of cell
the paging load of every cell obeys uniform distribution on the interval
the population size
The simulation program has been developed within the MATLAB programming environment.
We generate 10 groups of
The results of simulation experiment in Tables
In this paper, we propose an evolutionary algorithm with a novel coding method based on fourcolor theorem to solve an NPhard problem of LAP. The cell informationbased algorithm for LAP problem has proved its ability to significantly reduce signaling costs by computer simulation. In the process of initialization, we use the fuzzy clustering method based on the information of real situation to enhance exploration. The use of the new coding method based on fourcolor theorem increases the efficiency of the proposed algorithm. The constraints are adjusted during the update process using different strategies. The comparisons of the experimental results show the practical applicability of the algorithm.
This work was supported in part by the Natural Science Foundation of China (60974077), the Natural Science Foundation of Guangdong Province (S2011030002886, S2012010008813), the projects of Science and technology of Guangdong Province (2012B091100033), the projects of science and technology of the department of education of Guangdong province (2012KJCX0042), and Zhongshan projects of science and technology (20114A223).