Overview of Possible Solvers
An optimisation algorithm in combination with propagation models, allows network plan- ners to develop automatic planning strategies for wireless systems. The goal of an opti- misation algorithm is to determine a solution that satisfies the coverage constraints and to minimise the objective function. In Chapter 1, different types of techniques and algorithms used for solving the problem at hand were discussed.
The problem at hand is a continuous optimisation problem in nature, because APs can be placed anywhere in the area to obtain maximum coverage. Therefore, we are using a continuous optimisation technique to solve the problem. For this reason, algorithms based on a discrete search will not be discussed.
For solving a real world optimisation problem, the choice of the solver is crucial. The following issues can be noticed when considering problems (3.2), (3.3), and (3.6) formu- lated in Section 3.2.
Although the initial problems (3.2) and (3.3) is an integer programming problem, for each N , to verify the feasibility it is necessary to solve a continuous optimisation problem (3.6).
The objective function of (3.6) is discontinuous [1, 23–25, 30, 42, 65, 68, 72, 75] as it is based on path loss function (3.1). As a result, it is not possible to apply any of the classical methods based on the local properties of the functions and their derivatives (such as Newton-based or bundle methods [45, 55]), nor even the methods based on Lipschitz
To confront the discontinuity of the functions met in the telecommunications network design due to obstacles, a few authors [1, 2, 41, 65, 81, 82] have used genetic algorithms. Due to their heuristic nature, it is quite easy to adapt these methods for solving the type of problem under consideration. However, genetic algorithms are very dependant on the initial population. The complex structure of the functions (these functions have a large number of local minima) result in the necessity of having a large population size. Even for a simple real world problem, the evaluation of the objective function is computationally very demanding. Hence, it is very unpractical to use evolutionary algorithms for real world applications.
Other heuristic approaches, such as simulated annealing [10, 28, 41, 42, 90] or neural networks, present the same drawback: although these methods can be easily implemented, they would perform poorly or be too slow for the problem at hand.
In contrast to the above mentioned approaches, the AGOP [56, 57] finds a good solu- tion using a relatively small number of function evaluations. Its operation is explained in Subsection 1.4.3. The algorithm is modified to make it particulary suitable for solving the problem at hand.
Modification of the AGOP
The AGOP is a global solver, which can be run out of the box for finding the solution to problem (3.6). However, since the evaluation of the objective function is very computa- tionally intensive, it is necessary to modify the procedure of the AGOP in order to run the program within a reasonable time [88].
The first modification to the algorithm makes use of the known lower bound of the problem: the value of the function is nonnegative. What is more, the function is zero when the solution is found. If there exists a coverage by the given number of APs, then in many cases the set of optimisers is quite large. As a result, a function value of zero may be found very early on by the algorithm. In such a case, it is not necessary to continue searching, and the algorithm can be exited.
The second modification is based on the sequence of problems that are being solved during the execution of the Main Scheme. It also takes into consideration the geographical nature of the problem: a solution x is structured as a set of geographical points ¯a ∈ Rm.
Network Planning – Proposed Optimisation Model 3.3. Testing
is constructed as follows: Ω = {x1, . . . , xq}, where xi = (x ∗,n−1 0 , . . . , x
∗,n−1
q−m , y1, . . . , ym), x∗,n−1is the solution reached at iteration N − 1, and y ∈Rmis an initial point constructed from the boundaries of the geographical area.
This allows us to reduce the initial size of the set Ω, and therefore to accelerate the execution of the AGOP. Furthermore, it also generates initial points that are potentially closer to the set of optimisers, which is reached faster.
3.3
Testing
In order to provide a facility for determining the value of path loss at each point in the design area, and to be able to compare this value with the maximum allowable path loss,
test points or potential users are distributed in the buildings. This distribution depends on
the given design specification, which could be the areas where the mobile users are expected to congregate to perform their activities (lecture theatres, meeting rooms), or all parts of the design area. To explain the concept of coverage issue better, we use the word user instead of test points or potential users.
To show the validity and effectiveness of our model and algorithm, we examine two buildings used by other researchers for testing. For easier references, we call these factory and office buildings. Their specifications are described in the next subsection. Two build- ings at the Mount Helen campus of the University of Ballarat are also used to check the effect of WLAN parameters and size of the building on the number of APs. Our results are described in Chapter 5.