tilting the basic elements of wireless networks such as radio base stations (RBSs). .... to capture good solutions even thought their mathematical features differ.
OPTIMIZATION APPROACHES FOR WIRELESS NETWORK PLANNING L. Brunetta1 , B. Di Chiara2 , F. Mori1 , M. Nonato3 , R. Sorrentino4 , M. Strappini4 , L. Tarricone2 1
University of Padova
2
University of Lecce
3
University of Ferrara 4 University of Perugia
Abstract
Several optimization approaches have been compared to optimally locate and power size the RBSs of a GSM network, in order to efficiently dimension the transceiver properties without exceeding radioprotection limits. Such optimization tools are embedded within a planning system where several modules interact, exploiting each other’s features. In particular, the optimum planning tools exploit the available radiopropagation models implemented to take into account the different features of current scenarios. Tabu Search methods appear as robust and viable tools for this purpose.
I. Introduction In planning modern telecommunication systems, critical parameters are related to locating, power sizing and tilting the basic elements of wireless networks such as radio base stations (RBSs). Network designers need to rely on sophisticated and user friendly tools in order to accurately estimate the electromagnetic (EM) field levels in complex environments and to cheaply and efficiently dimension the transceiver properties without exceeding radioprotection limits. In order to meet such requirements, a planning system, based on a EM predictioning (EMP) tool, enclosing several radiopropagation models, interconnected with a optimum planning tool (OPT) in which several optimization routines are embedded, was developed ([1]-[3]). In this paper we focus on optimization approaches, comparing metaheuristics such as Tabu Search (TS) and Genetic Algorithms (GA), with analytical methods provided by commercial software packages, solving Mixed Integer Linear and Non Linear Programming Models (MIL/NLPM). In section II the implemented radiopropagation approaches are summarized. Section III is devoted to optimization strategies while computational results are discussed in section IV and conclusions are drawn in section V. II. EM field level estimation and radiopropagation models EM field estimation can be performed by means of several radiopropagation models depending on the geographical properties of the observed scenario. Friis’s formula works well in line-of-sight (LOS) condition, but when applied to urban scenarios, it often results in the over-estimation of actual values. More accurate results can be obtained by more sophisticated approaches, such as the empirical COST 231 Okumura-Hata model ([4]-[5]), the semi-empirical COST 231 Walfisch-Ikegami model ([6]-[7]), and a simple ray-tracing algorithm in order to perform EM field calculation in small geographic areas, in which is considered the first reflection contribution and/or the “single-knife-edge” effects. III. Optimization approaches Different optimization approaches to the network planning problem (NPP) are embedded into the optimization module. Sophisticated metaheuristics, such as GA and TS, are very attractive because of their high flexibility in dealing with objective functions and constraints which can not be easily modeled by algebraic expressions; however, such methods do not guarantee global optimality ([8], [9]). On the other hand, in algebraic analytical models all decisions to be taken are mapped by continuous or integer variables and a given objective function has to be optimized over all values complying with constraints which restrict variables values to those corresponding to feasible solutions in the real problem. Ad hoc analytical methods are used within commercial software for solving such models, providing global optimality under some given conditions, i.e. when dealing with convex problems. As shown, NPP can be modeled as a non linear non convex problem with continuous and binary variables, for which no state-of-the-art package is able to find a guaranteed global optimum. Therefore we study a linear relaxation of such model which can be solved to optimality. Before going into further details, let us formalize NPP. The optimal siting of RBSs and their power sizing (similar arguments can be used for optimum tilting) can be formalized as a combinatorial optimization problem as follows. A finite set of possible RBS sites is given; the maximum radiated power of the antennas is given and intermediate levels are discretized; a finite set of receiver test sites (RTSs) is given, and EM emissions are measured on a subset of grid points (GPs). In order to simplify the notation, we assume that RTSs and GPs coincide, however the model and all solution approaches can be easily generalized. The problem consists of selecting a set of RBSs sites, and assigning a power level to each antenna of the cells of each RBS (zero if the RBS is not selected), so that a given objective function defined on the electric field value at the GPs is minimized, subject to the following constraints: let |ETk| be the total
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r.m.s. electric field value at the kth RTS, |Egk| the r.m.s. electric field value radiated by the gth provider, and |Egk(i,j)| the one due to antenna j of RBS i of provider g. For each k and for a given g, the network coverage bond requires a minimum value for |Egk|, and a minimum value for |Egk(i,j)| for at least one pair (i,j). |ETk| is bounded from above by the radioprotection bond, which takes into account the interaction of multi-provider RBSs in the same scenario. The power sizing f of the antenna of cell j of RBS i is modeled by binary variables yfij, f{1,..,F}, among which at most one can be 1. All such constraints, along with any of the five objective functions, give rise to a non linear non convex mixed-integer programming model. III.1 Objective Functions Since solution quality cannot be translated into a single mathematical function, several optimization criteria were considered and compared. The following five objective functions were implemented: sum minimization of |ETk| over 30% of the RTSs with minimum intensity (Min30%Min); sum minimization of |ETk| over 30% of the RTSs with maximum intensity (Min30%Max); minimization of Min30%Min–Min30%Max (MinSlack); sum minimization of |ETk| over all RTSs (MinSum); minimization of the maximum level of |ETk| over all RTSs (MinMax). III.2 Metaheuristics: Genetic Algorithms and Tabu Search GAs, as already addressed in [1]-[3], rely on the encoding of each feasible solution as a chromosome, that is, in this case, a binary string modeling power level sizing and location of each RBS. A given set of solutions, the population, is iteratively randomly sampled, crossed, and filtered to obtain a new population with better quality individuals. Mutation is realized by flipping single bits in the string on about one third of the population. In TS frameworks ([9]), at each iteration the search focuses on the current solution and on its neighborhood, i.e. a set of solutions obtained by operating a “move” from the current solution. The neighborhood is inspected, looking for the best not forbidden (tabu) solution, which in turn becomes the next current solution and so on. The search does not necessarily follow a descent path, since it is designed to escape local optima. Several kinds of memories are exploited to guide the search: short term memories are used to avoid the visit of already known solutions by forbidding moves operated in the recent past; long term memories are employed either to intensify the search in promising sub-regions of the feasible space, or to diversify the search and leave the current sub-region. In our TS algorithm, the basic move consists of increasing and/or decreasing of one level the power of a single antenna. As a long term memory, each local optimum encountered along the search is recorded, as well as every improving solution. Intensification and diversification are based on identifying power levels common to several such solutions and either encouraging or discouraging moves reaching such power levels. Path re-linking strategies have also been applied. III.3 Mixed Integer Linear and Non Linear Models Since the non linear, non convex model for NPP given above can not be solved to optimality, and low quality local optima are returned when commercial software packages are fed with such models, a linear relaxation has been studied in order to devise a bound. Such relaxation, with enlarged feasible region and whose objective function underestimates the original objective function, has been obtained by approximating the r.m.s. of the electric field, that is the Euclidean norm L2 of its vector, by the sum of the absolute value of the vector components, i.e. its norm L1, exploiting inequality L1(v)/¥2d L2(v)d L1(v) v Rn. IV. Computational Results The proposed methods have been validated on a real case, where a single provider is present. We considered the GSM network of a suburban area of the province of Perugia (Italy) of about 25 km2, with 4 RBSs equipped with 24 antennas. EM field emissions have been measured in 25 RTSs evenly distributed in the area. Results for each approach are shown in Table 1 for each objective function. The relaxed model was implemented only for functions MinSum and MinMax, and solved under GAMS [10]; since it is a relaxation of the original problem, feasibility is not guaranteed, even though in this specific test both solutions turned out to be feasible and were evaluated according to all objective functions. In this case, then, this approach can be seen as another heuristic. The optimal values returned by the model for MinSum and MinMax (reported in bold) bound from below the value of the optimal solution for the respective functions, and allow the evaluation of the quality of any heuristic solution. TS always dominates GA in terms of solution quality, and is within 2,5% from the optimum for function MinMax, and within 19% for MinSum for which the bound is rather loose due to the approximation of L2 by L1. Indeed, the model solutions evaluated by the EMP module, are always worse than TS solution. Moreover, while the MinSum model solution gets a good
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evaluation under most functions (it ranks in between GA and TS), the MinMax model solution is much worse. Therefore, we suggest to use the linear model for the bound but do not rely upon it for problem solving. Concerning computing time of the heuristics, it is related to the number of calls to the EMP module, one for each solution. Linear models were solved by GAMS, with computing time of about 1 second on an Athlon1800, 512Mb under Linux RedHat7.1. The EMP module was used to compute model coefficients. Althought in this case metaheuristics are consistently slower than the optimization tool, it must be mention that while the computing time of the former reasonably grows with instance size, the latter’s one grows exponentially. TS GA LM-MinSum LM-MinMax 0.152 Min30%Min 0.048 / 44 / 46*105 0.062 / 75 / 86*105 0.048 0.370 Min30%Max 0.229 / 57 / 65*105 0.348 / 64 / 73*105 0.262 5 5 0.117 / 58 / 68*10 0.253 / 78 / 88*10 0.202 0.213 MinSlack 0.428 / 48 / 47*105 0.644 / 61 / 71*105 0.458 (0. 341) 0.874 MinSum 0.039 / 130 / 15*106 0.082 / 63 / 75*105 0.078 0.076 (0.038) MinMax Table 1: Computational results for the five objective functions; for GA and TS function value, Cpu time (in seconds on a Pentium III, 128 Mb) and number of calls to the EMP module are reported
V. Conclusions Different optimization approaches are proposed to the optimal power sizing and location of RBSs. Experimental analysis on real data confirm that sophisticated metaheuristics such as Tabu Search are viable tools to deal with optimal network planning. In particular, they easily accommodate any objective function or any additional constraint that technology may require. The proposed relaxed linear model can not guarantee feasible solutions and does not perform very well for MinMax function. Alternative relaxations are at present under investigation, in order to exploit all potentials of state-of-the-art commercial softwares. However, especially when requiring large sets of RTPs, the computing time of such optimization tools consistently deteriorates. In those cases the linear model could still be used to provide a bound. Concerning objective functions, they all appear to be able to capture good solutions even thought their mathematical features differ substantially: the first three functions can not be modeled in MIL/NL models, while MinMax functions are difficult to optimize. In Picture1 the resulting electric field is depicted, and compared to the actual values. It clearly shows the potentials of exploiting sophisticated integrated modeling tools for network planning. Actual values
Min30%Min and MinSum
Min30%Max
MinSlack
MinMax
Picture 1: Comparison among actual field configuration and field distributions obtained with the different objective functions; the used radiopropagation model is free space loss in an area of 25Km2 (all maps refer to the same scale).
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