JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.1 (1-21) Journal of Computer and System Sciences ••• (••••) •••–••• Contents lists available at ScienceDirect Journal of Computer and System Sciences www.elsevier.com/locate/jcss Soft computing methods for WiMax Network Planning on 3D Geographical Information Systems Le Hoang Son a,b , Pham Huy Thong c,d,∗ a Institute of Research and Development, Duy Tan University, Danang, Viet Nam VNU University of Science, Vietnam National University, Hanoi, Viet Nam Division of Data Science, Ton Duc Thang University, Ho Chi Minh City, Viet Nam d Faculty of Information Technology, Ton Duc Thang University, Ho Chi Minh City, Viet Nam b c a r t i c l e i n f o Article history: Received October 2015 Received in revised form May 2016 Accepted 20 June 2016 Available online xxxx Keywords: 3D GIS Particle Swarm Optimization Parallel Random Forest Soft computing WiMax Network Planning a b s t r a c t In this paper, we present an application of soft computing methods for the problem of WiMax Network Planning on 3D Geographical Information Systems (3D GIS) that optimizes both performance of the network (Coverage and Quality-of-Service) and investment costs (the number of base stations and sectors) A pre-processing procedure using latest results of parallel Random Forest classification algorithm to determine valid positions of base stations on a terrain of 3D GIS is proposed Based upon those positions, we design a generalized mathematical model taking into account 3D obstacles in path loss calculation process In order to generate optimal solutions of the model, a hybrid algorithm between greedy BTP and improved Particle Swarm Optimization incorporated with parallel computing is presented Experimental validation of the proposed method in comparison with other relevant ones is performed © 2016 Elsevier Inc All rights reserved Introduction With the growing demands of Internet accesses for scientific and commercial uses nowadays, there is a strong need of high-quality network infrastructures that ensure large bandwidths and high network speeds between access points in a geographic area like a town or a province The problem of WiMax Network Planning on 3D Geographical Information Systems (3D GIS) is indeed one of the most popular topics in current researches This problem aims to determine the optimal number of WiMax Base Stations (BSs), their suitable locations on a given terrain of 3D GIS and their configurations of sectors for the best results of both the performance of the network (i.e Coverage and Quality-of-Service) to all fixed users on the terrain and the investment costs (i.e the minimal numbers of BSs and sectors) It is a multi-objective optimization problem that involves some network parameters and geographic constraints of BS Several articles about soft computing methods and mathematical modeling for this problem were found in the literature Wahl, Stabler & Wolfle [21] presented a 2D model for the path loss calculation of WiMax network planning in a hybrid environment between urban and indoor The prediction concept does not rely solely on the direct ray like empirical models and does not consider hundreds of rays for a single radio link like ray tracing, but focuses on the most dominant path between transmitter and receiver, allowing the computation of the transition from an urban to an indoor scenario and vice versa Admed, Mughni & Akhtar [1] took an overview of some available path loss models such as Okumura & Hata, ECC – 33 and * Corresponding author E-mail addresses: sonlh@vnu.edu.vn (L.H Son), phamhuythong@tdt.edu.vn (P.H Thong) http://dx.doi.org/10.1016/j.jcss.2016.06.009 0022-0000/© 2016 Elsevier Inc All rights reserved JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.2 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• SUI Similarly, the works of Amaldi, Bosio, Malucelli & Yuan [2], Eisenblatter & Geerd [7], Nawrocki, Dohler & Aghvami [13] and Tsourakis & Voudouris [20] introduced several methods for the mathematical modeling of WiMax networks Regarding the soft computing methods, Taplin et al [18] introduced some automatic WiMax network planning on a 2D map such as FTP, BTP, Hill Climb (HC) and Tabu Search (TS) The two first algorithms belong to the greedy approach While FTP tries to increase or decrease the number of sectors in each BS for the best performance of the network, BTP sorts all BSs in the ascending order and puts sectors on them one after another to ensure that each user is served by its best BS The two last ones are neighborhood-based searching algorithms The experiments on three different maps showed that TS achieves the best results among all algorithms, but it has high computational complexity On the contrary, BTP can generate acceptable results in a reasonable time Carneiro et al [4] developed a planning tool that allows the WiMax network planning in the graphical environment of ArcGIS since the objective of this research was to relate the geographical data of a given location, to the number of Base Stations needed to cover that location, in terms of power and capacity The planning tool, based on local search methods, allows users to choose several parameters, from the number of base stations to the number of users, making the network operate in different environments, providing similarities to daily situations Zhang [22] offered a comprehensive explanation on how to design, plan, and optimize WiMax networks by heuristic methods involving the topology, capacity, congestion control, medium access control, scheduling and Quality-of-Service (QoS) Hu, Chen & Banzhaf [10] used the adaptive-population-size Genetic Algorithm with individual representation and genetic variation operations being re-constructed to enhance the search capability of the algorithm Simulation results showed that this algorithm is robust to different scenarios and has better search process than a conventional fixed population size scheme Hurley, Allen, Ryan & Taplin [11] introduced a mathematical model for the automated design of fixed wireless access networks through the automatic selection and configuration of base station sites, and presented a stochastic optimization algorithm to generate the fixed wireless access network infrastructure design Sebastiao et al [15] designed a genetic algorithm-based WiMax planning tool, which provides planners with practical and useful information through quick coverage/capacity based procedures, and outputs the number and position of the base stations and an estimation of the total cost of implementation, based on data provided by different equipment manufacturers It was applied for the zone of Covilhã, Portugal, where GIS are used for representation of rural and sparse urban areas Similar to Carneiro et al [4], Sapumohotti et al [14] also presented Network Planning Cell Tool (NPCET), a network planning tool that was designed for planning rural WiMax networks in Malaysia involving wireless propagation, GIS and maps, network planning and programming Even though the relevant soft computing methods and mathematical models were available, they contain some limitations that should be improved further Firstly, most existing mathematical models did not count for restricted locations of BSs on a terrain so that a BS could be put on unsuitable regions such as lakes and rivers Even if restricted locations are calculated, the models were constructed solely on the basis of flat plane (two-dimensions) but not on 3D so that the calculation from a path loss model is not accurate As such, the available soft computing methods working on those models resulted in less accuracy and ineffectiveness Secondly, some soft computing algorithms limited the objective function to either the performance of the network or the investment costs so that the planning is just semi-automatic Thus, it makes sense to design a generalized 3D mathematical model that takes into account the restricted locations of BSs and an automatic multi-objective WiMax network planning method on 3D GIS Our major ideas in the new algorithm that we call WNPA-3DT are summarized below Firstly, we present a pre-processing procedure using the latest results of parallel Random Forest classification algorithm (Thong, Son & Hoa, [19]) to determine valid positions of BSs on a terrain of 3D GIS Secondly, based upon those positions, we design a generalized mathematical model for the considered problem, taking into account 3D obstacles in the path loss calculation process Thirdly, a hybrid algorithm between the greedy BTP and improved Particle Swarm Optimization (Gong et al., [9]) incorporated with parallel computing is presented in order to generate optimal solutions of the model Experimental validation of the proposed method in comparison with other relevant ones is intensively performed Those ideas are all our contributions in this article The rests of this paper are organized as follows Section introduces the novel method – WNPA-3DT Section validates the proposed approach through various datasets and relevant works Finally, we give some conclusions and outline future works in the last section The proposed algorithm In this section, we describe the proposed algorithm, named as Multi-Objective WiMax Network Planning Algorithm on 3D Terrains (WNPA-3DT) It includes the determination of valid positions of BSs on the terrain, the mathematical modeling of our considered problem, the description of the path loss calculation used in the model and the details of the hybrid algorithm, which is used to specify the optimal planning solutions Those parts are presented in some sub-sections below, respectively 2.1 Terrain classification Given a terrain in the Digital Elevation Model (DEM) format represented by a matrix of height values The aim of this sub-section is to determine some basic objects in that terrain such as mountains, plateaus, hills, flat lands, rivers and lakes From this classification, the valid positions of BSs on the terrain are totally specified This problem belongs to the class of Image Classification which divides the collection of terrain data into two groups: the Training and the Testing sets While the Training set is used to construct a classification model, the Testing is designed for the verification of the performance JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.3 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• of the model In this part, we use the parallel Random Forest (Thong, Son & Hoa, [19]) for both the construction and the verification of the model In the construction step, some criteria for the classification of six basic objects above are defined Each terrain in the Training set is scanned with those criteria to detect objects Those objects are stored in a database according to the DEM Markup Language (DML) standard serving for the construction of the classification model Descriptions of the criteria are shown below • Boundary: a collection of boundary points of an object Basically, the number of those points should be as many as possible for the best prediction Nonetheless, in order to reduce the computational complexity, we keep a certain number of points and use the adaptive BandWidth algorithm (Dagher, [5]) to approximate them into the boundary lines of the object with a given error threshold θ • Inner values: ◦ The maximal and minimal heights of the object Note that an object is a collection of height values (a.k.a z values) since we are working with the DEM standard ◦ The angle between the line connecting the highest and the lowest points of the object and its projection ◦ The coordinate (x, y ) of the highest point of the object ◦ The number of height values of the object • Neighborhood: information about four neighbored objects in the directions: South, North, East and West This characteristic reflects the spatial relationships in a terrain, e.g plateaus are more likely to be near to flat lands than rivers • Reference parameters: some geographic values such as the projection, the code of the terrain in WGS84 Reference System, etc are used to transform all above characteristics into a unique standard Due to this transformation, all objects of terrains in the Training set can be synchronized in the DML standard In order to construct the classification model from the database, the parallel Random Forest algorithm (Thong, Son & Hoa, [19]) is used Random Forest, originated by Breiman [3], is a machine learning algorithm based on the decision tree approach for the classification of satellite and remote sensing images The advantages of Random Forest are the independence of the selection of Training sets, high accuracy, robust to outliers and supported by several useful tools such as the calculations of variables importance and classification errors In essence, Random Forest includes a collection of decision trees, constructed from a random subset of the original dataset, and the final classification result depends on which result is the most appearing in all trees The trees construction process in Random Forest does not require the tree-pruning, which is mostly used in the traditional decision tree algorithm such as ID3 Even though the classification accuracy of Random Forest is better than those of the traditional decision tree algorithms, its computational time is still a major weakness (Breiman, [3]) The parallel Random Forest algorithm (Thong, Son & Hoa, [19]) was designed to accelerate the whole computational process using parallel computing with multiple processors Each processor is responsible to generate some decision trees from small, random subsets of the Training set and to look for the most appearing class in all its trees for a dataset in the Testing set, which is sent from the Master processor Thus, the computational time of both the construction and the verification steps are significantly reduced The pseudo-code below shows some steps to generate decision trees in a processor Thanks to the design of the parallel Random Forest, the verification step is paralleled performed in all processors The Master processor synthesizes all results from other processors in charge and finds the most appearing class among them Using the parallel Random Forest, we can determine the valid positions of BSs on the terrain Those positions are included in the model for our problem, which will be presented in sub-section 2.2 2.2 The mathematical modeling The WiMax Network Planning on 3D GIS problem is described as follows Assume that we have a terrain including some valid positions of BSs and fixed positions of users Each BS can be attached by some sectors in different directions, and the performance of a BS to users depends on the position of BS and the angles of sectors Determine an optimal planning solution that satisfies the following constraints: • • • • The coverage to all users is larger than a threshold The QoS to all users is larger than a threshold β Maximal overload capacity of a sector is γ The numbers of BSs and sectors are minimal α We formulate the problem as follows Input: • Terrain & users data: a terrain T whose sizes are Ncols × Nrows and the cell’s distances are (height , width) Valid positions of BSs, the number and positions of users are determined from sub-section 2.1 JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.4 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• Input: DML data – X ( N , r ) where N is the number of records and r is the dimension of data; N trees is the number of decision trees; Output: N trees decision trees Random Forest Construction Algorithm: 1: No_trees = 2: Repeat 3: No_trees = No_trees +1 4: N child = rand(2, N ); rchild = rand(2, r ) 5: Choose N child random data and rchild random attributes from the Training set 6: Assume that the current table is S Split each attribute and its class in S into small tables S i where i is an attribute of S 7: Calculate Entropy value of S: Entropy ( S ) = − p j log2 p j , j where p j is the number of datasets in relation with class j in S j 8: j j Divide S i into subsets S i in terms of domain of values and calculate Entropy ( S i ) as in Step If the domain is discrete then S i refers to possible values of attribute i Otherwise, choose a random number k and divide the domain into k equal parts whose average values j representing for S i ; j = 1, k Calculate Gain( S , i ) where i is an attribute of S by the formula below: 9: Gain( S , A ) = Entropy ( S ) − v ∈ V alues( A ) |S v | Entropy ( S v ), |S| where V alues( A ) is the set of possible values of attribute A, and S v is a subset of S containing data whose attribute is A and value is v Choose attribute i having maximal value of Gain( S , i ) as the root node Divide S into sub-tables D j in terms of domain of values according to the attribute i If Entropy ( D j ) = then the class of all data in D j is the leaf node Remove attribute i from S and perform the similar steps from Step to Step 12 for the new table until the Entropy values of all sub-table are equal to zero Until No_trees = N trees 10: 11: 12: 13: 14: • Network parameters: The height of BS and the thresholds (α , β, γ ) Assume that the heights of BSs are equal, and all sectors are of the same types M A X S E R V is the maximal emitted power of a sector and M A X Q o S is the maximal signal strength that a user can receive from a sector Output: • The performance of the network including Coverage and QoS • The investment costs consisting of the numbers of BSs and sectors • The configuration of sectors on BSs such as the angles and the directions Modeling: • B = {b1 , b2 , b3 , , b N } is a set of BSs i b i = O {xbi , y bi , zbi , h i , N max _ sec }, ∀i ∈ [1, N] (1) ◦ ◦ ◦ ◦ N is the number of used BSs for the planning xbi , y bi , zbi is the position of BS b i on the terrain h i is the height of b i i N max _ sec is the maximal number of sectors that can be attached to b i • U = {u , u , , u M } is a set of users u i = O {xiU , y iU , ziU }, ∀i ∈ [1, M] (2) ◦ M is the number of users ◦ xiU , y iU , ziU is the position of user u i on the terrain • S = {s1 , s2 , , s N sec } is a set of sectors si = { P i , b j , θi , ϕi , f eqi , P rec (si , uk , T ), G i , L Fade }, i ∀i ∈ [1, N sec ] , ∀ j ∈ [1, N] , ∀k ∈ [1, M] ◦ ◦ ◦ ◦ ◦ N sec is the number of sectors P i is the broadcast capacity of sector si , measured in dB b j is the BS containing sector si θi is the angle of antenna, whose value falls into (60, 90, 120, 180) degrees ϕi is the direction of antenna, whose value is from to 360 degrees (3) JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.5 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• ◦ ◦ ◦ ◦ f eqi is the frequency of sector si , measured in MHz G i is the antenna gain of sector si , measured in dB L Fade is the attenuation coefficient of sector si , measured in dB i P rec (si , uk , T ) is the signal strength of sector si to user uk on terrain T , measured in dB and calculated by the equation below (Hurley, Allen, Ryan & Taplin, [11]): P rec (si , uk , T ) = P i − q(si , uk , T ) + G i − L Fade i (4) ◦ q(si , uk , T ) is the attenuation from sector si to user uk on terrain T , measured in dB and calculated by the path loss model in sub-section 2.3 • The coverage value is: if s P rec (s, u , T ) > 0 other, δcover (u ) = u δcover (u ) M coverage = M (5) (6) • The QoS value is: ui M Q oS = Maxs j P rec (s j , u i , T ) M × M A X Q oS , ∀ j ∈ [1, N sec ] , ∀i ∈ [1, M] (7) • The overload capacity of a sector is: ui l (s j ) = P rec (s j , u i , T ) M AX SERV , ∀ j ∈ [1, N sec ] , ∀i ∈ [1, M] (8) The WiMax Network Planning on 3D GIS problem is stated below F = d1 × (1 − F ) + d2 × (1 − F ) + d3 × F + d4 × F −→ Min F = M coverage F2 = M Q oS F3 = N N Max N sec F4 = N j =1 (9) , j N max _ sec where N Max is the maximal number of BSs and d1 , d2 , d3 , d4 are the weights Some constraints for (9) are: M coverage ∈ [α , 1] , (10) M Q o S ∈ [β, 1] , (11) l(s j ) ∈ [0, γ ] , ∀ j ∈ [1, N sec ] , (12) N i N max _ sec , N sec ≤ (13) i =1 xbi − xbj + ybi − ybj + zbi − zbj > DBS, ∀i , j ∈ [1, N] , i = j , (14) where D B S is the minimal distance between BSs on the terrain In the models (1)–(14), we defined the sets of users – U and BSs – B with sectors – S and then determined single objectives: F (measures the coverage quality from B to U ), F (measures the QoS value from B to U ), F (measures the number of BSs) and F (measures the number of sectors) with the equivalent formulae The multi-objective optimization problem is then given in equations (9)–(14) whose constraints relate to network parameters (10)–(13) and 3D terrain (14) Some special cases of the problem (9)–(14) are: • If constraints (10)–(12), (14) are not provided and S does not exist and y i = in bi and y j = in u j then we get the model of Taplin et al [18] Additionally, if equation (9) is not included then we receive the model of Carneiro et al [4] • If constraints (10)–(12), (14) are not provided and equation (9) is not included and S does not exist and y i = in b i then we get the models of Admed, Mughni & Akhtar [1] and Sapumohotti et al [14] Additionally, if the objective changes to F = F + F −→ Max in equation (9) then we receive the models of Sebastiao et al [15] • Without constraint (14) and S does not exist and y i = in bi and y j = in u j then we get the model of Hu, Chen & Banzhaf [10] JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.6 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 2.3 Path loss calculation In sub-section 2.2, we have already known that q(si , uk , T ) is the attenuation from sector si to user uk on terrain T But how can we calculate this quantity? This part mentions a model for the path loss calculation, which is the answer for that question Since we are working on a terrain, the number and positions of obstacles between BS b i (∀i ∈ [1, N]) containing sector si and user u j (∀ j ∈ [1, M]) must be pre-determined Based upon those obstacles, the model for the path loss calculation is specified Firstly, the problem of determining how many obstacles exist between a BS and a user refers to the Line-of-Sight or Visibility problem (Ghosh, [8]) Our previous work in Son [16] presented a method for this problem by dividing the line connecting the BS and the user by multiple splitting points The length of two consecutive splitting points is (height + width) /2 where (height , width) are the cell’s distances of terrain T Those points are verified whether the bounding rectangles of terrain T consisting of them are the obstacles or not If so, the positions and the number of obstacles are marked This verification is performed by parallel computing, and the coordinates of the bounding rectangle consisting of the splitting point (x, z) are shown below x1 = (x ÷ height ) × height z1 = ( z ÷ width) × width, (15) x2 = x1 + height z2 = z1 , (16) x2 = x1 z2 = z1 + width, (17) x2 = x1 + height z2 = z1 + width, (18) where ÷ is the integer division L ke + L f reespace if RAD(si , u j , θi , ϕi ) = +∞ other, θi RAD(si , u j , θi , ϕi ) = |ϕi − | ≤ q (s i , u j , T ) = (19) (20) Secondly, we present the improved model of Deygout [6] for the path loss calculation between BS b i and user u j in the environment with obstacles as in equations (19) and (20) In equation (19), L ke ( L f reespace ) is the attenuation in the environment with (without) obstacles In equation (20), is the slope of the line connecting si and u j L f reespace is calculated by the formula below L f reespace = 32.4 + 20 ∗ log( R i ) + 20 ∗ log( f eqi ) (21) R i is the Euclidean distance from BS b i to user u j , measured in kilometers, and f eq i is the frequency of sector si , measured in MHz In order to calculate L ke , let us examine the one-obstacle model in Fig In this figure, h1 , h2 , h3 are the heights of BS b i , the obstacle and user u j , respectively d1 , d2 are the Euclidean distances from b i to the obstacle and from the obstacle to user u j , respectively h2 is the relative height from the obstacle to the BS and the user, h2 = d1 (h2 − h3 ) + d2 (h2 − h1 ) (h1 − h3 )2 + (d1 + d2 )2 (22) d1 , d2 are the relative heights from the BS to the obstacle and from the obstacle to the user, d1 = d21 + (h2 − h1 )2 − h2 d2 = d22 + (h2 − h3 )2 − h2 , (23) (24) Thus, L ke in case of one-obstacle model is calculated as follows L ke = −20 log 0.225 v (25) , v = v (d1 , d2 , h2 ) = h2 d1 + d2 , λ ∗ d1 ∗ d2 where λ is the wavelength, and v is the Fresnel reflection coefficient of the obstacle (26) JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.7 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• Fig The one-obstacle model In case of many-obstacles model, the strategy below is applied to calculate L ke • • • • Determine the main obstacle that has highest height among all Calculate L main of the one-obstacle model including BS b i , the main obstacle and user u j ke Determine the second main obstacle, which has highest height among all, between BS b i and the main obstacle le f t Calculate L ke of the one-obstacle model including BS b i , the second main obstacle and the main obstacle right • Perform the similar steps and receive L ke L ke in case of many-obstacles model is calculated below le f t right L ke = L main + Lke + Lke ke (27) Using these rules, we can calculate the attenuation from a sector to a user on a 3D terrain Thus, all parameters in the model of sub-section 2.2 are totally specified 2.4 The hybrid algorithm for the determination of Optimal Planning Solutions This sub-section presents a hybrid method to determine the optimal planning solutions for our problem Our ideas are using the improved Particle Swarm Optimization (Gong et al., [9]), integrated with BTP algorithm (Taplin et al., [18]) and parallel computing Particle Swarm Optimization (PSO), introduced by Kennedy & Eberhart [12], is a stochastic, swarm-based optimization algorithm that simulates the food-looking behaviors of birds PSO was successfully applied to many optimization problems such as Graph Coloring, Traveling Salesman, etc Nonetheless, three main problems of PSO that can affect the performance of the algorithm are the initialization, the convergence to local optima and the computational time Gong et al [9] pointed out that bad initialization of particles may result in the quality of the solutions, and PSO tends to converge to local optima The large computational time of PSO is another problem if the number of iterations increases Thus, an improvement of PSO that can handle those limitations is required for our problem In this sub-section, we consider the uses of the parallel BTP algorithm for the initialization problem Being introduced in Section 1, BTP can generate acceptable results based on the greedy approach in a reasonable time A parallel version of this algorithm is introduced both to accelerate the computing process, especially in cases of very large terrain data, and to achieve the initial solutions of PSO In order to tackle with the convergence to local optima problem, we use the ideas of Gong et al [9] about the mutation operator An improvement of traditional PSO algorithm incorporating with the mutation operator and some additional techniques such as parameters updating and parallel computing scheme is designed to handle the two last problems of PSO The mechanism of the proposed approach – WNPA-3DT is described in Fig Let us make a deeper analysis about this mechanism Firstly, the parallel BTP algorithm is used to generate the initial solutions of PSO This algorithm divides the terrain into small grids whose sizes are η1 × η2 (height ≤ η1 ; width ≤ η2 ) and puts BSs to all grids’ nodes Each BS is equipped with four sectors in all directions such as South, North, East and West The algorithm checks those BSs and sectors and tries to remove some of them until the constraints (10)–(14) not hold Outputs of this algorithm are the numbers and positions of BSs and sectors as well as their configurations such as the direction of antennas The pseudo-code of the parallel BTP algorithm is shown below After we receive the outputs of the parallel BTP algorithm then use the improved PSO to generate the final solutions In the Init procedure, PSO algorithm is initialized by N pop particles – P = p , p , , p N pop whose first one is inherited from the parallel BTP and the rest are randomly initialized with the maximal number of BSs being the number of BSs from the parallel BTP (M A X_B S) Each p i (i = 1, N pop ) is encoded as follows • X idj ( V idj ) is the position (velocity) of b j in p i ( j = 1, M A X_B S, i = 1, N pop , d = 1, 2) • onkij is the checking variable for the possibility of putting sector sk on BS b j ( j = 1, M A X_B S, i = 1, N pop , k = 1, N sec ) Its domain of values is {0, 1} JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.8 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• Fig The mechanism of the proposed approach (WNPA-3DT) • θikj is the updating velocity of direction of sector sk on BS b j ( j = 1, M A X_B S, i = 1, N pop , k = 1, N sec ) Its domain of values is [0, 360] • ϕikj is the direction of sector sk on BS b j ( j = 1, M A X_B S, i = 1, N pop , k = 1, N sec ) Its domain of values is [0, 360] • pbest idj is the best position of b j in p i ( j = 1, M A X_B S, i = 1, N pop , d = 1, 2) • onkpbest_i j is the checking variable for the possibility of putting sector sk on BS b j ( j = 1, M A X_B S, i = 1, N pop , k = 1, N sec ) in relation with pbest idj • ϕ kpbest_i j is the direction of sector sk on BS b j ( j = 1, M A X_B S, i = 1, N pop , k = 1, N sec ) in relation with pbest idj i • f pbest is the best fitness value of p i (i = 1, N pop ) i Example Suppose that we have a solution: M A X_B S = 5, N max _ sec = (i = 1, 5) and the configuration of putting sectors on BSs as in equation (28) We clearly recognize that the number of BSs used for this solution is: N i = since there is no sector that is put on i BS b5 The maximal number of sectors is 20, and the total number of used sectors is: N sec = ⎛ ⎞ ⎛ b1 ⎜ b2 ⎟ ⎜ ⎜ ⎟ ⎜ b = ⎜ b3 ⎟ = ⎜ ⎝b ⎠ ⎝0 b5 1 0 0 0 ⎞ 1⎟ ⎟ ⎟ ⎠ 0 (28) The Evaluation procedure is designed to calculate the fitness values of all particles The fitness function is shown in equation (29) It allows the optimization by multiple objectives such as the numbers and positions of BSs and sectors, the coverage and QoS values f i = d1 × (1 − M coverage ) + d2 × (1 − M Q o S ) + d3 × where Ni M A X_B S i N sec + d4 × j bj N max _ sec , i = 1, N pop , (29) JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.9 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• Input in sub-section 2.2 and η1 , η2 parameters The number of BSs (M A X_B S), the positions of BSs (xi , y i , zi ) (i = 1, M A X_B S), the number of sectors (numSec) and the direction of antenna on each BS (ϕ ) BTP Algorithm: 1: BS b // BS array 2: Sector s // sector array 3: Int m = // number of BSs 4: Int nr, nc // number of splitting points 5: Int numSec = 6: Double x, y, z, ϕ // Position of BS and the direction of antenna 7: nc = ( width ì Ncols) ữ nr = (height × Nrows) ÷ η1 8: For i = to nr // Step to 22 are performed parallel in all processors 9: For j = to nc 10: m++ 11: xm = height × j zm = width × i 12: ym = get Height (xm , zm ) // Elevation interpolation 13: bm set P osition(xm , ym , zm ) m 14: For k = to N max _ sec 15: numSec + + m 16: ϕk = 360 × k/ N max _ sec 17: snumSec add Arc (ϕk ) 18: bm addSec(snumSec ) 19: End For 20: End For 21: End For 22: Calculate M coverage of all sectors by equations (4)–(6) where the attenuation values from sectors to users are paralleled computed from the path loss model 23: Sort the coverages in the ascending order 24: Add BSs and sectors to the current solution 25: For i = to numSec 26: Remove sector si from the current solution 27: If conditions (10)–(14) = f alse 28: Add sector si to the current solution 29: End If 30: End For 31: For i = to nr × nc 32: If b i NumSec = then 33: m−− 34: Remove BS b i from the current solution 35: End If 36: End For 37: M A X_B S = m Input: Output: • • • • • • • dk (k = 1, 4) are the weights of the coverage and QoS values, the numbers of BSs and sectors, respectively N i is the number of BSs in p i M A X_B S is the maximal number of BSs in p i i N sec is the number of sectors in p i j N max _ sec is the maximal number of sectors that can be put on BS b j (∀ j ∈ [1, N]) M coverage is the coverage value M Q o S is the QoS value Based upon the fitness values of particles, the best values of particles (pBest) and the swarm (gBest) are determined accordingly Some notions below show these best values: • gbest dj is the best position of b j in all particles ( j = 1, M A X_B S, d = 1, 2) • onkgbest_ j is the checking variable for the possibility of putting sector sk on BS b j ( j = 1, M A X_B S, k = 1, N sec ) in relation with gbest dj • ϕ kgbest_ j is the direction of sector sk on BS b j ( j = 1, M A X_B S, k = 1, N sec ) in relation with gbest dj • f gbest is the best fitness value of all particles The updating processes of pBest and gBest are shown in equations (30) and (31), respectively For i = to N pop i If f pbest > f i then i f pbest = fi; pbest dij = X idj ; JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.10 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 10 onkpbest_i j = onkij ; ϕ kpbest_i j = ϕikj ; End If ( j = 1, M A X_B S , k = 1, N sec ) End For, (30) For i = to N pop i If f gbest > f pbest then i f gbest = f pbest ; gbest dj = pbest dij ; onkgbest_ j = onkpbest_i j ; ϕ kgbest_ j = ϕ kpbest_i j ; End If ( j = 1, M A X_B S , k = 1, N sec ) End For (31) In Fig 2, we use k processors in a parallel computing system for the calculations of fitness and pBest values of all particles Those pBest values are synchronized at the Master processor for the calculation of gBest This value is then sent to all processors for the updating of other parameters, which is described as follows • Update the possibility of putting sector sk on BS b j ( j = 1, M A X_B S, k = 1, N sec ) Denote rands(0/1) as the random function whose output is or If rands(0/1) + onkij + onkpbest_i j + onkgbest_ j > rand(0, 1) then onkij = Else onkij = End If (i = 1, N pop ) (32) • Update the direction and the velocity of direction of sector sk on BS b j ( j = 1, M A X_B S, k = 1, N sec ) If onkij = then θikj = θikj + c ϕ kpbest_i j − ϕikj + c ϕ kgbest_ j − ϕikj ϕikj = ϕikj + θikj End If (i = 1, N pop ) (33) c and c are the coefficients of ϕ kpbest_i j and ϕ kgbest_ j , respectively • Update the position and the velocity of b j in p i ( j = 1, M A X_B S, i = 1, N pop , d = 1, 2) V idj = V idj + c 1∗ pbest dij − X idj + c 2∗ gbest dj − X idj , X idj = X idj + V idj (34) c 1∗ and c 2∗ are the coefficients of pbest idj and gbest dj , respectively Being mentioned in some first lines of this sub-section, the mutation operator is applied to our proposed algorithm after all parameters have been updated It changes the position and the velocity of b j in p i , the direction and the velocity of direction of sector sk on b j (i = 1, N pop , j = 1, M A X_B S, k = 1, N sec ) in a certain extent V idj = V idj + rand (− , X idj = X idj ), (35) + rand (− , ) , (36) θikj = θikj + rand − k ij k ij , ϕ = ϕ + rand − , ( (37) , ) is the changing coefficients of the position (direction) (38) JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.11 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 11 Fig The uniform distribution of 4000 users on the small terrain The proposed algorithm stops if one of the following conditions satisfies: • After L consecutive iterations, f gbest values are unchanged In the other words, (t +1) (t ) f gbest − f gbest ≤ ε , t = 1, L − (39) • Max_P S O iterations are reached Using this approach, the optimal numbers and positions of BSs and sectors as well as their configurations can be determined Experiments 3.1 Experimental environment We implemented the proposed algorithm – WNPA-3DT in addition to Genetic Algorithm – GA (Hu, Chen & Banzhaf, [10]) and BTP, Tabu Search (Taplin et al., [18]) in MPI/C programming language and executed them on a Linux cluster system containing eight nodes with the total computing power being 51.2 GFlops Each node includes two Intel Xeon dual core 3.2 GHz and GB Ram The experimental datasets are two benchmark DEM terrains of Bolzano–Bozen, Italy (Son, Thong, Linh, Cuong & Hoa, [17]) whose sizes are 8500 × 6500 m2 (small terrain) and 18000 × 28000 m2 (large terrain), respectively Two simulated distributions of users based on the uniform and Gaussian distributions are generated on that terrain Thus, we have datasets, namely “the dataset with the uniform distribution of users on the small terrain” (Fig 3, Scenario 1), “the dataset with the uniform distribution of users on the large terrain” (Fig 4, Scenario 2), “the dataset with the Gaussian distribution of users on the small terrain” (Fig 5, Scenario 3) and “the dataset with the Gaussian distribution of users on the large terrain” (Fig 6, Scenario 4) The objectives of the experiments are: i) to compare WNPA-3DT with some best-known planning algorithms such as Tabu Search and GA; ii) to verify whether or not the integration between the parallel BTP and improved PSO in the proposed approach is better than the standalone BTP; iii) to compare the solution of WNPA-3DT with the optimal one The evaluation criteria are: the investment costs (the numbers of BSs and sectors), the performance of the network (Coverage and QoS values) and the computational time of algorithms According to the relevant articles and our experiments, some parameters below are set up for the best performance of those algorithms • The maximal number of BSs (sectors) in Scenario and are: N = 30 and N sec = 120 Those values in Scenario and are: N = 500 and N sec = 2000 • The height of BS is 20 meters D B S = 15 meters • The maximal number of sectors that can be put on a BS is JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.12 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 12 Fig The uniform distribution of 10000 users on the large terrain Fig The Gaussian distribution of 4000 users on the small terrain • Parameters of a sector: P i = 46 dB, f eqi = 3500 MHz, G i = dB, L Fade = dB, ∀i ∈ [1, N sec ], M A X S E R V = 26182 dB, i M A X Q o S = 46 dB • Weights: d1 = d2 = d4 = 1; d3 = 10 • Wavelength: λ = 3.3356 m The BTP algorithm: • η1 = 1.5 × height and η2 = × width The PSO algorithm: • N pop = 1000; L = 20; ε = 0.01; Max_P S O = 500 JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.13 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 13 Fig The Gaussian distribution of 10000 users on the large terrain • c 1∗ = c 2∗ = c = c = • = 10 and = The GA algorithm: • The number of population: N pop_G A = 1000 • Mutation probability: ξ = 0.5 • The maximal number of iterations: Max_G A = 500 The Tabu Search (a.k.a Tabu): • The size of memory: sizetabu = 20 • The changing coefficients of the position of BS (the direction of sector): ϑ = 10 (ϑ = 90) • The maximal number of iterations: Max_T S = 500 In the next sub-sections, the experiments on different scenarios from Scenario to Scenario are shown 3.2 Scenario Table describes the comparative results of all algorithms by various parameters The results clearly show that WNPA3DT obtains better performance (Coverage and QoS values) and investment costs (the numbers of BSs and sectors) than GA, Tabu and BTP algorithms For example, in the first case (α , β, γ ) = (0.8, 0.65, 1.2) that means the minimal values of Coverage and QoS are 0.8 and 0.65, respectively and the maximal overload capacity of a sector is 1.2, the numbers of BSs and sectors produced by WNPA-3DT are 13 and 21, respectively Meanwhile, the results of GA, BTP and Tabu are (12, 29), (19, 25) and (18, 27), respectively Obviously, WNPA-3DT uses less number of sectors than other algorithms The Coverage and QoS values of WNPA-3DT are 0.841 and 0.809, respectively Those values of GA, BTP and Tabu are (0.805, 0.781), (0.803, 0.815) and (0.813, 0.707), respectively Clearly, the Coverage value of WNPA-3DT is larger than those of other algorithms, i.e 1.044 times larger than that of GA, 1.047 times larger than that of BTP and 1.034 times larger than that of Tabu We calculate the fitness values of all algorithms in the first case Based on the minimum criterion, WNPA-3DT is proven to be the second best algorithm of all We also measure the serial computational time of all algorithms The results show that WNPA-3DT is the second fastest algorithm of all, which consumes around 28 minutes for this case In order to achieve the optimal results, GA and Tabu require 78 and 65 minutes, respectively The fastest algorithm – BTP takes 11 minutes only since it is a greedy algorithm Through the first case, we recognize that WNPA-3DT is more effective than other best-known planning algorithms such as BTP and Tabu Do those remarks still hold when the parameters (α , β, γ ) change? We have made other test cases that verify the efficiency of the proposed algorithm with different values of parameters For example, the second test case (α , β, γ ) = (0.9, 0.6, 1.2) evaluates all algorithms when the value of α in the first case increases and the value of β decreases Similarly, JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.14 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 14 Table Comparative results of algorithms by different parameters in Scenario Criteria Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.8, 0.65, 1.2) (α , β, γ ) = (0.9, 0.6, 1.2) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 1694 13 21 0.841 0.809 5.087 4674 12 29 0.805 0.781 5.018 650 19 25 0.803 0.815 7.044 3896 18 27 0.813 0.707 6.855 8933 18 31 0.923 0.782 6.726 9151 18 45 0.948 0.796 6.881 367 22 81 0.902 0.764 8.588 4300 20 45 0.901 0.782 7.546 (α , β, γ ) = (0.7, 0.8, 1.2) Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.8, 0.65, 1.3) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 4776 14 34 0.855 0.807 5.612 8217 18 43 0.786 0.809 7.003 670 18 23 0.781 0.856 6.682 3021 17 26 0.723 0.812 6.514 6613 14 22 0.808 0.772 5.48 8174 14 28 0.803 0.793 5.571 649 19 25 0.803 0.815 7.044 3532 19 23 0.853 0.806 6.977 (α , β, γ ) = (0.9, 0.8, 1.1) Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.6, 0.6, 1.4) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 4391 21 43 0.920 0.811 7.781 6715 21 47 0.904 0.807 7.849 465 22 83 0.902 0.817 8.558 4833 23 50 0.912 0.833 8.465 4569 0.676 0.799 3.144 9015 15 0.623 0.714 3.288 845 11 15 0.614 0.792 4.602 2803 10 18 0.639 0.693 4.451 when the value of α in the first case decreases and the value of β increases, we have the third test case – (α , β, γ ) = (0.7, 0.8, 1.2) The fourth test case – (α , β, γ ) = (0.8, 0.65, 1.3) investigates the role of γ when its value increases When the values of (α , β) increase and the value of γ decreases, we have the fifth test case – (α , β, γ ) = (0.9, 0.8, 1.1) Finally, the last test case – (α , β, γ ) = (0.6, 0.6, 1.4) examines the situation that decreases the values of (α , β) and increases the value of γ The experimental results in those cases show that the performance and the investment costs of WNPA-3DT are better than those of other algorithms However, these results vary considerably among these cases In the second case, the Coverage and QoS values of WNPA-3DT are little smaller than those of GA, i.e (0.923, 0.782) vs (0.948, 0.796), respectively However, the number of sectors used in GA is much larger than that of WNPA-3DT, i.e 45 vs 31 sectors, respectively The fitness values show that WNPA-3DT obtains better performance and investment costs than GA and other algorithms WNPA-3DT is the second slowest algorithm in this case which takes 2.5 hours to find the solutions The fastest algorithm – BTP takes minutes only, but its total result is far from optimum in comparison with WNPA-3DT and GA The remark for this case is that when the value of α increases and the value of β decreases, the Coverage and QoS values of WNPA-3DT tend to be smaller than those of GA, and it takes much time for WNPA-3DT to find the solutions Nonetheless, the performance and the investment costs (a.k.a the total result) of WNPA-3DT in this case are still better than those of other algorithms In the third case, the total number of used sectors in WNPA-3DT is 34, which is larger than those of BTP and Tabu, i.e 23 and 26, respectively The QoS value of WNPA-3DT is also smaller than those of BTP and Tabu However, the Coverage value and the number of BSs produced by WNPA-3DT are much better than those of BTP and Tabu, and indeed WNPA-3DT is more effective than other algorithms as proven by the fitness values This test case clearly shows that when the value of α decreases and the value of β increases, some results of WNPA-3DT are not as good as those of BTP and Tabu, but the total result of WNPA-3DT is better than those of other algorithms In the fourth test case, the Coverage and QoS values of Tabu are better than those of WNPA-3DT, i.e (0.853, 0.806) vs (0.808, 0.772), respectively Nevertheless, the numbers of BSs and sectors in Tabu are much larger than those of WNPA-3DT, i.e (19, 23) vs (14, 22), respectively Thus, the same remark with that of the second test case is achieved The fifth test case is similar to the fourth one The QoS value of Tabu in this case is the largest value of all However, the values of other criteria of Tabu are worse than those of WNPA-3DT Thus, the total result of WNPA-3DT is the best of all The last test case shows the superiority of WNPA-3DT whose values are better than those of other algorithms Through those cases, WNPA-3DT is proven to be the best algorithm of all Even though the performance and the investment costs of WNPA-3DT are better than those of other algorithms, the computational time of that algorithm is a major challenge We compute the average serial computational time of all algorithms through all cases in Table and get the results: 86 minutes (WNPA-3DT), 128 minutes (GA), 10 minutes (BTP) and 62 minutes (Tabu) This means that WNPA-3DT takes 86 minutes to generate the optimal solutions on average This number is larger than those of BTP and Tabu In order to reduce the computational time, we run WNPA-3DT with many processors and get the results in Table This table shows the parallel computational time of WNPA-3DT with different number of processors and various test cases According to those results, the computational time of WNPA-3DT using processors in the first case – (α , β, γ ) = (0.8, 0.65, 1.2) is 693 seconds, which is approximate to 41% of the serial computational time JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.15 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 15 Table The parallel computational time of WNPA-3DT in Scenario (sec) Number of processors 12 16 20 24 28 (α , β, γ ) (0.8, 0.65, 1.2) (0.9, 0.6, 1.2) (0.7, 0.8, 1.2) (0.8, 0.65, 1.3) (0.9, 0.8, 1.1) (0.6, 0.6, 1.4) 1694 693 394 277 232 187 165 277 8933 3611 1992 1311 1199 834 741 997 4776 1831 1077 893 740 495 487 577 6613 2908 1716 1021 976 756 609 656 4931 2107 1442 904 803 605 558 573 4569 1789 906 712 492 407 351 371 Fig The comparison of fitness values between Scenario and Scenario When we use 24 processors for WNPA-3DT in the first case, the computational time reduces to 165 seconds, approximate to 10% of the serial computational time Similar situations are occurred in other test cases Obviously, using WNPA-3DT with many processors helps reduce the computational time of the algorithm If we remember the computational time of other algorithms, especially BTP, then those achieved parallel computational time of WNPA-3DT are significant However, how many processors are enough for our computation? If we scan the results in Table 2, we can recognize that using too many processors sometimes increases the computational time of WNPA-3DT For example, in the first case, the computational time of WNPA-3DT using 28 processors is larger than that using 24 processors As such, the determination of the optimal number of processors required for the best results of WNPA-3DT is necessary Through the analysis on the performance of WNPA-3DT in Scenario 1, we obtain some major remarks as follows • • • • WNPA-3DT is more effective than GA and Tabu The integration between the parallel BTP and improved PSO in WNPA-3DT is really better than the traditional BTP WNPA-3DT is stable through various values of parameters (α , β, γ ) Using WNPA-3DT with many processors, especially four processors helps reduce the computational time of the algorithm 3.3 Scenario In this scenario, we verify the proposed algorithm in terms of the large terrain and more number of users than Scenario The results are shown in Table Based on the fitness values, we recognize that WNPA-3DT achieves better performance and investment costs than other algorithms Thus, the first conclusion drawn from this scenario is that the changing of the sizes of terrains and the number of users not affect the effectiveness of the proposed algorithm However, how the large terrain and the large number of users influence on the performance of WNPA-3DT? We have made a comparison of fitness values in all cases and depicted the results in Fig Interestingly, the fitness values in Scenario are smaller than those in Scenario This means that the total result of Scenario is better than that of Scenario Fig clearly points out that the case that has the maximal fitness value in both scenarios is This means that the large values of (α , β) and the small value of γ in Case degrades the performance of the proposed algorithm the most Similarly, the case having minimal fitness value in both scenarios is Case 6, which decreases the values of (α , β) and increases the value of γ Similar to Table 2, we also measure the parallel computational time of WNPA-3DT with many processors and describe the results in Table From this table, we recognize that using parallel computing with many processors helps reduce the large serial computational time of WNPA-3DT For example, in the first case, the serial computational time of WNPA-3DT JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.16 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 16 Table Comparative results of algorithms by different parameters in Scenario Criteria Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.8, 0.65, 1.2) (α , β, γ ) = (0.9, 0.6, 1.2) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 55578 112 138 0.801 0.721 3.026 26810 121 274 0.809 0.754 3.423 36025 389 1550 0.801 0.828 9.147 28816 281 632 0.823 0.861 6.498 57655 141 175 0.9 0.787 3.443 22214 204 419 0.904 0.777 4.912 28486 440 1758 0.901 0.827 10.071 33121 293 584 0.921 0.753 6.684 (α , β, γ ) = (0.7, 0.8, 1.2) Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.8, 0.65, 1.3) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 63930 152 215 0.928 0.801 3.665 20457 264 576 0.947 0.798 6.08 33936 411 1641 0.843 0.828 9.547 39021 221 341 0.753 0.812 5.241 35210 108 130 0.8 0.723 2.938 40282 110 248 0.8 0.76 3.204 12703 389 1550 0.801 0.828 9.147 26532 251 431 0.813 0.831 5.805 (α , β, γ ) = (0.9, 0.8, 1.1) Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.6, 0.6, 1.4) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 89793 175 216 0.902 0.8 4.106 32631 267 560 0.962 0.8 6.102 36251 440 1758 0.90 0.827 10.072 28833 323 647 0.913 0.812 7.236 44341 53 64 0.6 0.736 2.026 18567 65 138 0.6 0.728 2.503 38445 54 66 0.612 0.736 2.038 19803 53 112 0.614 0.623 2.351 Table The parallel computational time of WNPA-3DT in Scenario (sec) Number of processors 12 16 20 24 28 (α , β, γ ) (0.8, 0.65, 1.2) (0.9, 0.6, 1.2) (0.7, 0.8, 1.2) (0.8, 0.65, 1.3) (0.9, 0.8, 1.1) (0.6, 0.6, 1.4) 55587 26413 24761 14716 13335 12679 10403 9703 57655 26671 24651 17877 15474 12604 10438 9798 63930 25513 23976 14358 14282 11997 10432 9938 35210 26401 24490 16495 15327 12663 9724 9692 89793 39109 23944 14822 14197 11996 9724 9933 44341 26392 24299 16847 15098 12721 10413 9707 is 15.4 hours Using processors, the computational time is 6.88 hours only The results of using 24 and 28 processors are 2.89 and 2.69 hours, respectively Similar results appeared in other cases By the speedup and efficiency coefficients, we find out that the optimal number of processors in Scenario is four Fig highlights the comparisons of the average computational time in terms of cases between Scenario and Scenario This figure shows that the computational time of Scenario is much larger than that of Scenario 1, i.e 14.6 times on average Some major conclusions in this scenario are: • • • • The conclusions about the effective of WNPA-3DT in Scenario are kept intact Large terrains results in the smaller fitness values of WNPA-3DT than those in Scenario Case (6) degrades (accelerates) the performance of the proposed algorithm the most The parallel computational time of WNPA-3DT is much larger than that in Scenario 3.4 Scenario In this section, we verify whether the remarks of sub-section 3.2 hold or not in cases of a different distribution of users The experiments were performed with 4000 users determined by the Gaussian distribution on the small terrain (Fig 5) The results are shown in Table The experimental results show that the performance and the investment costs of WNPA-3DT are better than those of other algorithms In order to comprehend the impact of the Gaussian distribution to the experimental results, we compare the results of WNPA-3DT in Scenario and Scenario in terms of: the numbers of BSs and sectors, the Coverage and QoS values (Fig 9) From this figure we recognize that the numbers of BSs and sectors in Scenario are less than those in Scenario The mean value of all cases in term of the number of BSs (sectors) in Scenario is 8.3 (20.1) whilst that in Scenario is 14.5 (26.5) JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.17 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 17 Fig The comparison of average time by cases between Scenario and Scenario Table Comparative results of algorithms by different parameters in Scenario Criteria Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.8, 0.65, 1.2) (α , β, γ ) = (0.9, 0.6, 1.2) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 4244 10 21 0.801 0.736 4.321 6768 11 27 0.803 0.768 4.709 824 13 23 0.803 0.815 5.158 2096 13 20 0.813 0.707 5.198 4513 11 31 0.923 0.782 4.666 4613 13 28 0.934 0.808 5.13 632 12 29 0.903 0.764 4.937 2321 13 28 0.901 0.782 5.189 (α , β, γ ) = (0.7, 0.8, 1.2) Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.8, 0.65, 1.3) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 5026 34 0.855 0.807 4.282 4554 11 20 0.768 0.810 4.543 892 13 20 0.948 0.803 4.967 6021 12 24 0.823 0.812 4.865 4781 0.831 0.792 2.494 4069 16 0.806 0.749 3.617 782 13 22 0.803 0.815 5.138 2332 13 23 0.843 0.776 5.157 (α , β, γ ) = (0.9, 0.8, 1.1) Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.6, 0.6, 1.4) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 4982 11 17 0.920 0.811 4.322 6876 15 30 0.933 0.801 5.766 565 12 21 0.902 0.817 4.719 2833 12 24 0.912 0.833 4.755 3982 0.654 0.813 2.429 7040 0.6145 0.7208 2.781 753 10 16 0.614 0.792 4.327 1303 14 0.639 0.693 3.772 The Coverage and QoS values in Scenario are also smaller than those in Scenario Again, the mean value in term of Coverage (QoS) in Scenario is 0.831 (0.790) which is smaller than that in Scenario – 0.837 (0.797) The explanation for this fact is that under the Gaussian distribution, some locations on the terrain have high concentration of BSs, and indeed users who are not located in those locations have low signal strength Even though the total result is not high in comparison with that in Scenario 1, the difference between them is small and acceptable In Table 6, the parallel computational time of WNPA-3DT in Scenario is described Similar to the results in Scenario 1, using WNPA-3DT with many processors reduces the computational time of the algorithm Interestingly, the parallel computational time in Scenario is even smaller than that in Scenario For example, using four processors in Scenario takes 1684 seconds on average whilst this number in case of Scenario is 2156 seconds Fig 10 depicts this fact in details Finally, by using the speed up and efficiency values, we determine the optimal number of processors in Scenario is four Some major remarks extracted from this scenario are shown below • The conclusions about the effective of WNPA-3DT in Scenario are kept intact • The performance and the numbers of BSs and sectors are smaller than those in Scenario • The parallel computational time of WNPA-3DT, especially using four processors, are smaller than that in Scenario JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.18 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 18 Fig The comparison between Scenario and Scenario Table The parallel computational time of WNPA-3DT in Scenario (sec) Number of processors 12 16 20 24 28 (α , β, γ ) (0.8, 0.65, 1.2) (0.9, 0.6, 1.2) (0.7, 0.8, 1.2) (0.8, 0.65, 1.3) (0.9, 0.8, 1.1) (0.6, 0.6, 1.4) 4424 1374 815 682 535 483 404 401 4513 1595 954 768 579 425 515 523 5026 1958 1177 893 708 595 537 574 4781 1908 1216 891 775 656 519 746 4982 2097 1442 1003 893 720 658 695 3982 1172 675 512 432 375 301 363 Fig 10 The comparison of the average time by cases in Scenario and Scenario JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.19 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 19 Table Comparative results of algorithms by different parameters in Scenario Criteria Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.8, 0.65, 1.2) (α , β, γ ) = (0.9, 0.6, 1.2) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 57585 30 42 0.801 0.799 1.35 16503 86 173 0.8 0.751 2.672 33032 136 497 0.8 0.812 4.022 27321 94 231 0.804 0.812 2.878 56292 56 62 0.9 0.76 1.737 22222 129 279 0.912 0.782 3.427 36431 266 1062 0.901 0.828 6.589 38121 162 332 0.902 0.822 4.028 (α , β, γ ) = (0.7, 0.8, 1.2) Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.8, 0.65, 1.3) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 7773 98 287 0.811 2.881 15736 156 320 0.927 0.8 3.906 2212 98 287 0.811 2.881 25213 123 241 0.895 0.802 3.253 36799 38 53 0.8 0.717 1.592 16063 82 175 0.822 0.767 2.585 31619 136 500 0.803 0.812 4.024 19532 98 231 0.813 0.826 2.91 (α , β, γ ) = (0.9, 0.8, 1.1) Time (s) BS Sector Coverage QoS Fitness (α , β, γ ) = (0.6, 0.6, 1.4) WNPA-3DT GA BTP Tabu WNPA-3DT GA BTP Tabu 42181 98 287 0.811 2.881 24440 179 383 0.94 0.8 4.375 32465 98 287 0.811 2.881 28833 195 309 0.912 0.833 4.551 43588 10 16 0.609 0.724 1.267 8120 41 85 0.615 0.76 1.963 33073 11 23 0.603 0.747 1.393 17803 48 93 0.659 0.683 2.102 Fig 11 The comparison of fitness values in all scenarios 3.5 Scenario Now, we discuss the last scenario in the experiment According to sub-section 3.1, this scenario mentions a dataset whose distribution of users and sizes of the terrain are different with those in three previous scenarios Thus, a comparison between these scenarios is a must From the fitness values of all algorithms in Table 7, we recognize that WNPA-3DT obtains better performance and investment costs than other algorithms In order to compare the experimental results of four scenarios, we put their fitness values into Fig 11 This figure clearly shows that the fitness values of Scenario are not only smaller than those of Scenario but also smaller than those of Scenario and Scenario For example, in the first case, the fitness values of all scenarios from Scenario to Scenario are 5.087, 3.026, 4.321 and 1.35, respectively The maximal (minimal) difference of fitness values between Scenario and Scenario is 4.989 (1.877) Those numbers in cases of Scenario and Scenario are 1.706 (0.759) and 2.971 (0.902), respectively This means that using the Gaussian distribution of users and the large terrain results in better performance and investment costs than using other configurations of users and terrains Table shows the parallel computational time of WNPA-3DT Since we are working on the large terrain, the computational time is quite large, for example, from 2.16 to 15.99 hours to generate the solutions as illustrated in Table The reduction of the computational time by parallel computing is necessary in this situation A similar remark with those in previous scenarios about the effectiveness of using WNPA-3DT with many processors is achieved Through calculations of the speedup and efficiency coefficients, the optimal number of processors is determined as four Finally, the comparison of computational time between Scenario and is depicted in Fig 12 JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.20 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 20 Table The parallel computational time of WNPA-3DT in Scenario (sec) Number of processors 12 16 20 24 28 (α , β, γ ) (0.8, 0.65, 1.2) (0.9, 0.6, 1.2) (0.7, 0.8, 1.2) (0.8, 0.65, 1.3) (0.9, 0.8, 1.1) (0.6, 0.6, 1.4) 57585 18502 9892 8724 8456 7102 5762 5270 56292 17833 9894 9868 8334 7116 5813 5219 40893 17120 9352 8188 7142 6229 5383 5124 36799 13220 9886 9513 8409 7088 5780 5235 42181 12172 9055 7675 7100 6203 5338 5145 43588 13305 9758 9489 8342 7073 5756 5245 Fig 12 The comparison of average time by cases between Scenario and Scenario Table Comparative results with the optimal solution Criteria Time (s) BS Sector Coverage QoS 1250 × 1250 m2 DEM 2500 × 2500 m2 DEM WNPA-3DT Optimal solution WNPA-3DT Optimal solution 0.362 0.964 0.823 40235 0.983 0.832 1.052 0.938 0.654 157902 0.976 0.764 Some major remarks extracted from this scenario are shown below • The conclusions about the effective of WNPA-3DT in Scenario are kept intact • Using the Gaussian distribution of users and the large terrain results in better performance and investment costs than using other configurations of users and terrains • The parallel computational time of WNPA-3DT is larger than that in Scenario 3.6 The comparison with optimal solutions In order to compare the solution of WNPA-3DT with the optimal one, we conduct the experiments to find the optimal solution based on brute forte’s strategy which places BSs at all grid points of the terrain and turns on or off sectors on a BS in a sequence with the azimuth being 0, 90, 180 and 270, respectively Two neighbor locations of BSs are set as 10 meters distant Small terrains with sizes being 1250 × 1250 and 2500 × 2500 m2 , 500 users in the uniform distribution and (α , β, γ ) = (0.6, 0.6, 1.4) are set up accordingly to reduce the computational cost The comparison between the solution of WNPA-3DT and the optimal one by criteria is demonstrated in Table It has been observed that two algorithms find the same numbers of BSs and sectors for the WiMax network while the coverage and QoS values of WNPA-3DT are approximate to those of the optimal solution The computational time of WNPA-3DT is much smaller than that of the optimal one Therefore, we realize the efficiency of WNPA-3DT for producing good solutions especially when the size of terrain and other parameters are larger JID:YJCSS AID:2998 /FLA [m3G; v1.183; Prn:25/07/2016; 13:19] P.21 (1-21) L.H Son, P.H Thong / Journal of Computer and System Sciences ••• (••••) •••–••• 21 Conclusions In this paper, we have presented a novel soft computing method for the WiMax Network Planning on 3D GIS problem It integrated some latest results of parallel Random Forest classification, the parallel BTP algorithm and improved Particle Swarm Optimization on the basis of the generalized mathematical model including 3D path loss calculation The experimental evaluations showed that the proposed method achieved better results of the performance of the network and the investment costs in comparison with some best-known planning algorithms such as BTP, Tabu Search and Genetic Algorithm Moreover, using the hybrid algorithm between the parallel BTP and improved Particle Swarm Optimization is more effective than using the standalone BTP The last finding of this paper is that the proposed algorithm works stably following by various numbers of processors and parameters Further works of this theme will expand the proposed method for mobile users and investigate the hybrid Wifi/WiMax Network Planning Acknowledgments The authors are greatly indebted to the Center for High Performance Computing, VNU University of Science, Vietnam National University for executing this program on the IBM Cluster 1350 system References [1] Y Admed, W Mughni, P Akhtar, Development of a GIS tool for WiMax network planning, in: Proceeding of the 2010 International Conference on Information and Emerging Technologies, Karachi, Pakistan, 2010, pp 1–5 [2] E Amaldi, S Bosio, F Malucelli, D Yuan, Solving nonlinear covering problems arising in WLAN design, Oper Res 59 (1) (2011) 173–187 [3] L Breiman, Random forests, Mach Learn 45 (1) (2001) 5–32 [4] H Carneiro, et al., Software planning tool for WiMax networks, in: 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[19] P.H Thong, L.H Son, N.D Hoa, A parallel random forest algorithm for digital elevation model classification, in: Proceeding of the 6th International Conference on GeoInformatics for Spatial-Infrastructure... Son, An exploratory study about spatial analysis techniques in three dimensional maps for SGIS -3D systems, in: Proceedings of the 2010 IEEE International Conference on Electronics and Information. .. introduced several methods for the mathematical modeling of WiMax networks Regarding the soft computing methods, Taplin et al [18] introduced some automatic WiMax network planning on a 2D map such