W-CDMA network design. Joakim Kalvenes. Jeffery Kennington and Eli V. Olinick

Transcription

1 Int. J. Mobile Network Design and Innovation, Vol. X, No. Y, XXXX 1 W-CDMA network design Qibin Cai* Verizon Business, Network Engineering Systems, Richardson, TX, USA kevin.cai@verizonbusiness.com *Corresponding author Joakim Kalvenes School of Management, University of Texas at Dallas, Richardson, TX, USA kalvenes@utdallas.edu Jeffery Kennington and Eli V. Olinick School of Engineering, Southern Methodist University, Dallas, TX, USA jlk@engr.smu.edu olinick@engr.smu.edu Abstract: In this investigation, the W-CDMA network design problem is modelled as a discrete optimisation problem that maximises revenue net the cost of constructing base stations, Mobile Telephone Switching Offices (MTSOs) and the backbone network to connect base stations through MTSOs to the Public Switched Telephone Network (PSTN). The formulation results in a very large scale integer programming problem with up to 18,000 integer variables and 20,000 constraints. To solve this large scale integer programming problem, we develop a pair of models, one for the upper bound and one for the lower bound. The upper bound model relaxes integrality on some of the variables while the lower bound model uses a 5% optimality gap to achieve early termination. Additionally, we develop a heuristic procedure that can solve the largest problem instances very quickly with a small optimality gap. To demonstrate the efficiency of the proposed solution methods, problem instances were solved with five candidate MTSOs servicing some 11,000 simultaneous cellular phone sessions on a network with up to 160 base stations. In all instances, solutions guaranteed to be within 5% of optimality were obtained in less than an hour of CPU time. Keywords: mobile networks; network design; integer programming; W-CDMA. Reference to this paper should be made as follows: Cai, Q., Kalvenes, J., Kennington, J. and Olinick, E. (XXXX) W-CDMA network design, Int. J. Mobile Network Design and Innovation, Vol. X, No. Y, pp.xxx XXX. Biographical notes: Qibin Cai graduated from Southern Methodist University with a PhD in Operations Research in August His research interest focuses on the design and analysis of mathematical optimisation models for telecommunication network design problems. Advised by Dr. Jeffery Kennington, he presents a set of optimisation-based design tools for W-CDMA cellular networks in his PhD dissertation. He is now employed by Verizon Business as an Operations Research Analyst in Richardson, Texas, developing and supporting optimisation algorithms that underlie a suite of domestic and global network planning systems. He holds Masters and Bachelors degrees in Transportation Planning from Shanghai Maritime University. Joakim Kalvenes is a Visiting Scholar at the University of Texas at Dallas. He received an MSc in Applied Mathematics from the Norwegian Institute of Technology in 1989, an MA in Economics from the Norwegian School of Economics and Business Administration in 1992, an MBA in Finance and Operations Management and a PhD in Management Information Systems in 1994 and 1996, respectively, from Vanderbilt University. His research interests include analysis and design of mobile communication systems, revenue management in communication services and design of mechanisms for providing transaction privacy and integrity in electronic market Copyright XXXX Inderscience Enterprises Ltd.

2 2 Q. Cai et al. places. His research has appeared in journals such as Operations Research, Management Science, Information Systems Research, INFORMS Journal on Computing, Production and Operations Management and Wireless Networks. Jeff Kennington received a BSIE from the University of Arkansas in 1968, an MSIE in 1970 and a PhD in 1973 from the Georgia Institute of Technology. He joined Southern Methodist University in June 1973 and has served in several Departments. He served as Chair of the Department of Operations Research and Engineering Management for 12 years and Chair of the Department of Computer Science and Engineering for five years. He serves as an Associate Editor for Networks and the INFORMS Journal on Computing and is on the Editorial Board of Telecommunication Systems and Computational Optimisation and Applications. Eli V. Olinick is an Associate Professor in the Department of Engineering Management, Systems and Information at Southern Methodist University. He completed his BS inapplied Mathematics at Brown University and earned his MS and PhD in Industrial Engineering and Operations Research at the University of California at Berkeley where he wrote his PhD thesis on OptimisationAlgorithms for Survivable Network Design Problems. His research interests are in applied optimisation and network design problems. He currently serves as the Treasurer of the INFORMS Technical Section on Telecommunications. 1 Introduction Third generation mobile communication systems currently under development promise to provide subscribers with high-speed data services at rates up to a hundred times that of second generation voice channels. There are two accepted major standards for third generation mobile systems (W-CDMA and CDMA2000, respectively), both of which are based on Code Division Multiple Access (CDMA) technology. This paper presents a comprehensive model of the wideband CDMA network design problem. Model features include Mobile Telephone Switching Office (MTSO) and base station (tower) site selection, backbone network design and customer service assignment to selected towers. CDMA network design problems differ considerably from other wireless network design problems in that channel allocation is not an explicit issue. In each cell, all of the bandwidth available to the service provider can be used. The features in CDMA making this possible are stringent power control of all system devices (including user handsets) and the use of orthogonal codes to ensure minimal interference between simultaneous sessions. Instead, however, the network design must take into consideration the system-wide interference generated by the mobile users in the service area. Previous work on CDMA system design has focused on base station location and customer assignment. Galota et al. (2001) proposed a profit maximisation model for base station location and customer service assignment based on a limited interference model. Similarly, Mathar and Schmeink (2001) developed a budget-constrained system capacity maximisation model, in which the interference model accounted for base stations utilised instead of the number of customers serviced by each respective base station. Amaldi et al. (2001a) provided a cost minimisation model that explicitly considers the signal-to-interference conditions generated by the base station location and customer service assignment choices by means of a penalty term in the objective function. Building upon this work, Kalvenes et al. (2006) developed a profit maximisation model in which the signal-to-interference requirements are enforced as constraints in the mathematical programming model. In another stream of work, researchers have modelled the selection of MTSOs and the assignment of base stations to MTSOs. Merchant and Sengupta (1995) developed a cost minimisation model that includes base station to MTSO wiring cost and handoff cost for given traffic volume at the base stations. The same concept was refined by Li et al. (1997). Neither investigation includes cost for connecting the MTSOs to one another or to the Public Switched Telephone Network (PSTN). This investigation expands previous work in the area by combining the tower location and the backbone design problems described above into a single, comprehensive model of W-CDMA network design. The resulting model includes the selection of base stations and MTSO locations, the assignment of customer locations to base stations and the design of a spanning tree to connect the base stations, MTSOs and the PSTN gateway. The selection of base stations and MTSOs combined with the design of the spanning tree is equivalent to a Steiner tree problem (see, for instance, Beasley 1989). The model presented in this paper enhances the current industry approach to wireless network design in two ways. Firstly, prevailing practice seeks to space towers uniformly in Euclidean space. Then, modifications of this tower location choice are based on ex-post calculations of coverage and capacity. In our approach, the tower location decision incorporates coverage and capacity. Secondly, prevailing practice considers the backbone design problem only after the tower locations have been selected. In our approach, the cost of the backbone design is solved simultaneously with the tower location problem so that the potential opportunity cost of problem partitioning is eliminated. The disadvantage of using a comprehensive model as proposed here comes primarily in the form of potentially high computational cost.

3 W-CDMA network design 3 The contributions of this work are several. Firstly, we provide the first comprehensive discrete optimisation model for the W-CDMA network design problem. The model maximises the net revenue of service provisioning to mobile subscribers and takes into account the cost of tower construction, MTSO location, tower to MTSO connection and MTSO to PSTN gateway connection. When selecting base station locations, the revenue potential of each tower is balanced with its cost of installation and operation while simultaneously ensuring sufficient quality of service. The selected base stations are then connected to a network of MTSOs that is generated based on the cost of MTSO location and the cost of wiring from the towers to the MTSO locations. Secondly, we develop a unique solution strategy that involves the application of discrete models to obtain both upper bounds and good feasible solutions. The solution procedures exploit the problem structure through the addition of valid inequalities to the model formulation. Thirdly, we develop a new heuristic procedure that substantially reduces the computational burden for the most difficult problem instances while resulting in only small reductions in objective function value. Finally, we demonstrate the efficiency of our solution procedures by solving 40 randomly generated test cases and seven test cases from the North Dallas area, comparing the three solution procedures proposed in this paper. The very reasonable computational times and the quality of the obtained solutions are very encouraging. The software implementation of both our exact solution procedure and our heuristic procedures have been placed in the public domain so that both practitioners and other research groups can experiment with our software and compare computational results. 2 W-CDMA network design model Our model differs from previous work in that it simultaneously selects base station and MTSO locations, connects the towers and MTSOs to the PSTN access point and provides service assignment of customer locations to base stations based on a realistic interference model. Thus, this is the first comprehensive planning model for W-CDMA network design. 2.1 Sets used in the model Let L denote the set of candidate locations for tower construction. In practice, candidate tower locations are identified through examiniation of databases listing existing towers designated to other uses, examination of local maps and review of city zoning restrictions. There is a set of subscriber locations, M. In practice, subscriber locations are found in census data to determine demand from residential areas and examination of local maps to determine demand from mobile customers and business premises. The set C m L is the set of candidate towers that are able to service customers in location m M, as determined by the maximum handset transmission power. For every l L, P l M is the set of customer locations that can be serviced by tower l. Each selected tower location will be connected to an MTSO. The set of candidate MTSO locations is K. In addition, there is a gateway to the PSTN which is labelled location 0. The union of the PSTN gateway and the set K is denoted K Constants used in the model The demand for service in customer area m M is denoted by d m. In practice, the demand parameters are determined in conjunction with the subscriber locations m M. This value is the number of channel equivalents 1 required to service the population in the area at an acceptable service level (call blocking rate). Let r denote the annual revenue (in $) generated by each channel equivalent utilised in a customer area. The cost (amortised annually) of building and operating a tower at location l L and connecting it to the backbone network is given by the parameter a l. Operating cost includes the cost of transmission power, marketing, accounting, customer aquisition and retention and any other cost that is contingent upon operating a tower. When a subscriber in location m is serviced by tower l, the subscriber s handset must transmit with sufficient power so that the tower receives it at the target power level P target. Due to attenuation, the signal transmitted weakens over the path from the handset to the tower based on the relative location of the origin and destination (depending on distance, topography, local conditions, etc.). The attenuation factor from subscriber location m to tower location l is given by the parameter g ml. To ensure proper received power, P target, at the tower location, the handset will transmit with power level P target /g ml. At each tower location, signals are received from many subscriber handsets in the surrounding neighbourhood. In order for the voice packets to be processed with a reasonable error rate, the signal to interference ratio for any active session must be more than the threshold value SIR min. The selected towers will be connected to an MTSO. The MTSOs are limited in the number of base stations they can service. This limit is given by the parameter α. The annualised cost of providing a link between tower location l L and MTSO hub location k K is given by c lk, while h jk is the annualised cost of providing a link from hub location j K to hub location k K 0. Finally, b k is the annualised cost of building and operating an MTSO in location k K. 2.3 Decision variables used in the model The decision variables in this model include general integer and binary variables. The decision to build a tower at a candidate location is represented by variable y l, which is one if a tower is built at location l L; and zero, otherwise. The integer variable x ml represents the capacity assignment (in channel equivalents) to tower l L for servicing of customers in location m M. In other words, m M x ml represents the instantaneous communication capacity of tower location l L. The variables are related so that x ml 1 only if y l = 1, that is, customers in location m can be assigned to tower l for service only if tower l is built. If an MTSO is established in location k K 0, the variable z k is one; and zero, otherwise. Each tower must be linked to an MTSO. If tower l L is connected to MTSO k K, then s lk is one; otherwise, it is zero. Finally, each MTSO location must have

4 4 Q. Cai et al. a path to the PSTN gateway. We use a flow formulation to create a path from every selected MTSO location to the PSTN gateway. The integer variable u jk denotes the units of traffic flow on the link between MTSO location j K and MTSO location k K 0. If there is any flow from MTSO location j K to MTSO location k K 0, then a link between j and k has to be established. The variable w jk is one if a link is established between locations j K and k K 0 ; and zero, otherwise. 2.4 Quality of service constraint In spread-spectrum system design, it is customary to express quality of a communication link in terms of a signal-to-interference ratio. A derivation of the signal-to-interference ratio based on the available bandwidth and the link quality requirements can be found in Kalvenes et al. (2006). The total received power at tower location l, Pl TOT, from all mobile users in the service area is given by P TOT l = P target m M j C m g ml g mj x mj (1) In this expression, the signal level from customers assigned to tower l is P target, while it is P target g ml /g mj from customers assigned to some other tower j. From a single customer s perspective, the signals from other customers represent interference. Thus, for each session assigned to tower l, Pl TOT P target represents interference, while P target is the signal strength associated with the session (Amaldi et al., 2001b). Consequently, a quality of service constraint based on the threshold signal to interference ratio for each session assigned to tower l is given by P target Pl TOT SIR min (2) P target provided that tower l is constructed. Since the tower is built only if y l = 1, this constraint can be written as follows: where and g ml x mj g m M j C mj m SIR min + (1 y l )β l l L (3) β l = m M d m { ( gml max j C m \{l} g mj max m C m \{l} ( gml g mj )} ) = 0 if C m \{l} = The second term on the right-hand side is zero when a tower is built (y l = 1), so that the signal-to-interference requirement must be met at tower l. When y l = 0, the right-hand side is so large that the constraint is automatically satisfied. 2.5 Mathematical formulation The base station and MTSO location with backbone network design problem is formulated as follows. max r x ml a l y l b k z k m M l C m l L k K }{{}}{{}}{{} Subscriber revenue Tower cost MTSO cost c lk S lk h jk w jk (4) l L k K j K k K }{{} 0 \{j} }{{} Connection cost Backbone cost There are 16 sets of constraints that define the model. The first set ensures that customers can be serviced only if there are towers that cover the demand area. x ml d m y l m M,l C m (5) The next set of constrains ensures that one cannot serve more customers in a location than there is demand for service. x ml d m m M (6) l C m The next set of constraints enforce the quality of service restrictions on received signal quality at the towers. m M j C m g ml g mj x mj SIR min + (1 y l )β l l L The following two sets of constraints ensure that each base station is connected to an MTSO and that an MTSO is installed if there is a base station connected to it. s lk = y l l L (8) k K (7) s lk z k l L, k K (9) The capacity constraint on the number of base stations that can be serviced by an MTSO is given by the set of constraints s lk αz k k K (10) l L The selected MTSO locations must be connected to the PSTN gateway either directly or indirectly via another selected MTSO location. We use a flow formulation that results in a spanning tree with the PSTN gateway as its root. The first set of constraints provides a link between MTSOs j K and k K 0 if there is any flow over the link. The second set of constraints ensures that there can be traffic flow from MTSO location j to MTSO location k (or the PSTN gateway) only if MTSO location k is constructed. If z k = 1, then one unit of flow will be generated at MTSO location k. The third set of constraints represents flow conservation where the flow-out minus the flow-in equals the flow generated at each MTSO location k. The last constraint ensures that the flow into the

5 W-CDMA network design 5 PSTN gateway equals the number of MTSOs selected. u jk K w jk j K, k K 0 \{j} (11) u jk K z k j K,k K 0 \{j} (12) (u kj u jk ) + u k0 = z k k K (13) j K\{k} u k0 = z k (14) k K k K The next constraint states that the PSTN gateway is always present. z 0 = 1 (15) The last five sets of constraints provide the domains for the variables. s lk {0, 1} l L, k K (16) u jk N j K,k K 0 \{j} (17) x ml N m M,l L (18) y l {0, 1} l L (19) z k {0, 1} k K 0 (20) 2.6 Model properties In this section, we prove that our problem is NP-hard by showing that it includes the Steiner tree problem as a special case. Recall that in the Steiner tree problem, which is known to be NP-hard, one is given a graph G = (V, E) wih edge costs for each edge (i, j) E and the problem is to find a minimum cost tree that connects a given subset of the nodes U V. The tree may include any of the Steiner nodes V \U, but is not required to do so. Proposition 1: The problem (4) (20) is NP-hard. Proof: Consider the set of instances of our CDMA problem where the input is restricted to cases where L = M, C m = {m}, a m = 0, b m = 0, d m = 1 m M, SIR min < M,α > M and r > m M k M c mk + j K k K 0 \{j} h jk. Restrict the input further to cases where L K and c lk = 0ifl = k and c lk > r if l = k. Observe that for these problems each unit of demand is economically attractive to serve since the revenue per channel equivalent is larger than the cost of building a tower to provide the service and connecting that tower to the backbone network. Therefore, an optimal solution will serve all of the demand and profit is maximised by finding a minimum cost backbone. The backbone cost is minimised by connecting tower 1 to MTSO 1, tower 2 to MTSO 2 and so forth, and connecting the PSTN gateway and MTSOs 1, 2,..., M to each other via a minimum cost tree network which may possibly include some of the other MTSOs. That is, these problems correspond to the set of all Steiner tree problem instances; G is the graph induced by the MTSOs and the PSTN gateway, U ={0, 1, 2,...,M} and the cost of edge (i, j) = h ij. In terms of the formulation (4) (20), x mm = y m = s mm = 1 is optimal. Constraints (5) (7) and (10) are trivially satisfied, while (18) and (19) are redundant. The problem reduces to min c mk s mk h jk w jk (21) m M k K j K k K 0 \{j} s.t. s mk = 1 m M (22) k K s mk z k m M, k K (23) u jk K w jk j K, k K 0 (24) {j} u jk K z k j K, k K 0 (25) {j} (u kj u jk ) + u k0 = z k k K (26) j K\{k} u k0 = z k k K k K (27) z 0 = 1 (28) s mk {0, 1} m M,k K (29) u jk N j K, k K 0 (30) {j} z k {0, 1} k K 0 (31) This is the flow formulation of the Steiner tree problem where the tower locations and the PSTN gateway location represent the customer locations and the candidate MTSO locations represent the Steiner nodes. The Steiner tree problem is known to be NP-hard. Kalvenes et al. (2006) showed that in the CDMA network design problem, customers are always assigned to the nearest tower that is constructed so as to minimise overall system interference levels. That is, the following set of valid constraints can be added to the formulation: x ml d m (1 y j ) m M, l, j C m : g ml <g mj (32) In order to improve computational performance, we add a set of valid inequalities to speed up the pruning of the branch-and-bound tree in CPLEX. Constraint (7) limits the total received signal power at tower l, regardless of the signal source. A subset of the total received power comes from customers assigned to tower l for service, that is, those customer locations m for which x ml 1. Thus, if (7) is satisfied, so is the following set of constraints: x ml l L (33) SIR m P min l Also note that, in the formulation (4) (20), the variable s is integer. However, constraints (8) (10) together with the objective function ensure that s is either 0 or 1 even if the integrality restriction is relaxed. In our computational procedure, we therefore use 0 s lk 1 l L, k K instead of (16). 3 Empirical analysis Our model is implemented in software using the AMPL modelling language (Fourer et al., 2003) with a direct link to the solver in CPLEX. All test runs are made on a Compaq AlphaServer DS20E with dual EV 6.7(21264A)

6 6 Q. Cai et al. 667 MHz processors and 4096 MB of RAM. Upper and lower bound models are applied to obtain provably near-optimal solutions for realistic-sized problem instances. The computational times increase substantially as the number of candidate towers increases from 40 to 160. Therefore, we also implemented a heuristic solution procedure to solve the largest problems instances. 3.1 Solution procedure with error guarantee Our solution procedure generates a feasible solution and an upper bound to demonstrate the quality of the feasible solution. The upper bound procedure solves to optimality the problem (4) (33) with the integrality constraint on variables x,y and s relaxed. In the lower bound procedure, integrality is imposed, but an optimality gap of 5% is permitted. We created two series of test problems for the empirical evaluation of our proposed solution method. Both series of test problems were based on the parameters listed in Table 1. While these data do not represent any service provider s actual system, we have conferred with local service provider engineers to confirm the validity of the parameter value ranges. Table 1 Parameters used in the computational experiments Parameter Value or Range Description r $4,282 Annual revenue for each customer channel equivalent serviced a l U[$70,000, $100,000] Annualised cost for installing a base station in location l b k U[$300,000, $375,000] Annualised cost for installing an MTSO in location k f $1.00 Annualised cost per foot of wiring α 225 Maximum number of base stations that can be connected to an MTSO In the first series of test problems, customer demand points and candidate tower locations were drawn from a uniform distribution over a 13.5 km 8.5 km rectangular service area. The number of demand points was 1,000 and 2,000, respectively, while the number of candidate tower locations was 40, 80, 120 or 160. Six candidate MTSO locations (including the PSTN gateway) were drawn from a uniform distribution over a 1.5km 1.0 km rectangular area centered on the 13.5km 8.5 km service area. Each demand point had demand drawn from a uniform distribution of integers between 1 and 10 channel equivalents. With a mean of 5.5 units of demand in each customer location, the mean demand over the entire service area was 5,500 and 11,000, respectively. The attenuation factors g ml were then calculated based on Hata s path loss model (Hata, 1980). A tower location l that was close enough to provide service to customer point m (given by the requirement that g ml > ) would be included in the set C m. Depending on the number of towers in the service area, the average size of the sets C m varied between 2.0 and 8.4. The test problem data are listed in Table 2 and problem instance R500 is displayed in Figure 1 with demand locations uniformly distributed over the service area (circles are subscriber locations, triangles are candidate tower locations and squares are candidate MTSO locations). Table 2 Problem data for uniformly distributed subscribers (d m U[1, 10]) Problem Number of Number of Total Average name candidate customer number of size of towers locations customers C m R R R R R R R R R R R R R R R R R R R R R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , The computational results for the 40 test problems with randomly distributed customer locations are displayed in Table 3. The table shows that out solution procedure can find very high quality solutions for realistic-sized problems with reasonable computational effort. The solution times varied from less than 30 sec for the smaller problem instances (R110 R150) to less than 60 min for the larger problem instances (R460 R500). Thus, when we increased the number of customer locations from 1,000 to 2,000 and the number of candidate tower locations from 40 to 160 (implying a larger number of possible tower selections for each customer location), the computational effort increased by less than two orders of magnitude. In the smaller problem instances, one MTSO was selected, while two MTSOs were in the solution for most of the larger problem instances.

7 W-CDMA network design 7 Table 3 Empirical results for test problems with uniformly distributed subscribers (d m U[1, 0]) Problem name Upper bound Best feasible solution (mipgap = 5%) MTSOs Towers Customers Profit CPU time MTSOs Towers Customers Profit CPU time optimality built built serviced (%) ($M) (hh:mm:ss) built built serviced (%) ($M) (hh:mm:ss) gap % R :00: :00: R :00: :00: R :00: :00: R :00: :00: R :00: :00: R :08: :01: R :05: :01: R :10: :01: R :05: :01: R :10: :01: R :43: :08: R :26: :07: R :24: :05: R :24: :06: R :45: :07: R :57: :15: R :40: :30: R :44: :21: R :00: :17: R :40: :15: R :00: :01: R :00: :00: R :00: :00: R :00: :01: R :00: :00: R :10: :03: R :07: :03: R :09: :04: R :10: :04: R :34: :04: R360 N/A N/A N/A N/A 02:00:00 a :14: R370 N/A N/A N/A N/A 02:00:00 a :16: R380 N/A N/A N/A N/A 02:00:00 a :17: R390 N/A N/A N/A N/A 02:00:00 a :21: R :59: :15: R460 N/A N/A N/A N/A 02:00:00 a :56: R470 N/A N/A N/A N/A 02:00:00 a :35: R480 N/A N/A N/A N/A 02:00:00 a :36: R490 N/A N/A N/A N/A 02:00:00 a :41: R500 N/A N/A N/A N/A 02:00:00 a :49: a terminated due to 2-hr time limit. Figure 1 Graphical representation of problem R500 The upper bound problem was solved to optimality, while the best feasible procedure was terminated when a solution was found that was less than 5% less than the upper bound generated by the branch-and bound tree in CPLEX. Comparing the upper bound solution to the best feasible solution, we observe that the optimality gap did not increase significantly as the problem size increased. For nine of the ten largest problem instances (R360 R400 and R460 R500), the upper bound procedure could not find a solution within 2 hr of CPU-time. In these cases, we reported the error tolerance (mipgap) of the best feasible solution procedure (which was 5%). Figure 2 illustrates the solution for test problem R500. Next, we solved seven problem instances with data from the North Dallas service area. We created sample

8 8 Q. Cai et al. problems with demand points concentrated along the major thoroughfares. In addition, we created three hot spots of demand in the downtown district, the Galleria area and the DFW airport. Residual demand was drawn from a uniform distribution over the service area. In each customer location, demand was drawn from a uniform distribution with values between one and ten simultaneous users. In these problem instances, there are six candidate MTSO locations, 120 candidate tower locations and 2000 customer locations with the number of simultaneous calls in each location distributed uniformly between one and ten. Problem ND700 is illustrated in Figure 3 with demand locations concentrated along four major thoroughfares and in three hotspots in the North Dallas area. Figure 2 Solution to problem R500 The solution to these seven problems are presented in Table 4. We note that the quality of the solutions as well as the computational times are comparable to those for the random problem instances in Table 3. The solution to test problem ND700 is illustrated in Figure 4. Examining this figure, a network engineer may find that coverage using the 82 selected towers is insufficient in certain areas. To remedy this problem, the network engineer can add candidate tower locations and solve the problem again using the current solution as a starting point. In this example, we added six towers to the current solution and resolved the customer allocation problem with these 88 towers fixed. The CPU time for the modified problem was 1 sec and the resulting solution is displayed in Figure 5. In the modified solution, the coverage increased from 84.7% to 89.8% and the net revenue increased from $31.93 to $33.73 million (or 5.6%). It is possible that this solution is not optimal given the full set of 126 candidate tower locations. However, a network engineer can use our solution method in an interactive fashion and when satisfied with the options for candidate tower locations, can solve the entire problem to optimality. A network engineer can also consciously choose to add towers in an area where it is not profitable in anticipation of future expansion needs. Thus, our tool provides considerable flexibility to the network engineer. Figure 4 Solution to problem ND700 Figure 3 Graphical representation of problem ND Heuristic procedures Based on our experience with the computational procedure presented in the section above, we observed that solution times increase substantially as the average number of towers Table 4 Problem name Empirical results for North Dallas test problems Upper bound Best feasible solution (mipgap = 5%) MTSOs Towers Customers Profit CPU time MTSOs Towers Customers Profit CPU time Optimality built built serviced (%) ($M) (hh:mm:ss) built built serviced (%) ($M) (hh:mm:ss) Gap (%) ND :39: :02: ND :55: :03: ND :13: :03: ND :45: :03: ND :40: :03: ND :46: :06: ND :53: :03:31 1.4

9 W-CDMA network design 9 that can service a customer area increases. This observation led us to design two heuristic procedures that capitalise on limiting the number of towers to which a customer area can be assigned. The first heuristic solves the problem (4) (20) with the valid constraints (32) and (33) but with C m limited to the nearest tower in the set L. The modified test problem data are displayed in Table 5. Since some customer areas are too far from any tower to receive service, the average number of towers per demand area is slightly below one. Figure 5 Modified solution to problem ND700 Table 6 gives the computational results for Heuristic 1 compared to the feasible solution procedure presented in the previous section. We observe that Heuristic 1 performs well on the smaller problem instances, but that the optimality gap increases substantially for larger problem instances. The reason is that Heuristic 1 will add too many towers to the solution in order to service the customers. While it is better to service these customers from a larger number of towers than not serving them at all, using such a large number of towers is inefficient. It is interesting to note, though, that Heuristic 1 performs better on problem instances with high demand density per tower (i.e. the optimality gap is smaller for the problem instances with 2000 customer locations than for those with 1000 customer locations with the same number of candidate tower locations). This result stems from the fact that in the high density demand problem instances, a higher percentage of the candidate towers will be constructed in the optimal solution, resulting in a smaller difference in solution between optimum and the solution obtained with Heuristic 1. Over all, we conclude that Heuristic 1 is too restrictive in the solution space considered to be of any significant practical use. In the second heuristic, we restrict the set of permissible tower assignments to at most two for each customer area. Table 7 displays the modified test data for Heuristic 2. Again, some of the customer service areas are not within the range of two towers (or not within the range of any tower) and thus, the average number of towers considered per demand area is slightly smaller than two. The computational results for Heuristic 2 compared to the feasible solution procedure are displayed in Table 8. Since the upper bound procedure failed to produce a solution within two hr of computational time for problem instances R360 R390 and R460 R500, we used the objective function Table 5 Problem data for uniformly distributed subscribers (d m U[1, 10]) for Heuristic 1 with C m 1 Problem Number of Number of Total Average name candidate customer number of size of towers locations customers C m R R R R R R R R R R R R R R R R R R R R R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , value obtained with the best feasible solution procedure in the previous section, divided by 0.95 (1-mipgap) to generate an upper bound. Thus, the gap reported for Heuristic 2 in Table 8 may be larger than the actual gap. However, the solutions based on Heuristic 2 are not quite as good as the solution obtained with the best feasible solution procedure. We note that the computational times are shorter for Heuristic 2 than for the best feasible solution procedure from the previous section, in particular for the larger problem instances. At the same time, the difference in solution quality is less than 5% in all problem instances. Thus, although Heuristic 2 does not provide an error guarantee, it is robust enough to produce good feasible solutions within reasonable computational times for very large problem instances. This is particularly true for the higher demand density problem instances with 160 towers (R460 R500), for which the best feasible solution procedure requires the most computational

10 10 Q. Cai et al. Table 6 Problem name Empirical results for Heuristic 1 applied to test problems with uniformly distributed subscribers (d m U[1, 0]) Best Feasible Solution (mipgap = 5%) Heuristic 1( C m 1) MTSOs Towers Customers Profit CPU time Optimality MTSOs Towers Customers Profit CPU time Optimality built built serviced (%) ($M) (hh:mm:ss) gap % built built serviced (%) ($M) (hh:mm:ss) gap % R :00: :00: R :00: :00: R :00: :00: R :00: :00: R :00: :00: R :01: :00: R :01: :00: R :01: :00: R :01: :00: R :01: :00: R :08: :00: R :07: :00: R :05: :00: R :06: :00: R :07: :00: R :15: :00: R :30: :00: R :21: :00: R :17: :00: R :15: :00: R :01: :00: R :00: :00: R :00: :00: R :01: :00: R :00: :00: R :03: :00: R :03: :00: R :04: :00: R :04: :00: R :04: :00: R :14: :00: R :16: :00: R :17: :00: R :21: :00: R :15: :00: R :56: :00: R :35: :00: R :36: :00: R :41: :00: R :49: :00: Table 7 Problem data for uniformly distributed subscribers (d m U[1, 10]) for Heuristic 2 with C m 2 Problem name Number of candidate towers Number of customer locations Total number of customers Average size of C m R R R R R R R R R R R

11 W-CDMA network design 11 Table 7 Problem data for uniformly distributed subscribers (d m U[1, 10]) for Heuristic 2 with C m 2(Continued) Problem name Number of candidate towers Number of customer locations Total number of customers Average size of C m R R R R R R R R R R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , R , Table 8 Problem name Empirical results for Heuristic 2 applied to test problems with uniformly distributed subscribers (d m U[1, 10]) Best Feasible Solution (mipgap = 5%) Heuristic 1( C m 2) MTSOs Towers Customers Profit CPU time Optimality MTSOs Towers Customers Profit CPU time Optimality built built serviced (%) ($M) (hh:mm:ss) gap % built built serviced (%) ($M) (hh:mm:ss) gap % R :00: :00: R :00: :00: R :00: :00: R :00: :00: R :00: :00: R :01: :00: R :01: :01: R :01: :00: R :01: :00: R :01: :01: R :08: :01: R :07: :01: R :05: :01: R :06: :01: R :07: :01: R :15: :00: R :30: :01: R :21: :00: R :17: :00: R :15: :01: R :01: :00:45 0.4

12 12 Q. Cai et al. Table 8 Problem name Empirical results for Heuristic 2 applied to test problems with uniformly distributed subscribers (d m U[1, 10]) (Continued) Best Feasible Solution (mipgap = 5%) Heuristic 1( C m 2) MTSOs Towers Customers Profit CPU time Optimality MTSOs Towers Customers Profit CPU time Optimality built built serviced (%) ($M) (hh:mm:ss) gap % built built serviced (%) ($M) (hh:mm:ss) gap % R :00: :00: R :00: :00: R :01: :00: R :00: :00: R :03: :01: R :03: :01: R :04: :01: R :04: :01: R :04: :01: R :14: :03: R :16: :05: R :17: :04: R :21: :04: R :15: :05: R :56: :06: R :35: :09: R :36: :07: R :41: :05: R :49: :07: time (35 60 minutes compared to 5 10 min for Heuristic 2). We conclude that Heuristic 2 is a viable solution procedure for very large problem instances with high demand density. 4 Conclusions In this investigation, the W-CDMA network design problem is modelled as a discrete optimisation problem. The model maximises revenue from customers serviced by the network net the cost of towers, switching facilities and backbone network connecting the towers and switching facilities to the PSTN. The resulting integer programme is very large and standard commercial software packages cannot obtain optimal solutions to realistic-sized problem instances. Therefore, we developed a solution method based on a pair of models, one for the upper bound and one for the lower bound. The solution method was implemented in software using the AMPL/CPLEX system. We tested our solution method on 40 large test problems with 1,000 2,000 customer locations with an average of 5.5 customers in each location, while the candidate tower locations varied between 40 and 160 and the number of candidate switching locations was 5. We solved all of these test problems to within a guaranteed 5% of optimality using very reasonable computational effort. The largest test problems required up to 60 min of CPU time. In an effort to reduce the computational times for the largest and most difficult problem instances, we developed and tested two heuristic procedures. One of these procedures proved efficient for the largest test problems, reducing the computational effort by one order of magnitude at a penalty of less than 5% of the objective function value. We also tested our solution method on seven test problems based on the infrastructure and travel patterns in the North Dallas area. The results for these test problems were on par with those for the randomly generated test problems. Additionally, we provided an example of how our tool can be used in an interactive fashion in which a network engineer can manually modify the solution to expand the number of candidate towers or to make use of specific parts of the network infrastructure. Modifications to a solution can be evaluated in seconds with our solution method. Thus, it provides network engineers with significant flexibility when analysing a network provisioning plan. Acknowledgement This research was supported in part by the Office of Naval Research Award Number N References Amaldi, E., Capone, A. and Malucelli, F. (2001a) Discrete Models and algorithms for the capacitated location problems arising in UMTS network planning, Proceedings of the 5th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, Rome. ACM, pp.1 8. Amaldi, E., Capone, A. and Malucelli, F. (2001b) Improved models and algorithms for UMTS radio planning. IEEE 54th Vehicular Technology Conference Proceedings, IEEE, pp Beasley, J.E. (1989) An SST-based algorithm for the steiner problem in graphs, Networks, Vol. 19, No. 1, pp.1 16.