Technical Report 03-EMIS-02 W CDMA Network Design Qibin Cai 1 Joakim Kalvenes 2 Jeffery Kennington 1 Eli Olinick 1 1 {qcai,jlk,olinick}@engr.smu.edu School of Engineering Southern Methodist University Dallas, TX 75275-0122 2 kalvenes@tecom.cox.smu.edu Edwin L. Cox School of Business Southern Methodist University Dallas, TX 75275-0333 December 2003 This research was supported in part by the Office of Naval Research Award Number N00014-96-1-0315.
Abstract In this investigation, the W CDMA network design problem is modeled as a discrete optimization problem that maximizes revenue net the cost of constructing base stations, mobile telephone switching offices, and the backbone network to connect base stations through mobile telephone switching offices to the public switched telephone network. 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 mobile telephone switching offices 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.
1 Introduction Third generation mobile communication systems currently under development promise to provide its 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 manuscript presents a comprehensive model of the wideband CDMA network design problem. Model features include mobile 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 maximization 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 maximization model, in which the interference model accounted for base stations utilized instead of the number of customers serviced by each respective base station. Amaldi et al. (2001a) provided a cost minimization 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. (2003) developed a profit maximization model in which the signal-to-interference requirements are enforced as constraints in the mathematical programing model. In another stream of work, researchers have modeled the selection of MTSOs and the assignment of base stations to MTSOs. Merchant and Sengupta (1995) developed a cost minimization 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 extends the basic ideas presented in Kalvenes et al. (2003) and provides a comprehensive model of W CDMA network design, including 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 contributions of this work are several. First, we provide the first comprehensive discrete optimization model for the W CDMA network design problem. The model maximizes 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 2
of MTSOs that is generated based on the cost of MTSO location and the cost of wiring from the towers to the MTSO locations. Second, 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. Third, 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 manuscipt. 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 at http://www.engr.smu.edu/ jlk/publications/publications.htm 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. 3
2.1 Sets Used in the Model Let L denote the set of candidate locations for tower construction. There is a set of subscriber locations, M. 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 a mobile telephone switching office (MTSO). The set of candidate MTSO locations is K. In addition, there is a gateway to the public switched telephone network which is labeled location 0. The union of the PSTN gateway and the set K is denoted K 0. 2.2 Constants Used in the Model The demand for service in customer area m M is denoted by d 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 utilized in a customer area. The cost (amortized 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 1 CDMA does not utilize channles to allocate bandwidth to sessions, but an equivalent maximum transmission bitrate is allocated to sessions through the use of orthogonal spreading codes. 4
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 neighborhood. 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 annualized 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 annualized cost of providing a link from hub location j K to hub location k K 0. Finally, b k is the annualized cost of locating 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 5
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 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. (2003). 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, P TOT l P target represents interference, while P target is the signal strength associated with the session (Amaldi et al. 2001b). Consequently, a quality of service con- 6
straint based on the threshold signal to interference ratio for each session assigned to tower l is given by P target P TOT l P target SIR min, (2) provided that tower l is constructed. Since the tower is built only if y l = 1, this constraint can be written as follows: m M where β l = m M d m g ml 1 x mj 1 + + (1 y l )β l l L, (3) g j C mj SIR min m { max m Cm\{l} ( gml g mj ) } ( ) and max gml j Cm\{l} 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 m M l C m x ml }{{} Subscriber revenue l L a l y l }{{} Tower cost k K b k z k }{{} MTSO cost l L k K c lk s lk }{{} Connection cost j K k K 0 \{j} h jk w jk. }{{} Backbone cost (4) There are 16 sets of constraints that define the model. The first set ensures that 7
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 constraints ensures that one cannot serve more customers in a location than there is demand for service. l C m x ml d m m M. (6) The next set of constraints enforce the quality of service restrictions on received signal quality at the towers. m M g ml 1 x mj 1 + + (1 y l )β l l L. (7) g mj SIR min j C m 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 s lk z k l L, k K. (9) The capacity constraint on the number of base stations that can be serviced by an 8
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 public switched telephone network 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 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) 9
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) with 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 = 0 if l = 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 10
profit is maximized by finding a minimum cost backbone. The backbone cost is minimized 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 subject to min c mk s mk h jk w jk (21) m M k K j K k K 0 \{j} 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 \ {j}, (24) u jk K z k j K, k K 0 \ {j}, (25) (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) 11
s mk {0, 1} m M, k K, (29) u jk N j K, k K 0 \ {j}, (30) 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. (2003) showed that in the CDMA network design problem, customers are always assigned to the nearest tower that is constructed so as to minimize 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 such that 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, i.e., those customer locations m for which x ml 1. Thus, if (7) is satisfied, so is the following set of constraints: m P l x ml 1 + 1 SIR min l L. (33) 12
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 modeling language (Fourer et al. 2003) with a direct link to the solver in CPLEX (http://www.cplex.com). All test runs are made on a Compaq AlphaServer DS20E with dual EV 6.7(21264A) 667 MHz processors and 4,096 MB of RAM. Upper and lower bound models are applied to obtain provably nearoptimal 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 I. While these data do not represent any service provider s actual system, we have conferred 13
with local service provider engineers to confirm the validity of the parameter value ranges. Table I about here. 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 by 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 1.5 km by 1.0 km rectangular area centered on the 13.5 km by 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 > 10 15 ) 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 II and problem instance R500 is displayed in Figure 1. Table II about here. Figure 1 about here. The computational results for the forty test problems with randomly distributed customer locations are displayed in Table III. The table shows that our solution procedure can 14
find very high quality solutions for realistic-size problems with reasonable compuational effort. The solution times varied from less than thirty seconds for the smaller problem instances (R110 R150) to less than sixty minutes 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. Table III about here. 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 hours of CPUtime. 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. Figure 2 about here. Next, we solved seven problem instances with data from the North Dallas service area. We created sample 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 15
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 2,000 customer locations with the number of simultaneous calls in each location distributed uniformly between one and ten. Problem ND700 is illustrated in Figure 3. Figure 3 about here. The solution to these seven problems are presented in Table IV. 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 III. Table IV about here. 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 re-solved the customer allocation problem with these 88 towers fixed. The CPU time for the modified problem was 1 second 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 million 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 16
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 about here. Figure 5 about here. 3.2 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 that can service a customer area increases. This observation lead us to design two heuristic procedures that capitalize 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 V. 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. Table V about here. Table VI 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 17
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 2,000 customer locations than for those with 1,000 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. Table VI about here. In the second heuristic, we restrict the set of permissible tower assignments to at most two for each customer area. Table VII 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. Table VII about here. The computational results for Heuristic 2 compared to the feasible solution procedure are displayed in Table VIII. Since the upper bound procedure failed to produce a solution within 2 hours of computational time for problem instances R360 R390 and R460-R500, we 18
used the objective function 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 VIII 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 solution 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 time (35 60 minutes compared to 5 10 minutes for Heuristic 2). We conclude that Heuristic 2 is a viable solution procedure for very large problem instances with high demand density. Table VIII about here. 4 Conclusions In this investigation, the W CDMA network design problem is modeled as a discrete optimization problem. The model maximizes 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 public switched telephone network. The resulting integer program is very large and standard commercial software packages cannot obtain optimal solutions to 19
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 to 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 minutes 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 analyzing a network provisioning plan. 20
References Amaldi, E., A. Capone, and F. Malucelli, 2001a. Discrete Models and Algorithms for the Capacitated Location Problems Arising in UMTS Network Planning. In Proceedings of the 5th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, pp. 1 8, Rome. ACM. Amaldi, E., A. Capone, and F. Malucelli, 2001b. Improved Models and Algorithms for UMTS Radio Planning. In IEEE 54th Vehicular Technology Conference Proceedings, pp. 920 924. IEEE. Beasley, J. E., 1989. An SST-Based Algorithm for the Steiner Problem in Graphs. Networks 19(1), 1 16. Fourer, R., D. M. Gay, and B. W. Kernighan, 2003. AMPL: A Modeling Language for Mathematical Programming. Brooks/Cole Thomson Learning, Pacific Grove, CA, 2nd edn. Galota, M., C. Glasser, S. Reith, and H. Vollmer, 2001. A Ploynomial-Time Approximation Scheme for Base Station Positioning in UMTS Networks. In Proceedings of the 5th International Workshop on Discrete Algorithms and Methods for Mobile Computing and Communications, pp. 52 59, Rome. ACM. Hata, M., 1980. Empirical Formula for Propagation Loss in Land Mobile Radio Service. IEEE Transactions on Vehicular Technology 29, 317 325. 21
Kalvenes, J., J. Kennington, and E. Olinick, 2003. Base Station Location and Service Assignment in W CDMA Networks. Technical Report 02-EMIS-03, School of Engineering, Southern Methodist University, Dallas, TX, http://engr.smu.edu/ olinick/cdma.pdf. Li, J., H. Kameda, and H. Itoh, 1997. Balanced Assignment of Cells in PCS Networks. In Proceedings of the 1997 ACM Symposium on Applied Computing, pp. 297 301, San Jose, CA. ACM Press. Martello, S., and P. Toth, 1990. Knapsack Problems: Algorithms and Computer Implementations. Wiley, Chichester, England. Mathar, R., and M. Schmeink, 2001. Optimal Base Station Positioning and Channel Assignment for 3G Mobile Networks by Integer Programming. Annals of Operations Research 107, 225 236. Merchant, A., and B. Sengupta, 1995. Assignment of Cells to Switches in PCS Networks. ACM/IEEE Transactions on Networking 3(5), 521 526. Pisinger, D., 1999. Core Problems in Knapsack Algorithms. Operations Research 47, 570 575. 22
Parameter Value or Range Description r $4,282 Annual revenue for each customer channel equivalent serviced a l U[$70, 000, $100, 000] Annualized cost for instaling a base station in location l b k U[$300, 000, $375, 000] Annualized cost for installing an MTSO in location k f $1.00 Annualized cost per foot of wiring α 225 Maximum number of base stations that can be connected to an MTSO Table I: Parameters used in the computational experiments. 23
Problem Number of Number of Total Average Name Candidate Customer Number of Size of Towers Locations Customers C m R110 40 1,000 5,620 2.1 R120 40 1,000 5,637 2.0 R130 40 1,000 5,638 2.0 R140 40 1,000 5,626 2.1 R150 40 1,000 5,625 2.2 R160 80 1,000 5,609 4.2 R170 80 1,000 5,598 4.2 R180 80 1,000 5,608 4.1 R190 80 1,000 5,593 4.2 R200 80 1,000 5,608 4.2 R210 120 1,000 5,665 6.4 R220 120 1,000 5,676 6.4 R230 120 1,000 5,686 6.3 R240 120 1,000 5,669 6.4 R250 120 1,000 5,695 6.4 R410 160 1,000 5,612 8.4 R420 160 1,000 5,630 8.4 R430 160 1,000 5,628 8.3 R440 160 1,000 5,642 8.3 R450 160 1,000 5,650 8.3 R260 40 2,000 11,020 2.0 R270 40 2,000 11,031 2.1 R280 40 2,000 11,031 2.1 R290 40 2,000 11,036 2.1 R300 40 2,000 11,051 2.1 R310 80 2,000 11,038 4.3 R320 80 2,000 11,051 4.3 R330 80 2,000 11,041 4.3 R340 80 2,000 11,033 4.3 R350 80 2,000 11,033 4.3 R360 120 2,000 11,042 6.4 R370 120 2,000 11,030 6.4 R380 120 2,000 11,033 6.4 R390 120 2,000 11,015 6.4 R400 120 2,000 11,010 6.4 R460 160 2,000 11,001 8.4 R470 160 2,000 11,003 8.4 R480 160 2,000 11,001 8.4 R490 160 2,000 10,998 8.4 R500 160 2,000 10,987 8.4 Table II: Problem data for uniformly distributed subscribers (d m U[1, 10]). 24
25 Problem Upper Bound Best Feasible Solution (mipgap=5%) Name MTSOs Towers Customers Profit CPU Time MTSOs Towers Customers Profit CPU Time Optimality Built Built Serviced ($M) (hh:mm:ss) Built Built Serviced ($M) (hh:mm:ss) Gap R110 1 35.6 92.6% 18.33 00:00:02 1 37 92.8% 18.22 00:00:20 0.6% R120 1 35.7 92.4% 18.37 00:00:03 1 36 92.2% 18.30 00:00:21 0.4% R130 1 33.4 88.4% 17.63 00:00:02 1 35 88.8% 17.53 00:00:16 0.6% R140 1 34.0 86.4% 17.03 00:00:29 1 35 85.9% 16.80 00:00:25 1.3% R150 1 30.9 85.0% 17.09 00:00:03 1 31 84.8% 16.99 00:00:22 0.6% R160 1 42.0 92.2% 17.55 00:08:43 1 39 87.5% 16.74 00:01:40 4.6% R170 1 42.0 91.4% 17.36 00:05:41 1 43 89.3% 16.75 00:01:50 3.6% R180 1 41.0 92.4% 17.79 00:10:34 1 43 90.4% 17.10 00:01:38 4.0% R190 1 42.7 92.8% 17.61 00:05:40 1 44 91.1% 17.07 00:01:46 3.2% R200 1 43.5 92.1% 17.41 00:10:58 1 43 89.7% 16.87 00:01:50 3.2% R210 1 50.0 94.2% 17.66 00:43:18 1 51 91.5% 16.97 00:08:48 4.1% R220 1 51.1 94.7% 17.58 00:26:28 1 53 92.6% 16.92 00:07:32 3.9% R230 1 50.1 94.9% 17.88 00:24:44 1 48 91.0% 17.15 00:05:05 4.3% R240 1 48.0 94.0% 17.81 00:24:33 1 51 93.0% 17.27 00:06:29 3.1% R250 1 48.1 94.3% 17.91 00:45:44 1 48 92.2% 17.42 00:07:54 2.9% R410 1 53.1 93.1% 16.81 00:57:02 1 53 90.3% 16.21 00:15:07 3.7% R420 1 53.7 93.1% 16.78 01:40:11 1 54 90.2% 16.04 00:30:43 4.6% R430 1 52.3 92.7% 16.85 01:44:26 1 53 90.2% 16.23 00:21:03 3.9% R440 1 54.0 93.2% 16.86 01:00:32 1 55 90.8% 16.25 00:17:25 3.7% R450 2 55.8 94.2% 16.84 01:40:19 1 55 91.4% 16.16 00:15:53 4.2% R260 1 37.0 65.3% 26.72 00:00:14 1 38 65.3% 26.60 00:01:17 0.4% R270 1 37.6 67.6% 27.80 00:00:14 1 39 67.7% 27.70 00:00:50 0.4% R280 1 37.3 69.7% 28.85 00:00:17 1 38 69.7% 28.75 00:00:52 0.4% R290 1 37.4 68.2% 28.25 00:00:17 1 39 68.3% 28.12 00:01:19 0.5% R300 1 38.2 66.8% 27.63 00:00:14 1 40 67.0% 27.53 00:00:59 0.4% R310 1 62.4 87.6% 34.93 00:10:04 1 65 86.8% 34.33 00:03:51 1.8% R320 1 62.0 87.0% 34.79 00:07:44 1 57 83.7% 33.70 00:03:32 3.2% R330 1 64.0 88.3% 35.14 00:09:17 1 68 87.9% 34.50 00:04:12 1.9% R340 1 65.1 89.1% 35.32 00:10:38 1 66 87.7% 34.59 00:04:16 2.1% R350 1 64.5 88.6% 35.15 00:34:46 1 64 87.2% 34.58 00:04:20 1.7% R360 N/A N/A N/A N/A 02:00:00 1 75 93.4% 36.42 00:14:52 5.0% R370 N/A N/A N/A N/A 02:00:00 2 81 94.5% 36.38 00:16:22 5.0% R380 N/A N/A N/A N/A 02:00:00 2 80 92.2% 35.27 00:17:14 5.0% R390 N/A N/A N/A N/A 02:00:00 2 78 93.6% 36.15 00:21:53 5.0% R400 2 76.7 94.1% 37.16 00:59:47 2 79 94.1% 36.26 00:15:29 2.5% R460 N/A N/A N/A N/A 02:00:00 2 88 93.7% 35.24 00:56:40 5.0% R470 N/A N/A N/A N/A 02:00:00 1 84 91.4% 34.61 00:35:20 5.0% R480 N/A N/A N/A N/A 02:00:00 2 88 93.6% 35.25 00:36:12 5.0% R490 N/A N/A N/A N/A 02:00:00 2 92 94.7% 35.31 00:41:08 5.0% R500 N/A N/A N/A N/A 02:00:00 2 87 93.1% 34.95 00:49:15 5.0% terminated due to 2-hour time limit. Table III: Empirical results for test problems with uniformly distributed subscribers (d m U[1, 10]).
26 Problem Upper Bound Best Feasible Solution (mipgap=5%) Name MTSOs Towers Customers Profit CPU Time MTSOs Towers Customers Profit CPU Time Optimality Built Built Serviced ($M) (hh:mm:ss) Built Built Serviced ($M) (hh:mm:ss) Gap ND100 2 75.4 82.6% 31.55 00:39:05 2 77 82.0% 31.13 00:02:12 1.4% ND200 2 81.3 85.5% 32.33 00:55:18 3 83 84.8% 31.71 00:03:35 2.0% ND300 3 84.2 87.3% 33.00 01:13:05 2 82 85.2% 32.17 00:03:15 2.6% ND400 2 79.5 86.3% 32.92 00:45:29 2 80 85.6% 32.45 00:03:04 1.5% ND500 2 81.5 87.0% 32.93 00:40:46 2 82 85.0% 32.00 00:03:10 2.9% ND600 2 80.7 86.1% 32.86 00:46:06 2 82 85.9% 32.63 00:06:00 0.7% ND700 2 81.4 85.6% 32.36 00:53:35 2 82 84.7% 31.93 00:03:31 1.4% Table IV: Empirical results for North Dallas test problems.
Problem Number of Number of Total Average Name Candidate Customer Number of Size of Towers Locations Customers C m R110 40 1,000 5,620 0.94 R120 40 1,000 5,637 0.94 R130 40 1,000 5,638 0.90 R140 40 1,000 5,626 0.88 R150 40 1,000 5,625 0.87 R160 80 1,000 5,609 0.96 R170 80 1,000 5,598 0.96 R180 80 1,000 5,608 0.96 R190 80 1,000 5,593 0.96 R200 80 1,000 5,608 0.96 R210 120 1,000 5,665 0.99 R220 120 1,000 5,676 0.99 R230 120 1,000 5,686 1.00 R240 120 1,000 5,669 1.00 R250 120 1,000 5,695 1.00 R410 160 1,000 5,612 1.00 R420 160 1,000 5,630 1.00 R430 160 1,000 5,628 1.00 R440 160 1,000 5,642 1.00 R450 160 1,000 5,650 1.00 R260 40 2,000 11,020 0.90 R270 40 2,000 11,031 0.90 R280 40 2,000 11,031 0.90 R290 40 2,000 11,036 0.87 R300 40 2,000 11,051 0.81 R310 80 2,000 11,038 0.97 R320 80 2,000 11,051 0.98 R330 80 2,000 11,041 0.98 R340 80 2,000 11,033 0.97 R350 80 2,000 11,033 0.97 R360 120 2,000 11,042 0.98 R370 120 2,000 11,030 0.99 R380 120 2,000 11,033 0.99 R390 120 2,000 11,015 0.99 R400 120 2,000 11,010 0.99 R460 160 2,000 11,001 1.00 R470 160 2,000 11,003 1.00 R480 160 2,000 11,001 1.00 R490 160 2,000 10,998 1.00 R500 160 2,000 10,987 1.00 Table V: Problem data for uniformly distributed subscribers (d m U[1, 10]) for Heuristic 1 with C m 1. 27
28 Problem Best Feasible Solution (mipgap=5%) Heuristic 1 ( C m 1) Name MTSOs Towers Customers Profit CPU Time Optimality MTSOs Towers Customers Profit CPU Time Optimality Built Built Serviced ($M) (hh:mm:ss) Gap Built Built Serviced ($M) (hh:mm:ss) Gap R110 1 37 92.8% 18.22 00:00:20 0.6% 1 40 93.5% 18.09 00:00:01 1.3% R120 1 36 92.2% 18.30 00:00:21 0.4% 1 39 92.6% 18.08 00:00:01 1.6% R130 1 35 88.8% 17.53 00:00:16 0.6% 1 39 89.5% 17.28 00:00:01 2.0% R140 1 35 85.9% 16.80 00:00:25 1.3% 1 38 86.6% 16.69 00:00:01 2.0% R150 1 31 84.8% 16.99 00:00:22 0.6% 1 40 86.3% 16.48 00:00:01 3.6% R160 1 39 87.5% 16.74 00:01:40 4.6% 1 67 92.4% 15.05 00:00:01 14.2% R170 1 43 89.3% 16.75 00:01:50 3.6% 1 67 91.4% 14.90 00:00:01 14.2% R180 1 43 90.4% 17.10 00:01:38 4.0% 1 70 94.0% 15.21 00:00:01 14.5% R190 1 44 91.1% 17.07 00:01:46 3.2% 1 69 93.2% 15.06 00:00:01 14.4% R200 1 43 89.7% 16.87 00:01:50 3.2% 2 69 92.5% 15.95 00:00:01 8.3% R210 1 51 91.5% 16.97 00:08:48 4.1% 2 94 93.0% 13.03 00:00:01 26.2% R220 1 53 92.6% 16.92 00:07:32 3.9% 2 98 95.0% 13.10 00:00:01 25.5% R230 1 48 91.0% 17.15 00:05:05 4.3% 2 96 94.0% 13.20 00:00:01 26.2% R240 1 51 93.0% 17.27 00:06:29 3.1% 2 98 94.7% 13.02 00:00:01 26.9% R250 1 48 92.2% 17.42 00:07:54 2.9% 2 95 94.5% 13.40 00:00:01 25.2% R410 1 53 90.3% 16.21 00:15:07 3.7% 1 94 83.9% 10.53 00:00:01 37.4% R420 1 54 90.2% 16.04 00:30:43 4.6% 2 102 86.4% 10.42 00:00:01 37.9% R430 1 53 90.2% 16.23 00:21:03 3.9% 2 104 87.4% 10.63 00:00:01 36.9% R440 1 55 90.8% 16.25 00:17:25 3.7% 2 106 89.4% 10.79 00:00:01 36.0% R450 1 55 91.4% 16.16 00:15:53 4.2% 2 104 88.0% 10.62 00:00:01 36.9% R260 1 38 65.3% 26.60 00:01:17 0.4% 1 40 65.3% 26.38 00:00:03 1.3% R270 1 39 67.7% 27.70 00:00:50 0.4% 1 40 67.0% 27.29 00:00:03 1.8% R280 1 38 69.7% 28.75 00:00:52 0.4% 1 40 69.7% 28.55 00:00:05 1.0% R290 1 39 68.3% 28.12 00:01:19 0.5% 1 40 68.0% 27.91 00:00:03 1.2% R300 1 40 67.0% 27.53 00:00:59 0.4% 1 40 66.6% 27.35 00:00:02 1.0% R310 1 65 86.8% 34.33 00:03:51 1.8% 1 79 89.8% 33.90 00:00:02 2.9% R320 1 57 83.7% 33.70 00:03:32 3.2% 2 77 87.7% 33.62 00:00:02 3.4% R330 1 68 87.9% 34.50 00:04:12 1.9% 2 80 89.6% 34.15 00:00:02 2.8% R340 1 66 87.7% 34.59 00:04:16 2.1% 2 80 90.3% 34.47 00:00:02 2.4% R350 1 64 87.2% 34.58 00:04:20 1.7% 2 78 89.8% 34.27 00:00:03 2.5% R360 1 75 93.4% 36.42 00:14:52 5.0% 2 113 96.5% 34.39 00:00:01 10.3% R370 2 81 94.5% 36.38 00:16:22 5.0% 2 113 96.6% 34.34 00:00:01 10.3% R380 2 80 92.2% 35.27 00:17:14 5.0% 2 117 96.7% 33.96 00:00:01 8.5% R390 2 78 93.6% 36.15 00:21:53 5.0% 2 117 96.5% 33.86 00:00:01 11.0% R400 2 79 94.1% 36.26 00:15:29 2.5% 2 115 96.8% 34.11 00:00:01 8.2% R460 2 88 93.7% 35.24 00:56:40 5.0% 2 141 96.4% 31.51 00:00:01 15.1% R470 1 84 91.4% 34.61 00:35:20 5.0% 2 140 95.7% 31.26 00:00:01 14.2% R480 2 88 93.6% 35.25 00:36:12 5.0% 2 140 96.0% 31.39 00:00:01 15.4% R490 2 92 94.7% 35.31 00:41:08 5.0% 2 138 95.7% 31.46 00:00:01 15.4% R500 2 87 93.1% 34.95 00:49:15 5.0% 2 142 96.4% 31.20 00:00:01 15.2% Table VI: Empirical results for Heuristic 1 applied to test problems with 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 R110 40 1,000 5,620 1.62 R120 40 1,000 5,637 1.61 R130 40 1,000 5,638 1.51 R140 40 1,000 5,626 1.48 R150 40 1,000 5,625 1.50 R160 80 1,000 5,609 1.81 R170 80 1,000 5,598 1.81 R180 80 1,000 5,608 1.83 R190 80 1,000 5,593 1.81 R200 80 1,000 5,608 1.85 R210 120 1,000 5,665 1.95 R220 120 1,000 5,676 1.95 R230 120 1,000 5,686 1.96 R240 120 1,000 5,669 1.95 R250 120 1,000 5,695 2.00 R410 160 1,000 5,612 1.97 R420 160 1,000 5,630 1.96 R430 160 1,000 5,628 1.97 R440 160 1,000 5,642 1.97 R450 160 1,000 5,650 1.97 R260 40 2,000 11,020 1.49 R270 40 2,000 11,031 1.52 R280 40 2,000 11,031 1.51 R290 40 2,000 11,036 1.48 R300 40 2,000 11,051 1.42 R310 80 2,000 11,038 1.84 R320 80 2,000 11,051 1.85 R330 80 2,000 11,041 1.87 R340 80 2,000 11,033 1.88 R350 80 2,000 11,033 1.88 R360 120 2,000 11,042 1.95 R370 120 2,000 11,030 1.95 R380 120 2,000 11,033 1.95 R390 120 2,000 11,015 1.95 R400 120 2,000 11,010 1.94 R460 160 2,000 11,001 1.98 R470 160 2,000 11,003 1.97 R480 160 2,000 11,001 1.97 R490 160 2,000 10,998 1.96 R500 160 2,000 10,987 1.96 Table VII: Problem data for uniformly distributed subscribers (d m U[1, 10]) for Heuristic 2 with C m 2. 29
30 Problem Best Feasible Solution (mipgap=5%) Heuristic 2 ( C m 2) Name 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 R110 1 37 92.8% 18.22 00:00:20 0.6% 1 37 92.8% 18.22 00:00:14 0.6% R120 1 36 92.2% 18.30 00:00:21 0.4% 1 36 92.2% 18.30 00:00:15 0.4% R130 1 35 88.8% 17.53 00:00:16 0.6% 1 35 88.8% 17.53 00:00:10 0.6% R140 1 35 85.9% 16.80 00:00:25 1.3% 1 35 86.3% 16.90 00:00:14 0.7% R150 1 31 84.8% 16.99 00:00:22 0.6% 1 34 85.6% 16.90 00:00:13 0.6% R160 1 39 87.5% 16.74 00:01:40 4.6% 1 47 90.4% 16.53 00:00:33 3.6% R170 1 43 89.3% 16.75 00:01:50 3.6% 1 52 91.7% 16.44 00:01:00 3.2% R180 1 43 90.4% 17.10 00:01:38 4.0% 1 49 91.7% 16.87 00:00:46 3.1% R190 1 44 91.1% 17.07 00:01:46 3.2% 1 49 90.9% 16.52 00:00:33 4.0% R200 1 43 89.7% 16.87 00:01:50 3.2% 1 53 92.0% 16.47 00:01:00 3.3% R210 1 51 91.5% 16.97 00:08:48 4.1% 1 67 93.9% 15.88 00:01:14 4.1% R220 1 53 92.6% 16.92 00:07:32 3.9% 2 66 94.0% 16.06 00:01:34 3.0% R230 1 48 91.0% 17.15 00:05:05 4.3% 1 66 94.6% 16.11 00:01:42 3.8% R240 1 51 93.0% 17.27 00:06:29 3.1% 1 63 92.7% 16.08 00:01:08 3.6% R250 1 48 92.2% 17.42 00:07:54 2.9% 1 65 93.4% 15.99 00:01:15 4.6% R410 1 53 90.3% 16.21 00:15:07 3.7% 1 76 92.1% 14.26 00:00:54 14.3% R420 1 54 90.2% 16.04 00:30:43 4.6% 1 77 92.3% 14.11 00:01:30 15.9% R430 1 53 90.2% 16.23 00:21:03 3.9% 2 77 91.8% 14.20 00:00:51 15.7% R440 1 55 90.8% 16.25 00:17:25 3.7% 1 75 91.7% 14.32 00:00:49 15.1% R450 1 55 91.4% 16.16 00:15:53 4.2% 1 77 93.6% 14.69 00:01:11 12.8% R260 1 38 65.3% 26.60 00:01:17 0.4% 1 38 65.3% 26.60 00:00:45 0.4% R270 1 39 67.7% 27.70 00:00:50 0.4% 1 39 67.7% 27.71 00:00:35 0.4% R280 1 38 69.7% 28.75 00:00:52 0.4% 1 38 69.7% 28.75 00:00:36 0.4% R290 1 39 68.3% 28.12 00:01:19 0.5% 1 39 68.2% 28.06 00:00:45 0.7% R300 1 40 67.0% 27.53 00:00:59 0.4% 1 40 67.0% 27.53 00:00:36 0.4% R310 1 65 86.8% 34.33 00:03:51 1.8% 1 65 86.5% 34.21 00:01:52 1.8% R320 1 57 83.7% 33.70 00:03:32 3.2% 1 65 86.4% 34.17 00:01:21 1.4% R330 1 68 87.9% 34.50 00:04:12 1.9% 1 67 87.5% 34.46 00:01:35 1.8% R340 1 66 87.7% 34.59 00:04:16 2.1% 1 68 88.2% 34.61 00:01:24 1.9% R350 1 64 87.2% 34.58 00:04:20 1.7% 1 64 86.6% 34.30 00:01:58 2.3% R360 1 75 93.4% 36.42 00:14:52 5.0% 1 85 94.3% 35.98 00:03:36 6.1% R370 2 81 94.5% 36.38 00:16:22 5.0% 2 87 94.3% 35.73 00:05:14 6.7% R380 2 80 92.2% 35.27 00:17:14 5.0% 2 83 93.6% 35.65 00:04:16 4.0% R390 2 78 93.6% 36.15 00:21:53 5.0% 2 83 93.9% 35.69 00:04:15 6.2% R400 2 79 94.1% 36.26 00:15:29 2.5% 2 84 93.6% 35.63 00:05:03 4.1% R460 2 88 93.7% 35.24 00:56:40 5.0% 2 100 93.3% 33.99 00:06:23 8.4% R470 1 84 91.4% 34.61 00:35:20 5.0% 2 103 94.8% 34.35 00:09:43 5.7% R480 2 88 93.6% 35.25 00:36:12 5.0% 2 101 92.9% 33.69 00:07:02 9.2% R490 2 92 94.7% 35.31 00:41:08 5.0% 2 104 93.7% 33.75 00:05:48 9.2% R500 2 87 93.1% 34.95 00:49:15 5.0% 2 101 93.0% 33.52 00:07:51 8.9% Table VIII: Empirical results for Heuristic 2 applied to test problems with uniformly distributed subscribers (d m U[1, 10]).
95 99 49 40 87 10 18 144 23 122 124 20 68 118 155 73 147 133 39 114 76 58 111 129 29 101 69 93 121 2 84 13 4 26 59 71 41 6 142 27 82 106 7 38 92 25 135 107 83 136 45 157 21 3 77 16 54 50 88 112 5 119 70 44 143 140 14 134 0 57 102 65 24 123 9 159 130 36 153 96 116 145 85 158 1 89 63 11 117 31 139 74 148 62 90 79 154 66 37 15 2 91 151 55110 94 17 1 12 81 131 109 125 160 43 35 78 149 152 5 60 75 51 3 67 120 52 19 100 22 32 86 28 53 98 104 127 103 42 126 72 30 132 138 97 47 64 80 8 108 46 4 48 150 146 137 33 115 56 156 34 128 61 141 113 105 Figure 1: Graphical representation of problem R500 with demand locations uniformly distributed over the service area (circles are subscriber locations, triangles are candidate tower locations and squares are candidate MTSO locations). 31
0 1 3 2 4 5 3 6 9 10 12 13 14 15 16 18 19 21 22 24 27 28 29 30 32 33 37 38 39 40 41 42 45 46 48 51 53 57 58 60 61 65 66 70 71 72 75 77 80 81 85 89 90 95 97 99 100 101 102 105 106 107 109 110 111 112 113 114 115 116 117 119 120 122 127 129 130 131 132 134 136 139 142 143 146 147 148 150 151 152 154 156 160 1 2 4 5 7 8 11 17 20 23 25 26 31 34 35 36 43 44 47 49 50 52 54 55 56 59 62 63 64 67 68 69 73 74 76 78 79 82 83 84 86 87 88 91 92 93 94 96 98 103 104 108 118 121 123 124 125 126 128 133 135 137 138 140 141 144 145 149 153 155 157 158 159 Figure 2: Graphical representation of the solution to problem R500. 32
73 75 88 112 82 109 96 29 5 114 28 48 53 51 49 52 46 50 64 21 4 20 86 77 85 27 47 44 2 2 1 6 7 8 9 10 11 12 13 19 71 100 81 41 37 80 42 116 76 45119 97 1043 87 26 106 120 107 18 36 43 65 38 103 1 70 35 101 62 25 17 39 93 68 95 34 0 40108 24 16 115 33 78 99 67 72 84 90 66 91 111 118 117 69 4 102 92 94 32 5 63 31 74 58 30 56 98 6 23 15 61 60 22 14 113 54 57 3 59 83 89 105 79 55110 Figure 3: Graphical representation of problem ND700 with demand locations concentrated along four major thoroughfares and in three hotspots in the North Dallas area. 33