w{cdma network design - lyle school of engineeringlyle.smu.edu/~olinick/cdma2.pdf · 2005-01-15 ·...

Technical Report 03-EMIS-02

W–CDMA Network Design∗

Qibin Cai1

Joakim Kalvenes2

Jeffery Kennington1

Eli Olinick1

1 {qcai,jlk,olinick}@engr.smu.edu

School of Engineering

Southern Methodist University

Dallas, TX 75275-0122

2 [email protected]

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 pro-

gramming 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 in-

stances 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 genera-

tion mobile systems (W–CDMA and CDMA2000, respectively), both of which are based on

code division multiple access (CDMA) technology. This manuscript presents a comprehen-

sive 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 inter-

ference 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 cus-

tomer 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) pro-

vided 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 pro-

vides 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 maxi-

mizes 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 ensur-

ing sufficient quality of service. The selected base stations are then connected to a network

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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 pro-

posed 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 Cm ⊂ 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, P` ⊂ M is the set of customer locations that can be serviced by tower `. 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 K0.

2.2 Constants Used in the Model

The demand for service in customer area m ∈ M is denoted by dm. This value is the

number of channel equivalents1 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 and connecting it to the backbone network is given

by the parameter a`. 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 `, the subscriber’s

handset must transmit with sufficient power so that the tower receives it at the target power

level Ptarget. Due to attenuation, the signal transmitted weakens over the path from the

1CDMA does not utilize channles to allocate bandwidth to sessions, but an equivalent maximum trans-mission 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 ` is given by the parameter gm`. To ensure proper received

power, Ptarget, at the tower location, the handset will transmit with power level Ptarget/gm`. 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

SIRmin. 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 and MTSO hub location

k ∈ K is given by c`k, while hjk is the annualized cost of providing a link from hub location

j ∈ K to hub location k ∈ K0. Finally, bk 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`, which is one if a tower is

built at location l ∈ L; and zero, otherwise. The integer variable xm` represents the capacity

assignment (in channel equivalents) to tower ` ∈ L for servicing of customers in location

m ∈ M . In other words,∑

m∈M xm` represents the instantaneous communication capacity

of tower location ` ∈ L. The variables are related so that xm` ≥ 1 only if y` = 1, that is,

customers in location m can be assigned to tower ` for service only if tower ` is built. If an

MTSO is established in location k ∈ K0, the variable zk is one; and zero, otherwise. Each

5

tower must be linked to an MTSO. If tower ` ∈ L is connected to MTSO k ∈ K, then s`k

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 ujk denotes the units of traffic flow on the link

between MTSO location j ∈ K and MTSO location k ∈ K0. If there is any flow from MTSO

location j ∈ K to MTSO location k ∈ K0, then a link between j and k has to be established.

The variable wjk is one if a link is established between locations j ∈ K and k ∈ K0; 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 `, P TOT

` , from all mobile users in the service

area is given by

P TOT

` = Ptarget

∑

m∈M

∑

j∈Cm

gm`

gmj

xmj. (1)

In this expression, the signal level from customers assigned to tower ` is Ptarget, while it

is Ptargetgm`/gmj 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 `, P TOT

` − Ptarget represents interference, while Ptarget 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

` is given by

Ptarget

P TOT

` − Ptarget

≥ SIRmin, (2)

provided that tower ` is constructed. Since the tower is built only if y` = 1, this constraint

can be written as follows:

∑

m∈M

∑

j∈Cm

gm`

gmj

xmj ≤ 1 +1

SIRmin

+ (1 − y`)β` ∀` ∈ L, (3)

where β` =∑

m∈M dm

{

maxm∈Cm\{`}

(gm`

gmj

) }

and maxj∈Cm\{`}

(gm`

gmj

)

= 0 if Cm \ {`} = ∅.

The second term on the right-hand side is zero when a tower is built (y` = 1), so that the

signal-to-interference requirement must be met at tower `. When y` = 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

∑

`∈Cm

xm`

︸︷︷︸

−∑

`∈L

a`y`

︸︷︷︸

−∑

k∈K

bkzk

︸︷︷︸

−∑

`∈L

∑

k∈K

c`ks`k

︸︷︷︸

−∑

j∈K

∑

k∈K0\{j}

hjkwjk

︸︷︷︸

.

Subscriberrevenue

Towercost

MTSOcost

Connectioncost

Backbonecost

(4)

There are 16 sets of constraints that define the model. The first set ensures that

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customers can be serviced only if there are towers that cover the demand area.

xm` ≤ dmy` ∀m ∈ M, ` ∈ Cm. (5)

The next set of constraints ensures that one cannot serve more customers in a location than

there is demand for service.

∑

`∈Cm

xm` ≤ dm ∀m ∈ M. (6)

The next set of constraints enforce the quality of service restrictions on received signal quality

at the towers.

∑

m∈M

∑

j∈Cm

gm`

gmj

xmj ≤ 1 +1

SIRmin+ (1 − y`)β` ∀` ∈ L. (7)

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.

∑

k∈K

s`k = y` ∀` ∈ L, (8)

s`k ≤ zk ∀` ∈ 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

∑

`∈L

s`k ≤ αzk ∀k ∈ K. (10)

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 ∈ K0 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 zk = 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.

ujk ≤ |K|wjk ∀j ∈ K, k ∈ K0 \ {j}, (11)

ujk ≤ |K|zk ∀j ∈ K, k ∈ K0 \ {j}, (12)

∑

j∈K\{k}

(ukj − ujk) + uk0 = zk ∀k ∈ K, (13)

∑

k∈K

uk0 =∑

k∈K

zk. (14)

The next constraint states that the PSTN gateway is always present.

z0 = 1. (15)

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The last five sets of constraints provide the domains for the variables.

s`k ∈ {0, 1} ∀` ∈ L, k ∈ K, (16)

ujk ∈ N ∀j ∈ K, k ∈ K0 \ {j}, (17)

xm` ∈ N ∀m ∈ M, ` ∈ L, (18)

y` ∈ {0, 1} ∀` ∈ L, (19)

zk ∈ {0, 1} ∀k ∈ K0. (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 , Cm = {m}, am = 0, bm = 0, dm = 1 ∀m ∈ M , SIRmin < |M |, α > |M |,

and r >∑

m∈M

∑

k∈M cmk +∑

j∈K

∑

k∈K0\{j}hjk. Restrict the input further to cases where

L ⊂ K, and c`k = 0 if ` = k and c`k > r if ` 6= 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

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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) = hij.

In terms of the formulation (4)–(20), xmm = ym = smm = 1 is optimal. Constraints

(5)–(7) and (10) are trivially satisfied, while (18) and (19) are redundant. The problem

reduces to

min∑

m∈M

∑

k∈K

cmksmk

∑

j∈K

∑

k∈K0\{j}

hjkwjk (21)

subject to

∑

k∈K

smk = 1 ∀m ∈ M, (22)

smk ≤ zk ∀m ∈ M, ∀k ∈ K, (23)

ujk ≤ |K|wjk ∀j ∈ K, k ∈ K0 \ {j}, (24)

ujk ≤ |K|zk ∀j ∈ K, k ∈ K0 \ {j}, (25)

∑

j∈K\{k}

(ukj − ujk) + uk0 = zk ∀k ∈ K, (26)

∑

k∈K

uk0 =∑

k∈K

zk, (27)

z0 = 1, (28)

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smk ∈ {0, 1} ∀m ∈ M, k ∈ K, (29)

ujk ∈ N ∀j ∈ K, k ∈ K0 \ {j}, (30)

zk ∈ {0, 1} ∀k ∈ K0. (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 sys-

tem interference levels. That is, the following set of valid constraints can be added to the

formulation:

xm` ≤ dm(1 − yj) ∀m ∈ M, `, j ∈ Cm such that gm` < gmj. (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 `, regardless of the signal source. A subset of the total

received power comes from customers assigned to tower ` for service, i.e., those customer

locations m for which xm` ≥ 1. Thus, if (7) is satisfied, so is the following set of constraints:

∑

m∈P`

xm` ≤ 1 +1

SIRmin

∀` ∈ L. (33)

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Also note that, in the formulation (4)–(20), the variable s is integer. However, con-

straints (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`k ≤ 1 ∀` ∈ 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 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

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 loca-

tions 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 (includ-

ing 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 gm` were then

calculated based on Hata’s path loss model (Hata 1980). A tower location ` that was close

enough to provide service to customer point m (given by the requirement that gm` > 10−15)

would be included in the set Cm. Depending on the number of towers in the service area, the

average size of the sets Cm 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 cus-

tomer 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 ef-

fort. 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 selec-

tions 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 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 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.


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.



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 Cm 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 so-

lution 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

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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 prob-

lem 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 Heuris-

tic 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 opti-

mization 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 lo-

cations 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 rea-

sonable 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 prob-

lem 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. Net-

works 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

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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.

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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 Im-

plementations. Wiley, Chichester, England.

Mathar, R., and M. Schmeink, 2001. Optimal Base Station Positioning and Channel As-

signment 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 Net-

works. 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 equiva-lent serviced

a` U [$70, 000, $100, 000] Annualized cost for instaling a base station in lo-cation `

bk U [$300, 000, $375, 000] Annualized cost for installing an MTSO in locationk

f $1.00 Annualized cost per foot of wiringα 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 Cm

R110 40 1,000 5,620 2.1R120 40 1,000 5,637 2.0R130 40 1,000 5,638 2.0R140 40 1,000 5,626 2.1R150 40 1,000 5,625 2.2R160 80 1,000 5,609 4.2R170 80 1,000 5,598 4.2R180 80 1,000 5,608 4.1R190 80 1,000 5,593 4.2R200 80 1,000 5,608 4.2R210 120 1,000 5,665 6.4R220 120 1,000 5,676 6.4R230 120 1,000 5,686 6.3R240 120 1,000 5,669 6.4R250 120 1,000 5,695 6.4R410 160 1,000 5,612 8.4R420 160 1,000 5,630 8.4R430 160 1,000 5,628 8.3R440 160 1,000 5,642 8.3R450 160 1,000 5,650 8.3R260 40 2,000 11,020 2.0R270 40 2,000 11,031 2.1R280 40 2,000 11,031 2.1R290 40 2,000 11,036 2.1R300 40 2,000 11,051 2.1R310 80 2,000 11,038 4.3R320 80 2,000 11,051 4.3R330 80 2,000 11,041 4.3R340 80 2,000 11,033 4.3R350 80 2,000 11,033 4.3R360 120 2,000 11,042 6.4R370 120 2,000 11,030 6.4R380 120 2,000 11,033 6.4R390 120 2,000 11,015 6.4R400 120 2,000 11,010 6.4R460 160 2,000 11,001 8.4R470 160 2,000 11,003 8.4R480 160 2,000 11,001 8.4R490 160 2,000 10,998 8.4R500 160 2,000 10,987 8.4

Table II: Problem data for uniformly distributed subscribers (dm ∼ U [1, 10]).

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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 (dm ∼ U [1, 10]).

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

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.

26





Table V: Problem data for uniformly distributed subscribers (dm ∼ U [1, 10]) for Heuristic 1with |Cm| ≤ 1. 27

Problem Best Feasible Solution (mipgap=5%) Heuristic 1 (|Cm| ≤ 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 (dm ∼ U [1, 10]).

28





Table VII: Problem data for uniformly distributed subscribers (dm ∼ U [1, 10]) for Heuristic2 with |Cm| ≤ 2. 29

Problem Best Feasible Solution (mipgap=5%) Heuristic 2 (|Cm| ≤ 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 (dm ∼ U [1, 10]).

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Figure 3: Graphical representation of problem ND700 with demand locations concentratedalong four major thoroughfares and in three hotspots in the North Dallas area.

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Figure 4: Graphical representation of the solution to problem ND700.

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Figure 5: Graphical representation of the modified solution to problem ND700.

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w{cdma network design - lyle school of engineeringlyle.smu.edu/~olinick/cdma2.pdf · 2005-01-15 ·...

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