Application of mathematical programming to the fixed channel assignment problem in mobile radio networks

A.I.Giortzis L. F. Tu r ne r

Indexing terms: Mobile radio netwosk, Channel assignment, Integer lineur programming, Bsanch and bound algorithm

Abstract: The fixed channel assignment problem (CAP) is formulated as an integer linear programming problem with compatibility and requirement constraints. The proposed formulation is general and has been extended for the case of maximum packing fixed channel assignment problems. For the solution of the resulting formulation a special branch and bound algorithm has been used. The exploitation of the problem’s special structure can improve the computational efficiency of the algorithm used. The model has been applied to a number of different benchmark problems that have appeared in the literature. The examples presented show that using the proposed formulation and a specially designed branch and bound algorithm, it is possible to solve optimally and efficiently fairly large channel assignment problems.

1 Introduction

In recent years the demand for mobile telephone services has increased rapidly; however, the frequency band allocated to mobile telephone services is limited. Consequently, a major challenge for future mobile communication systems is to improve their spectrum utilisation efficiency so as to achieve optimum performance and maximum capacity utilisation. One way of increasing the traffic capacity of cellular systems is by introduction of digital technologies and more efficient modulation schemes. However, optimum utilisation of the available spectrum is needed for present and future mobile communication systems. This radio resource allocation problem is referred to as the channel assignment problem (CAP) in mobile radio networks.

The channel assignment problem involves the efficient assignment of channels to each cell (or base station) in a cellular radio network so as to increase the capacity of mobile units, subject to various constraints

0 IEE, 1997 IEE Proceedings online no. 19971249 Paper first received 16th July 1996 and in revised form 7th February 1997 The authors are with the Department of Electrical and Electronic Engineering, Imperial College of Science, Technology and Medicine, London SW7 2BT, UK

IEE Psoc.-Commun., Vol. 144, No. 4, August 1997

being satisfied. The term ‘channel’ is general. This channel (or resource) could be a fixed radio frequency (FDMA), a specific time slot within a frame (TDMA), or a particular code (CDMA), depending on the multiple access technique used by the system.

For the fixed channel assignment (FCA) problem examined in this paper, each base station is assigned a fixed number of channels according to the expected traffic in the cell served by this base station. The quality of the communication links is guaranteed by a sufficiently large co-channel reuse distance, D. Estimation of D depends on several factors, including the multiple-access technique, the modulation scheme, the mobile environment and the acceptable voice quality. In the more general form this co-channel reuse distance varies from cell to cell over the area of the network according to the terrain and the topography. Calculating the radio frequency propagation for the specific terrain of the network results in compatibility constraints which determine which cells may use the same channel, or adjacent channels, at one time.

The channel assignment problem consists of assigning the required number of channels to each cell such that the constraints described above are satisfied. Equivalently, the CAP can be formulated as a discrete combinatorial optimisation problem. The general form of the problem can be stated in a compact way as follows: ‘given a number of base stations in a specific area where a given number of operating channels are to be assigned, find a channel assignment that satisfies various constraints and which minimises (or maximises) the value of a given objective function’.

Fig. 1 example

Layout and numbering of the 12 cells used in the application

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In the CAP usually there are two kinds of con- straints, namely electromagnetic compatibility con- straints and requirement (or traffic) Constraints. When only the co-channel constraint is considered the CAP is known to be equivalent to a graph colouring problem [I]. In the channel assignment case, colours represent channels (Fig. l), nodes represent base stations and links connect pairs of base stations that cannot be assigned the same channel (Fig. 2). The objective is to find a solution with the minimum number of colours required to colour all the nodes. For a radio network this chromatic number [2] is equal to the minimum number of compatible base station sets, according to the co-channel constraint. In the language of complex- ity theory, the graph colouring problem is nonpolyno- mial (NP) complete. The time needed to solve this type of optimisation problems grows exponentially with the size of the problem.

c=

Fig.2 , Network interference graph Circles indicate nodes (or cells), lines indicate links (or constraints); numbers denote the node index

-2 1 1 1 0 0 0 0 0 0 0 0- 1 2 0 1 1 0 0 0 0 0 0 0 1 0 2 1 0 1 1 0 0 0 0 0 1 1 1 2 1 0 1 1 0 0 0 0 0 1 0 1 2 0 0 1 1 0 0 0 0 0 1 0 0 2 1 0 0 1 0 0 0 0 1 1 0 1 2 1 0 1 1 0 0 0 0 1 1 0 1 2 1 0 1 1 0 0 0 0 1 0 0 1 2 0 0 1 0 0 0 0 0 1 1 0 0 2 1 0 0 0 0 0 0 0 1 1 0 1 2 1 -0 0 0 0 0 0 0 1 1 0 1 2-12x12

In order to solve the fixed channel assignment problem a number of different techniques have been applied. The first attempts to solve the problem were based on graph theory due to the close relation between the FCA problem and the graph colouring problem [l, 3-61. Alternatively, an approach based on the formulation of the CAP as a discrete optimisation problem has been proposed by a number of authors. In order to solve the optimisation problem, a number of mainly heuristic techniques have been applied in the literature. These techniques produce, generally, approximate solutions which it is hoped will be close to the optimum. This is, however, not guaranteed. The techniques may take different forms, including simulated annealing [7, 81 neural networks [9-111 and, more recently, genetic algorithms [12, 131. The common limitation of these techniques lies in the fact that they do not provide a theoretical guarantee that the given solution is optimal, and they are usually trapped in local optima depending on the starting conditions [ 141. Furthermore, they rely heavily on finding values for the appropriate parameters that lead to convergence.

In contrast, the present paper is of fundamental importance since it provides the first known general analytical approach to the CAP, based on a mathemat- ical programming formulation and an exact solution technique.

2 Problem description

In this Section a general description will be given for the CAP using matrices to describe and explain the

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main parameters of the problem. The problem constraints can be represented compactly by the compatibility matrix, the requirement vector and the channel service matrix [15]. These main parameters of the problem are discussed in more detail below.

2. I Compatibility matrix The electromagnetic compatibility constraints in an N- cell radio network are described by an N x N symmet- ric matrix called the 'compatibility matrix' C [5]. Each nondiagonal element cJy in G represents the minimum allowable separation distance in the frequency domain between a channel assigned to cell j and a channel assigned to cell j ' .

When cJjr = 0, there is no restriction on the channels i and i' used by cells j and j ' , respectively. In particular, i and i' may be the same. In this case, the channel i is reused by cellj'. On the other hand, if cjY > 0, reuse of certain channels from cells j and j ' is prohibited. The special case clY = 1 is often called the co-channel constraint. If cJY = 2, the use of adjacent channels (i.e. li ~ i'l = 1) by cells j and j' cannot be tolerated (adjacent channel constraint). The required separation among channels used by the same cell is expressed by the diagonal elements cJj of matrix G. This is the so called co-site constraint, where the condition cJy 2 1 is always satisfied.

Based on the previous comments, a general formula- tion of the compatibility matrix G is

cell 1 and cell 2 since these are adjacent cells and consequently cI2 = 1.

When planning real radio networks the channel assignment problem may involve a large number of cells which implies that the size of the compatibility matrix G could be large. However, in general the elements ell, of the compatibility matrix can take only a very limited number of values, depending on the compatibility constraints considered in the specific problem. When only the co-channel constraint is considered the compatibility matrix is a binary matrix.

The use of the compatibility matrix implies a priori knowledge of the interference conditions in a radio network. This approach follows a hard interference decision indicating whether or not the use of the same or adjacent channels by two cells is allowed. However, any hard decision is questionable since it is based on a number of random variables (weather conditions, fading, mobility of users, etc.) which have an effect on the interference situation for a real world application. In this case an interference adaptive approach based on a propagation model and an updated compatibility matrix is more suitable.

-0 1 1 0 0 0 0 1 0 1 0 0- 0 0 0 1 0 1 0 0 1 0 1 0 1 0 0 0 1 0 1 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0

s =

- 0 0 0 0 0 0 0 1 0 0 0

2.2 Requirement vector The channel requirements for each cell in an N-cell radio network are described by an N-element vector with nonnegative integer elements, which is called the 'requirement vector', R:

R = [ T I r2 . . . rJ . . . .NIT (3) Each element R(j) = ri in R represents the number of required channels to be assigned to cell j . The values of the elements in R are usually estimated in advance according to the expected traffic in each cell. This esti- mate of spatial traffic distribution depends on factors such as the population density, the size and the loca- tion of each cell. Usually, the network statistics kept by the base stations and the network management system, are used to estimate the value of R. When there is no existing cellular network in an area it is a marketing task to predict the expected traffic and to pass it to the network designer who is responsible for solving the resulting CAP [16]. The value of this requirement in a real system is generally a function of time due to the new calls, call termination and transfer of existing calls between adjacent cells (handovers). However, in fixed channel assignment problems the value of R is assumed to be constant with time.

For the illustrative example shown in Fig. 1, if it is assumed that one channel is required per cell, the requirement vector, R, will be

R = [ 1 1 1 1 1 1 1 1 1 1 1 1]* (4) Generally, the distribution of traffic is nonuniform over the area for a real network.

( 7 )

o-6x12

2.3 Channel service matrix In a radio network with A4 available channels and N base stations (or cells), the channel service matrix S is an A4 x N binary matrix which contains the current channel assignment in the network. The elements s of the channel service matrix S are binary variabes according to the rule: sq = 0, if channel i is not assigned at base station j sv = 1, if channel i is assigned at base station j For instance, for the requirement vector given by

IEE P I ~ L - C o r m " , Vol 144, No 4, August 1997

Given:

(1) the compatibility matrix G (2) the requirement vector R

(3) and the number of available channels A4

Determine:

the channel service matrix S so as to satisfy

(1) the compatibility constraints (described by 6) (2) and the requirement (traffic constraints (given by R)

3 Mathematical formulation

3. I Fixed channel assignment problem In this Section the FCA problem will be formulated as an integer linear programming (ILP) problem [15]. The problem statement is summarised in Table 1. The aim of the solution technique is to obtain the resulting channel service matrix S needed to satisfy both the compatibility and requirement constraints. The compatibility constraints ensure that in any

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incompatible pair of base stations (j, j’) , according to the compatibility matrix G (i.e. cjJr O), each channel i and certain channels adjacent to I can be used only once. Using the M x N binary variables sV defined earlier, this constraint can be written as

where 3 is the set of incompatible assignments, given a priori by the compatibility matrix G: S = {(i, j), (i ’ , j ’ ) where: Ii - i’ I < c j j ~ , i 5 i ’ , (i, j ) f (z’, j ’ ) }

with cJjf being the C(j, j ’ ) element of the compatibility matrix.

The resulting channel service matrix S must satisfy, if possible, the requirement for each base station as well. The requirement constraint can be written as

(9)

M

a = 1

where r, is the R(j) element of the requirement vector. The above equation implies that the number of ones in any column of matrix S has to be less than or equal to the base station’s requirement which corresponds to this column.

Since the objective for matrix S is to satisfy the max- imum requirement, the following objective function can be considered:

Note that Z(S) reaches its minimum value of zero if all the requirements are satisfied. In this case the inequal- ity in eqn. 10 can be replaced by equality and an alter- native objective function can be proposed according to the optimisation aim. Usually in the literature there is no distinction between the two solutions satisfying the compatibility and requirement constraints. According to this approach any feasible solution is an optimal solution of the FCA problem. This means that in some cases the FCA problem is a constraint satisfaction problem or a feasibility problem from the optimisation point of view. Alternatively, the objective from the optimisation point of view is to find just one, out of a very large number of solutions, which satisfies all the compatibility and requirement constraints. Otherwise the resulting channel assignment satisfies the maximum possible requirement according to the compatibility constraints imposed by C.

The proposed formulation for the FCA problem is summarised below: Problem PI:

subject to:

M

a = l

ss3 , S d 3 / E ( 0 , l} vi , j , i ’, 3’ (13) Problem P1 is an integer linear programming problem where all the variables involved are binary and can be

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solved using standard branch and bound methods [17]. Problem P1 can be extended so as to find, if they exist, more than one different optimal solution for a given FCA problem. The idea is to use an iterative solution procedure which excludes from future solutions all the current solutions. One simple way to achieve this is to make all the current solutions infeasible using special constraints called ‘integer cuts’ [17].

3.2 Maximum packing - fixed channel assignment problem (MP-FCA) Everitt and Macfadyen [3] introduced the term maxi- mum packing (MP) in order to describe dynamic chan- nel assignment strategies where the number of channels needed to satisfy all the problem constraints is mini- mum. In this Section, the term MP-FCA is used to describe an alternative formulation in order to solve the FCA problem with the minimum number of required channels. The problem is the same as previ- ously except that in this case the minimum number of required channels M has to be determined, together with the channel service matrix S, according to the problem constraints. This version of the problem is useful in cases where the number of available channels is not fixed but variable.

The aim of the optimisation is to minimise the number of channels needed to satisfy both the require- ment and the compatibility constraints. For this reason a superset, F, of available channels is used in order to determine the subset containing the minimum number of required channels and to prevent the problem from becoming infeasible. Generally, the cardinality, M,, of this superset of channels is in the order of the product of the diagonal elements e,, in the compatibility matrix 6, and the maximum value in the requirement vector R. For each channel i in the set F, a binary variable I , is defined according to the rule: 1, = 1, if channel i is the last channel used in S out of

1, = 0, otherwise This definition can be expressed mathematically by two constraints. The first one is written as:

the M, channels

s,!? 5 1 - 1, vi,z’(> i ) , j (14)

(15)

This expression implies that after the last used channel i, no other channel i‘ is used, or equivalently:

The second constraint s,rJ = 0 Vi’ > t , ~

Ms

p , = 1 (16) a=1

imposes the requirement that one channel must be the last channel used. In the maximum packing FCA the objective is to satisfy the compatibility and the require- ment constraints with the minimum number of chan- nels. Consequently the following objective function can be considered:

z= 1

For a given FCA problem the value of this objective function will be equal to the ordinality of the last chan- nel used in the channel service matrix S.

The resulting formulation for the maximum packing FCA problem including all the constraints is

IEE Proc -Commun , Vol 144, No 4, August 1997

summarised below: Problem P2:

i = I

subject to

S,r3 5 1 - I , VZ,Z’(> i ) , j

i=l

2=1

z Z > s 2 3 7 s 2 r 3 ’ E (O7I) v 2 7 j > 2 ’ > j ’ (20) Problem P2 is also an ILP problem that can be solved using the same methodology as Problem P1.

4 Solution strategy

4.1 Computational difficulty of FCA problems In the case of the FCA problem, the solution space consists of the total 2M x N different combinations, or channel service matrices, that need to be considered. Nevertheless, some of these solutions can be discarded because they violate the problem constraints. The solu- tion space can be represented by a unit hypercube in an M x N multidimensional space. Since S is a binary matrix only the corners of the unit hypercube can be considered as meaningful solutions for the examined FCA problem. The problem size is defined by the size of the channel service matrix M x N.

Therefore, each time the number of binary variables is increased by one, the number of possible combina- tions is doubled. This property is referred to as the exponential growth of the solution difficulty with the size of the problem. For example, when A4 x N = 10, there are more than a thousand possible combinations (1024); with M x N = 20, there are more than a million (1 048 576); with M x N = 30, there are more than a billion and so on. Consequently, even using the fastest computers, exhaustive enumeration by checking each solution for feasibility and, if feasible, calculating the value of the objective function is out of the question for problems with more than a few tens of variables.

Fortunately, sophisticated algorithms for ILP problems, which have been developed during recent years, have a better performance. In practice, many ILP problems frequently have some special structure that can be exploited in order to simplify the problem. For this reason it is possible, sometimes, to solve successfully very large versions of these problems having thousands of binary variables. Special purpose algorithms designed specifically to exploit certain kinds of special structures is the key to solving large ILP problems.

From the previous comments it is clear that the two primary determinants of the computational difficulty of an FCA problem are: (1) the number of binary variables; (2) the structure of the problem.

IEE Proc -Commun , Vol 144, No 4, August 1997

Since the values of A4 and N , which determine the total number of binary variables in an FCA problem, are usually fixed, the only possibility left is to take advan- tage of the problem’s special structure by developing a special algorithm.

4.2 A special branch and bound algorithm for the CAP In order to solve the ILP problems P1 and P2 described in the previous sections a branch and bound algorithm has been used [18]. The main principle of any branch and bound algorithm is the division of a given original problem into a number of intermediate subproblems of smaller size. At the first step a relaxed version of the original problem is solved. Usually this relaxed linear problem (RLP) is derived from the origi- nal problem without the constraint of having integer values for the problem variables. This first solution provides a bound for the subproblems to be solved in later steps. Each of the subproblems is generated by the inclusion of one or more constraints. Consequently, following this process the original ‘large’ problem is divided into smaller and smaller subproblems. These problems are easier to solve directly.

This partition process is repeatedly applied to the generated subproblems at each level and creates a search solution tree. The size of this tree depends on the size of the original problem. This process continues until each unexamined subproblem is decomposed, solved, or shown not to lead to an optimal solution of the original problem. In order to achieve this at each level in the solution tree, the algorithm compares the partial solutions with the best known solution which provides a bound for this level. This strategy reduces the search space considerably since the algorithm avoids examining a large number of feasible solutions (or branches in the solution tree) that cannot be opti- mal or cannot lead to an optimal solution.

In the case of an FCA problem formulated as prob- lem P1 or P2, all the variables involved are binary. The relaxed forms of problems P1 and P2 exclude the con- straints described by eqn. 13 and eqn. 20, respectively. Therefore the variables sEI and 1, are assumed to be con- tinuous in the relaxed problems. The first solution under this assumption gives a lower bound (in the case of minimisation) for the value of the objective function Z(S). Following this the ‘branching’ procedure com- mences with the algorithm fixing the possible values of the binary variables to be zero or one. At each level in the search solution tree the value of one variable is fixed and the generated subproblems are solved. It is possible for a number of nodes not to be expanded fur- ther owing to the problem constraints. The reduction in the size of the solution tree depends on the relative position of these nodes within the tree. Obviously, the higher the level of such a node the greater the size of the solution tree.

As a small example consider only the effect of the compatibility constraints given by eqn. 8 on the branching procedure (Fig. 3) . For this example it is assumed that c12 = 4 and for i = 1 , j = 1 , ~ ’ = 2 ~ 2 ’ = 1 , 2 , 3 , 4 (8) ’ (s incecl2=4)

Eqn. 8 results in the following four single constraints:

s11 + s 1 2 5 1 (21)

SI1 + s 2 2 5 1 (22) 261

s11 + 532 5 1

s11 + s42 5 1

( 2 3 )

(24) The order of the branching variables for this example is taken to be

sll + level 1, s12 + level 2, s22 + level 3 , ~ 3 2 + level 4, S42 + level 5 (etc.)

1 (RLP) ....................... 511'1 A s11:O

level: 0 [start)

l eve l : l

leve l :2

level: 3

0

level: L

t t t S Q = O S42=0 542'0

(e tc ) level: M x N

(end) Fig. 3 Branch and bound algorithm search solution tree

The ellipses in the figure include the nodes that are not examined because of the compatibility constraints alone. An additional number of nodes are not considered because of other problem constraints, such as the requirement constraints. This results in a very significant reduction in the size of the solution tree and, equivalently, a significant reduction in the complexity of the problem that has to be solved. For the example in Fig. 3, the total number of nodes up to the level 3 is 15, whereas the number of nodes that need to be examined up to level 3 is only 10. The arrows at the bottom of the Figure show the possible directions of branching. The search solution tree will be simplified further by using the optimal solution bound at each level.

Following this discussion it is clear that the order of branching variables determines the shape and size of the solution tree. A smaller tree has fewer nodes for examination and results in an easier problem. The most effective heuristic method for improving the solution performance of the branch and bound algorithm is to control the branching procedure by assigning a branch- ing priority to as many binary variables as possible.

Variables with higher priorities will be branched on to variables with lower priorities. This guidance for the search tree often dramatically reduces the number of nodes processed, since it eliminates wasted computa- tion time given over to exploring uninteresting parts of the search tree.

The branching priorities should be assigned based on a knowledge of the problem's special structure. In the case of the CAP the structure is described by the pair (C, R). For instance, the fact that some cells require more channels than other cells, and consequently must be given some priority, is a special property of the problem. For the FCA problem the objective is to con- trol the branching procedure based on the prior exami- nation of the binary variables that correspond to the cells with the largest number of required channels and compatibility conflicts. A heuristic expression for calcu- lating the priority, p y , for each of the binary variables sII is given by:

This expression suggests that the binary variables cor- responding to the more difficult assignments must be examined first in the solution tree. For the MP-FCA problem, maximum priority must be given to the binary variables I , according to their ordinality. Then the branching will continue normally using the proce- dure described above. The partition process is done automatically by the CPLEX MILP solver based on the branching priorities used in a specific problem.

5 Application examples

5, I Fixed channel assignment problem To test the formulation of the FCA problem as an integer linear programming problem the model has been applied into a number of different benchmark problems that have appeared in the literature. Modelling and optimisation were performed in the GAMS [ 191 (General Algebraic Modelling System) modelling environment using CPLEX [20] as an integer linear programming solver.

The problem specification is given in Table 2, where the total number of base stations varies from 4 to 58 and the total number of available channels varies from 5 to 73. Problems 1 and 5 are taken from [lo] (problem 5 was initially introduced in [9]), problems 2 and 3 are taken from [la] and problem 4 is taken from Ell]. Table 2 also summarises the computational results for the set of test problems examined for both cases, with- out and when using branching priorities in the CPLEX solver. For each problem, the number of search solu- tion tree nodes processed by the branch and bound

Table 2: Problem specification and computational results

CPU time (s) Number CPU time ( s ) of Number nodes Problem Number Number of of nodes number of BS ( M channels (M)

1 4 11 6 0.133 8 0.217 2 12 5 11 0.267 16 0.333 3 6 16 56 2.980 155 8.267 4 58 29 168 403.433 147 356.850 5 25 73 >>IO00 - 107 331 . I 67

without branching priorities using branching priorities

262 IEE Pvoc -Commun., Vol. 144, No. 4, August 1997

algorithm, and the reported CPU time (in seconds) are listed. The computations were carried out on a SPARC- 10 workstation.

The computational results confirm that the solution difficulty of FCA problems depends mainly on the structure of the problem, as represented by the compatibility matrix. The problem size is also important. The computational results show also that the use of branching priorities leads to a major improvement in computational efficiency of the branch and bound algorithm used for the large test problems examined. Most of the large ‘benchmark problems’ in the literature are ‘artificial problems’ without any connection to a real network planning CAP. These problems tend to have symmetries which reduce significantly the solution difficulty. Generally, in these problems, it is possible to fix in advance the channel assignment in one or more specific cells, usually those with the largest number of required channels. This reduces considerably the size of the problem to be solved and consequently the computational time required to solve it.

Table 3: Optimum channel assignment for problem 5 (Helsinki problem)

1 2 3 4 5 6 7 8 9 10 1 1 12 13 14 15 16 17 18 19 20 21 22 23 24 25

BS Number of channels

10

Assigned channels

26,29, 34, 39,42, 44, 46, 54, 57, 65 1 1 9 5 9 4 5 7 4 8 8 9 10 7 7 6 4 5 5 7 6 4 5 7 5

2, 7, IO, 14, 20, 25, 28, 47, 50, 60, 66 4, 6, 11, 13, 15, 18,22, 24,32 7, 14, 25, 29, 64 40, 51, 53, 56, 62 ,67, 69, 71, 73 2, 7, 19,60 56, 58,67, 69,71 40,45,51, 53, 55,62,73 1, IO, 21, 28 3, 5, 30, 36, 43, 48, 61, 63 2, 8, 17, 20, 37, 49, 52, 59 8, 17, 33, 35, 37, 41,49, 52, 59 1, 9, 12, 16, 19, 21, 23, 27, 31, 38 45, 55, 58, 64, 68, 70, 72 4, 6, 13, 18, 24, 35, 41 11, 15, 22, 25, 47, 60 7,32,36, 50 26, 28, 30, 67,71 29, 33,48, 54,66 3, 5, 7, IO, 14,36, 43 11, 15, 28, 39, 50, 65 1, 12, 27, 32 9, 16,42, 53, 67 4, 6, 34, 39, 56, 58,70 3, 24, 57, 61, 68

However, in the general case it is not possible to make an advance channel assignment, since there is then no guarantee that any algorithm will find an optimal solution. This class of problems is considered to be more difficult. Problem 5 in Table 2 refers to the area around Helsinki, Finland, and belongs to this class of problems. Using the approach presented in the previous Sections together with branching priorities, this problem has been solved easily. Table 3 presents

IEE Proc.-Commun., Vol. 144, No. 4, August 1997

one optimum channel assignment for this problem. The elements in the Table can be used to confirm that the algorithm has correctly assigned the 7 3 available channels to the 25 base stations, subject to the problem constraints described in 19, 101.

5.2 Maximum packing - fixed channel assignment problem (MP-FCA) Using the formulation described in Section 3.2 it is pos- sible to solve easily and efficiently all the application examples given in [12], with the minimum number of channels. Since the size of these problems is small, to draw more meaningful conclusions the MP-formulation has been applied to the example given in [ 111 for the area East Anglia. In this problem it is required to assign four channels from a possible choice of 29 at 58 base stations. Each base station is assumed to receive co-channel interference from base stations that have adjacent coverage areas. At each base station the assignment of adjacent channels (co-site constraints) is prohibited.

Using our formulation, without the constraint, to assign each of the available channels the same number of times, it is found to be possible to solve the East Anglia problem using only 16 of the 29 available channels. The CPU time reported for this problem is 310 s and the branch and bound algorithm processed 192 nodes in order to find the solution. This optimal solution is given in Table 4. The elements of the Table can be used to confirm that there is no violation of the previously described compatibility and requirement constraints.

Table 4: MP-optimum channel assignment for problem 4 (East Anglia problem) using 16 channels

Assigned Assigned Bs Assigned BS channels Bs channels channels

1 5,9, 11, 13 21 1, 11, 13, 15 41 4, 11, 14, 16 2 3,7, IO, 12 22 4, IO, 12, 16 42 1,3, 9, 12 3 2, 4, 6, 8 23 3, 5, 7, 14 43 2, 5, 7, 15 4 9, 11, 13, 15 24 2.4, 6,8 44 6,8, IO, 13 5 1,8, IO, 12 25 IO, 12, 14, 16 45 1.4, 11, 14 6 2,6,9, 13 26 9, 11, 13, 15 46 6, 8, IO, 13 7 5,7, 11, 15 8 3,9, 14, 16 9 7,9, 11, 13 10 2,4,6, 8 1 1 1, IO, 12, 15 12 5, IO, 12, 14 13 5, 10, 12, 14 14 2, 4,6,8 15 3,7, 11, 13 16 2,4,6,8 17 9, 12, 14, 16 18 5,7,9, 15 19 2, 4, 6, 10 20 1, 3,8, 14

27 28 29 30 31 32 33 34 35 36 37 38 39 40

2, 4, 11, 15 IO, 12, 14, 16 1, 3, 5,7 2, 4, 12, 16 5, 7, IO, 15 1, 8, 11, 13 2,4,9, 1 1 1,3,5,7 6,8, 12, 16 4,9, 11, 14 1,3, 5,7 2, IO, 13, 15 3, 5,7,9 8, 10 ,13, 16

47 48 49 50 51 52 53 54 55 56 57 58

3,5,7,9 2,4, 11, 15 1, 6, 8, 13 3, 12, 14, 16 5, 7, IO, 15 2,4, 6 8 5,7, IO, 12 3,9, 11, 14 2,4,6,8 1,9, 11, 13 3,7, IO, 12 1, 9, 11, 13

6 Conclusions

The CAP has been formulated as an integer linear pro- gramming problem including compatibility and requirement constraints. The approach put forward in

263

the paper results in a major reduction in the complexity of the problem. The proposed formulation is general and has been extended to maximum packing fixed channel assignment problems.

For the solution of the resulting formulation a spe- cial branch and bound algorithm has been used. The algorithm can take advantage of the problem’s special structure, which leads to a significant improvement in its computational efficiency. The model has been applied to different benchmark problems that have appeared in the literature. It has been found that the difficulty of solution for a given problem depends mainly on the structure of the compatibility matrix, rather than on the actual size of the problem. Using a special type of constraint and an iterative solution pro- cedure it is possible to find more than one optimal solution.

Application of the maximum packing formulation to a specific example showed that it is possible to solve optimally and efficiently channel assignment problems with a considerably smaller number of channels. This leads to a great improvement on the system’s spectrum utilisation efficiency. The computational results from the optimisation show that using the proposed formu- lation and a special branch and bound algorithm, it is possible to solve optimally and efficiently relatively large channel assignment problems. The approach pro- posed in the paper can be extended to the case of adap- tive or scheduled FCA schemes. In this case a series of FCA problems, based on long-term statistical data relating to G and R, can be solved offline to arrive at a scheduled set of solutions which can then be followed in adapting/modifying the actual mobile communica- tion system.

7 Acknowledgments

The authors gratefully acknowledge the helpful discus- sions with Dr A. Manikas, Dr E.N. Pistikopoulos, Dr M.G. Ierapetritou and V.D. Dimitriadis. The Centre for Process Systems Engineering at Imperial College made available some of its computer facilities and this support is most gratefully acknowledged. A.I. Giortzis gratefully acknowledges the Greek State Scholarships

Foundation (IKY) for the award of a scholarship.

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