Variations on Matrix Balancing for Telecommunication Demand Forecasting

Matrix balancing is a term that describes the process of altering the elements of a matrix to make it conform to known regularity conditions while still remaining close to the original matrix Often the regularity conditions relate only to the row and col umn sums which yields problems having a desirable network structure We describe an application in telecommunication demand forecasting that additionally requires the matrix to be symmetric Although this requirement destroys the network constraint structure we show that for certain objectives row and column sum conditions are su cient to ensure symmetry In addition we describe a rounding procedure for generating heuristic integer solutions The problem of adjusting the elements of a matrix so that they satisfy certain con sistency requirements but still remain close to the original matrix is generically referred to as matrix balancing Matrix balancing problems arise in a wide range of practical con texts that include accounting transportation and demographics These and several other applications are reviewed in an excellent overview by Schneider and Zenios In a typical matrix balancing problem we have a matrix that estimates certain quanti ties of interest but these estimates do not satisfy consistency requirements that the actual values are known to satisfy An example might be estimating the elements of a transition probability matrix which we know to be doubly stochastic Consistency with the doubly stochastic property requires that the rows and columns sum to one The doubly stochas tic matrix is an example of one of two types of matrix balancing problems discussed by Schneider and Zenios They are adjusting the elements of a matrix so that the row and column sums equal certain prescribed values adjusting the elements of a square matrix so that its row and column sums are equal to each other but not necessarily to any prescribed values Both the application we consider and the doubly stochastic condition mentioned above yield matrix balancing problems of the rst type The conditions imposed on the row and column sums are called balance conditions and a matrix that satis es the balance conditions is said to be balanced In the applications considered by Schneider and Zenios the balance conditions relate only to row and column sums More generally the balance conditions can be restrictions on the sums of various combinations of matrix elements See for example Censor and Zenios The fair representation problem considered by Balinski and Demange is one example In the most general case the balance conditions can be any set of linear restrictions on the matrix entries For a particular set of balance conditions there may be a large number of balanced matrices but in matrix balancing we seek a balanced matrix that is close to the original matrix Schneider and Zenios review several methods for obtaining such matrices These methods typically fall into one of two categories scaling methods and mathematical programming methods Of the two approaches the mathematical programming based methods are more exible in the sense that they are easily adapted to accomodate changes in the underlying model such as including bounds or objective weights The mathematical programs that result from many matrix balancing problems are nonlinear network ow problems that are solved quite e ciently using general purpose nonlinear network solvers such as that of Mulvey and Zenios However the applicability of network ow models depends upon the type of balance conditions that are imposed In this paper we consider a problem that arises in telecommunication demand fore casting and use it to motivate discussion of variations on matrix balancing that incorporate symmetry and integrality conditions The symmetry requirement essentially changes the constraints of our model from those of a network ow problem to those of a matching prob lem However we show that for certain intuitively appealing objective functions standard matrix balancing formulations do in fact yield optimal continuous solutions Ultimately our goal is to produce symmetric integer matrices This additional requirement transforms our problem to one of nding a perfect b matching that optimizes a convex separable non linear objective While it may be the case that this problem is infeasible we can always identify an almost perfect b matching that suits the needs of this particular application The Telecommunications Demand Forecasting Problem Our application arises in forecasting demands between wirecenters in a telecommunication network These forecasts are used in a variety of network planning activities Placing and sizing links and switches would be among the most important The particular demands that we consider are for symmetric broadband services or nar rowband services like ordinary voice telephone These services involve symmetric two way communication so meaningful forecasts for such products should likewise be symmetric A symmetric forecast requires that the forecasted demand from wirecenter A to wirecenter B be the same as that from B to A It is often the case that there are no historical measurements of inter wirecenter traf c but there are measurements or forecasts of the aggregate amount of demand at each wirecenter Given the aggregate demands at the wirecenters either measured or forecast econometric models are used to disaggregate these values to obtain forecasts for the inter wirecenter demands While these econometric forecasts contain valuable information they are not typically symmetric so they cannot be used directly for planning This is the point at which our application begins We are given a matrixM that contains inter wirecenter forecasts derived from econometric models An element mij represents the forecast for the demand from wirecenter i to wirecenter j The elements along the diagonal are forecasts of within wirecenter demand The forecasts are all integer and the sum of the forecasts along any row is equal to the forecast of the aggregate demand at the associated wirecenter Our goal is to nd another matrix that is close to M that preserves desirable properties that M already possesses but is also symmetric The particular properties of M that we wish to preserve are integrality its row sums and its ratio between intra and inter wirecenter tra c The reason for imposing the integrality requirement is simply that these demands would necessarily be integer amounts If our model does not produce integer forecasts they would likely be obtained by some ad hoc means We preserve the ratio between intra and inter wirecenter tra c only because there may be some inherent di erence in the forecasts for instance we may have better data on intra wirecenter tra c Preserving this ratio is not really a hard constraint but is more a guideline to prevent altering the character of the solution too much As a result this particular condition can be and ultimately is relaxed in a controlled way to obtain an integer solution Modify inter-wirecenter regularity conditions forecasts to satisfy Aggregate aggregate forecasts Generate forecasts at each wirecenter of aggregate demand Disaggregate Forecasts Forecasts Inter-wirecenter Final Forecasts Figure Overview of the demand forecasting process An overview of the demand forecasting process is provided in Figure The rst two steps produce the aggregate and disaggregate forecasts respectively The metholodology underlying these processes is described in detail in This paper addresses only the nal step in which the demands are modi ed to conform to known regularity conditions This nal step seeks a matrix that is close to the matrix M computed in the second step and has the following properties it has all integer entries its row sums are equal to those of M it is symmetric and its diagonal is the same as that of M Note that given the second condition xing the diagonal is equivalent to xing the ratio between intra and inter wirecenter demand at each wirecenter The last three conditions could be considered our balance conditions One nal point relating to our application is that the techniques we develop will ul timately be imbeded in decision support tools used by network planners This brings up several issues that may not be of concern in more theoretical treatments of this problem First the procedures are likely to be used in tools that are somewhat interactive so they should be fast on moderate sized problems of perhaps several hundred nodes Second the software employed should be as general as possible without sacri cing e cieny so that it is adaptable to variations in the underlying model Third the algorithm should always produce a solution to a reasonable instance of the problem In this case we consider a problem instance to be reasonable whenever a balanced continuous solution exists Our application requires that we provide an integer solution for any reasonable instance Of course it s simple to demonstrate that even when a balanced matrix exists a bal anced integer matrix may not A simple example su ces Suppose our matrix of forecasts is