Abstract
We address the problem of assigning probabilities at discrete time instants for routing toll-free calls to a given set of call centers to minimize a weighted sum of transmission costs and load variability at the call centers during the next time interval.
We model the problem as a tripartite graph and decompose the finding of an optimal probability assignment in the graph into the following problems: (i) estimating the true arrival rates at the nodes for the last time period; (ii) computing routing probabilities assuming that the estimates are correct. We use a simple approach for arrival rate estimation and solve the routing probability assignment by formulating it as a convex quadratic program and using the affine scaling algorithm to obtain an optimal solution.
We further address a practical variant of the problem that involves changing routing probabilities associated with k nodes in the graph, where k is a prespecified number, to minimize the objective function. This involves deciding which k nodes to select for changing probabilities and determining the optimal value of the probabilities. We solve this problem using a heuristic that ranks all subsets of k nodes using gradient information around a given probability assignment.
The routing model and the heuristic are evaluated for speed of computation of optimal probabilities and load balancing performance using a Monte Carlo simulation. Empirical results for load balancing are presented for a tripartite graph with 99 nodes and 17 call center gates.
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Servi, L.D., Humair, S. Optimizing Bernoulli Routing Policies for Balancing Loads on Call Centers and Minimizing Transmission Costs. Journal of Optimization Theory and Applications 100, 623–659 (1999). https://doi.org/10.1023/A:1022642624300
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DOI: https://doi.org/10.1023/A:1022642624300