Innovative Applications of O.R.
Modeling latent sources in call center arrival data

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Abstract

In this paper, we discuss issues that arise in the analysis of call center arrivals that are mostly linked to individual ads. More specifically, we consider the case where there is no complete linkage between the calls and the advertisements that led to the calls. The ability to model and infer such latent call arrival sources is important from a marketing as well as an operations point of view since knowledge of the linkage improves forecasting performance of the model. We pose this as a missing data problem and develop a data augmentation algorithm for the Bayesian analysis. We implement the proposed algorithm to simulated and actual call center arrival data and discuss its performance.

Section snippets

Introduction and overview

In this paper, we consider modeling call center arrival data that are typically linked to individual advertisements. As pointed out by Soyer and Tarimcilar (2008), analysis of such data requires models that allow for advertisement specific analysis of call arrivals and poses new modeling challenges. One of the challenging problems is to model and analyze such data when there is incomplete information about the sources of the call arrivals. Our aim is to develop a modeling strategy to address

Modulated poisson process model with latent variables

Following Soyer and Tarimcilar (2008) we define Ni(t) as the number of calls arrived during a time interval of length t as response to the ith advertisement and Zi as a p×1 vector of covariates that describe the characteristics of the ith advertisement. Typically, the covariate vector Zi will consist of media expense (in $’s), venue type (monthly magazine, daily newspaper etc.), ad format (full page, half page, color, etc.), offer type (free shipment, payment schedule etc.) and seasonal

Bayesian analysis of the latent variable model

In the modulated NHPP model with missing links we assume a Dirichlet prior on p̲j=(p1j,,pmj) with parameters (α1j,α2j,,αmj) which is independent across the intervals. Thus, for the jth interval we assume a Dirichlet prior asπ(p̲j)p1jα1j-1pmjαmj-1.It follows from (3.1) thatE[pij]=αijk=1mαkj.We can specify the prior parameters proportional toαijexp{-δ(tj-Ti)}1[Ti,)(tj)to reflect the fact that, for δ>0, larger number of calls during the early phases of the life of an ad will be followed by

Example using simulated data

We consider data simulated from a modulated nonhomogeneous Poisson process with cost of the advertising as the single covariate and with baseline cumulative intensity is a power law function. Thus, the cumulative intensity function for ad i is given byΛi(t,Zi)=γtαeβZi,where β is a scalar and Zi is the cost of the ith advertisement. Data was generated for 10 different ads starting at the same time assuming γ=10,α=0.5 and β=0.1. The costs of the ads changed between 1 and 10 units and 20 time

Concluding remarks

In conclusion, our experience with the proposed Bayesian approach for modeling latent sources and the corresponding data augmentation algorithm within the Gibbs sampler have shown a lot of promise. The proposed approach provided very close posterior inference results for the model parameters and actual arrivals when it is compared with complete source model results. The inferences about Yij’s when compared to actual values were found to be sufficiently close.

Furthermore, we note that in our

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