Wout Bakker MSc, Giuseppe Catanese MSc, Ger Koole PhD, Siqiao Li PhD, Alex Roubos PhD
CCmath bv, Amstelveen, Netherlands
Introduction
In call centers forecasts have to be at the 15-minute level to staff the right number of people at the right time. Daily forecasts are divided over the day using intra-day patterns. To compute these daily patterns, actuals of a number of representative days are averaged. This is a painful exercise: on one hand, it is hard to find sufficiently many representative days, on the other hand, the patterns are in general not very smooth due to unpredictable variability.
CCmath has adopted a new approach which we present in this white paper. It has a proven higher precision and works perfectly well with as little data as a single day.
Splines
Intra-day patterns tend to go up and down smoothly. Thus, we need a method that takes neighboring intervals into account. Smoothing splines is a method from statistics that is particularly fit to do this: it strikes a balance between an accurate fit and a smooth pattern. A comparison of the different methods can be found in Figure 1 for a particular day in a call center that is open for 14 hours resulting in 56 quarters. In black, we see the actuals of three days, as percentages of total volume during the day. We see quite bumpy patterns due to random fluctuations.
The standard approach to construct an intra-day pattern to be used in forecasting is taking the average over a number of days. The result, when using five days, is the red line in Figure 1. It still has quite a bumpy shape. To make it smoother we need (much) more representative data, which is often not available. Such a bumpy shape results in a low forecasting accuracy and bad agent schedules with highly fluctuating service levels.
Figure 1 The spline and the mean patterns compared for real call center data
Smoothing splines takes the neighboring intervals into account and produces a much smoother fit. Applying it to the data results in the green line of Figure 1. Indeed, it follows the overall pattern but in a much smoother way. But does this really produce more accurate forecasts? To test this hypothesis we set up the following experiment.
Experiments
To determine whether splines really give a better forecast we undertook the following experiment. We determined the mean and the spline using a number of days and tested it on a number of different days. This we repeated several times with different days to make sure the results are statistically significant (this method is called Monte Carlo cross-validation). To measure the accuracy of the patterns we used the WAPE (the weighted absolute percentage error), a common and in our opinion the best method to calculate forecast accuracy.
We also varied the number of days used to construct the pattern, from 1 to 15. For each, we calculated the WAPE based on the mean, the spline and also the median. The results are in Figure 2. We see that splines consistently give a lower WAPE, and also that using more representative days results in a lower WAPE. Splines perform very well, even using 1 day to determine the spline. Using 2 days gives almost optimal performance.
Figure 2 WAPE of the mean, the median, and smoothing spline compared
Similar experiments with data from other call centers give comparable results. Only extreme day patterns show outliers that reduce the added value of the spline method. These extreme values, in the rare cases we observed, occurred in the same interval each day and are caused by actual customer behavior.
Conclusion
The spline method for predicting intra-day patterns outperforms all other methods based on (trimmed) means and medians. It gives a better forecast with less data, and for that reason it is also suitable for situations where customer behavior changes quickly. In exceptional cases outliers have to be handled separately. We advise to use it in all situations using only a few representative days.
More information
The spline method is implemented in the CCmath CCforecast forecasting software. More information? Contact Wout Bakker at wout@ccmath.com or book a demo here.