Particular small actual values bias the MAPE.
If any true values are very close to zero, the corresponding absolute percentage errors will be extremely high and therefore bias the informativity of the MAPE (Hyndman & Koehler 2006). The following graph clarifies this point. Although all three forecasts have the same absolute errors, the MAPE of the time series with only one extremely small value is approximately twice as high as the MAPE of the other forecasts. This issue implies that the MAPE should be used carefully if there are extremely small observations and directly motivates the last and often ignored the weakness of the MAPE.
Jan also points out a couple of things people criticize MAPE for incorrectly, but several things for which it is actually guilty. It’s not a bad measure if you can make certain data assumptions, but Jan has a few alternatives which tend to be better than MAPE.