  Measurements of cosmic-ray (CR) positron fraction by PAMELA and other
experiments have found an excess above 10 GeV relative to the standard
predictions for secondary production in the interstellar medium (ISM). Although
the excess has been mostly suggested to arise from some primary sources of
positrons (such as pulsars and or annihilating dark matter particles), the
almost constant flux ratio of $e^{+}/ \bar{p}$ argues for an alternative
possibility that the excess positrons and antiprotons up to the highest
energies are secondary products generated in hadronic interactions. Recently,
Yang \& Aharonian (2019) revisit this possibility by assuming the presence of
an additional population of CR nuclei sources. Here we examine this secondary
product scenario using the \texttt{DRAGON} code, where the radiative loss of
positrons is taken into account consistently. We confirm that the CR proton
spectrum and the antiproton data can be explained by assuming the presence of
an additional population of CR sources. However, the corresponding positron
spectrum deviates from the measured data significantly above 100 GeV due to the
strong radiative cooling. This suggests that, although hadronic interactions
can explain the antiproton data, the corresponding secondary positron flux is
still not enough to account for the AMS data. Hence contribution from some
primary positron sources, such as pulsars or dark matter, is non-negligible.
