Sir Alan, 61, was angered by the team's decision to hire weathergirl Sian Lloyd, 49, to play a mum in the commercial, because she doesn't have any kids.
And team leader Raef was blasted for focusing more on the cinematography than getting the brand's message across.
'You made the biggest error going,' said Sir Alan. 'I don't know what your bloody advert was about. You don't mention tissues once in the voiceover.
'Raef, you put yourself across as Mr Elegance, Mr Perfect. You've got all the words. But yet I feel that you've been lucky that you've only been in this boardroom once. I've come to the conclusion that, with all due respect, you are a lot of hot air. Raef, you're fired!'
Raef says he has no hard feelings.
'We got carried away artistically,' he tells the Daily Mail. 'Our [team's] product was superior but, where it mattered commercially, it failed.
'I think I didn't come across as gritty enough. The others are more in your face, and that's how Sir Alan got to where he is today.'