I would propose a few changes:

1: Somebody earning Minimum Wage full time should not be taxed on their income, the amounts they are earning should be what is considered the minimum reasonable so why should they be losing money through tax? Nor should there be any payroll etc taxes on the employer at (or below, eg Part Time) this statutory Minimum level.
2: Welfare payments should be capped below the amount earned through full time Minimum Wage work - the unemployed should not earn more than the full time employed.
3: The loss of welfare should be phased such that for a part-time or even full-time Minimum Wage worker, the more they earn the more they keep. Work should always pay, this may mean still having welfare payments to the full time low-paid, but if so tax should not start yet (point 1).
4: Welfare payments (and thus Minimum Wage and the start of taxation) should be sufficient so we don't have anyone in real poverty.

I would consider these 4 premises to be "fair" and reasonable.