I've skimmed the responses but haven't really seen a detailed view from the management perspective. They have job requirements that are reasonable and directly related to the job.
Let's say that you have two employees who must occasionally travel to clients and/or work sites fifteen miles away. One employee drives and reaches their destination in thirty minutes or less. The other rides a bike and takes an hour to reach the same destination. Foul weather will extend the time traveling more for the cyclist than for the driver. The driver may also be able to use a highway while the cyclist may be restricted to a less direct route adding more of a time difference. The bottom line...one employee takes twice as long (or more) to do the same work as the other. You're the manager. Which employee is more productive?
In America, those using personal cars usually get reimbursed for on the job use of their personal vehicle (IRS mileage costs, etc.). This may be different in Cyprus. Those suggesting that the company buy or provide a car are ridiculous. That means that if I want a new car I should just get a job delivering pizzas and I'm set. Again, ridiculous.