The supervision works started with certificate of the effective day
ahead of five months before commencement day of the contractor.
In this case,
Is this possible to request for the Employer to compensate 5 months remuneration of Supervisors due to difference between the effective day and commencement day in a legal way ?
The Effective Date is the date the contract came into existence, The Commencement Date is the date on which you are required to commence your works following a notice by the other party. If you are therefore required to start your works earlier than planned, then your route to recovery of costs would likely be by an extension of time to your services.