The VFS agent is wrong. You can check this at the official Schengen calculator. If you enter your dates and click "calculate," you will see that it counts this as 16 days.
The method of counting days of presence is specified in the Schengen Borders Code, Article 6(2):
...the date of entry shall be considered as the first day of stay on the territory of the Member States and the date of exit shall be considered as the last day of stay on the territory of the Member States....
There is another consideration, however. The Schengen visa application form asks the applicant, at number 25, to indicate a specific duration of stay. If you entered 15 for this item, then it is your mistake and you are unlikely to get it corrected except by submitting a fresh application and paying the application fee again. This still probably better than changing your travel plans.
If you in fact entered the correct figure 16 for that item, you should file a formal complaint with VFS via their customer experience page. Explain that the 15-day visa is not adequate for the proposed trip and does not match the 16 days you requested in your application, and that you would like to have the visa corrected. I would probably mention both the Schengen visa calculator and Art. 6(2) of the Schengen Borders Code.
If VFS formally rejects your complaint, you should try to escalate it to the Swiss authorities. If VFS will not bring it to the attention of the Swiss authorities, you can try to find a way of complaining to the Swiss embassy in Saudi Arabia, but I was not able to find one. There is, however, contact information for the State Secretariat for Migration in Switzerland. These people, at least, should understand that your proposed trip requires a 16-day visa. If you explain your problem to them, they may be able to help you solve it, whether by instructing VFS to accept your complaint or by other means.