Learn Turkish: Personel kendi programını düzenlesin. - Let the staff organize their own schedule.

The verb base is düzenle (organize). Adding the 3rd-person imperative suffix -sin makes it a command or strong suggestion for someone else (he, she, it, or a group). In English, we usually translate this form using "Let him..." or "Let them...".

The English translation says "let them", so shouldn't the verb be the plural düzenlesinler?

Grammatically, the Turkish word personel (staff) is a singular noun, even though it refers to a group of people in real life. Because the subject is grammatically singular, the verb takes the singular 3rd-person imperative suffix -sin instead of the plural -sinler.

Why are there two ı suffixes at the end of programını?

The word breaks down as program + ı (his/her/its possessive) + n (buffer letter) + ı (accusative case). The first ı connects to kendi to mean "their own schedule". The second ı is required because the schedule is the specific direct object being organized.

Do we have to use kendi here? What does it add?

Using kendi (own/self) emphasizes that the staff should organize the schedule themselves, rather than relying on management or someone else to do it. Without it, Personel programını düzenlesin simply means "Let the staff organize their schedule".