The problem:
 
I'm not able to autofill the program look-up field that is built on the appointments (custom-object) with the corresponding program (custom-object) record.
 
Background:
 
X is a medical clinic that runs weight loss programs for obesity patients. X offers three types of programs; VLCD (Very Low Calorie Diet), Stepwise, Maintenance. Each of the programs length is 6 month.
 
Patients normally schedule appointments either weekly, bi-monthly, or monthly for six month period.
 
Objective:
 
My goal to write a code that tie every appointment record to the unique correspondent program for the patient. A program may have zero or many appointments. (This is what I'm trying to resolve here)
 
What I've built:
 
I have built three objects:
 
1. Patients (Accounts; standard-object)
 
2.Programs (Custom object; each record represents a program that is unique for the patient)
* There is a master-detail relationship between the patients object and program. Patients may have zero or many programs.
 
3.Appointments: (Custom object; each record represents an appointment that the patient has made).
* I've built a look-up relationship between the programs object and the appointments object. I want to tie every appointment to a unique program. A program may have zero or many appointments. (This is what I'm trying to resolve here)
 
I've tried these two methods (Process builder and workflow automation rules, and they both failed):
*I was told by SF support that this only can be done through APEX code since its "objects are falling under Dynamic Relationship"
 
I tried to do it through creating workflow automation rules and process builder by setting this condition and immidate actions below. When formula evaluates to true:
 
AND(Program__r.Program_start_date__c < appointments__r.
scheduled_time__c,
appointments__r.scheduled_time__c < Program__r.Program_end_date__c)
 
Fire Workflow action -- Update look-up field: Program name: appointemnts__r.patient__r.Program__c
 
Results: Appointments_r.Program__c remained NULL.

*I was told by SF support that this only can be done through APEX code since its "objects are falling under Dynamic Relationship"

Can someone please help me automate this look-up field update through APEX ? Clear guidance would be appericated since I have zero experince with APEX coding.

P.S:

All the fields (scheduled date, programs..etc) metioned above are autofiled either through APIs calls from our other internal systems or by being calculated fileds.