It looks like your father is having thoracic outlet syndrome
It is a condition where the blood vessels ( artery
and veins ) and nerves are compressed between the 2 bones namely the clavicle
and the first rib.
with increasing age, the shoulders sag, thus putting more pull on the neurovascular (artery, veins and nerves ) structures; indeed drooping shoulders alone may cause this condition and symptoms are characteristically posture-related.
Things to do-
surgical intervention is the treatment of choice if its compression causing pain and weakness of the limb.
In mean time conservative treatment can be opted:
physiotherapy:exercises to strengthen the shoulder girdle muscles,
instruction in work practices and
ways of preventing shoulder droop and muscle fatigue
Pain killer if required.