Hi
We had the same thing with my dad a couple of years back. He was really keen to go for walks around the neighbourhood alone, but me and mum were always dead worried in case he got lost, walked too far and ran out of steam, fell over, went into someones house with them or decided to hop on a passing bus (only happened once and him suddenly speeding up to 20 miles an hour gave the game away!) . I did a lot of research and eventually bought a Spytrack Nano, which we /he would turn on and slip into his jacket pocket as he left the house. You buy the little unit, then pay a very reasonable rolling subscription which allows you to see exactly where the tracker is on a map (like google maps), whether the tracker is moving, if it is stationary - and for how long, the speed and distance covered etc. I used to renew the subscription every 3 months so that if he stopped going out, was too ill or whatever, we wouldn't be paying for a service we weren't using. The tracker itself had a pretty good battery life (rechargeable). Dad's account page had an area where it saved past routes and we got to know his favourite walks, so if he nipped out without the tracker we had a good idea of where he might have gone, should he not have returned. It was set up so that I got a message to my mobile when it was activated, then anytime I wished I could click on the website (from my phone or laptop) and the map would come up showing me where he was in realtime. I could zoom right in to the extent that I could pinpoint the bench he was sitting on to have a breather! I had read some blogs etc. saying that trackers were an invasion of a dementia patient's privacy and comments along that theme but, for my dad, it gave him freedom (and space from me and mum), a chance to be responsible for himself and make his own choices, and it gave us peace of mind. He's too frail to go out alone at present so I've let the subscription lapse, but I would wholeheartedly recommend getting your dad a tracker.
There are several on the market but Spytrack Nano by Rewire worked well for us.
All the best.