[Huawei P10 Plus Questions & Answers] paarnoutse:
Titanium Backup schedules not executed (or interrupted after being started) and/or its related notifications not being shown appears to be an issue that is caused by crazy additional 'features' as implemented at some devices to improve battery life. This especially happens at OnePlus devices running Oreo (Android 8), but may possibly also appear at devices of other brands (Xiaomi?). Therefore it actually isn't a shortcoming of Titanium Backup itself, but the result of some stupid app handling and killing behaviors that may be active at your device.

Also other apps that should permanently be kept running in background may be impacted by this behavior (being killed for the same reason) like Tasker, Sync-to-cloud apps, etc.

Especially at OnePlus devices running Oreo, following implemented crazy 'optimization' behaviors are playing a role here:

After a user has set Battery optimisation OFF for a specific app, it may automatically be set ON again after a reboot or after the app was updated

Background services may be killed EVEN when for the related app the Battery optimisation was set OFF

The 'Autostart' attribute may be ignored, so apps may be NOT automatically started in background after rebooting your device

SOLUTION

1. Make sure the 'Advanced optimisation' options are disabled

Go to Settings > Battery > Battery optimisation

Click the 3 dots in the upper-right corner and select 'Advanced optimisation'

Verify that the options 'Advanced optimisation' and 'Sleep standby optimisation' are both

disabled

2. Set Battery optimisation OFF for any app that should continue to run in background

Go to Settings > Battery > Battery optimisation

Select the app that should never be killed, so in this example: 'Titanium Backup'

Select 'Don't optimize' and click DONE

Nb.

Above is not needed for the 2nd entry 'Titanium Backup PRO', which is just related to your PRO key payment and not the running Titanium Backup process itself.
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However, disabling Battery optimization for an app is NOT enough for preventing that it could be killed when running in background at OnePlus devices!

This not only because the 'Don't optimize' setting may automatically be reverted to 'Optimize' after a reboot or after an update of the app, but EVEN when Battery optimisation is actually disabled it's nevertheless still candidate for being killed at OnePlus devices running Oreo! Therefore also next action is required in addition.

3. Lock the app in your 'Recent Apps' list

Open and then close Titanium Backup (or just touch the Home button at your device after opening the app)

Touch the 'Recent Apps' button at your device

This shows your recent apps, where Titanium Backup will now be included (as last / most recent entry)

Toggle the 'Lock' icon in the upper right corner of that Titanium Backup entry, to switch it to a closed lock icon symbol

(see attached screenshot)

This last action not only just prevents that the app could be removed from your 'Recent Apps' list, but it ALSO prevents the issues as described above, so:

The app will not be killed anymore when running in background

The 'Battery optimisation' setting for that app will not be reverted anymore to 'Optimize' after a reboot or after an update of the app

When done, Titanium Backup schedules should now run as defined. Also, its completion messages shouldn't disappear from your notification entries anymore, but kept until you remove them.

OTHER APPS

Because it's not just a specific Titanium Backup issue but the result of the stupid behavior as implemented at OnePlus devices running Oreo, it's highly recommended to repeat the steps above for any app that should never be killed when running in background (eg. Tasker, Sync-to-cloud apps, Alarm/Alert apps, etc.). Also apps that are making use of scheduled actions, or should awake at incoming messages, chats, mails, alarms, or Security/Anti-virus apps etc. could be candidate of not being able to react properly because they were killed when running in background and/or not started at boot. When such apps doesn't always correctly react/awake at your device, just perform same actions as described above to prevent them to be killed. When an app still doesn't work because it wasn't auto-started at boot, you could use Tasker to create a Tasker-profile that executes a Tasker-task to launch such apps at boot. Start-at-boot via Tasker appears to be not required for Titanium Backup, but for some other apps this was my only option to ensure that they were auto-started after a reboot of my device.

Nb.

Some apps have options included in their internal settings like 'Keep active in background', 'Run in Foreground' or 'Turn off battery optimisation'. Even when these types of internal app settings are selected, they could still be killed at some device brands (eg. OnePlus) when the actions as described above aren't implemented in addition for such apps.

OnePlus: Never settle, but also never be so stubborn...