Appointment availability considering multiple chairs/rooms and a provider who is only required for a portion of the time

I need help thinking through the logic and how optimize scheduling which allowing 1 provider to see customers in 2 different chairs with overlapping start/end times. The challenge is that a provider may only be required for a portion of a given appointment.

Resources: 2 Chairs, 1 Provider
In the example below, we have 4 appointments. The Customer (CUST #) will occupy the chair for the duration of the appointment; however, provider is only occupied for for a portion of the time. The (X) denotes the 15 minute increments the provider is needed and otherwise unavailable and the (/) denotes 15 minute increments the provider is not needed and is available to see other customers.

Now, let’s say I’m a user looking to book a 60 minute appointment in which we only need the provider for the third 15 minute increment (pattern of (/)(/)(X)(/)). We know we need a chair that’s available for 60 minutes. However, the challenge I’m finding is ensuring the provider is available during that 15 minute increment in which they are needed. See below for a visual break down. I can’t seem to wrap my mind around the logic required.

APPOINTMENT TO BOOK: 60 Minutes where the provider is needed for a 15 minute portion as denoted by the (X) in the following pattern: (/)(/)(X)(/)

Expected availability based on current schedule:

• 10:30AM-11:30AM
• 11:15AM-12:15PM
• 11:30AM-12:30PM

CURRENT SCHEDULE:

INPUTS:

1. List of existing scheduled appointment objects which include the appointment start time, chair, and pattern as I’ve defined above.
2. List of provider schedule objects which is their working hours for a chair or chairs.
3. List of blockout objects for lunch, breaks, etc. for a chair or chairs.
4. Appointment to be scheduled. An object with a pattern as defined above.

If it matters, I’m using Javascript with moment.js.