Cycle time is the time it take to complete your task or piece of the process (washing, loading or drying).
Takt time is the pace at which you need to produce to meet customer demand. It is available working time divided by customer demand in that period of time. Say if you have 5 working hours available to do laundry and you have 10 customers a day that need your services, your takt time is 5/10=. So you need to deliver a clean load every .5 hrs, or every 30 minutes. It is solely a calculated time based on your available working time and customer demand. It has nothing to do with your cycle time which is how long it takes you to perform a task.
The time to wash might be 60 minutes or might be, 90 minutes. In which case you can use your cycle time to calculate how many machines you should have or people washing (if hand washing).
In this case you would CALCULATE the number of resources you need:
(cycle time) / (Takt time) = # of resources needed. In this case 60 / 30 = 2. So you would need 2 machines to meet your takt time.
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