Different types of products are transported to be packaged using special fully automated packaging machines. The packaging machines must be reprogrammed by an operator between two successive packaging processes, this is because eac process packages different types of products with different shapes, sizes, and the types of used carton. The operator requires $8$ minutes to reprogram the machine. Then the packaging machine operates automatically for $35$ minutes. The operator must replenish the machine with empty cartons of suitable size; he can do this at any time during the last $25$ minutes of the machine's automatic run time replenishment of empty cartons requires $6$ minutes. Travel time between machines plus data entry in computer system requires $3$ minutes.

$1-$ What is the maximum number of packaging machines one operator can serve without creating idle time for the packaging machines?

$2-$What is the cycle time in this case?

$3-$ What is the total time the operator is ideal?

$4-$ What is the cost of packaged products during each repeating cycle in this case?

$5-$ If Co $$70/$hr$.$ And Cim $=$S$8/$hr$.,$ what assignment of packaging machines to operators will minimize the cost per packaged batch?