$1.$ Consider the assembly line balancing problem described in the table below:

Task Task Time $($min$.)$ Immediate Predecessors

$110-$

$2141$

$381$

$453$

$5262$

$692$

$7114,5$

$8136,7$

$9228$

$10214$

$11199,10$

$12174,11$

a$.$ Develop a network for this assembly operation.

b$.$ Using the ranked positional weight technique, determine the resulting balance using a cycle time of $40$ minutes.

c$.$ Determine the number of work stations needed and the total idle times from part $($b$).$