Problem $3$

Consider the following tasks that must be assigned to $4$ workers on a conveyor$-$paced assembly line $($i$.$e$.,$ a machine$-$paced line flow$).$ Each worker must perform at least $1$ task. There is unlimited demand.

$\backslash$table$[[,\backslash$table$[[$Time to Complete$],[$Task $($seconds$)]]],[$Task $1,30],[$Task $2,25],[$Task $3,15],[$Task $4,20],[$Task $5,15],[$Task $6,20],[$Task $7,50],[$Task $8,15],[$Task $9,20],[$Task $10,25],[$Task $11,15],[$Task $12,20]]$

The current conveyor$-$paced assembly$-$line configuration assigns the workers in the following way:

Worker $1$: Tasks $1,2,3$

Worker $2$: Tasks $4,5,6$

Worker $3$: Tasks $7,8,9$

Worker $4$: Tasks $10,11,12$

$2$

a$.$ What is the capacity of the current line?

b$.$ What is the direct labor content?

c$.$ What is the average labor utilization $($do not consider any transient effects such as the line being emptied before breaks or shift changes$)?$

The firm is hiring a fifth worker. Assume that tasks are allocated to the $5$ workers to maximize capacity of the line, subject to the conditions that $($i$)$ a worker can only perform adjacent operations and $($ii$)$ all tasks need to be done in their numerical order.

d$.$ What is the capacity of this line now?

Again, assume the firm has hired a fifth worker. Assume further that tasks are allocated to maximize capacity of the line and that tasks can be performed in any order.

e$.$ What is the maximum capacity that can be achieved?

f$.$ What is the minimum number of workers that could produce at an hourly rate of $72$ units? Assume the tasks can be allocated to workers as described in the beginning $($i$.$e$.,$ tasks cannot be done in any order$).$