Problem $2(35$ points$)$

$R_2=\frac{48}{60}=0\mathrm{.}8$

$R_3=\frac{128}{60}=2\mathrm{.}13$

Consider the following tasks that must be assigned to three workers on a worker paced $($bottleneck$-$paced$)$ assembly line. Each worker must perform at least one task, and there is unlimited demand.

$w_1=30$ seconds

$u_2=60$ seconds

${}_3=70$ seconds

The current worker$-$paced $($bottleneck paced$)$ assembly line configuration assigns the workers in the following way:

Worker $1$: Tasks $1,2$ Worker $2$: Tasks $3,4,5$ Worker $3$: Tasks $6,7,8$

$(1)$ What is the capacity of the current line?

$(2)$ What are the direct labor content and the average labor utilization?

$(3)$ How long does it take to produce $30$ units, starting with an empty system?

For parts $(4)$ and $(5)$ assume that the firm hires a fourth worker and the tasks are allocated to four workers to maximize the capacity.

$(4)$ What is the maximum capacity that can be achieved if $($i$)$ the tasks are not divisible, $($ii$)$ a worker can only perform adjacent tasks and $($iii$)$ all tasks need to be done in numerical order?

$(5)$ What is the maximum capacity that can be achieved if the tasks are perfectly divisible? How does the worker utilizations change compared to part $(4),$ why?

For parts $(6)$ and $(7),$ assume that the firm combines all the tasks in a work$-$cell where a single worker performs all the tasks. In this case,

$(6)$ What is the capacity of the system with a single worker and what is the worker utilization?

$(7)$ What is the minimum number of workers $($i$.$e$.,$ work$-$cells$)$ needed to achieve a capacity of $100$ units$/$hour$?$