Problem $1(20$ points$)$

Consider a process consisting of three resources:

$\backslash$table$[[$Resource$,$Processing Time $[$Min$./$Unit$],$Number of Workers$],[1,10,2],[2,6,1],[3,16,3]]$

where the existence of the number of workers at each resource$($station$)$ means they are working in parallel.

$(1)$What is the bottleneck?

$(2)$What is the process capacity?

$(3)$What is the flow rate if demand is eight units per hour?

$(4)$What is the utilization of each resource if demand is eight units per hour?

Problem $2(35$ points$)$

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.

$\backslash$table$[[$Time to Complete Task $($seconds$),],[$Task $1,20],[$Task $2,10],[$Task $3,15],[$Task $4,25],[$Task $5,20],[$Task $6,15],[$Task $7,15],[$Task $8,40]]$

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

$1$

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$?$

Problem $3(15$ points$)$

During a typical Friday, the West End Donut Shop serves $2400$ customers during the $10$ hours it is open. A customer spends, on average, $5$ minutes in the shop. On average, how many customers are in the shop simultaneously?

Problem $4(15$ points$)$

A call center receives $25$ callers per minute on average. On average, a caller spends $1$ minute on hold and $3\mathrm{.}5$ minutes talking to a service representative. On average, how many callers are $"$in$"$ the call center $($meaning that they are either on hold or talking to a service representative$)?$

Problem $5(15$ points$)$

A fast$-$food restaurant is interested in studying its arrival of customers. During the busy lunch period they have observed an average of $20$ customers arriving per hour. We assume the interarrival time are exponential random variables.

$(1)$ If a customer has just entered the store, what is the probability of another arrival in the next $10$ minutes?

$(2)$ What is the expected number of customers arriving in a five minute window?