Problem $1)(12$ points$)$

A call center has analyzed the calls received in the last month and found that, on average, $40$ calls were received per hour. Assume that call arrivals follow Poisson distribution and come from an infinite population. Service times follow an exponential distribution with an average of $75$ seconds per call served. Assume the queue length can be infinite with FCFS discipline.

a$)$ What is the average number of people in line? $(2\mathrm{.}5$ points$)$

b$)$ What is the average number of people in the system? $(2\mathrm{.}5$ points$)$

c$)$ What is the average amount of time that a person can expect to spend in line? $(2\mathrm{.}5$ points$)$

d$)$ On average, how much time will a person spend in the system? $(2\mathrm{.}5$ points$)$

e$)$ During the weekends, the call arrival rate can be expected to increase to over $40$ per hour. What effect will this have on the average number of people waiting in line? $(2$ points$)$

Problem $2)(18$ points$)$

Astro Industries of Minneapolis, MN$,$ sends weekly shipments to $24$ customers in Dallas. Each customer's order weighs, on average, $1,500$ pounds. A direct truck shipment from Minneapolis to Dallas costs $\backslash (\backslash$$ $2,250\backslash ).$ The maximum load per truck is $40,000$ pounds.

a$)$ How much would it cost Astro to make direct, single$-$order shipments to all of its customers each week? What would the utilization level for the trucks look like? $(5$ points$)$

b$)$ Suppose a Dallas$-$based warehousing firm has agreed to run a break$-$bulk warehousing operation for Astro at a cost of $\backslash (\backslash$$ $125\backslash )$ per hundred$-$weight. Local deliveries to each customer would tack on another $\backslash (\backslash$$ $200\backslash )$ per customer per week. How much money could Astro save by going with the break$-$bulk solution? What would the utilization level for truck be in this case? $(8$ points$)$

c$)$ How high would the warehousing cost $($currently at $\backslash (\backslash$$ $125\backslash )$ per hundred$-$weight$)$ have to be before break$-$bulk warehousing is no more attractive than direct shipments? $(5$ points$)$ Problem $3)(12$ points$)$

U$.$S$.$ Airfilter has hired you as a supply chain consultant. The company makes air filters for residential heating and air$-$conditioning systems. These filters are made in a single plant located in Louisville, Kentucky, in the United States. They are distributed to retailers through wholesale centers in $100$ locations in the United States, Canada, and Europe. You have collected the following data relating to the value of inventory in the U$.$S$.$ Airfilter supply chain:

a$)$ What is the average inventory turnover $($quarterly and yearly$)$ for the firm? $(10$ points$)$

b$)$ If you were given the assignment to increase inventory turnover, what would you focus on$?$ Why? $(2$ points$)$ Problem $4)(8$ points$)$

a$)$ A barbecue sauce producer sells their product in a $75$ ounces bottle. Their historical process mean has been $75$ ounces with a standard deviation of $2\mathrm{.}50$ ounces. If their tolerance limits are set at $75$ ounces plus or minus $8$ ounces, what is the process capability ratio of the bottle filling process? $(4$ points$)$

b$)$ A barbecue sauce producer sells their product in a $75$ ounces bottle. Their current process mean is $74$ ounces with a standard deviation of $2\mathrm{.}5$ ounces. If their tolerance limits are set at $75$ ounces plus or minus $8$ ounces, what is the process capability index of the bottle filling process? $(4$ points$)$ Bonus Problem$)(8$ points$)$

MountainMole Foods has decided to use the perfect order measurement approach to track its logistics performance. According to MountainMole, a perfect order is one that $(1)$ is delivered on time, $(2)$ arrives in one complete shipment, $(3)$ arrives undamaged, $(4)$ is correctly billed. MountainMole has the following performance figures for the past four years: $($Assume that there is at most one error per order.$)$

a$)$ Calculate performance for each of the four years. What is the overall trend in the performance, if any? What factors explain the results? $(6$ points$)$

b$)$ If you were looking to improve MountainMole's logistics performance, what areas might you concentrate on based on these results? $(2$ points$).$