Operations Management $($POM $3430)$

Quantitative Problems for Constraint Management and Capacity Planning

November $12,2024$

$(1)$ Bruce Lee, Inc. is a manufacturer of a variety of windows and doors. It makes two brands of specialty windows $($Basic and Mid$-$line$)$ in one of its facilities. Both are processed through three machines $($workcenters$)$ of the same type, but the processing requirements at each workcenter varies depending on the product being made. The processing requirements at the respective workcenters in minutes are as follows:

Basic Brand $($WC$1$: $4\mathrm{.}0,$ WC$2$: $2\mathrm{.}0,$ WC$3$: $3\mathrm{.}5)$

Mid$-$line Brand $($WC$1$: $5\mathrm{.}0,$ WC$2$: $4\mathrm{.}0,$ WC$3$: $2\mathrm{.}5)$

Weekly demand for the windows are $600$ units for the Basic brand and $400$ units for the Midline brand. Their contribution margins are $$340/$unit and $$\frac{375}{?}$ unit respectively. Given that the company is open each day Monday through Saturday for a single $10-$hour production shift per day, how many units of each brand should the company make based on:

$($a$)$ the conventional approach of giving priority to products that have the higher contribution margins?

$($b$)$ the principle of the theory of constraints?

How do the two results compare to each other?

$[$Note: Assume that set$-$up times are negligible compared to the processing times$].$

$(2)$ The School of Business Administration's $($SBA$\text{'}$s$)$ undergraduate students advising office at Oakland University is open for walk$-$in advising sessions once a week $($for example$)$ from $1$:$00$ pm $-4\mathrm{.}00$ pm $.$ Assume that on a certain day, only one adviser is on duty, and that on average, one student walks in every $6$ minutes during the walk$-$in session. If a student's request takes an average of $5$ minutes to be processed, and if students' arrival rate and service times are assumed to have Poisson and exponential distributions respectively, answer the following questions:

$($a$)$ What are the students' arrival rate and the service rate? $($students$/$hr$)$

$($b$)$ How much of the walk$-$in time duration is the adviser idle? $($minutes$)$

$($c$)$ How much time, on average, does a student wait in line? $($minutes$)$

$($d$)$ How many students, on average, are waiting in line?

$($e$)$ What is the probability that there are more than two students in the system?

$($f$)$ If two advisers were to attend to the students on $2$ separate queues, by what percentage would student average waiting time be reduced? $($Assume that both advisers have the same service rate, and that students have no particular preference regarding which adviser to discuss with.$)$