$1.(10$pt$)$ Suppose throughput TH is near capacity $($bottleneck rate$)$ rb $.$ Using Little$$s law, relate $($a$)$ WIP and cycle time $($CT$)$ in a production line. $($b$)$ Finished goods inventory and time spent in finished goods inventory. $($c$)$ The number of cars waiting at a toll booth and the average waiting time $2.(10$pt$)$ Is it possible for a line to have the same throughput with both high WIP with high cycle time and low WIP with low cycle time? Which would you rather have? Why? $3.(10$ pts$)$ Can the critical WIP level W $0$ ever exceed the number of machines in the line? Why? No$.4.(10$ pts$)$ Suppose process times on a machine re exponentially distributed with a mean of $20$ mins. $($a$)$ A part is currently been processed on that machine for $18$ mins. What is the expected time until completion of the job? $($use memoryless property$)($b$)$ A part is currently been processed on that machine for $30$ mins. What is the expected time until completion of the job? $5.(25$ pt$)$ Computer the capacity $($bottleneck rate$)$ in parts per hour of the following: $(1)$ A balanced line with single$-$machine stations, all with average processing times of $15$ minutes. $(2)$ A four$-$station line with single$-$machine stations, where the average processing times are $15,20,10,12$ minutes, respectively for stations $1,2,3$ and $4.(3)$ A four$-$station line with multimachine stations, where the number of $($parallel$)$ machines at stations $1,2,3,4$ is $2,6,10,3,$ respectively. The average processing times at stations are $10,24,40,18$ minutes, respectively. $(4)$ A three$-$station with ample equipment $($i$.$e$.,$ such that operators are never prevented from processing a job by a lack of equipment$)$ staffed by $6$ operators who are identical with regard to average processing time and require $10,15$ and $5$ minutes, respectively on stations $1,2$ and $3.(5)$ The same line as in the above case except that station $2$ consists of only $2$ parallel machines. All other stations still have ample capacity. $6.(15$ pt$)$ A powder metal $($PM$)$ manufacturing line produces bushings in three steps, compaction, sinter$-$harden, and rough$/$finish turn, which are accomplished at three single$-$machine stations with average processing times of $12,10$ and $6$ minutes, respectively. However, while compaction and sinter$-$harden are dedicated to the bushing line, the rough$/$finish turn station also processes bearings from another line; the average processing times for bearings are $14$ mins. $($a$)$ If the production volumes of bushings and bearings are the same, what is the bottleneck of the PM line $($b$)$ If the volume of bearings is $(1)/(2)$ of that bushings, what is the bottleneck of the PM line? $7.(20$ pt$)$ Review the Penny Fab II example $($imbalanced line$)$ in the lecture notes: $(1)$ Increase the number of machines in station II by $1,$ and recalculate the station rate $($job$/$hr$).$ Find the bottleneck station, bottleneck rate, total cycle time, and critical WIP. $(2)$ Compare the above quantities with those in the original Penny Fab II example and analyze whether this change in machine count impacts the overall line performance. Calculate the performance gain or loss resulting from this change. $(3)$ What would you suggest to improve the system performance other than the proposed solution?