Atlas Incorporated is a toy bicycle manufacturing company producing a five$-$inch small version of the bike that Lance Armstrong rode to win his first Tour de France. The assembly line at Atlas Incorporated consists of seven work stations, each performing a single step. Stations and processing times are summarized here:

Step $1(35$ seconds$)$: The plastic tube for the frame is cut to size.

Step $2(25$ seconds$)$: The tube is put together.

Step $3(40$ seconds$)$: The frame is glued together

Step $4(25$ seconds$)$: The frame is cleaned.

Step $5(35$ seconds$)$: Paint is sprayed onto the frame.

Step $6(40$ seconds$)$: Wheels are assembled.

Step $7(45$ seconds$)$: All other parts are assembled to the frame.

Under the current process layout, workers are allocated to the stations as shown here:

Worker $1$: Steps $1$ and $2$

Worker $2$: Steps $3$ and $4$

Worker $3$: Step $5$

Worker $4$: Step $6$

Worker $5$: Step $7$

Assume the workers are paid $$20$ per hour. Each bicycle is sold for $$7$ and includes parts that are sourced for $$1.$ The company has fixed costs of $$225$ per hour. Despite Lance Armstrong$$s doping confession, there exists substantially more demand for the bicycle than Atlas can supply.

a$.$ What is the cost of direct labor for the bicycle?

b$.$ How much profit does the company make per hour?

c$.$ What would be the profits per hour if Atlas would be able to source the parts $11$ percent cheaper $($$$0\mathrm{.}89$ for the parts of one unit$)?$

d$.$ What would be the profits per hour if Atlas were able to reduce fixed costs by $12$ percent $($to $$198$ per hour$)?$

e$.$ What would be the profit per hour if Atlas were able to reduce the processing time at the bottleneck by $4$ seconds per unit $($assume unlimited demand$)?$