Problem $1$:

Consider a serialized process of a production consisting of $4$ stages, A$,$ B$,$ C$,$ and D$.$ Initially there

is one machine at each stage. All units must go through all stages, the time or rate given below is

constant.

Stage A is cleaning process; each unit takes $1$ min and $30$ seconds to finish.

Stage B is a welding process; the output rate is $50$ units per hour.

Stage C is a painting station; each unit takes $2$ minutes to finish.

Stage D is a packing station; each unit takes $1$ minute to finish.

Market demand for this product is $41$ units per hour $(9,840$ units per month$).$ The unit profit margin

associated with the sales of the product is $$7.$ Also assume the followings:

$$ Additional machines can be acquired and added to any stage.

$$ Additional machine is identical to the existing one in the stage. Example: If one machine

is added to Stage A $($initially $40$ units per hour$),$ the output rate of Stage A will increase by

$40$ units per hour to $40+40=80$ units per hour.

$$ Cash flows compound monthly to perpetuity; annual cost of capital is $12\%$; $8$ hour daily

operation; production is run $30$ days per month.

Part A$.$ Which stage is the bottleneck? What is the initial output rate of the process in units per

hour?

Part B$.$ A vendor that services the cleaning machine offers an upgrade to your existing machine at

station A$.$ After the upgrade, the machine will simply clean faster. The time spent for each unit will

be reduced to $1$ minute per minute. If the upgrade cost is a one$-$time charge of $$9,250.$ Would you

consider this upgrade?

Part C$.$ How much are you willing to pay for an additional machine at the bottleneck stage?

Part D$.$ Suppose the output at stage C is $60/$hour$.$ Now, you have an option to rent an add$-$on

equipment which will improve the output rate of stage A to $55/$hour$.$ The rental cost is $$15/$hour;

would you consider renting this machine for stage A$?$ Why?