Oxley, Inc. manufactures three products $($Alpha$,$ Beta, and Gamma$)$ utilizing, in one of the processes, a single machine for all products.

Data on the individual products follow:

The single machine used for all three products has a maximum capacity of $480,000$ minutes per period.

Required

a$.$ How many units of each product should be produced each period?

b$.$ Oxley has received a proposal from one of its vendors that the vendor could increase the capacity on the machine. What is

the maximum Oxley would be willing to pay the vendor for an increase of $1,200$ machine minutes?

c$.$ Suppose the maximum capacity on the machine was $300,000$ minutes $($instead of $480,000$ minutes$).$ What is the maximum

Oxley would be willing to pay for an increase of $1,200$ machine minutes?

d$.$ At what capacity $($in machine minutes$)$ would the machine no longer be a bottleneck?