Consider the product mix problem summarized in the table with $3$ products $($A$,$ B$,$ and C$)$ and $5$ types of resources $(1,2,3,4,5).$ There may be multiple workers working in parallel for each resource. Assume that all workers have $8$ hours per day. The number of workers in each resource and the times required for each activity $($product$)$ in each resource are given in Table $1.$

Table $1$ process tifores for sub products

$\backslash$table$[[\backslash$table$[[$Work$],[$stations$]],\backslash$table$[[$Number of$],[$workers$]],\backslash$table$[[$Activity time for A$],[($Min$/$Unit$)]],\backslash$table$[[$Activity time for B$],[($Min$/$Unit$)]],\backslash$table$[[$Activity time for C$],[($Min$/$Unit$)]]],[1,5,5,5,5],[2,4,3,4,5],[3,4,15,0,0],[4,6,0,4,3],[5,8,10,10,10]]$

For the given setting answer the following questions:

a$.$ What is the maximum flow rate if demand must be served in the mix determined by the demand rate $(7$ units of $A,5$ units of $B,$ and $4$ units of $C)?$

b$.$ What is the bottleneck workstation,

c$.$ What is the utilization of workstations?

Assume that the activity times given in Table $1$ are for experienced workers and the time requirements are $\%50$ more for inexperienced workers. Also, assume that our workforce is composed of $12$ experienced and $15$ inexperienced workers that can be assigned to any workstation. Formulate and solve a linear program $($with your favorite mathematical solver$)$ to find the optimal workforce assignment to workstations to maximize the flow rate.