$(2)$ Solve the ILP model to determine an optimal weekly phlebotomist schedule template for the phlebotomists currently available in the hospital laboratory. $(3)$ Using the ILP model developed in Part $($a$),$ determine the minimum number of phlebotomists needed in this the hospital laboratory in order to handle all blood collections in time.$$Hospital operates $247,$ with $34$ phlebotomists working $5$ days week in $8-$hour shifts across $10$ possible shifts per day, ensuring no morning shift after night shift, handling STAT and regular test orders with collection times of $1$ and $3$ hours respectively.$$The ILP model to determine phlebotomist shift schedule :Sets :$$I: Set of all phlebotomists $(1$ to $34)$J: Set of all shifts $(1$ to $10)$K: Set of all days $(1$ to $7)$T: Set of all hours $(1$ to $24)$O: Set of order types $("$STAT$"$ and "regular"$)$Parameters :$$W: Number of working days per week $(5)$H: Working hours per day $(8)$Sjk: Number of phlebotomists required in shift j on day k$$Tot: Average number of orders of type o received in hour t$$P: Time $($in minutes$)$ to collect blood for one order$$Decision Variables :$$Xijk: Binary variable. $1$ if phlebotomist i is assigned to shift j on day k$,0$ otherwise.$$Objective Function :$$Minimize :$$ijkXijk $($Maximize workload balance$)$Constraints :$$Shift Requirement : iXijk$=$Sjk$$j$,$k$$This ensures required number of phlebotomists are assigned to each shift on each day.$$Phlebotomist Workload : jkXijkW$$i$$Limits number of shifts phlebotomist can be assigned per week.$$Night Shift Constraint : jNight$$Shifts$$Xi$($k$+1)1$jMorning$$ShiftsXijk$$i$,$k$$Prevents assigning phlebotomist to morning shift day after working night shift.$$Order Fulfillment : ijXijk$\backslash$times HPotTot$$k$,$t$$Ensures enough phlebotomists are assigned to each day and hour to handle expected number of orders of both types within time limit$.$Binary Variables : Xijk$\{0,1\}$i$,$j$,$k