An airline operates four daily flights between three airports $($A$,$ B$,$ and C$).$ The flight timetable and the parameters of the fleet are shown in Table Q$3-1$ and Q$3-2,$ respectively. The yield is expected to be $8\mathrm{.}0$cen$\frac{t}{R}PK.$ The airline wants to determine the allocation of aircraft types to the flights.

Table Q$3-1.$ Flight timetable

$\backslash$table$[[$Flight$,$Departure Time,Arrival Time,Average Demand,Distance $($KM$)],[(A,B),11$:$00,14$:$00,100,3000],[(B,A),9$:$00,12$:$00,150,3000],[(A,C),9$:$00,14$:$00,240,5000],[(C,A),15$:$00,20$:$00,250,5000]]$

Table Q$3-2.$ Fleet Parameters

$\backslash$table$[[$Aircraft type,$\backslash$table$[[$Fleet$],[$size$]],$Seats,$\backslash$table$[[$Maximum$],[$Flight$],[$Duration $($H$)]],\backslash$table$[[$Unit Cost$],[($cent$/$ASK$)]],\backslash$table$[[$Turnaround Time$],[$ mins$)]]],[A310,2,300,9,5\mathrm{.}3,40],[A320,1,160,4,4\mathrm{.}6,30]]$

Q $3\mathrm{.}1$ Construct the time$-$space network for the flights and list the set of nodes, the set of flight arcs, and the set of ground arcs.

Q $3\mathrm{.}2$ List the following subsets of arcs:

$(1)$ Set of arcs that can be covered by A$320$ aircraft.

$(2)$ Set of arcs that are active at $12$:$00$;

$(3)$ Set of arcs that are active at $20$:$00$;

$(4)$ Set of arcs originating from node $($A$,12)$;

$(5)$ Set of arcs terminating at node $($A$,12)$;

Q $3\mathrm{.}3$ Define the decision variables for the problem and write down the objective function for maximizing the profit.

Q $3\mathrm{.}4$ Write down the constraints for the balance of aircraft in and out of node $($A$,12)$ for both aircraft types.

Q $3\mathrm{.}5$ Write down the constraints for the limit on the numbers of aircraft for both aircraft types at $12$:$00.$