Imagine that you are responsible for an evening at a music festival with three bands and five different stages $($each with its own time$-$slot to fill$).$ Each stage is equipped with different equipment and may not accommodate all bands. Each stage can only be occupied by one band at a time.

The stages are:

Stage $1$ Center stage show from $8$:$00$pm$-9$:$00$pm

Stage $2$ East stage show from $8$:$30$pm$-9$:$30$pm

Stage $3$ West stage show from $9$:$00$pm$-10$:$00$pm

Stage $4$ South stage show from $9$:$00$pm$-10$:$00$pm

Stage $5$ North stage show from $10$:$30-11$:$30$pm

The bands and their restrictions are:

Band A: Can play on stages $1,2$ and $5$

Band B: Can play on stages $3,4$ and $5$

Band C: Can play on stages $1,3$ and $4$

a$)$ Formulate the problem as a constraint satisfaction problem $($CSP$)$ in which there is one variable per stage. Start by listing the constraints. Be formal in your constraint definitions, which should be stated implicitly. $($HINT: Your constraints should say which shows can or can$$t occur at the same time; domains should show who can play on what stage.$)$

b$.)$ List the domains for each of your variables:

c$.)$ Draw the constraint graph associated with your CSP formulation.

d$.)$ Perform arc consistency on the nodes after assigning Band B to Stage $4$ and show the remaining domains for each variable.

e$.)$ Examining your answer to $($d$),$ what would you do after performing arc$-$consistency?occupied by one band at a time.

The stages are:

Stage $1-$ Center stage show from $8$:$00$pm$-9$:$00$pm

Stage $2-$ East stage show from $8$:$30$pm$-9$:$30$pm

Stage $3-$ West stage show from $9$:$00$pm$-10$:$00$pm

Stage $4-$ South stage show from $9$:$00$pm$-10$:$00$pm

Stage $5-$ North stage show from $10$:$30-11$:$30$pm

The bands and their restrictions are:

Band A: Can play on stages $1,2$ and $5$

Band B: Can play on stages $3,4$ and $5$

Band C: Can play on stages $1,3$ and $4$

can or can't occur at the same time; domains should show who can play on what stage.$)$

b$.)$ List the domains for each of your variables:

c$.)$ Draw the constraint graph associated with your CSP formulation.

d$.)$ Perform arc consistency on the nodes after assigning Band B to Stage $4$ and show the remaining domains for each variable.

e$.)$ Examining your answer to $($d$),$ what would you do after performing arc$-$consistency?