You have n items with sizes a$1,...,$an to be packed into some bins. Each bin hascapacity V $.($a$).$ Formulate an integer program to find the packing method that uses the least number of bins. $($Hint: Use Ij to denote whether bin j is used; use xij to denote whether item i should be placed in bin j$.$ This is called bin packing problem.$)($b$).$ Now consider a real case where you have items with size $4,4,5,7,8$ and the capacity of each bin is $12.$ Formulate the IP in that case. What are the optimal solutions to the IP and its LP relaxation? Is there an integrality gap in this case?