Consider a single machine$-$sequencing problem with setups. Specifically, there are N jobs that need to

be processed on a single machine. Processing times, denoted by t$,$ are the same for all jobs, but a setup

time sij is incuned if job i is processed immediately before job j$.$ The objective is to complete processing

all the jobs as soon as possible $($or equivalently to minimize the sum of:all setup times$).$

a$)$ Fonnulate the problem as an integer program. Make sure that your fonnulations do not allow for

cycles $($a cycle occurs if your solution allows for a job i to precede j$,$ job j to precede job k$,$ but job k to

precede job i$).$

b$)$ Consider now the problem where there are two identical machines on which jobs can be canied out.

The objective is still to complete processing all the jobs as soon as possible. Formulate the problem as an

integer program. Suggest a heuristic for solving the problem