Consider two projects, $1$ and $2,$ to be completed by a team of experts, and they can do only one project

at a time. It takes $7$ days to complete project $1,$ whereas the time of project $2$ is $4$ days. If the project

is switched from $1$ to $2,$ then the set$-$up takes $2$ days. On the other hand, if the project is switched

from $2$ to $1,$ then the set$-$up reduces to $1$ day. The projects cannot start before Day $5$ and both projects

must be completed on or before Day $20.$ Let x$1$ and x$2$ be the start date in days for projects $1$ and $2.$

Two projects will not be processed concurrently:

$$ If $1$ precedes $2,$ then the start date of $2$ must be greater than or equal to start date of $1$ plus the

working time of $1$ and the set$-$up time.

$$ If $2$ precedes $1,$ then the start date of $1$ must be greater than or equal to start date of $2$ plus the

working time of $2$ and the set$-$up time.

Use indicator variable$($s$)$ representing the order of working of projects and mathematically formulate

the constraint$($s$)$ that are verbally written above using indicator variable$($s$)$ and big M$($s$).$ Calculate the

minimum value of M that would work.