The linear programming example of reserve crew allocation discussed above is American Airline's own program. They also use a standard linear program to schedule their flight crews on aircraft and for training assignments. This is a considerably larger and more complicated program.

Exercises

$1.$ Suppose there are $4$ crews available. Crew $1$ is on duty days $1$ and $2.$ Crew $2$ is on duty days $1$ and $3.$ Crew $3$ is on duty days $2$ and $3.$ Crew $4$ is on duty days $1,2,$ and $3.$ A minimum of $2$ crews is required on days $1$ and $2,$ and $3$ crews on day $3.$ Use the simplex method to determine which crews should be selected and what the cost is$.$

$2.$ For this application, xi $=1$ if i is used and xi $=0$ if crew i is not used. What happens if the simplex method returns a value of $2$ or $1/2$ for one of the variables?