In $1862,$ during the second year of the Civil War, General Thomas J$.$ "Stonewall" Jackson fought a brilliant military campaign in the Shenandoah Valley in Virginia. One of his victories was at the Battle of McDowell. Using the following figure and your imagination, determine the shortest path and how long it will take $($in days$)$ for General Jackson to move his army from Winchester to McDowell to fight the battle. The network is shown in the figure below, and copy paste friendly set of branches are collected in the table below.a$)$ Formulate a binary programming model to determine the shortest path and how long it will take $($in days$)$ for General Jackson to move his army from Winchester to McDowell to fight the battle. $($i$.$e$.,$ Write down a mathematical model$)($Note that you still need to provide the mathematical model in Part a if you choose to apply the algorithm in Part b$)(10$ pts$)$bSolve the binary programming model by using Excel. $(40$ pts$)$WeeMow Lawn Service

WeeMow Lawn Service provides lawn service, including mowing lawns and lawn care, landscaping, and lawn maintenance, to residential and commercial customers in the Draper town community. In the summer WeeMow has three teams that it schedules daily for jobs. Team I has

five members, team $2$ has four members, and team $3$ has three members. On a normal summer day WeeMow will have approximately $14$ jobs. Each team works $10$ hours a day, but because of the heat plus work breaks, each team actually works only $45$ minutes out of every hour. The following table shows the times $($in minutes$)$ and costs $($in dollars$)$ required for each team to complete the $14$ jobs for a specific day.

$\backslash$table$[[$Job$,$Team $1,$Team $2,$Team $3],[$Time $($min$.),$Cost $($$$),$Time $($min$.),$Cost $($$$),$Time $($min$.),$Cost $($$$)],[A,45,48,65,55,78,50],[$B$,67,70,72,60,85,55],[$C$,90,94,105,84,125,75],[$D$,61,65,78,65,97,60],[E,75,80,93,75,107,66],[F,48,55,70,65,95,60],[$G$,65,70,83,70,110,67],[H,67,72,84,74,100,65],[$I$,95,100,110,90,130,80],[J,60,65,78,65,95,62],[K,47,55,64,57,84,56],[$L$,I14,118,135,110,155,95],[M,85,90,107,98,125,75],[N,63,67,81,72,102,68]]$

The WeeMow manager wants to develop a schedule of team assignments to the jobs for this day. Formulate and solve a linear programming model to determine the assignments of teams to jobs that will minimize total job time for the day, given a daily budget of $$1,000.$ Indicate how

many minutes each crew will work during the day. Next, reformulate and solve this model if cost minimization is the objective. Of these two models, which one do you think the manger should select and why?