Reconsider Case 3-5. The Springfield School Board still has the policy of providing busing for all middle school students who must travel more than approximately a mile. Another current policy is to allow splitting residential areas among multiple schools if this will reduce the total busing cost. (This latter policy will be reversed in Case 7-3.) However, before adopting a busing plan based on part a of Case 3-5, the school board now wants to conduct some what-if analysis.

a. If you have not already done so for part a of Case 3-5, formulate and solve a linear programming model for this problem on a spreadsheet.

b. Use Solver to generate the sensitivity report. One concern of the school board is the ongoing road construction in area 6. These construction projects have been delaying traffic considerably and are likely to affect the cost of busing students from area 6, perhaps increasing costs as much as 10 percent.

c. Use the sensitivity report to check how much the busing cost from area 6 to school 1 can increase (assuming no change in the costs for the other schools) before the current optimal solution would no longer be optimal. If the allowable increase is less than 10 percent, use Solver to find the new optimal solution with a 10 percent increase.

d. Repeat part c for school 2 (assuming no change in the costs for the other schools).

e. Now assume that the busing cost from area 6 would increase by the same percentage for all the schools. Use the sensitivity report to determine how large this percentage can be before the current optimal solution might no longer be optimal. If the allowable increase is less than 10 percent, use Solver to find the new optimal solution with a 10 percent increase. The school board has the option of adding portable classrooms to increase the capacity of one or more of the middle schools for a few years. However, this is a costly move that the board would only consider if it would significantly decrease busing costs. Each portable classroom holds 20 students and has a leasing cost of $2,500 per year. To analyze this option, the school board decides to assume that the road construction in area 6 will wind down without significantly increasing the busing costs from that area.

f. For each school, use the corresponding shadow price from the sensitivity report to determine whether it would be worthwhile to add any portable classrooms.

g. For each school where it is worthwhile to add any portable classrooms, use the sensitivity report to determine how many could be added before the shadow price would no longer be valid (assuming this is the only school receiving portable classrooms).

h. If it would be worthwhile to add portable classrooms to more than one school, use the sensitivity report to determine the combinations of the number to add for which the shadow prices definitely would still be valid. Then use the shadow prices to determine which of these combinations is best in terms of minimizing the total cost of busing students and leasing portable classrooms. Use Solver for finding the corresponding optimal solution for assigning students to schools.

i. If part h was applicable, modify the best combination of portable classrooms found there by adding one more to the school with the most favorable shadow price. Use Solver to find the corresponding optimal solution for assigning students to schools and to generate the corresponding sensitivity report. Use this information to assess whether the plan developed in part h is the best one available for minimizing the total cost of busing students and leasing portables. If not, find the best plan.