The Springfield School Board still has the policy of providing bussing for all middle school students who must travel more than approximately 1 mile. Another current policy is to allow splitting residential areas among multiple schools if this will reduce the total bussing cost. (This latter policy will be reversed in Case 12.4.) However, before adopting a bussing plan based on parts (a) and (b) of Case 4.3, the school board now wants to conduct some postoptimality analysis.
斯普林菲尔德学校董事会仍然有为所有必须行驶超过 1 英里的中学生提供校车的政策。目前的另一项政策是允许在多所学校之间分配住宅区，如果这将降低校车总成本。（后一种策略将在案例 12.4 中反转。然而，在采用基于案例 4.3 的 （a） 和 （b） 部分的公交计划之前，学校董事会现在希望进行一些最优性分析。
(a) If you have not already done so for parts (a) and (b) of Case 4.3, formulate and solve a linear programming model for this problem. (Either an algebraic or a spreadsheet formulation is acceptable.)
（a） 如果您尚未对案例 4.3 的 （a） 和 （b） 部分执行此操作，请为此问题制定并求解线性规划模型。（代数或电子表格公式都可以。
(b) Generate a sensitivity analysis report with the same software package as used in part (a).
（b） 使用与 （a） 部分相同的软件包生成敏感性分析报告。

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 bussing students from area 6 , perhaps increasing them as much as 10 percent.
学校董事会的一个担忧是 6 区正在进行的道路建设。这些建设项目大大延误了交通，并可能影响 从第 6 区接送学生的巴士成本 ，可能会增加多达 10%。
© Use the report from part (b) to check how much the bussing 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, re-solve to find the new optimal solution with a 10 percent increase.
© 使用第 （b） 部分的报告来检查在当前最优解决方案不再是最佳解决方案之前，从区域 6 到学校 1 的校车成本可以增加多少（假设其他学校的成本没有变化）。如果允许的增加小于 10%，则重新求解以找到增加 10% 的新最优解。
(d) Repeat part © for school 2 (assuming no change in the costs for the other schools).
（d） 为学校 2 重复部分©（假设其他学校的费用没有变化）。
(e) Now assume that the bussing cost from area 6 would increase by the same percentage for all the schools. Use the report from part $(b)$$(b)$(b)(b) to determine how large this percentage can be before the current optimal solution might no longer be optimal. If the allowable increase is less than 10 percent, re-solve to find the new optimal solution with a 10 percent increase.
（e） 现在假设所有学校从区域 6 出发的校车费用将增加相同百分比。使用部分 $(b)$$(b)$(b)(b) 报表可确定在当前最优解可能不再是最优之前，此百分比可以有多大。如果允许的增加小于 10%，则重新求解以找到增加 10% 的新最优解。

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 consider only if it would significantly decrease bussing costs. Each portable classroom holds 20 students and has a leasing cost of $\mathrm{\$}2,500$$\mathrm{\$}2,500$$2,500\$ 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 bussing costs from that area.
学校董事会可以选择增加便携式教室，以增加一所或多所中学的容量，为期几年。然而，这是一项代价高昂的举措，只有在能够显着降低公交成本的情况下，董事会才会考虑。每个移动教室可容纳 20 名学生，每年的租赁费用为 $\mathrm{\$}2,500$$\mathrm{\$}2,500$$2,500\$ 2,500 。为了分析此选项，学校董事会决定假设 6 区的道路建设将在不显着增加该区域的公交车成本的情况下结束。
(f) For each school, use the corresponding shadow price from the report obtained in part (b) to determine whether it would be worthwhile to add any portable classrooms.
（f） 对于每所学校，使用 （b） 部分中获得的报告中的相应影子价格来确定是否值得增加任何移动教室。
(g) For each school where it is worthwhile to add any portable classrooms, use the report from part (b) 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).
（g） 对于值得增加任何移动教室的每所学校，请使用 （b） 部分的报告来确定在影子价格不再有效之前可以增加多少（假设这是唯一一所接受移动教室的学校）。
(h) If it would be worthwhile to add portable classrooms to more than one school, use the report from part (b) 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 bussing students and leasing portable classrooms. Re-solve to find the corresponding optimal solution for assigning students to schools.
（h） 如果值得为多所学校增加移动教室，请使用 （b） 部分的报告来确定要添加的数字组合，影子价格肯定仍然有效。然后使用影子价格来确定这些组合中的哪一种在最大限度地降低校车学生和租赁便携式教室的总成本方面是最好的。重新求解以找到将学生分配到学校的相应最佳解决方案。
(i) If part ( $h$$h$hh ) was applicable, modify the best combination of portable classrooms found there by adding one more to the school with the most favorable shadow price. Find the corresponding optimal solution for assigning students to schools and generate the corresponding sensitivity analysis report. Use this information to assess whether the plan developed in part $(h)$$(h)$(h)(h) is the best one available for minimizing the total cost of bussing students and leasing portable classrooms. If not, find the best plan.
（i） 如果 （ $h$$h$hh ） 部分适用，修改那里找到的移动教室的最佳组合，为学校增加一个影子价格最优惠的教室。找到相应的最优解，将学生分配到学校，并生成相应的敏感性分析报告。使用此信息来评估部分 $(h)$$(h)$(h)(h) 制定的计划是否是最小化校车学生和租赁移动教室总成本的最佳计划。如果没有，请找到最佳计划。