The Springfield school board has made the decision to close one of its middle schoo (sixth, seventh, and eighth grades) at the end of this school year and reassign all ( next year’s middle school students to the three remaining middle schools. The schor district provides bussing for all middle school students who must travel more than af proximately a mile, so the school board wants a plan for reassigning the students th: will minimize the total bussing cost. The annual cost per student of bussing from eac of the six residential areas of the city to each of the schools is shown in the followin table (along with other basic data for next year), where 0 indicates that bussing is nc needed and a dash indicates an infeasible assignment.
斯普林菲尔德学校董事会已决定在本学年结束时关闭其一所中学（六年级、七年级和八年级），并将明年的初中学生重新分配到剩下的三所中学。schor 区为所有必须行驶超过一英里的中学生提供校车服务，因此学校董事会希望制定一个重新分配学生的计划：将最大限度地减少校车总成本。下表显示了从该市六个住宅区的 eac 到每所学校的校车的每个学生的年度费用（以及明年的其他基本数据），其中 0 表示需要 nc 校车，破折号表示不可行的作业。

The school board also has imposed the restriction that each grade must constitute between 30 and 36 percent of each school’s population. The above table shows the percentage of each area’s middle school population for next year that falls into each of the three grades. The school attendance zone boundaries can be drawn so as to split any given area among more than one school, but assume that the percentages shown in the table will continue to hold for any partial assignment of an area to a school.
学校董事会还施加了一项限制，即每个年级必须占每所学校人口的 30% 至 36%。上表显示了明年每个地区属于三个年级的初中人口的百分比。可以绘制学校出勤区边界，以便将任何给定区域划分为多所学校，但假设表中所示的百分比将继续适用于将区域部分分配给学校。

You have been hired as an operations research consultant to assist the school board in determining how many students in each area should be assigned to each school.
您已被聘为运筹学顾问，以协助学校董事会确定每个领域的学生人数应分配给每所学校。
(a) Formulate a linear programming model for this problem.
（a） 为此问题制定线性规划模型。
(b) Solve the model. （b） 求解模型。
© What is your resulting recommendation to the school board?
© 您向学校董事会提出的建议是什么？

After seeing your recommendation, the school board expresses concern about all the splitting of residential areas among multiple schools. They indicate that they “would like to keep each neighborhood together.”
在看到您的建议后，学校董事会对多所学校之间的住宅区划分表示担忧。他们表示他们“希望让每个社区团结在一起”。
(d) Adjust your recommendation as well as you can to enable each area to be assigned to just one school. (Adding this restriction may force you to fudge on some other constraints.) How much does this increase the total bussing cost? (This line of analysis will be pursued more rigorously in Case 12.4.)
（d） 尽可能调整您的建议，使每个区域仅分配给一所学校。（添加此限制可能会强制您忽略其他一些约束。这会使总线总成本增加多少？（在案例 12.4 中将更严格地进行这一分析。

The school board is considering eliminating some bussing to reduce costs. Option 1 is to eliminate bussing only for students traveling 1 to 1.5 miles, where the cost per student is given in the table as $\mathrm{\$}200$$\mathrm{\$}200$$200\$ 200. Option 2 is to also eliminate bussing for students traveling 1.5 to 2 miles, where the estimated cost per student is $\mathrm{\$}300$$\mathrm{\$}300$$300\$ 300.
学校董事会正在考虑取消一些公交车以降低成本。选项 1 是仅取消旅行 1 到 1.5 英里的学生的公交车，其中每个学生的成本在表中给出为 $\mathrm{\$}200$$\mathrm{\$}200$$200\$ 200 。选项 2 是也取消旅行 1.5 到 2 英里的学生的公交车，每个学生的估计费用为 $\mathrm{\$}300$$\mathrm{\$}300$$300\$ 300 .
(e) Revise the model from part (a) to fit Option 1, and solve. Compare these results with those from part ©, including the reduction in total bussing cost.
（e） 修改 （a） 部分中的模型以适合选项 1，然后求解。将这些结果与 part © 的结果进行比较，包括总线总成本的降低。
(f) Repeat part ( $e$$e$ee ) for Option 2.
（f） 对选项 2 重复部分 （ $e$$e$ee ）。

The school board now needs to choose among the three alternative bussing plans (the current one or Option 1 or Option 2). One important factor is bussing costs. However, the school board also wants to place equal weight on a second factor: the inconvenience and safety problems caused by forcing students to travel by foot or bicycle a substantial distance (more than a mile, and especially more than 1.5 miles). Therefore, they want to choose a plan that provides the best trade-off between these two factors.
学校董事会现在需要在三种备选公交计划（当前计划或选项 1 或选项 2）中进行选择。一个重要因素是公交车成本。然而，学校董事会也希望对第二个因素给予同等重视：强迫学生步行或骑自行车旅行相当长的距离（超过一英里，尤其是超过 1.5 英里）所造成的不便和安全问题。因此，他们希望选择一个在这两个因素之间提供最佳权衡的计划。
(g) Use your results from parts $(c),(e)$$(c),(e)$(c),(e)(c),(e), and $(f)$$(f)$(f)(f) to summarize the key information related to these two factors that the school board needs to make this decision.
（g） 使用您从 parts $(c),(e)$$(c),(e)$(c),(e)(c),(e) 中获得的结果，并 $(f)$$(f)$(f)(f) 总结与学校董事会做出此决定所需的这两个因素相关的关键信息。
(h) Which decision do you think should be made? Why?
（h） 您认为应该做出哪个决定？为什么？