Springfield School District Consolidation
The Springfield School District has made the decision to close one of its middle schools (6th, 7th, and 8th grades) at end of this school year and reassign all of next year’s middle school students to the three remaining middle schools in the district. The school district provides busing for all middle school students who must travel more than approximately a mile, so the school board wants a plan for reassigning the students that will minimize the total busing cost. The annual cost per student for busing from each of the six residential areas of the city to each of the schools is shown in the following table (along with other basic data for next year), where 0 indicates that busing is not needed and a dash indicates an infeasible assignment.
The school board also has imposed the restriction that each grade must constitute between 30% and 36% of each school’s population. The table below 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.
Busing Cost per Student
Number of Percentage Percentage Percentage Area
Students in 6th
Grade in 7th
Grade in 8th
Grade
School 1
School 2
School 3
1 450 32 38 30 $300 $0 $700
2 600 37 28 35 – $400 $500
3 550 30 32 38 $600 $300 $200
4 350 28 40 32 $200 $500 –
5 500 39 34 27 $0 – $400
6 450 34 28 38 $500 $300 0
School Capacity: 900 1,100 1,000
You have been hired as a management science consultant team to assist the school board in determining how many students in each area should be assigned to each school.
a) Formulate and solve a linear programming model for this problem.
b) What is your resulting recommendation to the school board?
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.”
c) 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 busing cost?
The school board is considering eliminating some busing to reduce costs. Option 1 is to only eliminate busing for students traveling 1 to 1.5 miles, where the cost per student is given in the table as $200. Option 2 is to also eliminate busing for students traveling 1.5 to 2 miles, where the estimated cost per students is $300.
d) Revise the model from part a) to fit Option 1, and solve. Compare these results with those from part b), including the reduction in total busing cost.
e) Repeat part d) for Option 2.
The school board now needs to choose among the three alternative busing plans (the current one or Option 1 or Option 2). One important factor is busing costs. However, the school board also wants to place qual 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.
f) Use your results from parts b), d), and e) to summarize the key information related to these two factors that the school board needs to make this decision.
g) Which decision do you think should be made? Why?
Project Submission Detail
This project focuses on modeling and techniques for solving problems where decisions are made simultaneously. I am specifically interested to understand your mastery of the various linear programming topics.
- Please submit your project report in either MS Word or pdf format.
- You can use the following outline for your project report:
a. Introduction: introduce the problem, provide some background
b. Problem Statement: describe the problem you are proposing to solve,
c. State the assumptions (if any)
d. Data Sources: list the sources of data you expect to extract data for developing the analysis. Provide any references you used.
e. Data Description: describe the data (explain any terms that are specific to your dataset
f. Methods: discuss your modelling approach, what is the objective function? What is the nature of the objective function (linear, non-linear, etc). What are your constraints? What is the nature of your constraints (linear, integer, binary)? What are your decision variables? What solver method did you adopt to develop the portfolio investment recommendation?
g. Results: discuss the outcome of your analysis.
h. Conclusion: Do you have a recommendation?
- Include tables and visuals to support your analysis.
- Please submit only ONE report for each group / team.
- Project #2 is due on.
- Please submit your Project #2 via D2L. CLASS SCHEDULE > CLASS #14 > TO DO
Project #2
- Your project report should be 5-7 pages.
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