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Assigning Students to Schools

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Assigning Students to Schools;The Springfield School Board (SSB) has made the decision to close one of its;middle schools (sixth, seventh, and eighth grades) at the end of this school year;and reassign all of next year?s middle school students to the three remaining;middle schools. 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.;Busing Cost per Student;Area Number of Percentage Percentage Percentage in School 1 School 2 School 3;Students in 6th Grade in 7th Grade 8th Grade;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;1;450;32;38;30;$300;$ 0;$700;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.;You have been hired as a decision analyst/management science consultant to;assist the SSB in determining how many students in each area should be;assigned to each school. Please work on this problem to answer the following;1;questions and prepare a managerial report to the SSB. Please do provide all;your model formulations and solutions.;a. Formulate and solve a linear programming model for this problem.;b. What is 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? (This line of analysis will be pursued more rigorously by formulating an;Integer Programming model (Module 9, Chapter 10), which won?t be pursue;in the assignment.);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;student 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 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.;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?;Note: The SSB can consider many additional questions which will be better;answered with an Integer Programming Model discussed in Chapter 10;(Module 9)).

 

Paper#20988 | Written in 18-Jul-2015

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