Case 3-5 Assigning Students to Schools

The Springfield School Board 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 student 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	# of 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

The school board also has imposed the restriction that each grade must constitute between 30 and 36% of each school's population. The above table shows the % 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 % shown in the table will continue to hold for any partial assignment of an area to a school.

You have been hired as a management science consultant to assist the school board in determining how many students in each area should 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 expressed 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 in Case 7-3.)

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 result from parts b, d and e to summarize the key information related to these two factors that the school board need to make this decision.

g. Which decision do you think should be made? Why?