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 years middle school students to the three remaining middle schools. The school district provides bussing 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 bussing cost. The annual cost per student of bussing 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 bussing 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 percent of each schools population. The below table shows the percentage of each areas 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 an operations research consultant to assist the school board in determining how many students in each area should be assigned to each school.

Formulate a linear programming model for this problem and show the decision variables.

No. of Area Students 1 450 2 600 3 550 4 350 5 500 6 450 Percentage Percentage Percentage Bussing Cost per Student in 6th in 7th In 8th Grade Grade Grade School 1 School 2 School 3 32 38 30 $300 0 $700 37 28 35 $400 $500 30 32 38 5600 $300 5200 28 40 32 3200 $500 39 34 27 $400 34 28 38 $500 $300 0 WN School capacity: 900 1,100 1,000