Problem -
The Hills County, Michigan, superintendent of education is responsible for assigning students to the three high schools in his county. He recognizes the need to bus a certain number of students, because several sectors, A-E, of the county are beyond walking distance to a school. The superintendent partitions the county into five geographic sectors as he attempts to establish a plan that will minimize the total number of student miles traveled by bus. He also recognizes that if a student happens to live in a certain sector and is assigned to the high school in that sector, there is no need to bus him because he can walk to school. The three schools are located in sectors B, C, and E.
The accompanying table reflects the number of high-school age students living in each sector and the distance in miles from each sector to each school:
|
Distance To School
|
|
Sector
|
School in Sector B
|
School in Sector C
|
School in Sector E
|
Number of Students
|
|
A
|
5
|
8
|
6
|
700
|
|
B
|
0
|
4
|
12
|
500
|
|
C
|
4
|
0
|
7
|
100
|
|
D
|
7
|
2
|
5
|
800
|
|
E
|
12
|
7
|
0
|
400
|
|
|
|
|
|
2,500
|
Each high school has a capacity of 900 students. Hans Magelssen/Shutter stock
a) Set up the objective function and constraints of this problem using linear programming so that the total number of student miles traveled by bus is minimized.
b) Solve the problem.