MGT 780 Goal Programming

8-10) The Arden County, Maryland, 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, for
several sectors of the county are beyond walking distance to a school. The superintendent partitions the
county into five geographic sectors as he establishes 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 that student because he or she can
walk to school. The three schools are in sectors B, C, and E. The following table reflects the number of
high school-age students living in each sector and the busing distance in miles from each sector to each
school:

Sector Distance to School No. of Students
School in Sector B School in Sector C School in Sector E
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
2500

Each high school has a capacity of 900 students. Set up the objective function and constraints of this
problem using LP so that the total number of student miles traveled by bus is minimized. Then solve the
problem
Variables:
AB = No of student from Sector A go to School B
AC = No of student from Sector A go to School C
AE = No of student from Sector A go to School E
BB = No of student from Sector B go to School B
BC = No of student from Sector B go to School C
BE = No of student from Sector B go to School E
CB = No of student from Sector C go to School B
CC = No of student from Sector C go to School C
CE = No of student from Sector C go to School E
DB = No of student from Sector D go to School B
DC = No of student from Sector D go to School C
DE = No of student from Sector D go to School E
EB = No of student from Sector E go to School B
EC = No of student from Sector E go to School C
EE = No of student from Sector E go to School E


Objective:
To Minimize travel distance to school

Total Travel Distance = 5(AB) + 8(AC) + 6(AE) + 0(BB) + 4(BC) + 12(BE) + 4(CB) + 0(CC) + 7(CE) +
7(DB) + 2(DC) + 5(DE) + 12(EB) + 7(EC) + 0(EE)

, Subject to:
1) Condition 1; Student at school B: AB + BB + CB + DB + EB ≤ 900
2) Condition 2; Student at school C: AC + BC + CC + DC + EC ≤ 900
3) Condition 3; Student at school E: AE + BE + CE + DE + EE ≤ 900
4) Student from Sector A: AB + AC + AE = 700
5) Student from Sector B: BB + BC + BE = 500
6) Student from Sector C: CB + CC + CE = 100
7) Student from Sector D: DB + DC + DE = 800
8) Student from Sector E: EB + EC + EE = 400
All Variables ≥ 0
Evaluation:
Linear Programming Result




Ranging Result