Question

The Mecklinburg County is faced with the challenge of designing a transportation program for the students in its three school districts to achieve socio-economic balance in its three schools. The basic requirement is that each and every student must be assigned to a school. It is important to transport these students with minimum cost to the county while ensuring that each school gets a representative mix of poor and non-poor students.

The following table contains the transportation cost and mileage data for every district-to-school combination and shows the number of students of each category in each district as well as the total school population capacities of the three schools.

	School 1		School 2		School 3		No of Pupils
	Transport. Cost($)       Miles		Transport. Cost($)         Miles		Transport. Cost($)        Miles		Poor	Non-poor
District A	39	4	71	8	103	12	450	80
District B	47	5	63	7	119	14	250	100
District C	87	10	31	3	47	5	20	500
Capacity	400		500		500		1400

The mileage figures represent the average miles traveled by a student within the county on the shortest route basis from strategic community bus stops to schools. The cost figures are based on the assumption that there is a fixed transportation cost of $15.00 per student per year and an additional annual cost of $8.00 per student for every mile traveled beyond the initial mile (e.g., 4 miles: $15+3*$8=$39.)

The board would like to get an idea of what it would be the minimum cost to transport these students without considering socio-economic balance.

a) Formulate an LP model to minimize the total transportation cost of the program. (6 pts)

Answer 1

Since we already have the calculated value of transportation cost per student, we can ignore the mileage value for this question. Next, we also will ignore the socio economic condition and then consider the transportation model. Looking at total students and total capacity, we know it is a balanced transportation problem and thus we can use equal sign instead of inequalities.

The formulation will be

Minimize 39A1 + 71A2 + 103A3 + 47B1 + 63B2 + 119B3 + 87C1 + 31C2 + 47C3

ST

A1 + A2 + A3 = 530

B1 + B2 + B3 = 350

C1 + C2 + C3 = 520

A1 + B1 + C1 = 400

A2 + B2 + C2 = 500

A3 + B3 + C3 = 500

A1, A2,…..C2, C3 >= 0

The model is shown below

The formulas are shown below

The solver parameters are shown below

The solution is shown below (highlighted in yellow). The total cost is shown separately.