2. Each year, a local school district contracts with a private bus company for the transportation of students in the primary grades to school. The district's annual payment is equal to $1 times the number of "kid-miles" the bus company carries. (For example, transporting 10 kids two miles each amounts to 20 kid-miles, or transporting 5 kids 4 miles each also equals 20 kid-miles.) The school district has four schools and draws students from four distinct geographic neighborhoodsNorth, East, West and South. The district's planning department has come up the following figures on the distance from a particular neighborhood to a particular school (distance is in miles):

North East West South

School 1 2.4 3.2 1.7 3.3

School 2 3.1 2.4 2.0 2.0

School 3 3.7 1.5 2.6 2.5

School 4 4.1 5.1 4.1 5.5

The capacities for Schools 1,2,3 and 4 are 322, 388, 254, and 96, respectively. The number of students in each district which are to be transported to school is 252 in North, 138 in East, 403 in West, and 196 in South.

a. The district's objective is to minimize the cost of transporting students to school while satisfying the school capacity and neighborhood constraints. Formulate the linear programming problem.

b. Find the cost-minimizing solution using EXCEL's Solver. Hand in copies of the answer report and the sensitivity report.

c. Suppose it costs $1 to add another unit school capacity at School 1. Is it desirable to add another unit of capacity at this school? Explain with reference to the sensitivity report.

d. Explain the value of School 1's shadow price with reference to the changing pattern of student transportation if School 1 had one more unit of capacity available.