Mathematics

5-12

The Oconee County, South Carolina, superintendent of education is responsible for assigning students to the three high schools in his county. A certain number of students have to travel to and from school by bus, as several sectors of the county are beyond walking distance from 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. Of course, 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 located in sectors B, C, and E. The table below reflects the number of high school age students living in each sector and the distance, in miles, from each sector to each school. Assuming that each high school has a capacity of 1,100 students, set up and solve Oconee County’s problem as a transportation model.

DISTANCE TO SCHOOLS

IN MILES

SECTOR SECTOR

B

SECTOR

C

SECTOR

E

NUMBER OF STUDENTS
A 6 7 11 800
B 0 3 10 600
C 9 0 6 400
D 8 3 5 700
E 15 8 0 500

5-22

A supply chain consists of three plants (A, B, and C), three distributors (J, K, and L), and three stores (X, Y, and Z). The relevant supply, demand, and unit shipping cost information are given in the table below. Set up and solve the transshipment model to minimize total shipping costs.

PLANT SUPPLY STORE DEMAND
A 400 X 400
B 500 Y 325
C 350 Z 400

TO

FROM J K L
A $4 $7 $5
B $8 $5 $4
C $5 $6 $7

FROM

FROM X Y Z J K L
J $6 $4 $8 $6 $5
K $3 $6 $7 $6 $7
L $2 $4 $5 $5 $7

5-28

Kelly Spaugh, course scheduler of a technical college’s business department, needs to assign instructors to courses next semester. As a criterion for judging who should teach each course, Kelly reviews the student evaluations of teaching for the past two years. Because each of the four professors taught each of the four course at one time or another during the two year period, Kelly is able to determine a course rating for each instructor. These ratings are shown in the table below.

COURSE

PROFESSOR STATISTICS MANAGEMENT FINANCE ECONOMICS
Straubaugh 70 60 80 75
Kelley 80 60 80 75
Davidson 65 55 80 60
Merkle 95 40 65 55

5-32

A security firm needs to connect alarm systems to the firm’s main control site from five potential trouble locations. Since the system must be fail-safe, the cables must be run in special pipes. These pipes are very expensive but large enough to simultaneously handle five cables (the maximum that might be needed). Use the minimal spanning tree model to find the minimum length of pipes needed to connect the location shown in figure 5.22. Node 6 represents the main control site.

5-35

The network in Figure 5.25 shows the pipeline transportation system for treated water from the treatment plant (node 1) to a city water supply system (node 14). The arc capacities represent millions of gallons per hour. How much water can be transported per hour from the plant to the city using this network?

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