A. Formulate a goal programming model to help the board with its dilemma. B. Solve the goal

Question:

A. Formulate a goal programming model to help the board with its dilemma.

B. Solve the goal programming model by using the computer.

The Oakdale County School Board was meeting in special session. A federal judge had ordered the board to present an acceptable busing plan for racially balancing the four high schools in Oakdale County within a week. The judge had previously given the school board several opportunities to informally present a plan, but the members had been unable to agree among themselves. Every time they met and started to develop a plan to bus students from one high school district to another, an argument would arise before they got past the first busing move, and they would adjourn the meeting. This time, however, they knew the judge had lost patience and they had to agree on something.

Of the four schools, only West High School was racially balanced, with 500 white students and 500 black students. North High School had 1,000 white students and only 300 black students; East High School was almost as bad, with 1,050 white students and 400 black students. South High School was predominantly black, with 800 black students and 450 white students. Overall, of the 5,000 students in Oakdale County, 60% were white and 40% black.

"Look," said John Connor, a school board member from the West district, "rather than starting off by trying to shift students from one district to another, why don't we try to establish what we want to accomplish-you know, what our goals are?"

Several of the other members nodded in agreement, and Fred Harvey, the board chairman, said, "Good idea, John."

"Okay, the first goal seems pretty evident to me," John said. "Sixty percent of our students are white and 40%

are black, so that's what we need our schools to be, 60% and 40%." "That's okay for you to say, John," Betty Philips argued, "because your district has those proportions already-so you won't have any busing. But my district in the North is a long way from that ratio, and we would have to bus a lot of our students to achieve a 60%/40% ratio." "I'm not saying it, Betty," said John. "That is basically what Judge Barry has been saying for 6 months."

"John's right, Betty, and we're not busing students yet; we're just putting down our objectives," said Fred. "I think that has to be our highest-priority objective. How about the rest of you?"

They all nodded their agreement, even Betty Philips, reluctantly.

"Since we know we're going to have to bus students to achieve this ratio at each school, I think we ought to try to minimize the amount of traveling the students will have to do," suggested Mickey Gibboney, a member from the South district. Fred Harvey noted that page 10 of their handout had a chart showing the average mileage a student in one district would have to travel on a bus to the high school in each of the other districts. The chart looked like this: