Chapter 3, Problem 18RP

Explanation of Solution

To find an assignment of students to schools:

• Assume the following,
  • Let “${\text{S}}_1$” be the school “Cooley High”.
  • Let “${\text{S}}_2$” be the school “Walt Whitman High”.
  • Let “${\text{m}}_{\text{pq}}$” be the number of minority students, who live in district “p” and will enroll school “q”.
• The Linear Programming (LP) to find an assignment of students to schools that minimizes the total distance that the students should travel to school is as follows,

  $\text{min}{\text{z=(m}}_{\text{11}}{\text{+Nm}}_{11}{\text{)+2(m}}_{\text{12}}{\text{+Nm}}_{12}{\text{)+2(m}}_{\text{21}}{\text{+Nm}}_{21}{\text{)+(m}}_{22}{\text{+Nm}}_{22}{\text{)+(m}}_{\text{31}}{\text{+Nm}}_{31}{\text{)+(m}}_{32}{\text{+Nm}}_{32}\text{)}$

  $\begin{array}{l}\text{S}\text{.T}\text{.}\\ {\text{m}}_{\text{11}}{\text{+m}}_{\text{12}}\text{=50}\\ {\text{m}}_{\text{21}}{\text{+m}}_{\text{22}}\text{=50}\\ {\text{m}}_{\text{31}}{\text{+m}}_{\text{32}}\text{=100}\\ {\text{Nm}}_{\text{11}}{\text{+Nm}}_{\text{12}}\text{=200}\\ {\text{Nm}}_{\text{21}}{\text{+Nm}}_{\text{22}}\text{=250}\\ {\text{Nm}}_{\text{31}}{\text{+Nm}}_{\text{32}}\text{=150}\end{array}$

• The following is the blending constraint for “${\text{S}}_1$”.

  $0.20\le \frac{{\text{m}}_{\text{11}}+{\text{m}}_{\text{21}}+{\text{m}}_{\text{31}}}{{\text{m}}_{\text{11}}+{\text{m}}_{\text{21}}+{\text{m}}_{\text{31}}+{\text{Nm}}_{\text{11}}+{\text{Nm}}_{\text{21}}+{\text{Nm}}_{\text{31}}}\le 0.30$

• The above constraint produces two LP constraints as below,

  $\begin{array}{l}\text{0}{\text{.8m}}_{\text{11}}+0.8{\text{m}}_{\text{21}}+0.8{\text{m}}_{\text{31}}-\text{0}{\text{.2Nm}}_{\text{11}}-\text{0}{\text{.2Nm}}_{\text{21}}-\text{0}{\text{.2Nm}}_{\text{31}}\ge 0\\ \text{0}{}_{}\end{array}$