Consider Problem $1$ again, but this time suppose you have access to the following additional data

provided in the "Problem $2"$ sheet of the Excel workbook SupplementalCA$-$Data.xIsx:

$c_i=$ the number of economically disadvantaged students $($out of $b_i)$ in the middle school cohort

who are zoned to elementary school $i=1,2,$dots,$9.$

Considering this new information, how would you assign elementary schools to middle school

campuses?

For this problem, you must use the seven$-$step method for ethical decision$-$making to recommend a

decision and justify your recommendation on the basis of both operational and ethical considerations.

Although the problem is intentionally open$-$ended, you must do the following to receive full credit:

$(1)$ Conduct relevant background research. $($For example: Why is equity in education important?

How have other communities handled similar decisions?$)$ There is no minimum required

number of references, but you should seek to identify high$-$quality sources that will be helpful in

forming your recommendation.

$(2)$ Generate at least five alternative solutions. For each alternative solution, clearly indicate which

elementary schools are assigned to which middle school campuses. At least one of your

alternative solutions must involve solving an optimization model that differs from the one you

used in Problem $1.$

$(3)$ Evaluate each alternative solution with respect to the following metrics:

a$.$ The enrollment$-$weighted distance, summed across all assignments of elementary

schools to middle school campuses $($smaller values of this metric are preferred$)$

b$.$ The difference between the maximum number of economically disadvantaged students

assigned to any one middle school and the minimum number of economically

disadvantaged students assigned to any one middle school $($smaller values of this metric

are preferred$)$

$(4)$ Synthesize your analysis by preparing a scatter plot to compare your alternative solutions with

respect to the two metrics. Each point on your plot should represent one of the alternative

solutions you generated.

$(5)$ Recommend a solution.

$(6)$ Justify your recommendation on the basis of your analysis and background research.

You are welcome to use as many pages as you need to answer this problem; however, I would anticipate

that most solutions would use no more than a few pages. Make sure you clearly explain your rationale

and the process you used to arrive at your decision.

Decision Variables

We let $x_{ij}$ be a binary variable to represent whether elementary school $i$ is assigned to

middle school $j.$

$i=1,2,3,$dots,$9$ and $j=1,2,3$

$x_{ij}=1$ if elementary school $i$ is assigned to middle school $j.$

$x_{ij}=0$ otherwise.

Parameters

$b_i$ is the number of students in elementary school $i.$

$d_{ij}$ is the average distance from elementary school $i$ to middle school $j.$

$u_j$ is the capacity of middle school $j.$

Objective Function

Minimize total enrollment$-$weighted distance $={\displaystyle \underset{i=1}{\overset{9}{}}}{\displaystyle \underset{j=1}{\overset{3}{}}}(b_i**x_{ij}**d_{ij})$

Constraints

${\displaystyle \underset{j=1}{\overset{3}{}}}{x}_{ij}=1$ for $i=1,2,$dots,$9($each elementary school is assigned to one middle school$)$

${\displaystyle \underset{i=1}{\overset{9}{}}}{b}_i**x_{ij}{u}_j$ for $j=1,2,3($capacity constraint for each middle school$)$

$x_{ij}=$ Binary; $b_i,d_{ij},u_{j}0($non$-$negative and binary constraints$)$

Solution

Elementary schools $1$ and $3$ will be sent to middle school $1$ where the capacity will be $376$

students.

Elementary schools $2,4,$ and $9$ will all be sent to middle school $2$ where the capacity will be

$534$ students.

Elementary schools $5,6,7,$ and $8$ will all be sent to middle school $3$ where the capacity will

be $756$ students.

This solution gives us the minimum total enrollment weighted distance at $4446\mathrm{.}4$ miles.