NCAA Class Madness Scheduling $2024$ Version $($V$\mathrm{.}1)$

Each year, there are many complaints about how the NCAA Men$$s Basketball Committee seeds

and schedules teams in the $68-$team single elimination tournament to determine the National

Champion. It is inevitable that somebody is unhappy. We will approach a portion of the task by

using our modeling skills to generate an alternative schedule that may or may not differ from the

actual $$ your task is twofold $$ both to generate a schedule using OUR parameters in a LP$/$IP

model and then compare this to the actual tournament bracket $($using a variety of metrics$).$

Your task: Create an LP$/$IP model that assigns teams to regions $($Sweet $16)$ and sub$-$regions

$($Round $1$ and Round $2$ games$)$ using $2024$ data. You will be using just the $16$ teams which are

the #$1$ thru #$4$ seeds, but all $4$ regions and $8$ sub$-$regions. The criterion for your model:

scheduling teams to minimize the sum of distance traveled by all the teams involved, counting

both the distance to their assigned regions and the distance to their assigned $$sub$-$regions$.$

Distance data is provided on a separate spreadsheet, approximating the distance from campus to

arena. The data file shows the actual NCAA assignments $$ in yellow$/$orange$.$

$($NOTE: each team is assigned to exactly one region and exactly one sub$-$region$).$

Restrictions$/$requirements for the assignments:

$1)$ Each of the four regions $($East$,$ West, South and Midwest $$ Boston, LA$,$ Dallas, Detroit$)$ will

have exactly one #$1$ seed assigned, exactly one #$2$ seed assigned, exactly one #$3$ assigned, and

exactly one #$4$ assigned. This is true for the NCAA schedule.

$2)$ Each sub$-$region MUST have either a #$1$ seed or a #$2$ seed assigned to it $($but exactly $1$ total$).$

Also, each sub$-$region MUST have exactly $1$ #$3$ or #$4$ seed assigned to it$.($The NCAA typically

doesn$$t consider that even though host sites of the sub$-$regionals have complained$).$ The sub$-$

regionals are held at sites in Brooklyn, Charlotte, Indianapolis, Memphis, Omaha, Pittsburgh,

Salt Lake City and Spokane.

$3)$ Teams from the same conference cannot be assigned to the same region. Conferences are

shown on the data file. Do not be concerned with conference affiliation at the sub$-$region level.

Also, don$$t worry about conferences which do not have multiple teams $($for obvious reasons and

model size reasons$).$

$4)$ A team cannot be assigned to a region and sub$-$region such that the selections choose BOTH

the $$closest$$ region AND the $$closest$$ sub$-$region. So if Kansas$$s closest region is Dallas and

its closest sub$-$region is Omaha $,$ Kansas can be assigned either to Dallas OR Omaha OR neither

one, but cannot be assigned to BOTH Dallas AND Omaha.

$5)$ Historically, teams have not been allowed to be assigned to their home court or even in a

home city so as not to give them an incredible home advantage. This only applies for Creighton

being assigned to the Omaha sub$-$region. Your model should not allow this assignment.

Distance is shown as $999999$ in the matrix.

Part A $-$ THE MODEL $-$ Implement an appropriate integer linear programming model that

assigns the $16$ teams to Regions and Sub$-$regions, minimizing the sum of overall distances

subject to the items $1)$ through $5)$ listed above. Suggestion: Attack modularly AND model

efficiently. What do we mean $$efficiently$?$ This problem approaches the EXCEL limits of $(200$

variables and $100$ constraints$).$ If you are $$sloppy$$ with your constraints, or include unnecessary

or duplicative constraints, you may exceed these limits$.$ Model carefully.

NOTE: If you $$run out$$ of space for constraints, calculate what algebraic constraints you have

and contact me$.$ You should be able to model all these requirements with $100$ algebraic

constraints or less. Algebraic is the actual algebra, not the families of constraints we add to the

Solver.

Part B $-$ THE COMPARISON: Compare how your solution differs from the actual schedule.

Keep in mind no model is necessarily $$better$$ than another as each approach uses different

criteria. NOTE: Don$$t just forget this part. At least $25\%$ of your grade will be based on a

thorough comparison of your model solution to the NCAA actual bracket.

Specifically, measure the following for your LP$/$IP model solution and compare to the actual

$$bracket$($obviously$,$ only the $16$ teams of interest$).$ Highlight differences between your IP$/$LP

results and the actual assignments.

$-$ Miles for #$1$ seeds, #$2$ seeds, #$3$ seeds and #$4$ seeds. This should be broken down by

region and by sub region.

$-$ The number of teams assigned to both the closest region and closest subregion. $($your

solution should have a measure of $0,$ this is requirement #$4).$

$-$ The number of regions where teams are assigned from the same conference. $($your

solution should have a measure of $0,$ this is requirement #$3).$

$-$ The number of subregions not assigned a $1$ or a $2$ seed $($your solution should have a

measure of $0,$ this is requirement #$2).$

Summarize the team assignments from the output of your mo