Editor's Note: To see the current rankings for all 119 teams, go to http://web.ineg.uark.edu/faculty/cassady/.

FAYETTEVILLE, Ark. — Does Florida — which lost to Auburn, which in turn was manhandled by Arkansas, which was crushed by USC, which was beaten by UCLA, which lost five games this season — deserve to play for the national championship in the Bowl Championship Series? According to a novel computer ranking system developed by a University of Arkansas researcher, yes, the University of Florida Gators earned the right to face Ohio State on Jan. 8 in Glendale, Ariz.

Richard Cassady

Richard Cassady, associate professor of industrial engineering, and Lisa Maillart, assistant professor of industrial engineering at the University of Pittsburgh, have developed a mathematical optimization model called a quadratic assignment problem. Unlike the six computer-based analyses currently used as part of the BCS formula, the approach used by these researchers to rank the 119 Division I-A teams can be tailored to suit decision-makers’ definitions of the degree of victory for each game and the relative distance between all sequentially ranked positions -- 1 and 2, 3 and 4, 89 and 90, for example.

“We believe that unbiased ranking systems do not exist,” Cassady said. “All systems include some degree of individual opinion as to the factors that drive the ranking. Therefore, we do not propose an unbiased system, but our approach is flexible and inherently fair because it requires decision makers to quantify their biases and uses mathematical programming to apply these biases across all competitors.”

The researchers’ method considers every game played between Division I-A teams. The system uses a five-step, numerical strategy for specifying the degree-of-victory parameter set. In the following order, it computes the margin of victory of each game; adjusts the margin of victory if the game ended in overtime; adjusts for home-field advantage; truncates degree of victory at an upper bound; and adjusts to give greater weight to games played late in the season. Games played by Division I-A teams against non-Division I-A teams are not considered. If two teams play twice in one season, and one team wins both games, only the second game is considered. If each team won a game, then both games are considered.

Cassady said their approach works best when the location and date of games are considered, and the margin of victory of individual games is not considered.

“Voters are often thought to be greatly influenced by margin of victory,” Cassady said, “We believe our system works best when margin of victory is not considered. The BCS computer ranking systems are not permitted to consider margin of victory.”

As a purely mathematical system, the method does not favor teams in traditionally powerful conferences. However, because it can be customized, decision makers who use the system could easily incorporate conference-strength parameters into their own degree-of-victory calculations.

Depending on the size of a problem -- in this case, the 119 teams — solution procedures for quadratic assignment problems can produce exact (optimal) or heuristic (near-optimal) results. Recent mathematical advances in the optimal solution of quadratic assignment problems include exact solutions for problems equivalent to a league with 36 teams. Because the number of teams in their problem is much higher, Cassady and Maillart depended on Sinan Salman, a UA graduate student in industrial engineering, to develop a procedure for a heuristic solution based in part on a genetic algorithm. With Salman’s help, the researchers’ solution procedure results in a ranking system that closely matches the biases specified by the user.

Over the years, Cassady and Maillart improved their method many times and tested it each football season. Under their original system, only the pairs of teams that played in actual games were related via the degree-of-victory parameter set. As such, fewer than 700 of the approximately 7,000 pairs of teams were directly related. Recently, Cassady and Kelly Sullivan, a UA undergraduate student in industrial engineering, modified the system to relate teams connected by common opponents, such as the complicated scenario above that connects Florida to UCLA, the team that catapulted the Gators into the championship game by knocking off heavily favored USC in their last regular-season game. Now, with the current system, more than one-third of the pairs of teams are directly related.

“With so few non-conference games, it is often difficult to relate teams that are not in the same conference,” Cassady said. “By directly relating teams with common opponents, we are able to connect more pairs of teams. For example, Ohio State defeated Michigan, who defeated Notre Dame. Under our original system, Ohio State and Michigan would have been directly related, and Michigan and Notre Dame would have been directly related, but Ohio State and Notre Dame would not have been directly related. With the revised system, we give Ohio State half-credit for defeating Notre Dame, since Ohio State defeated a team that defeated Notre Dame.

“Quite often, especially within conferences, you have situations like the one this year in which Arkansas beats Auburn, who beat LSU and Florida, both of whom beat Arkansas,” Cassady said. “How do you make sense of these contradictions? We simply assign the full and half-credit degree-of-victory values, define the relative distance between ranking positions, and then let the optimization do the sorting.”

As for this season, the researchers’ system agreed with the two teams selected for the national championship game. Their final ranking: No. 1 Florida, No. 2 Ohio State, No. 3 Michigan. How does Cassady explain Florida being ranked No. 1?

“The quality of their wins offset their one loss, which was to a good team relatively early in the season,” he said.

Cassady and Maillart began work on the system while graduate students at Virginia Tech more than 10 years ago, before the National Collegiate Athletics Association started using the Bowl Championship Series formula to rank Division I-A football teams.