DOC《1996 Problem B》

1996 Problem B When determining the winner of a competition like the Mathematical Contest in Modeling, there are generally a large number of papers to judge.

Let's say there are P=100 papers. A group of J judges is collected to accomplish the judging. Funding for the contest contains both the number of judges that can be obtained and amount of time that they can judge. For example if P=100, then J=8 is typical. Ideally, each judge would read paper and rank-order them, but there are too many papers for this. Instead, there will be a number of screening rounds in which each judge will read some number of papers and give them scores. Then some selection scheme is used to reduce the number of papers under consideration: If the papers are rank-ordered, then the bottom 30% that each judge rank-orders could be rejected. Alternatively, if the judges do not rank-order, but instead give them numerical score (say, from

1 to 100), then all papers below some cut-off level could be rejected. The new pool of papers is then passed back to the judges, and the process is repeated. A concern is then the total number of papers that judge reads must be substantially less than P. The process is stopped when there are only W papers left. There are the winners. Typically for P=100, W=3. Your task is to determine a selection scheme, using a combination of rank-ordering, numerical scoring, and other methods, by which the final W papers will include only papers from among the "best" 2W papers. (By "best", we assume that there is an absolute rank-ordering to which all judges would agree.) For example, the top three papers. Among all such methods, the one that required each judge to read the least number of papers is desired. Note the possibility of systematic bias in a numerical scoring scheme. For example, for a specific collection of papers, one judge could average

70 points, while another could average

80 points. How would you scale your scheme to accommodate for changes in the contest parameters (P, J, and W)? 1996年B 竞赛评判问题 在确定像数学建模竞赛这种形式的比赛的优胜者时,常常要评阅大量酌答卷.譬如说,有P=100份答卷,一个由J位评团人组成的小组来完成评阅任务,基于竞赛资金对于能够聘请的评阅人数量和评阅时间的限制,如果P=l00;

通常取J=8.理想的情况是每个评阅人看所有的答卷,并将它们一一排序,但这种方法工作量太大.另一种方法是进行一系列的筛选,在一次筛选中每个评阅人只看一定数量的答卷,并给出分数.为了减少所看答卷的数量,考虑如下的筛选模式:如果答卷是被排序的,则在每个评闯人给出的排序中排在最下面的30%答卷被筛除;

如果答卷被打分(譬如说从1分到10分),则某个截止分数线以下的答卷被筛除.这样,通过筛选的答卷重新放在一起返回给评阅小组,重复上述过程.人们关注的是,每个评阅人看的答卷总数要显著地小于P.评阅过程直到剩下W份答卷时停止,这些就是优胜者.当J=100 时通常取W=3. 你的任务是利用排序、打分及其它方法的组合,确定一种筛选模式,按照这种模式,最后选中的W份答卷只能来自"最好的"2W份答卷(所渭"最好的"是指,我们假定存在着一种评阅人一致赞同的答卷的绝对排序).例如,用你给出的方法得到的最后3份答卷将全部包括在"最好的"6份答卷中.在所有满足上述要求的方法中,希望位能给出使每个坪阅人所看答卷份数最少的一种方法. 注意在打分时存在系统偏差的可能.例如,对于一批答卷,一位评阅人平均给70分,而另一位可能给80分.在你给出的模型中如何调节尺度来适应竞赛参数(P,J和W)的变化? 1996年两道题都是由Daniel Zwillinger Zwillinger&Associates、Arlington,MA,USA提供的.