{"title":"线性排序问题:最优解的算法","authors":"A. Mushi","doi":"10.4314/AJST.V6I1.55167","DOIUrl":null,"url":null,"abstract":"In this paper we describe and implement an algorithm for the exact solution of the Linear Ordering problem. Linear Ordering is the problem of finding a linear order of the nodes of a graph such that the sum of the weights which are consistent with this order is as large as possible. It is an NP - Hard combinatorial optimisation problem with a large number of applications, including triangulation of input - output matrices in Economics, aggregation of individual preferences and ordering of teams in sports. We implement an algorithm for the exact solution using cutting plane and branch and bound procedures. The program developed is then applied to the triangulation problem for the input - output tables. We have been able to triangulate input - output matrices of size up to 41 x 41.","PeriodicalId":7641,"journal":{"name":"African Journal of Science and Technology","volume":"38 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2010-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"The linear ordering problem: an algorithm for the optimal solution\",\"authors\":\"A. Mushi\",\"doi\":\"10.4314/AJST.V6I1.55167\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper we describe and implement an algorithm for the exact solution of the Linear Ordering problem. Linear Ordering is the problem of finding a linear order of the nodes of a graph such that the sum of the weights which are consistent with this order is as large as possible. It is an NP - Hard combinatorial optimisation problem with a large number of applications, including triangulation of input - output matrices in Economics, aggregation of individual preferences and ordering of teams in sports. We implement an algorithm for the exact solution using cutting plane and branch and bound procedures. The program developed is then applied to the triangulation problem for the input - output tables. We have been able to triangulate input - output matrices of size up to 41 x 41.\",\"PeriodicalId\":7641,\"journal\":{\"name\":\"African Journal of Science and Technology\",\"volume\":\"38 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2010-06-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"African Journal of Science and Technology\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.4314/AJST.V6I1.55167\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"African Journal of Science and Technology","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.4314/AJST.V6I1.55167","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 2
摘要
本文描述并实现了线性排序问题精确解的一种算法。线性排序是寻找图中节点的线性顺序的问题,使得与该顺序一致的权重之和尽可能大。它是一个具有大量应用的NP - Hard组合优化问题,包括经济学中输入-输出矩阵的三角化,个人偏好的聚合和体育运动中的团队排序。我们使用切平面和分支定界法实现了精确解的算法。然后将所开发的程序应用于输入-输出表的三角剖分问题。我们已经能够三角化大小为41 x 41的输入-输出矩阵。
The linear ordering problem: an algorithm for the optimal solution
In this paper we describe and implement an algorithm for the exact solution of the Linear Ordering problem. Linear Ordering is the problem of finding a linear order of the nodes of a graph such that the sum of the weights which are consistent with this order is as large as possible. It is an NP - Hard combinatorial optimisation problem with a large number of applications, including triangulation of input - output matrices in Economics, aggregation of individual preferences and ordering of teams in sports. We implement an algorithm for the exact solution using cutting plane and branch and bound procedures. The program developed is then applied to the triangulation problem for the input - output tables. We have been able to triangulate input - output matrices of size up to 41 x 41.