GERT网络的矩阵式表达及求解模型  

Matrix representation model and its solution of GERT network

作　　者：陶良彦[1,3] 刘思峰[2,3] 方志耕[1,3] 陈顶[1,3] 

机构地区：[1]南京航空航天大学经济与管理学院,江苏南京210016 [2]英国De Montfort大学计算智能研究中心,莱斯特LE19BH [3]南京航空航天大学灰色系统研究所,江苏南京210016

出　　处：《系统工程与电子技术》2017年第6期1292-1297,共6页Systems Engineering and Electronics

基　　金：欧盟第7研究框架玛丽居里国际人才引进计划Fellow项目(FP7-PIIF-GA-2013-629051);国家自然科学基金(91324003,71671090,71671091);国家社科基金重点项目(12AZD102);中央高校基本科研业务费专项资金(NJ20140032,NP2015208);江苏省普通高校研究生科研创新计划项目(KYZZ15_0092)资助课题

摘　　要：图示评审技术(graphic evaluation and review technique,GERT)解析法一般利用信号流图的拓扑特征(梅森公式)和矩母函数进行求解,但当GERT网络节点较多且结构复杂(回路众多)时,拓扑结构特征的分析十分困难,易出现错判或遗漏情况。针对此问题,将GERT网络用矩阵形式进行表征,分析了以梅森公式为基础的解析法与矩阵变换的关系,设计了两类基于矩阵的GERT求解算法。首先给出GERT网络与信号流图增益矩阵、流图增益矩阵一一对应关系,分析增益矩阵行列式变换与信号流图求解公式的对应关系,设计GERT网络的增益矩阵行列式变换求解算法。另外,研究GERT网络(信号流图)化简操作(消除自环、消除节点)在信号流图增益矩阵上的变换形式,提出了GERT网络解析的矩阵变换方法。最后用两个例子说明矩阵表征及求解模型的简便性和正确性,为GERT解析的计算机操作奠定基础。The typical analytical algorithm for graphic evaluation and review technique （GERT） is based on the topological properties of the signal flow graph （Mason formula） and the moment generating function, whereas it is tremendously difficult to analyze the topological characteristics of the GERT network when the network consists of a large number of nodes and complex structure（including many loops）. The complexity of GERT network may lead to misjudge and false negative of the loops. For this problem, the matrix representation of the GERT network is explored, the corresponding relationship between the Mason formula-based algorithm and the matrix transform is analyzed, and two kinds of algorithms based on matrix for GERT network are designed. The first method is to give the gain matrix of the signal flow graph and gain matrix of the flow graph for a given GERT network firstly, and then to study the relationship between the determinant of the gain matrix and the Mason formula, and to design the resolving algorithm finally. The other method is to utilize the trans form operators on the matrix to represent the simplification operators of the signal flow graph including eliminating self-loop and some unconcerned nodes. As a consequence, the algorithm based on matrix transform is intro duced. Finally, two illustrative examples are presented to demonstrate the convenience and accuracy of the pro posed methods, which may provide a tool for the computer calculation of the GERT network.

关 键 词：图示评审技术 矩阵表达 矩阵行列式解析法 矩阵变换 

分 类 号：C93[经济管理—管理学]