以廣義反矩陣的特性推導路徑非負流量之研究

A Study of Estimating Nonnegative Path Flow by The Properties of Generalized I nverse Matrix

路段與路徑之關係可以用路段╱路徑投引矩陣（Arc/Pathincidencematrix）表示記為△，而路段與路徑流量分別以f與h表示，則路段流量與路徑流量之關係可表為△h=f。當△的反矩陣存在，則路徑流量可以由h=△¯1f求得。由網路的特性可知，△的反矩通常不存在。儘管如此，仍可透過廣義反矩陣的特性求出路徑流量，令△¯為△之廣義反矩陣，k為任意向量，則路徑流量為h=△¯f+（I-△¯△）k，但是此結果無法確定h為正流量。因此本研究提出一演算法，以確保可求出一組正流量解，提供相關研究應用。

ArC/Path incidence, matrix denoted as △=〔△ 〕where △ =1 if arc ais in path p,O otherwise. Arc flow and path flow denoted as f and hrespectively. The relationship between arc flow and path flow can be shown as △h = f. If △¯１exists, path flow h equals to △¯1f. Unfortunately, △¯1 does not always exist. But if △¯ denoted as generalized inverse of △ and kis a column vector, then path flow can be shown as h = △¯f +（ I-△¯ △）k.In this research we will provide an algorithm to find a nonnegative path flow.