Abstract
In this study we discuss the use of the simplex method to solve allocation problems whose flow matrices are doubly stochastic. Although these problems can be solved via a 0 - 1 integer programming method, H. W. Kuhn [1] suggested the use of linear programming in addition to the Hungarian method. Specifically, we use the existence theorem of the solution along with partially total unimodularity and nonnegativeness of the incidence matrix to prove that the simplex method facilitates solving these problems. We also provide insights as to how a partition including a particular unit may be obtained.
Highlights
The type of allocation problems in which flow matrices are doubly stochastic can be solved via 0 - 1 integer programming, which, is generally not solvable in polynomial time
We use the existence theorem of the solution along with partially total unimodularity and nonnegativeness of the incidence matrix to prove that the simplex method facilitates solving these problems
We examine the use of the simplex method for this type of problems by using the existence theorem of the solution along with partially total unimodularity and nonnegativeness of the incidence matrix
Summary
The type of allocation problems in which flow matrices are doubly stochastic can be solved via 0 - 1 integer programming, which, is generally not solvable in polynomial time. To address this issue, Kuhn [1] proposed the Hungarian ( ) algorithm which can be recently computed in O n3 time, and suggested that it be used along with the simplex method.
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