This paper proposes a primal-dual method for solving the minimum cost flow problem with an additional linear constraint. Each branch of a given network has two dimensional distance associated with it. The first element is related to the cost, and the second to the coefficient of the additional constraint. The lexico-shortest route is defined as the route with the minimum distance in the lexicographical ordering. A loop with negative distance is called a lexico-negative loop. A lexico-negative loop such that the first element of its distance is negative, never appears. When there exists a lexico-negative loop such that the first element is zero, and that the second is negative, the current primal solution (flow) is improved by changing the flow around the loop. Otherwise, the. lexico-shortest route exists and the current dual solution is improved. Our algorithm is a pure network algorithm in the meaning that we need not know what is the basis for the current solution. It is very suitable to the cases when degeneracies often occur.

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In this paper, we introduce lower-truncated transversal polymatroids, and develop efficient algorithms of network-flow type for those polymatroids. The lower-truncated transversal polymatroid contains, as special cases, a variety of useful matroids such as cycle matroids of graphs, matroids in plane skeletal structures, etc. We present simple and powerful theorems which enable us to solve various combinatorial optimization problems for those polymatroids by means of network-flow algorithms. Especially, we can solve greedy-type optimization problems concerning those polymatroids in a remarkably efficient manner. As greedy-type problems, we take up the problem of finding a maximum-weight independent vector, that of finding the principal partition and that of covering and packing, and give efficient solutions for them. Applying general algorithms for lower-truncated transversal polymatroids to cycle matroids of graphs and matroids in plane skeletal structures, we obtain various new results. From the viewpoint of applications, Iower-truncated transversal polymatroids are essentially related to discrete systems with internal degrees of freedom which arise in many fields of engineering, so that the algorithms for those polymatroids developed in this paper give efficient methods to analyze such systems in a unifying manner.

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In this paper, we analyze approximation algorithms for two types of scheduling problems. The first is the n jobs scheduling problem with due dates on m identical machines to minimize the maximum lateness. For this problem n/m/1/L_<max>, we propose two approximation algorithms and derive their worst case bounds. The second is the 2 × n flow shop scheduling problem with due dates to minimize the maximum lateness. For this problem n/2/F/L<max>, we first give a solvable case in the sense that the optimal schedule can be easily found. Then we again propose an approximation algorithm for general n/2/F/L_<max> and derive its worst case bound.

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The following two-machine flow-shop scheduling problem is solved. Given jobs are classified into groups, and each machine needs some setup before the first job in a group is started processing. If a job in a group is started its process, all jobs in the group must be finished before a job in another group is processed. Each job may have a specified lag time between machines. Moreover, a series-parallel precedence relation may be specified among groups. Find inter- and intra-group schedules minimizing the total elapsed times on both machines. The proposed method for this problem is based on a new definition of composite jobs and Sidney's theory about series-parallel algorithms. The proposed method can also solve a version of the problem where each job has setup times not included in the processing times and a series-parallel precedence relation is specified among jobs in a group.

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While the number of advocates of system dynamics has been gaining, many of the original criticisms leveled against it, valid or otherwise, persist to this date. These criticisms can be separated into three basic types. They consist of criticisms created by (i) the application, or rather mis-application, of competing sets of theories, standards and procedures on system dynamics, (ii) the underdevelopment of the theories, standards and procedures for assessing system dynamic model objectively, and (iii) the lack of rigor and elegance in the processes of model building, analysis and validation. Each modeling school defines a particular way of looking at the real-world system and provides a set of tools for working on particular kinds of problem. None is comprehensive enough to encompass all that might be observed about the real-world system or to solve all problems. And, of course, very many observations and problems fall far outside the range of any formal modeling method. Therefore, the essential points of important criticisms of system dynamics do not consist in its methodological character but the relentlessly native form of, and the lack of precision in, the ideas that it uses. We may then ask whether we cannot, by a refinement of our geometric intuition, resolve system dynamics criticisms with a stock of ideas and procedures subtle enough to give satisfactory qualitative representations to a given problem. This report studies the following problems,, First, we intend to expose, as the axiom system of system dynamics, the mathematical principles which should guide judgement in modeling of systems. Second, we attempts to adequately express the essential mathematics of system dynamics. Accordingly, we investigate, with the tools of the Catastrophe Theory, the structural stability, the classification and the discontinuities of dynamic behaviours of systems. Finally, we explain that, by adopting our line of reasoning, we can lay the groundwork of a theoretical investigation of a means to systematize any results in the past articles of system dynamics.

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