We consider the maximum balanced flow problem of a two-terminal network N, i.e.,a maximum flow problem with an additional constraint described in terms of a balancing rate function α : A → R_+ - {0}, where A is the arc set of N and R_+ is the set of nonnegative reals. In this paper, we propose a polynomial time algorithm for the maximum balanced flow problem, on condition that all given functions in N are rational. The proposed algorithm, which is composed of a binary search algorithm and Dinic's maximum flow algorithm with a parameter, requires O(max{log(c*), mlog(η*), nm}T(n, m)) time, where c* = max{c^o(a) : a ∈ A} for positive integral arc-capacities (c^o(a) : a ∈ A) and η* = max{η(a) : a ∈ A} for α(a) ≡ ζ(a)/η(a) ≤ 1 such that ζ (a) and η(a) are positive integers, and T(n, m) is the time for the maximum flow computation for a network with n vertices and m = |A| arcs.

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The essence of AHP is to evaluate objects in terms of the eigen vector of the comparison matrix. But when the number of objects, n, is too large, it causes often worse reliability for an observer to evaluate all paired comparisons at a time. So it is necessary to decompose the whole set of pairs into several classes, and for each class to be evaluated by one observer. We propose the decomposition by BIBD (balanced incomplete block design) well known in the field of experimental design or combinatorics. We show by simulation experiments that our method gives better evaluations than the ordinary AHP. In connection with these, we show that the logarithmic least square method, which has been proposed by several authors, gives very good approximation to the eigen vector method when n is rather small, and that the former completely coincides with the latter when n &ie; 3, surprisingly.

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Asymptotic behavior of some of the interior point methods for Linear Programming is investigated without assuming nondegeneracy of constraints. A detailed analysis is given to the fundamental pair of vectors, the Newton direction leading to the center of the problem "d_C" and the direction of the affine-scaling method "d_<AF>". The quadratic convergence property of the iteration by - d_C is demonstrated. Step-sizes for the direction - d_C are also given to maintain feasibility without sacrificing the quadratic convergence. The sequence generated by - d_<AF> is pulled towards the central trajectory while. it converges linearly to the optimal solution. Local convergence properties of Iri and Imai's Multiplicative Barrier Function method and Yamashita's method (a variation of the projective scaling methods) are also discussed. It is shown that the search direction of the method by Iri and Imai converges to - d_C, whereas the direction by Yamashita converges to d_<AF>. A proof is given for the quadratic convergence property of Iri and Imai's method with an exact line search procedure in the case where constraint is degenerate.

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Minimum variance problem may arise when we want to allocate a given amount of resource as fairly as possible to a finite set of activities under certain constraints. More formally, it is described as follows. Given a finite set E, a subset S of R^E and a function h_e(x(e)) from a certain domain to R for each e ∈ E (which represents the profit resulting from allocating x(e) amount of resource to activity e), the problem seeks to find x = {x(e) : e ∈ E} ∈ S that minimizes the variance of the vector {h_e(x(e)) : e ∈ E}. Here the variance of {h_e(x(e)) : e ∈ E} is defined as the summation over e ∈ E of the square of difference between the profit h_e(x(e)) and the mean value of profits of all activities. Such problem is called minimum variance problem This paper first presents a parametric characterization of optimal solutions. Based on this, for a class of problems satisfying certain assumptions, we shall develop an ε-approximation scheme which requires to solve the corresponding parametric problem a number of times polynomial in the input length and 1/ε. We shall then present three special cases for which such ε-approximation scheme becomes a fully polynomial time approximation scheme. The first case is that h_e is linear and increasing for each e, and the feasible set S is described by the set of linear equalities and/or inequalities containing the constraint such that the sum of x(e) over all e ∈ E is a fixed constant, and the second one is that h_e is linear and increasing for each e, and the feasible set S is the set of integral or real bases of submodular systems. The third one is that h_e is a certain nonlinear function and the feasible set S is the set of integral or real bases of a polymatroid. Finally we shall give a pseudopolynomial time algorithm if x(e) is an integer with lower and upper bounds on it, and the sum of x(e) over all e ∈ E is a fixed constant.

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The purpose of this paper is to present an O(n^3L) algorithm for a linear complementarity problem with a positive semi-definite matrix. The algorithm is superior to other O(n^3L) algorithms in the point that it is able to start from any initial feasible point whose components lie between 2^<-O(L)> and 2^<0(L)>. The algorithm is based on the O(n^<3.5>L) method presented by Mizuno [11]. In order to decrease the running time, we use the rank one update technique proposed by Karmarkar [5]. We evaluate the running time in an original way.

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In considering cost reductions, it is natural to postpone equipment replacement until innovative equipment is made available, if it will appear in the near future. On the other hand, if new equipment with these characteristics is not made available in the near term, existing equipment is upgraded with similar products. This study discusses equipment replacement where innovative technological advances occur together with gradual technological advances. It aims to clarify theoretically and experimentally the above equipment replacement behavior by using "Control Limit Policy" . Next, we give a practical replacement decision method using control limit policy when the appearance time of innovative equipment is forecasted. This study is significant in the sense of being able to show that control limit policy can propose not only a convenient method in actual replacement decisions, but also an effective method to explain the qualitative aspects of equipment replacement behavior.

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