This analysis explores the periodic property of congestion in the streetcars arriving at a station with an equal time interval, assuming the Poisson arrivals of passengers. We obtain a stationary condition and some properties of the spectral density about the number of passengers on board and consider the periodicity of the congestion in terms of numerical examples which show the existences of three types of spectral density.

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In this paper, we consider the multiobjective optimization problems (MOP) where all the coefficients of the objective functions and the constraints are linear with respect to the decision variables and can be interpreted as L-R fuzzy numbers. In order to cope with the fuzziness in such MOP, we introduce a new solution concept called "τ-Pareto optimal solutions" which simultaneously reflects the fuzziness of both the objective functions and the constraints by extending the well-known concept of Pareto optimal solutions. τ-Pareto optimal solutions thus defined in this paper can be obtained corresponding to the Pareto optimal solutions for the MOP where the coefficients of the objective functions and the constraints are set at the most possible values. In order to clarify the properties of τ-Pareto optimal solutions, we consider the influence on the solution of the MOP caused by the fuzziness of the coefficients of the objective functions and the constraints. After deriving the reasonable conditions between. the fuzziness of the coefficients and the solutions in the MOP, it is shown that τ-Pareto optimal solutions satisfy these conditions. Moreover, we give the trade-off rates between the spread parameters of the L-R fuzzy numbers which are used to express the fuzzy coefficients and τ-Pareto optimal solutions by applying the sensitivity theorem. Finally, the preferable features of τ-Pareto optimal solutions are illustrated by the numerical example.

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The worst-case performance of heuristics with bucketing techniques and/or spacefilling curves for the planar matching problem and the planar traveling salesman problem is analyzed. Two types of heuristics are investigated, one is to sequence given points in a spacefilling-curve order and the other is to sequence the points in the order of buckets which are arranged according to the spacefilling curve. The former heuristics take O(n log n) time, while the latter ones run in O(n) time when the number of buckets is O(n). It is shown that the worst-case performance of the former and that of the latter are the same if a sufficient number of O(n) buckets are provided, which is investigated in detail especially for the heuristics based on the Sierpinski curve. The worst-case performance of the heuristic employing the Hilbert curve is also analyzed.

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In this paper the problem of obtaining the Voronoi diagram which approximates a given tessellation of the plane is formulated as the optimization problem, where the objective function is the discrepancy of the Voronoi diagram and the given tessellation. The objective function is generally non-convex and nondifferentiable, so we adopt the primitive descent algorithm and its variants as a solution algorithm. Of course, we have to be content with the locally minimum solutions. However the results of the computational examples suggest that satisfactory good solutions can be obtained by our algorithm. This problem includes the problem to restore the generators from a given Voronoi diagram (i.e., the inverse problem of constructing a Voronoi diagram from the given points) when the given diagram is itself a Voronoi diagram. We can get the approximate position of the generators from a given Voronoi diagram in practical time; it takes about 10s to restore the generators from a Voronoi diagram generated from thirty-two points on a computer of speed about 17 MIPS. Two other practical examples are presented where our algorithm is efficient, one being a problem in ecology and the other being one in urban planning. We can get the Voronoi diagrams which approximate the given tessellations (which have 32 regions and are defined by 172 points in the former example, 11 regions and 192 points in the latter example) within 10s in these two examples on the same computer.

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