Keywords: optimal allocation, distance measurement, expressway network, graph theory, genetic algorithm.

The present study deals with the optimal allocation of transportation facilities or interchange-and-exits on a expressway network with loops under various kinds of constraints. Such a problem in general belongs to the combinatorial optimality problems frequently with a result in enormous amount of generate-and-test in exponential increase of searching space, which requires necessary establishment of the heuristic rules or the problem oriented rules to prune futile alternatives as effectively as possible [3]. Recent results show efficiency of the genetic algorithm (GA), though without rigorous guarantee of optimality. However, many cases for engineering requirements by GA could be satisfied [8]. A continuous treatment on the present optimality problem can be realized by traditional mathematical methods including the iteration technique or application of the first derivatives to accelerate approximation to optimality. The present optimal allocation problem can be described continuously, which belongs substantially to Voronoi allocation problem [4,5]. Focusing on the transportation problems, a spacing of bus-stops for many to many travel demand is analyzed [1,4]. Such optimal allocation of bus-stops on routes is methodologically accomplished[10]. Sequential location-allocation of public facilities in 2D plane is also dealt with [6]. Practically, it is important to evaluate constraints to optimality, which include various factors as a density distribution, such as of population, industrial output, or environmental effects. An equivalent density distribution in terms of distance from such factors could simplify complicatedness to a large extent. Between many synthesis techniques of evaluation, the maximum entropty criteria could provide rather consistent results.

Thus, a general formulation of the present optimal allocation problem or the original problem is, herein, accomplished with a result in simplified extension problem, which can provide consistently physical background together with easier numerical procedure by the iteration technique. Some numerical examples are compared to a practical design result with considerable rationality.

1

S.C. Wirasinghe, and N.C. Goneim, "Spacing of bus-stops for many to many travel demand", Transportation Science, Vol.15, No.3, 1981, pp.210-221. doi:10.1287/trsc.15.3.210

2

E. Polak, "On the mathematical foundations of nondifferentiable optimization in engineering design", SIAM Rev., Vol.29, No.1, 1987. doi:10.1137/1029002

3

A. Miyamura, Y. Kohama, and T. Takada, "A combinatorial problem: failure load analysis of rigid frames", Advances in Engineering Software, 27(1996), pp.137-143. doi:10.1016/0965-9978(96)00012-9

4

A. Okabe, A. Fujii, K. Oikawa, and T. Yoshikawa, "The statistical analysis of a distribution of activity points in relation to surface-like elements", Environment and Planning A, Vol.20, 1988, pp.609-620. doi:10.1068/a200609

5

A. Okabe, and F. Miki, "A conditional nearest-neighbor spatial association measure for the analysis of conditional locational independence", Environment and Planning A, Vol.16, 1984, pp.107-114. doi:10.1068/a160163

6

T. Suzuki, Y. Asami, and A. Okabe, "Sequential location-allocation of public facilities in one- and two- dimensional space: Comparison of several policies", Mathematical Programming, 52(1991), pp.125-146. doi:10.1007/BF01582883

7

T. Takada, Y. Kohama, and A. Miyamura, "Optimization of shear wall allocation in 3D frames by branch-and-bound method", Proc., OPTI2001, Bologna, pp.1-10.

8

Y. Kohama, T. Takada, A. De Stefano, A. Miyamura, and T. Aoki, "Limit analysis of masonry wall with opennings by GA", Proc., The fifth Int. Conf. on Computational Structure Technology, Leuven, 2000.

9

A. Miyamura, Y. Kohama, and T. Takada, "Optimal Allocation of Shear Wall at 3D Frame by Genetic Algorithm Application", Proc., The fifth Int. Conf. on Computational Structure Technology, 1996, Budapest

10

R.J. Vaughan, and E.A. Cousins, "Optimum location of stops on a bus route", Proc. of the Seventh InternationalSymposium on Transportation and Traffic Theory, 1977, pp.697-716

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