Urban and Regional Planning Review Vol. 5, 2018 | 1 Urban Form that Minimizes the Total Travel Cost Assuming Multiple Floors in a Three-Dimensional City with Two-Stage Hierarchical Bases Takehiro KONDO* and Tohru YOSHIKAWA** Abstract: This paper calculates the optimal urban form for a given total floor area with minimization of the total travel cost and studies the impact of changing various factors, by assuming the presence of multiple floors in three-dimensional cities having a hierarchical space structure. In this paper the optimal urban form is defined as the urban form offering the minimum travel cost, from an arbitrary point in a city to a specific city center site. For vertical transfers, a model is formulated by assuming the presence of multiple floors. This assumption makes the model more realistic and closer to actual cities than conventional models where multiple floors are disregarded. The results show that the optimal urban form changed; retaining similarity when the total floor area was changed, but without doing so when the vertical travel cost or travel cost by short-distance transportation changed. The paper also compares the city model and an actual city form to clarify the difference. The optimal urban form derived from the model calculation has a far smaller horizontal size than the actual city and hence includes many high-rise buildings, which differs significantly from the form of the actual city. Consequently, the optimal urban form with respect to the travel cost was that in which ultra high-rise buildings were concentrated in the city center. Keywords: Compact City System, Floor Height, Multiple Floors, District Centers, Hierarchy, Tama City 1. Background and objectives This paper obtains the optimal form of a three-dimensional city with a hierarchical space structure with ease of transfer in mind and clarifies how it differs from the actual urban form. Here, the optimal urban form is defined as the urban form offering the minimum travel cost, from an arbitrary point in a city to a specific city center site. For vertical transfers, a model is created assuming the presence of multiple floors. This is to make the model more realistic and closer to actual cities than conventional models, where multiple floors are disregarded. Many high-rise buildings have been constructed in Japan as well as in large cities worldwide and construction has accelerated in recent years1). Plans for ultra-large buildings incorporating a city function, such as hyper buildings, have also emerged2). However, it takes time to move vertically in a city full of high-rise buildings and moving from place to place is not always convenient. Japan and other nations have been planning and establishing models of cities to visualize compact cities3). Achieving such a compact city requires a quantitative analysis of the * Mitsubishi Electric Information Systems Corporation ** Department of Architecture and Building Engineering, Tokyo Metropolitan University (C) 2018 City Planning Institute of Japan http://dx.doi.org/10.14398/urpr.5.1 Urban and Regional Planning Review Vol. 5, 2018 | 2 effects of the city and is crucial for future implementation of city planning policy based on social agreement4). This paper provides suggestions for this purpose. The following studies on optimal urban form focus on minimizing travel cost. Many articles including that of Holden et al.5) analyzed the relation between compact city indicators such as population density and costs such as travel cost in real cities. There also exist many articles including that of Oliveira Panao et al.6) on the form and arrangement of urban buildings optimizing the energy efficiency in terms of building facility energy consumption. Morimoto7) studied the effect of making an actual city compact, but generalizing these effects is difficult, since it is strongly linked to city-specific situations. Conversely, studies on the optimal urban form of the model city would be useful to determine the generality of the effect of a compact city. Studies of this type include those by Koshizuka8) and Kurita9), who showed an optimal three-dimensional urban form assuming a rectangular parallel-piped urban model. Suzuki10) obtained an optimal urban form, assuming an urban model of stacked rectangular parallel-piped boxes, but such studies still face the following problems. In the former model, in which an urban form was approximated with a single rectangular parallel-piped structure, the difference in building height between the urban and surrounding areas, namely the difference in residential density, cannot be reflected. In the latter model, where an urban form was approximated with stacked rectangular parallel-piped boxes, the difference can be reflected to a certain extent. However, the number of rectangular parallel-piped boxes is restricted due to computational limitations, while the urban form is limited by the shape and size of the rectangular parallel-piped boxes. Moreover, given the uncertainty of what city component the rectangular parallel-piped box represents, it is difficult to discuss the obtained optimal urban form. In addition, these models assume either direct movement from any point to a single base or movement between any two points. For this reason, these models overlooked the impact of the hierarchical space structure observed in actual large cities on the urban form. Therefore, the influence of hierarchical bases on the optimal urban form has not been clarified. In response, Kondo et al.11) derived an optimal urban form by introducing hierarchical bases and means of transportation. The optimal inter-base distance was calculated using travel cost ratios of pedestrian, bus and elevator routes fixed to realistic values. Consequently, they revealed how creating a compact city with ultra high-rise buildings was not always efficient, while developing the city horizontally and exploiting certain means of transportation could be efficient in some cases. However, this ignored the fact that actual city spaces consist of the vertical division of a building into multiple floors with constant distance or floor height; instead treating buildings as horizontally and vertically continuous spaces. Namely they assumed a population distributed uniformly over any of the three dimensions, i.e. a uniform density of citizens floating within a three-dimensional city space. The key advantage of this assumption was the chance to simplify calculation because calculus, a very effective means of calculation, could be used to calculate the cost of travel both vertically and horizontally. However, the population distribution and the vertical travel cost in this model differ from the model in actual cities, which comprises buildings with many floors and hence the optimal urban form obtained could differ from the correct one. In the case of introducing multiple floors, each floor can be assumed to be a rectangular cuboid with the height of the floor as the vertical height. Based on this concept, cross-sections of both models with equal volumes are shown in Figure-1. The figure is based on the one by Kondo et al.12), though the location of the urban envelop when not considering the floors is different. It is because in the three-dimensional case the urban Urban and Regional Planning Review Vol. 5, 2018 | 3 models with the same volume should be compared, while in the two-dimensional case the urban models with the same length are compared. First, the urban model with floors and that without them differs as to the population distribution at the end of the city. When introducing multiple floors, the entire population concentrates on the floor, and the population density per unit floor is constant. In contrast, without considering multiple floors, the edge of the city has an inclined shape like an attic and the population density at the edge is lower, since the height of the city is regarded as the population density. Furthermore, the end of each floor of the model without considering multiple floors moves inward compared to the case considering the floors (Figure-1). This means that the urban model without considering multiple floors has a different population density at the edge of the urban area compared to actual cities and that the horizontal movement is overestimated. Also, in the case of considering multiple floors, the population distributed in the air between floors in the case without considering multiple floors is distributed on the nearest downward floor. Thus, the vertical travel distance is reduced by about half of the floor height as shown by the downward arrows in Figure-1. This shows that the urban model without considering multiple floors overestimates the travel cost in the vertical direction. These factors cause the difference in urban forms. This difference is expected to be conspicuous especially when the city size is small. Therefore, there is a danger that the optimum city shape obtained may differ from the actual one. Figure-1. Difference in population distribution according to the existence of multiple floors The above facts suggest that assuming multiple floors has the following advantages: Firstly, in an urban model comprising buildings with multiple floors, there would be less difference in the above-mentioned travel cost between the model and actual cities compared with conventional models. Secondly, there is scope to analyze in more detail how a change in various factors such as floor height, which has increased in recent years in Japan, will affect the urban form. Thirdly, not only does using a continuous function reduce the amount of computation, it is also expected to circumvent the difficulty in interpreting the travel cost and the calculated urban form arising from the above problem in preceding studies to a certain extent. This is because assuming multiple floors equates to describing city borders with a continuous rather than discrete function, given the minimum height in the actual cities as the height of stacked rectangular parallel-piped boxes in the previous study described above.